a concept exploration method for product family design

A Concept Exploration Method for Product Family Design

A Thesis

Presented to

the Academic Faculty

By

Timothy W. Simpson

In Partial Fulfillment

of the Requirements for the Degree of

Doctor of Philosophy in Mechanical Engineering

Georgia Institute of Technology

September 1998

Copyright © 1998 by Timothy W. Simpson

ii

A Concept Exploration Method for Product Family Design

Approved:

Farrokh Mistree, ChairProfessor, Mechanical Engineering

Janet K. AllenSenior Research Scientist, MechanicalEngineering

Jonathan S. ColtonProfessor, Mechanical Engineering

Russell G. HeikesAssociate Professor, Industrial and SystemsEngineering

William H. ReadProfessor, Public Policy

David W. RosenAssociate Professor, Mechanical Engineering

Daniel P. SchrageProfessor, Aerospace Engineering

Date Approved

iii

ACKNOWLEDGMENTS

“If I have seen far, it is because I have stood on the shoulders of giants.”

— Sir Isaac Newton

This statement is a testament to all of those who contributed, in one form or

another, to this research. Consequently, there are many I would like to acknowledge. To

begin, I would like to thank God for giving me the strength, perseverance, and ability to

achieve what I have done, and I would like to thank my family for their continual love,

support, and encouragement. I also extended my sincerest thanks to Sterling Odom for her

love and support throughout this process as well; she has provided immeasurable

inspiration and motivation to help me get through it all.

I would like to thank my committee members for their comments and suggestions.

In particular, I would like to say a special thanks to Dr. Russell Heikes for first alerting me

to kriging and for our many discussions on experimental designs and metamodeling, Dr.

William Read for several enlightening discussions on mass customization and the emerging

knowledge-based economy, and Dr. Jonathan Colton for his meticulous editorial comments

and feedback. A special thanks also goes to Dr. Kwok-Leung Tsui for his assistance and

interest in the kriging and experimental design study in Chapter 5. Thanks also to Dr. Dave

Rosen and Dr. Janet Allen for our countless interactions over the past four years. Finally, I

would like to thank my advisor, Farrokh Mistree; I cannot even begin to express the

gratitude I feel for all that he has done for me.

Thanks to my colleagues in the Systems Realization Laboratory (SRL) at the

Georgia Institute of Technology, particularly, Matt Bauer, Reid Bailey, and Scott

iv

McDermott. Thanks also to those in the SRL who have gone before me and were always

willing to lend a hand: Pat Koch, Jesse Peplinski, Kemper Lewis, Wei Chen, and Stewart

Coulter; I cherish the time we spent together. Also, special thanks to Gabriel Hernandez,

Zahed Siddique, Mark McIntosh, Marc McLean, Yao Lin, and Kiran Krishnapur for

sharing their thoughts and ideas for the future work section in Chapter 8.

I would like to thank Dr. Thomas Zang from the NASA Langley Research Center

for his interest in my research and for making my summer internship there possible.

During my brief stay, I am indebted to Dr. Tony Giunta for his helpful comments and

discussion about kriging and for letting me use, modify, and improve on his kriging code

so that I did not have to start from scratch. Furthermore, I extend my gratitude to Dr. John

J. Korte, Timothy M. Mauery, and Jack Dunn for all of their help and assistance with the

aerospike nozzle example in Chapter 4. Finally, I wish to acknowledge Bill Goffe for

making a copy of his simulated annealing algorithm available for use in the fitting portion

of my kriging code.

Special thanks to Jonathan Maier for his help with the universal electric motor

example in Chapter 6; without him, the example would have paled in comparison to what

we were able to accomplish. I also wish to acknowledge Douglas Dawson, David

Schooley, Dr. Rhett T. George (from Duke University), Gerry Rescigno, Dick Walter, Al

Lehnerd, and Henry Klein for their helpful comments and insight into the example.

For the Design of Experiments study in Chapter 5, I would like to acknowledge the

contributions of the following people who were kind enough to provide their own codes:

• Orthogonal array design generator: Dr. Art Owen, Department of Statistics,

Stanford University, Palo Alto, CA.

• Maximin Latin hypercube design generator: Kurt D. Palmer, Industrial and Systems

Engineering Department, Georgia Institute of Technology, Atlanta, GA.

v

• Optimal Latin hypercube design generator: Dr. Jeong-Soo Park, Statistics

Department, Chonnam National University, Korea.

• Orthogonal array-based Latin hypercube design generator: Dr. Boxin Tang,

Department of Mathematical Sciences, University of Memphis, Memphis, TN.

• Orthogonal Latin hypercube design generator: Kenny Qian Ye, Department of

Statistics, University of Michigan, Ann Arbor, MI.

• Hammersley Sample Sequence design generator: Dr. Urmila Diwekar, Engineering

and Public Policy, Carnegie Mellon University, Pittsburgh, PA.

Funding for this research has been provided through a Graduate Research

Fellowship from the National Science Foundation. Additional financial support was

received through a Presidential Fellowship and a Woodruff Teaching Fellowship awarded

by the Georgia Institute of Technology. Financial support from NSF Grant DMI-96-12327

is also acknowledged. The work on the aerospike nozzle example was supported by the

National Aeronautics and Space Administration under NASA Contract No. NAS1-19480

while in residence at the Institute for Computer Applications in Science and Engineering,

Mail Stop 403, NASA Langley Research Center, Hampton, VA 23681-0001. Finally, the

cost of computer time was underwritten by the Systems Realization Laboratory at the

Georgia Institute of Technology.

vi

TABLE OF CONTENTS

ACKNOWLEDGMENTS iii

TABLE OF CONTENTS vi

LIST OF TABLES xv

LIST OF FIGURES xix

NOMENCLATURE xxvi

SUMMARY xxxi

CHAPTER 1 FOUNDATIONS FOR PRODUCT FAMILY AND

PRODUCT PLATFORM DESIGN 1

1.1 FRAME OF REFERENCE: PRODUCT FAMILY AND PRODUCT

PLATFORM DESIGN 2

1.1.1 Engineering Examples of Successful Product Families 4

1.1.2 Opportunities in Product Family Design and Product

Platform Design 9

1.2 FOUNDATIONS FOR DESIGNING SCALABLE PRODUCT

PLATFORMS FOR A PRODUCT FAMILY 15

1.2.1 Decision-Based Design, the Decision Support Problem

Technique, and the Compromise Decision Support Problem 15

1.2.2 The Robust Concept Exploration Method 18

1.3 RESEARCH FOCUS IN THE DISSERTATION 21

1.3.1 Research Questions and Hypotheses in the Dissertation 21

vii

1.3.2 Contributions from the Research 25

1.4 OVERVIEW OF THE DISSERTATION 26

CHAPTER 2 A LITERATURE REVIEW: PRODUCT FAMILY

AND PRODUCT PLATFORM DESIGN, ROBUST

DESIGN, AND METAMODELING 31

2.1 WHAT IS PRESENTED IN THIS CHAPTER 32

2.2 PRODUCT FAMILY AND PRODUCT PLATFORM DESIGN

TOOLS AND METHODS 34

2.2.1 Attention Directing Tools for Product Family and Product

Platform Design 34

2.2.2 Product Platform Assessments and Cost Models 41

2.2.3 Engineering Methods for Product Family Design 45

2.3 ROBUST DESIGN 52

2.3.1 Implementation of Robust Design: Taguchi’s Method, in the

Robust Concept Exploration Method, and with Design

Capability Indices 56

2.3.2 Robust Design for Product Family Design: Scale Factors 64

2.4 METAMODELING TECHNIQUES 68

2.4.1 Limitations of Response Surface Approaches 73

2.4.2 The Kriging Approach to Metamodeling 75

2.4.3 Classical and Space Filling Experimental Designs 82

2.5 A LOOK BACK AND A LOOK AHEAD 90

CHAPTER 3 THE PRODUCT PLATFORM CONCEPT

EXPLORATION METHOD 92

3.1 OVERVIEW OF THE PPCEM AND RESEARCH HYPOTHESES 93

3.1.1 Step 1 - Create the Market Segmentation Grid 94

3.1.2 Step 2 - Classify Factors and Ranges 95

3.1.3 Step 3 - Build and Validate Metamodels 97

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3.1.4 Step 4 - Aggregate Product Platform Specifications 99

3.1.5 Step 5 - Develop Product Platform Portfolio 101

3.1.6 Infrastructure of the PPCEM 107

3.2 RESEARCH HYPOTHESES AND POSITS FOR THE PPCEM 108

3.2.1 Relationship of the Research Hypotheses to the RCEM 110

3.2.2 Supporting Posits for the Research Hypotheses 111

3.3 STRATEGY FOR VERIFICATION AND TESTING OF THE

RESEARCH HYPOTHESES 115

3.3.1 Testing Hypothesis 1 and Sub-Hypotheses 1.1-1.3 118

3.3.2 Testing Hypotheses 2 and 3 120


CHAPTER 4 INITIAL KRIGING FEASIBILITY STUDY:

DESIGN OF AN AEROSPIKE NOZZLE 123

4.1 OVERVIEW OF KRIGING MODELING AND A 1-D EXAMPLE 124

4.2 AEROSPIKE NOZZLE DESIGN PROBLEM 130

4.2.1 Metamodeling of the Aerospike Nozzle Problem 134

4.2.2 Error Analysis of Response Surface and Kriging Models 136

4.2.3 Graphical Comparison of Response Surface and Kriging

Models 137

4.2.4 Optimization using the Response Surface and Kriging

Metamodels 140

4.2.5 Lessons Learned from the Aerospike Nozzle Example 143


CHAPTER 5 THE UTILITY OF KRIGING AND SPACE

FILLING EXPERIMENTAL DESIGNS 145

5.1 OVERVIEW OF KRIGING/DOE STUDY AND PROBLEM

TESTBED 146

ix

5.1.1 Overview of Testbed Problems 148

5.1.2 Factors and Levels for Kriging/DOE Experiment 153

5.1.3 Experimental Design Choices for Test Problems 155

5.1.4 Responses for the Kriging/DOE Experiment 160

5.2 PRECURSORY KRIGING/DOE DATA ANALYSIS AND ANOVA 161

5.2.1 Analysis of Variance of Kriging/DOE Study 163

5.2.2 Correlation of Error Measures 164

5.3 TESTING HYPOTHESIS 2: THE UTILITY OF KRIGING 167

5.3.1 Effect of Correlation Function on Kriging Model Accuracy 168

5.3.2 Effects of Equation Type on Kriging Model Accuracy 171

5.4 TESTING HYPOTHESIS 3: THE UTILITY OF SPACE FILLING

EXPERIMENTAL DESIGNS 176

5.4.1 Comparison of Designs for Two Variable Problems 177

5.4.2 Comparison of Designs for 3 Factors 179

5.4.3 Comparison of Designs for 4 Factors 182

5.4.4 Lessons Learned from Experimental Design Study 185


CHAPTER 6 DESIGN OF A FAMILY OF UNIVERSAL

ELECTRIC MOTORS 190

6.1 OVERVIEW OF THE UNIVERSAL MOTOR PROBLEM 192

6.1.1 Physical Description, Schematic, and Nomenclature for the

Universal Motor Problem 192

6.1.2 Relevant Analyses for Universal Motor Problem 195

6.2 STEPS 1 AND 2: CREATE MARKET SEGMENTATION GRID

AND CLASSIFY FACTORS FOR UNIVERSAL MOTOR

PLATFORM 206

6.3 STEP 4: AGGREGATE PRODUCT SPECIFICATIONS AND

FORMULATE UNIVERSAL MOTOR PLATFORM

COMPROMISE DSP 210

x

6.4 STEP 5: DEVELOP THE UNIVERSAL MOTOR PLATFORM 212

6.5 RAMIFICATIONS OF THE RESULTS OF THE ELECTRIC

MOTOR EXAMPLE PROBLEM 216

6.5.1 Development of a Benchmark Universal Motor Family 216

6.5.2 Comparison between the Benchmark Universal Motor

Family and the PPCEM Motor Family 219

6.5.3 Improvements to the PPCEM Motor Family and Lessons

Learned 221


CHAPTER 7 DESIGN OF A FAMILY OF GENERAL AVIATION

AIRCRAFT 234

7.1 STEP 1: DEVELOPMENT OF THE MARKET SEGMENTATION

GRID 236

7.1.1 Overview of the General Aviation Aircraft Example Problem 236

7.1.2 Brief Overview of Aircraft Design 237

7.1.3 The Market Segmentation Grid for the GAA Example

Problem 241

7.2 STEP 2: GAA FACTOR CLASSIFICATION 244

7.3 STEP 3: BUILD AND VALIDATE METAMODELS 247

7.4 STEP 4: AGGREGATE PRODUCT SPECIFICATIONS AND

FORMULATE GAA PLATFORM COMPROMISE DSP 251

7.5 STEP 5: DEVELOP THE GAA PLATFORM PORTFOLIO 255

7.5.1 Results of the GAA Compromise DSP for the Family of

Aircraft 256

7.5.2 Instantiation of the Family of General Aviation Aircraft 259

7.6 VERIFICATION OF GAA PRODUCT PLATFORM RESULTS 262

7.6.1 GAA Product Platform Compromise DSP Verification 263

7.6.2 Comparisons of Kriging Predictions and GASP 265

7.6.3 Comparison of PPCEM Results to Benchmark Aircraft 266

xi

7.6.4 Product Variety Tradeoff Study 278

7.7 LESSONS LEARNED: A LOOK BACK AND A LOOK AHEAD 288

CHAPTER 8 CLOSURE: ACHIEVEMENTS AND

RECOMMENDATIONS 293

8.1 CLOSURE: ANSWERING THE RESEARCH QUESTIONS 294

8.2 ACHIEVEMENTS: REVIEW OF RESEARCH CONTRIBUTIONS 300

8.3 CRITICAL ANALYSIS: LIMITATIONS OF THE RESEARCH 302

8.4 RECOMMENDATIONS: AVENUES OF FUTURE WORK 308

8.4.1 Potential Avenues of Future Work in Metamodeling 308

8.4.2 Future Work in Product Family and Product Platform

Design 311

8.4.3 Additional Verification of the PPCEM and Kriging

Metamodels through the Concurrent Design of an Engine

Lubrication System 313

8.4.4 Configuration Design of Common Automotive Platforms 314

8.4.5 Integrated Product and Process Design of Product Families

and Mass Customized Goods 316

8.4.6 Product Family Mappings and “Ideality” Metrics 317

8.4.7 Modeling the Value of Reuse and Remanufacturing in a

Product Family 318

8.5 CONCLUDING REMARKS 320

APPENDIX A KRIGING ALGORITHMS AND SOURCE CODE 321

A.1 BUILDING, VALIDATING, AND IMPLEMENTING A KRIGING

MODEL 322

A.1.1 Building a Kriging Model 322

A.1.2 Validating a Kriging Model 323

A.1.3 Using a Kriging Model 325

A.2 KRIGING SOURCE CODE 326

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A.2.1 The mlefinder.f Algorithm 326

A.2.2 The krigit.f Algorithm 335

A.2.3 Kriging Algorithm README File 348

A.2.4 Sample Parameter and Data Input Files and Kriging Output 352

APPENDIX B EXPERIMENTAL DESIGN DESCRIPTIONS 354

B.1 CLASSICAL EXPERIMENTAL DESIGNS 355

B.1.1 Central Composite Designs 355

B.1.2 Box-Behnken Designs 356

B.2 SPACE FILLING EXPERIMENTAL DESIGNS 358

B.2.1 Random Latin Hypercubes 358

B.2.2 Random Orthogonal Arrays 359

B.2.3 IMSE Optimal Latin Hypercubes 360

B.2.4 Maximin Latin Hypercubes 361

B.2.5 Orthogonal-Array Based Latin Hypercubes 362

B.2.6 Uniform Designs 363

B.2.7 Orthogonal Latin Hypercubes 365

B.2.8 Hammersley Sequence 366

APPENDIX C A MINIMAX LATIN HYPERCUBE DESIGN

GENERATOR USING A GENETIC ALGORITHM 368

C.1 WHY A MINIMAX LATIN HYPERCUBE DESIGN? 369

C.2 A GENETIC ALGORITHM FOR GENERATING MINIMAX

LATIN HYPERCUBE DESIGNS 370

C.3 EXAMPLE MINIMAX LATIN HYPERCUBE DESIGNS AND

CONVERGENCE STUDIES 375

APPENDIX D KRIGING TESTBED PROBLEMS 380

D.1 DESIGN OF A TWO-BAR TRUSS 381

xiii

D.2 DESIGN OF A SYMMETRIC THREE-BAR TRUSS 382

D.3 DESIGN OF A HELICAL COMPRESSION SPRING 384

D.3.1 Shear Stress 386

D.3.2 Free Length 386

D.3.3 Preload Deflection 387

D.3.4 Combined Deflections 387

D.3.5 Deflection Requirement 387

D.3.6 Geometric Constraints 388

D.4 DESIGN OF A TWO-MEMBER FRAME 388

D.5 DESIGN OF A WELDED BEAM 391

D.5.1 Weld Stress 392

D.5.2 Bar Bending Stress 393

D.5.3 Bar Buckling Load 393

D.5.4 Bar Deflection 393

D.6 DESIGN OF A PRESSURE VESSEL 394

APPENDIX E SUPPLEMENTAL INFORMATION FOR

KRIGING/DOE STUDY 397

E.1 CULLING THE DATA TO REMOVE POTENTIAL OUTLIERS 398

E.1.1 STEP 1: Culling the Data Based on RMSE.RANGE 398

E.1.2 STEP 2: Culling the Data Based on MAX.RANGE 400

E.1.3 STEP 3: Culling the Data Based on CVRMSE.RANGE 402

E.2 ANALYSIS OF VARIANCE RESULTS 406

E.3 INTERACTION OF DOE AND NSAMP 410

APPENDIX F SUPPLEMENTAL INFORMATION FOR GAA

EXAMPLE PROBLEM 413

F.1 GAA DESIGN VARIABLE DESCRIPTION 414

xiv

F.2 GAA KRIGING METAMODELS: SAMPLE POINTS, RESPONSE

VALUES, AND FITTED MLE PARAMETERS 416

F.3 ANALYSIS OF VARIANCE OF GAA RESPONSE MEANS 422

F.4 ADDITIONAL DESIGN SCENARIOS FOR STUDY GAA

PRODUCT PLATFORM USING CDK FORMULATION 424

F.4.1 PPCEM Cdk Values for Design Scenarios 1-8 425

F.4.2 PPCEM Aircraft Instantiations 429

F.5 CONVERGENCE HISTORIES OF INDIVIDUALLY DESIGNED

BENCHMARK TWO, FOUR, AND SIX SEATER AIRCRAFT 430

F.6 GAA PRODUCT VARIETY TRADEOFF STUDY INFORMATION 432

F.6.1 Rank Ordering of Design Variables 432

F.6.2 Product Variety Tradeoff Study Results for Scenario 2 434

F.6.3 Product Variety Tradeoff Study Results for Scenario 3 438

F.7 SAMPLE GAA DSIDES FILES FOR PPCEM FAMILY 442

F.7.1 Sample DSIDES File: gasp.oa64.cdk.s1h.dat 442

F.7.2 Sample DSIDES File: gasp.oa64.cdk.s1h.f 443

F.8 SAMPLE GAA DSIDES FILES FOR BENCHMARK AIRCRAFT 461

F.8.1 Sample DSIDES File: gasp5s3.dat 461

F.8.2 Sample DSIDES File: gasp5s3lo.f 462

REFERENCES 467

VITA 484

xv

LIST OF TABLES

Table 1.1 Product Family Examples: Approach and Available Support 13

Table 1.2 Relationship Between Hypotheses and Dissertation Sections 25

Table 2.1 Summary of Correlation Functions 78

Table 2.2 Error Measures for Kriging Metamodels 81

Table 3.1 Relationship Between Hypothesis Testing and Chapters 117

Table 4.1 Sample Points for 1-D Example 124

Table 4.2 Error Analysis of One Variable Example 130

Table 4.3 Model Diagnostics of Response Surface Models 135

Table 4.4 Theta Parameters for Kriging Models of Aerospike Nozzle 136

Table 4.5 Error Analysis of Aerospike Nozzle Approximation Models 137

Table 4.6 Aerospike Nozzle Optimization Problem Formulations 141

Table 4.7 Aerospike Nozzle Optimization Results Using Metamodels 142

Table 5.1 Factors and Level for Kriging/DOE Study 154

Table 5.2 Experimental Designs for Two Factor Test Problems 157

Table 5.3 Experimental Designs for Three Variable Test Problems 158

Table 5.4 Experimental Designs for Four Variable Test Problems 159

Table 5.5 Additional Random Points Used to Assess Model Accuracy 161

Table 5.6 Kriging Test Problem Model Summary 161

Table 5.7 Summary of ANOVA Results for Kriging/DOE Study 163

Table 5.8 Summary of Equations Accurately Modeled by Kriging 174

Table 6.1 Nomenclature for Universal Motors 194

Table 6.2 Constraints for Universal Motor Product Family 209

xvi

Table 6.3 Goal Targets for Universal Motor Platform 209

Table 6.4 Universal Motor Product Platform Solution 213

Table 6.5 Universal Motor Product Family PPCEM Instantiations 216

Table 6.6 Benchmark Universal Motor Specifications 218

Table 6.7 Comparison of the Responses between the Benchmark Motor

Family and the PPCEM Motor Family 220

Table 6.8 New PPCEM Universal Motor Instantiations with Varying Numbers

of Turns, Wire Cross-Sectional Areas, and Stack Lengths 222

Table 6.9 Comparison of Benchmark Designs and New PPCEM Instantiations

with Varying Numbers of Turns, Wire Cross-Sectional Areas, and

Stack Lengths 223

Table 6.10 Third PPCEM Universal Motor Family with Varying Numbers of

Turns and Stack Lengths 225

Table 6.11 Efficiency and Mass of Benchmark Motors and PPCEM Motor

Platform Families 226

Table 7.1 Hypotheses Tested in Chapter 7 235

Table 7.2 Performance of the Baseline Model in GASP 243

Table 7.3 Baseline Model Specifications 243

Table 7.4 Constraints for the Two, Four, and Six Seater GAA 246

Table 7.5 Goal Targets for the Two, Four, and Six Seater GAA 246

Table 7.6 Error Analysis of GAA Kriging Models 251

Table 7.7 GAA Product Platform Compromise DSP Design Scenarios 256

Table 7.8 Summary of GAA Family Compromise DSP Results 257

Table 7.9 Deviation Functions of Individual Aircraft for Each Scenario 261

Table 7.10 Instantiations of the PPCEM Product Platform Based on Kriging

Metamodels 262

Table 7.11 Performance of PPCEM Platform Instantiations in GASP 265

xvii

Table 7.12 Approximation Errors for Individual Aircraft 266

Table 7.13 Design Scenarios for Designing GAA Benchmark Aircraft 268

Table 7.14 Individual PPCEM and Benchmark Aircraft for Scenario 1 269

Table 7.15 Individual PPCEM and Benchmark Aircraft for Scenario 2 273

Table 7.16 Individual PPCEM and Benchmark Aircraft - Scenario 3 275

Table 7.17 Relative Importance of Design Variables 279

Table 7.18 Product Variety Tradeoff Study - Scenario 2 281

Table 7.19 Rank Ordering of Effects on Means of Responses 284

Table 7.20 Product Variety Tradeoff Study - Scenario 3 286

Table 8.1 Applicability of PPCEM to Product Families from Section 1.1.1 304

Table 8.2 Contributions to Future Work Discussion 313

Table A.1 Error Measures for Approximation Models 324

Table B.1 Uniform Design Generating Vectors for Small Sample Sizes 364

Table C.1 Summary of Minimax Latin Hypercube Convergence Study 376

Table F.1 64 Point Orthogonal Array Used to Build Kriging Metamodels for

GAA Example Problem 418

Table F.2 Response Values for 1 Passenger GAA for 64 Point OA 419



Table F.5 MLE Values for Kriging Metamodels Parameters 422

Table F.6 Main Effects ANOVA Results for GAA Response Means 423

Table F.7 GAA Product Platform Design Scenarios 425

Table F.8 PPCEM Solutions - Goal Cdk Values 426

Table F.9 PPCEM Platform Specifications for Scenarios 1-8 429

Table F.10 PPCEM Instantiations in GASP from Scenarios 1-8 430

Table F.11 Viewpoints for Pairwise Comparisons for GAA Design Variables 433

xviii

Table F.12 Summary of Pairwise Comparisons and Resulting Relative

Importance of GAA Design Variables 433

Table F.13 PPCEM Platform Designs for Scenario 2 434

Table F.14 Product Variety Tradeoff Study Results for Scenario 3, Allowing 1

Design Variable to Vary Between Aircraft 435





Table F.17 Initial PPCEM Platform Design for Scenario 3 438




Design Variables to Vary Between Aircraft 439


Design Variables to Vary Between Aircraft 441

xix

LIST OF FIGURES

Figure 1.1 Nippondenso Panel Meter Components (from Whitney, 1993) 6

Figure 1.2 Rolls-Royce RTM322 Engine (Rothwell and Gardiner, 1990) 8

Figure 1.3 Mathematical Form of a Compromise DSP (Mistree, et al., 1993) 17

Figure 1.4 Steps and Tools of the RCEM (adapted from Chen, et al., 1996a) 19

Figure 1.5 RCEM Computer Infrastructure (adapted from Chen, et al., 1996a) 20

Figure 1.6 Overview of Dissertation Chapters 28

Figure 1.7 Pictorial Overview of the Dissertation 30

Figure 2.1 Transition of Literature Review in Chapter 2 32

Figure 2.2 Product Family Map (adapted from Meyer and Utterback, 1993) 35

Figure 2.3 Generic Product Development Map (adapted from Wheelwright and

Sasser, 1989) 36

Figure 2.4 Product Platform Market Segmentation Grid (adapted from Meyer,

1997) 37

Figure 2.5 Platform Leveraging in the Market Segmentation Grid (adapted from

Meyer, 1997) 39

Figure 2.6 V2OC Rating vs. Commonality (from Martin and Ishii, 1997) 44

Figure 2.7 Representing a Family of Office Chairs (Erens, 1997) 46

Figure 2.8 Product Variety Decomposed into Systems, Modules, and Attributes

(from Fujita and Ishii, 1997) 47

Figure 2.9 Robust Designs (from Rothwell and Gardiner, 1990) 50

Figure 2.10 Rolls Royce RB211 Engine Family (from Rothwell and Gardiner,

1990) 51

Figure 2.11 P-Diagram of a Product/Process in Robust Design (adapted from

Phadke, 1989) 53

xx

Figure 2.12 Two Types of Robust Design (adapted from Chen, et al., 1996a) 55

Figure 2.13 A Motivating Example for Design Capability Indices 60

Figure 2.14 Implementation of Design Capability Indices for Robust Design

Applications 61

Figure 2.15 Compromise DSP Formulation with Design Capability Indices 63

Figure 2.16 Type III Robust Design: Scale Factors for Product Platforms 66

Figure 2.17 General Approach to Metamodeling (Koch, et al., 1997) 69

Figure 2.18 Techniques for Metamodeling 72

Figure 2.19 Sample Two-Variable Second-Order Response Surfaces (adapted

from Box and Draper, 1987) 74

Figure 2.20 Example Classical and Space Filling Experimental Designs 83

Figure 2.21 Central Composite Design 84

Figure 2.22 Box-Behnken Design 84

Figure 2.23 Latin Hypercube Design 86

Figure 2.24 Orthogonal Array Design 86

Figure 2.25 OA-Based Latin Hypercube 87

Figure 2.26 Orthogonal Latin Hypercube 87

Figure 2.27 Maximin Latin Hypercube 87

Figure 2.28 IMSE Optimal Latin Hypercube 88

Figure 2.29 Hammersley Design 88

Figure 2.30 Uniform Design 88

Figure 2.31 Example 9, 11, and 14 Point Minimax Latin Hypercubes 89

Figure 2.32 Pictorial Review of Chapter 2 and Preview of Chapter 3 91

Figure 3.1 Steps and Tools of the PPCEM 93

Figure 3.2 Step 1 - Create the Market Segmentation Grid 94

Figure 3.3 Step 2 - Factor Classification 95

xxi

Figure 3.4 Relationship of Scale Factors to the Market Segmentation Grid 97

Figure 3.5 Step 3 - Metamodel Development 98

Figure 3.6 Step 4 - Aggregating the Product Platform Specifications 100

Figure 3.7 Step 5 - Developing the Product Platform Portfolio 102

Figure 3.8 Estimating the Dissimilarity Index for a Product Family 104

Figure 3.9 Product Non-Commonality Versus Performance 105

Figure 3.10 Sample Product Variety Tradeoff Study 106

Figure 3.11 Infrastructure of the PPCEM 107

Figure 3.12 Relationships Between Hypotheses and RCEM Modifications 110


Figure 4.1 One Variable Example Problem 125

Figure 4.2 MLE Objective Function for 1-D Example 127

Figure 4.3 One Variable Example of Response Surface and Kriging Models 129

Figure 4.4 VentureStar RLV with Aerospike Nozzle (Korte, et al., 1997) 131

Figure 4.5 Aerospike Components and Flow Field Characteristics (Korte, et

al., 1997) 131

Figure 4.6 Multidisciplinary Domain Decomposition for Aerospike Nozzle

(Korte, et al., 1997) 132

Figure 4.7 Nozzle Geometry Design Variables (Korte, et al., 1997) 133

Figure 4.8 Sample Points of 25 Point Orthogonal Array 134

Figure 4.9 Response Surface and Kriging Models for Thrust and Weight 138

Figure 4.10 Response Surface and Kriging Models for GLOW 139

Figure 5.1 Pictorial Overview of Kriging/DOE Study 147

Figure 5.2 Two-Bar Truss 150

Figure 5.3 Three-Bar Truss 150

Figure 5.4 Compression Spring 151

xxii

Figure 5.5 Two-Member Frame 151

Figure 5.6 Welded Beam 152

Figure 5.7 Pressure Vessel 152

Figure 5.8 Correlation Between RMSE.RANGE and MAX.RANGE 165

Figure 5.9 Correlation of RMSE.RANGE and CVRMSE.RANGE 166

Figure 5.10 Correlation of MAX.RANGE and CVRMSE.RANGE 167

Figure 5.11 Effect of Correlation Function Type on RMSE.RANGE 169

Figure 5.12 Effect of Correlation Function Type on MAX.RANGE 170

Figure 5.13 Effect of EQN of RMSE.RANGE for Testbed Problems 172

Figure 5.14 Effect of Equation Type on MAX.RANGE 173

Figure 5.15 Effect of 9 and 13 Point DOE on RMSE.RANGE 178

Figure 5.16 Effect of 9 and 13 Point DOE on MAX.RANGE 179

Figure 5.17 Effect of 13, 15, 21, and 23 Point DOE on RMSE.RANGE 180

Figure 5.18 Effect of 13, 15, 21 and 23 Point DOE on MAX.RANGE 182

Figure 5.19 Effect of 25 and 33 Point DOE on RMSE.RANGE 183

Figure 5.20 Effect of 25 and 33 Point DOE on MAX.RANGE 184


Figure 6.1 Schematic of a Universal Motor (adapted from G.S.Electric, 1997) 193

Figure 6.2 Comparison of the Torque-Speed Characteristics of a Universal

Motor Rated at 1/4 Hp and 8000 rpm when Operating on AC and

DC Power Supplies (Martin, 1986) 195

Figure 6.3 An Idealized DC Motor (Chapman, 1991) 202

Figure 6.4 Model Geometry for a Universal Motor 204

Figure 6.5 Relative Permeability Versus Magnetizing Intensity for a Typical

Piece of Steel (Chapman, 1991) 205

Figure 6.6 Universal Motor Market Segmentation Grid 207

xxiii

Figure 6.7 P-Diagram for the Universal Motor Example 210

Figure 6.8 Universal Motor Product Platform Compromise DSP Formulation

for Use with OptdesX 211

Figure 6.9 Convergence Plots for the Universal Motor Product Platform 213

Figure 6.10 Compromise DSP Formulation for Instantiating the PPCEM

Platform for Use with OptdesX 215

Figure 6.11 Compromise DSP Formulation for Benchmark Universal Motor

Family for Use with OptdesX 217

Figure 6.12 Convergence Plots for 0.25 Nm Benchmark Motor 219


Figure 7.1 Phases of Aircraft Design (Schrage, 1992) 238

Figure 7.2 GAA Market Segmentation Grid 241

Figure 7.3 Pictorial Representation of Baseline Aircraft 242

Figure 7.4 GAA Mission Profile 244

Figure 7.5 P-Diagram for GAA Example Problem 247

Figure 7.6 Product Array Approach for Constructing GAA Kriging Models 249

Figure 7.7 GAA Product Platform Compromise DSP Formulation 254

Figure 7.8 Convergence History of GAA Family C-DSP for Scenario 1 263

Figure 7.9 Convergence History for Scenarios 2 and 3 264

Figure 7.10 GAA Compromise DSP for Individual Aircraft 267

Figure 7.11 Graphical Comparison of Benchmark Aircraft and PPCEM Family

for Scenario 1 271


for Design Scenario 2, Priority Level 1 Only 274


for Design Scenario 3, Priority Level 1 Only 277

Figure 7.14 Scenario 2 Product Variety Tradeoff Study Results 282

xxiv

Figure 7.15 Pareto Plots for GAA Response Means 284

Figure 7.16 Scenario 3 Product Variety Tradeoff Study Results 287

Figure 8.1 Pictorial Overview of the Dissertation 295

Figure 8.2 The PPCEM as a Platform for Other Platform Design Methods 312

Figure A.1 Fitting a Kriging Model 322

Figure A.2 Optimization Using a Kriging Model 325

Figure B.1 Central Composite Design for 3 Factors 355

Figure B.2 Box-Behnken Design for 3 Factors 357

Figure B.3 Box-Behnken Design Matrices for 4, 5, and 6 Factors (from Box

and Behnken, 1960) 357

Figure B.4 Three Nine Point Latin Hypercube Designs for 2 Factors 359

Figure B.5 Orthogonal Array of Strength 2 for 3 Factors 360

Figure B.6 Optimal Latin Hypercube Designs for 2 Factors 361

Figure B.7 Maximin Latin Hypercube Designs for 2 Factors 362

Figure B.8 Six Point U Design and Latin Hypercube Design 363

Figure B.9 Uniform Designs for 2 Factors with 7, 9, and 11 Points 365

Figure B.10 Nine Point Orthogonal Latin Hypercube Designs for 2 Factors 366

Figure B.11 HSS Designs for 2 Factors with 7, 9, and 11 Points 367

Figure C.1 Genetic Algorithm for Creating Minimax Latin Hypercubes 371

Figure C.2 Expanding Concentric Circles of Equidistant Points 372

Figure C.3 Genetic Algorithm-Based Minimax Latin Hypercube Generator 374

Figure C.4 Convergence of 2 Factor mMlhd with 9 Sample Points 377





xxv


Figure D.1 Two-Bar Truss 381

Figure D.2 Symmetric Three-Bar Truss 383

Figure D.3 Helical Compress Spring (from Siddall, 1982) 385

Figure D.4 Two-Member Frame 389

Figure D.5 Welded Beam 392

Figure D.6 Pressure Vessel 395

Figure E.1 Original Distribution of RMSE.RANGE of Data 399

Figure E.2 Resulting Distribution of RMSE.RANGE after Culling 400

Figure E.3 Distribution of MAX.RANGE of Culled Data Set 401

Figure E.4 Resulting Distribution of MAX.RANGE after Culling 402

Figure E.5 Distribution of CVRMSE.RANGE of Culled Data Set 403

Figure E.6 Distribution of CVRMSE.RANGE of Final Culled Data Set 404

Figure E.7 Distribution of RMSE.RANGE of Final Culled Data Set 405

Figure E.8 Distribution of MAX.RANGE of Final Culled Data Set 406

Figure E.9 Effect of DOE and Sample Size on RMSE.RANGE for the Two

Variable Problems 411

Figure E.10 Effect of DOE and Sample Size on RMSE.RANGE for the Three


Figure E.11 Effect of DOE and Sample Size on RMSE.RANGE for the Four


Figure F.1 Convergence of Individual Benchmark Aircraft - Scenario 1 431



xxvi

NOMENCLATURE

ANOVA Analysis of Variance

β Underlying constant in a kriging model

Cdk Design Capability Index

Control factor Design variables which can be freely specified by a designer

CVRMSE Cross Validation Root Mean Square Error

di+, d i

- Positive and negative deviation variables in compromise DSP

DBD Decision-Based Design

Derivative Product A specific instantiation of a product platform within a product family

which possesses unique form features and function(s) from other

members in the product family

DOE Design of Experiments

DSIDES Decision Support In the Design of Engineering Systems (computer

software for solving the compromise Decision Support Problem)

DSP Decision Support Problem

GAA General Aviation Aircraft

GASP General Aviation Synthesis Program

GRG Generalized Reduced Gradient algorithm in OptdesX optimization

software (Parkinson, et al., 1998)

IMSE Integrated Mean Square Error

JMP® Statistical analysis and experimental design software

Kriging Metamodeling technique from geostatistics which relies on the spatial

correlation of sample data to predict new values of a response, see,

e.g., (Cressie, 1993)

xxvii

LRL Lower Requirement Limit

Market

Segmentation Grid An attention directing tool used to map product platform leveraging

strategies within a product family

MAX Maximum absolute error

µ Mean value

Metamodel A “model of a model” which provides a surrogate approximation for

the actual response (Kleijnen, 1987)

MLE Maximum Likelihood Estimate

MSE Mean Square Error

MDO Multidisciplinary Design Optimization

NCI Non-Commonality Index

Noise factors Parameters in a system over which a designers has no control or are

too difficult or expensive to control

PDI Performance Deviation Index

Product Family A group of products which share common form features and

function(s), targeting one or multiple market niches

Product Platform The common set of design variables around which a family of

products can be developed. In more general terms, a product platform

is the common technological base from which a product family is

derived through modification and instantiation of the product platform

to target specific market niches.

PPCEM Product Platform Concept Exploration Method

Product Variant Synonym for derivative product

R Correlation matrix in a kriging model

R2, R2adj The ratio of the model sum of squares to the total sum of squares, and

that ratio adjusted for the number of parameters in the model

RCEM Robust Concept Exploration Method, see, e.g., (Chen, et al., 1996a)

xxviii

Response Performance parameter of the system, i.e., a system constraint or goal

RMSE Root Mean Square Error

RS Response surface

S-PLUS4® Statistical analysis software (Mathsoft, 1997)

Scalability The capability of a product platform to be “scaled,” “stretched,” or

“leveraged” to satisfy specific market niches

Scale factor A factor (i.e., design variable) around which a product platform can

be “scaled” or “stretched” to realize derivative products within a

product family

SN Signal-to-Noise ratio in robust design

σ Standard deviation

θk Spatial correlation parameters used to fit a kriging model

URL Upper Requirement Limit

x Design variable or control factor

y response value

ˆ y estimated response value

Z Deviation function in compromise DSP formulation

Variables used in Aerospike Nozzle Example in Chapter 4

a Thruster (starting) angle on the rocket nozzle

GLOW Gross Lift-Off Weight

h Base height of the rocket nozzle

l Base length of the rocket nozzle

Variables used in Kriging/DOE Study in Chapter 5

NSAMP Number of samples

CORFCN Correlation function

xxix

EQN Equation number

MAX.RANGE Maximum absolute error normalized by the sample range

RMSE.RANGE Root mean square error normalized by the sample range

Experimental Design Types

bxbnk Box-Behnken Design

CCD, ccdes Central Composite Design

CCF, ccfac Central Composite Face-Centered Design

CCI, ccins Central Composite Inscribed Design

hamss Hammersley Sampling Sequence Design

mnmxl Minimax Latin Hypercube

mxmnl Maximin Latin Hypercube

OA, oarry Orthogonal Array

oalhd Orthogonal Array-Based Latin Hypercube

oplhd IMSE Optimal Latin Hypercube

rnlhd Random Latin Hypercube

unifd Uniform Design

yelhd Orthogonal Latin Hypercube

Variables used in General Aviation Aircraft Example in Chapter 7

AF Engine Activity Factor

AR Aspect Ratio

CSPD Aircraft Cruise Speed

DOC Direct Operating Cost

DPRP Propeller Diameter

LDMAX Maximum Lift to Drag Ratio

xxx

NOISE Take-off Noise

PAX Number of Passengers

PURCH Aircraft Purchase Price

RANGE Maximum Cruise Range

ROUGH Ride Roughness Factor

VCRMX Maximum Cruise Speed

WEMP Aircraft Empty Weight

WFUEL Aircraft Fuel Weight

WL Wing Loading Estimate

WS Seat Width

(Variables used in the universal electric motor example are listed separately in Chapter 6)

xxxi

SUMMARY

Today’s competitive and highly volatile markets are redefining the way companies

do business. Companies are being faced with the challenge of providing as much variety

as possible for these markets with as little variety as possible between products.

Developing a family of products—a group of related products derived from a common

product platform—provides an efficient and effective means to realize sufficient product

variety to satisfy customer demands. Product families based on derivatives of a scalable

product platform are investigated in this dissertation. In particular, the Product Platform

Concept Exploration Method (PPCEM) is developed, presented, and tested as a Method

which facilitates the synthesis and Exploration of a common Product Platform Concept

which can be scaled into an appropriate family of products. The PPCEM consists of five

steps which prescribe how to formulate the problem and describe how it can be solved.

Finally, the PPCEM facilitates generating common product platform alternatives and their

corresponding product families but is not necessarily used to select one of them.

Testing and verification of the method occurs through two example problems:

• the design of a universal electric motor platform which is scaled around the stack

length of the motor to realize a family of motors capable of satisfying a variety of

torque and power requirements, and

• the design of a General Aviation aircraft platform which is scaled into two, four,

and six seat configurations to realize a family of aircraft capable of satisfying a

variety of performance and economic requirements.

Furthermore, indices of commonality and performance of a product family based on a

common, scalable product platform are proposed, enabling product variety tradeoff studies

to be performed. As a demonstration, the indices are employed in the General Aviation

xxxii

aircraft example to assess the compromise between commonality and performance of the

individual aircraft within the family.

In addition to developing a method to facilitate the design of a scalable product

platform for a product family, metamodeling techniques to facilitate concept exploration,

and the implementation of robust design within the PPCEM are investigated. Specifically,

the utility of kriging and space filling experimental designs for building inexpensive-to-run

surrogate approximations of deterministic computer analyses is examined. An initial 1-D

example is used first to familiarize the reader with the mathematics of kriging and the

differences between a kriging model and a response surface. Then a simple, yet realistic,

engineering example—the design of an aerospike nozzle—is used to compare and contrast

kriging models with second-order response surface models, the current standard for

building surrogate approximations in engineering design.

After the usefulness of kriging is demonstrated in the aerospike nozzle example, a

testbed of structural and mechanical engineering problems is introduced to test further the

utility of kriging and space filling experimental designs. Of the five correlation functions

studied for the kriging models, the Gaussian correlation function yields the most accurate

kriging models, on average, when building metamodels for the testbed of problems, and

the kriging models, in general, yield sufficiently accurate results for more than three

quarters of the functions investigated. Furthermore, the space filling and classical

experimental designs yield comparable results for the two variable problems in the testbed,

but the space filling designs tend to produce more accurate kriging models as the number of

variables increases and more so as the number of sample points increases. Finally, a

minimax Latin hypercube design also is developed in this dissertation specifically for use

with kriging metamodels; when compared to the other experimental designs in the problem

testbed, it is consistently among the best designs.

1

1. CHAPTER 1

FOUNDATIONS FOR PRODUCT FAMILY ANDPRODUCT PLATFORM DESIGN

The principal objective in this dissertation is to develop the Product Platform

Concept Exploration Method (PPCEM) to facilitate the design of a common scalable

product platform for a product family. As the title of this chapter implies, the foundations

for developing this method are presented here. The heart of the chapter lies in Section 1.3

wherein the research objectives, hypotheses, and contributions for the work are described;

this sets the stage for the chapters that follow, culminating in the development of the

PPCEM in Chapter 3. Sections 1.1 and 1.2 contain the motivation, foundation, and

context for investigating the proposed research and serve to establish context for the reader.

Specifically, in Section 1.1 the concepts of product family and product platform design are

introduced and defined, and opportunities for advancing this nascent research area are

identified. In Section 1.2, the foundations for the work—Decision-Based Design and the

Robust Concept Exploration Method—are presented. Finally, Section 1.4 contains an

overview of the dissertation.

2

1.1 FRAME OF REFERENCE: PRODUCT FAMILY AND PRODUCTPLATFORM DESIGN

Today’s competitive and highly volatile market is redefining the way companies do

business. “Customers can no longer be lumped together in a huge homogeneous market,

but are individuals whose individual wants and needs can be ascertained and fulfilled”

(Pine, 1993). Companies are being called upon to deliver better products faster and at less

cost for customers who are more demanding in a market which is characterized by words

such as mass customization and rapid innovation. Even government agencies like NASA

are re-examining the way they operate and do business, adopting slogans such as “better,

faster, cheaper.”

Erens (1997) refers to the market as a “buyer’s market” in which manufacturing

companies must satisfy individual customer requirements; he writes as follows:

“The sellers’ market of the fifties and sixties was characterized by high demand and

a relative shortage of supply. Firms produced large volumes of identical products,

supported by mass production techniques. ... The buyer’s market of the eighties

and beyond is forcing companies making specific high-volume products to

manufacture a growing range of products tailored to individual customer’s needs at

the cost of standard mass-produced goods.”

So why the growing concern for satisfying the individual customer? Stan Davis,

the person who coined the term mass customization, captures it best: “The more a company

can deliver customized goods on a mass basis relative to their competition, the greater is

their competitive advantage” (Davis, 1987). Simply stated, companies which offer

customized goods at minimal extra cost have a competitive advantage over those that do

not. Pine (1993) attributes the increasing attention on product variety and customer

demand to the saturation of the market and the need to improve customer satisfaction:

“Today, demand for new products frequently has to be diverted from older ones. It

is therefore important for new products to meet customer needs more completely, to

3

be of higher quality, and simply to be different from what is already in the

marketplace.”

Similar themes pervade the texts by Wortmann, et al., (1997) who examine industry’s

response in Europe to the “customer-driven” market, and Anderson (1997) who examines

the role of agile product development for mass customization.

This increasing need to distinguish and differentiate products from competitors is

further evidenced by Hollins and Pugh (1990):

"The customer now has plenty of choice for almost every product within a price

range. With this increased choice, consumers have become more aware of the good

and bad features of a product...they select the product that most closely fulfills their

opinion of being the best value for the money. This is not just price but a wide

range of non-price factors such as quality, reliability, aesthetics..."

Chinnaiah, et al. (1998) also examine the trend toward mass customized goods, citing more

demanding customers and market saturation as impetus for the shift. Uzumeri and

Sanderson (1997) state that “The emergence of global markets has fundamentally altered

competition as many firms have known it” with the resulting market dynamics “forcing the

compression of product development times and expansion of product variety.” The study

by Womack, et al. (1990) of the automobile industry in the 1980s provides just one of

numerous examples of this trend.

Since many companies typically design new products one at a time, Meyer and

Lehnerd (1997) have found that the focus on individual customers and products results in

“a failure to embrace commonality, compatibility, standardization, or modularization among

different products or product lines.” Similarly, Erens (1997) states that “If sales engineers

and designers focus on individual customer requirements, they feel that sharing

components compromises the quality of their products.” The end result is a

“mushrooming” or diversification of products and parts with proliferating variety and

4

costs. Mather (1995) states that “Rarely does the full spectrum of product offerings get

reviewed at one time to ensure it is optimal for the business.”

Consequently, companies are being faced with the challenge of providing as much

variety as possible for the market with as little variety as possible between products.

Toward this end, the approach advocated in this dissertation and by many strategic

marketing/management researchers and designers/engineers alike is to design and develop a

family of products with as much commonality between products as possible with minimal

compromise in quality and performance. Several engineering examples are presented in the

next section to provide context and foster a better understanding of the product family

concept and how product families have been successfully developed and realized.

Research opportunities in product family and product platform design then are discussed in

Section 1.1.2.

1 .1 .1 Engineering Examples of Successful Product Families

The following examples from Sony, Lutron, Nippondenso, Black & Decker,

Canon, and Rolls-Royce exemplify successful product families and have been studied as

such. Additional examples which might interest the reader include: Swiss army knives and

Swatch watches (Ulrich and Eppinger, 1995), Xerox copiers (Paula, 1997), Anderson

windows (Stevens, 1995), Hewlett-Packard printers (see, e.g., Lee, et al., 1993), the

Boeing 747 family of aircraft (see, e.g., Rothwell and Gardiner, 1990), and the Kodak

single use camera (see, e.g., Clark and Wheelwright, 1993).

Sony - The Sony Walkman

The design of the Sony Walkman is a classic example of managing the design of a product

family (Sanderson and Uzumeri, 1997). Sony first introduced the Walkman in 1979,

which has dominated the personal portable stereo market for over a decade, and has

remained the leader both technically and commercially despite fierce competition from

5

world-class competitors, e.g., Matushita, Toshiba, Sanyo and Sharp. Sony built all of

their Walkman models around key modules and platforms and used modular design and

flexible manufacturing to produce a wide variety of quality products at low cost.

Incremental design changes accounted for only 20-30 of the 250+ models Sony introduced

in the U.S. in the 1980s. “The remaining 85% of Sony's models were produced from

minor rearrangements of existing features and cosmetic redesigns of the external

case...topological changes [such as these] can be made with little cost or risk” (Sanderson

and Uzumeri, 1995). The basic mechanisms in each platform were refined continually

while following a disciplined and creative approach of focusing its families on clear design

goals while targeting models to distinct market segments.

Lutron - Electronic Lighting Control Systems

When engineers at Lutron design a new product line, they begin with a fairly standard

product with very few options (see, e.g., Spira, 1993). They then work with individual

customers to extend the product line until they eventually have a hundred or so models

which customers can purchase. Then engineering and production work together to

redesign the product line with 15-20 standardized components that can be configured into

the same hundred models from which customers could initially chose. Additional

customization work can be performed to meet individual customer requirements; in its

electronic lighting systems line, used in conference rooms, ballrooms, and hotel lobbies,

Lutron has rarely shipped the same system twice (Spira, 1993).

Nippondenso - Automotive Panel Meters

Nippondenso Co. Ltd. makes automotive components for Toyota, other Japanese car

makers, and car makers in other countries. They design their panel meters using a

combinatoric strategy as illustrated in Figure 1.1. A panel meter is composed of six parts

(in rare cases, only five), and in order to reduce inventory and production costs, each type

6

of part has been redesigned so that its mating features to its neighbors are identical across

the part type. This was done by standardizing the design (denoted by SD in the figure) in

an effort to reduce the number of variants of each part. Inventory and manufacturing costs

were reduced without sacrificing the product offering. Each zigzag line on the right hand

side of Figure 1.1 represents a valid type of meter, and as many as 288 types of meters can

be assembled from 17 different components (Whitney, 1993).

Figure 1.1 Nippondenso Panel Meter Components (from Whitney, 1993)

Black & Decker - Universal Motor Platform

The most common component in all power tools is the universal motor which Black &

Decker redesigned in the early 1970s. The redesign was in response to the threat of

required double insulation on electrical devices to protect the user from electrical shock if

the main insulation system fails. Double insulation was incorporated into 122 basic tools

with hundreds of variations, from jig saws and grinders to edgers and hedge trimmers.

Through standardization of the product line, Black & Decker was able to produce all of its

power tools using a line of motors that varied only in the stack length and the amount of

7

copper wrapped within the motor. As a result, all of the motors could be produced on a

single machine with stack lengths varying from 0.8 in to 1.75 in and power output ranging

from 60 to 650 watts. Furthermore, new designs were developed using standardized

components such as the redesigned motor, which allowed products to be introduced,

exploited and retired with minimal expense related to product development; see (Lehnerd,

1987) for additional information.

Canon - Copiers

Canon has successfully dominated the low volume end of the copier market since the mid

1980s. Canon's copiers offer a wide range of functions and market uses, including: 500-

70,000 copies per month, 8-200 copies per minute, 35-400% reduction/enlargement,

fixed/variable/automatic exposure control, single sheets to double-sided, stapled, collated

copies in either black and white or as many as six different colors. To provide this variety,

Canon has a number of different series (base models or platforms) from which variant

derivatives are created to cover most of the customer's economic and technical

requirements. About 80 percent of the components of these copiers are standard; the

remaining 20 percent are altered and modified to produce product variants within the

product family, see (Rothwell and Gardiner, 1990) for additional information.

Rolls-Royce - Aircraft Engine Platforms

Rolls-Royce designs its aircraft engines around a common platform and then “derates” or

“upgrades” the platform to suit specific customer needs (cf., Rothwell and Gardiner,

1990). An example is the RTM322 engine which was designed to allow several versions

to be produced to cater to different market requirements and power outputs. As shown in

Figure 1.2, the RTM322 platform is common to multiple versions of the engine, namely,

the turboshaft, turbofan and turboprop. When the RTM322 engine is scaled by a factor of

8

1.8, the engine platform becomes the core for the RB550 series which is produced in two

versions: turboprop and turbofan .

Figure 1.2 Rolls-Royce RTM322 Engine (Rothwell and Gardiner, 1990)

In light of these examples, the following definitions for product family, product

platform, and derivatives and product variants are offered to provide context for the

remainder of the dissertation.

A product family is a group of products which share common form features and

function(s), targeting one or multiple market niches. Here, form features refer

generally to the shape and characterizing features of a product; function refers

generally to the utilization intent of a product. The Sony Walkman product family

is one such example; it contains a variety of models with different features and

functions, e.g., graphic equalizer, auto-reverse, and waterproof casing, to target

specific market niches.

A product platform, in this dissertation, is the common set of design variables

around which a family of products can be developed. In general terms, a product

platform is the common technological base from which a product family is derived

through modification and instantiation of the product platform to target specific

market niches (cf., Erens, 1997; McGrath, 1995; Meyer and Lehnerd, 1997). The

9

universal motor platform developed by Black & Decker is an example of a

successful product platform. Product platforms are also prevalent in the automobile

industry, for example, where several car models are typically derivatives of a

common platform (cf., Siddique and Rosen, 1998); Kobe (1997) and Naughton

(1997) describe GM’s and Honda’s global platform strategies, respectively.

A derivative or product variant is a specific instantiation of a product platform

within a product family which possesses unique form features and function(s) from

other members in the product family. Paper copiers are good examples of products

derived from a common product platform; in addition to the Canon example

discussed previously, Xerox’s 1090 copier is a derivative of its 1075 model while

both copiers are part of Xerox’s 10 series of copiers (Jacobson and Hillkirk, 1986).

Furthermore, the Boeing 747-200, 747-300, and 747-400 are derivatives of the

Boeing 747 (Rothwell and Gardiner, 1990).

A single product or individual product is a unique product that has no pre-defined

relationships to other products; any resemblance to other products is strictly through

coincidence or producer’s preference (Erens, 1997). A single product contrasts a

derivative product that has similarities to other products in the product family

having been derived from the same product platform.

In light of these examples and definitions, opportunities for making contributions in

product family and product platform design are discussed in the next section.

1 .1 .2 Opportunities in Product Family Design and Product Platform Design

To understand some of the research opportunities in product family and product

platform design, a closer look at the previous examples is needed. The examples from

Lutron, Nippondenso, and Black & Decker exemplify an a posteriori or bottom-up

approach to product family design. Each company redesigned or consolidated a group of

distinct products to create a more “efficient and effective” product family. Here, efficient

and effective refers to the increased economies of scale each company was able to realize by

standardizing components to reduce manufacturing and inventory costs without

significantly compromising product quality and performance.

10

The main drivers for this type of approach are as follows:

• simplify the product offering and reduce part variety by

• standardizing components so as to

• reduce manufacturing costs and inventory costs and

• reduce manufacturing variability (i.e., the variety of parts that are produced in a

given manufacturing facility) and thereby

• improve quailty and customer satisfaction.

While the cost savings in manufacturing and inventory begin almost immediately from this

type of approach, the rewards are typically long-term since the capital investments and

redesign costs can be significant. Black & Decker, for example, estimated that it would

take seven years to reach the break-even point when they redesigned their universal motor

platform for Double Insulation. As Lehnerd (1987) discusses, between capital

expenditures, development, and tooling, Black & Decker spent $17M to redesign their

motors; however, by paying attention to standardization and exploiting platform scaling

around the motor stack length, all of their motors could be produced on the same machines.

As a result, material costs dropped from $0.77 to $0.42 per motor while labor costs fell

from $0.248 to $0.045 per motor, yielding an annual savings of $1.82M per year. The

cost of Black & Decker tools decreased by as much as 62%, boosting sales, increasing

production volumes, and further improving savings.

But must a company spend millions of dollars in costly redesign to achieve a good

product family? The answer is obviously no, and the examples from Rolls Royce, Canon,

and Sony demonstrate such an approach. These three companies exemplify an a priori or

top-down approach to product family design, i.e., strategically manage and develop a

family of products based on a common platform and its derivatives. McGrath (1995) states

that “A clear platform strategy leverages the resulting products, enabling them to be

11

deployed rapidly and consistently.” Furthermore, Wheelwright and Clark (1992) write as

follows:

“Companies target new platforms to meet the needs of a core group of customers

but design them for easy modification into derivatives through the addition,

substitution, or removal of features. Well-designed platforms also provide a

smooth migration path between generations so neither the customer nor the

distribution channel is disrupted.”

Finally, commonality and standardization across product families allow new designs to be

introduced, exploited, and retired with minimal expense related to product development

(Lehnerd, 1987).

As discussed in Section 1.1.1, Sony and Canon have been able to dominate their

respective markets despite serious local and global competition through a well managed

product platform implementation strategy. The Sony Walkman has been the leader in the

personal stereo market for decades; Sanderson and Uzumeri (1995) studied the success of

the Sony Walkman, commenting as follows:

“Sony's strategy employed a judicious mix of design projects, ranging from large

team efforts that produced major new model 'platforms' to minor tweaking of

existing designs. Throughout, Sony followed a disciplined and creative approach

to focus its sub-families on clear design goals and target models to distinct market

segments. Sony supported its design efforts with continuous innovation in features

and capabilities, as well as key investments in flexible manufacturing.”

Similiarly, Canon was able to steal, and henceforth dominate, the low-end copier market

from Xerox through careful development and realization of a family of products derived

from common platforms (Jacobson and Hillkirk, 1986). Companies like Xerox now are in

the process of re-engineering their product development processes to facilitate the design

and development of new families of copiers in record time (Paula, 1997). Along these

same lines, Rolls Royce can boast similar success. By scaling the RTM322 engine

platform to satisfy a range of thrust and power requirements, Rolls Royce was able to (a)

12

reduce manufacturing and inventory costs by using similar modules and components from

one engine to the next and, more importantly, (b) facilitate the costly certification phase of

its engine development process.

Good product platforms do not just come off the shelf; they must be carefully

planned, designed, and developed. This requires intimate knowledge of customer

requirements and a thorough understanding of the market. However, as discussed in the

literature review in Section 2.2.1, many of the tools and methods which have been

developed to facilitate the management and development of effective product platforms and

product families are at too high of a level of abstraction to be useful to engineering

designers particularly for modeling and design synthesis. Meanwhile, engineering design

methods and tools for synthesizing product families and product platforms are limited or

slowly evolving. Consider the brief summary in Table 1.1 of the product family examples

from Section 1.1.1 and the availability of design support. The majority of the examples

from Section 1.1.1 require modular design to facilitate upgrading and derating product

variants through the addition and removal of modules; a survey of these many of

approaches is offered in Section 2.2.3. In additioin, clustering approaches have been

developed to reduce variability within a product family and facilitate redesigning product

families to improve component commonality, see Section 2.2.3. Meanwhile, little to no

attention has been paid to platform scaling issues for product family design. The notion of

a “scalable” or “stretchable” product platform is introduced by Rothwell and Gardiner

(1990) and may be loosely defined as follows:

Scalable refers to the capability of a product platform to be “scaled,” “stretched,” or

“leveraged” to satisfy specific market niches. For example, the Boeing 747 is a

scalable product platform. It has been “scaled up” and “scaled down” to create the

Boeing 747-200, 747-300, and 747-400 to satisfy different market niches based on

number of passengers, flight range, etc. (Rothwell and Gardiner, 1990). The

13

Rolls-Royce RTM322 aircraft engine and the Black & Decker universal motor

examples discussed in Section 1.1.1 heavily exploit platform scaling.

Table 1.1 Product Family Examples: Approach and Available Support

Examplefrom §1.1.1

Top-Down orBottom-Up

Product FamilyComposition

Availability ofDesign Support

Sony: Walkman Top-DownProduct platform withpredominantly modular

design innovationsModular design

Lutron: LightingControl Systems Bottom-Up

Combinatoric strategybased on modular designand part standardization

Modular design andclustering approaches

Nippondenso:Panel Meters Bottom-Up Similar to Lutron Clustering approaches

and modular design

Black & Decker:Universal Motor Bottom-Up Product platform scaled

around stack length

Modular design tostandardize interfaces; no

support for scalability

Canon: Copiers Top-DownProduct platform withpredominantly modular

design for varietyModular design

Rolls Royce:RTM322 Engine Top-Down

Product platform whichis both scaled and

modular for upgrading

Modular design for somecomponents; no support

for scalability

There are several reasons to investigate scalability in product platform design:

• While modular design has received considerable attention in engineering design

research, the design of parametrically scalable product platforms for a product

family has received little to none.

• In many product families, scalability can be exploited from both a technical

standpoint and a manufacturing standpoint to increase the potential benefits of

having a common product platform. The Rolls Royce RTM322 engine and the

Black & Decker universal motor are excellent examples of this.

• Finally, and perhaps most importantly, the concept of scalability and scalable

product platforms provides an excellent inroads into product family and product

platform design through the synthesis of current research efforts in Decision-Based

Design and the Robust Concept Exploration Method (described in Sections 1.2.1

14

and 1.2.2, respectively), robust design (described in Section 2.3) and tools from

marketing/management science (described in Section 2.2.1).

Consequently, the primary research question investigated in this dissertation is as follows:

How can a common scalable product platform be modeled and designed for a

product family?

To address this question, the Product Platform Concept Exploration Method

(PPCEM) is developed in this dissertation to provide a Method which facilitates the

synthesis and Exploration of a common Product Platform Concept which can be scaled into

an appropriate family of products. The PPCEM and its associated tools and steps are

introduced in Section 3.1. The underlying assumption behind the PPCEM is that a

common set of specifications (i.e., design variable settings) can be found for a product

platform which can then be scaled in one or more of its “dimensions” to realize a product

family. This product family can then satisfy a wide variety of customer requirements with

minimal compromise in individual product quality and performance even though the

product family is derived from a common platform through scaling. Although the PPCEM

is predominantly a method for parameteric or variant design, it is asserted that commonality

of product dimensions and specifications promotes commonality of components which

leads to reduced manufacturing and inventory costs through better economies of scale and

amortization of capital investment over a wider variety of derivative products based on the

common product platform. In special cases, such as the Rolls Royce RTM322 engine

platform mentioned earlier and the Boeing 747 series of aircraft, an added benefit of scaling

a common product platform is to expidite the testing and certification phase of development

(cf., Rothwell and Gardiner, 1990). The foundation for developing this approach is

15

presented in the next section. The specific research focus for the dissertation then is

outlined in Section 1.3.

1.2 FOUNDATIONS FOR DESIGNING SCALABLE PRODUCTPLATFORMS FOR A PRODUCT FAMILY

The technology base for the dissertation is described in this section. An overview

of Decision-Based Design, the design paradigm subscribed to in this dissertation, and the

compromise Decision Support Problem is given in Section 1.2.1. This is followed by an

overview of the Robust Concept Exploration Method (from which the Product Platform

Concept Exploration Method is derived) in Section 1.2.2.

1 .2 .1 Decision-Based Design, the Decision Support Problem Technique,and the Compromise Decision Support Problem

Decision-Based Design (DBD) is rooted in the notion that the principal role of a

designer in the design of an artifact is to make decisions (see, e.g., Muster and Mistree,

1988). This role is useful in providing a starting point for developing design methods

based on paradigms that spring from the perspective of decisions made by designers (who

may use computers) as opposed to design that is predicated on the use of computers,

optimization methods (computer-aided design optimization), or methods that evolve from

specific analysis tools such as finite element analysis.

The implementation of Decision-Based Design that is employed in this dissertation

is the Decision Support Problem (DSP) Technique (see, e.g., Bras and Mistree, 1991), a

technique that supports human judgment in designing systems that can be manufactured

and maintained. In the DSP Technique, designing is defined as the process of converting

information that characterizes the needs and requirements for a product into knowledge

about a product (Mistree, et al., 1990). This definition is extended easily to product family

design: the process of converting information that characterizes the needs and requirements

16

for a product family into knowledge about a product family, or as is the case of this work,

a common scalable product platform. A complete description of the DSP Technique can be

found in, e.g., (e.g., Mistree, et al., 1990).

Among the tools available within the DSP Technique, the compromise DSP

(Mistree, et al., 1993) is a general framework for solving multiobjective, non-linear,

optimization problems. In this dissertation, the compromise DSP is central to modeling

multiple design objectives and assessing the tradeoffs pertinent to product family and

product platform design. Examples of these tradeoffs are discussed in the context of the

two example problems in Chapters 6 and 7.

Mathematically, the compromise DSP is a multiobjective decision model which is a

hybrid formulation based on Mathematical Programming and Goal Programming (Mistree,

et al., 1993), see Figure 1.3. The compromise DSP is used to determine the values of the

design variables which satisfy a set of constraints and bounds and achieve as closely as

possible a set of conflicting goals. The compromise DSP is solved using the Adaptive

Linear Programming (ALP) algorithm which is based on sequential linear programming

and is part of the DSIDES (Decision Support in Designing Engineering Systems) software

(Mistree, et al., 1993).

In the compromise DSP, goals either may be weighted in an Archimedean solution

scheme or rank-ordered into priority levels using a preemptive approach to effect a solution

on the basis of preference. For the preemptive approach, the lexicographic minimum

concept (Ignizio, 1985) is used to evaluate different design scenarios quickly by changing

the priority levels of the goals to be achieved. The capabilities of the lexicographic

minimum concept are employed to develop the product platform portfolio as discussed in

Section 3.1.4, with further examples in Sections 6.4 and 7.5. Differences between the

Archimedean and preemptive deviation functions and a description of the ALP algorithm,

17

design and deviation variables, system constraints, goals, and bounds are discussed by,

e.g., Mistree, et al. (Mistree, et al., 1993).

GivenAn alternative to be improved. Assumptions used to model the domain of interest.The system parameters:n number of system variablesp + q number of system constraints (p equality constraints, q inequality constraints)m number of system goalsgi(x) system constraint functionfk(di) function of deviation variables to be minimized at priority level kth for the

preemptive case.Find

The values of the independent system variables:xi i = 1, …, n;

The values of the independent system variables:di

-, d i+ i = 1, …, m

SatisfySystem constraints (linear, nonlinear)

gi(x) = 0 for i = 1, .., p; gi(x) ≥ 0 for i = p+1, .., p+qSystem goals (linear, nonlinear)

Ai(x) + di- + di

+ = Gi i = 1, …, mBounds

ximin ≤ xi ≤ xi

max i = 1, …, ndi

-, d i+ ≥ 0 ; i = 1, …, m; d i

- . di+ = 0 ; i = 1, …, m

MinimizePreemptive deviation function (lexicographic minimum):

Z = [ f1(di-, d i

+), ..., fk(dk-, dk

+) ]

Figure 1.3 Mathematical Form of a Compromise DSP(Mistree, et al., 1993)

A solution to the compromise DSP is called a satisficing (Simon, 1996) solution,

because it is a feasible point that achieves the system goals to the “best” extent that is

possible. This notion of satisficing solutions is in philosophical harmony with the notion

of developing a broad and robust set of top-level design specifications. The efficacy of the

compromise DSP in creating ranged sets of top-level design specifications has been

demonstrated in both aircraft design (Lewis, et al., 1994; Simpson, et al., 1996) and ship

design (Smith and Mistree, 1994). Developing ranged sets of top-level design

18

specifications is generalized into the notion of developing a product platform portfolio

which is discussed in Section 3.1.5. By finding a “portfolio” of solutions rather than a

single point solution, greater design flexibility can be maintained during the design process.

Finally, the compromise DSP also provides the cornerstone of the Robust Concept

Exploration Method which is reviewed in the next section.

1 .2 .2 The Robust Concept Exploration Method

The Robust Concept Exploration Method (RCEM) has been developed to facilitate

quick evaluation of different design alternatives and generation of top-level design

specifications with quality considerations in the early stages of design (see, e.g., Chen, et

al., 1996a). It is primarily useful for designing complex systems and facilitating

computationally expensive design analysis. The RCEM is created by integrating several

methods and tools—robust design methods (see, e.g., Phadke, 1989), the Response

Surface Methodology (see, e.g., Myers and Montgomery, 1995), and Suh's Design

Axioms (Suh, 1990)—within the compromise DSP (Mistree, et al., 1993). A review of

the wide variety of applications that have successfully employed the RCEM is given in

(Simpson, et al., 1997b).

The RCEM is a four step process as illustrated in Figure 1.4. The corresponding

computer infrastructure is illustrated in Figure 1.5. The steps are described as follows.

Step 1 - Classify Design Parameters: Given the overall design requirements,

this step involves the use of Processor A, see Figure 1.5, to (a) classify different

design parameters as either control factors, noise factors, or responses following

the terminology used in robust design, and (b) define the concept exploration space.

Step 2 - Screening Experiments: This step requires the use of the point generator

(Processor B), simulation programs (Processor C), and an experiment analyzer

(Processor D) shown in Figure 1.5 to set up and perform initial screening

experiments and analyze the results. The results of the screening experiments are

19

used to (a) fit low-order response surface models, (b) identify significant main

effects, and (c) reduce the design region.

Step 3 - Elaborate the Response Surface Model: This step also requires the

use of the point generator (Processor B), simulation programs (Processor C), and

experiment analyzer (Processor D) to set up and perform secondary experiments

and analyze the results. The results from the secondary experiments are used to (a)

fit second-order response surface models (using Processor E) which replace the

original computer analyses, (b) identify key design drivers and the significance of

different design factors and their interactions, and (c) quickly evaluate different

design alternatives and answer "what-if" questions in Step 4.

Step 4 - Generate Top-Level Design Specifications with Quality

Considerations: Once accurate response surface models have been created, Step

4 involves the use of the compromise DSP (Processor F in Figure 1.5) to determine

top-level design specifications with quality considerations. The original analyses or

simulation programs are replaced by response surfaces which are functions of both

control and noise factors. Different quality considerations and multiple objectives

are incorporated in the compromise DSP which is then solved to determine robust,

top-level design specifications.

STEP 4Generate robust top-level design specifications

Overall Design Requirements

RCEM Steps: Methods, Tools, and Math Construct:

STEP 3Elaborate response surface models

STEP 2Conduct “screening experiments”

STEP 1Classify design parameters

Robust Design Principle /Techniques

Response Surface Methods / DOE/ANOVA Statistical Methods

Compromise Decision SupportProblem

Figure 1.4 Steps and Tools of the RCEM(adapted from Chen, et al., 1996a)

20

Robust, Top-LevelDesign Specifications

E. Response Surface Model

y=f(x, z)

µˆy = f(x,µz)

σ2ˆy=Σi=1

k∂f∂zi( )

2

σ2ˆz ii=1

l∂f∂xi( )

2

σ2ˆx i+Σ

Input and Output

Processor

Simulation Programs

D. Experiments Analyzer

Eliminate unimportant factorsReduce the design space to the region of

interestPlan additional experiments


A. Factors and Ranges

Product/ Process

Noise zFactors

yResponse

xControlFactors

B. Point Generator

Design of ExperimentsPlackett-Burman

Full Factorial DesignFractional Factorial DesignTaguchi Orthogonal ArrayCentral Composite Design

etc.

C. Simulation Programs

(Rigorous Analysis Tools)

F. The Compromise DSP

Find Control VariablesSatisfy Constraints Goals "Mean on Target" "Minimize Deviation" “Maximize the independence” BoundsMinimize Deviation Function

Figure 1.5 RCEM Computer Infrastructure(adapted from Chen, et al., 1996a)

The RCEM is taken as the foundation for the research work in this dissertation for

several reasons, namely,

• integration of robust design principles with the compromise DSP (see Section 2.3),

• demonstrated effectiveness for complex systems and robust design, see, e.g.,

(Chen, et al., 1997),

• increased computational efficiency achieved through metamodeling (see Section

2.4), and because it is a

• domain independent method which is particularized easily through specification of a

simulation program or analysis code, see, e.g., (Chen, et al., 1997).

The usefulness of these features of the RCEM to this research work are elaborated

throughout the dissertation, particularly in Sections 3.1 and 3.2 wherein the PPCEM is

introduced. The research objectives for the dissertation are described in the next section.

21

1.3 RESEARCH FOCUS IN THE DISSERTATION

The research focus in this dissertation is embodied as follows:

• a set of research questions that capture motivation and specific issues to be

addressed,

• a set of corresponding research hypotheses that offer a context by which the

research proceeds, defining the structure of the verification studies performed in

this work, and

• a set of resulting research contributions that embody the deliverables from the

research in terms of intellectual value, a repeatable method of solution, limitations,

and avenues of further investigation.

The research questions are presented in Section 1.3.1 along with the corresponding

research hypotheses. The research hypotheses (and supporting posits) are discussed in

more detail in Section 3.2 along with issues of verification and validation. The resulting

research contributions are introduced in Section 1.3.2.

1 .3 .1 Research Questions and Hypotheses in the Dissertation

The principal goal in this dissertation is the development of a method to facilitate the

design of a scalable product platform around which a family of products can be developed.

As discussed in the previous section, Decision-Based Design and the RCEM provide the

foundation on which this work is built. Given this foundation and goal, the motivation for

this research is embodied in the primary research question identified in Section 1.1.2 which

is repeated here.

Primary Research Question:

Q1. How can a common scalable product platform be modeled and designed for a

product family?

22

This research question is related directly to the principal goal in this research which

is to advance product family design through the development of a method to design a

scalable product platform for a product family. The following hypothesis is investigated in

this dissertation in response to the primary research question.

Hypothesis 1: The Product Platform Concept Exploration Method provides a

method for designing a common product platform which can be scaled to realize

a product family.

Since Question 1 is quite broad, three supporting research questions and sub-

hypotheses are proposed to facilitate the verification of Hypothesis 1. The supporting

questions and sub-hypotheses are stated as follows.

Q1.1. How can product platform scaling opportunities be identified from overall

design requirements?

Q1.2. How can robust design principles be used to facilitate designing a common

scalable product platform?

Q1.3. How can individual targets for product variants be aggregated and modeled

for product platform design?

Sub-Hypothesis 1.1: The market segmentation grid can be utilized to help

identify scale factors for a product platform.

23

Sub-Hypothesis 1.2: Robust design principles can be used to facilitate the

design of a common scalable product platform by minimizing the sensitivity of a

product platform to variations in scale factors.

Sub-Hypothesis 1.3: Individual targets for product variants can be aggregated

into an appropriate mean and variance and used in conjunction with robust

design principles to effect a common product platform for a product family.

There is a one-to-one correspondence between each supporting question and sub-

hypothesis. The sub-hypotheses are stated here primarily to provide context for the

literature review in the next chapter and the development of the PPCEM in Section 3.1.

The strategy for verification and testing of the hypotheses is presented in Section 3.3.

In addition to the primary research question related to the design of scalable product

platforms, two secondary research questions are also investigated in this dissertation.

Secondary Research Questions:

Q2. Is kriging a viable metamodeling technique for building approximations of

deterministic computer analyses?

Q3. Are space filling designs better suited for building approximations of

deterministic computer analyses than classical experimental designs?

As discussed in Section 1.2.2, metamodeling techniques—design of experiments

and response surface models—are employed in the RCEM to facilitate concept exploration

and the implementation of robust design. As is discussed in Section 2.4, alternative

metamodeling techniques such as kriging may be better suited for building approximations

24

of deterministic computer analyses than the response surface models currently employed in

Steps 2 and 3 of the RCEM (see Section 1.2.2). Moreover, the traditional or “classical”

experimental designs which are typically used to sample the design space by querying the

computer code to generate data to build these approximations, may not be well-suited for

deterministic computer analyses either; hence, alternative “space filling” designs also are

investigated as part of the research in this dissertation. The specific hypotheses, which are

investigated in response to the secondary research questions, entail affirmative answers to

each question.

Hypothesis 2: Kriging is a viable metamodeling technique for building

approximations of deterministic computer analyses.

Hypothesis 3: Space filling experimental designs are suited better for building

metamodels of deterministic computer experiments than classical experimental

designs.

The motivation for these last two research questions and hypotheses is discussed in

Section 2.4 wherein the limitations of response surface modeling and design of

experiments techniques within the RCEM are discussed in greater detail. It is worth noting

that Hypotheses 2 and 3 are related to Hypothesis 1 but have implications which extend

beyond product family and product platform design, see Section 3.2.2.

The relationship between the hypotheses and the various sections of the dissertation

are summarized in Table 1.2. The hypotheses are elaborated more in the literature review

in the next chapter in the sections listed in the table and revisited in Chapter 3 after the

Product Platform Concept Exploration Method is presented. Verification and validation

issues are discussed in Section 3.3, and testing of the individual hypotheses commences in

25

Chapter 4, lasting until Chapter 7. Although it is not noted in the table, Chapter 8 contains

a review of the hypotheses and their verification. The resulting contributions from these

hypotheses are described in the next section to provide context for the development of the

research in the dissertation.

Table 1.2 Relationship Between Hypotheses and Dissertation Sections

HypothesisSections

DiscussedSectionsTested

H1 Product Platform Concept Exploration Method Chp 3 Chp 6 & 7

SH1.1 Usefulness of market segmentation grid §2.2.1, §3.1.1,§3.1.2, §3.2 §6.2, §7.1.3

SH1.2 Robust design of scalable product platform §2.3, §3.1.2,§3.1.4, §3.2

§6.3-6.5,§7.4-7.6

SH1.3 Aggregating product family specifications §2.3.3, §3.1.4,§3.2

§6.3-6.5,§7.4-7.6

H2 Utility of kriging for metamodelingdeterministic computer experiments

§2.4.1, §2.4.2,§3.1.3, §3.2

Chp 4, §5.2,§7.3

H3 Utility of space filling experimental designs §2.4.3, §3.1.3,§3.2 §5.3

1 .3 .2 Contributions from the Research

The hypotheses and sub-hypotheses, taken together, define the research presented

in this dissertation and hence the contributions from the research. As evidenced by the

principal goal in the dissertation and Hypothesis 1, the PPCEM is the primary contribution

in the dissertation. Additional contributions from the dissertation are the following:

Contributions related to Hypothesis 1 and Sub-Hypotheses 1.1-1.3:

• The notion of scale factors in product platform design and a means of identifying

them for a product platform: Sections 2.3, 3.1.1, 3.1.2, 6.2, and 7.1-7.2.

• An abstraction of robust design principles for realizing scalable product platforms

for product family design: Sections 2.3, 3.1.2, 3.1.4, 6.3-6.5, and 7.4-7.6.

26

• Non-commonality and performance deviation measures for performing product

variety tradeoff studies, see Sections 3.1.5 and 7.6.2.

Contributions related to Hypothesis 2:

• An algorithm to build, validate, and use a kriging model: Section 2.4.2, Chapters 4,

5, and 7, and Appendix A.

• A preliminary comparison of the predictive capability of second-order response

surface models and kriging models: Chapter 4.

• An investigation of the effect of five different spatial correlation functions on the

accuracy of a kriging model: Section 2.4.2 and Chapter 5.

Contributions related to Hypothesis 3:

• An investigation of the effect of eleven different experimental designs on building

an accurate kriging model: Section 2.4.3 and Chapter 5.

• An algorithm for generating minimax Latin hypercube designs: Section 2.4.3 and

Appendix C.

This being the first chapter of the dissertation, these contributions cannot be substantiated;

therefore, they are revisited in Section 8.1 after all of the research findings have been

documented and discussed. An overview of the dissertation is presented next.

1.4 OVERVIEW OF THE DISSERTATION

To facilitate this discussion, an overview of the chapters in the dissertation is

shown in Figure 1.6. Having lain the foundation by introducing the research questions and

hypotheses for the work in this chapter, the next chapter contains a literature review of

related research, elucidating the problems and opportunities in product family and product

platform design. Three research areas are reviewed: (1) product family and product

platform design with particular emphasis on scalability and sizing, (2) robust design and its

application in engineering design, and (3) statistical metamodeling and its role in

27

engineering design, see Sections 2.2, 2.3, and 2.4, respectively. A discussion of how

these disparate research areas relate to one another is offered in Section 2.1

The PPCEM is introduced in Chapter 3 as elements from Chapters 1 and 2 are

synthesized into a method for designing a scalable product platform for a product family.

The PPCEM and its associated steps are presented in Section 3.1. After the PPCEM is

presented, the research hypotheses are revisited in Section 3.2, and supporting posits are

stated and substantiated. Section 3.3 contains an outline of the strategy for verification and

testing of the hypotheses which includes a preview of Chapters 4 and 5—wherein

Hypotheses 2 and 3 are tested—and Chapters 6 and 7 wherein the PPCEM is applied to

two example problems, verifying Hypotheses 1 and Sub-Hypotheses 1.1 through 1.3.

Testing of the hypotheses begins in Chapter 4, but Chapters 4 and 5 entail a brief

departure from product platform design yet are an integral part of the development of the

PPCEM. In Chapter 4, an initial feasibility study of the usefulness of kriging is performed

to familiarize the reader with the method and to begin to verify Hypotheses 2 by comparing

the accuracy of kriging models to second-order response surface models, the current

standard in metamodeling. In Chapter 5, an extensive study of six engineering test

problems selected from the literature is conducted to determine the utility of kriging

metamodels and various experimental designs, testing and verifying Hypotheses 2 and 3.

Once the kriging/DOE study is completed in Chapter 5, the first of two examples

used to demonstrate the PPCEM and verify its associated hypotheses is given in Chapter 6:

the design of a family of universal electric motors. This first example employs the PPCEM

without any metamodeling, providing “proof of concept” that the method works. Then, in

Chapter 7 the PPCEM is applied to the design of a family of General Aviation aircraft,

making full use of the kriging metamodels and robust design capabilities. In each chapter,

an overview of the problem is given along with pertinent analysis information, the steps of

the PPCEM are performed, and the ramifications of the results are discussed.

28

• Introduction, motivation, andtechnical foundation

• Identify research objectives, hypotheses, and contributions

Relevance

• Elaborate opportunities in productfamily and platform design

• Extend robust design principles• Introduce metamodeling concerns:

kriging and experimental designs

• Introduce PPCEM and its steps• Review hypotheses and posits• Outline verification strategy

• Familiarize reader with kriging• Compare kriging and response

surface metamodels• Set the stage for Chapter 5 study

• Investigate utility of kriging and space filling DOE

• Make kriging metamodelingrecommendations

• Demonstrate implementation of PPCEM without metamodels

• Provide proof of concept and initial verification of method

• Demonstrate full implementationof PPCEM, including metamodels

• Provide further verification ofmethod

• Summarize research findings,contributions, and limitations

• Identify avenues of future work

Chapter 3 Product Platform

Concept Exploration Method

Chapter 4Initial Kriging

Feasibility Study

Chapter 5 Kriging/DOE Study

Chapter 6 Design of a Family of Universal Motors

Chapter 7 Design of a Family of

General Aviation Aircraft

Chapter 8 Closing Remarks

Chapter 2 Literature Review: Product

Family Design, Robust Design, and Metamodeling

Chapter 1Foundations for Product

Family and Product Platform Design

Pro

ble

m I

dentif

icatio

nM

eth

od

Hyp

oth

esi

s T

est

ing

Hypotheses

Verify H2

VerifyH2 & H3

VerifyH1; SH1.1,

SH1.2, & SH1.3

VerifyH1 & H2;

SH1.1, SH1.2, & SH1.3

Clo

sure

Revisit

Elaborate

Introduce

Summarize

Figure 1.6 Overview of Dissertation Chapters

29

Chapter 8 is the final chapter in the dissertation and contains a summary of the

dissertation, emphasizing answers to the research questions and resulting research

contributions in Sections 8.1 and 8.2, respectively. Possible avenues of future work are

discussed in Section 8.3. Finally, some closing remarks are given in Section 8.4.

There are six appendices which supplement the dissertation. Appendix A contains

a description of the kriging algorithm which is employed in Chapters 4, 5, and 7.

Appendix B contains detailed descriptions of the experimental designs investigated in the

kriging/DOE study in Chapter 5; the minimax Latin hypercube design, which is introduced

in Section 2.4.3 and investigated in Chapter 5, is described separately in Appendix C as it

is unique to this dissertation. Appendix D contains descriptions of the six engineering test

problems used in the kriging/DOE study, and supplemental information for the

kriging/DOE study is given in Appendix E. Supplemental information for the General

Aviation aircraft problem in Chapter 7 is given in Appendix F.

Finally, a pictorial overview of the dissertation is illustrated in Figure 1.7. It

proceeds from bottom to top, beginning with the foundation provided in this chapter:

Decision-Based Design and the Robust Concept Exploration Method. This figure provides

a road map for the dissertation, and it is referred to at the end of each chapter to help guide

the reader through the work as the research progresses from chapter to chapter.

30

Nozzle Design

Chp 8: Achievements and Recommendations

Family of General Aviation AircraftKriging/DOE Testbed

Family of UniversalElectric Motors

Metamodeling

ModelingMean andVariance

Robust Design Principles

ConceptualNoise

Factors

ScalableProductPlatform

MarketSegmentation

Grid

Product Family Design

KrigingSpaceFillingDoE

Chp 6

Chp 4 Chp 3

§2.2§2.3 §2.1

Chp 5

Chp 7

FOUNDATIONS: Decision-Based Design & the Robust Concept Exploration Method

Platform

Product Platform Concept Exploration Method

Figure 1.7 Pictorial Overview of the Dissertation

31

2. CHAPTER 2

A LITERATURE REVIEW: PRODUCT FAMILYAND PRODUCT PLATFORM DESIGN, ROBUST

DESIGN, AND METAMODELING

Given the research focus identified in Section 1.3, a survey of relevant work in

product family and product platform design, robust design, and metamodeling is presented

in this chapter in Sections 2.2, 2.3, and 2.4, respectively. A thorough description of what

is in this chapter and how these disparate fields of research relate to each other is offered in

Section 2.1. In Section 2.2, the tools and methods for designing product families and

product platforms introduced in Section 1.1.2 are discussed in more detail. Section 2.3

then contains a review of robust design principles, focusing on robust design opportunities

in product family and product platform design. This segues into a discussion of

metamodeling and approximation techniques in Section 2.4 to facilitate the implementation

of robust design. In particular, the kriging approach to metamodeling is introduced in

Section 2.4.2 as a viable alternative for building approximations of deterministic computer

experiments, and a variety of space filling experimental designs for querying a computer

code to build kriging models are described in Section 2.4.3. Section 2.5 concludes the

chapter with a summary of what has been presented and a preview of what is next.

32

2.1 WHAT IS PRESENTED IN THIS CHAPTER

In the preceding chapter, product families and product platforms were introduced

along with several illustrative examples. In this chapter, a literature review of tools and

methods which facilitate the development of product families and product platforms is

presented; the focus is on three areas: (1) approaches for product family and product

platform design, (2) robust design principles and their implementation, and (3)

metamodeling, in Sections 2.2, 2.3, and 2.4, respectively. At first glance, these three

research areas appear unrelated; however, transitional elements presented at the end of each

section preface the discussion in the section that follows as the literature review moves

from the general area of product family design to the specific area of metamodeling, see

Figure 2.1. The relevant hypotheses covered in each section are noted in Figure 2.1.

SpaceFillingDoE

Metamodeling




Kriging


ConceptualNoise

FactorsScalableProductPlatform

MarketSegmentation

Grid

§2.3§2.4 §2.2

SH1.1H1

SH1.2SH1.3

H2H3

Figure 2.1 Transition of Literature Review in Chapter 2

33

As shown in Figure 2.1, the discussion in Section 2.2 explores in greater depth

some of the tools and approaches for product family and product platform design including:

product family maps and the market segmentation grid (Meyer, 1997); approaches to

product family and product platform design; and finally, the notion of a scalable product

platform (Rothwell and Gardiner, 1990). The work by Rothwell and Gardiner then is used

to provide a transition to a discussion of robust design principles in Section 2.3 by relating

Rothwell and Gardiner’s concept of “robust design” for product families to the idea of a

“conceptual noise factor” in a distributed design environment as introduced in (Chang and

Ward, 1995; Chang, et al., 1994). This notion of a “conceptual noise factor” then is

extended to scale factors within a scalable product platform, providing a means to abstract

robust design principles for application in product family design.

In Section 2.3, the focus also shifts from extending Taguchi’s robust design to its

implementation within the Robust Concept Exploration Method (RCEM), i.e., through the

use of metamodels and design capability indices. A brief overview of metamodeling is

presented in the beginning of Section 2.4, providing a transition from robust design to

utilizing metamodels to facilitate its implementation as alluded to in Figure 2.1. The general

approach to metamodeling also is discussed in the beginning of Section 2.4, followed by a

closer look at some of the limitations of second-order response surface models in

engineering design in Section 2.4.1. This discussion provides the impetus for a closer

look at two specific aspects of metamodeling—model selection and experimental

sampling—which also are investigated as part of this research. Specifically, kriging and

space filling experimental designs are examined as potential alternatives to the response

surface methods and classical design of experiments (DOE) currently employed in the

RCEM. Taken together, this literature review provides the necessary elements for the

development of the Product Platform Concept Exploration Method for designing scalable

34

product platforms for a product family as presented in Chapter 3. Toward this end, the

state-of-art in product family and product platform design is discussed in the next section.

2.2 PRODUCT FAMILY AND PRODUCT PLATFORM DESIGN TOOLSAND METHODS

As stated in Section 1.1, in order to provide as much variety as possible for the

market with as little variety as possible between products, many researchers advocate a

product platform and product family approach to satisfy effectively a wide range of

customer needs. In Section 2.2.1, several attention directing tools developed to facilitate

product family and product platform design are presented. In Section 2.2.2, metrics for

assessing product platform effectiveness are discussed. Finally, in Section 2.2.3, methods

for product family design are reviewed.

2 .2 .1 Attention Directing Tools for Product Family and Product PlatformDesign

A large portion of the work in strategic marketing and management is focused on

either categorizing or mapping the evolution and development of product families. These

maps typically are applied a posteriori to a product family but can be used a priori to

identify new directions for product development within the product family. Examples of

product family maps include the work by Meyer and Utterback (1993) and Wheelwright

and Sasser (1989); a brief description of each follows.

Meyer and Utterback (1993) use the Product Family Map shown in Figure 2.2 to

trace the evolution of a product family. In their map, each generation of the product family

employs a platform as the foundation for targeting specific products at different (or

complimentary) markets. Improved designs and new technologies spawn successive

generations, and cost reductions and the addition and removal of features can lead to new

products. Multiple generations can be planned from existing ones, expanding to different

35

markets or revitalizing old ones. A more formal map, with four levels of hierarchy in the

product family (i.e., product family, product platforms, product extensions, and specific

products) also is introduced in their work in an effort to assess the dynamics of a firm’s

core capabilities for product development; several examples can be found in their paper.

Platform Development Family A

Product 1

Product 2

Product 3

Product 4

Product 7

Product 8

Product 5

Product 6

Product 1’

Product 2’

Product 3’

Product 4’

Platform Development Family B

Plan Multiple GenerationsCost Reduction and New Features

New Niches

Time

Adaptation of Core Technologies to New Markets

New Generation Platform Family A

Figure 2.2 Product Family Map (adapted from Meyer and Utterback, 1993)

In related work, Wheelwright and Sasser (1989) have developed the Product

Development Map to trace the evolution of a company’s product lines, see Figure 2.3. In

addition to mapping the evolution of the product line, they also categorize a product line

36

into “core” and “leveraged” products, dividing leveraged products into “enhanced,”

“customized,” “cost reduced,” and “hybrid” products.

“These distinctions—core, hybrid, and the others—are immediately useful because

they give managers a way of thinking about their products more rigorously and less

anecdotally. But the various turns on the product map—the various “leverage

points”—also serve as crucial indicators of previous management assumptions

about the corporate strengths and market forces shaping product evolutions.”

(Wheelwright and Sasser, 1989, p. 114)

Enhanced

Cost-reduced

Customized

Hybrid

• • • • • • • • •Prototype

Time

Core

Core

Incr

easi

ng F

unct

iona

lity,

Val

ue, P

rice

Figure 2.3 Generic Product Development Map(adapted from Wheelwright and Sasser, 1989)

As shown in Figure 2.3., the core product, typically derived from an engineering

prototype, provides the engineering platform upon which further enhancements are made.

37

Enhanced products are developed from the core by adding distinctive features to target

specific market niches; enhanced products are typically the first products leveraged from the

core product. Enhanced products can be customized further to provide more choice if

necessary. Cost-reduced products are “scaled” or “stripped” down versions (e.g., less

expensive materials and fewer features) of the core which are targeted at price-sensitive

markets. Finally the hybrid product is an entirely new design, resulting from the

combination of characteristics of two or more core products. As an example, the evolution

of three generations of a family of vacuum cleaners is mapped and discussed in their article.

These product family maps are useful attention directing tools for product family

design and development but offer little direction for designing a scalable product platform.

Toward this end, the market segmentation grid developed by Meyer (1997) facilitates

identifying leveraging strategies for a product platform, see Figure 2.4.

What Market NichesWill Your Product Platforms Serve?

High CostHigh Performance

Mid-Range

Low CostLow Performance

Segment A Segment B Segment C

Product Platform

Derivative Products

Figure 2.4 Product Platform Market Segmentation Grid(adapted from Meyer, 1997)

38

In a market segmentation grid, the major market segments serviced by a company’s

products are listed horizontally in the grid. The vertical axis reflects different tiers of price

and performance within each market segment. Several example instantiations of this grid

can be found in (Meyer, 1997; Meyer and Lehnerd, 1997) for companies such as Hewlett

Packard, Compaq, Steelcase, and Herman Miller.

This simple market segmentation grid can be used by firms to segment their

markets, helping to define a clear product platform strategy. For instance, a marketing

strategy which employs no leveraging is shown in Figure 2.5a. Companies which fail to

maintain a good platform leveraging strategy often have too many products that share too

little technology, resulting in a myriad of products, higher costs, and lower margins.

Three types of platform leveraging strategies can be identified within the market

segmentation grid: horizontal leveraging, vertical leveraging, and a beachhead approach as

shown in Figure 2.5b-d. All three leveraging strategies enable a more efficient and

effective product family to be developed. Examples of these leveraging strategies include

the following (Meyer, 1997):

Horizontally leveraging - subsystems of components within a product family are

leveraged from one market segment to the next within a given price/performance

tier. The main benefit of a horizontal leveraging strategy is to facilitate the

introduction of new products across a series of related market niches without

having to “reinvent the wheel.” Black & Decker often employs such a strategy:

“We don’t need to reinvent the power tool in every country, but rather, we have a

common product and adapt it to individual markets” states one of their top

executives (DiCamillo, 1988). Another example given by Meyer and his coauthors

is the Gillette Sensor-Excel razor which uses exactly the same razor cartridge in the

male and female market segments while the shape, color, and general design of the

handles are completely different for men’s and women’s.

Vertically leveraging - a product platform is leveraged to address a range of

price/performance tiers within a specific market segment. A company which excels

in the high-end segment of its market may scale down its platform into lower

39

price/performance tiers by removing functionality from its high-end platform to

achieve lower price products. The other option is to scale up a low-end platform by

adding more powerful component technologies or modules to meet the higher

performance demands for the higher tiers. The main benefit of this approach is the

capability of the company to leverage its knowledge about a particular market niche

without having to develop a new platform for each price/performance tier. The

Rolls Royce RTM322 engine and Canon’s low-end copiers discussed in Section

1.1.1 exemplify this approach.


Mid-Range




Mid-Range




Mid-Range




Mid-Range


Platform 1 Platform 2 Platform 3

Platform 5Platform 4


Platform A

Platform C

Scaled D

own S

cale

d U

p

High End Platform Leverage

Low End Platform Leverage

Platform

(a) No Leveraging (b) Horizontal Leveraging

(c) Vertical Leveraging (d) Beachhead Strategy

Figure 2.5 Platform Leveraging in the Market Segmentation Grid(adapted from Meyer, 1997)

Beachhead approach - combines horizontal and vertical leveraging to achieve

perhaps the most powerful platform leveraging strategy. In a beachhead approach,

a company develops a low-cost effective platform for a particular market segment

and then scales up the performance characteristics of the platform and adds other

features to target new market segments. The example of Compaq computers is

offered in (Meyer, 1997). Compaq entered the personal computer market in 1982

and, after establishing a foothold in the portable computer market niche, slowly

40

introduced a stream of new products for other market segments and different

price/performance tiers, including a line of desktop PCs for business and home use.

Of the examples discussed in Section 1.1.1, the Sony Walkman, Black & Decker’s

universal electric motor platform, and Lutron’s lighting systems also exemplify this

type of approach to platform leveraging. Sony initiated a beachhead approach from

the start with their Walkman product lines. The same is not true for Black &

Decker and Lutron. Both companies began with no leveraging strategy and only

after redesigning their product lines as discussed in Section 1.1.2 where they able

to achieve a more efficient and effective beachhead approach. Consequently, they

are now both leaders in their respective fields.

The market segmentation grid provides a useful attention directing tool to help map

and idenfity product platform leveraging opportunities within a product family, providing

an answer to the question:



Keep in mind, however, that the market segmentation grid is only an attention directing

tool; considerable engineering “know-how” and planning is needed to develop a successful

product leveraging strategy and exploit scaling opportunities within a product family. The

market segmentation grid is simply a way of representing that strategy, providing a clear

mapping of product leveraging opportunities within the product family. Use of the market

segmentation grid to help identify scaling opportunities within the Product Platform

Concept Exploration Method is further elaborated in Section 3.1.1. In the next section,

metrics for assessing product platforms are discussed.

41

2 .2 .2 Product Platform Assessments and Cost Models

Several metrics and cost models have been developed to assess either the efficiency

and effectiveness of a product platform or the commonality between a group of products

within a product family. Meyer, et al. (1997), in particular, define two metrics—platform

efficiency and platform effectiveness—to manage the research and development costs of

product platforms and product families. Platform efficiency is defined as follows:

Platform Efficiency = R & D Costs for Derivative Product

R & D Costs for Platform Version[2.1]

which assesses how much it costs to develop derivative products relative to how much it

costs to develop the product platform within the product family. The platform efficiency

metric can also be used to compare different platforms across different product families to

assess the capability of R&D teams to develop robust and efficient platforms.

Platform effectiveness is a ratio of the revenue a product platform and its derivatives

creates to the cost required to develop them and is measured as follows:

Platform Effectiveness = Product Sales

Product Development Costs[2.2]

where the effectiveness of the platform can be assessed at the individual product level or for

a group of products within distinct platform versions.

These metrics require costing and revenue information which is typically known

only after the product platform and its derivatives have been developed and reached the

market. These metrics prove useful for managing research and development within the

product family and determining when to renew or re-focus product platform efforts;

however, they offer little for designers during the concept exploration and design process.

42

A more relevant measure of the effectiveness of a product platform is to measure the

commonality of parts within a product family. Many commonality indices have been

proposed for assessing the degree of commonality within a product family. Products

which share more parts and modules within a product family achieve greater inventory

reductions, exhibit less part variability, improve standardization, and shorten development

and lead times because more parts are reused and fewer new parts have to be designed (cf.,

Collier, 1981). McDermott and Stock (1994) discuss the benefits of commonality on new

product development time, inventory, and manufacturing; they also cite several researchers

who have shown that part commonality across a range of products has reduced inventory

costs while maintaining a desired level of customer service. Particular measures for

assessing commonality includes the following:

• Collier (1981; 1982) proposes the Degree of Commonality Index, an analytical

measure based on a firm’s bills of materials, which can be applied to a single

product, a product line, or a product family. He uses regression analysis to relate

the Degree of Commonality Index to inventory carrying cost, set-up cost, total

costs, average work center load, and the variability in work center loads. He

concludes that component part commonality can have a significant impact on system

performance, reducing manufacturing costs, total costs, and delivery performance.

• Kota and Sethuraman (1998) introduce the Product Commonality Index for

determining the level of part commonality in a product family. Through the study

of a family of portable personal stereos, they illustrate methods to “measure and

eliminate non-value added variations, suggest robust design strategies including

modularity and postponement of product differentiation.” Their approach provides

a means to benchmark product families based on their capability to simultaneously

share parts effectively and reduce the total number of parts.

• Siddique, et al. (1998) propose a commonality index to aid in the configuration

design of common automotive platforms. They are working with an automobile

manufacturer to reduce the number of platforms they utilize across their entire range

of cars and trucks in an effort to reduce development times, costs, and product

43

variety. Ongoing research efforts for measuring the “goodness” of a common

platform are discussed in (Siddique, 1998).

Commonality measures such as these are based primarily on the ratio of the number

of shared parts, components, and modules to the total number of parts, components, and

modules in the product family. Taking this one step further, Martin and Ishii (1996) seek

to assess the cost of producing product variety through the measurement of three indices:

commonality, differentiation point, and set-up costs. The commonality index is similar to

that proposed by Collier (1981) and measures the percentage of common components

within a group of products in a product family. The second index measures the

differentiation point for product variety within an assembly or manufacturing process; the

idea being that the later the differentiation point can be postponed the lower the costs of

producing the necessary variety (cf., Lee and Billington, 1994; Lee and Tang, 1997).

Finally, the set-up cost index assesses the cost contributions needed to provide variety

compared to the total cost for the product. The indirect costs of providing product variety

then is taken as a weighted linear combination of these indices; the weightings for the

individual indices may vary from industry to industry. The direct costs of providing

product variety, they assert, are relatively straightforward to determine. Generalizations are

made regarding the costs of product variety based on these indices; however, there is no

work to substantiate their claims or the usefulness of the indices. The generalizations could

be made just as easily without the indices.

In later work, Martin and Ishii (1997) introduce a process sequence graph which

provides a qualitative assessment of the flow of a product through the assembly process

and its differentiation point. A product family of eighteen instrument panels is analyzed,

citing that differentiation for product variety begins in the second step in the assembly

process. This leads them to investigate process re-sequencing to improve component

commonality and postpone differentiation to reduce production costs and lead-times. The

44

end result is a graph of Variety Voice of the Customer (V2OC) versus percentage

commonality for the family of instrument panels as shown in Figure 2.6.

Figure 2.6 V2OC Rating vs. Commonality (from Martin and Ishii, 1997)

In Figure 2.6, commonality is a measure the number of common or standardized

assemblies shared between products to the total number of assemblies in the product family

and is again very similar to that of Collier (1981). The V2OC measure assesses “the

importance of a component’s variety to the aggregated market—not the individual buyer.

V2OC is a measure the importance of a component to a customer, as well as the

heterogeneity of the market with response to that component” (Martin and Ishii, 1997).

They do not describe how to measure V2OC or explain how the V2OC ratings for the

instrument panel family are created; consequently, V2OC does not provide a useful measure

for product variety. The resulting graph, however, is insightful and similar to the product

variety tradeoff graph which is introduced in Section 3.1.5 and illustrated in Section 7.6.2

in the context of the General Aviation aircraft example. The reasoning behind the target

region in the figure is not discussed in their paper either; however, intuition suggests that

components with low V2OC rating (i.e., are not important to the customer) can be common

from one product to the next while it is important to customize components (i.e., decrease

their commonality) that have a high V2OC. This idea is explored in greater depth in

45

Sections 3.1.5 and 7.6.2 wherein a non-commonality index based on the dissimilarity of a

family of products defined by a parameter set is introduced for assessing and studying

product variety tradeoffs. In the meantime, methods for designing families of products are

reviewed in the next section.

2 .2 .3 Engineering Methods for Product Family Design

The majority of engineering design research has been directed at improving the

efficiency and effectiveness of designers in the product realization process, and until

recently, the focus has been on designing a single product. For instance, Suh (1990)

offers his two axioms for design: (1) maintain independence of functional requirements,

and (2) minimize the information content of a design. Pahl and Beitz (1988; 1996) offer

their four phase approach to product design which involves the following: clarification of

the task, conceptual design, embodiment design, and detail design. Similarly, Hubka and

Eder (1988; 1996) advocate an approach which involves the following: elaboration of the

assigned problem, conceptual design, laying out, and elaboration. Pugh (1991) introduces

the notion of total design which has at its core market/user needs and demands, the product

design specification, conceptual design, detail design, manufacturing, and selling. In the

well-known review of mechanical engineering design research conducted by Finger and

Dixon (1989a; 1989b), scant trace of product family and product platform design is found.

Perhaps the most developed method for product family design which currently

exists is the work by Erens (1997). Erens, in conjunction with several of his colleagues

(Erens and Breuls, 1995; Erens and Verhulst, 1997; Erens and Hegge, 1994; McKay, et

al., 1996), develops a product modeling language for product variety. The primary focus

is on the product modeling language as an efficient representation of product architecture

and modularity; it offers little aid for design synthesis and analysis, only representation.

The product modeling language allows product families to be represented in three domains:

46

functional, technological, and physical. Use of the product modeling langurage is

demonstrated in the context of a family of office chairs, a family of overhead projectors,

and a family of cardio-vascular Xray machines. Excerpts from the family of office chairs

example is illustrated in Figure 2.7. The office chair itself is shown in Figure 2.7a, and the

variety of options from which to choose: upholstery, materials, colors, fixtures, etc., are

shown in Figure 2.7b. In Figure 2.7c, the general representation of the product

architecture for the office chair is depicted, and the hierarchy in the product variety model is

illustrated in Figure 2.7d. As illustrated in this example, the product modeling language

provides an effective means for representing product variety but offers little aid for design

synthesis and analysis.

(a) An office chair (b) Office chair options

(c) Office chair architecture (d) Office chair product variety model

Figure 2.7 Representing a Family of Office Chairs (Erens, 1997)

47

In other work, Fujita and Ishii (1997) outline a series of tasks—design specification

analysis, system structure synthesis, configuration, and model instantiation—for product

variety design as their foundation for a formal approach for the design and synthesis of

product families. They decompose product families into systems, modules, and attributes

as shown in Figure 2.8. Under this hierarchical representation scheme, product variety can

be implemented at different levels within the product architecture. For instance, two shared

modules and two sets of shared attributes are shown in Figure 2.8. A formal algorithm has

not yet been developed however.

System Modules Attributes

Shared

Con

figu

ratio

n/G

eom

etry

Arc

hite

ctur

e

Different

Shared Shared

Functional/Physical

Figure 2.8 Product Variety Decomposed into Systems, Modules, andAttributes (from Fujita and Ishii, 1997)

Clustering approaches based on similarity or commonality of products have been

investigated for product family design. Stadzisz and Henrioud (1995) cluster products

48

based on geometric similarities to obtain product families in order to decrease product

variability within a product family in order to minimize the required flexibility of the

associated assembly system. A similar Design for Mass Customization approach is

developed in (Tseng, et al., 1996) which groups similar products into families based on

product topology or manufacturing and assembly similarity and provides a series of steps

to formulate an optimal product family architecture based on grouping according to

fulfillment of functional requirements. Shirley (1990) describes a process for redesigning a

set of related products through similarity and clustering of common products around a

“core product concept,” i.e., a product platform. The resulting product family is composed

of a set of product variants which share characteristics in common with the core product;

the redesign of a family of hydraulic cylinders is used as an example.

Numerous researchers have focused on the implications of modularity on product

variety and in the context of a product platform and product family. Modularity greatly

facilitates the addition and removal of features to upgrade and derate a product platform

(cf., Ulrich, 1995). Ulrich and Tung (1991), Ulrich (1995), and Ulrich and Eppinger

(1995) investigate product architecture and modularity and its the impact on product

change, product variety, component standardization, product performance, and product

development management. Similarly, Uzumeri and Sanderson (1995) emphasize flexibility

and standardization as a means for enhancing product flexibility and offering a wide variety

of products. Meanwhile, Chen, et al. (1994) suggest designing flexible products which

can be adapted readily in response to large changes in customer requirements by changing a

small number of components or modules. Sanderson (1991) investigates how modular

designs reduce the cost of offering product variety. Rosen (1996) investigates the use of

discrete mathematics as a formal foundation for configuration design of modular product

architectures. He emphasizes, as do Ulrich and Eppinger (1995), that the design of

product architectures is “critical in being able to mass customize products to meet

49

differentiated market niches and satisfy requirements on local content, component carry-

over between generations, recyclability, and other strategic issues.” A Product Module

Reasoning System (Newcomb, et al., 1996) currently is being developed “to reason about

sets of product architectures, to translate design requirements into constraints on these sets,

to compare architecture modules from different viewpoints, and to directly enumerate all

feasible modules without generate-and-test or heuristic approaches” (Rosen, 1996).

Pahl and Beitz (1996) also discuss the advantages and limitations of modular

products to fulfill various overall functions through the combination of distinct modules.

Because such modules often come in various sizes, modular products often involve size

ranges where the initial size is the basic design and derivative sizes are sequential designs.

In the context of a scalable product platform, the initial size constitutes the product platform

and the derivative sizes are its product variants. Their approach for designing size ranges is

as follows (Pahl and Beitz, 1996):

• Prepare the basic design for the range either from a new or existing product;

• Use similarity laws to determine the physical relationships between geometrically

similar product ranges;

• Determine appropriate “theoretical” steps sizes within the desired size range;

• Adapt “theoretical” step sizes to overriding standards or technical requirements;

• Check the product size range against assembly layouts, checking any critical

dimensions; and

• Improve and document the design and prepare production drawings.

In the context of their approach, the method developed in this dissertation facilitates the

development of the basic design (i.e., the platform) and the sequential designs (i.e.,

derivative products) simultaneously.

The concept of sizing leads into an area of product platform design that has received

little attention—product platforms that can be “scaled” or “stretched” into derivative

50

products for a product family (in combination to being upgraded/degraded through the

addition/removal of modules). The implications of design “stretching” and “scaling” within

the context of developing a family of products are discussed first in (Rothwell and

Gardiner, 1988; 1990), see Figure 2.9. Rothwell and Gardiner (1988) use the term

“robust designs” to refer to designs that have sufficient inherent design flexibility or

“technological slack” to enable them to evolve into a design family of variants that meet a

variety of changing market requirements by “uprating,” “rerating,” and “derating” a

platform design as shown in Figure 2.9. The process of developing these designs is

shown in Figure 2.9 and consists of three phases, namely, composite, consolidated, and

stretched designs as explained in the bottom of each column.

Figure 2.9 Robust Designs (from Rothwell and Gardiner, 1990)

51

Rothwell and Gardiner (1990) provide several examples of successful robust

designs and discuss how they “allow for change because essentially they contain the basis

for not just a single product but rather a whole product family of uprated or derated

variants.” Consider the Rolls Royce RB211 engine family illustrated in Figure 2.10. The

original RB211 consisted of seven modules which could be easily upgrade or scaled down

to improve or derate the engine. For example, by replacing the large front low pressure fan

with a scaled down fan, the lower thrust, derated, 535C engine was derived. Further

improvements are made by scaling different components of the engine to improve fuel

consumption while increasing thrusts. Rolls Royce takes advantage of similar stretching

and scaling in its RTM322 engine which was discussed previously in Section 1.1.1.

Figure 2.10 Rolls Royce RB211 Engine Family(from Rothwell and Gardiner, 1990)

Several other products also have benefited from platform scaling. For example,

Black & Decker scales the stack length of their universal motor platform to vary the output

power of the motor for a wide variety of applications, see Section 1.1.1 and (Lehnerd,

1987). The Boeing 747-200, 747-300, and 747-400 are scaled derivatives of the Boeing

747 (Rothwell and Gardiner, 1990). Many automobile manufacturers also scale their

52

passenger car platforms to offer, for example, two-door coupes, two- and four-door

sedans, three- and five-door hatchbacks, and maybe a wagon which are all derived from

the same platform (Rothwell and Gardiner, 1990). Honda, for instance, is taking full

advantage of platform scaling to compete in today’s global market by developing two

scaled versions of their Accord for the U.S. and Japanese markets from one platform

(Naughton, et al., 1997). Siddique, et al. (1998) document efforts at Ford to improve the

commonality of their product platforms to capitalize on commonality and stretching within

their automotive product families.

Despite the apparent advantages of scalable product platforms, a formal approach

for the design and synthesis of stretchable and scalable platforms does not exist. Rothwell

and Gardiner state that it has “become increasingly possible to develop a robust design

which has the deliberate designed-in capability of being stretched;” however, they only

offer the process shown in Figure 2.9 as a guide to designers. Consequently, developing a

method to model and design scalable product platforms around which a family of products

can be developed through scaled derivatives of the product platform is the principal

objective in this dissertation. In an effort to realize such a method, an extension of robust

design principles is offered in the next section, providing a means to turn Rothwell and

Gardiner’s idea of “robust design” for scalable product platforms into a reality.

2.3 ROBUST DESIGN

Generally speaking, the fundamental motive underlying robust design, as originally

proposed by Taguchi, is to improve the quality of a product or process by not only striving

to achieve performance targets but also by minimizing performance variation. Taguchi’s

methods have been widely used in industry (see, e.g., Byrne and Taguchi, 1987; Phadke,

1989) for parameter and tolerance design. Reviews of such applications can be found in,

e.g., (Nair, 1992).

53

In robust design, the relationship between different types of design parameters or

factors can be represented with a P-diagram as shown in Figure 2.11, where P represents

either product or process (Phadke, 1989). The three types of factors which serve as inputs

to the P-diagram and that influence the (output) response y are as follows:

• Control Factors (x) – parameters that can be specified freely by a designer; the

settings for the control factors are selected to minimize the effects of noise factors

on the response y.

• Noise Factors (z) – parameters not under a designer’s control or whose settings

are difficult or expensive to control. Noise factors cause the response, y, to deviate

from their target and lead to quality loss through performance variation. Noise

factors may include system wear, variations in the operating environment, uncertain

design parameters, and economic uncertainties.

• Signal factors (M) – parameters set by the designer to express the intended value

for the response of the product; signal factors are those factors used to adjust the

mean of the response but which no effect on the variation of the response.

Product / Process

Noise Factors

ResponseSignal Factors

Control Factorsx

z

M y

µz , σz

µy , σy

Figure 2.11 P-Diagram of a Product/Process in Robust Design(adapted from Phadke, 1989)

This robust design terminology is used to classify design parameters and responses

and to identify sources of variability. The objective in robust design is to reduce the

variation of system performance caused by uncertain design parameters, thereby reducing

54

system sensitivity. Variations in noise factors, shown in Figure 2.11 as normally

distributed with mean µz and standard deviation σz, lead to variation in performance

responses, which are represented in Figure 2.11 as normally distributed with mean µy and

standard deviation σy.

In an effort to generalize robust design for product design, Chen, et al. (1996a)

develop a general robust design procedure based on two sources of variation:

Type I - Robust design associated with the minimization of the deviation of

performance caused by the deviation of noise factors (uncontrollable parameters).

Type II - Robust design associated with the minimization of the deviation of

performance caused by the deviation of control factors (design variables).

The idea behind the two major types of robust design applications are illustrated in Figure

2.12. As indicated by the P-diagrams for Type I and Type II applications, the deviation of

the response is caused by variations in the noise factor, z, the uncontrollable parameter in

Type I applications. Type II is different from Type I in that its input does not include a

noise factor. The variation in performance is caused solely by variations in control factors

or design variables in the region ±∆x.

The traditional Taguchi robust design method is of Type I as shown in the top half

of Figure 2.12. A designer adjusts control factors, x , to dampen the variations caused by

the noise factor, z . The two curves represent the performance variation as a function of

noise factor when x is at two different levels, x = a and x = b. If the design objective is to

achieve a performance as closely as possible to the target, M, the designs at both levels are

acceptable because their means are the target M. However, introducing robustness, when

x = a, the performance varies significantly with the deviation of noise factor, z ; however,

when x = b, the performance deviates much less. Therefore, x = b is more robust than x

55

= a as a design solution because x = b dampens the effect of the noise factors more than

when x = a.

y =

Res

pons

e

M =

Sig

nal F

acto

rs

x = Control Factors

z =Noise Factors

Type I

x = Control Factors

M =

Sig

nal F

acto

rs

Type II

y =

Res

pons

e

∆x∆x

y

xrobustµ

M

OptimalSolution

RobustSolution

DesignVariable

Objective orDeviationFunction

optx

Noise Factor, z

x = a

x = b

M

Control Factor


y

(x = b)(x = a)

Figure 2.12 Two Types of Robust Design(adapted from Chen, et al., 1996b)

The concept behind Type II robust design is represented in the lower half of Figure

2.12. For purposes of illustration, assume that performance is a function of only one

variable, x . In general, for this type of robust design, to reduce the variation of the

response caused by the deviations of design variables, a designer is interested in the flat

56

part of a curve near the performance target instead of seeking the peak or optimum value.

If the objective is to move the performance function towards target M and if a robust design

is not sought, then the point x = a is chosen. However, for a robust design, x = b is a

better choice. This is because if the design variable varies within ±∆x of its mean, the

resulting variation of response of the design at x = b is much smaller than that at x = a,

while the means of the two responses are essentially equal. Implementation of these two

types of robust design are discussed in the next section.

2 .3 .1 Implementation of Robust Design: Taguchi’s Method, in the RobustConcept Exploration Method, and with Design Capability Indices

Design of experiments, specifically orthogonal arrays (OA), are typically employed in

Taguchi’s robust design method to systematically vary and test the different levels of each

of the control factors. Taguchi advocates the use of an inner-array and outer-array

approach to implement robust design (cf., e.g., Byrne and Taguchi, 1987). The inner-

array consists of an OA which contains the control factor settings; the outer-array consists

of the OA which contains the noise factors and their settings which are under investigation.

The combination of the inner-array and outer-array constitutes the product array. The

product array is used to test various combinations of the control factor settings

systematically over all combinations of noise factors after which the mean response and

standard deviation may be approximated for each run using the equations:

• Response mean: y =1

nyi

i =1

n

∑ [2.3]

• Standard deviation: S =(y i − y )2

n −1i=1

n

∑ [2.4]

57

Preferred parameter values then can be determined through analysis of the signal-to-noise

(SN) ratio; factor levels that maximize the appropriate SN ratio are optimal. There are three

“standard” types of SN ratios (see, e.g., Phadke, 1989):

• Nominal the best (for reducing variability around a target):

SNT = 10logy 2

S2

[2.5]

• Smaller the better (for making the system response as small as possible):

SNL = −10log1

n

1

yi2

i =1

n

∑

[2.6]

• Larger the better (for making the system response as large as possible):

SNS =− 10log1

ny i

2

i =1

n

∑

[2.7]

Once all of the SN ratios have been computed for each run of an experiment, there

are two common options for analysis: Analysis of Variance (ANOVA) and a graphical

approach. ANOVA can be used to determine the statistically significant factors and the

appropriate setting for each. In the graphical approach, the SN ratios and average

responses are plotted for each factor against its levels. The graphs then are examined to

“pick the winner,” i.e., pick the factor levels which (1) best maximize SN and (2) bring the

mean on target (or maximize or minimize the mean, as the case may be).

There are many criticisms of Taguchi’s implementation of robust design through the

inner and outer array approach: it requires too many experiments, the analysis is statistically

questionable because of the use of orthogonal arrays, it does not accommodate constraints,

and the responses should be modeled directly instead of the SN ratios (see, e.g.,

58

Montgomery, 1991; Nair, 1992; Otto and Antonsson, 1993; Shoemaker, et al., 1991;

Tribus and Szonyi, 1989; Tsui, 1992). Consequently many variations of the Taguchi

method have been proposed and developed; a review of numerous robust design

optimization methods can be found in (Otto and Antonsson, 1993; Simpson, et al., 1997a;

Simpson, et al., 1997b; Su and Renaud, 1996; Tsui, 1992; Yu and Ishii, 1998).

To facilitate the implementation of robust design within the RCEM, second-order

response surface models are created and used to approximate the design space, replacing

the computer analysis code or simulation routine used to model the system. The major

elements of the response surface model approach for robust design applications are as

follows (see, e.g., Myers and Montgomery, 1995; Shoemaker, et al., 1991):

• combining control and noise factors in a single array instead of using Taguchi's

inner- and outer-array approach,

• modeling the response itself rather than expected loss, and

• approximating a prediction model for loss based on the fitted-response model.

Instead of using Taguchi’s orthogonal array as the combined array for experiments, central

composite designs are employed in the RCEM to fit second-order response surface models

for integration with Taguchi's robust design. The response surface model postulates a

single, formal model of the type:

ˆ y = f(x ,z) [2.8]

where ˆ y is the estimated response and x and z represent the settings of the control and

noise variables, respectively. In Equation 2.8, it is assumed that the noise variables are

independent. From the response surface model, it is possible to estimate the mean and

variance of the response. For Type I applications in which the deviations of noise factors

are the source of variation:

59

• Mean of response: µ ˆ y = f(x ,µz) [2.9]

• Variance of response: σ ˆ y 2 =

∂f

∂z i

2

i =1

m

∑ σ z i

2 [2.10]

where µ represents the mean values, m is the number of noise factors in the response

model, and σ z i is the standard deviation associated with each noise factor. In Type II

robust design, i.e., when the deviations of control factors are the source of variation, µz

and σ z i in Equations 2.9-2.10 are replaced by the mean and deviation of the variable

control factors. Using this approach, robust design can be achieved by having separate

goals for “bringing the mean on target” and “minimizing the deviation” within a

compromise DSP (cf., Chen, et al., 1996b).

When satisfying the design requirements and reducing the variation of system

performance are equally important, it is effective to model the two aspects of robust design

as separate goals in the compromise DSP. For instance, when designing a power plant, it

may be required to bring the power output as close as possible to its target value while at

the same time, reduce the variation of the system performance so that the power output

remains constant during operation. Moreover, setting an overall design requirement at a

specific value during the early stages of design may sometimes be crucial because a small

variation may require significant changes in other design requirements or incur substantial

costs in order to compensate for it. However, modeling the two aspects of robust design

as two separate goals may not be an effective approach when satisfying a range of design

requirements is the major concern, see Figure 2.13.

In Figure 2.13, the quality distributions of two different designs (I and II) are

illustrated. Both designs have the same mean value but different deviations. If the two

aspects of robust design are modeled as separate goals, obviously the design with the least

60

deviation (Design I) would be chosen because both designs have the same performance

mean. However, in this particular situation where the mean of the quality performance lies

outside the range of requirements, a smaller fraction of the performance falls inside the

upper and lower requirement limits (URL and LRL, respectively) with a thinner bell shape,

i.e., the shadowed area which is enclosed by A, B, and C is smaller than the area enclosed

by A', B' and C. This is acceptable in manufacturing when the process itself can be

manually shifted to bring the mean back on target, but when designing a system to

accommodate noise, this option is not always available.

URLLRL Target

mean, µ

Requirement Range

Dis

trib

utio

n

A

A’

BB’ C

Design II

Design I

Designs meeting

requirements

Figure 2.13 A Motivating Example for Design Capability Indices

Design capability indices have been developed with exactly this in mind. They are

based on process capability indices from statistical process control and apply in the same

manner to a design with variation as to a manufacturing process with variation.

Specifically, a design capability index (see Figure 2.14) is computed to assess the

capability of a family of designs to satisfy a ranged set of design requirements (Chen, et

al., 1996c; Simpson, et al., 1997a).

61

3 σ

LRL

C dlµ

- LRLµURL

C du

URL - µ

µ

3σ

LRL URLTarget

Cdl Cdu

- LRLµ URL - µ

3σ3 σ

C ≥ 1dk

µ

Larger is Better• Target above LRL• All variants above LRL• Desire C ≥ 1, C = C

Smaller is Better• Target below URL• All variants below URL• Desire C ≥ 1, C = Cdk

Nominal is Better• Target between URL and LRL• All variants between URL and LRL• Desire C ≥ 1, C = min {C , C }dk

C ≥ 1dkC = Cdudk

C ≥ 1dkC = Cdldk

dk du dk dk dldudldk

Figure 2.14 Implementation of Design Capability Indices for RobustDesign Applications

Assume that the system performance is normally distributed with mean, µ, and

standard deviation, σ. The design capability indices Cdl, Cdu , and Cdk measure the extent to

which a family of designs satisfies a ranged set of design requirements as specified by

upper and lower requirement limits (URL and LRL, respectively). As shown the figure,

when nominal is better, i.e., upper and lower design requirement limits are given, finding a

family of designs with Cdk ≥ 1 satisfies the design requirements. In this scenario, Cdk is

computed using Equation 2.11; Cdk is taken as the minimum of Cdl and Cdu .

Cdl =(ˆ µ − LRL)

3ˆ σ ;C du =

(URL − ˆ µ )

3ˆ σ ;Cdk = min{Cdl ,Cdu} [2.11]

When smaller is better (e.g., “the motors should weigh less than 0.5 kg”) designs

with a Cdk ≥ 1 are capable of satisfying the requirement. In this case, Cdk = Cdu as shown

in Figure 2.14, and designs with a Cdu < 1 do not meet this requirement because a portion

of the distribution falls outside of the URL. Similarly, when larger is better (e.g., “the

62

efficiency of these motors should be 30% or better”), designs with a Cdk ≥ 1 are capable of

meeting this requirement; Cdk = Cdl.

There are some assumptions associated with the use of Cdk . For example, Cdk = 1

implies only 99.73% of the designs conform to requirements assuming that the system

performance parameters—which are based on the range of designs—are normally

distributed. However, the type of distribution of system performance depends on the

actual system response and the statistical distribution of each design variable or uncertainty

parameter. When the system function is complex, it may be difficult to perform a judicious

evaluation to determine performance distribution. As an approximation, if it is assumed

that the uncertain parameters deviate by ±3σz (as is typical in a six sigma approach to

quality) around their nominal value µz, and that each system response varies by ±3σy

around its mean value, µy, which can be calculated by:

µy = y (µx) [2.12]

The standard deviation, σy, is approximated using a first order Taylor series expansion

(assuming σx is small):

ˆ σ y2 =

∂y

∂z i

2

i =1

m

∑ σ z2 [2.13]

Modifications to the process capability indices for different variances have been

proposed (see, e.g., Johnson, et al., 1992; Ng and Tsui, 1992; Rodriguez, 1992), and

design capability indices could be modified similarly. For example, if a uniform

distribution is used for each response instead of a normal distribution, then Cdk , Cdu , and

Cdl become as follows:

63

Cdl =(ˆ µ − LRL)

3ˆ σ ;C du =

(URL − ˆ µ )

3 ˆ σ ;Cdk = min{Cdl ,Cdu} [2.14]

where the standard deviation is computed using the following:

ˆ σ 2 =(b − a)2

12[2.15]

where a and b are the lower and upper limits of the range of y.

The compromise DSP mentioned in Section 1.2.1 is modified to implement Cdk as

shown in Figure 2.15. Design capability indices can be used for constraints and/or goals,

depending on whether satisfying a range of design requirements is a wish or a demand. In

all three cases—smaller is better, nominal is better, and larger is better—if a design

requirement is a wish, then making Cdk as close to one as possible is a goal in the

compromise DSP. When a requirement is a demand, then Cdk ≥ 1 is taken as a constraint.

Note that when a deviation function solely includes design capability indices, the negative

deviation variable, di-, is always minimized, and di

+ is always zero.

Given:• Functions y(x), including those ranged design requirements which are constraints,

gi(x), and those which are objectives, Ai(x)• Deviations of the uncontrollable variables, σZ • Target upper and lower design requirement limits, URLi and LRLi

Find:• Design variables, x

Satisfy:• Constraints: Cdk-constraints ≥ 1 (or use worst-case analysis)• Goals: Cdk-objectives + di

- - di+ = 1

• Bounds on the design variables• di

- , d i+ ≥ 0 ; di

- • di+ = 0

Minimize:• Deviation Function: Z = [f1(di

-, ..., d i+)...fk(di

-, ..., d i+)]

Figure 2.15 Compromise DSP Formulation with Design Capability Indices

64

Working with these two types of robust design implementations, it is possible to

develop an extension for product family design, specifically for scalable product platforms,

which addresses the question:

Q1.2. How can robust design principles be used to facilitate the design a common


Extensions of robust design for product family design are discussed in the next section.

2 .3 .2 Robust Design for Product Family Design: Scale Factors

There have been two known allusions to using robust design in product family

design. First, Lucas (1994) describes a way that the results of a robust design experiment

can be used to identify the need for product differentiation. When large effects are present

in the system, different product types can be sent to customers having different features as

opposed to designing one product which is robust over the entire range of effects. He

states that this is common practice in the chemical industry where, for example, different

polymer viscosities are desired by different customers and better results often are obtained

by customizing the product for its specific environment rather than delivering a single

robust product.

Second, Chang, et al. (1994) and Chang and Ward (1995) introduce the notion of

“conceptual robustness” which is pertinent to this research. The term “conceptual

robustness” is developed by Chang and his colleagues for mathematically modeling and

computationally supporting simultaneous (or concurrent) engineering in a distributed

design environment. By treating variations in the design proposed by other members of the

development team as “conceptual noise,” robust design principles can be used to make

“conceptually robust” decisions which are robust against these variations (Chang, et al.,

65

1994). The “conceptually robust” design of a two-axis CNC milling machine is used as an

illustrative example. In (Chang and Ward, 1995), this idea is applied to modular design

which is a “function-oriented design that can be integrated into different systems for the

same functional purpose without (or with minor) modifications.” The design of an air

conditioning system for ten different automobiles is used to demonstrate their approach.

It is this idea of a “conceptual noise factor” that enables the utilization of robust

design in the context of product family design, particularly in the design of a scalable

product platform. By identifying an appropriate “scale factor” for a scalable product

platform, robust design principles can be used to minimize the sensitivity of the product

platform to variations in a scale factor. In this regard, a “conceptually robust” product

platform can be realized which has minimum sensitivity to variations in the scale factor,

realizing a robust product family. For the work in this dissertation, then, a scale factor is

defined as follows:

• Scale factor - factor around which a product platform can be “scaled” or

“stretched” to realize derivative products within a product family.

In essence, a scale factor is a noise factor within a scalable product family or, to

borrow terminology from Chang, et al. (1994)), a “conceptual noise factor” around which

a “conceptually robust” product platform can be developed for a product family. Examples

of scale factors include the stack length in a motor, as in the Black & Decker universal

motor example (Lehnerd, 1987), the number of passengers on an aircraft ,as in the Boeing

747 family (Rothwell and Gardiner, 1990), or the number of compressor stages in an

aircraft engine, as in the Rolls Royce RTM322 example (Rothwell and Gardiner, 1990).

Scale factors may be either discrete or continuous; however, continuous scale factors are

66

investigated primarily in this dissertation. The specific relationship between different types

of scale factors and different platform leveraging strategies is discussed in Section 3.1.2.

Given the definition for a scale factor, a third type of robust design now can be

identified for product family design, complementing the two types of robust design

discussed previously:

Type III - Robust design associated with minimizing the sensitivity of a product

platform to variations in a scale factor.

As defined, Type III robust design is nearly identical to Type I robust design as shown in

Figure 2.16. Notice that the P-diagram on the left of the figure has been modified to

accommodate scale factors because essentially they are treated as noise factors in the

product platform design process.

y =

Res

pons

e

M =

Sig

nal F

acto

rs

x = Control Factors

S = ScaleFactor(s)

Type III

Scale Factor, S

x = a

x = b

M

Platform DesignVariable


y

z =NoiseFactors

Figure 2.16 Type III Robust Design: Scale Factors for Product Platforms

It should be noted that these scale factors are not the same “scaling/leveling factors”

shown in the P-diagram in (Taguchi and Phadke, 1986) or (Suh, 1990) which are used to

67

scale a response to achieve a desired value. Using the same diagram shown in Figure 2.12

for the Type I robust design, the idea behind Type III robust design is illustrated in the

right hand side of Figure 2.16. Given two possible settings (x = a and x = b) for one of

the design variables, x, which defines the platform, the setting x = b should be selected

because it minimizes the sensitivity of the product platform to variations in the scale factor.

Consequently, if a family of products can be leveraged or scaled around an

identifiable scale factor, then robust design can be used to minimize the sensitivity of the

product platform to changes in these scaling factors. In this manner, a scalable product

platform can be developed and instantiated to realize a family of products. This raises the

following question then.

Q1.3. How can individual targets for product variants be aggregated and modeled

for product platform design?

Using the concept of a scale factor for a product platform, it is now possible to

aggregate the individual targets for product variants within a product family around an

appropriate mean and a standard deviation. Robust design principles then can be used to

“match” the mean and standard deviation of the product family with the desired mean and

standard deviation of the performance requirements by either of the following as discussed

in Section 2.3.1:

• creating separate goals for “bringing the mean on target” and “minimizing the

deviation” of the product platform for variations in the scale factor within a

compromise DSP, or

• using design capability indices to assess the capability of a family of designs to

satisfy a ranged set of design requirements.

To demonstrate these implementations, the former approach is utilized in the universal

electric motor problem in Chapter 6; the latter is employed in the General Aviation aircraft

68

example in Chapter 7. The General Aviation aircraft example also makes use of

metamodels to facilitate the implementation of robust design and design capability indices

and expedite the concept exploration process as mentioned earlier. Metamodeling

techniques are discussed next.

2.4 METAMODELING TECHNIQUES

To facilitate the implementation of robust design, metamodeling techniques often

are employed to create approximations of the mean and variation of a response in the

presence of noise. A metamodel is a “model of a model” (Kleijnen, 1987) which is used as

a surrogate approximation for the actual analysis (i.e., computer code) during the design

process. The general approach to response surface modeling is shown in Figure 2.17. In

statistical terms, design variables are factors, and design objectives are responses; the

factors and responses to be investigated for a particular design problem provide the input

for the approach of Figure 2.17, and the solutions (improved or robust) are the output. To

identify these solutions, this approach includes three sequential stages: screening, modeling

building, and model exercising.

The first step (screening) is employed only if the problem includes a large number

of factors (usually greater than 10); screening experiments are used to reduce the set of

factors to those that are most important to the response(s) being investigated. Statistical

experimentation is used to define the appropriate design analyses which must be run to

evaluate the desired effects of the factors. Often two level fractional factorial designs or

Plackett-Burman designs are used for screening (cf., Myers and Montgomery, 1995), and

only main (linear) effects of each factor are investigated.

69

YESLarge # of Factors?

Run Screening Experiment

Reduce #Factors

Run Modeling Experiment(s)

Build Predictive Model ( )

Solutions

Search Design Space

NO

Given: Factors,

Responses

y

YES

NO

NoiseFactors?

Build Robustness Model ( )

Screening

Model Building

ModelExercising

(improved or robust)

µy , σy

Figure 2.17 General Approach to Metamodeling (Koch, et al., 1997)

In the second stage (model building) of the approach in Figure 2.17, response

surface models are created to replace computationally expensive analyses and facilitate fast

analysis and exploration of the design space. If little curvature appears to exist, a two level

fractional factorial experiment is designed, and the first-order polynomial of the form

y = β0 + βixi∑i=1

k

[2.16]

is used to approximate the response(s). If significant curvature exists, then a second-order

polynomial of the form:

70

y = β0 + βixi∑i=1

k

+ βiixi 2∑

i=1

k

+ βijxixj∑j

∑i

[2.17]

i<j

is commonly used. Among the various types of experimental design for fitting a second-

order response surface model, the central composite design (CCD) is probably the most

widely used experimental design for regularly shaped (spherical or cuboidal) design spaces

(cf., Myers and Montgomery, 1995). In the case of irregularly shaped design spaces, D-

optimal designs have been successfully employed to build second order response surface

models (see, e.g., Giunta, et al., 1994).

If noise factors are included for robust design, the mean and variance of each

response must be estimated, and predictive metamodels for both are constructed. As

discussed in (Koch, et al., 1998), there are essentially three approaches which can be

employed to construct metamodels for robust design applications:

1. Statistical expected value and Taylor series expansion approximations: A single

experimental design is created which contains both control and noise factors from

which a response surface model is built (see, e.g., Chen, et al., 1996b; Shoemaker,

et al., 1991). The mean value of a response is estimated by evaluating the response

surface at the mean of the noise factor, and the variance is estimated using a Taylor

series approximation. This is the approach currently employed in the RCEM as

described previously in Section 2.3.1.

2. DOE-based Monte Carlo simulation approach: A combined response surface and

Monte Carlo simulation approach is used to create mean and variance metamodels

(see, e.g., Mavris, et al., 1996; Mavris, et al., 1995). An initial experiment is

constructed to vary both control and noise factors and build response surface

models of each response similar to the first approach. A second experiment is

created to vary only the control factors, running a Monte Carlo simulation for each

experiment to vary the noise factors. Approximations for mean and variance are

constructed based on this second set of experiments.

71

3. Product array approach: Uses the inner- and outer-array approach advocated by

Taguchi (see, e.g., Montgomery, 1991; Phadke, 1989) to develop separate

approximations for the mean and variance of each response. The inner-array

prescribes settings for the control factors, and the outer-array prescribes settings for

the noise factors. This experimentation strategy leads to multiple response values

for each set of control factor settings from which a response mean and variance can

be computed from which metamodels can be constructed.

Of the three approaches, the product array approach typically yields the most accurate

approximations because the metamodels are built directly from the original analysis code

rather than from an estimate based on an approximation (cf., Koch, et al., 1998).

As shown in Figure 2.17 and as evidenced by the preceding discussion, building

approximations of computer analysis and simulation codes involves the following: (a)

choosing an experimental design to sample the computer code, (b) choosing a model to

represent the data, and (c) fitting the model to the observed data. There are a variety of

options for each of these steps as shown in Figure 2.18, and some of the more prevalent

approximation techniques have been identified. For example, response surface

methodology usually employs central composite designs, second-order polynomials, and

least squares regression analysis. The reader is referred to (Simpson, et al., 1997b) for a

recent review of numerous mechanical and aerospace engineering applications of many of

the metamodeling techniques shown in Figure 2.18 with particular emphasis on response

surface methodology, neural networks, inductive learning, and kriging.

By far the most popular technique for building metamodels these days is the

response surface approach which typically employs second-order polynomial models fit

using least squares regression techniques (Myers and Montgomery, 1995). These

response surface models replace the existing analysis code while providing the following:

• an understanding of the relationship between (input) design variables x and (output)

responses y ,

72

• easier integration of domain-dependent and/or geographically distributed computer

codes, and

• fast analysis tools for optimization and exploration of the design space.

Best LinearUnbiased Predictor

Realization of aStochastic Process

SAMPLE APPROXIMATION

TECHNIQUES

MODELFITTING

EXPERIMENTALDESIGN

MODELCHOICE

Kriging

NeuralNetworks

InductiveLearning

(Fractional) Factorial

Central Composite

Box-Behnken

Latin Hypercube

D-Optimal

Plackett-Burman

Hexagon

Hybrid

Polynomial(linear, quadratic)

Splines(linear, cubic)

Network ofNeurons

Rulebase orDecision Tree

Radial BasisFunctions

Kernel Smoothing

Least SquaresRegression

WeightedLeast Squares

Regression

Backpropagation

Entropy (info.-theoretic) Random Selection

Select By Hand

Log-Likelihood

G-Optimal

Orthogonal Array

Best LinearPredictor

Response Surface Methodology

Figure 2.18 Techniques for Metamodeling

An added advantage of response surfaces is that they can smooth the data in the

case of numerical noise which may hinder the performance of some gradient-based

optimizers (cf., Giunta, et al., 1994). This “smoothing” effect is both good and bad,

depending on the problem. Su and Renaud (1996) present an example where a second-

order response surface smoothes out the variability in a response so that the robust solution

is lost in the approximating function; a “flat region” does not exist in a second-order

response surface, only an inflection point. Su and Renaud’s example is investigated in

more detail in Section 4.1 wherein the kriging process is demonstrated step-by-step as it

73

applies to their example to familiarize the reader with kriging. In the meantime, additional

limitations of response surfaces are discussed in the next section, providing motivation for

investigating alternative metamodeling techniques for use in engineering design.

2 .4 .1 Limitations of Response Surface Approaches

Response surfaces typically are second-order polynomial models which make them

easy to use and implement; however, they have limited capabilities to model accurately non-

linear functions of arbitrary shape. Some two-variable examples of the types of surfaces

that a second-order response surface can model are illustrated in Figure 2.19. Obviously,

higher-order response surfaces can be used to model a non-linear design space; however,

instabilities may arise (cf., Barton, 1992), or it may be too difficult to take a sufficient

number of sample points in order to estimate all of the coefficients in the polynomial

equation, particularly in high dimensions. Hence, many researchers advocate the use of a

sequential response surface modeling approach using move limits (see, e.g., Toropov, et

al., 1996) or a trust region approach (see, e.g., Rodriguez, et al., 1997). More generally,

the Concurrent Sub-Space Optimization procedure uses data generated during concurrent

subspace optimization to develop response surface approximations of the design space

which form the basis of the subspace coordination procedure (Renaud and Gabriele, 1994;

Renaud and Gabrielle, 1991; Wujek, et al., 1995). The Hierarchical and Interactive

Decision Refinement methodology uses statistical regression and other metamodeling

techniques to recursively decompose the design space into subregions and fit each region

with a separate model during design space refinement (Reddy, 1996). Finally, the Model

Management Framework (Booker, et al., 1995; Dennis and Torczon, 1995) is being

developed collaboratively by researchers at Boeing, IBM, and Rice to implement

mathematically rigorous techniques to manage the use of approximation models in

optimization.

74

Many of the previously mentioned sequential approaches are being developed for

single objective optimization applications. Since much of engineering design is

multiobjective in nature, it is often difficult to isolate a small region of good design which

can be accurately represented by a low-order polynomial response surface model. Koch, et

al. (1997) discuss the difficulties encountered when screening large variable problems with

multiple objectives as part of the response surface approach. Barton (1992) states that the

response region of interest will never be reduced to a “small neighborhood” which is good

for all objectives during multiobjective optimization. Hence, there is a need to investigate

alternative metamodeling techniques which have sufficient flexibility to build accurate

global approximations of the design space and which are suitable for modeling computer

experiments which are typically deterministic, i.e., contain no random error or variability.

y x=80 -4x +12x -3x -12x -12 x1 2 12

22

1 2 y x=80 + 4x +8x -2x -12x -12 x1 2 12

22

1 2

y x=80 + 4x +8x -4x -12x -12 x1 2 12

22

1 2 y x=80 + 4x +8x -3x -12x -12 x1 2 12

22

1 2

x2 x1

x1

x1

x1

x2

x2

x2

Figure 2.19 Sample Two-Variable Second-Order Response Surfaces(adapted from Box and Draper, 1987)

75

The approach investigated in this dissertation is called kriging, and it is introduced

in the next section. This discussion of kriging is followed by a discussion of different

experimental designs which can be used to sample the design space in Section 2.4.3.

These two sections lay the foundation for the work in Chapters 4 and 5 wherein

Hypotheses 2 and 3 are tested explicitly to determine the utility of kriging and space filling

experimental designs for building approximations of deterministic computer experiments.

2 .4 .2 The Kriging Approach to Metamodeling

Kriging has its roots in the field of geostatistics—a hybrid discipline of mining

engineering, geology, mathematics, and statistics (cf., Cressie, 1993)—and is useful for

predicting temporally and spatially correlated data. Kriging is named after D. G. Krige, a

South African mining engineer who, in the 1950s, developed empirical methods for

determining true ore grade distributions from distributions based on sampled ore grades

(Matheron, 1963). Several texts which describe kriging and its usefulness for predicting

spatially correlated data (see, e.g., Cressie, 1993) and mining (see, e.g., Journel and

Huijbregts, 1978) exist. These metamodels are extremely flexible due to the wide range of

correlation functions which can be chosen for building the metamodel. Furthermore,

depending on the choice of the correlation function, the metamodel can either “honor the

data,” providing an exact interpolation of the data, or “smooth the data,” providing an

inexact interpolation (Cressie, 1993). In this work, as in most applications of kriging, the

concern is solely on spatial prediction; it is assumed that the data are not correlated

temporally.

These days, kriging goes by a variety of names including DACE (Design and

Analysis of Computer Experiments) modeling—the title of the inaugural paper by Sacks, et

al. (1989)—and spatial correlation metamodeling (see, e.g., Barton, 1994). There are also

several types of kriging (cf., Cressie, 1993): ordinary kriging, universal kriging,

76

lognormal kriging, and trans-Gaussian kriging. In this dissertation, ordinary kriging is

employed, following the work in, e.g., (Booker, et al., 1995; Koehler and Owen, 1996),

and only the term kriging is used.

Unlike response surfaces, however, kriging models have found limited use in

engineering design applications since introduction into the literature by Sacks, et al. (1989).

Consequently, the following question is addressed in this dissertation:



As introduced in Section 1.3.1, Hypothesis 2 is that kriging is a viable

metamodeling technique for building approximations of deterministic

computer analyses. Initial applications of kriging in engineering design include:

• Giunta (1997) and Giunta, et al. (1998) perform a preliminary investigation into the

use of kriging for the multidisciplinary design optimization of a High Speed Civil

Transport aircraft.

• Sasena (1998) compares and contrasts kriging and smoothing splines for

approximating noisy data.

• Schonlau, et al. (1997) use a global/local search algorithm based on kriging for

shape optimization of an automobile piston engine.

• Osio and Amon (1996) develop a multistage numerical optimization strategy based

on kriging which they demonstrate on the thermal design of embedded electronic

package which has 5 design variables.

• Booker (1996) and Booker, et al. (1996) using a kriging approach to study the

aeroelastic and dynamic response of a helicopter rotor during structural design.

77

Some researchers have also employed kriging-based strategies for numerical optimization

(see, e.g., Cox and John, 1995; Trosset and Torczon, 1997). A look at the mathematics of

kriging is offered next.

Mathematics of Kriging

Kriging postulates a combination of a polynomial model and departures of the following

form:

y(x) = f(x) + Z(x) [2.18]

where y(x) is the unknown function of interest, f(x) is a known polynomial function of x ,

and Z(x) is the realization of a stochastic process with mean zero, variance σ2, and non-

zero covariance. The f(x) term in Equation 2.18 is similar to the polynomial model in a

response surface, providing a “global” model of the design space. In many cases f(x) is

simply taken to be a constant term β (cf., Koehler and Owen, 1996; Sacks, et al., 1989;

Welch, et al., 1990). Only kriging models with constant underlying global models are

investigated in this work as well.

While f(x) “globally” approximates the design space, Z(x) creates “localized”

deviations so that the kriging model interpolates the ns sampled data points. The covariance

matrix of Z(x) which dictates the local deviations is as follows:

Cov[Z(x i),Z(x j)] = σ2 R([R(x i,x j)] [2.19]

where R is the correlation matrix, and R(x i,x j) is the correlation function between any two

of the ns sampled data points x i and x j. R is a ns x ns symmetric, positive definite matrix

with ones along the diagonal. The correlation function R(x i,x j) is specified by the user.

In this work, five different correlation functions are examined for use in the kriging

model, see Table 2.1. In all of the correlation functions listed in Table 2.1, ndv is the

78

number of design variables, θk are the unknown correlation parameters used to fit the

model, and dk = xki - xk

j which is the distance between the kth components of sample points

x i and x j. The correlation functions of Equations 2.20 and 2.21 are from (Sacks, et al.,

1989); the correlation functions of Equations 2.22-2.24 are from (Mitchell and Morris,

1992b).

These five correlation functions are chosen primarily because of the frequency with

which they appear in the literature; the Gaussian correlation function, Equation 2.20, is the

most popular one in use. Correlation functions with multiple parameters per dimension

exist; however, correlation functions with only one parameter per dimension are considered

in this dissertation to facilitate finding the maximum likelihood estimates (MLEs) or “best

guess” of the θk used to fit the model. As mentioned in Section 1.3.2, one of the

contributions in this work is to study the effects of these five different correlation functions

on the accuracy of a kriging model; this study is performed in Chapter 5.

Table 2.1 Summary of Correlation Functions

Name Spatial Correlation Function # Deriv. Eqn. #

Exponential exp(−θk dk )k=1

n dv∏ 1 [2.20]

Gaussian exp(−θk dk2)k=1

ndv∏ ∞ [2.21]

Cubic spline

1 − 6 θk dk( )2+ 6 θk dk( )3

2 1− θk dk( )3

0

θk dk <1

21

2≤ θk dk <1

θk dk ≥1

k=1

n dv∏ 1 [2.22]

Matérn linearfunction (1+ θk dk )exp( −θk dk )[ ]k=1

n dv∏ 1 [2.23]

Matérn cubicfunction

(1+ θk dk +θk

2 dk2

3)exp( −θk dk )

k=1

n dv∏ 2 [2.24]

79

Once a correlation function has been selected, predicted estimates, ˆ y (x), of the

response, y(x), at untried values of x are given by Equation 2.25:

ˆ y = ˆ β + rT(x)R −1 (y − fˆ β ) [2.25]

where y is the column vector of length ns (number of sample points) which contains the

values of the response at each sample point, and f is a column vector of length ns which is

filled with ones when f(x) in Equation 2.18 is taken as a constant. In Equation 2.25, rT(x)

is the correlation vector of length ns between an untried x and the sampled data points {x1,

x2, ..., xns} and is given by Equation 2.26.

rT(x) = [R(x ,x1), R(x ,x2), ..., R(x ,xns)]T [2.26]

Finally, the ˆ β in Equation 2.25 is estimated using Equation 2.27.

ˆ β = (f TR−1f)−1 fTR −1y [2.27]

When f(x) is assumed to be a constant, then ˆ β is a scalar which simplifies the calculation

of Equation 2.27 and all others involving ˆ β .

The estimate of the variance, ˆ σ 2 , from the underlying global model (not the

variance of the randomness in the observed data itself) is as follows:

ˆ σ 2 =(y − f ˆ β )T R−1(y − fˆ β )

ns

[2.28]

where f is again a column vector of ones because f(x) is assumed to be a constant. The

maximum likelihood estimates (i.e., “best guesses”) for the θk used to fit the model are

found by maximizing Equation 2.29 over θk > 0 (Booker, et al., 1995):

80

−[n s ln( ˆ σ 2 ) + ln|R|]

2[2.29]

Both ˆ σ 2 and |R| are functions of θk. While any values for the θk create an interpolative

approximation model, the “best” kriging model is found by solving the k-dimensional

unconstrained nonlinear optimization problem given by Equation 2.29; this process is

discussed further in the next section. It is worth noting that in some cases using a single

correlation parameter gives sufficiently good results (see, e.g., Booker, et al., 1995; Osio

and Amon, 1996; Sacks, et al., 1989). In this work, however, a unique θ value for each

dimension always is considered based on past difficulties with scaling the design space to

[0,1]k during the model fitting process. The algorithms used in this dissertation to build

and predict with a kriging model are included in Appendix A.

Once the MLEs for each theta have been found, the final step is to validate the

model. Since a kriging model interpolates the data, residual plots and R2 values—the usual

model assessments for response surfaces (cf., Myers and Montgomery, 1995)—are

meaningless because there are no residuals. Therefore, validating the model using

additional data points is essential if they can be afforded. If additional validation points can

be afforded, then the maximum absolute error, average absolute error, and root mean

square error (RMSE) for the additional validation points can be calculated to assess model

accuracy. These measures are summarized in Table 2.2. In the table, nerror is the number of

random test points used, then yi is the actual value from the computer code/simulation, and

ˆ y i is the predicted value from the approximation model.

81

Table 2.2 Error Measures for Kriging Metamodels

Name Error Measure Eqn. #

max. abs. error max. |y i − ˆ y i | i = 1, ..., nerror [2.30]

avg. abs. error1

n error

y i − ˆ y ii =1

nerror∑ [2.31]

RMSE(y i − ˆ y i )

2

i =1

nerror∑n error

[2.32]

However, sometimes taking additional validation points is not possible due to the

added expense of running additional experiments on the computer code/simulation; thus, an

alternative model assessment which requires no additional points is needed. One such

approach is the leave-one-out cross validation (Mitchell and Morris, 1992a). In this

approach, each sample point used to fit the model is removed one at a time, the model is

rebuilt without that sample point, and the difference between the model without the sample

point and actual value at the sample point is computed for all of the sample points. The

cross validation root mean square error (cvrmse) is computed using Equation 2.33:

cvrmse = (y i − ˆ y i )

2

i =1

ns∑ns

[2.33]

The MLEs for the θk are not re-computed for each model; the initial θk MLEs based on the

full sample set are used. Mitchell and Morris (1992a) describe an approach which

facilitates cross validation since it can be time consuming depending on the sample size.

Before a kriging metamodel (or any metamodel for that matter) can be created, the

design space must be sampled in order to obtain data to fit the model. Hence, an important

step in any metamodeling approach is the selection of an appropriate sampling strategy,

82

i.e., an experimental design by which the computer analysis or simulation code is queried.

In the next section, space filling and classical experimental designs are discussed.

2 .4 .3 Classical and Space Filling Experimental Designs

Many researchers (see, e.g., Currin, et al., 1991; Sacks and Schiller, 1988) argue

that classical experimental designs, such as the central composite designs and Box-

Behnken designs, are not well-suited for sampling deterministic computer experiments.

Sacks, et al. (1989) state that the “classical notions of experimental blocking, replication

and randomization are irrelevant” when it comes to deterministic computer experiments that

have no random error; hence, designs for deterministic computer experiments should “fill

the space” as opposed to possess properties for estimating the variability in the data.

Booker (1996) summarizes the difference between classical experimental designs

and new space filling designs well. In the classical design and analysis of physical

experiments, random variation is accounted for by spreading the sample points out in the

design space and by taking multiple data points (replicates), see Figure 2.20a. In

deterministic computer experiments, replication at a sample point is meaningless; therefore,

the points should be chosen to fill the design space. One approach is to minimize the

integrated mean square error over the design region (cf., Sacks, et al., 1989); the space

filling design illustrated in Figure 2.20b is an example of such a design.

As part of the research in this dissertation, the following question is addressed.



As stated in Section 1.3.1, Hypothesis 3 is that space filling designs are

better suited for building approximations of deterministic computer

83

analyses than classical experimental designs. In an effort to test this hypothesis,

an investigation into the utility of several classical and space filling experimental designs is

conducted in Chapter 5. Eleven different types of experimental designs investigated in this

dissertation: two classical experimental designs and nine space filling experimental designs.

The different designs are described next; detailed descriptions of each design are also given

in Appendix B and C.

0.0x2

0.5

1.0

-0.5

-1.0

x1

0.0 0.5-0.5-1.0 1.0

0.0x2

x1

0.0

0.5

1.0

-0.5

-1.0

0.5-0.5-1.0 1.0

(a) Classical design w/replicates (b) Space filling design w/o replicates

Figure 2.20 Example Classical and Space Filling Experimental Designs

Classical Experimental Designs

Classical experimental designs are so named because they have been developed for what

are considered to be the more “classical” applications of response surface metamodeling:

physical experiments which are plagued by variability and random error (see, e.g., Box

and Draper, 1987; Myers, et al., 1989; Myers and Montgomery, 1995). Among these

designs, the central composite and Box-Behnken designs are well known and easily

generated; hence, they are employed in this work to serve as a basis for comparison against

the sampling capability of space filling designs. A brief description of these two types of

designs follows.

84

A central composite design (CCD) is a

combination of 2k factorial points, 2k star points, and a

center point for k factors as shown in Figure 2.21. CCDs

are the most widely used experimental design for fitting

second-order response surfaces (Myers and Montgomery,

1995). Different CCDs are formed by varying the

distance from the center of the design space to the star

points; in this work, three types of CCDs are considered:

X1

X2

X3Star pts

Center ptFactorial pts

Figure 2.21 CentralComposite Design

• ordinary central composite design (CCD) - star points are placed a distance of ±α

(α > 1) from the center with the cube points placed at ±1 from the center,

• face centered central composite (CCF) design - star points are positioned on the

faces of the cube, and

• inscribed central composite (CCI) design - star points are positioned at ±1/α from

the center with the cube points placed at ±1.

In addition, combinations of the CCD and CCF, and CCI and CCF are investigated based

on the suggestions and observations discussed in (Koch, 1997).

A Box-Behnken design is formed by

combining 2k factorial and incomplete block designs and

are an important alternative to central composite designs

because they require only three levels of each factor (Box

and Behnken, 1960). However, Myers and Montgomery

(1995) warn that these designs should not be used when

accurate predictions at the extremes (i.e., the corners) are

important. An example 13 point Box-Behnken design for

three factors is shown in Figure 2.22.

X1

X2

X3

Figure 2.22 Box-Behnken Design

85

Space Filling Experimental Designs

Numerous space filling experimental designs have been developed in an effort to provide

more efficient and effective means for sampling deterministic computer experiments. For

instance, Koehler and Owen (1996) describe several Bayesian and Frequentist types of

space filling experimental designs, including maximin and minimax designs, maximum

entropy designs, integrated mean squared error (IMSE) designs, orthogonal arrays, Latin

hypercubes, scrambled nets and randomized grids. Latin hypercube designs were

introduced in (McKay, et al., 1979) for use with computer codes and compared to random

sampling and stratified sampling. Minimax and maximin designs were developed by

Johnson, et al. (1990) specifically for use with computer experiments. Sherwy and Wynn

(1987; 1988) and Currin, et al. (1991) use the maximum entropy principle to develop

designs for computer experiments. Similarly, Sacks et al. (1989) discuss entropy designs

in addition to IMSE designs and maximum mean squared error designs for use with

deterministic computer experiments. Finally, a review of several Bayesian experimental

designs for linear and nonlinear regression is given in (Chaloner and Verdinelli, 1995).

Comparisons of the different types of space filling experimental designs are few;

often the novel space filling design being described is compared against Latin hypercube

designs and random sampling (see, e.g., Kalagnanam and Diwekar, 1997; Park, 1994;

Salagame and Barton, 1997), but rarely is it compared against other space filling designs.

An exception are the maximin Latin hypercubes (Morris and Mitchell, 1995) which are

compared against maximin designs (Johnson, et al., 1990) and Latin hypercubes; the

authors conclude by means of an example that maximin Latin hypercube designs are better

than either maximin or Latin hypercube designs alone. In this dissertation, one of the

contributions is to compare and contrast a wide variety of space filling designs against

themselves and classical experimental designs. Toward this end, nine space filling

experimental designs are investigated: Latin hypercubes, orthogonal arrays, orthogonal

86

Latin hypercubes, orthogonal array-based Latin hypercubes, maximin Latin hypercubes,

minimax Latin hypercubes, optimal Latin hypercubes, Hammersley point designs, and

uniform designs. An overview of each of these designs follows; complete descriptions can

be found in Appendix B and C.

A Latin hypercube is a matrix of n rows and k

columns where n is the number of levels being examined

and k is the number of design variables. Each column

contains the levels 1, 2, ..., n, randomly permuted, and

the k columns are matched at random to form the Latin

hypercube (McKay, et al., 1979). These designs are the

earliest space filling experimental design intended for use

with computer experiments. An example two factor, nine

point Latin hypercube for is shown in Figure 2.23.

X1

X2

Figure 2.23 LatinHypercube Design

An orthogonal array (OA) is a matrix of n rows

and k columns with every element being one of q

symbols: 0, ..., q-1 (Owen, 1992). An orthogonal array

has an associated strength t depending on the number of

combinations of l levels appearing in any of the r columns

of the OA. The orthogonal arrays used in this work are

limited to q2 runs where q is a prime power. An example

nine point OA in three dimensions is shown in Figure

2.24.

X1

X2

X3

Figure 2.24 OrthogonalArray Design

87

Orthogonal array-based Latin hypercube

designs combine the orthogonality properties of an

orthogonal array with the good stratification capabilities

of a Latin hypercube (Tang, 1993). Tang provided an

algorithm to generate these designs. An example of a six

point OA-based Latin hypercube is shown in Figure 2.25.

X1

X2

Figure 2.25 OA-BasedLatin Hypercube

An orthogonal Latin hypercube is a Latin

hypercube which has orthogonal columns. These designs

retain the orthogonality of traditional experimental designs

(e.g., CCDs) while attempting to maintain a good spread

of points throughout the design space. These designs are

constructed from purely algebraic means using the

process described in (Ye, 1997). An example nine point

orthogonal Latin hypercube for two factors is shown in

Figure 2.26.

X1

X2

Figure 2.26 OrthogonalLatin Hypercube

A maximin Latin hypercube is a Latin

hypercube design which provides a good compromise

between the maximin criterion (which maximizes the

minimum distance between any two sample points

(Johnson, et al., 1990)) and the good projection

properties of Latin hypercube designs. The simulated

annealing algorithm in (Morris and Mitchell, 1995) is

used to construct these designs for varying sample sizes.

An example seven point design is shown in Figure 2.27.

X1

X2

Figure 2.27 MaximinLatin Hypercube

88

An optimal Latin hypercube is found by

imposing a specific criterion on a Latin hypercube in the

same fashion as the maximin Latin hypercubes. In this

work, integrated mean square error (IMSE) optimal Latin

hypercubes (Park, 1994) are employed. Park has

provided his algorithm to generate these designs. An

example eight point IMSE optimal Latin hypercube design

for two factors is shown in Figure 2.28.

X1

X2

Figure 2.28 IMSEOptimal Latin Hypercube

A Hammersley sampling sequence design

is a good design for placing n points in a k-dimensional

hypercube (Kalagnanam and Diwekar, 1997), providing

better uniformity properties over a k-dimensional space

than Latin hypercubes. An example nine point

Hammersley design for two factors is shown in Figure

2.29. Diwekar (1995) has provided an algorithm to

generate HSS designs for use in this dissertation.

X1

X2

Figure 2.29 HammersleyDesign

A uniform design is a design based strictly on

number-theoretic methods; these designs are based on

generating vectors of good lattice points based on the

mean square error criterion (Fang and Wang, 1994). All

of the uniform designs considered in this work have q

levels which is equal to the number of sample points, n

where n is strictly an odd number. An example nine point

uniform design for two factors is shown in Figure 2.30.

X1

X2

Figure 2.30 UniformDesign

89

In addition to the maximin Latin hypercube designs from (Morris and Mitchell,

1995), a minimax Latin hypercube design is introduced in this dissertation. Only a brief

description of this unique design is given here; a detailed description of the design an a

discussion of how it is generated are included in Appendix C. From an intuitive stand

point, because prediction with kriging relies on the spatial correlation between data points,

a design which minimizes the maximum distance between the sample points and any point

in the design space should yield an accurate predictor. Such a design is referred to as a

minimax design (Johnson, et al., 1990). While the minimax criterion ensures good

coverage of the design space by minimizing the maximum distance between points, it does

not ensure good stratification of the design space (i.e., when the sample points are

projected into 1-dimension, many of the points may overlap (cf., Johnson, et al., 1990)).

Meanwhile, because a Latin hypercube ensures good stratification of the design space,

combining it with the minimax criterion provides a good compromise between the two

much as the maximin Latin hypercubes developed by Morris and Mitchell (1995) does.

Example 9, 11, and 14 point maximin Latin hypercube designs are shown in Figure 2.31.

The specifics of the genetic algorithm used to generate these minimax Latin hypercube

designs are detailed in Appendix C.

X1

X2

X1

X2

X1

X2

Figure 2.31 Example 9, 11, and 14 Point Minimax Latin Hypercubes

90

In Chapter 5, the minimax Latin hypercube designs are compared against the other

classical and space filling experimental designs discussed in this section. A look ahead to

that and the next chapters is offered in the next section.

2.5 A LOOK BACK AND A LOOK AHEAD

Through the review of the literature which is presented in this chapter, the

necessary elements for a method to model and design a scalable product platform for a

product family have been identified by elucidating the research questions (and hypotheses)

introduced in Section 1.3.1. In the next chapter, these constitutive elements are integrated

to create the Product Platform Concept Exploration Method (PPCEM), providing a Method


which can be scaled into an appropriate family of products. The relationship between the

individual sections in this chapter and the PPCEM developed in the next chapter are

illustrated in Figure 2.32. In particular, the market segmentation grid is revisited in Section

3.1.1 as it applies to the PPCEM. The concept of a “conceptual noise factor” is formalized

into a scale factor in Section 3.1.2 which is fundamental to the utilization of the PPCEM.

Metamodeling techniques within the PPCEM are discussed in Sections 3.1.3. Aggregation

of the individual product specifications into an appropriate comproimse DSP formulation

for the product family is described in Section 3.1.4, and development of the product

platform portfolio for the product family is explained in Section 3.1.5.

In addition to the presentation of the PPCEM, the research hypotheses are revisited

in Section 3.2 in the next chapter. Supporting posits are stated for each hypothesis, and the

verification strategy for testing the hypotheses is elaborated in Section 3.3. The discussion

in Section 3.3 sets the stage for the example problems which are presented in Chapters 4

through 7 to verify the hypotheses.

91

Metamodeling



ConceptualNoise

Factors


MarketSegmentation

Grid



Chp 3

§2.2§2.3 §2.1


Platform


§3.1.5 §3.1.1§3.1.2§3.1.4§3.1.3§3.1.3

Figure 2.32 Pictorial Review of Chapter 2 and Preview of Chapter 3

92

3. CHAPTER 3

THE PRODUCT PLATFORM CONCEPTEXPLORATION METHOD

Platform


In this chapter, the elements of the previous chapters are synthesized to meet the

principal objective in this dissertation, namely, to develop the Product Platform Concept

Exploration Method (PPCEM) for designing a common scalable product platform for a

product family. An overview of the PPCEM and its associated steps and tools is given in

Section 3.1 with each step of the PPCEM and its constituent elements elaborated in

Sections 3.1.1 through 3.1.5; the resulting infrastructure of the PPCEM is presented in

Section 3.1.6. In Section 3.2, the research hypotheses are revisited from Section 1.3.1

and supporting posits are identified. Section 3.3 follows with an outline of the strategy for

verification and testing of the research hypotheses. Section 3.4 concludes the chapter with

a recap of what has been presented and a look ahead to the metamodeling studies in

Chapters 4 and 5 and the example problems in Chapters 6 and 7 which are used to test the

research hypotheses and demonstrate the application of the PPCEM.

93

3.1 OVERVIEW OF THE PPCEM AND RESEARCH HYPOTHESES

As stated in Section 1.3.2, the principal contribution in this dissertation is the

Product Platform Concept Exploration Method (PPCEM) for designing a common scalable

product platform for a product family. As the name implies, the PPCEM is a Method


which can be scaled into an appropriate family of products. The steps and associated tools

(with relevant sections noted in parentheses) in the PPCEM are illustrated in Figure 3.1.

Tools

Market Segmentation Grid (§ 2.2)


(§ 2.3)

Metamodeling Techniques

(§ 2.4)

Compromise Decision Support Problem (§1.2)

Step 1Create Market Segmentation Grid

Step 2Classify Factors and Ranges

Step 3Build and Validate Metamodels

Step 4Aggregate Product Platform Specifications

Step 5Develop Product Platform Portfolio

PPCEM Steps


Product Platform Portfolio

Figure 3.1 Steps and Tools of the PPCEM

94

There are five steps to the PPCEM as illustrated in Figure 3.1. The input to the

PPCEM are the overall design requirements, and the output of the PPCEM is the product

platform portfolio which is described in Section 3.1.5. The tools utilized in each step of

the PPCEM are shown on the right hand side of Figure 3.1; their involvement in the

various steps of the PPCEM is elaborated further in Sections 3.1.1 through 3.1.5 wherein

the implementation of each step of the PPCEM is described. These steps prescribe how to

formulate the problem and describe how to solve it; the actual implementation of each step

is liable to vary from problem to problem.

3 .1 .1 Step 1 - Create the Market Segmentation Grid

Given the overall design requirements, Step 1 in the PPCEM is to create the market

segmentation grid as shown in Figure 3.2. As discussed in Section 2.2.1, the market

segmentation grid provides a link between management, marketing, and engineering design

to help identify and map which type of leveraging can be used to meet the overall design

requirements and realize a suitable product platform and product family. In the PPCEM,

the market segmentation grid serves as an attention directing tool to help identify potential

opportunities for horizontal leveraging, vertical leveraging, or a beachhead approach to

product platform design. Examples of this step are given in Sections 6.2 and 7.1.3.

A. Market Segementation Grid

IDENTIFYLEVERAGING:

1. vertical 2. horizontal 3. beachead


Platform

Figure 3.2 Step 1 - Create the Market Segmentation Grid

95

3 .1 .2 Step 2 - Classify Factors and Ranges

Once the market segmentation grid has been created, Step 2 of the PPCEM is to

classify factors as illustrated in Figure 3.3. Factors are classified in the following manner:

yResponse

Noise zFactors

xControlFactors Scale

s Factors

B. Factors and Ranges


ProductPlatform


IDENTIFYLEVERAGING:


Platform

Figure 3.3 Step 2 - Factor Classification

• Responses are performance parameters of the system; in the problem formulation,

they may be constraints or goals or both and are identified from the overall design

requirements and the market segmentation grid.

• Control factors are variables that can be freely specified by a designer; settings

of the control factors are chosen to minimize the effects of variations in the system

while achieving desired performance targets and meeting the necessary constraints.

Signal factors also are lumped within control factors because it is often difficult to

know, a priori, which design variables are control factors and can be used to

minimize the sensitivity of the design to noise variations and those which are signal

factors and have no influence on the robustness of the system.

• Noise factors are parameters over which a designer has no control or which are

too difficult or expensive to control.

96

• Scale factor is a factor around which a product platform is leveraged either

through vertical scaling, horizontal scaling, or a combination of the two.

Appropriate ranges for the control and noise factors are identified during this step, and

constraints and goal targets for the responses are also identified.

The relationship between different leveraging capabilities and types of scale factors

considered in this dissertation are illustrated in Figure 3.4. As discussed in Section 2.2.1,

three types of leveraging can be mapped using the market segmentation grid: (1) vertical

leveraging, (2) horizontal leveraging, and (3) a beachhead approach which is a combination

of vertical and horizontal leveraging. Correspondingly, two types of scale factors can be

identified—parametric and conceptual/configurational—related to the type of

scaling which is identified in conjunction with the market segmentation grid as shown in

Figure 3.4.

As shown in Figure 3.4, the relationship between each type of scale factor and the

three types of leveraging are as follows:

• Vertical leveraging: parametric scale factors, such as the length of a motor to

provide varying torque as demonstrated in Chapter 6 or the number of compressor

stages in an aircraft engine as in the Rolls Royce RTM322 engine example (see

Rothwell and Gardiner, 1990) mentioned in Section 1.1.1.

• Horizontal leveraging: conceptual/configurational scale factors, such as the

size of evaporator in a family of air conditioners (see Chang and Ward, 1995) or the

number of passengers carried by the Boeing 747 family of aircraft (Rothwell and

Gardiner, 1990) and as demonstrated in the General Aviation aircraft example in

Chapter 7.

• Beachhead approach: combination of parametric, conceptual, and/or

configurational scale factors as needed.

97


Mid-Range



Platform A

Platform C

Scaled D

own S

cale

d U

p

(a) Vertical


Mid-Range



Platform

(c) Beachhead


Mid-Range



High End Platform Leverage

Low End Platform Leverage

(b) Horizontal

Leveraging Scale Factors

Scale factors are:parametric, e.g., • the length of a motor to

provide varying torque• number of compressors

in an engine

Scale factors are:conceptual and/or configurational, e.g., • number of persons

on an aircraft• size of evaporator in a

family of air conditioners

Scale factors are: a combination of• parametric, conceptual, • and/or configurational scaling factors

Figure 3.4 Relationship of Scale Factors to the Market Segmentation Grid

If known, an appropriate range—upper and lower limit—is identified for each scale factor

during this step of the PPCEM; otherwise, finding this range becomes part of the design

process. Examples of Step 2 are offered in Sections 6.2 and 7.3. Once the responses,

control, noise, and scale factors and corresponding constraints/targets and ranges have

been identified, the next step is to build metamodels for each response.

3 .1 .3 Step 3 - Build and Validate Metamodels

Step 3 in the PPCEM is to build and validate metamodels relating the control, noise,

and scale factors to the responses using the elements of the PPCEM shown in Figure 3.5.

98

The goal in Step 3 is to build surrogate approximations of the computer analyses or

simulation routines which are inexpensive to run. Because robust design principles are

being used, these metamodels are either functions of control, noise, and scale factors as

discussed in Sections 2.3.1 and 2.4, or approximate the mean and standard deviation of

each response for known variations in the noise and scale factors. If the analytic equations

or simulation routine are not sufficiently expensive to warrant metamodeling, this step can

be skipped provided that the mean and standard deviation of each response (as a result of

variation in the scale factor and any relevant noise factors) can be computed easily. The

universal motor example in Chapter 6 forgoes metamodel construction because the analyses

permit the mean and standard deviation of each response to be easily computed; however,

such is not the case in Chapter 7, the design of a family of General Aviation aircraft. In

Chapter 7, extensive use of metamodels occurs to facilitate the implementation of robust

design and search for a good aircraft platform.

yResponse

Noise zFactors


s Factors

E. Metamodeling

B. Factors and RangesE2. Model Fitting & Analysis

Analysis of VarianceEliminate unimportant factors

Reduce the design spaceto the region of interest

Plan additional experiments

D. Analysis orSimulation

Routine

C. Design of Experiments

CLASSICAL: Plackett-Burman,Fractional Factorial, CCD, etc.

SPACE FILLING: minimax & maximin Latin hypercubes,rnd orthogonal arrays, uniformdesigns, Hammersley pts, etc.

E1. Model Choice

Build metamodelsValidate metamodels

Use metamodels

ResponseSurface

Kriging

ProductPlatform

Figure 3.5 Step 3 - Metamodel Development

The steps for building and validating the metamodels follow the traditional

metamodeling approach described in Section 2.4.1. An experimental design is selected by

which the computer analysis or simulation program used to model the system is queried to

99

obtain data. The experimental design is used to sample the design space identified by the

ranges (i.e., bounds) on the control, noise, and scale factors. The resulting sample data

then is used to build a metamodel (e.g., a kriging model, Section 2.4.2) for each response.

Model accuracy then is assessed through additional validation points or otherwise

appropriate procedures for the specific type of metamodel employed.

If identifying an appropriate region of interest for constructing the metamodels is

difficult, a sequential approximation strategy with screening experiments may be employed.

Description of such an approach can be found in, e.g., (Chen, et al., 1997; Koch, et al.,

1997; Myers and Montgomery, 1995). The difficulty, then, lies in defining an appropriate

design space. In this dissertation, the design space for the General Aviation aircraft

example problem is known, making identification of a good design space appear trivial. In

reality it is not, and there is often great difficulty finding an appropriately good design

space. Identifying and quantifying a “good” design space is not addressed in this

dissertation; it is a possibility for future work (see Section 8.3).

3 .1 .4 Step 4 - Aggregate Product Platform Specifications

Once the necessary metamodels have been built and validated, Step 4 in the PPCEM

is to aggregate the product platform specifications. This involves formulating an

appropriate compromise DSP to model the necessary constraints and goals for the product

family and product platform based on the overall design requirements, the market

segmentation developed in Step 1, and the factor classification and ranges developed in

Step 2, see Figure 3.6. It is imperative that product constraints or goals given in the overall

design requirements that are not captured within the desired platform leveraging strategy be

included in the compromise DSP.

100

yResponse

Noise zFactors


s Factors



ProductPlatform


Find Control VariablesSatisfy Constraints Goals "Mean on Target" C "Minimize Deviation" values BoundsMinimize Deviation Function


IDENTIFYLEVERAGING:


Platform

dkor{ } }{

Figure 3.6 Step 4 - Aggregating the Product Platform Specifications

Two approaches for aggregating the product platform specifications are

demonstrated in this dissertation.

1. Separate goals for “bringing the mean on target” and “minimizing the variation” are

created (see Section 2.3.2) for the product family. This follows the implementation

of robust design principles which is traditionally used in the RCEM, except they are

being applied to a product family as opposed to a single product. The procedure is

as follows:

a. identify targets from market segmentation grid and overall design requirements

for each derivative product;

b. compute target means for the platform based by averaging individual targets;

c. Compute standard deviations for the platform based on individual targets by

dividing the range of each target by six, assuming a normal distribution with

±3σ variations, or by 12 , assuming a uniform distribution; and

d. create separate goals for “bringing the mean on target” and “minimizing the

variation” as necessary.

101

2. Design capability indices (see Section 2.3.2) to assess the capability of a family of

designs to satisfy a ranged set of design requirements. The procedure is as follows:

a. identify upper and lower requirement limits (URL and LRL, respectively) from

the market segmentation grid and overall design requirements for each

derivative product;

b. compute the mean and standard deviation as the average and standard deviation

of the individual instantiations of the product family for a given set of design

variables; and

c. formulate Cdk targets and constraints depending on whether “smaller is better,”

“nominal is best,” or “larger is better” (refer to Section 2.3.1).

The first approach is utilized in the universal motor example in Chapter 6 (see Section 6.3

in particular for more details on its implementation). Meanwhile, design capability indices

are employed in the General Aviation aircraft example in Chapter 7 (see Section 7.4).

The compromise DSP is used to determine the values for the control (design)

variables which best satisfy the product family goals (“bringing the mean on target” and

“minimizing the variation” in the first; making Cdk ≥ 1 in the second) while satisfying the

necessary constraints. Constraints can be either worse case scenario, evaluated on an

individual basis, or aggregated in a similar manner as the goals, constraining the mean and

deviation of the responses or the appropriate Cdk . The compromise DSP is exercised in

Step 5 of the PPCEM to obtain the product platform portfolio as described next.

3 .1 .5 Step 5 - Develop Product Platform Portfolio

Step 5 of the PPCEM is to solve the compromise DSP using the aggregate product

platform specifications to develop the product platform portfolio. This step makes use of

the metamodels created in Step 3 in conjunction with the compromise DSP and the

aggregate product specifications formulated in Step 4; design scenarios for exercising the

compromise DSP are abstracted from the overall design requirements, see Figure 3.7. The

resulting “pool” of solutions obtained by exercising the compromise DSP as such is

102

referred to as a solution portfolio. This portfolio of solutions can provide a wealth of

information about the appropriate settings for the design variables for the product platform

based on different design scenarios or robustness considerations; hence, the solution

portfolio is called the product platform portfolio.

ProductPlatformPortfolio

E. Metamodeling


E2. Model Fitting & Analysis




E1. Model Choice


Use metamodels

ResponseSurface

Kriging



dkor{ } }{

Figure 3.7 Step 5 - Developing the Product Platform Portfolio

The concept of a solution portfolio is not new to this research; it is simply a more

appropriate name for what has been previously referred to as a ranged set of specifications

(see, e.g., Lewis, et al., 1994; Simpson, et al., 1996; Simpson, et al., 1997a; Smith and

Mistree, 1994). The objective when using the PPCEM is to generate a variety of options

for product platforms; it is not necessarily used to evaluate these options or select one from

them. It facilitates generating these options with the end result being the product

platform portfolio, namely, the “pool” of solutions (i.e., design variable settings) which

103

should be maintained in order for the product platform to have sufficient flexibility to meet

the desired design scenarios in the event that one scenario is preferred to the next.

In addition to developing the product platform portfolio for the product family,

product variety tradeoff studies also can be performed by making use of two measures—

the Non-Commonality Index (NCI) and the Performance Deviation Index (PDI)—which

are described as follows:

Non-commonality Index (NCI): NCI is used to assess the amount of variation

between parameter settings of each product within a product family; the smaller the

variation, the smaller NCI, and the more common the products within the product

family. Computing NCI is perhaps best illustrated through example; consider the 3

products shown in Figure 3.8. Assume that each product is described by three

design variables: x1, x2, and x3 (if these three hypothetical products were electric

motors, then x1, x2, and x3 might be the outer radius of the motor, the length of the

motor, and the number of windings in the motor). First, the dissimilarity of each

design variable settings for each product within the family is computed as follows:

1. Compute the mean of each of the xj within the product family and the absolute

value of the difference between µj and xj for each of the i products.

2. Normalize each difference by the range of that particular design variable: [upper

bound (ubj) - lower bound (lbj)]; this measures the relative variation in the

values of the design variables to the total range for that design variable.

3. Compute the average of the resulting normalized differences; this value is

denoted DIi in the figure and is the dissimilarity of the settings of xi for the

group of products.

The scale factor around which the product family is derived is not included in this

computation. NCI is taken as a weighted sum of the individual DIi, where the

weights reflect the relative difficulty or cost associated with allowing each parameter

to vary. For an electric motor for instance, it may be easier or cheaper to allow the

number of windings (x3) to vary between different motor models but not so to

allow the outer radius to vary (x1). In this case, w1 would be much larger than w3

to reflect this within the product family.

104

2 3

µ1

10 20

µ2

0 100

µ3

Pro

duct

1

Pro

duct

2

Pro

duct

3

x1: 2.5 2.5 2.8

x2: 12.1 10.0 10.9

x3: 13.0 18.0 35.0

DI 3 =1

3

22 −13 + 22−18 + 22− 35

(100 − 0)

DI3 = 0.0867

DI j =1n

µ j − x i,j

ub j − lb j( )i=1

n

∑

j = 1, ..., # design variables

i = 1, ..., # products in family

DI 2 =1

3

11−12.1 + 11−10 + 11−10.9

(20−10)

DI 2 = 0.0733

Product Descriptions

Dissimilarity Index

Dissimilarity of x1

Dissimilarity of x2

Dissimilarity of x3

DI 1 =1

3

2.6 − 2.5 + 2.6 − 2.5 + 2.6 − 2.8

(3 −2)

DI1 = 0.133

DisIndx = w jDI j = 0.55(0.133)+j=1

#d.v.

∑ 0.3(0.0733) + 0.15(0.0867) = 0.108NCI =Non-commonality Index:

Figure 3.8 Estimating the Dissimilarity Index for a Product Family

Performance Deviation Index: Assuming that a market niche is defined by a set

of goal targets and constraints and that the necessary constraints are met for each

individual derivative product, then the deviation variables in the compromise DSP

are a direct measure of how well each derivative product meets its targets. The

Performance Deviation Index (PDI) for a product family thus is taken as a linear

combination (possibly weighted) of the deviation variables for each derivative

product within the product family as given by Equation 3.1:

PDI = wiZ ii=1

n

∑ [3.1]

where i = {1, ..., # products in family}, and Zi is the corresponding deviation

function for each derivative product within the product family. Weightings may be

used to bias the measure for certain products within the family.

105

Example computations of the NCI and PDI for a family of products are demonstrated in the

General Aviation aircraft example (see Section 7.6.2) and are explained in more detail in the

context of that example.

NCI and PDI are, in and of themselves, ad hoc measures for a product family,

similar to the commonality indices and platform efficiency and effectiveness measures

discussed in Section 2.2.2. However, having these measures for non-commonality and

performance deviation for a family of products allows product variety tradeoff studies to be

performed, see Figure 3.9 and Figure 3.10.

NCI

PD

I

low

high

low high

Best DesignsLow NCILow PDI

Worst DesignsHigh NCIHigh PDI

IndividuallyOptimizedDesigns

Designs basedon Common

Product Platform

Figure 3.9 Product Non-Commonality Versus Performance

By plotting NCI and PDI for a family of designs as illustrated in Figure 3.9,

regions of good and bad product family designs can be identified; the worst designs have

high NCI and PDI, while the best have low NCI and PDI. Individually optimized designs

within a product family, where commonality is not important, are liable to have a low PDI

106

but a high NCI for the resulting group of products. On the other hand, a product family

based on a common product platform is liable to have a low NCI; ideally, a low PDI is

desirable but may be difficult to achieve depending on the amount of commonality desired

between products within the resulting product family.

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5NCI

PD

I

∆NCIgain, ∆PDIlost =

gain in NCI andloss in PDI fromhaving commonproduct platform

PPCEMDesigns

BenchmarkDesigns

KEY:

∆PDIlost

∆NCIgain

∆PDIi

∆NCIi

∆NCIi, ∆PDIi =

change in NCI and PDI by allowing i variables to vary b/n each design

Figure 3.10 Sample Product Variety Tradeoff Study

NCI vs. PDI curves of the form shown in Figure 3.10 can be generated by trading

off product commonality for product performance and vice versa. By designing each

product individually, benchmark designs can be created which have a low PDI.

Meanwhile, the platform designs obtained by implementing the PPCEM have a low NCI.

What is of interest to study is the resulting ∆PDIlost and the ∆NCIgain to assess the tradeoff

between commonality and performance. If ∆PDIlost is too large, the non-commonality of

107

the designs can be increased (∆NCIi) in hopes of obtaining a large decrease in performance

deviation (∆PDIi). Ideally, traversing the front of the envelope provides the largest (∆PDIi)

with the smallest (∆NCIi). This curve is generated in the General Aviation aircraft example

(Section 7.6.3), and additional insights are offered in the context of each problem.

3 .1 .6 Infrastructure of the PPCEM

By assembling the various elements of the PPCEM, the complete infrastructure of

the PPCEM is shown in Figure 3.11. As illustrated in the figure, the PPCEM consists of

“Processors” A-F which are employed as the overall design specifications are transformed

into the product platform portfolio. As described in the previous sections, each step

employs one or more “processors” within this infrastructure.

yResponse

Noise zFactors


s Factors


E. Metamodeling



E2. Model Fitting & Analysis




D. Analysis orSimulation

Routine

C. Design of Experiments

CLASSICAL: Plackett-Burman,Fractional Factorial, CCD, etc.

SPACE FILLING: minimax & maximin Latin hypercubes,rnd orthogonal arrays, uniformdesigns, Hammersley pts, etc.

E1. Model Choice


Use metamodels

ResponseSurface

Kriging

ProductPlatform




IDENTIFYLEVERAGING:


Platform

dkor{ } }{

Figure 3.11 Infrastructure of the PPCEM

Step 1 - Create Market Segmentation Grid relies on human judgment and

engineering “know-how” as Processor A in the PPCEM to map the overall

108

design requirements into an appropriate market segmentation grid and

identify leveraging opportunities.

Step 2 - Classify Factors and Ranges relies on the human judgment and

Processor B in the PPCEM to map the overall design requirements and

market segmentation grid into appropriate control, noise, and scale factors

and identify corresponding ranges for each. The responses being

investigated also need to be identified in this step of the process.

Step 3 - Build and Validate Metamodels relies on human judgment and

Processors C, D, and E for construction and validation of the necessary

metamodels.

Step 4 - Aggregate Product Platform Specifications relies on human

judgment to formulate a compromise DSP, Processor F, using information

from Processors A and B and the overall design requirements.

Step 5 - Develop the Product Platform Portfolio involves exercising

Processor F in combination with the metamodels in Processor E to obtain

the product platform portfolio which best satisfies the given overall design

requirements.

Referring back to Figure 1.5, the structure of the PPCEM is very similar to the

RCEM; this is not a coincidence. In essence, the PPCEM is derived from the RCEM

through a series of modifications based on the research questions and hypotheses presented

in Section 1.3.1. The research hypotheses are revisited next.

3.2 RESEARCH HYPOTHESES AND POSITS FOR THE PPCEM

As stated in Section 1.3, there are three main hypotheses in this dissertation:

Hypothesis 1: The Product Platform Concept Exploration Method provides a

method for developing a scalable product platform for a product family.

109

Hypothesis 2: Kriging is a viable alternative for building metamodels of

deterministic computer analyses.

Hypothesis 3: Space filling experimental designs are better suited for building

metamodels of deterministic computer experiments than classical experimental

designs.

As discussed in Section 1.3.1, Hypotheses 2 and 3 help to support Hypothesis 1 by

increasing the efficiency of the PPCEM but have ramifications beyond the PPCEM itself.

To facilitate verification of Hypothesis 1, three sub-hypotheses have been formulated; these

sub-hypotheses are as follows:

Sub-Hypothesis 1.1: The market segmentation grid can be utilized to help

identify scale factors for a product platform.

Sub-Hypothesis 1.2: Robust design principles can be used to facilitate the

design of a common scalable product platform by minimizing the sensitivity of a

product platform to variations in scale factors.

Sub-Hypothesis 1.3: Individual targets for product variants can be aggregated

into an appropriate mean and variance and used in conjunction with robust

design principles to effect a common product platform for a product family .

It is upon these hypotheses that the PPCEM and the work in this dissertation are

grounded. The relationship between these hypotheses and the modifications to RCEM

which form the PPCEM are presented in the next section. This is followed in Section

3.2.2 with supporting posits for the research hypotheses.

110

3 .2 .1 Relationship of the Research Hypotheses to the RCEM

As the research hypotheses are addressed, modifications to the RCEM are made and

the PPCEM thus is realized. The relationships between the research hypotheses,

designated as H1, H1.1, ..., H3, and the specific elements of the RCEM are illustrated in

Figure 3.12. Addressing the first hypothesis provides an interface with marketing,

enabling the identification of scalable product platforms; this is accomplished through the

addition of a new module to the RCEM, namely, the market segmentation grid. Scale

factors, Sub-Hypotheses 1.1, then are identified through the use of the market

segmentation grid around which a scalable product platform is to be designed. Addressing

this hypothesis results in a modification to the factor classification step of the RCEM.

E. Response Surface Model

y=f(x, z)

µˆy = f(x,µz)

σ2ˆy=Σi=1

k∂f∂zi( )

2

σ2ˆz i i=1

l∂f∂xi( )

2

σ2ˆx i+Σ

Robust, Top-LevelDesign Specifications


A. Factors and Ranges

Product/ Process

Noise zFactors

yResponse

xControlFactors

B. Point Generator

Design of ExperimentsPlackett-Burman

Full Factorial DesignFractional Factorial DesignTaguchi Orthogonal ArrayCentral Composite Design

etc.

C. Simulation Programs

(Rigorous Analysis Tools)


Find Control VariablesSatisfy Constraints Goals "Mean on Target" "Minimize Deviation" “Maximize the independence” BoundsMinimize Deviation Function

D. Experiments Analyzer

Eliminate unimportant factorsReduce the design space to the region of

interestPlan additional experiments


SpaceFillingDoE

MarketSegmentationGrid

ClassifyScaleFactors

H1.1

H3

Kriging

H2

H1.3

RobustDesign ofScalablePlatforms

H1.2

Figure 3.12 Relationships Between Hypotheses and RCEM Modifications

111

Sub-Hypothesis 1.2 relates to designing the actual product platform; robust design

principles are abstracted for use in product family design by aggregating product family

targets and constraints into appropriate means and variances. The resulting formulation

allows robust design principles, already embodied in the RCEM in the form of separate

goals for “bringing the mean on target” and “minimizing the variation,” to be utilized when

solving the compromise DSP. Notice that the goal for “minimize the independence” is not

included in the compromise DSP formulation because the intent is to rely solely on robust

design principles to designing a suitable product platform which can be scaled into a

product family. As the research hypotheses are addressed and the RCEM is

correspondingly modified, the PPCEM is realized; compare Figure 3.12 to Figure 3.11.

Hypotheses 2 and 3 relate specifically to the metamodeling capabilities of the

RCEM with Hypothesis 2 affecting Processor E and Hypothesis 3 affecting Processor B as

shown in Figure 3.12. The intent is not to replace the current response surface and design

of experiments capabilities of the RCEM; rather, it is to augment the current capabilities

with kriging and novel space filling experimental designs, enabling better approximations

of deterministic computer experiments. The specific posits which support these claims and

the research hypotheses are detailed in the next section.

3 .2 .2 Supporting Posits for the Research Hypotheses

There are several posits which support the research hypotheses which have been

revealed during the literature review in Chapter 2 and in the discussion in Section 1.1. Six

posits support Hypotheses 1 and Sub-Hypotheses 1.1-1.3.; they are the following:

Posit 1.1: The RCEM provides an efficient and effective means for developing

robust top-level design specifications for complex systems design.

112

Posit 1.2: Metamodeling techniques, specifically, design of experiments and the

response surface methodology, can be used to facilitate concept exploration and

optimization, thus increasing a designer’s efficiency.

Posit 1.3: Robust design principles can be used to minimize the sensitivity of a

design to variations in uncontrollable (i.e., noise) factors and/or variations in

design parameters (i.e., control factors).

Posit 1.4: Robust design principles can be used effectively in the early stages of the

design process by modeling the response itself with separate goals for “bringing

the mean on target” and “minimizing the variation.”

Posit 1.5: The compromise DSP is capable of effecting robust design solutions

through separate goals for “bringing the mean on target” and “minimizing

variation” of noise factors and/or variations in the design variables.

Posit 1.6: The market segmentation grid can be used to identify opportunities for

platform leveraging in product family design.

These posits are founded on and substantiated by the following references.

• Posit 1.1 is substantiated in (Chen, 1995) by explicitly testing and verifying the

efficiency and effectiveness of the RCEM for developing robust top-level design

specifications for complex systems design.

• Posit 1.2 is substantiated through the numerous applications of design of

experiments and response surface models that have been used to illustrate the

usefulness of metamodeling techniques in design, facilitating concept exploration

and optimization and increasing a designer’s efficiency; over twenty-five such

applications are reviewed in (Simpson, et al., 1997b). Furthermore, Chen (1995)

explicitly tests and verifies this posit as part of the development of the RCEM.

• Posit 1.3, Posit 1.4, and Posit 1.5 are substantiated by the work in (Chen, et al.,

1996b); Chen and her coauthors describe a general robust design procedure which

can minimize the sensitivity of a design to variations in noise factors and/or design

113

parameters (Posit 1.3) by having separate goals for “bringing the mean on target”

and “minimizing the variation” (Posit 1.4) of each response in the compromise DSP

(Posit 1.5). These posits are further substantiated in (Chen, 1995) as part of the

development of the RCEM.

• Posit 1.6 is substantiated by the discussion in Section 2.2.1, i.e., the market

segmentation grid can be used as an attention directing tool to identify leveraging

opportunities in product platform design (cf., Meyer, 1997; Meyer and Lehnerd,

1997); identifying these leveraging opportunities provided the initial impetus for

developing the market segmentation grid.

These six posits help to support Hypothesis 1 and Sub-Hypotheses 1.1-1.3. The strategy

for testing and verifying these hypotheses is outlined in Section 3.3. Before the

verification strategy is outlined, however, posits for Hypotheses 2 and 3 are stated.

The following two posits have been identified in support of Hypothesis 2.

Posit 2.1: Building an interpolative kriging model is not predicated on the

assumption of underlying random error in the data.

Posit 2.2: Kriging provides very flexible modeling capabilities based on the wide

variety of spatial correlation functions which can be selected to model the data.

• Posit 2.1 is more fact than assumption; it is substantiated by, e.g., Sacks, et al.

(1989); Koehler and Owen (1996); and Cressie (1993).

• Posit 2.2 is substantiated by many researchers, most notably Sacks, et al. (1989);

Welch, et al. (1992); Cressie (1993); and Barton (1992; 1994).

Posits 2.1 and 2.2 both help to verify Hypothesis 2; the strategy for testing Hypothesis 2 is

outlined in Section 3.3.

Finally, the following two posits support Hypothesis 3:

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Posit 3.1: The experimental design conditions of replication, blockability,

rotatability have no significant meaning when sampling deterministic computer

experiments.

• Posit 3.1 is taken from Sacks, et al. (1989) who state that the “classical notions of

experimental blocking, replication, and randomization are irrelevant” for

deterministic computer experiments which contain no random error. Moreover, any

experimental design text (see, e.g., Montgomery, 1991) can verify that replication,

blockability, and rotatability are developed explicitly to handle and account for

random (measurement) error in a physical experiment for which classical

experimental designs have been developed.

Since kriging (using an underlying constant model) is being advocated in this

dissertation for metamodeling deterministic computer experiments, an additional posit in

support of Hypothesis 3 is the following.

Posit 3.2: Since kriging (with an underlying constant model) models rely on the

spatial correlation between data, confounding and aliasing of main effects and

two-factor interactions have no significant meaning when predicting a response.

• Posit 3.2 is substantiated by Sacks, et al. (1989); Currin, et al. (1991); Welch, et

al. (1990); and Barton (1992; 1994). In physical experimentation, great care is

taken to ensure that aliasing and confounding of main effects and two-factor

interactions do not occur to ensure accurate estimation of coefficients of the

polynomial response surface model (see, e.g., Montgomery, 1991).

The experimental procedure for testing Hypothesis 3 is introduced in the next section along

with the specific strategy for verification and testing of all of the other hypotheses.

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3.3 STRATEGY FOR VERIFICATION AND TESTING OF THERESEARCH HYPOTHESES

The question of whether testing the proposed hypotheses really answers the

research questions is a difficult one, and it raises the issues of verification and validation as

discussed by Peplisnki (1997). According to the Concise Oxford English Dictionary

(1982), to validate is to make valid, to ratify or confirm. The root, valid, is then defined

as:

• (of reason, objection, argument, etc.) sound, defensible, well-grounded;

• (law) sound and sufficient, executed with proper formalities (valid contract);

• legally acceptable (valid passport).

With respect to engineering design research, the intent of the validation process is to show

the research and its products to be sound, well grounded on principles of evidence, able to

withstand criticism or objection, powerful, convincing and conclusive, and provable.

In the Merriam-Webster hypertext dictionary under a discussion of the synonyms

for “confirm”, “verify” is used in the context of establishing the correspondence of actual

facts or details with those proposed or guessed at, while “validate” is used in the context of

establishing validity by authoritative affirmation or by factual proof. The boundary

between verification and validation is thus shifting and often open to interpretation; in many

cases the two words are used interchangeably.

In this research, definitions for “verification” and “validation” are applied that,

while not inconsistent with the general uses above, are more specific and tailored for efforts

in engineering design research. In practice, the verification and validation of design

methods is much more than a debugging process. Three primary phases can be identified:

firstly, problem justification; secondly, completeness and consistency checks of the

methodology, and thirdly, validation of performance. (This classification is based on a

discussion of the validation of expert systems by Ignizio (1990).) Verification then

116

refers to the second phase of the process and is focused primarily on internal consistency

and completeness, while validation as the third phase of the process is focused on

consistency with external evidence, ideally through testing the design method on actual case

studies. This validation of performance is perhaps the area most open to interpretation by

peers and experts in the field alike.

If what is to be validated is a closed form mathematical expression or algorithm, it

can be proven, or validated, in a traditional and formal mathematical sense. For example,

the case of showing a solution vector, x , belongs to the set of feasible solutions for a given

mathematical model is a closed problem. Alternatively, if the problem is open, if the

subject is dealing with some “heuristic,” non-precise scheme, the issue of validation

becomes one of “correctness beyond reasonable doubt.” The validation of design methods

falls into this category. In this case it is achieved ultimately by results and usefulness and

through a convincing demonstration to (and an acceptance and ratification by) one’s peers

in the field. An analogy with mathematics and the concept of “necessary” and “sufficient”

conditions can be drawn with respect to the validation of heuristics. Heuristics are aimed

toward satisfying the necessary conditions only; it is not possible to develop an absolute

proof for an open problem by definition.

As anticipated, the operations research literature provides some useful insight into

the validation of heuristics, in the context of heuristic programming. In discussing the

nature of problem solving by heuristic programming Lin (1975) makes the following

remarks:

We therefore define a valid heuristic algorithm (to solve a given problem) as anyprocedure which will produce a feasible solution acceptable to the designengineer, within limits of computing time, and consider the problem solved ifwe can construct a valid heuristic procedure to solve it. We see that in thedomain where a heuristic algorithm operates, there are elements of technique,experimentation, judgment and persuasion, as well as compromise.

The issue of justification is addressed by Ignizio, et al. (1972):

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Specific heuristic programs are justified, not because they attain an analyticallyverifiable optimum solution, but rather because experimentation has proven thatthey are useful in practice.

In summary, while noting that judgment is subjective and based on faith, the validation of a

heuristic, and therefore the validation of design methods, can be established if (Smith,

1992):

• the solutions are feasible and acceptable to the design engineer,

• the time and consumed resources are within reasonable limits, and

• the solutions are, above all, useful.

It is against these three issues that a verification and validation strategy is developed.

Meanwhile, verification and testing of the hypotheses has already begun by stating

and substantiating posits in support of each hypothesis. What is tested in the remainder of

the dissertation is the “intellectual leap of faith” required to jump from the posits to the

hypotheses. The relationships between the next four chapters and the individual research

hypotheses are summarized in Table 3.1.

Table 3.1 Relationship Between Hypothesis Testing and Chapters

Hypothesis Chp 4 Chp 5 Chp 6 Chp 7H1 Product Platform Concept Exploration

Method X XSH1.1 Usefulness of market segmentation

grid X XSH1.2 Robust design of scalable product

platform X XSH1.3 Aggregating product family

specifications X XH2 Utility of kriging for metamodeling

deterministic computer experiments X X XH3 Utility of space filling experimental

designs X X

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The relationships listed in Table 3.1 are elaborated further in the next two sections.

The strategy for testing Hypothesis 1 and the related sub-hypotheses is outlined in Section

3.3.1. The strategy for testing Hypotheses 2 and 3 is outlined in Section 3.3.2.

3 .3 .1 Testing Hypothesis 1 and Sub-Hypotheses 1.1-1.3

The PPCEM is hypothesized to provide a method for designing a scalable product

platform for a product family. To verify this, two example problems are utilized to

demonstrate the effectiveness of the PPCEM: the design of a family of universal electric

motors (Chapter 6) and the design of a family of General Aviation aircraft (Chapter 7).

These two examples have been chosen to demonstrate different capabilities of the PPCEM.

• Design of a family of universal electric motors is used to demonstrate the following:

- vertical scaling of a product platform (see Section 6.2),

- a parametric scale factor: stack length (see Section 6.2), and

- product family aggregated around mean and standard deviation of stack length

and separate goals for “bringing the mean on target” and “minimize the

variation” are employed to design the product platform (see Section 6.3).

Note that metamodels are not employed in this first example because mean and standard

deviation of the responses can be estimated directly from the relevant analysis equations

(see Section 6.3).

• Design of a family of General Aviation aircraft is used demonstrate the following:

- horizontal scaling of a product platform (see Section 7.1.3),

- a configurational scale factor: number of passengers (see Section 7.2),

- metamodels for mean and standard deviation of the GAA family to facilitate

implementation of robust design and development of the aircraft platform (see

Section 7.3), and

- design capability indices to assess quickly the capability of the family of aircraft

to satisfy the range of requirements (see Section 7.4).

119

The first example parallels Black & Decker’s vertical scaling strategy for its universal

motors (Lehnerd, 1987) discussed in Section 1.1.1 and is used to provide “proof of

concept” that the PPCEM works. The second example is based on a previous application

of the RCEM to develop a “common and good” set of top-level design specifications for a

family of General Aviation aircraft (see, e.g., Simpson, 1995; Simpson, et al., 1996). The

General Aviation aircraft problem is employed in this work to demonstrate further the

effectiveness of the PPCEM while exploring the problem itself in greater depth.

In each example, the product platform obtained using the PPCEM is compared to

(a) the initial baseline design to show improvement over the starting design, and (b)

individually designed, benchmark products which are aggregated into a product family to

provide a reference to compare against the PPCEM product family (i.e., design the family

of products with the PPCEM and without the PPCEM and discuss the differences in

product performance, computational expense, and usefulness). Product variety tradeoff

studies are also performed for the family of General Aviation aircraft, examining the

tradeoff between commonality of the aircraft and their corresponding performance for the

PPCEM family and the individually designed benchmark group of aircraft.

Testing of the sub-hypotheses related to Hypotheses 1 entail the following:

Testing Sub-Hypothesis 1.1 - The procedure for using the market segmentation

grid to identify scale factors for a product platform is shown in Figure 3.4 and

described in Section 3.1.2. Further verification of this sub-hypothesis requires

demonstrating that this procedure can be used to identify scale factors for a product

platform. In the universal motor example in Chapter 6, the market segmentation

grid is used to identify a vertical leveraging strategy and parametric scaling factor

(stack length); in the General Aviation aircraft example, a horizontal leveraging

strategy and configurational scale factor (number of passengers) are used.

Testing Sub-Hypothesis 1.2 - If appropriate scale factors can be identified for a

product platform (i.e., if Sub-Hypothesis 1.1 is true), then the principles of robust

design can be employed to develop a product platform which has minimum

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sensitivity to variations in the scale factor and is thus robust for the product family.

Verification of this sub-hypothesis requires implementation of the approach, and the

two examples provide such a demonstration.

Testing Sub-Hypothesis 1.3 - The procedure for aggregating the individual targets

of the product variants is outlined in Section 3.1.4. As with Sub-Hypothesis 1.1,

further verification of this sub-hypothesis requires demonstrating that this

procedure can be used to model and design a family of products; the approaches

outlined in Section 3.1.4 are used in the two examples to illustrate both methods for

aggregating product family specifications.

Verification of these sub-hypotheses also helps to support Hypothesis 1.

3 .3 .2 Testing Hypotheses 2 and 3

The testing of Hypotheses 2 and 3 occurs predominantly in Chapter 5; however, an

initial feasibility study of the utility of kriging is presented in Chapter 4 to familiarize the

reader with kriging. In Chapter 4, a simple yet realistic engineering example—the design

of an aerospike nozzle—is used to compare the predictive capability of a kriging model

against that of second-order response surfaces. The specific aspect of Hypothesis 2 being

tested in Section 4.2 is whether or not kriging, using an underlying constant global model

in combination with a Gaussian correlation function (one of the five being investigated in

this dissertation, see section 2.4.2) is as accurate of a predictor of the response values as a

full second-order response surface.

Chapter 5 continues from where Chapter 4 leaves off. To test the utility of kriging

and space filling designs (and thus Hypotheses 2 and 3) a testbed of six engineering test

problems is created to:

• test the effect of different correlation functions on the accuracy of the kriging model

for a wide variety of engineering analysis equations (linear, quadratic, cubic,

reciprocal, exponential, etc.);

• correlate the types of functions (analysis equations) which kriging models can and

cannot approximate accurately; and

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• test the effect of eleven different experimental designs on the accuracy of the

resulting kriging model.

Of the eleven experimental designs mentioned in the last bullet, two are classical designs—

central composite and Box-Behnken—and the remaining nine are space filling (see Section

5.1.3 and Appendices B and C for a description of each). In this manner, Hypothesis 3 is

explicitly tested by comparing the accuracy of the kriging model built from a space filling

experimental design against that of a classical experimental design. The first two bullets

relate to testing Hypothesis 2, and the particulars of that portion of the study are described

in Section 5.2. A detailed overview of the whole study is offered in Section 5.1.


The elements of the previous chapters are synthesized in this chapter to meet the

principal objective in this dissertation, namely, to develop the Product Platform Concept

Exploration Method (PPCEM) for designing common scalable product platforms for a

product family, see Figure 3.13. There are five steps to the PPCEM which prescribe how

to formulate the problem and describe how to solve it. As such, the PPCEM provides a

Method which facilitates the synthesis and Exploration of a common Product Platform

Concept which can be scaled into an appropriate family of products.

Testing and verification of the PPCEM is outlined in the previous section and takes

place predominantly in Chapters 6 and 7 of the dissertation. In the meantime, testing of

Hypothesis 2 (which has implications for Step 3 of the PPCEM) commences in the next

chapter wherein an initial feasibility study of the utility of kriging is given. At the end of

Chapter 4, several questions are posed which preface the kriging/DOE study in Chapter 5.

The implications of the results of the study on metamodeling within the PPCEM (Step 3)

are discussed at the end of Chapter 5.

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Metamodeling



ConceptualNoise

Factors


MarketSegmentation

Grid



Chp 4 Chp 3


Platform

Product Platform Concept Exploration MethodNozzle Design


123

4. CHAPTER 4

INITIAL KRIGING FEASIBILITY STUDY:DESIGN OF AN AEROSPIKE NOZZLE

Nozzle Design

In this chapter, the process of testing Hypothesis 2 and establishing kriging as a

viable alternative for approximating deterministic computer experiments commences. In

particular, the accuracy of simple kriging models is compared to that of second-order

response surface models through two examples. The first is a simple one-dimensional

example in Section 4.1 which is used to familiarize the reader with (a) the process of

creating a kriging model and (b) some of the differences between a kriging model and a

second-order response surface. The second example is a simple, yet realistic, engineering

problem—the design of an aerospike nozzle—which is given in Section 4.2. In the

aerospike nozzle example, a comparison of second-order response surface models and

kriging models is conducted by means of error analysis (Section 4.2.2), visualization

(Section 4.2.3), and optimization (Section 4.2.4) These examples establish that a simple

kriging model can compete with a second-order response surface, thereby setting the stage

for an extensive investigation into the utility of kriging and space filling experimental

designs in Chapter 5 to test Hypotheses 2 and Hypothesis 3.

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4.1 OVERVIEW OF KRIGING MODELING AND A 1-D EXAMPLE

Having presented the mathematics behind kriging in Section 2.4.2, a simple one

variable example best illustrates the difference between the approximation capabilities of a

second-order response surface model and a kriging model. This example comes from Su

and Renaud (1996) who fabricated this example to demonstrate some of the limitations of

using second-order response surface models, see Figure 4.1. The function is an eighth-

order function given by Equation 4.1.

f(x) = a i(x i − 900)(i − 1)

i=1

9

∑ [4.1]

a1 = -659.23a2 = 190.22a3 = -17.802a4 = 0.82691

a5 = -0.021885a6 = 0.0003463

a7 = -3.2446 x 10-6

a8 = 1.6606 x 10-8

a9 = -3.5757 x 10-11

A second-order response surface model is fit to five sample points within the region

of the optimum (x = 932) using least squares regression. The five sample points are given

in Table 4.1. The original function, the location of the five sample points, and the resulting

second-order response surface are shown in Figure 4.1.

Table 4.1 Sample Points for 1-D Example

No . x x (scaled) y1 922 0.00 43.9762 927 0.25 20.1433 932 0.50 13.9634 937 0.75 17.3305 942 1.00 22.698

125

10

20

30

40

50

60

70

80

915 925 935 945 955 965x

y

Su and Renaud (1996) Fcn

Sample Points

2nd Order RS Model

Figure 4.1 One Variable Example Problem

A kriging model using a constant for the global model and the Gaussian correlation

function of Equation 2.17 is fit to the same five points in order to compare a kriging model

against a second-order response surface model. The process of fitting a kriging model is

described step-by-step to foster a better understanding of what is involved in building a

kriging model.

In order to fit a kriging model to the five sample points, the x values are scaled to

[0,1] as shown in Table 4.1, and the response values are written as a column vector, yT =

{43.977, 20.143, 13.963, 17.330, 22.698}. Because a constant underlying global model

is selected for the kriging model, f is simply a column vector of ones: fT = {1, 1, 1, 1, 1}.

Using a Gaussian correlation function for the localized portion of the model, Equation

2.21, is particularized for this example as:

126

R(x i,x j) = exp(−θ |x i − x j |2) i,j = 1,2,3,4,5;i ≠ j1 i = j

[4.2]

The correlation function for each sample point is then computed as follows:

i = 1, j = 1, R(x1,x1) = 1

i = 1, j = 2, R(x1,x2) = Exp(-θ|0.0 - 0.25|2) = exp(-0.0625θ)

i = 1, j = 3, R(x1,x3) = Exp(-θ|0.0 - 0.5|2) = exp(-0.25θ)

i = 1, j = 4, R(x1,x4) = Exp(-θ|0.0 - 0.75|2) = exp(-0.5625θ)

i = 1, j = 5, R(x1,x5) = Exp(-θ|0.0 - 1.0|2) = exp(-θ)

M

i = 5, j = 5, R(x5,x5) = 1

The resulting correlation matrix is thus:

R =

1 e−0.0625θ e−0.25 θ e−0.5625θ e− θ

1 e−0.0625θ e−0.25 θ e−0.5625θ

1 e− 0.0625θ e− 0.25θ

sym 1 e−0.0625θ

1

where θ is the unknown parameter which is used to fit the kriging model to the data.

The constant portion of the global model is now estimated using Equation 2.27

which is repeated here as Equation 4.3:

ˆ β = (f TR−1f)−1 fTR −1y [4.3]

and is a function of θ. The value for ˆ β , once the maximum likelihood estimate for θ is

known, is essentially a weighted average of the sample points based on intersite distances.

127

In order to find the maximum likelihood estimate for θ, the variance of sample data

from the underlying constant global model must be estimated from Equation 2.28 which is

repeated here as Equation 4.4:

ˆ σ 2 =(y − f ˆ β )T R−1(y − fˆ β )

ns

[4.4]

where ns = 5. The MLE for θ is then found by maximizing Equation 4.5 which is the same

as Equation 2.29 given previously in Section 2.4.2.

Φ(θ) = −[n s ln(ˆ σ 2) + ln | R |]

2[4.5]

A plot of Φ(θ) is given in Figure 4.2. The MLE, or “best” guess, for θ is the point which

maximizes Φ(θ) from Equation 4.5.

-18

-17

-16

-15

-14

-13

-12

-11

-100 2 4 6 8 10 12 14 16 18 20

theta

ML

E o

bj

fcn

θ* = 6.924

Φ(θ

)

Figure 4.2 MLE Objective Function for 1-D Example

128

In this example, the MLE for θ is 6.924; hence, the “best” kriging model to fit these

five sample points when using a constant underlying global model and the Gaussian

correlation function is when θ = 6.924. Substituting this value into Equation 4.2, the

resulting correlation matrix is thus:

R =

1 0.649 0.177 0.020 0.0011 0.649 0.177 0.020

1 0.649 0.177sym 1 0.649

1

Now, new points are predicted using the scalar form of Equation 2.25 which is

repeated here as Equation 4.6:

ˆ y = ˆ β + rT(x)R−1(y − f ˆ β ) [4.6]

where rT(x) is the correlation vector of length 5 between an untried value of x and the

sampled data points {0.00, 0.25, 0.50, 0.75, 1.00}. The general form of rT(x) is given by

Equation 2.26 which is particularized for this example as follows:

rT(x) = { R(x,0.00), R(x,0.25), R(x,0.50), R(x,0.75), R(x,1.00) }

where R is the Gaussian correlation function. Notice that the x values for which a new y is

to be predicted are scaled to [0,1]; however, the predicted values of y are the actual values.

A plot of the resulting kriging model—using a Gaussian correlation function and an

underlying constant global model—is shown in Figure 4.3 along with the original function,

the second-order response surface, and the five sample points. Immediately evident from

the figure is fact that the kriging model interpolates the data points, approximating the

original function better than the second-order response surface model which represents a

129

least squares fit. In this example, the interpolating capability of the kriging model allows it

to predict an optimum which is much closer to the actual optimum.

10

20

30

40

50

60

70

80

915 925 935 945 955 965x

y

Su and Renaud (1996) Fcn

Sample Points

2nd Order RS Model

Kriging w/Gauss.

Figure 4.3 One Variable Example of Response Surface and Kriging Models

It is also important to notice that outside of the design space defined by the sample

points (920 ≤ x ≤ 945), neither model predicts as well as expected. The kriging model

returns to the underlying global model which is a constant in this example. This is typical

behavior for a kriging model; far from the design points, the kriging model returns to the

underlying global model because the influence of the sample points has “exponentially

decayed away” outside of the design space.

Sixteen, evenly spaced points (not including the sample points) are taken from

within the sample range (920 ≤ x ≤ 945) to assess the accuracy of the two approximations.

130

The maximum absolute error, the average absolute error, and the root mean square error

(MSE), Equations 2.30-2.32, for the 16 validation points are listed in Table 4.2. Both raw

values and percentages of actual values are listed in the table for ease of comparison.

Table 4.2 Error Analysis of One Variable Example

Raw Values As a % of Actual ValueError

Measures2nd OrderRS Model

KrigingModel

2nd OrderRS Model

KrigingModel

Max ABS(error) 3.134 2.507 18.34% 7.52%Avg ABS(error) 1.911 0.776 10.32% 3.59%root MSE 2.155 1.004 11.83% 4.16%

Based on this error analysis, the kriging model approximates the original function

better because it has a lower root MSE, average absolute error, and maximum absolute

error. A more involved example to compare further the predictive capability of second-

order response surface models and kriging models is presented in the next section.

4.2 AEROSPIKE NOZZLE DESIGN PROBLEM

The design of an aerospike nozzle has been selected as the preliminary test problem

for comparing the predictive capability of response surface and kriging models. The linear

aerospike rocket engine is the propulsion system proposed for the VentureStar reusable

launch vehicle (RLV) which is illustrated in Figure 4.4. The VentureStar RLV is one of the

concepts for the next generation space shuttles (Sweetman, 1996).

131

Figure 4.4 VentureStar RLV with Aerospike Nozzle (Korte, et al., 1997)

The aerospike rocket engine consists of a rocket thruster, cowl, aerospike nozzle,

and plug base regions as shown in Figure 4.5. The aerospike nozzle is a truncated spike or

plug nozzle that adjusts to the ambient pressure and integrates well with launch vehicles

(Rao, 1961). The flow field structure changes dramatically from low altitude to high

altitude on the spike surface and in the base region (Hagemann, et al., 1996; Mueller and

Sule, 1972; Rommel, et al., 1995). Additional flow is injected in the base region to create

an aerodynamic spike (Iacobellis, et al., 1967) which gives the aerospike nozzle its name

and increases the base pressure and contribution of the base region to the aerospike thrust.

Figure 4.5 Aerospike Components and Flow Field Characteristics(Korte, et al., 1997)

132

The analysis of the nozzle involves two disciplines: aerodynamics and structures;

there is an interaction between the structural displacements of the nozzle surface and the

pressures caused by the varying aerodynamic effects. Thrust and nozzle wall pressure

calculations are made using computational fluid dynamics (CFD) analysis and are linked to

a structural finite element analysis model for determining nozzle weight and structural

integrity. A mission average engine specific impulse and engine thrust/weight ratio are

calculated and used to estimate vehicle gross-lift-off-weight (GLOW). The

multidisciplinary domain decomposition is illustrated in Figure 4.6. Korte, et al. (1997)

provide additional details on the aerodynamic and structural analyses for the aerospike

nozzle.

Figure 4.6 Multidisciplinary Domain Decomposition for Aerospike Nozzle(Korte, et al., 1997)

For this study, three design variables are considered: starting (thruster) angle, exit

(base) height, and (base) length as shown in Figure 4.7. The thruster angle (a) is the

entrance angle of the gas from the combustion chamber onto the nozzle surface; the base

133

height (h) and length (l) refer to the solid portion of the nozzle itself. A quadratic curve

defines the aerospike nozzle surface profile based on the values of thruster angle, height,

and length.

Figure 4.7 Nozzle Geometry Design Variables (Korte, et al., 1997)

Bounds for the design variables are set to produce viable nozzle profiles from the

quadratic model based on all combinations of thruster angle, height, and length within the

design space. Second-order response surface models and kriging models are developed

and validated for each response (thrust, weight, and GLOW) in the next section;

optimization of the aerospike nozzle using the response surface and kriging models for

different objective functions is performed in Section 4.2.4.

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4 .2 .1 Metamodeling of the Aerospike Nozzle Problem

The data used to fit the response surface and kriging models is obtained from a 25

point random orthogonal array (Owen, 1992). The use of these orthogonal arrays in this

preliminary example is based, in part, on the success of the work by Booker, et al. (1995)

and the recommendations of Barton (1994). The actual sample points are illustrated in

Figure 4.8 and are scaled to fit the three dimensional design space defined by the bounds

on the thruster angle (a), base height (h), and length (l).

Angle

Height

Length

Figure 4.8 Sample Points of 25 Point Orthogonal Array

Response Surface Models for the Aerospike Nozzle Problem

The response surface models for weight, thrust, and GLOW are fit to the 25 sample points

using ordinary least squares regression techniques and the software package JMP® (SAS,

1995). The resulting second-order response surface models are given in the Equations 4.7-

4.9. The equations are scaled against the baseline design to protect the proprietary nature

of some of the data.

Weight = 0.810 - 0.116a + 0.121h + 0.152l + 0.065a2 - 0.025ah +0.0013h2 - 0.0539al - 0.0131hl + 0.0301l2 [4.7]

135

Thrust = 0.997 + 0.00031a + 0.0019h + 0.0060l - 0.00175a2 +0.00125ah - 0.0011h2 + 0.00125al - 0.00198hl - 0.00165l2 [4.8]

GLOW = 0.9930 - 0.0270a + 0.0065h - 0.0265l + 0.0307a2 -0.0163ah + 0.0100h2 - 0.0226al + 0.0151hl + 0.0195l2 [4.9]

The R2, R2adj, and root MSE values for each of these second-order response surface

models are summarized in Table 4.2. As evidenced by the high R2 and R2adj values and

low root MSE values, the second-order polynomial models appear to capture a large

portion of the observed variance.

Table 4.3 Model Diagnostics of Response Surface Models

ResponseMeasure Weight Thrust GLOW

R2 0.986 0.998 0.971R2

adj 0.977 0.996 0.953root MSE 1.12% 0.01% 0.25%

Kriging Models for the Aerospike Nozzle Problem

The kriging models are built from the same 25 sample points used to fit the response

surface models. In this preliminary example, a constant term for the global model and a

Gaussian correlation function, Equation 2.21, for the local departures are chosen.

Initial investigations revealed that a single θ parameter was insufficient to model the

data accurately due to scaling of the design variables (a similar problem is encountered in

(Giunta, et al., 1998)). Therefore, a simple 3-D exhaustive grid search with a refinable

step size is used to find the maximum likelihood estimates for the three θ parameters needed

to obtain the “best” kriging model. The resulting maximum likelihood estimates for the

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three θ parameters for the weight, thrust, and GLOW models are summarized in Table 4.4;

note that these values are for the scaled sample points.

Table 4.4 Theta Parameters for Kriging Models of Aerospike Nozzle

MLE ResponseValues Weight Thrust GLOW

θangle = 0.548 0.30 3.362θheight = 1.323 0.50 2.437θlength = 2.718 0.65 0.537

With these parameters for the Gaussian correlation function, the kriging models

now are specified fully. A new point is predicted using these θ values and the 25 sample

points as shown in combination with Equations 2.25-2.27. The accuracy of the response

surface and kriging models is examined in the next two sections.

4 .2 .2 Error Analysis of Response Surface and Kriging Models

An additional 25 randomly selected validation points are used to verify the accuracy

of the response surface and kriging models. Error is defined as the difference between the

actual response from the computer analysis, y(x), and the predicted value, ˆ y (x), from the

response surface or kriging model. The maximum absolute error, the average absolute

error, and the root MSE, see Equations 2.30-2.32, for the 25 randomly selected validation

points are summarized in Table 4.5.

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Table 4.5 Error Analysis of Aerospike Nozzle Approximation Models

2nd Order Response Surface ModelsError Measure Weight Thrust GLOWMax. ABS(error) 19.57% 0.032% 3.68%Avg. ABS(error) 2.44% 0.012% 0.53%root MSE 4.54% 0.015% 0.90%

Kriging Models (with Constant Term)Error Measure Weight Thrust GLOWMax. ABS(error) 17.23% 0.048% 3.43%Avg. ABS(error) 2.51% 0.012% 0.59%root MSE 4.37% 0.018% 0.89%

For the weight and GLOW responses, the kriging models have lower maximum

absolute errors and lower root MSEs than the response surface models; however, the

average absolute error is slightly larger for the kriging models. For thrust, the response

surface models are slightly better than the kriging models according to the values in the

table; the maximum absolute error and root MSE are slightly less while the average absolute

errors are essentially the same. It is not surprising that the response surface models predict

thrust better; it has a very high R2 value, 0.998, and low root MSE, 0.01%. It is

reassuring to note, however, that the kriging model, despite using only a constant term for

the underlying global model, is only slightly less accurate than the corresponding response

surface model. In summary, it appears that both models predict each response reasonably

well, with the kriging models having a slight advantage in overall accuracy because of the

lower root MSE values. A graphical comparison is presented in the next section to examine

the accuracy of the response surface and kriging models further.

4 .2 .3 Graphical Comparison of Response Surface and Kriging Models

In addition to the numerical error analysis of the previous section, a graphical

comparison of the response surface and kriging models is performed to visualize

differences in the two approximation models. In Figure 4.9-4.10, three dimensional

138

contour plots of thrust, weight, and GLOW as a function of thruster angle, length, and

height are given. In each figure, the same contour levels are used for the response surface

and kriging models so that the shapes of the contours can be compared.

(a) Thrust (b) Weight

Figure 4.9 Response Surface and Kriging Models for Thrust and Weight

In Figure 4.9a, the contours of thrust for the response surface and kriging models

are very similar. As evidenced by the high R2 and low root MSE values, the response

surface models should fit the data quite well, and it is reassuring to note that the kriging

models resemble the response surface models even through the underlying global model for

the kriging models is just a constant term. This demonstrates the power and flexibility of

the “local” deviations of the kriging model in general, and of the Gaussian correlation

function in particular.

The contours of the response surface and kriging models in Figure 4.9b are also

very similar, but the influence of the localized perturbations caused by the Gaussian

correlation function can be seen in the kriging model for weight. The error analysis from

the previous section indicated that the kriging model for weight is slightly more accurate

139

than the second-order response surface model which may result from the small non-linear

localized variations in the kriging model.

The general shape of the GLOW contours is the same in Figure 4.10; however, the

size and shape of the different contours, particularly along the length axis, are quite

different. The end view along the length axis in Figure 4.10b further highlights the

differences between the two models. Notice also in Figure 4.10b that the kriging model

predicts a minimum GLOW located within the design space centered around Height = -0.8,

Angle = 0, along the axis defined by 0.2 ≤ Length ≤ 0.8; this minimum was verified

through additional experiments and is assumed to be the minimum value for GLOW.

(a) GLOW - Isometric View (b) GLOW - End View

Figure 4.10 Response Surface and Kriging Models for GLOW

From the graphical and error analyses of the response surface and kriging models,

it appears that both models fit the data quite well. In the next section the accuracy of both

metamodels is put to the test. Four optimization problems are formulated and solved using

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each of the metamodels and the efficiency and accuracy of the results are compared as a

final test of model adequacy.

4 .2 .4 Optimization using the Response Surface and Kriging Metamodels

The true test of the accuracy of the response surface and kriging models comes

when the approximations are used during optimization. It is paramount that any

approximations used in optimization prove reasonably accurate, lest they lead the

optimization algorithm into regions of bad designs. Trust Region approaches (see e.g.,

Lewis, 1996; Rodriguez, et al., 1997) and the Model Management framework (see e.g.,

Alexandrov, et al., 1997; Booker, et al., 1995) have been developed to ensure that

optimization algorithms are not led astray by inaccurate approximations. In this work,

however, the focus has been on developing the approximation models, particularly the

kriging models, and not on the optimization itself.

Four different optimization problems are formulated and solved to compare the

accuracy of the response surface and kriging models, see Table 4.6: (1) maximize thrust,

(2) minimize weight, (3) minimize GLOW, and (4) maximize thrust/weight ratio. The first

two objective functions in Table 4.6 represent traditional single objective, single discipline

optimization problems. The second two objective functions are more characteristic of

multidisciplinary optimization; minimizing GLOW or maximizing the thrust/weight ratio

requires tradeoffs between the aerodynamics and structures disciplines. As seen in the

table, for each objective function, constraint limits are placed on the remaining responses;

for instance, constraints are placed on the maximum allowable weight and GLOW and the

minimum allowable thrust/weight ratio when maximizing thrust. However, none of the

constraints are active in any of the final results.

141

Table 4.6 Aerospike Nozzle Optimization Problem Formulations

Problem #1: Maximize Thrust Problem #2: Minimize WeightFind:

-1 ≤ a ≤ 1-1 ≤ h ≤ 1-1 ≤ l ≤ 1

Satisfy:Weight ≤ WeightmaxGLOW ≤ GLOWmaxThr/Wt ≥ (Thr/Wt)min

Maximize:Thrust = f(a,h,l)

Find:-1 ≤ a ≤ 1-1 ≤ h ≤ 1-1 ≤ l ≤ 1

Satisfy:Thrust ≥ ThrustminGLOW ≤ GLOWmaxThr/Wt ≥ (Thr/Wt)min

Minimize:Weight = f(a,h,l)

Problem #3: Minimize GLOW Problem #4: Maximize Thr/Wt RatioFind:

-1 ≤ a ≤ 1-1 ≤ h ≤ 1-1 ≤ l ≤ 1

Satisfy:Thrust ≥ ThrustminWeight ≤ WeightmaxThr/Wt ≥ (Thr/Wt)min

Minimize:GLOW = f(a,h,l)

Find:-1 ≤ a ≤ 1-1 ≤ h ≤ 1-1 ≤ l ≤ 1

Satisfy:Thrust ≥ ThrustminWeight ≤ WeightmaxGLOW ≤ GLOWmax

Maximize:Thr/Wt = f(a,h,l)

Each optimization problem is solved using: (a) the second-order response surface

models and (b) the kriging model approximations for thrust, weight, and GLOW. The

optimization is performed using the Generalized Reduced Gradient (GRG) algorithm in

OptdesX (Parkinson, et al., 1998). Three different starting points are used for each

objective function (the lower, middle, and upper bounds of the design variables) to assess

the average number of analysis and gradient calls necessary to obtain the optimum design

within the given design space. The same parameters (i.e., step size, tolerance, constraint

violation, etc.) are used within the GRG algorithm for each optimization. The optimization

results are summarized in Table 4.7. Design variable and response values have been scaled

as a percentage of the baseline design due to the proprietary nature of some of the data.

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Table 4.7 Aerospike Nozzle Optimization Results Using Metamodels

Avg. # of Avg. # of VerifiedAnalysis Gradient Optimum Design Predicted Optimum Optimum1 % Error2

Calls CallsMaximize Thrust

Angle 0.096 Thrust 1.0016 1.0013 0.02%RS 27 4 Height -0.433 Weight 0.9450 0.9476 -0.27%

Models Length 1.000 Thr/Wt 1.0141 1.0134 0.07%GLOW 0.9724 0.9759 -0.36%

Angle 0.656 Thrust 1.0016 1.0014 0.02%Kriging 62 5 Height -0.627 Weight 0.9385 0.9155 2.51%Models Length 1.000 Thr/Wt 1.0157 1.0210 -0.51%

GLOW 0.9690 0.9683 0.08%Minimize Weight

Angle 0.800 Thrust 0.9957 0.9957 -0.01%RS 29 3 Height -1.000 Weight 0.7584 0.7496 1.18%

Models Length -1.000 Thr/Wt 1.0533 1.0555 -0.21%GLOW 0.9936 0.9906 0.30%

Angle 1.000 Thrust 0.9965 0.9956 0.08%Kriging 43 4.67 Height -0.873 Weight 0.7725 0.7443 3.79%Models Length -1.000 Thr/Wt 1.0506 1.0568 -0.59%

GLOW 0.9824 0.9914 -0.90%Minimize GLOW

Angle 0.616 Thrust 1.0013 0.9957 0.56%RS 30.67 3.33 Height -1.000 Weight 0.8969 0.8617 4.09%

Models Length 1.000 Thr/Wt 1.0251 1.0286 -0.34%GLOW 0.9660 1.0146 -4.79%

Angle 0.764 Thrust 1.0009 1.0006 0.04%Kriging 57.67 6.33 Height -0.833 Weight 0.9060 0.8732 3.75%Models Length 0.676 Thr/Wt 1.0228 1.0302 -0.72%

GLOW 0.9675 0.9680 -0.05%Maximize Thrust/Weight Ratio

Angle 0.096 Thrust 1.0016 0.9959 0.57%RS 27 4 Height -0.433 Weight 0.9450 0.9073 4.16%

Models Length 1.000 Thr/Wt 1.0141 1.0173 -0.31%GLOW 0.9724 1.0228 -4.93%

Angle 0.656 Thrust 1.0016 1.0014 0.02%Kriging 62 5 Height -0.627 Weight 0.9385 0.9063 3.56%

Models Length 1.000 Thr/Wt 1.0157 1.0231 -0.73%GLOW 0.9690 0.9666 0.25%

1 The predicted optimum value is obtained by using the values of angle, height, and length (from theoptimum design) in the actual analysis code.2A (+) error term indicates that the model is over predicting; a (-) indicates that it is under predicting.

The following observations are made based on the data in Table 4.7.

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• Average number of analysis and gradient calls: In general, the response surface

models require fewer analysis and gradient calls to achieve the optimum than the

kriging models do. This can be attributed, in part, to the fact that the response

surface models are simple second-order polynomials; the kriging models are more

complex, non-linear functions as evidenced in Figure 4.9 and Figure 4.10.

• Convergence rates: Although not shown in the table, optimization using the

response surface models tends to converge more quickly than when using kriging

models. This can be inferred from the number of gradient calls which is one to

three calls fewer for the response surface models than the kriging models.

• Optimum designs: The optimum designs obtained from the response surface and

kriging models are essentially the same for each objective function, indicating that

both approximations send the optimization algorithm in the same general direction.

The largest discrepancy is the length for the minimize GLOW optimization;

response surface models predict the optimum GLOW occurs at the upper bound on

length (+1) while the kriging models yield 0.676. This difference is evident from

Figure 4.10. Furthermore, it has been verified through additional experiments that

the GLOW value obtained using the kriging models is the actual minimum.

• Predicted optima and prediction errors: To check the accuracy of the predicted

optima, the optimum design values for angle, height, and length are used as inputs

into the original analysis codes and the percentage difference between the actual and

predicted values is computed. The prediction error is less than 5% for all cases and

is 0.5% or less in three quarters of the results, indicating close agreement between

the metamodels and the actual analyses.

4 .2 .5 Lessons Learned from the Aerospike Nozzle Example

In summary, the response surface and kriging approximations yield comparable

results with minimal difference in predictive capability. It is worth noting that the kriging

models perform as well as the second-order response surface models even though the

global portion of the kriging model is only a constant. This helps to verify

Hypothesis 2 which states that kriging models are a viable metamodeling

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technique for building approximations of deterministic computer analyses;

however, many questions remain unanswered.

• Correlation function: A Gaussian correlation function is utilized in this example to

fit the data, but is this the best correlation function of the five being considered in

this dissertation?

• Experimental design: A 25 point random orthogonal array is used in this example

to sample the design space and provide data to fit both the kriging and response

surface models, but is this the best type of experimental design for sampling

deterministic computer codes such as the ones used in this example? Would an

alternative experimental design yield a more accurate predictor?

• Model validation: Because kriging models interpolate the data, R2 values and

residual plots cannot be used to assess model accuracy. In this example an

additional 25 validation points are employed to assess accuracy; however, other

validation approaches exist. One such approach which does not require additional

validation points is leave-one-out cross validation (Mitchell and Morris, 1992)

mentioned in Section 2.4.2. Does cross validation provide a sufficient assessment

of model accuracy?

A study of six engineering test problems is set up and performed in the next chapter to

answer these questions. In closing this chapter, a brief look ahead to that study is offered

in the next section.


In an attempt to determine the types of applications for which kriging is useful,

several engineering examples are introduced in the next chapter to serve as test problems to

establish the utility of kriging and verify Hypothesis 2. In addition to testing Hypothesis 2,

several classical and space filling experimental designs are compared and contrasted in an

effort to test Hypothesis 3 to determine if space filling experimental designs are better

suited for building metamodels of deterministic computer experiments.

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5. 5CHAPTER 5

THE UTILITY OF KRIGING AND SPACEFILLING EXPERIMENTAL DESIGNS

Kriging/DOE Testbed

In this chapter, Hypotheses 2 and 3 are tested explicitly, verifying the utility of

kriging and space filling experimental designs for building metamodels of deterministic

computer analyses. A pictorial overview and specific details of the study are given in

Section 5.1. Six engineering examples, introduced in Section 5.1.1, provide a testbed

of problems to benchmark kriging and space filling designs and verify

Hypotheses 2 and 3. In Sections 5.1.2, 5.1.3, and 5.1.4, the factors, experimental

designs, and responses in the study are explained. Analysis of variance of the data and

response correlation are presented in the precursory data analysis in Section 5.2. Section

5.3 contains the results of testing Hypothesis 2 and a discussion of the ramifications of the

results; the results and discussion regarding Hypothesis 3 follow in Section 5.4. A

summary of the study and its relevance to the development of the PPCEM is offered in

Section 5.5.

146

5.1 OVERVIEW OF KRIGING/DOE STUDY AND PROBLEM TESTBED

Consider the following scenario. Assume there is a “black-box” simulation which

is expensive to run and you desire to replace it with a metamodel, a kriging one in

particular. Assume that there are k design variables which you wish to include in the

metamodel. What is best type of experimental design you should use to query the

simulation to generate data to build an accurate kriging metamodel? How many sample

points should you use? What type of correlation function should you use to obtain the best

predictor? Lastly, how can you best validate the metamodel once you have constructed it?

The objective in this study is to answer precisely these questions. Given a series of

test problems (i.e., analyses), determine the best experimental design, sample size, and

correlation function to generate the most accurate model and determine how best to validate

it. Toward this end, a testbed of six engineering examples—the design of a three-bar truss,

a two-bar truss, a spring, a two-member frame, a welded beam, and a pressure vessel as

introduced in Section 5.1.1 and Appendix D—has been created to test the utility of kriging

and space filling experimental designs. A pictorial overview of the kriging/DOE study is

given in Figure 5.1; the figure is viewed from top to bottom.

Contained in these six engineering examples are a total of 26 different types of

equations which are used to test the utility of kriging at metamodeling deterministic

computer analyses. If a kriging metamodel yields an accurate approximation of all, or a

majority, of these equations, then Hypothesis 2 is considered to be verified. Moreover, for

each example, five correlation functions are used to construct five different kriging

metamodels in an effort to determine which is the best correlation on average.

Meanwhile, for each example several classical and space filling experimental

designs are used to construct each kriging metamodel. By analyzing the accuracy of the

resulting kriging metamodel, the experimental design which yields the most accurate

147

predictor, on average, can be determined. In this regard, Hypothesis 3 is tested explicitly

to verify that space filling experimental designs yield more accurate kriging metamodels

than do classical experimental designs. And while Hypothesis 2 and 3 are being tested, the

usefulness of cross validation root mean square error as a measure of accuracy of a kriging

metamodel is investigated.

ProblemTestbed(§5.1.1)

DOE(1-15)

CORFCN(1-5)

EQN(1-26)

NSAMP(§4.1.3)

3-bartruss

2-bartruss

2 Variable Problems

2-memberframe

spring

3 Variable Problems

weldedbeam

pressurevessel

4 Variable Problems

Factors& Levels(§5.1.2)

Responses(§5.1.4)

1-4 5, 6, 7 8-14 15-17 18-22 23-26

ccdaf •••bkbnk cciaf mnmxlhamss yelhd•••

Test Hypothesis 2Isolate CORFCN and EQN

(see §5.3)

Test Hypothesis 3Isolate DOE and NSAMP

(see §5.4)

mxmnl

7, 8, 9, 10, 11, 12, 13, 14

Exponential GaussianPiecewise

CubicLinearMatérn

QuadraticMatérn

13-25 20-41

Max. Abs. ErrorSample Range

RMSESample Range

CVRMSESample Range

Space Filling DesignsClassical Designs

Six TestProblems

ErrorMeasures

HypothesisTesting

For 2 Variables For 3 Variables For 4 Variables

Precursory Data Analysis and Analysis of Variance (see §5.2)

Figure 5.1 Pictorial Overview of Kriging/DOE Study

148

In total, 7905 kriging models are constructed: one for each correlation function

(CORFCN) for each experimental design (DOE) for each sample size (NSAMP) for each

equation (EQN) in each problem. As an example, the arrows in Figure 5.1 trace Equation

7 in the two-bar truss problem. For EQN 7, there are 15 possible experimental design

(DOE) choices; in this case, the minimax Latin hypercube design (mnmxl) is being

considered. For this design, there are several possible choices for NSAMP, ranging from

7-14, because this is a two variable problem. Using 10 sample points as an example, at the

next level there are five correlation functions (CORFCN) which can be used to build a

kriging model; the Gaussian correlation function is highlighted in this example. Finally,

three measures of model accuracy are computed for the kriging model resulting from this

particular combination of EQN, DOE, NSAMP, and CORFCN: max. abs. error, root mean

square error (RMSE), and cross validation root mean square error (CVRMSE) which are

“normalized” by the corresponding sample range so that responses of different magnitude

can be compared directly.

After a precursory analysis of the data in Section 5.2, these three error measures of

model accuracy are used to test Hypotheses 2 and 3 explicitly. As shown in Figure 5.1:

Hypothesis 2 is tested in Section 5.3 by isolating the effects of correlation function

(CORFCN) and equation (EQN) on the accuracy of the resulting kriging model as

assessed through the error measures.

Hypothesis 3 is tested in Section 5.4 by isolating the effects of experimental design

(DOE) and sample size (NSAMP) on the error measures of accuracy of the resulting

kriging model.

The test problems used in this study are introduced next.

5 .1 .1 Overview of Testbed Problems

Six test problems were selected from the literature to provide a testbed for assessing

the utility of kriging and several different space filling experimental designs. These

149

problems are not meant to be all inclusive; rather, they are taken as representative of typical

analyses encountered in mechanical design. The analysis of these problems is simple

enough not to warrant building kriging models of the responses; however, these problems

have been selected because:

a. they have been well studied and the behavior of the system and the underlying

analysis equations are known,

b. the corresponding “region of interest” is known in each problem, and

c. they have been used by other researchers to test their own metamodeling strategies

and algorithms.

Furthermore, the optimum solution for each problem is also known; however, a more

extensive error analysis is employed to assess the accuracy of the kriging models (see

Section 5.1.4).

In the following sections, each example is described along with its pertinent

constraints, design variable bounds, and the objective function; note that a kriging model is

constructed for each constraint and objective function in each problem. The values of the

parameters in the equations (i.e., all of the letters and symbols which are not explicitly

stated as being design variables) are given in the referenced sections of Appendix D which

contain the complete description of each problem.

150

Two Variable Problems

The two variable problems investigated are the design of a two-bar truss (Figure 5.2) and

of a symmetric three-bar truss (Figure 5.3). The problem formulations (objective

functions, constraints, and bounds) follow each figure. A complete description of the two-

bar and three-bar examples is given in Appendix D, Sections D.1 and D.2, respectively.

t

D

Section C-C’

H2P

B B

C

C’

N

x

P1P2

αααα

αα

A1 A2 A3

Figure 5.2 Two-Bar Truss

Find:• Tube diameter, D• Height of the truss, H

Satisfy:• Constraints:

g1(x) = π 2E(D 2 + T 2 )

8(B2 + H 2 )−

P(B2 + H 2 )1/2

πTDH ≥ 0

g2(x) = σy - P(B 2 + H 2 )1/2

πTDH ≥ 0

• Bounds: 0.5 in. ≤ D ≤ 5.0 in.5.0 in. ≤ H ≤ 50 in.

Minimize:Weight, W(x) = 2ρπDT(B2 + H2)1/2

For more information:• see, e.g., (Schmit, 1981)• see Appendix D, Section D.1

Figure 5.3 Three-Bar Truss

Find:• Cross section area, A1 = A3• Cross section area, A2


g1(x) = 20,000 - 1

A1

−A2

2A 1A 2 + 2A 12

≥ 0

g2(x) = 20,000 - 20,000 2 A1

2A 1A2 + 2A 12 ≥ 0

g3(x) = 15,000 −20,000A 2

2A 1A2 + 2A12 ≥ 0

• Bounds:0.5 in2 ≤ A1 = A3 ≤ 1.2 in2

0.0 in2 ≤ A2 ≤ 4.0 in2

Minimize:Weight, W(x) = ρN(2 2A1 + A2)

For more information:• see, e.g., (Schmit, 1981)• see Appendix D, Section D.2

151

Three Variable Problems

The three variable problems are the design of a compression spring (Figure 5.4) and two-

member frame (Figure 5.5). Complete descriptions of these problems are given in

Appendix D, Sections D.3 and D.4, respectively.

ht

d

(1)

(2)

(3)LL

z

P

x y

U1

U2 U3

Figure 5.4 Compression Spring

Find:• Number of active coils, N• Mean coil diameter, D• Wire diameter, d


g1(x) = S - 8CfFmaxD/(πd3) ≥ 0g2(x) = lmax - lf ≥ 0g3(x) = δpm - δ ≥ 0

g4(x) = (Fmax - Fload)/K - δw ≥ 0g5(x) = Dmax - D - d ≥ 0

g6(x) = C - 3 ≥ 0

• Bounds: 3 ≤ N ≤ 30

1.0 in. ≤ D ≤ 6.0 in.0.2 in. ≤ d ≤ 0.5 in.

Minimize:Volume, V(x) = π2Dd2(N + 2)/4

For more information:• see, e.g., (Siddall, 1982)• see Appendix D, Section D.3

Figure 5.5 Two-Member Frame

Find:• Frame width, d• Frame height, h• Frame wall thickness, t


g1(x) = (σ12 + 3τ2)1/2 ≤ 40,000

g2(x) = (σ22 + 3τ2)1/2 ≤ 40,000

• Bounds:2.5 in. ≤ d ≤ 10 in.2.5 in. ≤ h ≤ 10 in.0.1 in. ≤ t ≤ 1.0 in.

Minimize:Volume, V(x) = 2L(2dt + 2ht - 4t2)

For more information:• see (Arora, 1989)• see Appendix D, Section D.4

152

Four Variable Problems

The four variable problems being investigated are the design of a welded beam, Figure 5.6,

and design of a pressure vessel, Figure 5.7. The problem formulations follow each figure.

Complete descriptions are given in Appendix D, Sections D.5 and D.6, respectively.

B

A

L

F

h

F

b

l tR

TsTh

R

L

Figure 5.6 Welded Beam

Find:• Weld height, h• Weld length, l• Bar thickness, t• Bar width, b


g1(x) = [(τ’)2 + 2τ’τ’’cosθ + (τ’’)2]1/2 ≤ τdg2(x) = 6FL/(bt2) ≤ 30,000

g3(x) = 4.013 EIα

L2 [1− (t

2L)

EI

α] ≥ 6000

g4(x) = 4FL3/(Et3b) ≤ 0.25

• Bounds:0.125 in. ≤ h ≤ 2.0 in. 2.0 in. ≤ l ≤ 10.0 in. 2.0 in. ≤ t ≤ 10.0 in.0.125 in. ≤ b ≤ 2.0 in.

Minimize:F(x) = (1 + c3)h

2l + c4tb(L + l)

For more information:• see (Ragsdell and Phillips, 1976)• see Appendix D, Section D.5

Figure 5.7 Pressure Vessel

Find:• Cylinder radius, R• Cylinder length, L• Shell thickness, Ts• Spherical head thickness, Th


g1(x) = Ts - 0.0193R ≥ 0g2(x) = Th - 0.00954R ≥ 0

g3(x) = πR2L + (4/3)πR3 - 1.296E6 ≥ 0

• Bounds: 25 in. ≤ R ≤ 150 in. 25 in. ≤ L ≤ 240 in.

1.0 in. ≤ Ts ≤ 1.375 in.0.625 in. ≤ Th ≤ 1.0 in.

Minimize:F(x) = 0.6224T sRL + 1.7781ThR

2 +

3.1661Ts2L + 19.84Ts

2R

For more information:• see, e.g., (Sandgren, 1990)• see Appendix D, Section D.6

153

Taken together, these six problems provide a wide variety of functions to

approximate since a kriging model is built for each objective function and constraint for

each problem. In total, there are 26 different equations contained in these six problems,

ranging from simple linear functions to reciprocal square roots; some equations even

require the inversion of a finite element matrix (see Section D.4 for the analysis of the two-

member frame). With these six problems as the testbed for verifying Hypotheses 2 and 3,

the factors (and corresponding levels of interest) being studied are explained next.

5 .1 .2 Factors and Levels for Kriging/DOE Experiment

The three basic factors considered in this experiment are listed in Table 5.1:

CORFCN refers to the correlation function used in the kriging model, EQN refers to the

equation being approximated, and DOE refers to the type of experimental design being

utilized to sample the equation to provide data to fit the model. The corresponding levels

for each factor also are listed in the table and are explained as follows.

• CORFCN has 5 levels of interest based on the correlation functions being studied

(refer to Table 2.1, Equations 2.20-2.24); the correlation function associated with

each level is given in the first two columns of Table 5.1.

• EQN has 26 levels based on the total number of equations (i.e., objective functions

and constraints) in the six test problems; when showing the levels for EQN, the

objective function for each problem is singled out from the constraints, see the

middle two columns of Table 5.1.

• DOE has 15 levels based on all of the classical and space filling experimental design

introduced in Section 2.4.3 for investigation; the acronyms and corresponding

name of each design is listed in the last two columns of Table 5.1.

Every effort is made to ensure that the observations of each factor level in the

experiment are properly balanced; however, some factors (and levels) are beyond control.

Each level of CORFCN given in Table 5.1 occurs an equal number of times in each

problem; hence, it is easy to examine the effect of the different correlation functions on the

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overall accuracy of the kriging model (see Section 5.3.1). The factor EQN is used to

isolate the functions being considered and is utilized in Section 5.3.2 when the accuracy of

the kriging model is examined for each pair of problems. As such, both of these factors are

relatively well-balanced in the design. The levels of DOE, however, are not well-balanced

because the fifteen levels for DOE do not appear equally in each problem; for example,

there is no Box-Behnken experimental design for two variable problems.

Table 5.1 Factors and Level for Kriging/DOE Study

CORFCN EQN DOELevel Equation Level Equation Level Type of Design

1 Exponential Two Variable Problems Classical DOEEqn. 2.20 Three-Bar Truss bkbnk Box Behnken

2 Gaussian 1 W(x) ccdaf CCD + CCFEqn. 2.21 2-4 g1(x) - g3(x) ccdes CCD

3 Cubic ccfac CCFEqn. 2.22 Two-Bar Truss ccins CCI

4 Lin. Matérn 5 W(x) cciaf CCI + CCFEqn. 2.23 6-7 g1(x) - g3(x)

5 Quad. Matérn Space Filling DOEEqn. 2.24 Three Variable Problems hamss Hammersley

Spring Sequence8 V(x) mnmxl Minimax Lh

9-14 g1(x) - g6(x) mxmnl Maximin Lhoalhd Orthogonal

Two-Member Frame Array-Based Lh15 V(x) oarry Orthogonal

16-17 g1(x) - g2(x) Arrayoplhd Optimal Lh

Four Variable Problems rnlhd Random Lh Welded Beam unifd Uniform Design

18 F(x) yelhd Orthogonal Lh19-22 g1(x) - g4(x)

Pressure Vessel 23 F(x)

24-26 g1(x) - g3(x)

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To make things even more complicated, the number of sample points within each

design depends on the type of DOE considered and the number of variables in the problem.

For instance, a CCD for the two variable problems has 22 + 2•2 + 1 = 9 points while a

random Latin hypercube can have any number of sample points. Hence, great care must be

taken when analyzing the effects of DOE because of the biasing which occurs due to

unbalanced sample sizes in the experiment. This is discussed in more detail in the next

section which contains a complete listing of which experimental designs (and

corresponding sample sizes) are used in each of the two, three, and four variable problems.

5 .1 .3 Experimental Design Choices for Test Problems

A very important factor in the selection of an experimental design is the number of

points used. How is the number of points to be determined for a given design? For the

two types of classical designs utilized in this dissertation—CCDs and Box-Behnken

designs—the number of points essentially is fixed once the number of factors is specified.

Fractional factorial designs within a CCD are not considered for these problems because

they contain so few variables. Unlike the CCDs and Box-Behnken designs, for most space

filling designs the number of points is not dictated by the number of factors and can be any

number within reason.

Therefore, in order to determine the number of points used in a space filling design,

a CCD with the same number of factors is used to determine the baseline number of points,

e.g., for three factors, a CCD requires 15 points, and the number of points used in all

space filling designs for three factors would be selected to be as close to 15 as possible.

However, because some space filling designs can have a variable number of sample points,

a variety of sample sizes for each design are considered in order to see if fewer or slightly

more points provides an improved fit. As a guideline, an upper bound on the number of

points of about 1.5 times the number prescribed by the baseline CCD is employed. This

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factor of 1.5 is primarily based on the recommendations of Giunta, et al. (1994) who found

that for small problems (i.e., fewer than about five factors) the variance of a second-order

response surface model leveled off when the number of sample points was about 1.5 times

the number of terms in the polynomial model. This number serves as a guideline in this

work despite the fact that kriging models do not necessarily use a second-order polynomial.

How important is the number of design points when picking an experimental

design? The answer is very important. In order to compare the utility of different

experimental designs properly, it is important to use the same number of sample points

because a design with more sample points is expected to provide more information,

possibly resulting in a more accurate model. Therefore, when designs do not have the

same number of points, it is impossible to determine if an improvement in model accuracy

is from the design itself (i.e., spacing of the points in the design space) or from the number

of sample points. However, in some cases it is extremely difficult, if not impossible, to

have two different designs which have the same number of points. For instance, a three

factor CCD has 15 points, a three factor Box-Behnken has 13 (since replicates are not

used), and a strength 2 randomized OA has either 9, 16, or 25 points since it is restricted to

q2 points where q is the number of levels and is restricted to be a prime power. Despite

these difficulties, every effort is made to make the sample sizes overlap as much as possible

from one design to the next. The experimental designs and corresponding sample sizes for

each pair of problems are described in the following sections.

Two Variable Problems

For the two variable problems (the two-bar and three-bar trusses), nine types of

experimental designs are considered, see Table 5.2. Of these nine types of designs, there

are 51 unique designs because each design which has a different number of points is

considered a unique design. For instance, a seven point Latin hypercube and an eight point

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Latin hypercube are unique designs because they have different sample sizes even though

they are both Latin hypercube designs.

Table 5.2 Experimental Designs for Two Factor Test Problems

Type of Design # Points Type of Design # PointsCCD, CCF, CCI 9 CCI + CCF 13

Box-Behnken (bxbnk) NA CCD + CCF 13Minimax Latin

Hypercube (mnmxl)7-14 Randomized Orthogonal

Array (oarry)† NA

Maximin LatinHypercube (mxmnl)

7-14 Optimal Latin Hypercube(oplhd)

7-14

Random LatinHypercubes (rnlhd)† 7-14 Orthogonal Array-Based

Latin Hypercubes (oalhd)† 9

Orthogonal LatinHypercubes (yelhd)† 9 Hammersley Sampling

Sequence (hamss)7-14

Uniform Designs (unifd) 7-14† Each design is instantiated three times because it is based on a random permutation, and

the resulting error measures are averaged over all three randomizations for that design.

Some designs are based on random permutations of levels as indicated by the

superscript (†) in the table. To minimize the effects of this randomness, each of these

designs is randomized three times, and the resulting error measures are averaged over all

three randomizations for that specific design to prevent a design from yielding a poor model

because of its randomly chosen levels. As a result, there a total of 71 designs which are fit

for each of the two variable problems. Finally, notice that neither Box-Behnken designs

nor orthogonal arrays are included in these problems; there is no Box-Behnken design for

two factors, and a nine point orthogonal array for two factors is a 3 x 3 grid, the same as a

face-centered central composite design (CCF).

Three Variable Problems

Eleven types of experimental designs for a total of 63 unique designs are considered (as

shown in Table 5.3) for the three variable spring and a two-member frame test problems.

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In all, there are 92 total designs constructed for each three variable problem once the three

randomizations of the Latin hypercube, orthogonal Latin hypercube, orthogonal array, and

orthogonal array-based Latin hypercube designs are added to the study.

Table 5.3 Experimental Designs for Three Variable Test Problems


Box-Behnken (bxbnk) 13 CCD + CCF 23Minimax Latin

Hypercube (mnmxl)13-19,

21, 23, 25Randomized Orthogonal

Array (oarry)† 16, 25


13-15, 17, 19 Optimal Latin Hypercube(oplhd)

13-19,21, 23, 25

Random LatinHypercubes (rnlhd)†

13-19,21, 23, 25

Orthogonal Array-BasedLatin Hypercubes (oalhd)† 16, 25


Sequence (hamss)13-19,

21, 23, 25

Uniform Designs (unifd) 13, 15, 17, 19,21, 23, 25

† Each design is instantiated three times because it is based on a random permutation, andthe resulting error measures are averaged over all three randomizations for that design.

Notice that a 13 point Box-Behnken design is included in the set of designs for the

three variable problems along with two randomized orthogonal array designs: a 16 point

OA and a 25 point OA. One thing to note about these designs (and the orthogonal array-

based Latin hypercubes as well) is that the number of points in the design is limited to q2

sample points, where q is a power of a prime number. Thus, only q = 4 and q = 5 point

OAs are considered for the three variable problems in order to maintain a fairly consistent

number of points between the different designs.

159

Four Variable Problems

For the two, four variable problems, 66 unique designs from eleven types of experimental

designs are employed (see Table 5.4). Including the repetitions of the designs with random

permutations, a total of 102 designs are examined for each of these problems.

Table 5.4 Experimental Designs for Four Variable Test Problems


Box-Behnken (bxbnk) 25 CCD + CCF 41Minimax Latin

Hypercube (mnmxl)20-29, 31, 33 Randomized Orthogonal

Array (oarry)† 16, 25, 32


22, 25, 26,28

Optimal Latin Hypercube(oplhd)

20, 22, 25, 26,28, 29, 31, 33

Random LatinHypercubes (rnlhd)† 20-29, 31, 33 Orthogonal Array-Based

Latin Hypercubes (oalhd)† 16, 25


Sequence (hamss)20-29, 31, 33

Uniform Designs (unifd) 21, 23, 25, 27,29, 31

† Each design is instantiated three times because it is based on a random permutation, andthe resulting error measures are averaged over all three randomizations for that design.

Notice in Table 5.4 that only four maximin Latin hypercubes are considered: 22,

25, 26, and 28 point designs. This is because the simulated annealing (Morris and

Mitchell, 1995) used to create these designs is not very robust in generating large four

factor designs, and large four factor designs are not listed in (Morris and Mitchell, 1992).

In addition, three orthogonal arrays are employed: 16, 25, and 32 point designs. The 16

and 25 point designs are strength 2 designs; the 32 point OA design is a strength 3 design

with 2q3 points and levels 0, ..., q-1. So while there are more points with the 32 point OA

design than in the 25 point OA design, the number of unique factor levels being considered

in the 32 point OA design is actually less than in the 25 point OA design.

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In summary, the experimental designs and corresponding levels listed in Table 5.2

through Table 5.4 are used to generate data to build kriging models for each equation in

each problem. For each kriging model, the kriging model is cross validated and the

accuracy of the kriging model is further assessed using a set of validation points which is

independent of the design and number of samples. The end result is three measures of

model accuracy which provide the responses for this study as explained in the next section.

5 .1 .4 Responses for the Kriging/DOE Experiment

As shown in Figure 5.1, there are three responses in the kriging/DOE study:

1. cross validation root mean square error (CVRMSE), see Equation 2.33 in Section

2.4.2, of the kriging model;

2. maximum absolute error (MAX), see Equation 2.30 in Section 2.4.2, of the kriging

model; and

3. root mean square error (RMSE), see Equation 2.32 in Section 2.4.2 of the kriging

model.

The CVRMSE of the kriging model is based on the leave-one-out cross validation

procedure described in Section 2.4.2; it utilizes the sample data to validate the model and

does not require additional data for validation. As such, it is uncertain whether CVRMSE

provides an assessment of model adequacy; therefore, three sets of validation points are

used to compute MAX and RMSE. The average absolute error measure, Equation 2.27, is

not included in this study since it correlates well with RMSE and provides little additional

information beyond that obtained from analysis of RMSE. The number of validation points

used in each problem is listed in Table 5.5: 1000, 1500, and 2000 validation points for the

two, three, and four variable problems, respectively.

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Table 5.5 Additional Random Points Used to Assess Model Accuracy

Test Problem Name of Problem # Points2 variables Two-bar truss & Sym. three-bar truss 10003 variables Two-member frame & Helical spring 15004 variables Pressure vessel & Welded beam 2000

Rather than randomly pick these validation points, the points are obtained from a

random Latin hypercube to ensure uniformity within the design space. The predicted

values from each kriging model are compared against the actual values from the set of

validation points, and the error measures MAX and RMSE are computed. These measures

are then “normalized” as a percentage of the sample range, for the particular design under

investigation, in order to compare responses with different magnitudes. A precursory

analysis of the data is given in the next section.

5.2 PRECURSORY KRIGING/DOE DATA ANALYSIS AND ANOVA

In total, there are 11535 kriging models constructed as shown in Table 5.6 for the

six test problems—one kriging model for each equation for each design for each test

problem. For each of these models, there are three measures of model accuracy: MAX,

RMSE, and CVRMSE; hence, there are 34605 data points in the resulting data set.

Table 5.6 Kriging Test Problem Model Summary

ProblemName

No. ofVariables

No. ofResponses

No. ofUnique DOE

Total No.of DOE

No. ofCORFCN

No. ofModels

No. of TotalModels

2bar 2 3 51 71 5 765 10053bar 2 4 51 71 5 1020 1340

2mem 3 3 63 92 5 945 1380spring 3 7 63 92 5 2205 3220press 4 4 66 102 5 1320 2040weld 4 5 66 102 5 1650 2550

Grand Totals 7905 11535

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To facilitate analysis of the data set, the error measures of the designs which are

replicated—the orthogonal arrays, random Latin hypercubes, OA-based Latin hypercubes,

and orthogonal Latin hypercubes—are averaged to reduce the data set to 7905 models.

However, not all of these 7905 models are good, i.e., many contain outliers which bias the

results, and potential outliers must be removed. The cause of the outliers can be attributed

to incomplete convergence of the numerical optimization used to fit the model or

singularities in the data set which occur during model fitting, numerical round-off error, or

bad data resulting from transferring data from file to file, program to program, and

computer to computer.

Hence, the data set is culled to remove any potential outliers. Rather than first fit

the model and remove potential outliers based on the residuals, the data is culled based on

(a) potential RMSE outliers, (b) potential MAX outliers, and (c) potential CVRMSE

outliers since it is known that many outliers exist due to singularities in the data set which

occur during model fitting. The process is described in detail in Appendix E; density plots

are included in Appendix E to show the distribution of the resulting data for the two, three,

and four variable problems.

In this manner, the data set is reduced from 7905 models to 7578. This constitutes

a reduction of about 4% which is considered reasonable given the magnitude of the study

and the potential for errors. From this point forward, any reference to “the data set” refers

to the final culled data set with all of the potential outliers removed and not to the original

data set unless explicitly specified.

Analysis of variance is performed in the next section to determine which factors

have a significant effect on the accuracy of the resulting kriging model. This is followed in

Section 5.2.2 with an examination of the correlation of the resulting error measures.

163

5 .2 .1 Analysis of Variance of Kriging/DOE Study

Before Hypotheses 2 and 3 are tested, analysis of variance (ANOVA) is performed

to determine which factors have a significant effect on the accuracy of the resulting kriging

model (see, e.g., (Chambers, et al., 1992; Montgomery, 1991) for more on ANOVA).

The software package S-Plus4 (MathSoft, 1997) is used to analyze the data. The ANOVA

is performed separately for each pair of two, three, and four variable problems for all three

error measures. Furthermore, because of the size of the data set, only main effects and

two-factor interactions can be studied. The ANOVA results are given in Section E.2, and a

summary of the ANOVA results are given in Table 5.7. In the table, the factor main effects

and two-factor interaction effects are listed in the first column of the table; a colon between

factors (e.g., CORFCN:NSAMP) indicates a two-factor interaction. The abbreviations

“sig” and “not sig” are used to indicate whether or not the effect is significant. For

instance, all of the main effects and two-factor interactions except CORFCN:NSAMP are

significant for RMSE.RANGE and MAX.RANGE in the two and three variable problems.

Table 5.7 Summary of ANOVA Results for Kriging/DOE Study

2 Variable Problems 3 Variable Problems 4 Variable Problems RMSE MAX CVRMSE RMSE MAX CVRMSE RMSE MAX CVRMSE

Effect RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGEDOE sig sig sig sig sig sig sig sig sig

CORFCN sig sig sig sig sig sig sig sig sigNSAMP sig sig sig sig sig sig sig sig sig

EQN sig sig sig sig sig sig sig sig sigDOE:CORFCN sig sig sig sig sig sig sig not sig sig

DOE:NSAMP sig sig not sig sig sig sig sig sig not sigDOE:EQN sig sig sig sig sig sig sig sig sig

CORFCN:NSAMP not sig not sig sig not sig not sig sig not sig not sig not sigCORFCN:EQN sig sig sig sig sig sig sig not sig sig

NSAMP:EQN sig sig sig sig sig not sig sig sig not sig

As can be seen in Table 5.7, the majority of the effects are significant on all of the

error measures. It is not surprising to see that the main effects of the factors DOE,

164

CORFCN, NSAMP, and EQN are significant for all responses for all of the problems.

Likewise, the interaction between DOE and NSAMP is significant for all RMSE.RANGE

and MAX.RANGE values. The interaction between CORFCN and NSAMP is not

significant in the majority of cases since it is unlikely that these two factors would interact

to provide a more accurate model. The interaction between DOE and CORFCN is

significant in all but one case which is interesting to note. In summary, it appears that there

are many significant interactions and main effects to examine. Observations regarding

many of these interactions can be inferred from the appropriate graphs; however, the

commentary in Sections 5.3 and 5.4 focuses primarily on main effects.

5 .2 .2 Correlation of Error Measures

The two most important measures of model accuracy in this study are considered to

be RMSE and MAX. Why are these two particular measures the most important? RMSE

is used to gauge the overall accuracy of the model, and MAX is used to gauge the local

accuracy of the model. Ideally, RMSE and MAX would be zero, indicating that the

metamodel predicts the underlying analysis or model exactly; however, this is rarely the

case. Therefore, the lower the value of either error measure, the more accurate the model.

Both measures are important when using an approximation in an optimization or

robust design application because high values of RMSE can lead an optimization algorithm

into a region of bad design and high values of MAX prevent the optimization algorithm

from finding the true optimum solution. To see if the two measures are correlated, a plot of

RMSE.RANGE versus MAX.RANGE for the data set is given in Figure 5.8. Here and

henceforth, the acronyms RMSE.RANGE and MAX.RANGE are used to refer to the

values of RMSE and MAX when normalized against the sample range.

165

0.0 0.1 0.2 0.3 0.4 0.5 0.6rmse.range

0

2

4

6m

ax.r

ange

Figure 5.8 Correlation Between RMSE.RANGE and MAX.RANGE

Since the data is widely scattered in Figure 5.8, the two error measures do correlate

well. Models with low RMSE.RANGE values tend to have low MAX.RANGE values,

but models with moderate RMSE.RANGE values have any of a variety of MAX.RANGE

values. As such, it is important to analyze both RMSE.RANGE and MAX.RANGE when

drawing conclusions.

A plot of RMSE.RANGE versus CVRMSE.RANGE is given in Figure 5.9. Based

on the wide scattering of the data, RMSE.RANGE and CVRMSE.RANGE are not

correlated either. This means that the cross validation root mean square error is not a

sufficient measure of model accuracy because root mean square error provides the best

possible assessment of overall model accuracy. If CVRMSE.RANGE and

RMSE.RANGE had been correlated, then CVRMSE alone could be computed to assess

166

model accuracy without having to take any additional points to compute RMSE to validate

the model.

0.0 0.1 0.2 0.3 0.4 0.5 0.6rmse.range

0.0

0.2

0.4

0.6

0.8

1.0

cvrm

se.r

ange

Figure 5.9 Correlation of RMSE.RANGE and CVRMSE.RANGE

A plot of MAX.RANGE versus CVRMSE.RANGE is given in Figure 5.10. As

with Figure 5.9, there is a wide scattering of the data, and it appears that MAX.RANGE

and CVRMSE.RANGE are not well correlated either. Hence, there is no need to examine

CVRMSE.RANGE further because it does not provide a good assessment of model

accuracy since it does not correlate well with either MAX.RANGE or RMSE.RANGE. As

stated earlier, this finding is unfortunate because it means that additional validation points

must be taken in order to assess the accuracy of a kriging model properly; cross validating

the model using the sample data is not a sufficient measure of accuracy.

167

0 1 2 3 4 5 6max.range

0.0

0.2

0.4

0.6

0.8

1.0

cvrm

se.r

ange

Figure 5.10 Correlation of MAX.RANGE and CVRMSE.RANGE

Using only RMSE.RANGE and MAX.RANGE, the error of the resulting kriging

models can now be assessed by isolating a single factor (or pair of factors). The process

for analyzing the data in order to interpret specific results is identified at the beginning of

each section when Hypotheses 2 and 3 are tested. Hypothesis 2 is tested first in the next

section, and Hypothesis 3 is tested second in Section 5.4.

5.3 TESTING HYPOTHESIS 2: THE UTILITY OF KRIGING

Recall Hypothesis 2 from Section 1.3.1 and Section 3.3.2:

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Hypothesis 2: Kriging is a viable metamodeling technique for building

approximations of deterministic computer analyses.

In order to test this hypothesis, two factors are isolated to analyze the results further,

namely, CORFCN and EQN. Both factors were found to have a significant effect on the

accuracy of the resulting kriging in the ANOVA in Section 5.2.1. The effect of CORFCN

on RMSE.RANGE and MAX.RANGE is investigated in the next section. The effect of

EQN on RMSE.RANGE and MAX.RANGE is discussed in Section 5.3.2. Keep in mind

that all of these results are based strictly on averages of the data at a given level of a

particular variable; it is assumed that biasing due to unbalanced numbers of observations at

each level is negligible since such a large data set is being used.

5 .3 .1 Effect of Correlation Function on Kriging Model Accuracy

The effect of correlation function on model accuracy was found to be significant in

the ANOVA in Section 5.2.1, but it is uncertain which correlation function yields the best

results on average. Therefore, the effect of CORFCN on RMSE.RANGE aggregated over

all the problems and for each pair of problems is shown in Figure 5.11. The average

(mean) of RMSE.RANGE for each factor level is plotted on the vertical axis in the figure.

Meanwhile, the vertical bars within the figure are used for grouping purposes, showing the

range of effects of the different levels of the factor being considered (in this case,

CORFCN) for each problem group as indicated on the x-axis. The numbers 1, 2, 3, 4, and

5 in the figure indicate the level of correlation function as described in the key in the figure.

The horizontal dashed lines which cross each vertical bar indicate the group average of

RMSE.RANGE for that particular grouping; for instance, the mean RMSE.RANGE for all

of the problems is about 0.062. The arrows are used to indicate the effect a particular level

169

of CORFCN has on RMSE.RANGE; the same holds true regardless of the factor being

considered. For example, in Figure 5.11 the average effect of CORFCN = 1 in the two

variable problems is slightly less than 0.08 while the average effect of CORFCN = 4 in the

same problems is slightly greater than 0.06. Finally, lower values of RMSE.RANGE (and

MAX.RANGE) are better; so, the lower the arrow of a particular level on the vertical line,

the more accurate is the resulting kriging model.

mea

n of

rm

se.r

ange

0.05

0.06

0.07

0.08

1

2

345

1

23

4

5

1

2

3

4

5

1

2

34

5

Factor - CORFCN

2 variable problems 3 variable problems 4 variable problemsall problems

Key:1 = Exponential2 = Gaussian3 = Cubic4 = Linear Matérn5 = Qaud. Matérn

Figure 5.11 Effect of Correlation Function Type on RMSE.RANGE

Some observations regarding Figure 5.11 are as follows. The exponential

correlation function (CORFCN = 1) repeatedly is the worst. Overall, the Gaussian

correlation function (CORFCN = 2) provides the lowest RMSE.RANGE on average and

also for the three and four variable problems as well. The linear Matérn (CORFCN = 4)

170

yields the lowest average RMSE.RANGE for the two variable problems but yields

comparable results to the piece-wise cubic correlation function (CORFCN = 3) and the

quadratic Matérn (CORFCN = 5) correlation function otherwise. The piece-wise cubic

(CORFCN = 3) and quadratic Matérn (CORFCN = 5) correlation functions generally yield

worse results than the Gaussian correlation function (CORFCN = 2).

The effect of CORFCN on MAX.RANGE is shown in Figure 5.12. As in Figure

5.11, the exponential correlation function (CORFCN = 1) repeatedly is the worst but does

surprisingly well in the four variable case. Overall, the Gaussian correlation function

(CORFCN = 2) provides the lowest MAX.RANGE on average and for the three and four

variable problems as well.

mea

n of

max

.ran

ge

0.4

0.5

0.6

0.7

1

2

34

5

1

23

4

5

1

2

345

12

345

Factor - CORFCN

2 variable problems 3 variable problems 4 variable problemsall problems

Key:1 = Exponential2 = Gaussian3 = Cubic4 = Linear Matérn5 = Qaud. Matérn

Figure 5.12 Effect of Correlation Function Type on MAX.RANGE

171

As seen in Figure 5.12, the linear Matérn correlation function (CORFCN = 4)

yields the best MAX.RANGE for the two variable problems and the worst for the four

variable problems with on average results otherwise. The piece-wise cubic (CORFCN = 3)

and quadratic Matérn (CORFCN = 5) correlation functions yield comparable results, falling

somewhere in the middle of the spectrum in each problem and performing slightly better

than average overall.

In summary, the Gaussian correlation function (CORFCN = 2) is the best

correlation function to use for building kriging models. On average, it provides the lowest

RMSE.RANGE and MAX.RANGE, yielding the most accurate kriging models.

Furthermore, it also yields the best results in the three and four variable problems when

averaged over all designs, sample sizes, and equations; in the two variable problems its

performance is average but not far behind the linear Matérn correlation function (CORFCN

= 4).

5 .3 .2 Effects of Equation Type on Kriging Model Accuracy

In order to determine which types of equations are fit best, the factor EQN is used

to isolate which equations are well fit by the kriging models, thus explicitly testing

Hypothesis 2. A plot of the resulting RMSE.RANGE of the two, three, and four variable

problems for each level of the factor EQN is shown in Figure 5.13; the effect of each level

of EQN on the mean of RMSE.RANGE is averaged over all DOE, NSAMP, and

CORFCN for each problem. For clarity, dashed lines are used to indicate the 5% and 10%

RMSE.RANGE values. If a 5% level of model accuracy is used as a cut-off point, then 14

out of the 26 equations in this study are accurately modeled by kriging. If that cut-off is

raised to 10%, then 20 out of the 26 equations are accurately modeled by kriging.

172

mea

n of

rm

se.r

ange

0.0

0.15

1

2

3

4

5

6

7

8 9

1011 12

1314

15

16

17

1819

20

21

22

2324 2526

Factor - EQN

2 variable problems 3 variable problems 4 variable problems

5%

10%

Figure 5.13 Effect of EQN of RMSE.RANGE for Testbed Problems

Looking at the effect of equation type on MAX.RANGE in Figure 5.14, however,

fewer kriging models meet a 10% cut-off point. In Figure 5.14, only nine of the 26

equations fall within the 10% level of accuracy. If the level of accuracy is allowed to drop

to 20%, which is quite high, then five more equations may be considered modeled

accurately by kriging (EQN = 7, 8, 9, 20, and 22). The mean values for MAX.RANGE

for Equations 18, 19, and 21 are beyond the scale of the chart. For convenience, the

equations and corresponding level of EQN are listed in Table 5.8 which summarizes which

equations are fit well and which are not based on the 5% and 10% error levels used.

173

mea

n of

max

.ran

ge

0.0

0.2

0.4

0.6

0.8

1.0

1

2

3

4

5

6

7

Factor - EQN

8 9

1011 12

13

14 15

16

17

18 19

20

21

22

2324 2526

2 variable problems 3 variable problems 4 variable problems

10%

Figure 5.14 Effect of Equation Type on MAX.RANGE

The types of equations which are well fit by kriging are noted in Table 5.8. At the

5% level of RMSE.RANGE the linear combinations of the design variables (EQN = 1, 5,

8, 13, 15, 24, and 25) and most reciprocal equations (EQN = 7, 9, 20, and 22) are

modeled well. Some higher-order equations are also modeled well (EQN = 23 and 26). At

the 10% level, all of the equations in the three variable problems are modeled well (EQN =

8-17). At this level, the equations based on the finite element model of the two-member

frame (EQN = 16 and 17) also are accurately represented by the kriging models. Looking

at MAX.RANGE, however, the majority of the equations which meet the 10% cut-off are

linear combinations of the design variables (e.g., EQN = 1, 5, 13, 24, and 25) which may

involve higher-order terms or reciprocals (e.g., EQN = 14, 23, and 26).

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Table 5.8 Summary of Equations Accurately Modeled by Kriging

EQN Equation and Design Variables, x , RMSE.RANGE MAX.RANGE# for Each Problem %error < 5% %error < 10% %error < 10%

Three-bar truss: x = {A1, A2}1 W(x) = ρN(2 2A1 + A2) yes yes yes

2 g1(x) = 20,000 - 1

A1

−A2

2A 1A 2 + 2A 12

no no no

3 g2(x) = 20,000 - 20,000 2 A1

2A 1A2 + 2A 12

no no no

4 g3(x) = 15,000 −20,000A 2

2A 1A2 + 2A12 no no no

Two-bar truss: x = {D, H}5 W(x) = 2ρπDT(B2 + H2)1/2 yes yes yes

6 g1(x) = π 2E(D 2 + T 2 )

8(B2 + H 2 )−

P(B2 + H 2 )1/2

πTDHno yes no

7 g2(x) = σy - P(B 2 + H 2 )1/2

πTDHyes yes no

Spring: x = {N, D, d}8 V(x) = π2Dd2(N + 2)/4 yes yes no9 g1(x) = S - 8CfFmaxD/(πd3) yes yes no10 g2(x) = lmax - lf no yes no11 g3(x) = δpm - δ no yes no12 g4(x) = (Fmax - Fload)/K - δw no yes no13 g5(x) = Dmax - D - d yes yes yes14 g6(x) = D/d - 3 yes yes yes

Two-member frame: x = {d, h, t}15 V(x) = 2L(2dt + 2ht - 4t2) yes yes yes16 g1(x) = (σ1

2 + 3τ2)1/2 no yes no17 g2(x) = (σ2

2 + 3τ2)1/2 no yes noWelded Beam: x = {h, l, t, b}

18 F(x) = (1 + c3)h2l + c4tb(L + l) no no no

19 g1(x) = [(τ’)2 + 2τ’τ’’cosθ + (τ’’)2]1/2 no no no20 g2(x) = 6FL/(bt2) yes yes no

21 g3(x) = 4.013 EIα

L2 [1− (t

2L)

EI

α] no no no

22 g4(x) = 4FL3/(Et3b) yes yes noPressure Vessel: x = {R, L, Ts, Th}

23 F(x) = 0.6224TsRL + 1.7781ThR2 +

3.1661Ts2L + 19.84Ts

2R yes yes yes

24 g1(x) = Ts - 0.0193R yes yes yes25 g2(x) = Th - 0.00954R yes yes yes26 g3(x) = πR2L + (4/3)πR3 - 1.296E6 yes yes yes

175

Which types of functions are not modeled well by kriging? Based on the data in

Table 5.8, the equations which are not modeled well by kriging are the equations involving

reciprocals of combinations of the design variables in the three-bar truss problem (EQN =

2, 3, and 4) and two-bar truss problems (EQN = 6 and 7), and the majority of the welded

beam equations which include shear stress calculations, a cosine term which is a function

of the design variables, and a variety of reciprocals and square roots of terms which are

functions of the design variables. In addition, the finite element equations for the two-

member frame (EQN = 16 and 17) are not modeled well at the 5% level accuracy or at the

10% level of accuracy of MAX.RANGE. It is also interesting to note that the objective

function of the welded beam problem (EQN = 18) is one of the equations approximated

worst by the kriging. This is rather surprising considering it is very similar to the objective

function of the pressure vessel problem (EQN = 23) which is modeled well in all cases.

Perhaps these differences are due to the size of the design space as opposed to the

equations themselves; very few approximation methods will work well if the points are

sparsely scattered throughout the design space which may be the case in welded beam

problem and the three-bar truss problem.

In summary, it appears that kriging does provide an accurate approximation of a

variety of equations. Using a 5% level of accuracy, over half (14 out of the 26) of the

equations studied are accurately modeled over the entire design space as measured by

RMSE.RANGE; if a 10% level of accuracy is used instead, then over 3/4 of the equations

(20 out of 26) are accurately modeled. Unfortunately, only nine of the 26 meet the 10%

level of accuracy in MAX.RANGE; however, of the two measures, RMSE.RANGE is

considered to be more important from a design standpoint since accuracy over the entire

design space is more important during design space search than the maximum discrepancy

at any one given point. As such, Hypothesis 2 is verified. In the next section, Hypothesis

3 is tested.

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5.4 TESTING HYPOTHESIS 3: THE UTILITY OF SPACE FILLINGEXPERIMENTAL DESIGNS

Recall Hypothesis 3 from Section 1.1.3 and Section 3.3.2.

Hypothesis 3: Space filling experimental designs are better suited for building

metamodels of deterministic computer experiments than classical designs.

In order to test this hypothesis, the data set is analyzed by isolating the factor DOE which

was found to have a significant effect on the accuracy of the resulting kriging in the

ANOVA in Section 5.2.1. However, as the same number of sample sizes are not used in

each design as discussed in Section 5.1.3, the results also must be conditioned on sample

size (NSAMP) for a fair comparison between designs. Consider, for instance, the

combined CCD + CCF (ccdaf) design in the two variable problems which has 13 sample

points. It cannot be concluded that the combined CCD + CCF is the best by averaging over

all designs because its effect is biased by the fact that it has 13 sample points which is at the

upper end of the number of points in the two variable problems and is therefore expected to

yield good results because of the large number of sample points. Meanwhile, the effects of

all of the other designs with variable numbers of points (i.e., unifd, hamss, mxmnl,

mnmxl, and oplhd) are averaged over all sample sizes where the smaller the sample size,

the less accurate the model, and the worse the effect of these designs. The results for the

two, three, and four variable problems are discussed in Sections 5.4.1, 5.4.2, and 5.4.3,

respectively, by conditioning on both design type and sample size. As stated previously,

keep in mind that the results are based on averaging the data at a given level of a particular

variable; it is assumed that biasing due to unbalanced numbers of observations at each level

is negligible since such a large data set is being used.

177

5 .4 .1 Comparison of Designs for Two Variable Problems

For the two variable problems, the classical designs (CCI, CCF, and CCD) each

utilize nine sample points, and the combined CCF + CCI and CCF + CCD each have 13

points. Hence, the average effect of each design which has nine points and each design

which has 13 points on RMSE.RANGE is shown in Figure 5.15. The average effect on

MAX.RANGE of all the designs with these sample sizes is shown in Figure 5.16.

Looking first at the nine point designs in Figure 5.15, it is surprising to note that

both the CCD and CCI designs perform well with the minimax Latin hypercube (mnmxl)

design yielding the best RMSE.RANGE values; recall that the minimax Latin hypercube

designs are unique to this research (see Appendix C). The orthogonal Latin hypercubes

(yelhd), maximin Latin hypercubes (mxmnl), and the uniform designs (unifd) also perform

well. The worst designs are the Hammersley sampling sequence design, the CCF design

and the OA-based Latin hypercube. The random Latin hypercubes yield average results.

In the 13 point designs, the uniform design (unifd) yields the best results with the

maximin and minimax Latin hypercube designs giving equally good results which are only

slightly worse than that of the uniform design. The random Latin hypercubes (rnlhd)

continue to give average results with the combined CCD + CCF (ccdaf) and CCI + CCF

(cciaf) designs giving slightly better results but results which are still about 1% worse than

the maximin and minimax Latin hypercubes. The optimal Latin hypercube designs are the

worst with the Hammersley sampling sequence designs showing good improvement with

the extra four sample points.

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mea

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rm

se.r

ange

0.04

0.06

0.08

0.10

0.12

ccdes

ccfac

ccins

hamss

mnmxl

mxmnl

oalhd

oplhdrnlhd

unifd yelhd

Factor - DOE

ccdafcciaf

hamss

mnmxlmxmnl

oplhd

rnlhd

unifd

NSAMP = 9 NSAMP = 13

Figure 5.15 Effect of 9 and 13 Point DOE on RMSE.RANGE

Turning to the results of MAX.RANGE in Figure 5.16, the classical designs

perform quite well with the CCI design (ccins) being the best in the nine point designs and

the combined CCD + CCF (ccdaf) being the best in the 13 point designs. The nine point

minimax Latin hypercubes (mnmxl) and OA-based Latin hypercubes (oalhd) yield

comparable results to the CCD and CCF designs. The remaining space filling designs all

fair worse than the CCD with the Hammersley sampling sequence giving the worst

MAX.RANGE. In the 13 point designs, the combined CCI + CCF (cciaf) is the second

best design with the maximin Latin hypercube (mxmnl) coming in a close third. The

uniform, optimal Latin hypercube, minimax Latin hypercube, and random Latin hypercube

designs are slightly worse than the average, and the Hammersley sampling sequence yields

the worst results.

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mea

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0.2

0.4

0.6

0.8

ccdesccfac

ccins

hamss

mnmxl

mxmnl

oalhd

oplhd

rnlhdunifd

yelhd

ccdafcciaf

hamss

mnmxl

mxmnl

oplhd rnlhdunifd

Factor - DOENSAMP = 9 NSAMP = 13

Figure 5.16 Effect of 9 and 13 Point DOE on MAX.RANGE

Based on these results, the space filling designs do best in terms of RMSE.RANGE

while the classical designs yield the lowest MAX.RANGE. If RMSE.RANGE is taken as

the more important of the two measures of error, than the space filling DOE are better than

the classical DOE for the two variable problems considered in this dissertation. The results

for the three variable problems are discussed next.

5 .4 .2 Comparison of Designs for 3 Factors

In the three variable problems, there are four values of NSAMP which must be

considered due to differences in prescribed sample sizes. The Box-Behnken design has 13

points; the classic CCD, CCF, and CCI designs have 15; the CCI + CCF has 21; and the

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CCD + CCF has 23. The effects of these designs on RMSE.RANGE is shown in Figure

5.17; the effects on MAX.RANGE are plotted in Figure 5.18.

mea

n of

rm

se.r

ange

0.04

0.05

0.06

0.07

0.08

0.09

0.10

bxbnk

hamss

mnmxl

mxmnloplhd

rnlhd

unifd

Factor - DOENSAMP = 13

ccdes

ccfac ccins

hamss

mnmxl

mxmnl

oplhd

rnlhd

unifd

NSAMP = 15 NSAMP = 21 NSAMP = 23

cciaf

hamss

mnmxl

oplhd

rnlhd

unifd

ccdaf

hamss

mnmxloplhd

rnlhd

unifd

Figure 5.17 Effect of 13, 15, 21, and 23 Point DOE on RMSE.RANGE

In Figure 5.17, the Box-Behnken (bxbnk) design dominates the 13 point designs

with a RMSE.RANGE of about 5%. All of the space filling designs perform quite poorly

in fact with RMSE.RANGE values of about 8% or worse; it appears that these designs do

not fare well when relatively few sample points are taken in the design space. A similar

observation can be made regarding the 15 point designs also. The maximin Latin

hypercube (mxmnl) yields the best result, but the CCI and CCF designs are both almost as

good. The minimax Latin hypercube (mnmxl) design yields average results with the

uniform and optimal Latin hypercube design (oplhd) fairing slightly better but not as good

181

as the CCI and CCF. The CCD is the worse design with the random Latin hypercube

(rnlhd) and the Hammersley sampling sequence (hamss) yielding comparably poor results.

In the 21 and 23 point designs in Figure 5.17, the optimal Latin hypercube design

(oplhd) yields the lowest RMSE.RANGE with the random Latin hypercube design (rnlhd)

yielding the worst. The combined CCI + CCF (cciaf) is the second best 21 point design,

followed closely by the minimax Latin hypercube (mnmxl). The uniform design (unifd)

gives an average result in the 21 point case but is the second best design in the 23 point

case; the best design is the combined CCD + CCF (ccdaf). The optimal Latin hypercube

design (oplhd) is the worst of the 23 point designs with the minimax Latin hypercube

(mnmxl) and random Latin hypercube designs (rnlhd) yielding results which are worse

than the average.

In Figure 5.18, the effects of these different designs on MAX.RANGE are plotted.

The classical experimental designs consistently provided the lowest MAX.RANGE when

averaging over all other factors. The space filling designs do not perform well in any case

and yield particularly poor results in the 13 and 21 point designs. The minimax Latin

hypercube (mnmxl) design is no exception, giving near average results in 15, 21, and 23

point designs and the next to worst result among the 13 point designs.

As with the two variable problems, the space filling designs offer better results if

RMSE.RANGE is considered while the classical designs are better when it comes to

MAX.RANGE for the problems considered in this dissertation. The four variable

problems are examined in the next section to see if the same holds true.

182

mea

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max

.ran

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0.2

0.4

0.6

0.8

bxbnk

hamssmnmxl

mxmnl

oplhd

rnlhd

unifd

ccdes

ccfac

ccins

hamss

mnmxl

mxmnl

oplhd

rnlhd

unifd

Factor - DOENSAMP = 13 NSAMP = 15 NSAMP = 21 NSAMP = 23

ccdaf

oplhd

rnlhd

unifd

hamss

mnmxl

cciaf

hamss

mnmxl

oplhd

rnlhd

unifd

Figure 5.18 Effect of 13, 15, 21 and 23 Point DOE on MAX.RANGE

5.4 .3 Comparison of Designs for 4 Factors

For the four variable problems, there are two sample sizes to examine: NSAMP =

25 points and NSAMP = 33 points. Figure 5.19 contains the effects of DOE on

RMSE.RANGE for these sample sizes, and Figure 5.20 contains the effects on of DOE on

MAX.RANGE for these sample sizes. As seen in the figures, there are twelve designs

which use 25 sample points and only six with 33. The combined CCD + CCF is not

considered because it is the only design which uses 41 sample points.

Looking first at Figure 5.19, the minimax Latin hypercube (mnmxl) design

introduced in this dissertation yields the best results on average. The uniform design

(unifd) is a close second in the 25 point case, and the random Latin hypercube design

(rnlhd) is a close second in the 33 point case. Of the classical 25 point designs, the Box-

183

Behnken (bxbnk) design performs slightly better than average while the CCD (ccdes), CCF

(ccfac), and CCI (ccins) designs all do worse than average. Finally, the Hammersley

sampling sequence (hamss) designs perform poorly at both sample sizes.

mea

n of

rm

se.r

ange

0.02

0.03

0.04

0.05

0.06

0.07

0.08

bxbnk

ccdes = 0.14

ccfacccins

hamss

mnmxl

mxmnl

oalhd

oarry

oplhd

rnlhd

unifd

cciaf

hamss

mnmxl

oplhd

rnlhd

yelhd


Figure 5.19 Effect of 25 and 33 Point DOE on RMSE.RANGE

In Figure 5.20, the effects of the 25 and 33 point DOE on MAX.RANGE are

plotted. Unlike the two and three variable problems, the space filling designs yield the best

MAX.RANGE for the four variable problems. The randomized 25 point orthogonal

(oarry) produces the lowest MAX.RANGE with the 25 point minimax Latin hypercube

(mnmxl) a close second. The classical Box-Behnken (bxbnk), CCF (ccfac), and CCI

(ccins) designs and the space filling uniform design (unifd) and optimal Latin hypercube

184

design (oplhd) all yield comparable results which are only slightly worse than either the

minimax Latin hypercube or the randomized orthogonal array. The 25 point orthogonal

array-based Latin hypercube (oalhd) and the maximin Latin hypercube (mxmnl) produce

results which are close to the average effect. The Hammersley sampling sequence (hamss)

designs yield the worst MAX.RANGE in both the 25 and 33 point designs; the 25 point

CCD (ccdes) does not fair much better than the Hammersley sampling sequence design,

however.

mea

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max

.ran

ge

0.2

0.4

0.6

0.8

1.0

1.2

bxbnk

ccdes

ccfacccins

hamss

mnmxl

mxmnl

oalhd

oarry

oplhd

rnlhd

unifdcciaf

hamss

mnmxl

oplhd

rnlhd

yelhd


Figure 5.20 Effect of 25 and 33 Point DOE on MAX.RANGE

In the 33 point designs, the minimax Latin hypercube (mnmxl) yields the lowest

MAX.RANGE on average. The combined CCI + CCF (cciaf) and random Latin

hypercube designs (rnlhd) give comparable results which are slightly worse than the

185

minimax Latin hypercube. Finally, the 33 point orthogonal Latin hypercubes (yelhd) and

optimal Latin hypercubes (oplhd) are both slightly worse than the average.

5 .4 .4 Lessons Learned from Experimental Design Study

The space filling designs yield lower RMSE.RANGE values for all of the two,

three, and four variable problems considered in this dissertation. The classical

experimental designs yield lower MAX.RANGE values for the two and three variable

problems but do not perform as well as the space filling designs in the four variable

problems. In small dimensions, i.e., two and three variables, the classical designs spread

the points out equally well in the design space regardless if they are “space filling” designs

or not. However, based on the observed trends in the data, it appears that as the number of

design variables increases, the space filling designs perform better and better in terms of the

two error measurements used in this study. As their name implies, the space filling designs

do a better job at spreading out points in the design space and thus filling the space as the

number of variables increases. Hence, Hypothesis 3 is verified because the space filling

designs do perform better than the classical designs in terms of RMSE.RANGE which

provides the best assessment of overall accuracy of a metamodel. Furthermore, the larger

the number of design variables and the more sample points, the better is the accuracy of the

resulting kriging model from a space filling design.

Some additional comments about particular space filling designs are as follows.

• The minimax Latin hypercube designs introduced in this dissertation perform quite

well in these problems. With the exception of the 13 point minimax Latin

hypercube design for the three variable problems, these designs consistently are

among the best of the designs in terms of its effect on RMSE.RANGE and

MAX.RANGE.

• The Hammersley sampling sequence designs perform poorly in all of these

problems. The impetus for the Hammersley designs, though, are to provide good

stratification of a k-dimensional space (Kalagnanam and Diwekar, 1997); as such,

186

they are designed to perform well in large design space which may explain why

they perform so poorly in these relatively small problems.

• The random Latin hypercube designs provide average results as might be expected

because these designs simply rely on a random scattering of points in the design

space. By imposing additional considerations on these designs to “control” the

randomization of the points, the performance of these designs can be improved.

For instance, the orthogonal Latin hypercubes (Ye, 1997), the maximin Latin

hypercubes (Morris and Mitchell, 1995), the (IMSE) optimal Latin hypercubes

(Park, 1994), and the orthogonal-array based Latin hypercubes (Tang, 1993)

typically yield a more accurate kriging model than does the basic random Latin

hypercube. This observation is not new; rather, it supports the claims which the

creators of these designs when they introduced them.

• The uniform designs perform surprisingly well, considering they are based solely

on number-theoretic reasoning (Fang and Wang, 1994). Regardless, the

importance of these designs lies in that fact that uniformly spreading the points out

the design space yields an accurate kriging model; this is a new observation which

is obvious but not well documented in the literature.

Hypotheses 2 and 3 have now been verified as a result of this study. In the next

section, the relevance of these results with regard to the PPCEM are discussed.


In this chapter, 7905 kriging metamodels of six engineering test problems have

been constructed and validated using a variety of correlation functions and experimental

designs to test and verify Hypotheses 2 and 3. In closing this chapter, recall the questions

posed at the beginning of this study:

• What is best type of experimental design you should use to query the simulation to

generate data to build an accurate kriging metamodel? For problems containing

only two variables, either classical or space filling designs yield good results;

however, as the size of the design space increases (i.e., number of variables

increases), space filling experimental design tend to yield more accurate kriging

187

metamodels on average since they tend to spread the points out well in the design

space. In particular, the minimax Latin hypercube design, uniform designs, and

orthogonal arrays yield good results. Random Latin hypercubes also provide good

results, provided orthogonality or optimality (e.g., IMSE) are imposed to control

the randomization. Finally, of the designs considered, Hammersley point designs

are not recommended unless numerous sample points can be afforded.

• How many sample points should you use? The interaction between sample size and

experimental design type is examined in Section E.3 because this impact does not

directly impact testing Hypotheses 2 or 3 since the analysis is conditioned on

sample size. In general, the more sample points which can be afforded, the more

accurate the resulting model. However, as discussed in Section E.3, a

recommendation on the number of sample points cannot be made at this time

because a wide enough spread of points was not investigated.

• What type of correlation function should you use to obtain the best predictor?

Based on the results in Section 5.3.1, the Gaussian correlation function yields the

most accurate predictor on average of the five studied.

• Lastly, how can you best validate the metamodel once you have constructed it? As

discussed in Section 5.2.2, cross validation root mean square error is not a

sufficient measure of model accuracy since it does not correlate well with either root

mean square error or max. error. One possible explanation of this is that because

the sample sizes are relatively small, an insufficient number of points is available to

cross-validate the model properly. If more points were available, then cross

validation error may yield a reasonable assessment of model accuracy; however,

this has not been tested. In light of this result, then, it is imperative that additional

sample points be taken to validate a kriging model.

These results have a direct bearing on the metamodeling capabilities within the Product

Platform Concept Exploration Method (PPCEM) as depicted in Figure 5.21. In the event

that metamodels need to be constructed for a deterministic computer code or simulation

routine within the context of product platform design, then the best correlation function to

select—if kriging metamodels are to be utilized—is the Gaussian correlation function, and

the best design to use is a space filling experimental design if the problem has more than

188

two variables (if the problem only has two variables, then a classical experimental design

will suffice). Also, it is recommended to take as many sample points as possible, but keep

in mind that additional sample points are needed to validate the model since cross validation

does not appear to provide a sufficient assessment of model accuracy.


Chp 6

Metamodeling



ConceptualNoise

Factors


MarketSegmentation

Grid


Chp 3


Platform


§5.3 §5.2

MDO Example


Kriging/DOE Testbed


189

In the next chapter, the focus returns to the PPCEM, and the process of testing and

verifying Hypotheses 1 resumes. Specifically, in Chapter 6 the design of a family of

universal electric motors is offered as “proof of concept” that the PPCEM works and that it

is effective at facilitating the design and development of a scalable product platform for a

product family.

190

6. CHAPTER 6

DESIGN OF A FAMILY OF UNIVERSALELECTRIC MOTORS

In this chapter, the Product Platform Concept Exploration Method (PPCEM) is

implemented to verify its use for designing a family of universal motors around a common

scalable product platform. An overview of the universal motor problem is presented in

Section 6.1; a schematic of a typical universal motor is given in Section 6.1.1, and a

practical mathematical model for universal motors is derived in Section 6.1.2. Section 6.2

contains the implementation of Steps 1 and 2 of the PPCEM. A market segmentation grid

is created for the problem and relevant factors and responses for the universal motor

platform are identified. Section 6.3 follows with the implementation of Step 4 of the

PPCEM by aggregating the universal motor specifications and formulating a compromise

DSP; Step 3 of the PPCEM—building metamodels—is not utilized in this example because

analytical expressions for mean and standard deviation of the responses are derived

separately. Section 6.4 contains the development of the actual universal motor platform;

ramifications of the resulting universal motor platform and product family are analyzed in

Section 6.5. Through this example problem, Hypothesis 1 and Sub-Hypotheses 1.1-1.3

are tested and verified as follows:

191

Hypothesis 1 - All but Step 3 of the PPCEM is employed in this chapter to design a

family of universal motors based on a scalable product platform. The success of

the method as discussed in Section 6.5 provides an initial “proof of concept” for the

method and hence Hypothesis 1.

Sub-Hypothesis 1.1 - The market segmentation grid is utilized in Section 6.2 to

help identify the stack length as the scale factor around which the motor family is

vertically scaled to achieve the desired platform leveraging strategy; this supports

Sub-Hypothesis 1.1.

Sub-Hypothesis 1.2 - The scale factor for the family of universal motors is taken as

the stack length, following the footsteps of the example by Black & Decker

(Lehnerd, 1987). Robust design principles are used in this example to develop a

universal motor platform—defined by seven design variables—which is insensitive

to variations in the scale factor and is thus good for a family of motors based on

different instantiations of the stack length (the scale factor). The success of this

implementation helps to support Sub-Hypothesis 1.2.

Sub-Hypothesis 1.3 - Robust design principles of “bringing the mean on target”

and “minimizing the deviation” are utilized in this example to aggregate individual

targets and constraints and to facilitate the design of the family of motors.

Combining this formulation with the compromise DSP allows a family of motors to

be designed around a common, scalable product platform, verifying Sub-

Hypothesis 1.3.

Despite all of the work in the previous two chapters, Hypotheses 2 and 3 are not

tested in this example. Analytic expressions for mean and standard deviation of the

response are derived from the analysis equations themselves and used directly in the

compromise DSP for the PPCEM. In concluding the chapter, a brief look ahead to the

General Aviation aircraft example in Chapter 7 in which the PPCEM is tested in full is

offered in Section 6.6.

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6.1 OVERVIEW OF THE UNIVERSAL MOTOR PROBLEM

Universal electric motors are so named for their capability to function on both direct

current (DC) and alternating current (AC). Universal motors also deliver more torque for a

given current than any other kind of AC motor (Chapman, 1991). The high performance

characteristics and flexibility of universal motors understandably have led to a wide range

of applications, especially in household use where they are found in electric drills, saws,

blenders, vacuum cleaners, and sewing machines, to name a few examples (Martin, 1986).

In addition, many companies manufacture several products which use universal

motors; for example several companies offer a complete line of power tools, whereas

several others offer a line of kitchen appliances or yard care tools (cf., Lehnerd, 1987).

For these companies, it has already become common practice to utilize a family of universal

motors of similar physical dimensions to meet a range of performance requirements for a

group of products (Nasar, 1987). The advantages of this approach included increased

modularity with decreased manufacturing time and inventory costs. For example, Black &

Decker developed a family of universal motors for its power tools in the 1970s in response

to a need to redesign their tools as discussed in Section 1.1.1.

In this chapter the task is to identify a set of common physical dimensions for a

hypothetical family of universal motors to satisfy a range of performance needs, providing

initial “proof of concept” for the PPCEM. To begin, a physical description and schematic

of the universal motor is offered in the next section. In Section 6.1.2, relevant analyses for

modeling the performance of a universal motor are introduced.

6 .1 .1 Physical Description, Schematic, and Nomenclature for theUniversal Motor Problem

A universal motor is composed of an armature and a field which are also referred to

as the rotor and stator, respectively, see Figure 6.1. The motor depicted in the figure has

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two field poles, an attached cooling fan, and laminations in both the armature and the field.

Laminating the metal in both the armature and field greatly reduces certain kinds of power

losses (cf., Nasar, 1987).

The armature consists of metal shaft about which wire is wrapped longitudinally

around two or more metal slats, or armature poles, as many as thousands of times. The

field consists of a hollow metal cylinder within which the armature rotates. The field also

has wire wrapped longitudinally around interior metal slats, or field poles, as many as

hundreds of times.

Figure 6.1 Schematic of a Universal Motor(adapted from G.S.Electric, 1997)

For a universal motor, the wraps of wire around the armature and the field are

wired in series, which means that the same current is applied to both sets of wire. As

current passes through the field windings, a large magnetic field is generated, which passes

through the metal of the field, across an air gap between the field and the armature, then

through the armature windings, through the shaft of the armature, across another air gap,

and back into the metal of the field, thus completing a magnetic circuit.

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However when the magnetic field passes though the armature windings, which are

themselves carrying current, the magnetic field exerts a force on the current carrying wires,

which is in the direction of the cross product of the vector direction of the current in the

armature windings and the vector direction of the magnetic field. Because of the geometry

of the windings, current on one side of the armature always is passing in the opposite

direction to the current on the other side of the armature. Thus, the force exerted by the

magnetic field on one side of the armature is opposite to the force exerted on the other side

of the armature. Thereby a net torque is exerted on the armature, causing the armature to

spin within the field. The reader is referred to (Chapman, 1991) or any physics text book

(e.g., Tipple, 1991) to learn more about how an electric motor operates. The nomenclature

for the universal electric motor is listed in Table 6.1.

Table 6.1 Nomenclature for Universal Motors

a Number of current paths on the armature Nc Number of turns of wire on the armatureAa

Awa

Area between a pole and the armature [mm2]Cross-sectional area of the wires on the

Ns Number of turns of wire on the field, perpole

Awf

armature [mm2]Cross-sectional area of the wires on the

ωρ

Rotational speed [rad/sec]Resistivity of copper [Ohms/m]

Bfield [mm2]

Magnetic field strength (generated by theρcopper

ρsteel

Density of copper [kg/m3]Density of steel [kg/m3]

current in the field windings) [Tesla, T] parmature Number of poles on the armatureφ Magnetic flux [Webers, Wb] pfield Number of poles on the fieldℑ Magnetomotive force [Ampere⋅turns] P Gross Power Output [W]H Magnetizing intensity [Ampere⋅turns/m] ro Outer radius of the stator [m]I Electric current [Amperes] Ra Resistance of armature windings [Ohms]K Motor constant [n.m.u.] Rs Resistance in the field windings [Ohms]lr

lg

Diameter of armature [m]Length of air gap [m]

ℜ Total reluctance of the magnetic circuit[Ampere⋅turns/m]

lc Mean path length within the stator [m] ℜs Reluctance of the stator [Ampere⋅turns/m]L Stack length [m] ℜa Reluctance of one air gap [Ampere⋅turns/m]µsteel

µo

Relative permeability of steel [n.m.u.]Permeability of free space [Henrys/m]

ℜr Reluctance of the armature[Ampere⋅turns/m]

µair Relative permeability of air [n.m.u.] t Thickness of the stator [m]m Plex of the armature winding [n.m.u.] T Torque [Nm]M Mass [kg] Vt Terminal voltage [Volts, V]η Efficiency [n.m.u.] Z Number of conductors on the armature

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6 .1 .2 Relevant Analyses for Universal Motor Problem

A universal motor is the same as a direct current (DC) series motor; however, in

order to minimize certain kinds of power losses within the core of the motor when

operating on AC power, a universal motor is constructed with slightly thinner laminations

in both the field and the armature and less field windings. However, the governing

electromagnetic equations for the operation of a series DC motor and a universal motor

running on DC current are identical (Chapman, 1991). The performance at full-load torque

of a universal motor running on AC current is only slightly less than the performance of the

same motor running on DC current, see Figure 6.2. This discrepancy in performance is

due to losses caused by the inherent oscillation in alternating current (AC); for an overview

of the extra losses associated with AC operation, see, e.g., (Unnewehr, 1983).

Figure 6.2 Comparison of the Torque-Speed Characteristics of a UniversalMotor Rated at 1/4 Hp and 8000 rpm when Operating on AC and DC Power

Supplies (Martin, 1986)

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These extra losses incurred in AC operation of a universal motor are difficult, if not

impossible, to model analytically; thus, complicated finite element analyses are becoming

more popular for modeling motor behavior under AC current. Since such a detailed

analysis is beyond the scope of this work, the derived model for the performance of the

universal motor is for DC operation for which simple analytical expressions are known or

can be derived. Moreover, several texts indicate that the performance of universal motors

under AC and DC conditions is quite comparable and include diagrams such as the one

reproduced in Figure 6.2 (see, e.g., (Chapman, 1991; Martin, 1986; Shultz, 1992;

Unnewehr, 1983); Shultz (1992) states that “Universal motors...will operate either on DC

or AC up to 60 Hz. Their performance will be essentially the same when operated on DC

or AC at 60 Hz.” The sample torque-speed curves in Figure 6.2 graphically illustrate this,

showing that for one specific motor, the performance characteristics between AC and DC

operation do not deviate significantly until well past the full-load torque of the motor. For

this work, all motors are designed for operation at full-load torque. Thus, it is assumed

that designing a universal motor under DC conditions yields satisfactory performance under

AC conditions as well.

The model takes as input the design variables {Nc, Ns, Awa, Awf, ro, t, lgap, I, Vt, L}

and returns as output the power (P), torque (T), mass (M), and efficiency (η) of the motor.

To formulate the model, it is necessary to derive equations for P, T, M, and η as functions

of the design variables. The equations are based primarily on those given in (Chapman,

1991) and (Cogdell, 1996) for DC electric motors unless otherwise noted.

Power

The basic equation for power output of a motor is the input power minus losses:

P = P in - Plosses [6.1]

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where the input power is the product of the voltage and the current,

Pin = VtI [6.2]

and, for a universal motor, power is lost:

• in the copper wires as they heat-up (copper losses),

• at the interface between the brushes and the armature (brush losses),

• in the core due to hysteresis and eddy currents (core losses),

• in mechanical friction in the bearings supporting the rotor (mechanical losses),

• in heating up the core and copper wires which adversely effects the magnetic

properties of the core and the current carrying ability of the wires (thermal

losses), and

• due to stray losses (stray losses).

Simple analytic expressions only exist for the copper losses and the brush losses. Stray

losses usually are assumed to be no more than one percent, and thus can be neglected.

Mechanical losses can be minimized by an appropriate choice of bearing and housing

arrangement; however, these variables are beyond the scope of the motor model itself.

Hence mechanical losses are neglected. Core losses, especially those incurred by eddy

currents, can be minimized by the use of thin laminations in the stator and rotor; assuming

this is done, the core losses can be assumed to be small and thus can be neglected.

Thermal losses are in general non-negligible, but are highly dependent upon the external

cooling scheme (e.g., cooling fan and fins on the housing) applied to the motor. Because

an effective cooling scheme can keep the motor from running too hot, and as the setup of

the cooling configuration is beyond the scope of this model, thermal losses are neglected.

The combined effects of all the aforementioned neglected losses will, however, decrease

the output power and efficiency from the predicted value from the model. Nevertheless,

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the following equations serve as a sufficiently accurate model for the DC operation of a

universal motor. Consequently, the general equation for power losses reduces from,

Plosses = Pcopper + Pbrush + Pthermal + Pcore + Pmechanical + Pstray [6.3]

to a more manageable:

Plosses = Pcopper + Pbrush [6.4]

where

Pcopper = I2(Ra + Rs) [6.5]

and

Pbrush = αI [6.6]

where α is typically 2 volts. Substituting these expressions into the power equation yields:

P = VtI-I2(Ra + Rs) - 2I [6.7]

However, Ra and Rs, the resistances of the armature and field windings, can be specified

further as functions of the design variables. The resistances Ra and Rs can be computed

directly from the general equation that the resistance of any wire is given by:

Resistance = (Resistivity)(Length)

Areacross −section

[6.8]

Assuming that each wrap (i.e., turn) of wire on the armature is approximately the shape of

a rectangle with length L (the stack length of the motor) and width lr (the diameter of the

armature) then in terms of the physical dimensions of the motor, lr can be expressed as two

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times the radius of the armature, which is just the outer radius of the stator minus the

thickness of the stator minus the air gap length, or,

lr = 2(ro - t - lgap) [6.9]

so that the length of one wrap of wire on the armature is:

Lengthone wrap = 2L + 2lr = 2L + 4(ro - t - lgap) [6.10]

The total length of wire on the armature is the stack length, L, times the total number of

wraps on the armature, Nc, so that the resistance of the armature, Ra, is

Ra = ρ(2L + 4(r o−t − lgap))N c

A armature_wire

[6.11]

Similarly, assuming that each wrap of wire on the field is approximately the shape of a

rectangle with length L (the stack length of the motor) and width double the inner radius of

the stator (ro-t), then the resistance of the stator, Rs, is:

Rs = ρ(p field)(2L + 4(ro−t))Ns

A field_wire

[6.12]

However the purpose of the field windings is to create a magnetic field across the armature,

thus requiring two field poles, one for the “North” end of the magnetic field and one for the

“South” end. Thus, pfield is 2, and Equation 6.12 becomes Equation 6.13 which is:

Rs = ρ(2)(2L + 4(ro− t))N s

A field_wire

[6.13]

Now that Ra and Rs are expressed in terms of the design variables in Equations 6.11 and

6.13, the power equation is complete.

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Efficiency

The equation for efficiency can be computed directly from the equation for power. The

basic equation for efficiency, expressed as a decimal and not a percentage, is given by:

η = P/Pin [6.14]

where P and Pin are given by Equations 6.2 and 6.7.

Mass

For the purpose of estimating the mass of the motor, it is modeled as a solid steel cylinder

with length L and radius lr/2 for the armature and a hollow steel cylinder with length L,

outer radius ro and inner radius (ro-t) for the stator. The mass of the windings on both the

armature and the field are also included, where the length of each winding is the same as

those assumed for the derivation of the power equation, see Equation 6.10. Thus the

equation for mass is of the form:

Mass = Mstator + Marmature + Mwindings [6.15]

where:

Mstator = π(ro2 - (ro - t)

2)(L)(ρsteel) [6.16]

Marmature = π(ro - t - lgap)2(L)(ρsteel) [6.17]

Mwindings = (Nc(2L + 4(ro - t - lgap))Awa + 2Ns(2L + 4(ro - t))Awf)ρcopper [6.18]

Using Equations 6.16-6.18 for Mstator, Marmature, and Mwindings, the mass of the motor,

Equation 6.15, can be estimated from the design variables.

Torque

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The last equation to derive is an equation for torque. In general, the torque of a DC motor

is given by:

T = KφI [6.19]

where K is a motor constant, φ is the magnetic flux, and I is the current. For a DC motor,

K is computed as:

K = (Z)(parmature)

2πa[6.20]

where Z, the number of conductors on the armature, is:

Z = 2Nc [6.21]

and a, the number of current paths on the armature, is:

a = 2m = 2 [6.22]

assuming a simplex (m = 1) wave winding on the armature. Since the number of armature

poles on a universal motor is almost invariably two (cf., Martin, 1986), or,

parmature = 2 [6.23]

K can be reduced to:

K = 2N c(2)

2π(2) =

N c

π[6.24]

The derivation of the flux term, φ, is significantly more complicated. To begin,

consider the idealized DC motor shown in Figure 6.3a with its corresponding magnetic

circuit shown in Figure 6.3b. As shown in the figure, N is the number of turns on the

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stator (which is equal to 2Ns for the model being derived), I is the current, A is the cross-

sectional area of the stator, lr is the diameter of the armature, lg is the gap length, and lc is

the mean magnetic path length in the stator.

(a) Physical model (b) Magnetic circuit

Figure 6.3 An Idealized DC Motor (Chapman, 1991)

In general the equation for flux through a magnetic circuit is simply the

magnetomotive force, ℑ, divided by the total reluctance of the circuit, ℜ:

φ =ℑℜ

[6.25]

where the magnetomotive force, ℑ, is simply the number of turns around one pole of the

field times the current:

ℑ = NsI [6.26]

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The total reluctance, ℜ, is calculated from the magnetic circuit shown in Figure

6.3b. For a magnetic circuit, reluctances in series add just like resistors in series in an

electric circuit; therefore, the total reluctance in the idealized DC motor is the sum of the

reluctances of the stator, rotor, and two air gaps:

ℜ = ℜs + ℜr + 2ℜa [6.27]

where, in general, reluctance is calculated as:

ℜ = Length

(Permeability)(Area cross − section )[6.28]

When permeability, µ, is expressed as the relative permeability of the material times the

permeability of free space, µo, the reluctance of the stator, rotor, and air gaps are:

ℜs = l c

µ steelµoA s

, ℜr = l r

µ steelµoA r

, ℜa = l a

µ airµoAa

[6.29]

In order to approximate more closely a universal motor for this example, the

idealized DC motor geometry shown in Figure 6.3 is modified to be more representative of

a real universal motor. The resulting model geometry is shown in Figure 6.4a and is

described by the outer radius of the stator, ro, the thickness of the stator, t, the diameter of

the armature, lr, the length of the air gap, lgap, and the stack length, L. The resulting

magnetic circuit is shown in Figure 6.4b; notice that the magnetic circuit for the idealized

DC motor and the magnetic circuit for a universal motor are different, because in a

universal motor there are two paths which the magnetic flux can take around the stator, i.e.,

clockwise and counter-clockwise. These two paths are in parallel and thus are included in

the magnetic circuit as two parallel flux paths. Reluctances in parallel in a magnetic circuit

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act like resistors in parallel in an electric circuit, so that the combined reluctance of two

identical reluctances in parallel is simply one half the reluctance of either path. Therefore,

for a universal motor:

ℜs = l c

2µ steelµoAs

, ℜr = l r

µ steelµoA r

, ℜa = l a

µ airµoAa

[6.30]

so that Equation 6.27 for the total reluctance, ℜ, still holds.

(a) Physical model (b) Magnetic circuit

Figure 6.4 Model Geometry for a Universal Motor

In Equation 6.30, the mean magnetic path length in the stator, lc, is taken to be one

half the mean circumference of the hollow stator cylinder, or,

lc = π(2ro + t)/2 [6.31]

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The cross-sectional area of the stator, As, is taken to be the thickness of the stator times the

stack length, or,

As = (t)(L) [6.32]

The cross-sectional area of the armature is taken to be approximately the diameter of the

armature times the stack length:

At = (lr)(L) [6.33]

The cross-sectional area of the air gap is the length of the air gap times the stack length:

Aa = (lgap)(L) [6.34]

The last expression needed for the calculation of reluctance is the relative

permeability of the stator and the armature. For the purposes of this model, both the stator

and the armature are assumed to be made of steel with the relative permeability versus

magnetizing intensity curve for typical steel is shown in Figure 6.5.

Figure 6.5 Relative Permeability Versus Magnetizing Intensity for aTypical Piece of Steel (Chapman, 1991)

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The curve is divided into three regions, and each section is fit with an appropriate

numerical expression in order to include the curve shown in Figure 6.5 in the model. The

curve fits used are as follows:

µr = -0.2279H2 + 52.411H + 3115.8 H ≤ 220

µr = 11633.5 - 1486.33ln(H) 220 < H ≤ 1000

µr = 1000 H > 1000 [6.35]

where, from Ampere's Law, the magnetizing intensity, H, is given by,

H = N cI

l c + l r + 2lgap

[6.36]

The relative permeability of air, µair, is taken as unity, and the permeability of free space is a

constant, µo = 4 π 10-7. Now with expressions for K, φ, ℑ, ℜs, ℜr, ℜa, lc, lr, As, Ar, Aa,

and µsteel in terms of the design variables, the torque equation is complete.

This completes the mathematical model for the universal motor, and the PPCEM

now can be implemented to design a family of universal motors around a common product

platform. The initial steps of the PPCEM, Steps 1 and 2, are outlined in the next section.

6.2 STEPS 1 AND 2: CREATE MARKET SEGMENTATION GRID ANDCLASSIFY FACTORS FOR UNIVERSAL MOTOR PLATFORM

With a given set of performance requirements and the model derived in Section

6.1.3, the first step in implementing the PPCEM is to create the market segmentation grid

to identify and map which type of leveraging can be used to meet the overall design

requirements and realize the desired product platform and product family. The market

segmentation grid shown in Figure 6.6 depicts the desired leveraging strategy for this

universal motor example. The goal is to design a motor platform which can be leveraged

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vertically for different market segments which are defined by the torque needs of each

market, following in footsteps of the Black & Decker universal motor example from

(Lehnerd, 1987) which was discussed in Section 1.1.1. In this specific example, ten

instantiations of the motor are to be considered; moreover, in order to reduce cost, size, and

weight, it is supposed the best motor is the one that satisfies its performance requirements

with the least overall mass and greatest efficiency. Standardized interfaces will ensure

horizontal leveraging across market segments; however, only vertical leverage is

considered in this example.

Low End

Mid-Range

High End

Functionalparametricscale factor: torque = f(length)

PowerTools

Universal Motor Platform

Ver

tical

Sca

ling

KitchenAppliances

Lawn &Garden

Figure 6.6 Universal Motor Market Segmentation Grid

Having created the market segmentation grid and identified an appropriate

leveraging strategy and scale factor, Step 2 in the PPCEM is to classify the factors of

interest within the universal motor problem. The design variables (i.e., control factors) and

corresponding ranges of interest in this study are as follows:

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1. Number of turns of wire on the armature (100 ≤ Nc ≤ 1500 turns)

2. Number of turns of wire on each field pole (1 ≤ Ns ≤ 500 turns)

3. Cross-sectional area of the wire used on the armature (0.01 ≤ Awa ≤ 1.0 mm2)

4. Cross-sectional area of the wire used on the field poles (0.01 ≤ Awf ≤ 1.0 mm2)

5. Radius of the motor (0.01 ≤ ro ≤ 0.10 m)

6. Thickness of the stator (0.0005 ≤ t ≤ 0.10 m)

7. Current drawn by the motor (0.1 ≤ I ≤ 6.0 Amp)

The terminal voltage, Vt, is fixed at 115 volts to correspond to standard household

voltage, and the length of the air gap, lgap, is set to 0.7 mm which is taken to be the

minimum possible air gap length. The minimum air gap length is fixed because the

performance equations derived in Section 6.1.3 indicate that minimizing the air gap length

maximizes torque and minimizes mass with no effect on the other performance measures.

Following in the footsteps of the Black & Decker example, the stack length, L, is

the scale factor for the product family primarily because of its importance in the torque

equation, Equation 6.19, derived in Section 6.1.3, i.e., torque is directly proportional to

stack length. To increase torque across the platform, stack length of the motors is

increased while keeping the other physical parameters (e.g., the outer radius and the

thickness) unchanged. Furthermore, it is assumed that the greatest manufacturing costs

savings can be achieved by exploiting the fact that only the stack length of the motors varies

while still providing a variety of torque and power ratings. The initial range of interest for

stack length is taken to be 1 to 20 centimeters; specific instantiations are computed in Step 5

so as to meet the desired torque requirements for each platform derivative.

There are a total of six responses (i.e., goals and constraints) which are of interest

for each motor. The following constraint values—Table 6.2—and goal targets—Table

6.3—are assumed to define each market niche for each motor.

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Table 6.2 Constraints for Universal Motor Product Family

Constraints ValueMagnetizing intensity, H H < 5000Feasible geometry ro > tPower of each motor, P P = 300 WEfficiency of each motor, η η ≥ 0.15Mass of each motor, M M ≤ 2.0 kg

The constraint on magnetizing intensity ensures that the magnetic flux within the

motor does not exceed the physical flux carrying capacity of the steel (Chapman, 1991).

The constraint on feasible geometry ensures that the thickness of the stator does not exceed

the radius of the stator, since the thickness is measured from the outside of the motor

inward, as indicated in Figure 6.4a. The desired power for each motor is 300 W which is

treated as an equality constraint to ensure that design variable settings are selected to match

this requirement exactly. A minimum allowable efficiency of 15% and a maximum

allowable mass of 2.0 kg are assumed to define a feasible motor. The efficiency and mass

goal target for each motor are listed in Table 6.2 along with the desired torque requirement

for each motor.

Table 6.3 Goal Targets for Universal Motor Platform

GoalMotor Torque [Nm] Mass [kg] Efficiency

1 0.05 0.50 0.702 0.10 ò ò3 0.125 ò ò4 0.15 ò ò5 0.20 ò ò6 0.25 ò ò7 0.30 ò ò8 0.35 ò ò9 0.40 ò ò10 0.50 ò ò

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For the purpose of illustration, the relationship between the design variables, the

scale factor, and the responses, are shown in the P-Diagram in Figure 6.7.

X = Control Factors# wire turns on armature# wire turns on field pole

Afield wire

Aarmature wire

Motor radiusStator thicknessCurrent drawn

Y = ResponsesPowerTorqueMass

EfficiencyFeasible Geometry

Magnetizing Intensity S = stack length

UniversalMotorModel

Figure 6.7 P-Diagram for the Universal Motor Example

This concludes Steps 1 and 2 of the PPCEM. Step 3 is not utilized in this particular

example since it is possible to derive expressions for mean and variance of each motor due

to scaling the stack length as described in the next section. The next step, then, is Step 4

which is to formulate an appropriate compromise DSP for the family of universal motors.

6.3 STEP 4: AGGREGATE PRODUCT SPECIFICATIONS ANDFORMULATE UNIVERSAL MOTOR PLATFORM COMPROMISE DSP

The corresponding compromise DSP formulation for the universal motor product

platform is listed in Figure 6.8. In summary, there are nine design variables, seven

constraints, and two objectives. The two objectives (minimize mass to its target and

maximize efficiency to its target) are assumed to have equal importance to the design, and

are thus weighted equally in the problem formulation.

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Given:q Parametric (horizontal) scale factor: stack lengthq Universal motor model analysis equations, Section 6.1.2

Find:q The system variables, x:

• Number of turns on the armature, Nc • Thickness of the stator, t• Number of turns on each pole on the

field, Ns

• Current drawn by the motor, I• Radius of the motor, r

• Cross-sectional area of the wire onthe armature, Awa

• Mean of stack length, µL• Standard deviation of stack

• Cross-sectional area of the wire onthe field, Awf

length, σL

Satisfy:q The system constraints:

• Magnetizing intensity, H: Hmax ≤ 5000• Feasible geometry: t < ro• Power output, P: P = 300 W• Motor efficiency, η: η ≥ 0.15• Mass, M: M ≤ 2.0 kg

q Aggregated torque requirements:• Mean torque, µT: µT = 0.2425 Nm• Standard deviation torque, σT: σT = 0.13675 Nm

q The bounds on the system variables:100 ≤ Nc ≤ 1500 turns 0.5 ≤ t ≤ 10.0 mm

1 ≤ Ns ≤ 500 turns 0.1 ≤ I ≤ 6.0 A0.01 ≤ Awa ≤ 1.0 mm2 1.0 ≤ µL ≤ 10.0 cm0.01 ≤ Awf ≤ 1.0 mm2 0.0 ≤ σL ≤ 10.0 cm

1.0 ≤ ro ≤ 10.0 cm

Minimize:q Mean mass, target: M = 0.50 kg

Maximize:q Mean efficiency, target: η = 0.70

Figure 6.8 Universal Motor Product Platform Compromise DSPFormulation for Use with OptdesX

The aggregated mean torque, µT, and standard deviation, σT are calculated as the

sample mean and standard deviation of the set of torque requirements {0.05, 0.1, 0.125,

0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5} Nm assuming a uniform distribution. Power,

efficiency, and mass for the family are assumed to be uniformly distributed with respective

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means and standard deviations because it is assumed that the distribution of the demand for

the motors is uniform.

The mean power, mean efficiency, and mean mass are calculated as the power,

efficiency, and mass, respectively, for the mean length. The standard deviation of torque is

approximated using a first-order Taylor series expansion, assuming that the standard

deviation is small (Phadke, 1989):

σT ˙ = ∂T

∂µL

σ L [6.37]

Now that the compromise DSP for the family of universal motors is formulated, Step 5 of

the PPCEM is executed to develop the universal motor platform.

6.4 STEP 5: DEVELOP THE UNIVERSAL MOTOR PLATFORM

For this example problem, the compromise DSP formulated in Section 6.3 is solved

using the Generalized Reduced Gradient (GRG) algorithm in OptdesX. For a thorough

explanation of OptdesX and the GRG algorithm, see, e.g., (Parkinson, et al., 1998). The

OptdesX software package is used instead of DSIDES in this example since implementation

of the PPCEM is not algorithm dependent.

Note that in order to develop the product portfolio, the compromise DSP is

formulated with goals for mean torque, µT, and standard deviation of torque, σT, which

ensures that the product portfolio will be able to be instantiated for all ten values of torque

within the range of the scale factor specified by the mean and standard deviation, µL and σL,

for the stack length. Also note that the constraint on magnetizing intensity is formulated to

ensure that the entire product family will meet the constraint on magnetizing intensity

individually. This is accomplished by computing a maximum magnetizing intensity which

represents the magnetizing intensity for the largest instantiation of the product family and is

simply evaluated at the upper bound of current. The compromise DSP in Figure 6.8 is

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solved using three different starting points in OptdesX: the lower, middle, and upper

bounds of the design variables. The best design variable settings and responses for the

motor platform are listed in Table 6.3. The values for the number of armature turns and

field turns have been rounded to the nearest integer.

Table 6.4 Universal Motor Product Platform Solution

Design Variable Value Response ValueNumber of armature turns Nc 1062 Torque, mean [Nm] 0.2425Number of field turns Ns 54 Torque, std. dev. [Nm] 0.137Wire x-sect area, field [mm2] Awf 0.376 Power, mean [W] 300Wire x-sect area, armature [mm2] Awa 0.241 Efficiency, mean [%] 0.608Motor radius [cm] r 2.59 Mass, mean [kg] 0.751Stator thickness [mm] t 6.66Current drawn [Amp] I 4.29Stack length, mean [cm] µL 2.62Stack length, std dev [cm] σL 1.48

For the purpose of verifying the solution itself, convergence plots for mean mass

and mean efficiency are presented in Figure 6.10 for high, middle, and low starting points

used in OptdesX. Excellent convergence is shown in both plots.

(a) Mass, mean (b) Efficiency, mean

Figure 6.9 Convergence Plots for the Universal Motor Product Platform

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To develop the individual motors within the scaled product family using the product

platform specifications from Table 6.4, the compromise DSP given in Figure 6.8 is

modified such that Nc, Ns, Awa, Awf, r, and t are held constant at the values listed in Table

6.4, and only the current, I, and stack length, L, are allowed to vary to meet the original set

of torque requirements. Because the mean and standard deviation for stack length have

been found for the product platform, the initial range of interest for stack length now can be

discarded in favor of the range for the product platform. Using the assumption that length

is uniformly distributed, the minimum and maximum bounds on length can be estimated as:

µL - 3 σL ≤ L ≤ µL - 3 σL [6.38]

Substituting the values for mean and standard deviation of stack length shown in Table 6.4,

the new lower and upper bounds of interest for stack length are as follows:

0.057 ≤ L ≤ 5.18 cm

Note that because individual torque goals are being set, the goal for standard deviation of

torque is eliminated in the modified compromise DSP formulation. Also, the constraint on

magnetizing intensity is no longer imposed on any maximum magnetizing intensity but

rather on the individual magnetizing intensity and is evaluated at the current for each motor.

The product platform is instantiated by selecting appropriate values for the scale

factor (stack length) within the range specified by the mean and standard deviation in Table

6.4 for each desired set of torque and power requirements. The current also is being

allowed to vary since it is a dependent variable in the system, i.e., it is the amount of

current which is drawn by the motor such that the given torque and power requirements are

met for a given motor geometry. In terms of the principles of robust design, the values

shown in Table 6.4 for the product portfolio are found such that the goal for mean power is

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on target, while varying the current allows the standard deviation of power across the

instantiated product family to be zero. The modified compromise DSP for the product

family is shown in Figure 6.10 and is again solved using OptdesX while starting from

three different starting points.

Given:q Configuration scale factor = stack lengthq Universal motor model equationsq Platform settings for Nc, Ns, Awa, Awf, r, and t (Table 6.4)


• Stack length, L • Current drawn by the motor, I



q Individual torque requirements:• Torque, T: T = {0.05, 0.1, 0.125, 0.15, 0.2, 0.25,

0.3, 0.35, 0.4, 0.5} Nmq The bounds on the system variables:

0.1 ≤ I ≤ 6.0 A0.057 ≤ L ≤ 5.18 cm

Minimize:q Mass, target: M = 0.50 kg

Maximize:q Efficiency, target: η = 0.70

Figure 6.10 Compromise DSP Formulation for Instantiating the PPCEMPlatform for Use with OptdesX

The resulting values for current and stack length of each motor (PPCEM platform

instantiation) are listed in Table 6.5 along with the corresponding response values. Notice

that the stack length varies from 0.865 cm to 2.95 cm in order to meet the desired torque

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and power requirements. The resulting current drawn by the system ranges from 3.39

Amps to 5.82 Amps which is slightly high but acceptable for a motor with such a large

torque. Finally, notice that only three motors meet the desired efficiency target of 70%,

and these are all at the low-end, and only one motor achieves the mass target of 0.5 kg.

Table 6.5 Universal Motor Product Family PPCEM Instantiations

Product Specifications (Design Variables) ResponsesAwf Awa I r t L T P η M

Motor Nc Ns [mm2] [mm2] [Amp] [cm] [mm] [cm] [Nm] [W] [%] [kg]1 1062 54 0.376 0.241 3.39 2.59 6.66 0.865 0.05 300 76.8 0.3802 ò ò ò ò 3.62 ò ò 1.53 0.10 ò 72.2 0.5203 ò ò ò ò 3.73 ò ò 1.79 0.125 ò 70.0 0.5764 ò ò ò ò 3.85 ò ò 2.02 0.15 ò 67.9 0.6255 ò ò ò ò 4.08 ò ò 2.39 0.20 ò 63.9 0.7036 ò ò ò ò 4.33 ò ò 2.66 0.25 ò 60.2 0.7597 ò ò ò ò 4.59 ò ò 2.83 0.30 ò 56.8 0.7978 ò ò ò ò 4.87 ò ò 2.94 0.35 ò 53.6 0.8209 ò ò ò ò 5.16 ò ò 2.99 0.40 ò 50.5 0.83010 ò ò ò ò 5.82 ò ò 2.95 0.50 ò 44.8 0.820

It is uncertain whether the failure to achieve the desired mass and efficiency targets

is a property of the system itself or a result of using the PPCEM. Therefore, a family of

individually designed universal motors is developed in the next section to provide a

benchmark for comparison. The differences between this family of benchmark motors and

the PPCEM instantiations are then compared to verify the PPCEM solutions.

6.5 RAMIFICATIONS OF THE RESULTS OF THE ELECTRIC MOTOREXAMPLE PROBLEM

6.5 .1 Development of a Benchmark Universal Motor Family

In order to generate a family of benchmark motors to compare with the PPCEM

family of motors, the compromise DSP presented in Figure 6.10 is modified such that Nc,

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Ns, Awa, Awf, r, and t are all design variables variables in addition to I and L. The resulting

compromise DSP is shown in Figure 6.12. This compromise DSP is solved using

OptdesX for each of the ten power and torque ratings. Three different starting points—

lower, middle, and upper bounds—are used to solve the compromise DSP for each motor.

Given:q Universal motor model analysis equations, Section 6.1.2


• Number of turns on the armature, Nc • Thickness of the stator, t• Number of turns on each pole on the

field, Ns

• Current drawn by the motor, I• Radius of the motor, r

• Cross-sectional area of the wire onthe armature, Awa

• Mean of stack length, µL• Standard deviation of stack

• Cross-sectional area of the wire onthe field, Awf

length, σL



q Individual torque requirement:• Torque T = {0.05, 0.1, 0.125, 0.15, 0.2, 0.25,

0.3, 0.35, 0.4, 0.5} Nmq The bounds on the system variables:

100 ≤ Nc ≤ 1500 turns 1.0 ≤ ro ≤ 10.0 cm1 ≤ Ns ≤ 500 turns 0.5 ≤ t ≤ 10.0 mm

0.01 ≤ Awa ≤ 1.0 mm2 0.1 ≤ I ≤ 6.0 A0.01 ≤ Awf ≤ 1.0 mm2 0.057 ≤ L ≤ 5.18 m

Minimize:q Mean mass, target: M = 0.50 kg

Maximize:q Mean efficiency, target: η = 0.70

Figure 6.11 Compromise DSP Formulation for Benchmark UniversalMotor Family for Use with OptdesX

218

The resulting design variable settings and responses for each benchmark motor are

summarized in Table 6.6. Compared to the PPCEM solutions listed in Table 6.5, the

number of armature turns, Nc, is generally lower than the PPCEM platform specification

and the number of field turns, Ns, is slightly higher. The cross-sectional area of the field

wire, Awf, is lower than the PPCEM platform specification; however, the PPCEM platform

value for Awa, armature wire cross-sectional area, is contained within the range observed

for the benchmark motors. Similarly, the ranges for motor radius, r, and thickness, t, for

the benchmark motors both span the values of the PPCEM platform specifications. These

motors draw less current—a maximum of 4.71 Amps—compared to the PPCEM family of

motors which draw as much as 5.82 Amps for the equivalent motor. Finally, note that the

range of stack lengths of the benchmark motors are comparable to the range of stack

lengths found using the PPCEM.

Table 6.6 Benchmark Universal Motor Specifications


Motor Nc Ns [mm2] [mm2] [Amp] [cm] [mm] [cm] [Nm] [W] [%] [kg]1 730 45 0.205 0.203 3.65 3.62 9.69 0.998 0.05 300 71.4 0.5002 750 76 0.203 0.186 3.73 3.31 11.77 1.28 0.10 ò 70.6 0.5003 760 89 0.203 0.190 3.73 3.12 11.20 1.41 0.125 ò 70.0 0.5004 785 95 0.205 0.205 3.70 2.82 8.88 1.63 0.15 ò 70.5 0.5005 988 74 0.217 0.241 3.84 2.26 5.75 2.38 0.20 ò 67.9 0.5586 1007 73 0.224 0.246 4.02 2.35 6.17 2.61 0.25 ò 64.9 0.6397 1030 73 0.230 0.253 4.19 2.44 6.35 2.74 0.30 ò 62.2 0.7128 1056 73 0.237 0.260 4.36 2.51 6.46 2.81 0.35 ò 59.8 0.7779 1082 72 0.243 0.267 4.53 2.58 6.67 2.87 0.40 ò 57.7 0.83710 1087 72 0.247 0.284 4.71 2.71 7.15 3.16 0.50 ò 55.3 0.985

Regarding the performance of each motor, the desired torque and power

requirements are achieved by each motor; moreover, more of the benchmark motors

achieve the mass and efficiency targets of 0.5 kg and 70%. Unlike the PPCEM family of

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motors, four of the benchmark motors achieves the efficiency target of 70%, and four of

the motors achieves the mass target of 0.5 kg. A closer comparison of the performance of

the individual motors is offered in Table 6.7.

For the purpose of validating the solutions themselves, convergence plots for mass

and efficiency for the 0.25 Nm benchmark motor torque are presented in Figure 6.13 for

the high, middle, and low starting points. Fairly good convergence is observed in each

graph; however, the final value of efficiency from the high starting point is slightly worse

than the final values of efficiency from the low and middle starting points. Therefore, in

situtations where all three points do not converge to the same final value, only the best

solution is reported.

(a) Mass (b) Efficiency

Figure 6.12 Convergence Plots for 0.25 Nm Benchmark Motor

6 .5 .2 Comparison between the Benchmark Universal Motor Family and thePPCEM Motor Family

In the previous section, a family of individually designed benchmark motors is

created to compare against the performance of the PPCEM family of universal motors

found in Section 6.4. As shown in Table 6.5 and Table 6.6, both families meet their goals

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for both power and torque; however, their responses for efficiency and mass differ. The

efficiency and mass of each motor within the benchmark family and the PPCEM family are

repeated in Table 6.7 along with the percentage difference of each response from the

benchmark to the PPCEM. For efficiency, a positive change denotes an improvement from

the benchmark to the PPCEM; meanwhile, a negative change denotes an improvement in

the mass. Finally, note that a motor which has achieved its target mass (0.5 kg) and

efficiency (70%) is considered to be equivalent to a motor with a mass which is lower than

the target or an efficiency which is higher than the target.

Table 6.7 Comparison of the Responses between the Benchmark MotorFamily and the PPCEM Motor Family

Benchmark Motors PPCEM Motors Percent DifferenceMotor η [%] M [kg] η [%] M [kg] η M

1 71.4 0.500 76.8 0.380 equiv. equiv.2 70.6 0.500 72.2 0.520 equiv. 4.0%3 70.0 0.500 70.0 0.576 equiv. 15.2%4 70.5 0.500 67.9 0.625 -3.7% 25.0%5 67.9 0.558 63.9 0.703 -5.9% 26.0%6 64.9 0.639 60.2 0.759 -7.2% 18.8%7 62.2 0.712 56.8 0.797 -8.7% 11.9%8 59.8 0.777 53.6 0.820 -10.4% 5.5%9 57.7 0.837 50.5 0.830 -12.5% -0.8%10 55.3 0.985 44.8 0.820 -19.0% -16.7%

Average change: -6.74% 8.89%

Three of the PPCEM motors have equivalent efficiency ratings to the corresponding

benchmark motor which produces the same torque; however, only the motor with the

smallest torque (Motor #1) is considered to have a mass equivalent to its corresponding

benchmark motor. As tallied at the bottom of Table 6.7, the PPCEM motors lose 7% in

efficiency and weigh 9% more than the family of benchmark motors, on average.

Therefore, while the family of PPCEM motors based on a common product platform scaled

in stack length are able to achieve the desired range of torque and power requirements, they

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lose, on average, 10% in both efficiency and mass compared to an equivalent family of

individually designed benchmark motors. At this point, it is up to the judgment of the

engineering designers and managers to decide if the savings (in inventory, manufacturing,

etc.) from having a family motors based a common platform and scaled only in stack length

outweighs the sacrifice in mass and efficiency. Meanwhile, an attempt to improve the

performance of the PPCEM family of motors in relation to the benchmark family of motors

is offered in the next section.

6 .5 .3 Improvements to the PPCEM Motor Family and Lessons Learned

While investigating this example, it was learned that Black & Decker varies more

than just stack length when it scales its universal motors to meet a variety of power ratings.

In addition to increasing the stack length of the motor, they also allow the number of turns

in the field and armature and the cross-sectional area of the wires in the field and armature

to vary from one motor to the next. Careful inspection of any two motors from one of their

power tool lines (say, corded drills) reveals that this is indeed the case.

The question then becomes: how well do the PPCEM instantiations perform if the

number of field and armature turns (Ns and Nc) and the cross-sectional area of the field and

armature wires (Awf and Awa) are allowed to vary in conjunction with varying the stack

length (and current)? The results obtained by solving a new set of compromise DSPs for

each universal motor are listed in Table 6.8. These solutions are obtained by modifying the

compromise DSP in Figure 6.10 to allow Nc, Ns, Awf, and Awa to vary from their platform

settings of 1062, 54, 0.376 mm2, and 0.241 mm2, respectively.

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Table 6.8 New PPCEM Universal Motor Instantiations with VaryingNumbers of Turns, Wire Cross-Sectional Areas, and Stack Lengths


Motor Nc Ns [mm2] [mm2] [Amp] [cm] [mm] [cm] [Nm] [W] [%] [kg]1 970 41 0.306 0.221 3.49 2.59 6.66 1.18 0.05 300 74.7 0.3972 981 66 0.306 0.224 3.62 ò ò 1.37 0.10 ò 72.1 0.4563 986 74 0.306 0.225 3.67 ò ò 1.44 0.125 ò 71.1 0.4774 990 82 0.306 0.227 3.72 ò ò 1.51 0.15 ò 70.1 0.4995 999 84 0.307 0.230 3.86 ò ò 1.81 0.20 ò 67.5 0.5686 1064 80 0.359 0.239 4.03 ò ò 2.03 0.25 ò 64.6 0.6467 1135 76 0.309 0.257 4.19 ò ò 2.20 0.30 ò 62.2 0.7128 1166 75 0.282 0.268 4.35 ò ò 2.42 0.35 ò 59.9 0.7749 1195 72 0.280 0.277 4.51 ò ò 2.60 0.40 ò 57.7 0.83310 1242 67 0.286 0.293 4.85 ò ò 2.91 0.50 ò 53.8 0.941

Recall that the target for efficiency is 70%, and the target for mass is 0.5 kg. So as

discussed previously, even if a particular motor weighs less than 0.5 kg or has an

efficiency greater than 70%, it is still considered to be equivalent to a motor which is

exactly 0.5 kg or has 70% efficiency. With this in mind, the new family of PPCEM

motors (allowing the numbers of turns and wire cross-sectional areas to vary along with

stack length and current) and the family of benchmark motors are essentially identical in

terms of performance as can be seen in Table 6.9. In both families of motors, the

necessary torque and power requirements have been met, and the two sets of motors are

compared solely on their respective efficiencies and masses. The result is that four of the

ten motors are equivalent (identical) since they achieve the targets for mass and efficiency,

and the remaining six motors vary by less than 2%. The highest torque motor in this new

PPCEM family is slightly less efficient (-2.7%) than the corresponding benchmark motor,

but it weighs less (-4.5%). This tradeoff is really negligible since more wire can be

wrapped around the field or armature to improve the efficiency with only slight increase in

mass. Consequently, by allowing the numbers of wire turns and the wire cross-sectional

areas to vary while also scaling the stack length, the resulting family of motors obtained

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using the PPCEM is equivalent to the family of individually designed benchmark motors.

This is a very important observation because it indicates that the PPCEM can be used to

obtain a family of motors which sacrifices minimal performance even though the motors are

based, for the most part, on a common platform specification.

Table 6.9 Comparison of Benchmark Designs and New PPCEMInstantiations with Varying Numbers of Turns, Wire Cross-Sectional

Areas, and Stack Lengths

Efficiency (Target = 70.0%) Mass (Target = 0.5 kg)Benchmark PPCEM Percent Benchmark PPCEM Percent

Motor η [%] η [%] Difference M [kg] M [kg] Difference1 71.4 74.7 equivalent 0.500 0.397 equivalent2 70.6 72.1 equivalent 0.500 0.456 equivalent3 70.0 71.1 equivalent 0.500 0.477 equivalent4 70.5 70.1 equivalent 0.500 0.499 equivalent5 67.9 67.5 -0.6% 0.558 0.568 1.8%6 64.9 64.6 -0.5% 0.639 0.646 1.1%7 62.2 62.2 0.0% 0.712 0.712 0.0%8 59.8 59.9 0.2% 0.777 0.774 -0.4%9 57.7 57.7 0.0% 0.837 0.833 -0.5%10 55.3 53.8 -2.7% 0.985 0.941 -4.5%

From an engineering standpoint, does it make sense to let Nc, Ns, Awf, and Awa vary

along with the stack length (and current)? From a manufacturing perspective, it makes

perfect sense to allow the number of turns of wire in the armature and field (Nc and Ns,

respectively) to vary since it costs little extra to wrap more (or fewer) turns when the motor

is being produced. From a inventory perspective, however, it would appear that allowing

Awf and Awa to vary is not cost effective since it requires that multiple wire types (i.e.,

varying cross-sectional areas) must be kept in stock in order to produce the family of

motors. The justification for allowing Awf and Awa to vary is as follows. As the stack

length of the motor increases (with everything else being held constant), the torque on the

motor increases; however, the power output actually decreases because the copper losses,

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given by Equation 6.5, increase as since Ra and Rs increase (see Equation 6.8) as stack

length increases.

One way to compensate for this loss in power is to allow more current to be drawn

as is the case in this example. What is typically done, however, to compensate for this

decrease in power (as the stack length is increased) is to decrease the number of field and

armature turns while simultaneously increasing the field and armature wire cross-sectional

areas. This lowers the resistances Ra and Rs and reduces copper losses without having to

draw additional current in order to maintain the desired output power. An added benefit of

this approach is that the rotational speed of the motor will also increase. In reality, the

operating speed of the motor is a very important design consideration since power and

torque are related through the equation:

P = Tω [6.39]

where P is power, T is torque, and ω is the rotational speed of the motor. The speed of the

motor has been neglected in this example since it is fixed once power and torque have been

specified. Based on Equation 6.39, for a fixed power output, as torque increases, the

rotational speed of the motor must decrease. In many cases however, as power increases

and torque increases, maintaining a consistent operating speed for the motor is often

desired. The additional inventory costs incurred by stocking a wider variety of wire sizes

is offset by this combination of effects.

But what if Awf and Awa were held fixed and only Nc and Ns were allowed to vary

along with stack length (and current)? The answer is summarized in Table 6.10 wherein

the compromise DSP for the family of motors for instantiating the PPCEM platform,

Figure 6.10, has been modified a third time to allow only Nc, Ns, L and I to vary from the

platform specifications. Comparison of the data in Table 6.10 with Table 6.8 (the PPCEM

225

instantiations with Awf and Awa varying also) reveals that both families of motors are nearly

identical in terms of their performance characteristics (mass and efficiency).

Table 6.10 Third PPCEM Universal Motor Family with Varying Numbersof Turns and Stack Lengths


Motor Nc Ns [mm2] [mm2] [Amp] [cm] [mm] [cm] [Nm] [W] [%] [kg]1 1104 40 0.376 0.241 3.49 2.59 6.66 1.18 0.05 300 74.6 0.4232 1120 68 ò ò 3.61 ò ò 1.37 0.10 ò 72.2 0.4663 1126 79 ò ò 3.66 ò ò 1.44 0.125 ò 71.2 0.4834 1131 87 ò ò 3.72 ò ò 1.51 0.15 ò 70.2 0.5005 1119 84 ò ò 3.88 ò ò 1.81 0.20 ò 67.2 0.5756 1091 81 ò ò 4.03 ò ò 2.03 0.25 ò 64.6 0.6487 1060 77 ò ò 4.20 ò ò 2.20 0.30 ò 62.1 0.7178 1025 74 ò ò 4.38 ò ò 2.42 0.35 ò 59.6 0.7859 987 71 ò ò 4.56 ò ò 2.60 0.40 ò 57.2 0.85110 909 65 ò ò 4.97 ò ò 2.91 0.50 ò 52.4 0.979

The mass and efficiency of the four families of motors (the benchmark, the PPCEM

varying only L, the PPCEM varying L, Nc, and Ns, and the PPCEM varying L, Nc, N s,

Awf, and Awa) are summarized in Table 6.11 to facilitate comparison. The percentage

difference (% Diff.) listed in the table is a comparison of each PPCEM instantiation against

the corresponding benchmark motor, i.e., the performance characteristics of a motor which

has been individually designed and optimized. As stated previously, motors which achieve

their respective mass and efficiency targets of 0.50 kg and 70.0% are considered equivalent

solutions even if the motor has a lower mass or higher efficiency. In this regard, some

observations based on the data in Table 6.11 are as follows:

• As more variables are allowed to vary from the platform specifications, the better

the performance of the individual motors; the tradeoff is that less and less is

common between the motors within the product family. It then becomes a decision

of the engineering designers and management to evaluate the tradeoffs between

commonality and performance to determine the best family to pursue. This

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reinforces the statement that the PPCEM facilitates generating these options but is

not necessarily used to select the best one since it requires information which is

beyond the scope of this investigation.

Table 6.11 Efficiency and Mass of Benchmark Motors and PPCEM MotorPlatform Families

PPCEM PPCEM PPCEMBenchmark

MotorsVary Lonly

Vary L,Nc, Ns

Vary L, Nc,Ns, Awf, Awa

Motor η [%] η [%] % Diff. η [%] % Diff. η [%] % Diff.1 71.4 76.8 equiv. 74.6 equiv. 74.7 equiv.2 70.6 72.2 equiv. 72.2 equiv. 72.1 equiv.3 70.0 70.0 equiv. 71.2 equiv. 71.1 equiv.4 70.5 67.9 -3.7% 70.2 equiv. 70.1 equiv.5 67.9 63.9 -5.9% 67.2 1.0% 67.5 0.6%6 64.9 60.2 -7.2% 64.6 0.5% 64.6 0.5%7 62.2 56.8 -8.7% 62.1 0.2% 62.2 0.0%8 59.8 53.6 -10.4% 59.6 0.3% 59.9 -0.2%9 57.7 50.5 -12.5% 57.2 0.9% 57.7 0.0%10 55.3 44.8 -19.0% 52.4 5.2% 53.8 2.7%

Motor M [kg] M [kg] % Diff. M [kg] % Diff. M [kg] % Diff.1 0.500 0.380 equiv. 0.423 equiv. 0.397 equiv.2 0.500 0.520 equiv. 0.466 equiv. 0.456 equiv.3 0.500 0.576 equiv. 0.483 equiv. 0.477 equiv.4 0.500 0.625 25.0% 0.500 equiv. 0.499 equiv.5 0.558 0.703 26.0% 0.575 -3.0% 0.568 -1.8%6 0.639 0.759 18.8% 0.648 -1.4% 0.646 -1.1%7 0.712 0.797 11.9% 0.717 -0.7% 0.712 0.0%8 0.777 0.820 5.5% 0.785 -1.0% 0.774 0.4%9 0.837 0.830 -0.8% 0.851 -1.7% 0.833 0.5%10 0.985 0.820 -16.8% 0.979 0.6% 0.941 4.5%

• In this example, the PPCEM instantiations when Nc, Ns, Awf, and Awa are allowed

to vary in addition to the stack length yields an equivalent family of motors to the

family of individually designed benchmark motors. Varying only Nc and Ns in the

PPCEM family while holding Awf and Awa fixed at the platform specification also

yields a good family of motors with minimal sacrifice in performance (mass and

efficiency) when compared to the benchmark family of motors.

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In light of these observations, are the solutions obtained from the PPCEM useful?

The answer is undoubtedly yes. The initial family of motors obtained using the PPCEM

meets the range of torque and power requirements which have been specified for the

product family. However, because these motors are based on a common product platform

and vary only in stack length (and current), the motors lose, on average, 10% in both

efficiency and mass for the specified targets when compared to a family of individually

designed benchmark motors. In an effort to reflect a more realistic set of motors, by

allowing the number of turns in the armature and field to vary in addition to the stack

length, the family of motors obtained using the PPCEM are essentially identical to the

equivalent family of benchmark motors. The necessary torque and power requirements are

met with minimal sacrifice in performance (mass and efficiency). If the wire cross-

sectional areas of the wire in the field and armature are allowed to vary in addition to the

number of wire turns in each and the stack length, then the family of motors obtained using

the PPCEM are identical, for all intents and purposes, to the corresponding family of

benchmark motors. Thus, the PPCEM has greatly facilitated generating a variety of

options which the engineering designers and managers can select from based on what is

best for the company.

Are the time and resources consumed within reasonable limits? In general, fewer

analysis calls are required to obtain the PPCEM family of motors than the benchmark

family of motors. To obtain the benchmark motor family, ten optimization problems must

be solved where each optimization involves finding the best settings of eight design

variables. For the PPCEM platform instantiations, the initial family of motors requires

solving one optimization problem to find the values of nine design variables (which

includes mean and standard deviation of stack length for the platform) followed by solving

ten optimization problems involving as few as two (current, I, and stack length, L) and as

many as six (current, I, stack length, L, # armature turns, Nc, # field turns, Ns, cross-

228

sectional areas of the field wire, Awf, and armature wire, Awa) design variables where the

size of the subsequent optimization problems is dependent on the number of design

variables which are being instantiated for each motor from the platform design.

Because a gradient based algorithm—the GRG algorithm in OptdesX—is being

used to optimize the motor platform and individual motors, each iteration of the

optimization requires one analysis call to evaluate the current iterate and two evaluation calls

per design variable to estimate the gradient to determine the next iterate. For the family of

benchmark motors, the number of analysis calls is approximately:

(10 motors)n iterations

motor

1 analysis

iteration

+

2 analyses

(d.v.)(iteration)

8 d.v.( )

= 170n [6.40]

where d.v. is an abbreviation for design variable, and n is the average number of iterations

required to solve each optimization. For the PPCEM family of motors, the number of

analysis calls required to obtain the family of motors is approximately:

(m iterations)1 analysis

iteration

+

2 analyses

(d.v.)(iteration)

9 d.v.( )

+ (10 motors)•

k iterations

motor

1 analysis

iteration

+

2 analyses

(d.v.)(iteration)

2 d.v.( )

= 19m + 50k [6.41]

where m is the number of iterations required to find the PPCEM platform design, and k is

the number of iterations required to find each instantiation of the PPCEM platform. On

average, n ≈ 10, m ≈ 12, and k ≈ 5; therefore, the average number of analysis calls

required to obtain the benchmark motor designs is 170•10 = 1700 while the average

number of analysis calls required to find the PPCEM motor designs is 19•12 + 50•5 = 478,

a difference of 1222 analyses. So even if as many as six design variables are allowed to

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vary between PPCEM instantiations from the product platform, then by replacing (2 d.v.)

in Equation 6.41 with (6 d.v.), it would still only require about 19•12 + 130•5 = 878

analysis calls which is slightly more than half the analysis calls required to find the

benchmark motor designs, and this estimate does not even take into consideration the fact

that each optimization is solved from three different starting point. So by using the

PPCEM to first find a common motor platform design and then scaling the platform in the

stack length, an equivalently good family of motors can be obtained with fewer analysis

calls than if each motor were designed individually. Plus, there is the added benefit of the

family of motors found using the PPCEM have more in common with one another.

Is the work grounded in reality? The problem formulation has been developed from

an electric motor for a 3/8” variable speed, reversible, corded Black & Decker drill, model

#7190. The drill is rated at 288 W and 1200 rpm, drawing 3.5 Amps of current and is at

the low-end of their product line. The gear reduction on the motor is estimated to be 10:1;

therefore, the operating speed of the motor itself is 12,000 rpm. Using Equation 6.39, the

operating motor torque is computed as being 0.23 Nm. Assuming an input voltage of 115

V, the input power is 402.5 W when drawing 3.5 Amps of current (see Equation 6.2).

Since the output power is 288 W, the efficiency of the motor is computed using Equation

6.14 and is 71.6%. The mass of the motor is 0.496 kg. Consequently, the target values of

300 W power, 70% efficiency, 0.5 kg, and 0.05 Nm to 0.5 Nm of torque are built around

the performance ratings for this motor, and the motor from this drill is taken as the mid-

range motor in a family of universal motors for this example.

The pertinent motor specifications (design variable settings) for this torque and

power rating are as follows:

• Stack length, L = 2.84 cm • Air gap length, lgap = 0.7 mm

• Number of field turns, Ns = 135 • Stator thickness, t = 4.2 mm

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• Number of armature turns, Nc ≈ 700-800 • Motor radius, r = 2.85 cm

• Field wire cross-sectional area, Awf = 0.13 mm2

• Armature wire cross-sectional area, Awa ≈ 0.10 mm2

It is difficult to count the number of armature turns in the actual motor; the best guess is

around 750 turns. Can the analytical model be used to predict the performance of the actual

motor given these specifications? Unfortunately, the analytical model developed in this

chapter cannot be used to predict the performance to the actual motor given these

specifications. There are two discrepancies which arise between the model and the actual

motor. First, the real motor from the drill is not a true universal motor since it appears to

be designed for AC use only (as stated on the exterior of the box). Second, the number of

poles on the armature is twelve; in a real universal motor, the number of poles in the

armature is typically two which is an important assumption used when deriving the torque

equations (Equations 6.19-6.24) for the motor. However, these specifications can still be

used to gauge the solutions obtained from the analytical model.

In general, the values for stator thickness, motor radius, stack length, and current

are in close agreement with the values obtained using the PPCEM, see Table 6.5, Table

6.6, Table 6.8, and Table 6.10. The number of field turns is on the low end while the

number of armature turns is on the high end compared to this actual motor. Finally, the

wire cross-sectional areas in the actual motor are slightly smaller than the values obtained

using the analytical model for the motor. These discrepancies are discussed in more detail

in the context of the limitations and shortcomings of the model and problem formulation.

What are the limitations of the analytical model and problem formulation developed

in this chapter? There are two noteworthy shortcomings to the analytical model and

problem formulation presented in this chapter for the family of universal electric motors.

First, the speed of the motor has not been taken into consideration in the problem

formulation as discussed previously. By specifying power and torque requirements for

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each motor, the resulting rotational speed of the motor is fixed through Equation 6.40. It is

important to ensure that as the torque of the motor increases that the power also increases

so that the operating speed of the motor does not decrease significantly. For purposes of

this demonstration, this is not a major concern; however, a more realistic representation of

the motor problem formulation would take this into consideration.

The second notable shortcoming of the analytical model relates to the large numbers

of armature turns in each motor and the related discrepancies between wire cross-sections

and number of field turns. Can 1062 turns of 0.241 mm2 wire be packed into a cylindrical

volume with a radius of ≈ 2.0 cm (the motor radius minus the thickness of the PPCEM

platform listed in Table 6.4) and a length of 2.62 cm? The answer depends on how tightly

wires can be packed around the armature and how much steel is used within the poles of

armature. The complexity of such an analysis was considered to be beyond the scope of

this example; however, it is recommended to include these space considerations in future

studies in order to improve the fidelity of the model. Furthermore, decreasing the number

of armature turns is liable to increase the required number of field turns (which are

considered to be low given that the Black & Decker motor has 135 turns) in order to

maintain sufficient magnetic flux through the motor. Placing space constraints on the

amount of wire in the field and armature should also have the effect of decreasing Awf and

Awa in addition to making the number of field and armature windings more realistic.

Finally, do the benefits of the work outweigh the cost? The answer to this last

question is also affirmative. Regardless of whether the family of benchmark motors or the

PPCEM family was being designed, the analytical model would still have been constructed.

The only addition to the model required to use the PPCEM is deriving an expression for the

standard deviation of torque based on variations in the motor stack length. This is achieved

by means of a first-order Taylor series approximation, Equation 6.37, which requires

taking the derivative of the torque equation, Equation 6.19, with respect to stack length.

232

Once this is accomplished, using the PPCEM yields a family of motors with high

commonality and negligible performance losses in fewer analyses than an equivalent family

of individually designed benchmark motors. Thus, the PPCEM facilitates the exploration

of product platform concepts which can be scaled into an appropriate family of products,

providing initial “proof of concept” that the PPCEM does what it is intended to do. A look

ahead to the next example to verify further this observation is offered in the next section.


The design of a family of universal electric motors has been utilized to demonstrate

how the PPCEM can be applied to a fairly small-scale, parametrically scalable problem.

Furthermore, the application of the market segmentation grid to help identify and map

vertical leveraging of a product platform for a wide range of performance/price tiers within

a given market segment is illustrated in this example. From the simple analytical model

derived in Section 6.1.3, it has also been shown in this example that for small-scale

problems such as this one, Step 3 of the PPCEM—building metamodels—is not always

necessary provided that analytical expressions exist, or can be derived, to relate variations

in the scale factor (the stack length of the motor in this example) to variations in product

performance (goals and constraints). As evidenced by the discussion in the previous

section comparing the individually designed benchmark motors and the PPCEM platform

motors, the PPCEM has been implemented to design a family of universal motors (based

on a scalable product platform) which is capable of meeting a wide range of torque

requirements with minimal compromise in efficiency and mass. A summary of Chapter 6

and a preview of Chapter 7 is offered in Figure 6.13.

233

Family of General Aviation Aircraft

Platform



Chp 7Chp 6

Demonstrated:• vertical leveraging• parametric scale

factor: stack length• no metamodels• robust design

implementation:separate goals for“bringing mean ontarget” and “minimizethe variation”

• OptdesX solverTest:• H1, SH1.1, SH1.2,

and SH1.3

Demonstrate:• horizontal leveraging• configurational scale

factor: # passengers• kriging metamodels

to facilitate robustdesign and concept exploration

• robust designimplementation:design capabilityindices

• use DSIDESTest:• H1, SH1.1, SH1.2,

SH1.3, and H2


As shown in Figure 6.13, in Chapter 7 the PPCEM is applied to a larger, more

complex problem, namely, the design of a family of General Aviation aircraft. In the

General Aviation aircraft example, all of the steps in the PPCEM are implemented,

including metamodeling in Step 3. In addition, a product variety tradeoff study is

performed to verify further Hypotheses 1, its related sub-hypotheses, and Hypothesis 2.

The details of the General Aviation aircraft example are discussed at the beginning of the

next chapter.

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7. CHAPTER 7

DESIGN OF A FAMILY OF GENERALAVIATION AIRCRAFT

In this chapter, the PPCEM is applied in full to the design of a family of General

Aviation aircraft (GAA) for final verification of the method. The layout of this chapter

parallels that of the universal motor case study in the previous chapter. Motivation and

background for the GAA are given in Section 7.1, along with Step 1 of the PPCEM,

namely, creation of an appropriate market segmentation grid for the family of GAA to

accommodate the problem requirements. Based on the desired horizontal leveraging

strategy, the scale factor for the problem is taken as the number of passengers on the

aircraft as explained in Section 7.2; the control factors and responses for the family of GAA

also are described in Section 7.2 as part of Step 2 of the PPCEM. Kriging metamodels

then are created for the mean and standard deviation of each response for the family of

GAA in Section 7.3. After validating the accuracy of the metamodels, a compromise DSP

for the family of aircraft is formulated in Section 7.4 and exercised in Section 7.5 to

develop the GAA platform portfolio. Ramifications of the results and comparison of the

PPCEM solutions against individually designed benchmark aircraft are discussed in Section

7.6. In addition, a product variety tradeoff study is performed, making use of the non-

235

commonality indices (NCI) and performance deviation indices (PDI) to examine the

tradeoff between the two within the family of GAA.

As shown in Table 7.1, all but Hypothesis 3 are verified further in this chapter. As

mentioned in the preceding paragraph, the market segmentation grid is used to identify a

horizontal leveraging strategy for the GAA product family providing further verification of

Sub-Hypothesis 1.1. The use of design capability indices is demonstrated to aggregate the

product family specifications (SH1.3) and facilitate the development of a product platform

which is robust (SH1.2) to variations in the number of passengers on the aircraft, the scale

factor. Furthermore, kriging metamodels are exploited in this chapter to facilitate the

implementation of robust design within the PPCEM, providing further support for

Hypothesis 2. Space filling designs are utilized to create these metamodels; however, only

one type of design is used—an orthogonal array—and Hypothesis 3 is not tested.

Table 7.1 Hypotheses Tested in Chapter 7

Hypothesis Chp 6 Chp 7H1 Product Platform Concept Exploration

Method X XSH1.1 Usefulness of market segmentation

grid X XSH1.2 Robust design of scalable product

platform X XSH1.3 Aggregating product family

specifications X XH2 Utility of kriging for metamodeling

deterministic computer experiments X

A summary of the findings and lessons learned in this example are summarized at

the end of the chapter in Section 7.7. The objective in the summary is to describe how

Hypothesis 1 has been verified further through this example. A brief look ahead to Chapter

8 is given in this last section.

236

7.1 STEP 1: DEVELOPMENT OF THE MARKET SEGMENTATION GRID

Before developing the market segmentation grid, an overview of the General

Aviation aircraft example is given in the next section. This is followed in Section 7.1.2

with a brief overview of the phases of aircraft design. The market segmentation grid for

the General Aviation aircraft example is presented in Section 7.1.3 along with the baseline

design which provides the starting point for designing the family of GAA.

7 .1 .1 Overview of the General Aviation Aircraft Example Problem

What is a General Aviation aircraft? The term General Aviation encompasses all

flights except military operations and commercial carriers. Its potential buyers form a

diverse group that include weekend and recreational pilots, training pilots and instructors,

traveling business executives and even small commercial operators. Satisfying a group

with such diverse needs and economic potential poses a constant challenge for the General

Aviation industry because it is impossible to satisfy all of the market needs with a single

aircraft. The present financial and legal pressures endured by the General Aviation sector

makes small production runs of specialized models unprofitable. As a result, many

General Aviation aircraft are no longer being produced, and the few remaining models are

beyond the financial capability of all but the wealthiest buyers.

In an effort to revitalize the General Aviation sector, the National Aeronautics and

Space Administration (NASA) and the Federal Aviation Administration (FAA) recently

sponsored a General Aviation Design Competition (NASA and FAA, 1994). For this

work, a General Aviation aircraft (GAA) is defined as follows:

• a fixed-wing, single-engine, single-pilot, propeller driven aircraft,

• carries 2-6 passengers,

• cruises at 150-300 kts, and

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• has a range of 800-1000 nautical miles.

One solution to the GAA crisis is to develop a family of aircraft which can be

adapted easily to satisfy distinct groups of customer demands. Therefore, the purpose in

this example is to develop the following:

a family of aircraft scaled around the two, four, and six seater configurations using

the PPCEM. This family of General Aviation aircraft must be capable of satisfying

the diverse demands of the General Aviation buyers at an affordable price and

operating cost while meeting desired technical and economic considerations.

In order to realize this objective and demonstrate the application of the PPCEM, a brief

overview of aircraft design is given in the next section. This is followed in Section 7.1.3

with the development of the market segmentation grid—Step 1 of the PPCEM—for the

family of GAA.

7 .1 .2 Brief Overview of Aircraft Design

How does one go about designing an aircraft? Aircraft design traditionally is

divided into three phases, namely, conceptual, preliminary, and detailed design as

illustrated in Figure 7.1. If manufacturing design is considered as a part of the design

process, design for production can be added as a fourth phase. The first two phases of

aircraft design, the conceptual and preliminary phases, are sometimes combined and called

advanced design or synthesis in the aerospace industry, while follow-on phases are called

project design or analysis. More detailed descriptions of the decisions made in each phase

and the disciplines involved in aircraft design can be found in, e.g., (Bond and Ricci,

1992). Specifically, the efforts in this example are directed toward utilizing the computer

within the traditional conceptual phase of aircraft design as it is defined in (Nicolai, 1984).

Conceptual Design Phase: In this phase, the general size and configuration of the

aircraft is determined. Parametric trade studies are conducted using preliminary

estimates of aerodynamics and weights to determine the best wing loading, wing

238

sweep, aspect ratio, thickness ratio, and general wing-body-tail configuration.

Different engines are considered and the thrust loading is varied to obtain the best

match of airframe and engine. The first look at cost and manufacturing possibilities

is made at this time. The feasibility of the design to accomplish a given set of

mission requirements is established, but the details of the configuration are subject

to change.

Mission Requirements

Conceptual Baseline

Allocated Baseline

Production Baseline

Top-Level Design Specifications• General arrangement & performance• Representative configurations• General internal layout

• System specifications• Detailed subsystems• Internal arrangements• Process design

AdvancedDesign

ProjectDesign

Production & Support

Preliminary Design

Conceptual Design

Detailed Design

used to develop a scaleable aircraft platform

Platform


Figure 7.1 Phases of Aircraft Design (Schrage, 1992)

In general, in the early stages of aircraft design, the aircraft concept is synthesized

at the system level based on mission requirements or market opportunities. As a result, the

conceptual baseline is developed and represented by a set of top-level design specifications

as illustrated in Figure 7.1. Top-level design specifications are the descriptions of

system/subsystem concepts or the definitions of the complex system at the

239

system/subsystem level. Top-level design specifications are used as the starting point for

the preliminary design at the subsystem level, and form the basis for the specifications

(functional properties) that are developed during the preliminary design phase. These top-

level design specifications include variables such as aspect ratio, thickness ratio, and wing-

body-tail configuration. The top-level design specifications can be continuous (e.g., aspect

ratio = 7-11) or they can be discrete design concepts (e.g., single- or twin-engine, single,

number of propeller blades, high or low wing, and fixed or retractable landing gear). The

settings of the top-level design specifications define the conceptual baseline which becomes

the configuration input for preliminary design, where the system is decomposed for more

sophisticated analysis by discipline, subsystem, or component. The reader is referred to

(Koch, 1997) for more discussion on the resulting “requirements flowdown” process.

Several synthesis and analysis programs have been created to facilitate the

conceptual and preliminary design of aircraft and hence the development of these top-level

design specifications. One such program is entitled FLOPS (FLight OPtimization System

(McCullers, 1993)). FLOPS is a multidisciplinary system of computer programs for

conceptual and preliminary design and evaluation of advanced aircraft concepts. Another

program is called GASP (General Aviation Synthesis Program (NASA, 1978)). GASP is

a computer program which performs tasks specifically associated with the conceptual

design of General Aviation aircraft; consequently, it has been selected as the synthesis

program for use in this example.

What is GASP and how does it work? GASP is a synthesis and analysis computer

program which facilitates parametric studies of small aircraft. GASP specializes in small

fixed-wing aircraft employing propulsion systems varying from a single piston engine with

fixed pitch propeller through a twin turboprop/turbofan powered business or transport type

aircraft. GASP contains an overall control module and six technology submodules which

240

perform the various independent studies required in the design of General Aviation or small

transport type aircraft. The six technology submodules include the following:

• Geometry Module: By inputting parameters such as the number of passengers,

aspect ratio, taper ratio, sweep angles and thicknesses of wing and tail surfaces, the

dimensions of the aircraft components are calculated.

• Aerodynamics Module: Lift and drag coefficients, lift curve slope computation

due to aspect ratio, sweep angle, Mach number, and induced drag are determined.

• Propulsion Module: Reciprocating and rotating combustion engines,

turboprop, turbofan, turbojet are simulated. The results provide engine thrust and

fuel flow at any flight condition using performance data for the specific engine

used.

• Weight and Balance Module: Weight trend coefficients for gross weight,

payload, and aircraft geometry are used to estimate the center of gravity, travel of

the aircraft, fuel tank size, and compartment weight.

• Mission Performance Module: All of the mission segments such as taxi, take

off, climb, cruise and landing are analyzed including the total range. The program

also calculates the best rate of climb, high speed climb, and other characteristics.

• Economics Module: Manufacturing and operating costs are estimated based on

the date of inception of the program (i.e., in 1970's dollars).

Input variables for GASP are general indicators of aircraft type, size, and

performance. The numerical output from GASP includes many aircraft design

characteristics such as range, direct operating cost, maximum cruise speed, and lift-to-drag

ratio. For conceptual design of an aircraft, GASP is used to find appropriate settings for

the top-level design specifications that satisfy the design requirements. By utilizing GASP

as the simulation package within the PPCEM, the PPCEM can be used to develop a set of

top-level design specifications for a suitable aircraft platform which is good for the entire

family of aircraft as shown in Figure 7.1. The first step in the PPCEM is to develop the

market segmentation grid which is accomplished in the next section.

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7 .1 .3 The Market Segmentation Grid for the GAA Example Problem

As stated at the beginning of this section, the objective in this example is to design a

family of GAA around the two, four, and six seater configurations. The market

segmentation grid shown in Figure 7.2 depicts a potential leveraging strategy for the GAA

example. The goal is to design a low end aircraft platform which can be leveraged across

three different market segments which are defined by the capacity of the aircraft (i.e., two

people, four people, and six people) similar to the Boeing 747 series of aircraft (Rothwell

and Gardiner, 1990). Each aircraft could eventually be vertically scaled through the

addition and removal of features to increase its performance and attractiveness to a

customer base willing to pay a higher price. At this stage, however, only a low-end

platform is to be developed.

Low End

Mid-Range

High End

GAA Platform Design

2 Seater 4 Seater 6 Seater

Configurationalscale factor:fuselage length = f(# passengers)

Low End Platform Leveraging

Figure 7.2 GAA Market Segmentation Grid

The baseline configuration is derived from an existing General Aviation aircraft,

namely, the Beechcraft Bonanza B36TC. The Bonanza is a four-to-six seat, single-engine

business and utility aircraft as illustrated in Figure 7.3 and is one of the most popular GAA

sold in recent years. It is powered by a 300 horsepower, turbocharged engine and is

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capable of cruising at 25,000 ft with a speed of 200 knots and a minimum range of 956

nautical miles (at 79% of power). Furthermore, Bonanza’s mission and performance

characteristics are close to those specified in the GAA competition (NASA and FAA,

1994). Taking the Bonanza as the starting point, several calculations can be performed to

determine the GASP input data, specifically for aerodynamics, engine performance, and

stability control parameters, according to the mission profile. The aircraft performance

characteristics of the Beechcraft Bonanza B36TC specifications when used in GASP are

summarized in Table 7.2, and the corresponding top-level design specifications for this

baseline aircraft are listed in Table 7.3 where dimensions in bold are the design variables.

Figure 7.3 Pictorial Representation of Baseline Aircraft

243

Table 7.2 Performance of the Baseline Model in GASP

Maximum cruise speed 210 kts @ 25,000 ftMaximum range with maximum fuel 2423 n.m.Maximum range with maximum payload 715 n.m.Lift off distance with maximum payload 1310 ftAll engine distance to 50 ft with maximum load 2183 ftLanding distance from 50 ft 1120 ft

Table 7.3 Baseline Model Specifications

Group Key Product Characteristics DimensionFuselage Length 30.62 ft

Width 4.33 ftSeat Width 20 in.Tail Length to Diameter Ratio 3.09Wetted Area 325 ft2

Wing Aspect Ratio 7.88Area 186.5 ft2

Span 38.3 ftMean Chord 5.09 ft1/4 Chord Sweep 4.0°Taper Ratio 0.46Root Thickness 0.15Wing Loading 20.5 lb/ft2

Wing Fuel Volume 180 galHorizontal Tail Aspect Ratio 5.08

Area 45.2 ft2

Span 15.15 ftMean Chord 3.09 ftThickness 0.09

Vertical Tail Aspect Ratio 1.07Area 20.8 ft2

Span 4.71 ftMean Chord 4.61 ftThickness 0.07

Engine Power 350 HP turbochargedStatic Thrust/Wt 0.339Activity Factor 110

Propeller Diameter 6.30 ft# of Blades 3

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The flight mission for the family of GAA in this example is illustrated in Figure

7.4. As specified in the GAA competition guidelines (NASA and FAA, 1994), a General

Aviation aircraft is required to fly at 150-300 kts (Mach 0.24 to 0.48) for a range of 800-

1000 nautical miles. The mission profile shown in Figure 7.4 has a (baseline) cruise speed

of Mach 0.31 (≈ 200 kts) and a range of 900 n.m. (nautical miles). In the diagram, FAR

23 represents Part 23 of the Federal Aviation Requirement which designates acceptable

noise levels during aircraft takeoff and landing as determined by the FAA.

Takeoff @ sea level,FAR 23

Cruise speed Mach 0.31900 n.m. @ 7500 ft

Landing @ sea level,FAR 23

45 min. reserve fuel

Figure 7.4 GAA Mission Profile

Based on the GAA market segmentation grid in Figure 7.2, the scale factor for the

GAA platform is conceptual/configurational in nature. The aircraft platform is to be

“scaled” around the number of people on the aircraft; hence, the number of passengers is

the scale factor in this problem. The effect of the number of passengers on the length of the

fuselage and sizing of the aircraft is discussed more in the next section wherein the targets

and requirements for each aircraft and the design variables for this example are described.

7.2 STEP 2: GAA FACTOR CLASSIFICATION

Having created the market segmentation grid and identified an appropriate

leveraging strategy and scale factor, the next step in the PPCEM is to classify the factors

245

within the GAA problem. The general configuration of each aircraft has been fixed at three

propeller blades, high wing position, and retractable landing gear based on previous work

(Simpson, 1995). The design variables (i.e., control factors) and corresponding ranges of

interest in this study are as follows:

1. Cruise speed, CSPD ∈ [Mach 0.24, Mach 0.48]; baseline is Mach 0.31

2. Aspect ratio, AR ∈ [7, 11]; baseline is 7.88

3. Propeller diameter, DPRP ∈ [5.0 ft, 5.96 ft]; baseline is 6.3 ft

4. Wing loading, WL ∈ [19.0 lb/ft2, 25.0 lb/ft2]; baseline is 20.5

5. Engine activity factor, AF ∈ [85, 110]; baseline is 100

6. Seat width, WS ∈ [14.0 in, 20.0 in]; baseline is 20 in.

A brief description of the importance and effects of each of these variables is included in

Section F.1.

There are a total of nine responses (i.e., requirements and goals) which are of

interest for each aircraft: takeoff noise, direct operating cost, ride roughness, empty weight

and fuel weight, purchase price, and maximum cruise speed, flight range and lift/drag ratio.

In general, it is desired to find settings of the design variables which:

• lower direct operating cost, purchase price, empty weight and fuel weight to their

targets;

• raise maximum cruise speed, flight range, and lift/drag ratio to their targets; and

• meet constraints on the maximum takeoff noise, ride roughness, direct operating

cost, empty weight, and fuel weight, and minimum flight range.

The constraint values and target values for the goals employed in this example are listed in

Table 7.4 and Table 7.5, respectively. As such, these constraints and targets define each

market niche for each of the three aircraft. As shown in Table 7.4 the constraint values for

take-off noise, direct operating cost, ride roughness aircraft empty weight, and range are

246

the same for each aircraft within the family; only the fuel weight constraint varies for each

aircraft, allowing larger aircraft to carry more fuel.

Table 7.4 Constraints for the Two, Four, and Six Seater GAA

Constraints Acronym 2 Seater 4 Seater 6 SeaterMaximum take-off noise NOISE 75 db 75 db 75 dbMaximum direct operating cost DOC $80/hr $80/hr $80/hrMaximum ride roughness coefficient ROUGH 2.0 2.0 2.0Maximum aircraft empty weight WEMP 2200 lbs 2200 lbs 2200 lbsMaximum aircraft fuel weight WFUEL 450 lbs 475 lbs 500 lbsMinimum flight range RANGE 2000 nm 2000 nm 2000 nm

The goal targets which define each market niche are listed in Table 7.5. A

compromise DSP is used to determine the settings of the six design variables which lower

fuel weight, empty weight, direct operating cost, and purchase price to their targets or

below while raising maximum lift/drag, cruise speed, and range to their targets. The

compromise DSP formulation for the GAA problem is given in Section 7.4.

Table 7.5 Goal Targets for the Two, Four, and Six Seater GAA

Goal Targets Acronym 2 Seater 4 Seater 6 SeaterAircraft fuel weight [lbs] WFUEL 450 400 350Aircraft empty weight [lbs] WEMP 1900 1950 2000Direct operating cost [$/hr] DOC 60 60/hr 60Purchase price [$] PURCH 41,000 42,000 43,000Maximum lift/drag ratio LDMAX 17 17 17Maximum cruise speed [kts] VCRMX 200 200 200Maximum range [nm] RANGE 2500 2500 2500

Based on the leveraging strategy shown in Figure 7.2, the number of people in the

aircraft is taken as the scale factor in the design process, ranging from a minimum of 2 to a

maximum of 6. Furthermore, it is assumed that the demand for the aircraft is uniform;

247

therefore, the scale factor—the number of passengers—is assumed to be uniformly

distributed and so are the corresponding responses. Taking the number of passengers as a

scale factor, the length of the central portion of the fuselage of the aircraft is scaled

automatically within GASP to accommodate the necessary number of passengers (plus one

pilot). Because the length of the aircraft is fixed once the number of people is specified, the

mean and variance of the scale factor are known in this example unlike in the universal

motor example.

Based on this factor classification scheme, the P-diagram for the GAA example

problem is illustrated in Figure 7.5.

X = Control FactorsCruise speedAspect Ratio

Propeller DiameterWing Loading

Engine Activity FactorSeat Width

Y = ResponsesTakeoff Noise

Ride RoughnessEmpty WeightFuel Weight

Purchase PriceDirect Operating Cost

Maximum RangeMaximum Speed

Maximum Lift/dragS = # passengers

GASP

Figure 7.5 P-Diagram for GAA Example Problem

As shown in the figure, there are six control factors (design variables), one scale factor (the

number of passengers), and nine responses (constraints and goals). The process of

constructing kriging metamodels which relate the control and scale factors to each of the

responses is explained in the next section.

7.3 STEP 3: BUILD AND VALIDATE METAMODELS

The next step in the PPCEM is to build and validate metamodels of the

analysis/simulation routine, i.e., GASP. In particular, robustness models are constructed

248

for each of the nine responses, yielding a total of 18 metamodels: one metamodel for the

mean and a variance of each response. Why use metamodels in the GAA example? The

impetus is two-fold. First, GASP provides a “black-box” type analysis for sizing an

aircraft. The computation time for GASP is about 45 seconds which does not necessarily

warrant the use of metamodels; however, after multiplying this number by three—the

number of aircraft in the family—and considering the number of design scenarios that are

to be considered, the computational expense adds up quickly. Moreover, it is difficult to

estimate the mean and variance of each response for the family of aircraft without the

metamodels. It is much more efficient to build metamodels for the mean and deviation of

each response and use them to search the design space than it is to use GASP directly in the

search for a good platform design.

The product array approach is employed to build kriging metamodels of the mean

deviation of each response to variations in the number of passengers in each aircraft. This

approach is illustrated in Figure 7.6. The outer array is based on a randomized orthogonal

array of 64 points (n = 64). The use of the randomized orthogonal array is based,

primarily, on ease of generation and available sample sizes; it is also based, in part, on its

performance in the kriging/DOE study in Chapter 5 even though a six variable test problem

was not utilized in the study. To compare the sample size, a half-fraction CCD for six

factors would contain 45 points, and a full-fraction CCD would contain 77 points. The

kriging models employ the Gaussian correlation function, Equation 2.16, because this

correlation function yielded the lowest RMSE and max. error, on average, in the

kriging/DOE study in Chapter 5 (see Section 5.3 in particular).

249

Sampledesign space

#P

AX

11

23

35

# cspd ar dprp wl af ws

1 0.24 7.0 5.9 20 85 172 0.24 7.0 5.0 19 91 20

• •• •• •

n 0.48 11.0 5.0 20 85 17

µj,1 σj,1

µj,2 σj,2

• • •

µj,n σj,n

Inner Array

Out

er A

rray

yj,1,1 yj,1,3 yj,1,5

yj,2,1 yj,2,3 yj,2,5

•••

yj,n,1 yj,n,3 yj,n,5µ ˆ y = f(cspd,ar,dprp,wl,af,ws)σ ˆ y = f(cspd,ar,dprp,wl,af,ws)

Responses

j =

noisewempdoc

roughwfuelpurchrangevcrmxldmax

Kriging models for each response

j

j

PlatformScale Factor(s)

Figure 7.6 Product Array Approach for Constructing GAA Kriging Models

Because there is only one scale factor which has three possible settings, the inner

array shown in Figure 7.6 simply contains three runs, one for each possible value of the

scale factor. Hence, GASP is executed 3n times in order to build the kriging models for

the mean and deviation of the GAA responses. Notice that the variable PAX—the number

of passengers—varies from 1 to 5 in the figure. This is because the total number of people

on the aircraft is equal to the number of passengers plus 1 pilot; varying PAX from one to

five is the same as varying the number of people on the aircraft from two to six, allowing a

family of aircraft to be design around the two, four, and six seater configurations.

After varying the number of passengers for each combination of the design

variables as specified by the outer array, the mean and standard deviation of each response

are computed for each run using Equations 7.1 and 7.2 which are as follows:

• Mean: µy j,i=

y j,i,1 + y j,i,3 + y j,i,5

3, j = {noise, wemp, ..., ldmax}, i = {1, 2, ..., n} [7.1]

250

• Std. Dev.: σy j,i=

y j,i,1 − y j,i,5

12, j = {noise, wemp, ..., ldmax}, i = {1, 2, ..., n} [7.2]

Computation of the standard deviation assumes a uniform distribution of the response

because the number of passengers is assumed to vary uniformly over the design space. As

an example, the mean and standard deviation of the direct operating cost, DOC, for the 3rd

experimental design is estimated as follows:

µy DOC,3=

yDOC,3,1 + yDOC,3,3 + yDOC,3,5

3[7.3]

σyDOC,3=

yDOC,3,1 − yDOC,3,5

12[7.4]

It is in this manner that the means and deviations for each response for each experimental

run in the outer array are computed for a given experimental design. Kriging metamodels

then are constructed for the mean and deviation of each response, resulting in 18

metamodels. The kriging algorithm described in Section A.2.1 is used to fit the model; the

fitted values (MLE estimates) for the “best” kriging model for each response are listed in

Section F.2.

A set of 1000 validation points from a random Latin hypercube are used to assess

the accuracy of the GAA kriging models. The maximum error and root mean square error

(RMSE) based on the set of validation points for the kriging models based on the 64 point

orthogonal array are summarized in Table 7.6; both raw values and percentages (of the

sample range) are listed.

251

Table 7.6 Error Analysis of GAA Kriging Models

Raw Values As a Percent of Range

ResponseMaxError RMSE

MaxError RMSE

µNOISE 0.020 0.006 0.03% 0.01%µWEMP 4.630 0.942 0.24% 0.05%µDOC 16.318 2.889 16.87% 3.49%

µROUGH 0.015 0.004 0.76% 0.17%µWFUEL 7.201 1.620 2.17% 0.40%µPURCH 117.147 21.017 0.27% 0.05%µRANGE 116.924 26.094 4.14% 1.01%µVCRMX 0.913 0.158 0.46% 0.08%µLDMAX 0.103 0.030 0.64% 0.17%σNOISE 0.002 0.000 0.00% 0.00%σWEMP 1.684 0.260 0.09% 0.01%σDOC 2.895 0.348 3.23% 0.41%

σROUGH 0.002 0.001 0.11% 0.03%σWFUEL 1.689 0.368 0.40% 0.09%σPURCH 49.998 7.636 0.12% 0.02%σRANGE 39.702 8.797 1.37% 0.31%σVCRMX 0.298 0.065 0.16% 0.03%σLDMAX 0.024 0.002 0.16% 0.01%

With the exception of the maximum error for µDOC, all of the kriging metamodels

appear sufficiently accurate for this study; maximum errors are about 4% or less, and

RMSEs are 1% or less. Despite the large maximum error for µDOC, the RMSE for µDOC is

sufficiently low enough, however, to provide a reasonable approximation. As such, these

kriging metamodels are used throughout the rest of the GAA example. Thus, Step 3 of the

PPCEM is complete, and the compromise DSP for the family of aircraft is formulated in the

next section as Step 4 in the PPCEM.

7.4 STEP 4: AGGREGATE PRODUCT SPECIFICATIONS ANDFORMULATE GAA PLATFORM COMPROMISE DSP

In the universal motor example, separate goals for “bringing the mean on target”

and “minimizing the deviation” for variations in the stack length are used. In this example,

252

the compromise DSP for the family of GAA employs design capability indices (Cdk) to

assess the capability of a family of designs—composed of the three General Aviation

aircraft—to satisfy a ranged set of design requirements. Design capability indices are

formulated for both the constraints and goals for the family of GAA as defined by the

constraints and target values listed in Table 7.4 and Table 7.5, respectively.

The compromise DSP for the family of GAA is derived as follows:

C dk Constraint Formulations

❏ For the case where “Smaller is Better,” i.e., constraint ≤ maximum:

• NOISE ≤ 75 = URL Cdk,noise = Cdu,noise =(75 − µnoise )

3σnoise

[7.5]

• DOC < 80 = URL Cdk,doc = Cdu,doc =(80 − µdoc)

3σdoc

[7.6]

• ROUGH ≤ 2 = URL Cdk,rough = Cdu,rough =(2.0 − µrough )

3σ rough

[7.7]

• WEMP ≤ 2200 = URL Cdk,wemp = Cdu,wemp =(2200 − µwemp )

3σwemp

[7.8]

• WFUEL ≤ 450 = URL Cdk,wfuel = Cdu,wfuel =(450 − µwfuel )

3σwfuel

[7.9]

❏ For the case where “Larger is Better,” i.e., constraint ≥ minimum:

• RANGE ≥ 2000 = LRL Cdk,range = Cdl,range =(µrange − 2000)

3σ range

[7.10]

C dk Goal Formulations

❏ For the case where “Nominal is Best” for a goal:

• WFUEL:

Cdu,wfuel =(450 − µwfuel )

3σwfuel,C du,wfuel =

(µwfuel − 350)

3σwfuel

Cdk,wfuel = min{Cdu,wfuel,Cdl,wfuel} [7.11]

253

• WEMP:

Cdu,wemp =(2000 − µwemp)

3σwemp,Cdu,wemp =

(µwemp − 1900)

3σwemp

Cdk,wemp = min{Cdu,wemp,Cdl,wemp} [7.12]

• PURCH:

Cdu,purch =(43000 − µpurch )

3σpurch,Cdu,purch =

(µpurch − 41000)

3σpurch

Cdk,wfuel = min{Cdu,purch,Cdl,purch} [7.13]

❏ For goals where “Smaller is Better”:

• DOC: Cdk,doc = Cdu,doc =(60 − µdoc )

3σdoc

[7.14]

❏ For the case where “Larger is Better” for a goal:

• LDMAX: Cdk,ldmax = Cdl,ldmax =(µ ldmax −17)

3σ ldmax

[7.15]

• VCRMX: Cdk,vcrmx = Cdl,vcrmx =(µvcrmx − 200)

3σvcrmx

[7.16]

• RANGE: Cdk,range = Cdl,range =(µrange − 2500)

3σ range

[7.17]

The resulting compromise DSP for the GAA product platform using these Cdk

formulations is given in Figure 7.7. There are six design variables, six constraints, and

seven goals. Of the seven goals, three are related to the economic performance of the

aircraft—empty weight (WEMP), purchase price (PURCH), and direct operating cost

(DOC)—and the remaining four are related to the technical performance of the aircraft: fuel

weight (WFUEL), maximum lift/drag (LDMAX), maximum cruise speed (VCRMX), and

maximum flight range (RANGE).

254

Given:❏ Baseline aircraft configuration and mission profile❏ Configuration scale factor = # passengers (where total # seats = # passengers + 1 pilot)❏ Kriging models for mean and standard deviation of each response

Find:❏ The system variables, x:

• cruise speed, CSPD • wing loading, WL• wing aspect ratio, AR • engine activity factor, AF• propeller diameter, DPRP • seat width, WS

❏ The values of the deviation variables associated with G(x):• fuel weight Cdk , d1

-, d1+ • maximum lift/drag Cdk , d5

-, d5+

• empty weight Cdk , d2-, d2

+ • maximum speed Cdk , d6-, d6

+

• direct operating cost Cdk , d3-, d3

+ • maximum range Cdk , d7-, d7

+

• purchase price Cdk , d4-, d4

+

Satisfy:❏ The system constraints, C(x), based on kriging models:

• NOISE Cdk greater than 1: Cdk,noise(x) ≥ 1 [7.5]• DOC Cdk greater than 1: Cdk,doc(x) ≥ 1 [7.6]• ROUGH Cdk greater than 1: Cdk,rough(x) ≥ 1 [7.7]• WEMP Cdk greater than 1: Cdk,wemp(x) ≥ 1 [7.8]• WFUEL Cdk greater than 1: Cdk,wfuel(x) ≥ 1 [7.9]• RANGE Cdk greater than 1: Cdk,range(x) ≥ 1 [7.10]

❏ The system goals, G(x), based on kriging models:• WFUEL Cdk greater than 1: Cdk,noise(x) + d1

- - d1+ = 1.0 [7.11]

• WEMP Cdk greater than 1: Cdk,wemp(x) + d2- - d2

+ = 1.0 [7.12]• DOC Cdk greater than 1: Cdk,doc(x) + d3

- - d3+ = 1.0 [7.13]

• PURCH Cdk greater than 1: Cdk,purch(x) + d4- - d4

+ = 1.0 [7.14]• LDMAX Cdk greater than 1: Cdk,ldmax(x) + d5

- - d5+ = 1.0 [7.15]

• VCRMX Cdk greater than 1: Cdk,vcrmx(x) + d6- - d6

+ = 1.0 [7.16]• RANGE Cdk greater than 1: Cdk,range(x) + d7

- - d7+ = 1.0 [7.17]

❏ Constraints on deviation variables: di- • di

+ = 0 and di-, d i

+ ≥ 0.

❏ The bounds on the system variables:0.24 M ≤ CSPD ≤ 0.48 M 19 lb/ft2 ≤ WL ≤ 25 lb/ft2

7 ≤ AR ≤ 11 85 ≤ AF ≤ 1105.0 ft ≤ DPRP ≤ 5.96 ft 14.0 in ≤ WS ≤ 20.0 in

Minimize:❏ The sum of the deviation variables associated with:

• fuel weight Cdk , d1- • maximum lift/drag Cdk , d5

-

• empty weight Cdk , d2- • maximum speed Cdk , d6

-

• direct operating cost Cdk , d3- • maximum range Cdk , d7

-

• purchase price Cdk , d4-

Z = { f1(d1-), f2(d2

-), f3(d3-), f4(d4

-), f5(d5-), f6(d6

-), f7(d7-) }

Figure 7.7 GAA Product Platform Compromise DSP Formulation

255

Based on this GAA compromise DSP, the initial baseline design is infeasible in two

regards. First, the propeller diameter is too great as explained in Section 7.1 At 6.3 ft, the

speed of the propeller tip is above sonic speed, violating a tipspeed constraint which is not

explicitly modeled in the GAA compromise DSP; thus, the range for the propeller diameter

is set at 5-5.96 ft so that this constraint is always met. Second, the DOC violates the

$80/hr constraint which has been selected. The baseline design still represents a good

design; however, the GAA compromise DSP is being used to improve it as discussed in the

next section wherein the product platform portfolio is developed in Step 5 of the PPCEM.

7.5 STEP 5: DEVELOP THE GAA PLATFORM PORTFOLIO

In order to develop the GAA platform portfolio for the family of GAA to meet the

constraints and goals set forth in Section 7.2, three design scenarios are investigated (see

Table 7.7).

Overall Tradeoff Study: All of the goals are weighted equally in an effort to

develop a platform that simultaneously meets both economic and performance

requirements as best as possible.

Economic Tradeoff Study: Economic related goals (Cdk’s for empty weight,

purchase price, and direct operating cost) are given top priority to find a platform

which meets all of the economic requirements as best as possible; satisfying

performance goals is second priority.

Performance Tradeoff Study: Performance related goals (Cdk’s for fuel weight,

max. lift/drag, max. speed, and max. range) are placed at the first priority level to

develop a platform that satisfies all of the performance requirements as best as

possible; meanwhile, economic goals are given second priority.

The corresponding deviation function formulations for each scenario are listed in Table 7.7.

256

Table 7.7 GAA Product Platform Compromise DSP Design Scenarios

Deviation FunctionScenario PLEV1 PLEV2 Note:

1. Overall Tradeoff(d1

- + d2- + d3

-

+ d4- + d5

- + d6-

+ d7-)/7

d1- drives Cdk-wfuel to 1

d2- drives Cdk-wemp to 1

d3- drives Cdk-purch to 1

2. Economic Tradeoff (d2- + d3

- +d4

-)/3(d1

- + d5- + d6

-

+ d7-)/4

d4- drives Cdk-doc to 1

d5- drives Cdk-ldmax to 1

3. Performance Tradeoff (d1- + d5

- + d6-

+ d7-)/4

(d2- + d3

- +d4

-)/3d6

- drives Cdk-vcrmx to 1d7

- drives Cdk-range to 1

Three starting points are used when solving the GAA product platform compromise

DSP for each scenario: the lower, middle, upper bounds of the design variables; in a

situation where all three starting points do not converge to the same solution, the design

with the lowest deviation function value is taken as the best design (the reader is referred to

the convergence studies in Section 7.6.1). The resulting product platform specifications

obtained by solving the compromise DSP in Figure 7.7 are given in the next section. The

individual instantiations of the aircraft within the family based on the kriging metamodels

then are discussed in Section 7.5.2.

7 .5 .1 Results of the GAA Compromise DSP for the Family of Aircraft

The resulting product platform specifications for each design scenario are

summarized in Table 7.8. Recall that the target values for each Cdk is 1; values above one

indicate that the family of GAA has met the desired URL or LRL while values below one

indicate that the targets have not been met for that particular requirement. All solutions are

feasible, and the values for Cdk,rough and Cdk,noise have not been included because they have

no bearing on the deviation function (other than to make the solution infeasible). The Cdk

values for the initial baseline design have also been included in the table for the sake of

comparison. The PPCEM based family has an unfair advantage because the baseline

aircraft (the Beechcraft Bonanza B36TC presented in Section 7.1.3) is a six seater aircraft

257

and, as such, is not expect to perform well when scaled down to fit fewer passengers;

however, it still provides a reference point to compare against the family of aircraft

developed using the PPCEM.

Table 7.8 Summary of GAA Family Compromise DSP Results

Baseline ScenarioDesign 1 2 3

Des. Var.CSPD [Mach] 0.31 0.244 0.242 0.291AR 7.88 8.00 8.09 7.62DPRP [ft] 6.3 5.13 5.19 5.55WL [lb/ft2] 20.5 22.45 22.63 22.48AF 110 89.60 89.40 85.63WS [in] 20 18.60 18.72 18.70

GoalsCdk-wfuel P* -0.640 1.164 1.236 1.156Cdk-wemp E 0.074 0.810 0.903 0.806Cdk-doc E -670.476 -1.588 -1.312 -26.270Cdk-purch E -2.557 0.733 0.449 0.070Cdk-ldmax P -3.230 -4.474 -4.427 -4.964Cdk-vcrmx P -4.397 -4.303 -3.702 -2.017Cdk-range P -4.157 0.577 -0.672 0.429Dev. Fcn.

PLEV1 2.036 0.986 2.388PLEV2 2.950 9.4556

* P indicates Cdk is related to performance; E to economics—economicgoals rank first in Scenario 2; performance goals rank first in Scenario 3

Compared to the initial baseline design, the PPCEM designs have a lower cruise

speed, propeller diameter, engine activity factor, and seat width. Meanwhile, the wing

loading is slightly larger in general; and the aspect ratio fluctuates around the baseline

value. Comparing the design variables for Scenario 1 and 2, there is negligible difference.

This indicates that in the overall tradeoff study, the economic goals tend to dominate the

solution despite all goals being equally weighted. In an effort to achieve better performance

in Scenario 3 (at the sacrifice of the economic goal achievement), the cruise speed is

slightly higher, the propeller diameter is slightly larger, and the aspect ratio and engine

258

activity factor are slightly lower for this scenario than either Scenario 1 or 2. Thus, in

order to maintain sufficient flexibility to achieve all the design considerations in all three

scenarios, the resulting product platform is taken as follows:

• Cruise speed = Mach 0.242 or Mach 0.291 (if performance is first priority)

• Aspect ratio = 7.85 ± 0.24

• Propeller diameter = 5.34 ± 0.2 ft

• Wing loading = 22.54 ± 0.09 lb/ft2

• Engine activity factor = 87.61 ± 2.0

• Seat width = 18.66 ± 0.06 in

These values comprise the range of values that cruise speed, aspect ratio, etc. should be

allowed to take in order to meet the goals as best as possible in any of the three design

scenarios. It is these values which define the GAA product platform around which the

family of aircraft are created.

Before instantiating the individual aircraft to examine how well they perform given

these specifications, notice in Table 7.8 that very few Cdk goals achieve their target of 1;

only Cdk,wfuel is consistently larger than 1, indicating that the family of GAA are capable of

meeting the specified fuel weight targets. The empty weight Cdk and purchase price Cdk are

the second best with Cdk,range performing well in Scenarios 1 and 3. All Cdk values are

improved over the baseline design except for Cdk,ldmax which has decreased slightly. In

Scenario 2, the economic Cdk’s for direct operating cost and empty weight improve slightly

but at the expense of a slight decrease in the purchase price Cdk when compared to the value

obtained in Scenario 1. The big tradeoff between the economic and performance goals in

Scenarios 2 and 3 is best seen in Cdk,doc. In all three scenarios, the family of GAA is far

from achieving its target of $60/hr for the direct operating cost as indicated by the low

values for Cdk,doc; however, in Scenario 3 when achieving performance goals is given a

259

higher priority than economic goals, Cdk,doc is even worse, indicating the compromise

between a family of aircraft that has performs well versus one that is economical.

To study these compromises further, five more design scenarios are formulated (see

Section F.4) to determine whether it is the Cdk formulation that is performing poorly or that

the targets are difficult to achieve. The results are listed separately in Section F.4 and

discussed therein. The end result of examining all these design scenarios is learning that

significant tradeoffs are occurring in Scenarios 1, 2, and 3 where the economic and

performance Cdk goals are equally weighted at different priority levels. Only when a

particular Cdk is given first priority (i.e., placed at PLEV1) in the GAA product platform

compromise DSP can the target (Cdk ≥ 1) be achieved. Any other time, the solutions from

the GAA product platform compromise DSP represent the best possible compromise which

can be obtained for a particular design scenario, poor Cdk value or not. Furthermore, the

deviation function values shown in Table 7.8 are not much value in and of themselves

because they are based on how well the Cdk achieve their target of 1. Recall that Cdk is only

a means to end, i.e., to generate a family of aircraft which satisfies the given ranged set of

requirements as well as possible . What is important, however, is the resulting aircraft

which come from instantiating the PPCEM aircraft platform to accommodate two, four, and

six passengers.

7 .5 .2 Instantiation of the Family of General Aviation Aircraft

Unlike in the universal motor example, instantiation of the individual aircraft within

the GAA product family only requires specifying the number of passengers on the plane,

not solving another compromise DSP to find the best stack length to meet a particular

torque requirement. The individual constraints and goals for each aircraft must be

formulated first, however, based on the specifications given in Section 7.2. Based on the

260

requirements in Table 7.4, the individual constraints for each aircraft are given by the

following:

• noise [dbA]: NOISE(x) ≤ 75 dbA [7.18]

• direct operating cost [$/hr]: DOC(x) ≤ $80/hr [7.19]

• ride roughness: ROUGH(x) ≤ 2.0 [7.20]

• aircraft empty weight [lbs]: WEMP(x) ≤ 2200 lbs [7.21]

• aircraft fuel weight [lbs]: WFUEL(x) ≤ Ci,wfuel [7.22]

• maximum flight range [nm]: RANGE(x) ≥ 2200 nm [7.23]

where Ci,wfuel = {450 lbs, 475 lbs, 500 lbs} and i = {1, 3, 5} passengers. Meanwhile, the

individual goals based on the targets in Table 7.5 for each aircraft are given by:

• aircraft fuel weight [lbs]: WFUEL(x)/Ti,wfuel + d1- - d1

+ = 1.0 [7.24]

• aircraft empty weight [lbs]: WEMP(x)/Ti,wemp + d2- - d2

+ = 1.0 [7.25]

• direct operating cost [$/hr]: DOC(x)/60 + d3- - d3

+ = 1.0 [7.26]

• purchase price [$]: PURCH(x)/Ti,purch + d4- - d4

+ = 1.0 [7.27]

• maximum lift/drag: LDMAX(x)/17 + d5- - d5

+ = 1.0 [7.28]

• maximum cruise speed [kts]: VCRMX(x)/200 + d6- - d6

+ = 1.0 [7.29]

• maximum range [nm]: RANGE(x)/2500 + d7- - d7

+ = 1.0 [7.30]

where Ti,wfuel = {450 lbs, 400 lbs, 350 lbs}, Ti,wemp = {1900 lbs, 1950 lbs, 2000 lbs},

Ti,purch = {$41000, $42000, $43000} and i = {1, 3, 5} passengers. Based on these goals,

the deviation function for each aircraft is a combination of: d1+, d2

+, d3+, d4

+, d5-, d6

-, and d7-

because it is desired to lower fuel weight, empty weight, direct operating cost, and

purchase price to their targets and to raise maximum lift/drag, cruise speed, and range to

theirs. The resulting deviation function formulations for each aircraft for each scenario is

listed in Table 7.9. These deviation functions are identical to those listed in Table 7.7

except that di+ and di

- are for the individual goals of each aircraft and not the Cdk for the

family (which only uses di- to raise Cdk to its target of 1 as noted in Table 7.7).

261

Table 7.9 Deviation Functions of Individual Aircraft for Each Scenario


1. Overall tradeoff(d1

+ + d2+ +

d3+ + d4

+ + d5-

+ d6- + d7

-)/7

d1+ lowers fuel weight to target

d2+ lowers empty weight to target

d3+ lowers direct oper. cost to target

2. Economic tradeoff (d2+ + d3

+ +d4

+)/3(d1

+ + d5- + d6

-

+ d7-)/4

d4+ lowers purchase price to target

d5- raises max. lift/drag to target

3. Performance tradeoff (d1+ + d5

- + d6-

+ d7-)/4

(d2+ + d3

+ +d4

+)/3d6

- raises max. speed to targetd7

- raises max. range to target

The instantiations of the two, four, and six seater GAA for the family of aircraft

based on the PPCEM platform values are summarized in Table 7.10. These response

values are obtained by evaluating the kriging metamodels at the design variable values listed

in Table 7.8 for each scenario. Based on the low deviation function values listed in Table

7.10, it appears that the PPCEM based family of aircraft perform reasonably well on an

individual basis. The targets for fuel weight and empty weight are met in all cases. Despite

the poor showing of Cdk,doc, the DOC values for the individual aircraft in Scenarios 1 and 3

are within $2/hr of the target of $60/hr; meanwhile, the DOC values are near their

maximum permitted value ($80/hr) in Scenario 3 when economics takes second priority to

performance. The purchase price goals of {$41000, $42000, $43000} are within $1000 or

less of being met in all cases. The maximum lift/drag ratio (LDMAX) and cruise speeds

(VCRMX) do not meet their targets of 17 or 200 very well. The maximum range target

(2500 n.m.) is met in Scenario 1 and by all but the six seater in Scenario 3; the range values

for Scenario 2 are slightly below the target.

262

Table 7.10 Instantiations of the PPCEM Product Platform Based onKriging Metamodels

ResponseDesign No. of WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE Dev Fcn

Scenario Seats [lbs] [lbs] [$/hr] [$] [kts] [nm] PLEV1 PLEV22 447.70 1892.32 61.05 42078.2 16.16 193.56 2542.7 0.018

1 4 409.19 1929.50 62.31 42601.3 15.80 190.17 2517.3 0.0286 376.65 1959.37 62.54 43138.7 15.68 188.87 2502.8 0.0362 445.06 1895.36 60.52 42221.0 16.16 194.82 2497.3 0.013 0.019

2 4 405.92 1932.81 61.83 42749.0 15.81 191.51 2462.0 0.016 0.0366 373.48 1962.93 62.20 43296.0 15.68 190.20 2444.0 0.015 0.0542 446.36 1891.95 78.16 42402.6 16.04 198.16 2543.1 0.016 0.112

3 4 406.63 1929.39 79.30 42972.1 15.73 195.16 2509.0 0.029 0.1156 377.05 1959.14 78.28 43462.1 15.56 193.72 2482.9 0.050 0.105

So what does all this mean in terms of designing a scalable platform for a product

family? Considerable improvement has been made over the initial baseline design, but has

a good family of aircraft been designed? Answers to this question are offered in the next

section in which verification and the implications of the results are discussed.

7.6 VERIFICATION OF GAA PRODUCT PLATFORM RESULTS

To verify the results obtained from implementing the PPCEM, the following

questions are addressed.

Verify compromise DSP solutions - What do the convergence histories look like

for each scenario? Is the best solution being obtained?

Verify kriging predictions - How does the predicted performance of the individual

aircraft based on the kriging models compare to the actual performance in GASP?

Verify instantiations of PPCEM platform- How do the individual aircraft based

on the PPCEM platform compare to individually designed benchmark aircraft?

Verify PPCEM family - How does the family of aircraft based on the PPCEM

compare to the aggregate group of individually designed benchmark aircraft?

263

Each question is addressed in turn in the following sections.

7 .6 .1 GAA Product Platform Compromise DSP Verification

Convergence histories of the GAA compromise DSP solution for Scenario 1 for the

family of GAA is illustrated in Figure 7.8. As seen in the figure, all three starting points

converge to approximately the same solution, indicating that the best possible solution has

likely been obtained. The initial deviation function for the high starting point is quite large

(~15) while the initial design based on the middle starting point is slightly infeasible; hence,

the jump in iteration 2 and the increase in PLEV1.

0

2

4

6

8

10

12

14

16

0 5 10 15Iterations

PLEV

1

Low

Mid

Hi

Figure 7.8 Convergence History of GAA Family C-DSP for Scenario 1

The convergence histories for the PLEV1 and PLEV2 for Scenarios 2 and 3 are

illustrated in Figure 7.9. Similar to Figure 7.8, the three starting points for Scenarios 2 and

3 yield a wide range of initial deviation function values, but the model tends to converge at

264

the same or nearly similar solutions. This trend holds true at both priority levels in both

scenarios.

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16

Iterations

PLEV

1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25

Iterations

PLEV

1

(a) Scenario 2 - Priority Level 1 (b) Scenario 3 - Priority Level 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12 14 16

Iterations

PLEV

2

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

Iterations

PLEV

2

(c) Scenario 2 - Priority Level 2 (d) Scenario 3 - Priority Level 2

Figure 7.9 Convergence History for Scenarios 2 and 3

Furthermore, it is interesting to note the parity between PLEV1 in Scenario 2

(Figure 7.9a) and PLEV2 in Scenario 3 (Figure 7.9d) and between PLEV1 in Scenario 3

(Figure 7.9b) and PLEV2 in Scenario 2 (Figure 7.9c) since the same goals are equally

weighted at different levels in these scenarios. Comparing these graphs reveals the true

nature of the tradeoffs that occur between the economic goals and the performance goals.

When the economic goals are placed at the first priority level in Scenario 2, a much lower

value for the deviation function is capable of being achieved compared to when they are

placed at the second priority level as in Scenario 3. The same holds true for the

265

performance goals in the first priority level in Scenario 3 when compared to the second

priority level of Scenario 2.

7 .6 .2 Comparisons of Kriging Predictions and GASP

Previously, the performance of the individual aircraft have been based on

predictions from kriging metamodels (see Table 7.10). Therefore, the performance of the

individual aircraft is evaluated directly in GASP as opposed to being estimated from the

kriging metamodels. The results are summarized in Table 7.11 and can be compared

directly to the previous values listed in Table 7.10. The resulting approximation error

between the kriging model predictions and the actual aircraft instantiations in GASP are

summarized in Table 7.12.

Table 7.11 Performance of PPCEM Platform Instantiations in GASP

Design No. Response Dev FcnScenario Seats WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2

2 449.43 1887.15 61.98 41817.0 15.89 190.83 2491.0 0.0241 4 413.80 1921.71 63.31 42374.5 15.61 188.47 2436.0 0.038

6 388.49 1946.59 63.85 42827.0 15.53 187.61 2420.0 0.0512 447.25 1889.73 61.60 41959.4 15.91 192.24 2446.0 0.017 0.031

2 4 411.34 1924.56 62.85 42502.8 15.63 189.51 2393.0 0.020 0.0516 385.69 1949.77 63.38 42989.1 15.55 189.07 2377.0 0.019 0.0732 450.92 1886.97 77.43 42150.1 15.79 195.53 2499.0 0.024 0.106

3 4 415.34 1921.70 79.13 42727.0 15.52 193.32 2451.0 0.045 0.1126 389.84 1946.82 79.76 43190.3 15.44 192.46 2437.0 0.067 0.111

The errors are expressed as a percent of the actual value obtained from GASP; a

positive error indicates over-prediction of the response, and a negative error indicates

under-prediction. The maximum error occurs for RANGE of the six seater GAA (=

3.42%). In general, the kriging models over-predict PURCH, LDMAX, VCRMX, and

RANGE and under-predict WFUEL and DOC. The average percentage error for each

response also is listed in the table; values range from 0.62% to a high of 2.52%. In

266

summary, then, it appears that the kriging metamodel predictions are quite accurate based

on the error analysis in Table 7.12.

Table 7.12 Approximation Errors for Individual Aircraft

Design No. ResponseScenario Seats WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE

2 -0.39% 0.27% -1.51% 0.62% 1.73% 1.43% 2.08%1 4 -1.12% 0.41% -1.58% 0.54% 1.20% 0.90% 3.34%

6 -3.05% 0.66% -2.06% 0.73% 0.98% 0.67% 3.42%2 -0.49% 0.30% -1.74% 0.62% 1.60% 1.34% 2.10%

2 4 -1.32% 0.43% -1.62% 0.58% 1.14% 1.05% 2.88%6 -3.16% 0.67% -1.85% 0.71% 0.86% 0.60% 2.82%2 -1.01% 0.26% 0.94% 0.60% 1.56% 1.35% 1.76%

3 4 -2.10% 0.40% 0.21% 0.57% 1.36% 0.95% 2.37%6 -3.28% 0.63% -1.85% 0.63% 0.81% 0.65% 1.88%

Average%error = -1.77% 0.45% -1.23% 0.62% 1.25% 0.99% 2.52%

7 .6 .3 Comparison of PPCEM Results to Benchmark Aircraft

For further verification of the PPCEM aircraft, individual (benchmark) aircraft are

designed using GASP and DSIDES directly to compare to the aircraft obtained through the

implementation of the PPCEM. The compromise DSP for these benchmark aircraft is

shown in Figure 7.10 and is derived from Equations 7.16-7.28 for the individual

constraints and goals listed in Table 7.4 and Table 7.5, respectively. As in the individual

instantiations of the PPCEM platform, the deviation variables of interest are: d1+, d2

+, d3+,

d4+, d5

-, d6-, and d7

- because it is desired to lower fuel weight, empty weight, direct

operating cost, and purchase price to their targets and to raise maximum lift/drag, cruise

speed, and range to theirs.

267

Given:❏ Baseline aircraft configuration and mission profile❏ General Aviation Synthesis Program (GASP)

Find:❏ The system variables, x:

• cruise speed, CSPD • wing loading, WL• wing aspect ratio, AR • engine activity factor, AF• propeller diameter, DPRP • seat width, WS

❏ The values of the deviation variables associated with G(x):• fuel weight Cdk , d1

-, d1+ • maximum lift/drag Cdk , d5

-, d5+

• empty weight Cdk , d2-, d2

+ • maximum speed Cdk , d6-, d6

+

• direct operating cost Cdk , d3-, d3

+ • maximum range Cdk , d7-, d7

+

• purchase price Cdk , d4-, d4

+

Satisfy:❏ The system constraints, C(x), based on kriging models:

• noise [dbA]: NOISE(x) ≤ 75 dbA [7.18]• direct operating cost [$/hr]: DOC(x) ≤ $80/hr [7.19]• ride roughness: ROUGH(x) ≤ 2.0 [7.20]• aircraft empty weight [lbs]: WEMP(x) ≤ 2200 lbs [7.21]• aircraft fuel weight [lbs]: WFUEL(x) ≤ Ci,wfuel [7.22]• maximum flight range [nm]: RANGE(x) ≥ 2200 nm [7.23]

❏ The system goals, G(x), based on kriging models:• aircraft fuel weight [lbs]: WFUEL(x)/Ti,wfuel + d1

- - d1+ = 1.0 [7.24]

• aircraft empty weight [lbs]: WEMP(x)/Ti,wemp + d2- - d2

+ = 1.0 [7.25]• direct operating cost [$/hr]: DOC(x)/60 + d3

- - d3+ = 1.0 [7.26]

• purchase price [$]: PURCH(x)/Ti,purch + d4- - d4

+ = 1.0 [7.27]• maximum lift/drag: LDMAX(x)/17 + d5

- - d5+ = 1.0 [7.28]

• maximum cruise speed [kts]: VCRMX(x)/200 + d6- - d6

+ = 1.0 [7.29]• maximum range [nm]: RANGE(x)/2500 + d7

- - d7+ = 1.0 [7.30]

❏ Constraints on deviation variables: di- • di

+ = 0 and di-, d i

+ ≥ 0.

❏ The bounds on the system variables:0.24 M ≤ CSPD ≤ 0.48 M 19 lb/ft2 ≤ WL ≤ 25 lb/ft2

7 ≤ AR ≤ 11 85 ≤ AF ≤ 1105.0 ft ≤ DPRP ≤ 5.96 ft 14.0 in ≤ WS ≤ 20.0 in

Minimize:❏ The sum of the deviation variables associated with:

• fuel weight, d1+ • maximum lift/drag ratio, d5

-

• empty weight, d2+ • maximum speed, d6

-

• direct operating cost, d3+ • maximum range, d7

-

• purchase price, d4+

Z = { f1(d1+), f2(d2

+), f3(d3+), f4(d4

+), f5(d5-), f6(d6

-), f7(d7-) }

Figure 7.10 GAA Compromise DSP for Individual Aircraft

268

To design each benchmark aircraft, the compromise DSP in Figure 7.10 is

particularized with the appropriate targets and constraints and solved: Ci,wfuel = {450 lbs,

475 lbs, 500 lbs}, Ti,wfuel = {450 lbs, 400 lbs, 350 lbs}, Ti,wemp = {1900 lbs, 1950 lbs,

2000 lbs}, T i,purch = {$41000, $42000, $43000} and i = {1, 3, 5} passengers. The same

three design scenarios are used when designing each benchmark aircraft, see Table 7.13.

All three scenarios are tradeoff studies: Scenario 1 is an overall tradeoff with all goals

weighted equally; Scenario 2 has the economic goals weight equally at the first priority

level (PLEV1), and the performance goals weighted equally at the second priority level

(PLEV2); and Scenario 3 is the reverse of Scenario 2 with performance goals being ranked

first and economics second. The deviation function formulations for each scenario for the

benchmark aircraft are listed in the table. Notice that a combination of di+ and di

- are being

used in the deviation function and not just di-. This is because it is desired to lower fuel

weight, empty weight, direct operating cost, and purchase price to their targets and to raise

maximum lift/drag, cruise speed, and range to theirs. (With the Cdk formulation, the only

concern is to minimize di- in order to ensure that Cdk ≥ 1.)

Table 7.13 Design Scenarios for Designing GAA Benchmark Aircraft



+ + d2+ +

d3+ + d4

+ + d5-

+ d6- + d7

-)/7

d1+ lowers fuel weight to target

d2+ lowers empty weight to target

d3+ lowers direct oper. cost to target

2. Economic tradeoff (d2+ + d3

+ +d4

+)/3(d1

+ + d5- + d6

-

+ d7-)/4

d4+ lowers purchase price to target

d5- raises max. lift/drag to target

3. Performance tradeoff (d1+ + d5

- + d6-

+ d7-)/4

(d2+ + d3

+ +d4

+)/3d6

- raises max. speed to targetd7

- raises max. range to target

As before, three starting points—lower, middle, and upper values—are used when

designing each aircraft for each scenario; the best design(s) is then taken as the one with the

269

lowest deviation function value. Convergence plots for each aircraft for each scenario are

listed separately in Section F.5 and are similar to those observed for the PPCEM solutions

in 7.6.1. The final settings of the design variables for each aircraft for each of these three

design scenarios are listed in Table 7.14 through Table 7.16 for Scenarios 1-3,

respectively. Each set of results is discussed in turn and plotted graphically with the

corresponding PPCEM instantiations for a quick comparison of the results. The results for

Scenario 1 are listed in Table 7.14.

Table 7.14 Individual PPCEM and Benchmark Aircraft for Scenario 1

PPCEM Aircraft Benchmark AircraftDes. Var. 2 Seater 4 Seater 6 Seater 2 Seater 4 Seater 6 Seater

CSPD [Mach] ï 0.244 ð 0.240 0.240 0.240AR ï 8.00 ð 8.61 7.96 9.16DPRP [ft] ï 5.13 ð 5.00 5.00 5.00WL [lb/ft2] ï 22.46 ð 22.07 21.42 21.69AF ï 89.60 ð 85.25 95.00 86.37WS [in] ï 18.60 ð 18.04 18.35 19.45Responses

WFUEL [lbs] 449.43 413.8 388.49 449.67 408.35 349.97WEMP [lbs] 1887.15 1921.71 1946.59 1888.23 1926.33 1986.58DOC [$/hr] 61.98 63.31 63.85 61.6 63.9 64.02PURCH [$] 41817 42374.5 42827 41607.8 42240 43262.9LDMAX 15.89 15.61 15.53 16.54 15.8 16.4VCRMX [kts] 190.83 188.47 187.61 187.47 184.58 181.94RANGE [nm] 2491 2436 2420 2536 2672 2466Dev. Fcn.

PLEV1 0.0240 0.0377 0.0506 0.0187 0.0341 0.0303

As can be seen in the table, there is little variation between the design variable

settings for the benchmark aircraft even though they have been designed individually. The

benchmark aircraft share common settings for the cruise speed and propeller diameter

despite being individually designed. Aspect ratio and wing loading only vary slightly

between each aircraft, and the difference in seat widths (WS) for each aircraft is less than

1.5 in. The engine activity factor varies the most of the six design variables. It is

270

interesting to note that the PPCEM design variable values are quite close to the benchmark

designs. Seat width, aspect ratio, and engine activity factor all are contained within the

range of settings for the benchmark aircraft. The PPCEM values for cruise speed and

propeller diameter are only slightly larger than the corresponding values which are shared

between all three benchmark aircraft.

Despite the similarity of the design variable settings for the two families of aircraft,

only the 4 seater benchmark and PPCEM aircraft have similar deviation function values.

The two and six seater aircraft from the PPCEM are both slightly worse than the

benchmark designs as a result of having a common set of design variables for all three

aircraft. To see why this is and to facilitate comparison of the performance of the two

families of aircraft (the one based on the PPCEM and the group of benchmark aircraft),

plots of the individual goal achievements for each aircraft are given in Figure 7.11 for

Scenario 1. The idea of using a “spider” or “snowflake” plot to show goal achievement

comes from Sandgren (1989). In the spider plot, goal deviation values are plotted on the

axes of the web; the closer a mark is on its axis to the origin, the better that particular goal

has been achieved. In this manner, the shape of the polygon formed by connecting the

deviation values for each design can be used to compare designs quickly. In other words,

the two seater aircraft from the PPCEM platform and the benchmark design can be quickly

compared by plotting their goal achievement on the same spider plot as is done in Figure

7.11 for all three aircraft which comprise the GAA family.

271

PPCEM FamilyBenchmark

SCENARIO 1 - Overall Tradeoff Study

0.00

0.02

0.04

0.06

0.08WEMP

DOC

PURCH

WFUELLDMAX

VCRMX

RANGE

2 Seater

0.000.020.040.060.080.10

WEMP

DOC

PURCH

WFUELLDMAX

VCRMX

RANGE

4 Seater

0.00

0.05

0.10

0.15WEMP

DOC

PURCH

WFUELLDMAX

VCRMX

RANGE

6 Seater

Benchmark PPCEM Family 2 Seater 0.0187 0.0240 4 Seater 0.0341 0.0377 6 Seater 0.0303 0.0506

Deviation Functions: PLEV1

Figure 7.11 Graphical Comparison of Benchmark Aircraft and PPCEMFamily for Scenario 1

In the overall tradeoff study (Scenario 1) shown in Figure 7.11 all seven goals are

equally weighted. Some observations based on the graphs are as follows:

• In the two seater aircraft, the achievement of WEMP, DOC, PURCH, WFUEL,

and RANGE appear virtually equal. The PPCEM aircraft exhibits slightly better

achievement of the VCRMX target; however, the benchmark design has better

LDMAX than the PPCEM which can account for the difference between the

deviation functions for these aircraft.

272

• In the four seater aircraft, the PPCEM solutions perform slightly better at DOC and

VCRMX, but slightly worse with RANGE, WFUEL, and LDMAX. Both aircraft

designs achieve the target for empty weight (WEMP).

• In the six seater aircraft, both designs achieve the WEMP and PURCH targets.

DOC achievement is essentially equal for both aircraft. The PPCEM designs yield

slightly better VCRMX than the benchmark aircraft; however, the benchmark

design outperforms the PPCEM design in WFUEL, LDMAX, and RANGE. It

appears that the difference in achievement in LDMAX and WFUEL account for the

large discrepancy in the two deviations functions for the six seater aircraft because

the achievement of the other goals are comparable for both aircraft.

The results for Scenario 2 are summarized in Table 7.15. As seen in Scenario 1,

the cruise speeds for the PPCEM aircraft and the benchmark aircraft are essentially the

same. The aspect ratio for the PPCEM aircraft is contained within the range of the

benchmark designs but is on the low end. The propeller diameter for the PPCEM aircraft is

slightly higher than the benchmark aircraft, which have nearly identical propeller diameters

again despite being designed individually. The wing loading for the PPCEM aircraft is

about 2 lb/ft2 lower than that of the benchmark designs, whose value, only vary by about

0.7 lb/ft2 between all three aircraft. The engine activity factors for the benchmark aircraft

vary from 85 to a high of 109; the PPCEM aircraft have a value of 89.4 which falls within

the range of the benchmark designs. Finally, the seat widths for the benchmark designs are

lower than for the PPCEM aircraft and converging almost to the lower bound of 14 in.

273

Table 7.15 Individual PPCEM and Benchmark Aircraft for Scenario 2


CSPD [Mach] ï 0.242 ð 0.240 0.243 0.240AR ï 8.09 ð 10.00 8.50 7.96DPRP [ft] ï 5.20 ð 5.00 5.06 5.00WL [lb/ft2] ï 22.63 ð 24.49 24.27 24.91AF ï 89.40 ð 91.37 109.09 85.00WS [in] ï 18.72 ð 18.18 15.42 14.39Responses

WFUEL [lbs] 447.25 411.34 385.69 450.04 474.07 494.81WEMP [lbs] 1889.73 1924.56 1949.77 1891.95 1865.35 1843.77DOC [$/hr] 61.60 62.85 63.38 60.34 59.99 60.21PURCH [$] 41959.4 42502.8 42989.1 42049.5 41778.0 41106.4LDMAX 15.91 15.63 15.55 17.11 16.23 15.85VCRMX [kts] 192.24 189.51 189.07 193.46 196.63 194.29RANGE [nm] 2446.00 2393 2377 1997 2133 2058Dev. Fcn.

PLEV1 0.0167 0.0198 0.0188 0.0104 0.000 0.0011PLEV2 0.0311 0.0510 0.0728 0.0585 0.0986 0.1717

The resulting deviation functions for the two families of aircraft are comparable to

the benchmark designs having consistently lower PLEV1 (deviation function value at

priority level 1) but slightly larger PLEV2 than the PPCEM aircraft. In the preemptive

(i.e., lexicographic) case, however, having lower PLEV2 values does not matter unless

PLEV1 values are the same. The first level deviation function value for the four seater

benchmark design is zero, indicating that the design is capable of meeting all of its

designated targets. The 2 and 6 seater benchmark designs also both fare well at achieving

their targets, having PLEV1 values of 0.01 and 0.001, respectively. The PPCEM aircraft,

on the other hand, have PLEV1 values which are slightly worse, and the four seater

PPCEM design does not achieve all of its targets as did the benchmark design. A look at

the discrepancies between the goal achievement of the two families of aircraft can be seen in

the spider plot for the Scenario 2 shown in Figure 7.12.

274

0.00

0.02

0.04

0.06WEMP

DOC

PURCH

0.000.010.020.030.040.05

WEMP

DOC

PURCH

0.00

0.01

0.02

0.03WEMP

DOC

PURCH


SCENARIO 2 - Economic Tradeoff Study

2 Seater 4 Seater

6 Seater


Deviation Functions: PLEV1

Figure 7.12 Graphical Comparison of Benchmark Aircraft and PPCEMFamily for Design Scenario 2, Priority Level 1 Only

In Figure 7.12 the results of the economic tradeoff study (Scenario 2) are

illustrated. Only the three economic related goals which are considered in the first priority

level are shown: empty weight (WEMP), direct operating cost (DOC), and purchase price

(PURCH). Some observations based on the graphs are as follows:

• Both sets of aircraft achieve the desired targets for empty weight.

• The purchase price (PURCH) for the seater aircraft from the PPCEM is slightly

lower than that of the benchmark aircraft; however, the purchase price for the four

275

seater PPCEM design is slightly higher than its comparative benchmark. Both six

seater aircraft do equally well at achieving the target.

• The DOC target achievement for the PPCEM solutions are higher for all three

aircraft than the individually designed benchmark aircraft; the inability to achieve the

DOC target is the main cause for the large discrepancy in the deviation function

value (PLEV1) for the PPCEM aircraft.

Finally, the results for Scenario 3 are summarized in Table 7.16. Notice that the

PPCEM cruise speed values are larger than all three benchmark designs while the aspect

ratio is less. The propeller diameter again is slightly larger for the PPCEM designs than for

the benchmark aircraft. The wing loading for both families of aircraft are comparable, but

the engine activity factor for the PPCEM tends to be on the lower end of the benchmark

aircraft setting. The seat width for the PPCEM is within the range of seat widths found for

the benchmark designs and is nearly identical to that of the four seater benchmark aircraft

with the two seater being slightly smaller and the six seater slightly larger.

Table 7.16 Individual PPCEM and Benchmark Aircraft - Scenario 3


CSPD [Mach] ï 0.291 ð 0.257 0.270 0.240AR ï 7.62 ð 8.30 8.32 9.19DPRP [ft] ï 5.55 ð 5.03 5.12 5.00WL [lb/ft2] ï 22.48 ð 22.35 22.04 21.63AF ï 85.63 ð 101.11 95.86 85WS [in] ï 18.70 ð 18.22 18.88 19.42Responses

WFUEL [lbs] 450.92 415.34 389.84 449.25 399.57 350.17WEMP [lbs] 1887 1921.7 1946.82 1888.8 1937.72 1986.37DOC [$/hr] 77.43 79.13 79.76 68.03 72.92 64.19PURCH [$] 42150 42727 43190.3 41871 42699.8 43237.2LDMAX 15.79 15.52 15.44 16.32 16.05 16.44VCRMX [kts] 195.53 193.32 192.46 190.62 187.99 181.68RANGE [nm] 2499 2451 2437 2494 2497 2478Dev. Fcn.

PLEV1 0.0240 0.0446 0.0671 0.0223 0.0293 0.0335PLEV2 0.1062 0.1120 0.1112 0.0517 0.0772 0.0251

276

The deviation function values at priority level 1 (PLEV1) for the two families of

aircraft exhibit similar trends to those seen previously except this time it is the two seater

aircraft which have comparable goal achievement of their first level goals, not the four

seater aircraft as in the previous scenario. The PPCEM 4 seater aircraft deviation function

(PLEV1) is about 1.5 that of the benchmark design while the six seater PPCEM is about

twice that of the benchmark. Unlike in Scenario 2, however, the benchmark designs also

have lower PLEV2 compared to the PPCEM designs. To see the discrepancy between the

individual goal achievement at the first priority level, the deviation variables for the four

performance goals which are considered at the first priority level—fuel weight (WFUEL),

maximum lift to drag ratio (LDMAX), maximum cruise speed (VCRMX), and maximum

flight range (RANGE)—are plotted in Figure 7.13. Some observations based on these

spider plots are as follows:

• The fuel weight target is met by all three benchmark aircraft while the PPCEM

aircraft do not achieve their target. In fact, the PPCEM aircraft exhibit increasingly

worse achievement of the target as the aircraft is scaled to accommodate more

passengers which can account for the increase in PLEV1 for the four and six seater

PPCEM aircraft.

• Neither family of aircraft achieves the target for maximum lift/drag ratio well;

however, the benchmark aircraft consistently perform better.

• All three of the PPCEM aircraft do better at achieving the target for maximum cruise

speed than do the individually designed benchmark aircraft.

• The PPCEM aircraft have only slightly worse RANGE achievement than the

benchmark aircraft.

277


SCENARIO 3 - Performance Tradeoff Study

0.000.020.040.060.080.10WFUEL

LDMAX

VCRMX

RANGE

4 Seater


Deviation Functions: PLEV1 6 Seater

0.00

0.05

0.10

0.15WFUEL

LDMAX

VCRMX

RANGE

2 Seater

0.00

0.02

0.04

0.06

0.08WFUEL

LDMAX

VCRMX

RANGE

Figure 7.13 Graphical Comparison of Benchmark Aircraft and PPCEMFamily for Design Scenario 3, Priority Level 1 Only

In summary, in order to improve the commonality of the aircraft within the GAA

product family, the overall performance of the individual aircraft within the product family

decreases. This decrease in performance, however, varies from aircraft to aircraft and

scenario to scenario. The question that the designers/managers are now faced with is: how

much performance degradation are we willing to accept so that we can have as common a

product platform as possible? Ideally, minimal performance would have to be sacrificed to

278

increase commonality between derivative products, but it appears that a tradeoff does exist

as one might expect. In reality, however, it would not be known how much performance

was being sacrificed by designing a common platform for the product family because

benchmark designs would not necessarily exist (unless this was a redesign process).

Toward this end, a product variety tradeoff study is performed in the next section.

7 .6 .4 Product Variety Tradeoff Study

To assess the tradeoff between product commonality and product performance

within the family of GAA, a product variety tradeoff study is performed using the PDI and

NCI measures described in Section 3.1.5. Currently, there are two points on the PDI vs.

NCI graph: the family of aircraft based on the PPCEM solutions and the group (family) of

benchmark aircraft which have been individually designed. What is interesting to study is

the effect of allowing one or more design variables to vary in the PPCEM for each aircraft

while holding the remaining variables constant at the platform values found using the

PPCEM. In this manner, the PPCEM facilitates generating a variety of alternatives for the

product platform and corresponding product family. By allowing one or more variables to

vary between aircraft, the performance of the individual aircraft within the resulting product

family can be improved such that there is minimal tradeoff between product commonality

and performance. Before this tradeoff can be assessed, however, the relative importance of

the design variables is needed in order to compute NCI for each family of aircraft.

The weightings in NCI used in this study are based on rank ordering the design

variables with regard to relative ease/cost with which they can be allowed to vary—the

more costly it is to allow that variable to change, the more important it is to have that

variable stay the same across derivative products. For this example, the weightings listed

in Table 7.17 are used. Cruise speed (CSPD) is the easiest/cheapest variable to allow to

vary between designs because it is easy to vary the cruise speed throughout the mission

279

without having to make any modifications to the aircraft; meanwhile, seat width (WS) is the

most expensive to allow to vary because it is costly not to have the same fuselage width

(fuselage width being directly proportional to seat width) for all of the aircraft within the

GAA family. These weights are derived from a pairwise comparison of the design

variables; the justification for the pairwise comparison and computation of the rank

ordering and relative importance are explained in Section F.4.1.

Table 7.17 Relative Importance of Design Variables

DesignVariable

RankOrder†

RelativeImportance

AF 3 0.1429AR 5 0.2381

CSPD 1 0.0476DPRP 2 0.0952WL 4 0.1905WS 6 0.2857

† Larger numbers indicate preference.

For this product variety study, two design scenarios are considered: the economic

tradeoff study (Scenario 2) and the performance tradeoff study (Scenario 3) listed in Table

7.13. For these two scenarios, the individual PPCEM and benchmark aircraft are listed in

Table 7.15 and Table 7.16 for Scenarios 2 and 3, respectively. The resulting PDI and NCI

for each group of aircraft based on these design variable values are computed and listed in

Table 7.18 and Table 7.20 for Scenarios 2 and 3, respectively; remember that only the first

priority level is used when computing PDI, and the weightings used in the NCI are the

relative importances listed in Table 7.17 and do not include the variation in the scale factor.

Knowing the two extremes of the PDI vs. NCI curve, it is possible to work

“backward” along the curve from the PPCEM solutions toward the benchmark designs by

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allowing one or multiple design variables to vary between each aircraft while holding the

others fixed at the PPCEM platform values. This process proceeds as follows:

1. Starting with the individual PPCEM aircraft, vary one variable at a time for each

aircraft; for instance, hold {AF, AR, CSPD, DPRP, WL} at the settings prescribed

by the PPCEM platform and vary WS for each aircraft to improve the performance

of that aircraft as much as possible. This entails solving a compromise DSP for

each aircraft, with the PPCEM value for WS taken as the starting point in DSIDES.

All six variables are allowed to vary one-at-a-time from the PPCEM platform

values, solving a compromise DSP for each aircraft for each variable. NCI and

PDI are then computed for each of the six resulting aircraft families, e.g., the family

of aircraft which share common {AF, AR, CSPD, DPRP, WL} but varying WS.

2. Repeat Step 1, allowing any two variables to vary at a given time between aircraft

from the PPCEM platform values. There are 15 possible pairs of variables which

are varied two-at-a-time, and a compromise DSP is solved for each possible pair

with the PPCEM value taken as the starting point. NCI and PDI are computed for

each of the 15 resulting product families.

3. Repeat Step 1, allowing any three variables to vary at a given time between aircraft.

In order to reduce the number of combinations that must be examined, CSPD is not

varied from aircraft to aircraft because it is known not to change much between

aircraft in the group of benchmark designs. Hence, only AF, AR, DPRP, WL, and

WS are allowed to vary from their PPCEM platform values, resulting in 10

different combinations of the five variables taken three at a time. NCI and PDI are

computed for each of the 10 resulting product families.

In total, (6x3)+(15x3)+(10x3) = 93 compromise DSPs are solved in this product

variety study for each scenario. The resulting NCI and PDI are listed in Table 7.18 for

Scenario 2 and in Table 7.20 for Scenario 3. In the tables, the results are grouped by the

number of variables where the variables not listed are being held constant. For instance, in

Table 7.18 the NCI and PDI for the family of aircraft when allowing WL to vary from one

aircraft to the next are 0.0171 and 0.0155, respectively, with all over design variables fixed

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at the PPCEM values; when AF and WL are allowed to vary from one aircraft to the next,

the resulting NCI and PDI for the group of products are 0.0234 and 0.0150, respectively.

Table 7.18 Product Variety Tradeoff Study - Scenario 2

NCI PDIBenchmark Designs - Each aircraft is optimized;

all variables can vary0.1795 0.0038

PPCEM Designs using Cdk - Each aircraft isdesigned to have same variables

0.0000 0.0184

Allow 1 AF 0.0178 0.0181variable to AR 0.0040 0.0181

vary between CSPD 0.0000 0.0182aircraft from DPRP 0.0026 0.0179

PPCEM WL 0.0171 0.0155designs WS 0.0381 0.0117

AR 0.0147 0.0181CSPD 0.0178 0.0181

AF DPRP 0.0155 0.0175WL 0.0234 0.0150

Allow 2 WS 0.0509 0.0113variables to CSPD 0.0041 0.0181

vary between AR DPRP 0.0172 0.0175aircraft from WL 0.0230 0.0147

PPCEM WS 0.0559 0.0096designs DPRP 0.0027 0.0179

CSPD WL 0.0233 0.0152WS 0.0382 0.0117

DPRP WL 0.0249 0.0154WS 0.0528 0.0106

WL WS 0.0702 0.0086DPRP 0.0110 0.0181

AR WL 0.0370 0.0146Allow 3 AF WS 0.0848 0.0100

variables to DPRP WL 0.0495 0.0154vary between WS 0.0672 0.0147aircraft from WL WS 0.1068 0.0081

PPCEM AR DPRP WL 0.0405 0.0151designs WS 0.0572 0.0107

AR WL WS 0.0803 0.0068DPRP WL WS 0.0701 0.0075

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The gray shaded rows in Table 7.18 indicate the best increase in PDI which can be

achieved by allowing 1, 2, or 3 variables to vary at a given time. So, if only one variable is

allowed to vary between aircraft, then allowing WS to vary yields the best improvement in

Scenario 2; if two can vary, then WL and WS should be allowed to vary; if three can vary,

then varying AR, WL, and WS yields the best improvement. The complete set of results

for each scenario for each aircraft are listed in Section F.4. Plots of NCI versus PDI for

each scenario follow each table and are discussed in turn. The PDI and NCI values for

Scenario 2 are plotted in Figure 7.14.

∆PDIi = best change in PDIby allowing i variables tovary b/n eachaircraft design

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0.00 0.03 0.06 0.09 0.12 0.15 0.18

NCI - Weighted by Importance

PD

I

Scenario 2 - Economic Tradeoff

Cdk

Cdk-Vary1Cdk-Vary2Cdk-Vary3Benchmark

∆PDI1

∆PDI2

∆PDI3

KEY:

∆PDIlost

∆NCIgain

WL

WS

WL,WSDPRP,WL,WS

AR,WL,WS

AR,WS

Figure 7.14 Scenario 2 Product Variety Tradeoff Study Results

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Notice that the PPCEM solution using the Cdk formulation yields the top left point

in Figure 7.14; the individual benchmark designs provide the bottom right point with all of

the variations on the PPCEM Cdk solutions falling in between the two, creating an envelope

of possible combinations of NCI and PDI. As highlighted in the Table 7.18, varying

{WS}, {WS and WL}, and {AR, WL, and WS} yields the best improvement in PDI if 1,

2, and 3 variables are allowed to vary between each of the PPCEM aircraft; notice that these

points lay on the front of the product variety envelope. In general, as more design

variables are allowed to vary, greater ∆PDI can be achieved but NCI does increase.

Is there any way to move down this curve without having to look at all possible

combinations? As it turns out, the design variables that have the most impact on the

performance of aircraft are the ones that progress down the front of the curve. This

information can be obtained from a statistical Analysis of Variance (ANOVA) of the data

used to build the kriging metamodels in Step 3 of the PPCEM. The full ANOVA for the

family of GAA is given in Section F.3, and Pareto plots based on the results of the

ANOVA are illustrated in Figure 7.15. The Pareto plots provide a means of quickly

identifying which variables have the most impact on a particular response; the larger the

horizontal bar, the more influence a variable has on the response. In Figure 7.15, only the

effects of the design variables on the response means have been plotted because they

govern the average performance of the GAA family.

Based on these Pareto plots for the GAA response means, the effect of each factor

on each response can be ranked by order of importance, see Table 7.19. In the table, 1

indicates most important and 6 the least. So for example, the seat width (WS) has the

largest effect on the purchase price (PURCH) and cruise speed (CSPD) has the least. The

economic responses in the first priority level in Scenario 2 are shown in the top half of the

table; the performance responses which are in the first priority level in Scenario 3 are

shown in the bottom half of the table.

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TermWSARWLDPRPAFCSPD

Orthog Estimate 38.206370 20.124813-17.199147 4.408426 2.979447 2.248510

.2 .4 .6 .8 TermCSPDDPRPWLAFARWS

Orthog Estimate 6.6543136-0.8645803-0.6941198-0.5676188 0.1848961 0.0400254

.2 .4 .6 .8

(a) µWEMP (b) µDOC

TermWSWLARDPRPAFCSPD

Orthog Estimate-39.709244 18.061805-17.572171 -5.366124 -3.216492 -2.750743

.2 .4 .6 .8 TermWSARDPRPWLAFCSPD

Orthog Estimate 620.43621 449.97104 336.36582-142.43496 116.66169 44.55396

.2 .4 .6 .8

(c) µWFUEL (d) µPURCH

TermWLWSDPRPARAFCSPD

Orthog Estimate-349.65163 -96.81741 -84.83165 -42.94887 -30.40475 -7.67476

.2 .4 .6 .8 TermWLDPRPWSARCSPDAF

Orthog Estimate 3.5626436 2.9891508-2.7846897 0.6015649-0.3211545 0.2301845

.2 .4 .6 .8

(e) µRANGE (f) µVCRMX

TermARWLWSCSPDAFDPRP

Orthog Estimate 1.0556939-0.3815507-0.3732309 0.3281411-0.0144062 0.0030519

.2 .4 .6 .8 KEY: DOC = direct oper costAF = eng act factor LDMAX = max lift/dragAR = aspect ratio PURCH = purchase priceCSPD = cruise spd RANGE = max flight rangeDPRP = prop diam VCRMX = max speedWL = wing loading WEMP = empty weightWS = seat width WFUEL = fuel weight

(g) µLDMAX

Figure 7.15 Pareto Plots for GAA Response Means

Table 7.19 Rank Ordering of Effects on Means of Responses

Importance on Economic Related GoalsResponse 1 2 3 4 5 6DOC CSPD DPRP WL AR WS AFWEMP WS AR WL DPRP AF CSPDPURCH WS AR DPRP WL AF CSPD

Importance on Performance Related GoalsResponse 1 2 3 4 5 6LDMAX AR WL WS CSPD DPRP AFRANGE WL WS DPRP AF AR CSPDWFUEL WS WL AR DPRP AF CSPDVCRMX WL DPRP WS AR CSPD AF

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Returning to the tradeoff study for Scenario 2, the design variables that shape the

front of the envelope when allowed to vary are as follows: WS, WL, {AR,WS},

{WS,WL}, {WS, WL, DPRP}, and {AR, WL, WS}. Looking at the rank ordering of

importance in Table 7.19, it can be seen that the variables in these combinations are

variables that have the largest effect on the responses. WS has the largest effect on WEMP

and PURCH, two of the three economic responses in Scenario 2; {WS, AR} are the two

most important factors both of these economic responses; {AR, WL, WS} are among the

top three variables that are most important to the three economic responses in Scenario 2.

Thus, by allowing the design variables with the most impact to vary between aircraft while

keeping the others fixed, substantial improvements in performance can be obtained.

To see if the same holds true in Scenario 3, the NCI and PDI values for Scenario 3

are listed in Table 7.20 and plotted in Figure 7.16. As highlighted in the table, the best

improvement in PDI can be obtained by allowing AR to vary between aircraft if only one

variable is allowed to vary. Notice in Table 7.19 that AR is most important to LDMAX.

Recall from Figure 7.13 that the largest discrepancy between the PPCEM family of aircraft

and the benchmark group of aircraft is the achievement of LDMAX. By allowing AR to

vary between aircraft within the PPCEM family, each aircraft is able to achieve better

LDMAX, resulting in a lower PDI for the PPCEM product family. Meanwhile, if two

variables are allowed to vary, then AF and WL yield the best improvement in PDI because

WL has a large impact on both RANGE and VCRMX. In the three variable case, varying

DPRP, WL, and WS yield the best improvement; notice that all three of these variables are

among the most influential variables on the performance responses as shown in Table 7.19.

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Table 7.20 Product Variety Tradeoff Study - Scenario 3

NCI PDIBenchmark Designs - Each aircraft is optimized;

all variables can vary0.0918 0.0284

PPCEM Designs using Cdk - Each aircraft isdesigned to have same variables

0.0000 0.0452

Allow AF 0.0267 0.04521 variable AR 0.0059 0.0434to vary CSPD 0.0000 0.0453

b/n aircraft DPRP 0.0013 0.0452from Cdk WL 0.0010 0.0437designs WS 0.0068 0.0443

AR 0.0269 0.0430CSPD 0.0000 0.0453

AF DPRP 0.0193 0.0450WL 0.0119 0.0390

Allow WS 0.0159 0.04402 variables CSPD 0.0017 0.0452

to vary AR DPRP 0.0101 0.0428b/n aircraft WL 0.0082 0.0402from Cdk WS 0.0183 0.0404designs DPRP 0.0000 0.0453

CSPD WL 0.0049 0.0397WS 0.0093 0.0449

DPRP WL 0.0150 0.0401WS 0.0081 0.0446

WL WS 0.0071 0.0422DPRP 0.0530 0.0430

AR WL 0.0085 0.0394Allow AF WS 0.0361 0.0399

3 variables DPRP WL 0.0272 0.0388to vary WS 0.0158 0.0442

b/n aircraft WL WS 0.0349 0.0408from Cdk AR DPRP WL 0.0135 0.0414designs WS 0.0203 0.0397

AR WL WS 0.0154 0.0417DPRP WL WS 0.0194 0.0361

The PDI and NCI values for Scenario 3 are plotted in Figure 7.16. As in the

previous graph for Scenario 2, the PPCEM solution using the Cdk formulation yields the

top left point; the individual benchmark designs provide the bottom right point. Notice that

the combinations of design variables which move the family of aircraft down the front of

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the product variety envelope are, in general, the ones which are rank ordered highest in

Table 7.19. Notice also that more than half of ∆PDIlost can be gained back if {DPRP, WL,

WS} are allowed to vary between aircraft with only a minimal increase in NCI.

0.025

0.030

0.035

0.040

0.045

0.050

0.00 0.02 0.04 0.06 0.08 0.10

NCI - Weighted by Importance

PD

I

Scenario 3 - Performance Tradeoff

∆PDI1

∆PDI2∆PDI3

∆PDIi = best change in PDIby allowing i variables tovary b/n eachaircraft design

Cdk

Cdk-Vary1Cdk-Vary2Cdk-Vary3Benchmark

KEY:

∆PDIlost

∆NCIgain

WL

AR

DPRP,WL,WS

CSPD,WL

AF,WL

Figure 7.16 Scenario 3 Product Variety Tradeoff Study Results

In closing, despite the tradeoff between commonality and performance observed in

comparing the benchmark and PPCEM aircraft in Section 7.1.2, considerable improvement

in the performance of the PPCEM family of aircraft can be obtained by allowing 1 or more

variables to vary between aircraft while holding the remainder of the variables at the

platform setting. In the product variety study performed in this section, it has been shown

how statistical analysis of variance can be used to traverse the front of the product variety

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tradeoff envelope, maximizing the gains in PDI with minimal loss in commonality. It is

now up to the discretion of the designers/managers to evaluate the implications of this

tradeoff on inventory, production, and sales to decide the appropriate compromise between

commonality and performance. A closer look at some of the lessons learned from this

example is offered in the next section along with a summary of the chapter.

7.7 LESSONS LEARNED: A LOOK BACK AND A LOOK AHEAD

In this chapter, the PPCEM is applied in full to the design of a family of General

Aviation aircraft. The GAA family is based on a common scalable product platform which

is scaled around the number of passengers in much the same way that Boeing has scaled

their 747 series of aircraft around the capacity and flight range (cf., Rothwell and Gardiner,

1990). The market segmentation grid has been used to help identify an appropriate

leveraging strategy for the family of aircraft based on the initial problem statement, i.e.,

horizontally leverage the family of GAA to satisfy a variety of low-end market segments.

Each aircraft eventually could be vertically scaled as well through the addition and removal

of features as technology improves to increase its performance and attractiveness to a mid-

range or high-end customer base.

Particularization of the PPCEM for this example occurs through GASP, the General

Aviation Synthesis Program, which is used to model and simulate the performance of each

aircraft mathematically . Kriging metamodels for response means and variances are

employed within the PPCEM to facilitate the implementation of robust design based on

GASP analyses. These kriging metamodels then are used in conjunction with design

capability indices and a GAA compromise DSP to synthesize a robust aircraft platform

which is scalable into a family of aircraft.

Three different design scenarios are used to exercise the GAA compromise DSP to

create alternative product platforms and the product platform portfolio. Instantiation of the

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individual aircraft within the PPCEM family reveals that the PPCEM provides an effective

means for designing a common scalable aircraft platform for the family of GAA.

However, upon comparison with individually designed benchmark aircraft, a tradeoff is

found to exist between having a common set of design variables which define the aircraft

platform and the performance of the scaled derivatives based on that platform. To examine

the extent to which this tradeoff occurs, a product variety tradeoff study is performed using

the PPCEM to demonstrate the ease with which alternative product platforms and product

families can be generate and to make use of the NCI and PDI measures proposed in Section

3.1.5. It is observed that considerable improvement can be made by allowing one or more

variables to vary between each aircraft based on the original PPCEM platform; however,

commonality between the aircraft is sacrificed. To determine which variables to vary,

ANOVA of the data used to build the kriging metamodels in Step 3 of the PPCEM can be

used to determine the variables that have the largest effect on each response, allowing the

front portion of the product variety envelope to be traversed for maximum improvement in

PDI with minimal loss of commonality. The implications of this tradeoff on inventory,

production, and sales must be considered in order to decide upon an appropriate

compromise between commonality and performance. However, the purpose of the

PPCEM is to facilitate generating a variety of alternatives for the common product platform

and corresponding product platform and not to select one from them.

A summary of the hypotheses tested in this example is as follows:

Hypothesis 1 - The PPCEM is employed in this chapter to design a family of aircraft

based on a common scalable product platform. The success of the method to

improve the baseline design and generate a variety of alternatives for the GAA

platform as discussed in Sections 7.5 and 7.6 provides further verification of

Hypothesis 1.

Sub-Hypothesis 1.1 - The market segmentation grid is utilized in Section 7.1.3 to

help identify an appropriate (horizontal) scale factor for the family of GAA—the

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number of passengers—in order to achieve the desired platform leveraging based

on the problem objectives; this further supports Sub-Hypothesis 1.1.

Sub-Hypothesis 1.2 - The scale factor for the GAA product family is the number of

passengers, see Sections 7.1.3 and 7.2. Robust design principles then are used in

this example to develop an aircraft platform—defined by six design variables—

which is insensitive to variations in the scale factor and is thus good for the family

of General Aviation aircraft based on the two, four, and six seater configurations.

The success of this implementation helps to support Sub-Hypothesis 1.2.

Sub-Hypothesis 1 .3 - Design capability indices are utilized in this example to

aggregate individual targets and constraints and to facilitate the design of a family of

General Aviation aircraft. Combining this formulation with the compromise DSP

allows a family of GAA to be designed around a common, scalable product

platform, further verifying Sub-Hypothesis 1.3.

Hypothesis 2 - Kriging metamodels are utilized to facilitate the implementation of the

design capability indices and expedite the search for a suitable product platform for

the family of GAA. Validation of the kriging metamodels in Section 7.3 and 7.6.2

indicates that the kriging models are sufficiently accurate to benefit the PPCEM.

So, are the solutions obtained from the PPCEM useful? The PPCEM has been

used to generate a variety of feasible options for the GAA platform and corresponding

family of aircraft. While there is some tradeoff between the performance of the individual

aircraft based on the PPCEM platform when compared to a family of individually designed

benchmark aircraft, the increased commonality between the design specifications of each

aircraft (i.e., aspect ratio, seat width, propeller diameter, etc.) should generate sufficient

savings to offset the minimal loss in performance. Regardless of whether it does or not,

the family of aircraft obtained using the PPCEM yields considerable improvement over the

initial family of aircraft based on the baseline Beechcraft Bonanza design, see Table 7.8 and

the discussion thereafter.

Are the time and resources consumed within reasonable limits? Basically, the

PPCEM has been used to design a family of three aircraft almost as efficiently as a single

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aircraft. The initial start up costs to use the PPCEM in this example are about one day

which is the time it takes to sample the GAA design space and construct kriging

metamodels to approximate GASP. Once this is accomplished, the computational savings

resulting from using the PPCEM are significant in two regards:

• computational efficiency gained by using kriging metamodels instead of GASP, and

• design efficiency gained by using the PPCEM to design a family of aircraft

simultaneously around a common scalable platform.

As a result, the computational savings are comparable to, if not greater than, those obtained

in the universal electric motor example in Chapter 6. Consider, for instance, that it requires

approximately 45 seconds to complete one “run” of GASP on a Unix-based SparcServer

670 MP. Meanwhile, the kriging metamodels require approximately 0.25 seconds to run

after about a minute of “pre-processing” which is done automatically prior to optimization.

The savings from using metamodels in the PPCEM are substantial when one considers the

large numbers of design scenarios and tradeoff studies used in this chapter and in Appendix

F, not to mention the fact that multiple starting points are used in all cases. The cost

savings (in terms of number of analysis) are not as clear cut as they are in the universal

motor example in Chapter 6 (see Section 6.5.3); therefore, they are not estimated.

However, the discussion in (Simpson, 1995) regarding the cost savings of using

approximations to replace GASP sheds some light on the magnitude of these savings.

Is the work grounded in reality? As stated in Section 7.1.3, the baseline design

(i.e., starting point) for the GAA product family is the Beechcraft Bonanza B36TC

presented in Section 7.1.3. While the Beechcraft Bonanza is only a six seater aircraft, its

specifications are employed in GASP to provide a family of baseline to compare with the

PPCEM family of aircraft based on a common scalable platform. Discussion of these

results in Section 7.5.1 reveals that the PPCEM solutions are able to improve upon both the

technical and economic performance characteristics of this family of baseline aircraft

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significantly. While these improvements are slightly less than the improvements obtained

by individually designing each aircraft (i.e., the benchmark designs), the time savings

resulting from using the PPCEM to design the family of three aircraft simultaneously can

be used to “tweak” the individual designs as needed to ensure adequate performance and

product quality. It still stands, however, that the PPCEM solutions, even with all six

design variables held at the common product platform specifications, yield improvement

over the baseline design. The results of the product variety tradeoff studies discussed in

Section 7.6.4 provides several options to improve the PPCEM family of aircraft.

Finally, do the benefits of the work outweigh the cost? The true benefit from using

the PPCEM in a problem like this is the wealth of information that is obtained during its

implementation. The PPCEM greatly facilitates the generation of a variety of alternatives

for a common product platform and its corresponding scaled derivative products. Use of

the PPCEM permits the product platform and the scaled product family to be designed

simultaneously, thus increasing the commonality of specifications across the products

within the family. Product variety tradeoff studies can be easily performed using the

PPCEM (and NCI and PDI metrics) to evaluate the compromise between commonality and

individual product performance, yielding a variety of options for the company to pursue.

This concludes the second, and final, example for testing and verifying the PPCEM

having demonstrated the full implementation of the PPCEM to design a family of products

and facilitate product variety tradeoff studies. In the next and final chapter, a summary of

achievements and contributions from the work is offered along with critical review of the

research and a discussion of possible avenues of future work.

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8. CHAPTER 8

CLOSURE: ACHIEVEMENTS ANDRECOMMENDATIONS

In this dissertation, a method has been developed, presented, and tested to facilitate

the design of a scalable product platform for a product family. The development and

presentation of this method is brought to a close in this chapter. In Section 8.1, closure is

sought by returning to the research questions posed in Chapter 1 and reviewing the

answers that have been offered. The resulting contributions are then summarized in

Section 8.2. Limitations of the research are discussed in Section 8.3, and possible avenues

of future work are described in Section 8.4. Concluding remarks are given in Section 8.5,

closing this chapter and the dissertation.




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8.1 CLOSURE: ANSWERING THE RESEARCH QUESTIONS

As stated in the introduction to Chapter 1, the principal objective in this dissertation

is to develop the Product Platform Concept Exploration Method (PPCEM) to facilitate the

design of a common product platform which can be scaled to realize a product family. In

particular, the concept of platform scalability is introduced and exploited in the context of

the following motivating research question.

Q1. How can a common scalable product platform be modeled and designed for a

product family?

Two secondary research questions are also offered in Section 1.3.1 for investigation in this

dissertation in conjunction with the primary research question.





To address these questions, research hypotheses and posits are introduced and

identified in support of achieving the principal objective for the dissertation. Their

elaboration and verification have provided the context in which the research work has

proceeded. The end result is a synthesis of engineering design, operations research,

applied statistics, and strategic management methods and tools to form the Product

295

Platform Concept Exploration Method. Its development has been portrayed pictorially

using Figure 8.1 which depicts the flow of the research throughout the dissertation.

Nozzle Design




Metamodeling



ConceptualNoise

Factors


MarketSegmentation

Grid



Chp 6

Chp 4 Chp 3

§2.2§2.3 §2.1

Chp 5

Chp 7


Platform


Figure 8.1 Pictorial Overview of the Dissertation

296

Answering Question 1: Question 1 is the primary research question posed for

the work in this dissertation and its answer is embodied by the Product Platform Concept

Exploration Method: a Method which facilitates the synthesis and Exploration of a common

Product Platform Concept which can be scaled into an appropriate family of products. The

method consists of a prescription for formulating the problem and a description for solving

it. Application of the method is demonstrated by means of two examples, namely,

• the design of a universal electric motor platform which is (vertically) scaled around

the stack length of the motor to realize a family of electric motors capable of

satisfying a variety of torque and power requirements (Chapter 6), and

• the design of a General Aviation aircraft platform which is (horizontally) scaled into

a two, a four, and a six seat configuration to realize a family of aircraft capable of

satisfying a variety of performance and economic requirements (Chapter 7).

While only demonstrated for these two examples, it is asserted that the method is generally

applicable to other examples in this class of problems: parametrically scalable product

platforms whose performance can be mathematically modeled or simulated. Other

examples which have taken advantage of this type of scaling include the design of a family

of oil filters (Seshu, 1998) and the design of a family of absorption chillers for a variety of

refrigeration capacities (Hernandez, et al., 1998). Both examples integrate nicely within

the framework of the PPCEM.

In support of the primary research question and objective, three additional questions

also are offered in Section 1.3.1. Answers to these questions are summarized as follows.



297

In this research, the market segmentation grid (Meyer, 1997) is employed to help

identify platform scaling opportunities based on overall design requirements. Its success as

an attention directing tool for mapping scaling opportunities within a product family is

discussed in Section 2.2.1 and then demonstrated in both examples. In the universal motor

example in Chapter 6, the market segmentation grid is used to identify vertical scaling

opportunities within the desired product family to realize a range of torque and power

ratings for different price/performance tiers within the market; standardization of the motor

interfaces will provide horizontal leveraging opportunities of this family of motors into

other market segments in a manner similar to Black & Decker’s response to Double

Insulation in the 1970s (Lehnerd, 1987). Meanwhile, in the General Aviation aircraft

example in Chapter 7, a horizontal leveraging strategy is identified by means of the market

segmentation grid, resulting in a family of three aircraft based on a two, four, and six seater

configuration leveraged about a common product platform. Opportunities for vertical

scaling of the resulting family of aircraft through engine upgrades, add-on features, and

technological advancements also are discussed; however, none of these features are

implemented in this example.

Q1.2. How can robust design principles be used to facilitate designing a common


By identifying “conceptual noise” factors around which a family of products can be

scaled, robust design principles can be abstracted for use in product family and product

platform design. Consequently, the idea of a scale factor is introduced in Section 2.3.2 as

a factor around which a product platform can be “scaled” or “stretched” to realize derivative

products within a product family. Scale factors are, in essence, noise factors for a scalable

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product platform, and robust design principles can be used accordingly to minimize the

sensitivity of the product platform to variations in these scale factors. Implementation of

this approach is demonstrated through the two examples. In the universal motor example

in Chapter 6, the stack length of the motor is taken as the (parametric) scale factor around

which a family of motors is created. In the General Aviation Aircraft example, the number

of passengers is the (configurational) scale factor around which a family of three aircraft are

developed. In both cases, robust design principles are employed to develop a common set

of design variables which are robust with respect to variations in the scaling factor as the

product platform is scaled and instantiated to realize the product family. A product variety

tradeoff study also is performed in the General Aviation aircraft example (see Section

7.6.2) to further verify this approach.

Q1.3. How can individual targets for derivative products be aggregated and

modeled for product platform design?

Through the identification of appropriate scaling factors during the product family

design process, the individual targets for derivative products can be aggregated into a mean

and variance around which the product family can be simultaneously designed either by

having separate goals for “bringing the mean on target” and “minimizing the variation” or

through the formulation and implementation of appropriate design capability indices to

measure the capability of a family of designs to satisfy a ranged set of design requirements.

The former approach is utilized to design the universal electric motor platform in Chapter 6.

Goals for “bringing the mean on target” and “minimizing the variation” caused by

variations in the scale factor (stack length) are used within a compromise DSP to effect a

platform design which matches the target mean and variation for the aggregated product

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family. In Chapter 7, design capability indices are employed to design a family of General

Aviation aircraft around a common platform which is instantiated to seat 2, 4, and 6

passengers.

Answering Question 2: Since its introduction into the literature as a useful

metamodeling tool for engineering design by Sacks, et al. (1989), kriging has received little

attention by the engineering community building surrogate models. Perhaps this is because

of the added complexity of fitting the model or using it or of the inability to glean useful

information directly from the MLE parameters used to fit the model. Whatever the reason,

the research in this dissertation has been directed at improving the ease with which kriging

models can be built, validated, and used. Moreover, the initial feasibility study and

comparison of kriging models—with a global underlying constant—with second-order

response surface models in Chapter 4 and the extensive kriging/DOE investigation in

Chapter 5 is aimed at familiarizing the reader with kriging and making it a viable alternative

for building surrogate metamodels of deterministic computer experiments. Its utility was

tested extensively in Chapter 5 wherein it was concluded that the Gaussian correlation

function provides the most accurate kriging predictor, on average, and that kriging can

accurate model a wide variety of functions typical of engineering analysis. While the study

is not all inclusive, nor is it intended to be, it has provided valuable insight into the utility of

kriging metamodels for engineering design. Potential avenues of future work to extend this

promising metamodeling alternative are discussed in Section 8.4.1.

Answering Question 3: As discussed in Section 2.4.3, many researchers argue

that classical experimental designs are not well suited for sampling computer experiments

which are deterministic; rather, points should be chosen to “fill the space,” providing good

coverage of the design space since replicate sample points are not needed. In an effort to

verify the utility of space filling experimental designs, a comparison of nine space filling

and two classical experimental designs is performed in Chapter 5 (see Section 5.4 in

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particular) to address this third research question. The eleven experimental designs are

compared on the basis of their capability to produce accurate kriging metamodels for the

testbed of six engineering problems used in this dissertation. For the sample sizes

investigated in this study, it was observed that the space filling experimental designs

yielded more accurate kriging models in the larger design spaces (3 and 4 variables) while

the classical experimental designs (CCDs) performed well in the two dimensional design

space for the reasons discussed at the end of Section 5.4.4. Prior to this investigation, few

researchers had compared their experimental designs against one another, or to classical

designs for that matter. As such, the findings in the kriging/DOE study in Chapter 5

represent unique contributions from the research. A summary of the research contributions

is offered in the next section.

8.2 ACHIEVEMENTS: REVIEW OF RESEARCH CONTRIBUTIONS

The contributions offered in this dissertation are introduced in Section 1.3.2 and

realized throughout the dissertation. As stated at the beginning of Chapter 1, the primary

contribution from this work is embodied in the Product Platform Concept Exploration

Method which provides a method to identify, model, and synthesis scalable product

platforms for a product family. The other contributions can be summarized as follows:

Contributions Related to Hypothesis 1 and Sub-Hypotheses 1.1-1.3:

• A procedure for identifying scale factors for a product platform, see Sections 3.1.1

and 3.1.2.

• An abstraction of robust design principles for realizing scalability in product family

design, see Sections 3.1.2 and 3.1.4.

• Non-commonality and performance deviation indices for performing product

variety tradeoff studies, see Sections 3.1.5 and 7.6.2.

Contributions Related to Hypothesis 2:

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• An algorithm to build, validate, and use a kriging model, see Section 2.4.2,

Chapter 4 and 5, and Appendix A.

• A preliminary comparison of the predictive capability of second-order response

surfaces and kriging models in the design of a rocket nozzle, see Section 4.2.

• An extensive investigation of the effect of five different spatial correlation functions

on the accuracy of a kriging model, see Sections 2.4.2 and Chapter 5.

Contributions Related to Hypothesis 3:

• An extensive investigation of the effect of eleven different sampling strategies on

building an accurate kriging model, see Sections 2.4.3 and Chapter 5.

• An algorithm for generating minimax Latin hypercube designs, see Section 2.4.3

and Appendix C.

What is the value of these contributions? These contributions must be of sufficient

worth to be either an addition to the fundamental knowledge of the field or a new and better

interpretation of the facts already known. The contributions associated with kriging

represent a new interpretation of facts already known. Kriging has been around since the

1960s (see, e.g., Cressie, 1993; Matheron, 1963) when it was developed originally for

mining and geostatistics applications; however, it has received limited attention in the

engineering design community until recently. The kriging algorithm presented is not totally

unique to this dissertation; however, the use of a simulated annealing algorithm (see

Appendix A for more details on its use in the maximum likelihood estimation for the

kriging metamodels) to find the “best” kriging model is. Moreover, the comparison of the

accuracy of different correlation functions on the resulting kriging model had never been

performed in such depth. Likewise, the comparison of space filling and experimental

designs represents a new and better interpretation of facts already known because such an

extensive study has never been undertaken. With the exception of the minimax Latin

hypercube design, the experimental designs investigated in this dissertation are the result of

years of research work by statisticians and mathematicians. The minimax Latin hypercube

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design, however, represents an addition to the fundamental knowledge of the field of

experimental design.

The contributions made in the area of product family design, specifically the method

of designing scalable product platforms, represents an addition to the fundamental

knowledge of the field. While other product family design strategies and methods have

been slowly evolving, the investigation of a method for platform scaling is previously

unrecorded. The incorporation of the market segmentation grid into the engineering design

process provides a new interpretation of facts already known, demonstrating how the

market segmentation grid becomes a useful attention directing tool for identifying platform

leveraging strategies in product family and, with a little engineering knowledge, appropriate

scale factors for the intended scalable platform. In this regard, the concept of scale factors

in product family design and extending robust design to product family and product

platform design is unique to this dissertation as are the NCI and PDI measures for product

family non-commonality and performance deviation. The measures are not of significant

value in and of themselves, however, the product variety tradeoff studies which these

indices make possible provide significant insight into the tradeoffs of product family

design. Taken together, the resulting Product Platform Concept Exploration Method for

designing scalable product platforms for a product family provides an addition to the

fundamental knowledge of the nascent field of product family design. However, the

PPCEM is by no means a panacea for product platform and product family design nor is it

without its limitations. Toward this end, a critical evaluation of the work is offered in the

next section followed by recommendations for future work in Section 8.4.

8.3 CRITICAL ANALYSIS: LIMITATIONS OF THE RESEARCH

This section comprises the confessional portion of the dissertation wherein the

research itself is critically evaluated. Already the PPCEM has been critically evaluated as it

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pertains to the two example problems in Chapters 6 and 7, see Sections 6.5 and 7.6.5. In

this section, the critical evaluation is applied to the work as a whole.

So what is really necessary for the PPCEM to be applied to the design of a product

family based on a common scalable product platform? There are two basic requirements

which must be met in order for the PPCEM to be applicable to the design of a scalable

product platform. First, the concept of scalability must be exploitable within the product

family; exploited in the sense that having one or more scale factors provides a means to

realize a variety of performance requirements while also facilitating the manufacturing

process. For instance, in the electric motor example in Chapter 6, the motor could have

just as easily been scaled in the radial dimension as it was in the axial direction (i.e., stack

length) to achieve the necessary torque requirements; however, the underlying assumption

in the choice of stack length as the scale factor is that it can be exploited from both a

technical sense and a manufacturing sense. As Lehnerd (1987) alludes to in his article on

Black & Decker and their universal motor platform, by varying only the stack length of the

motor, all of the motors—ranging from 60 Watts to 660 Watts—could be produced on the

same machine simply by stacking more laminations onto the field and armature. Had the

radius of the motor been scaled instead of the stack length, different machines and tooling

configurations would have been required to produce the family of motors since varying the

radius of the motors is more than a stacking operation. Consequently, it is very important

that one or multiple scale factors be identified for the product family and that it be capable

of being exploited from both a technical standpoint and a manufacturing standpoint in order

for the PPCEM to yield useful results.

The second consideration when applying the PPCEM is that the performance of the

product family must be able to mathematically modeled, simulated, or quantified in order

for the PPCEM to be employed. It would be extremely difficult, if not impossible, for the

PPCEM to be utilized to design a common scalable automotive body platform based solely

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on aesthetic considerations for instance. Consider the examples discussed in Section 1.1.1;

to which of these examples could the PPCEM be applied and why (or why not)? For the

sake of brevity, the answers are summarized in Table 8.1.

Table 8.1 Applicability of PPCEM to Product Families from Section 1.1.1

Examplefrom §1.1.1

WouldPPCEMApply?

Why or Why Not?

Sony: Walkman NoTheir platform strategy involves modular design andstandardization of components; few, if any, scaling

issues are present within the product family.

Nippondenso:Panel Meters No

They employ a combinatoric strategy to realize thenecessary product variety based on a few well-designed,standardized parts; few, if any, scaling issues are present.

Lutron: LightingControl Systems No Same reasoning as Nippondenso.

Black & Decker:Universal Motor Yes

The platform is scaled around the stack length of motor andan attempt was made to recreate their family of motors as

the initial “proof of concept” for the PPCEM in Chapter 6.

Canon: Copiers Notreally

The majority of copier design involves modular designof components and assemblies; however, some scaling

issues may arise to accommodate different print volumes,paper sizes, etc.

Rolls Royce:RTM322 Engine

Yes, insome

aspects

The RTM322 was scaled to create a new product platform,but modularity of engine components facilitated vertical

scaling of the platform to upgrade and derate engine.

As stated in Section 1.1.2, the types of problems to which the PPCEM is readily

applicable (given that the previous two conditions regarding scalability and quantifiability

are met) typically involve parametric or variant design. The fact that the PPCEM is

intended primarily for parametric or variant design raises another important issue, namely,

successful implementation of the PPCEM assumes that the basic concept or configuration

on which the product platform is being based is good for the entire product family. In

order for the PPCEM to be employed, a good underlying concept or configuration must

have already been established in order to obtain the full benefit of the method. In the GAA

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example in Chapter 7, for instance, if the three blade, high wing position, retractable

landing gear configuration had not been a suitable concept for the two, four, and six seater

aircraft, then no matter what parameter settings were obtained from using the PPCEM, the

performance of the family of aircraft would have been poor regardless because the

underlying concept was not good for all three aircraft. An attempt to identify a good

configuration for the family of GAA is discussed in (Simpson, 1995) but the existence of

such a concept is assumed to already exist in this work. Incorporation of the conceptual

and configurational design of the product family along with the parametric scaling of the

product platform is a fertile area for future work.

Furthermore, it is important to keep in mind that the PPCEM facilitates generating

options for common product platforms which can be scaled into an appropriate product

family. The PPCEM is not necessarily intended to be used to evaluate these options or

select one of them. The idea behind the product platform portfolio—the output from

applying the PPCEM—is to maintain sufficient design flexibility to accommodate a wide

variety of customer requirements for as long as possible. As the product platform design

progresses into the detailed stages of design, this design freedom is reduced; however,

during the early stages of the design process, formulating and answering a variety of "what

if" type questions and examining a wide variety of design scenarios is important to the

product platform design process.

Meanwhile, the NCI and PDI measures introduced in Section 3.1.5 and employed

in Section 7.6.4 represent an attempt to provide a means to evaluate different product

platforms and their respective product families. Ultimately the non-commonality of a set of

parameters would be linked to corresponding savings in manufacturing costs and

performance deviation to losses in customer sales; however, this is extremely difficult to

accomplish without sufficient industry input. Modeling the process and manufacturing

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aspects of product platforms and product families is another fertile research area which has

yet to be explored.

As far as the scale factors themselves goes, the concept of a scale factor—while

discussed in Section 2.3.2 and 3.1.2—is still not fully understood. In the motor example

in Chapter 6, for instance, the mean and standard deviation of the motor stack length was a

scale factor which was treated much like a design variable. Meanwhile, in the GAA

example in Chapter 7, the scale factor was the number of passengers which was treated as a

design parameter which varied from two to six, i.e., its permissible range of values was

known a priori based on the intended leveraging strategy. In any event, when metamodels

are to be utilized within the PPCEM, an initial range for each scale factor is necessary in

order to construct these metamodels. This follows in the same manner that a permissible

range of any noise factor is expected to be known before robust design principles can be

applied to a problem (cf., Phadke, 1989). It is important to examine the concept of scale

factors further, finding more examples of scaled product platforms to understand the

manner in which they have been scaled and, more importantly, how those scale factors are

identified during the design process.

This brings to light another shortcoming of the PPCEM, namely, the use of the

market segmentation grid to “identify” scale factors around which the product platform is

leveraged within a product family. As stated in Section 2.2.1, the market segmentation

grid is only an attention directing tool and considerable engineering “know-how” and

problem insight are required before a successful platform leveraging strategy can be

identified. Then, only after a suitable platform leveraging strategy is identified, can

engineers hope to find (and be able to exploit) scaling opportunities within the product

family to realize the necessary product variety. The market segmentation grid is the end

result of this process and is really only useful for mapping the resulting platform leveraging

strategy. The two examples used in this dissertation trivialize this process when in reality it

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is extremely difficult, if not impossible, to identify one or more scaling factors which can

be exploited within a product family. Developing tools and methods to facilitate the

process of identifying scale factors is one potential avenue for further investigation.

Part of understanding scale factors better involves understanding their effect on

product performance and how scale factors can be used effectively to satisfy a wide variety

of customer requirements. If scale factors induce too much variability in product

performance, then it might not be possible to apply the PPCEM to develop a common

product platform which does not significantly compromise the performance of the product

family over the range of interest. In such a case, it might be necessary to “split” the design

space into two or more product platforms and corresponding product families rather than

compromise product performance and quality by having one single product platform which

is scaled over the entire range of performance. The work in (Chang and Ward, 1995;

Lucas, 1994; Rangarajan, 1998; Seshu, 1998) further investigates and discusses these

types of issues. Lucas (1994) in particular presents interesting remarks on how to resolve

these types of issues using concepts from robust design as mentioned in Section 2.3.2.

Turning to specific implementation issues within the PPCEM, it may not have been

sufficiently clear that kriging, while part of the PPCEM, is not an integral part of the

PPCEM since it is not the only metamodeling technique which can be used within the

PPCEM. Response surfaces, neural nets, radial basis functions, etc. are all viable

metamodeling options for use in engineering design and with the PPCEM. The extensive

literature review of metamodeling applications in engineering design in (Simpson, et al.,

1997b) supports this. The most important consideration when using metamodels as

surrogate approximations in engineering design is that they are sufficiently accurate for the

task at hand.

The investigations into kriging in this dissertation are primarily intended to shed

light on alternative metamodeling techniques which offer some advantages to response

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surface models which are typically employed. The case for investigating alternatives to

response surfaces has been made in Section 2.4.1 and is also discussed in (Simpson, et al.,

1998; Simpson, et al., 1997b). The objective in this research is not to prove that kriging

metamodels are better than response surface models; rather, it is to demonstrate that kriging

metamodels are a viable alternative for building surrogate approximations of deterministic

computer analysis.

Similarly, the use of space filling experimental designs as opposed to classical

designs are not mandated by this research. The investigation served to gain a better

understanding of the different sampling strategies which exist and the associated

advantages and disadvantages of each. If one experimental design type had proven

superior in every example, then perhaps only that design should be considered in the

future. However, that was not the case, and the results of this study are by no means

generalizable to all types of engineering design problems. Very few engineering problems,

for instance, involve only two to four variables, and the availability of codes to generate

these space filling designs, the computation expense of them, and the nature of the

underlying analyses are just a few of the key factors that influence the decision of how to

sample a design space efficiently and effectively. Recommendations for future work in the

areas of experimental design and kriging are discussed in more detail in the next section.

8.4 RECOMMENDATIONS: AVENUES OF FUTURE WORK

8.4 .1 Potential Avenues of Future Work in Metamodeling

While extensive in nature, the experimental design study offered in Chapter 5 is by

no means complete nor is it intended to be. Obviously, a wider variety problems should be

considered in order to obtain more generalizable recommendations. Additional space filling

and classical experimental designs which have not been considered include the following:

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Classical Experimental Designs: fractional factorial designs and small central

composite designs (see, e.g., Box and Draper, 1987); D-optimal designs (see, e.g.,

Box and Draper, 1971; Giunta, et al., 1994; Mitchell, 1974; St. John and Draper,

1975); I-, A-, E-, and G-optimal designs (see, e.g., Hardin and Sloane, 1993;

Myers and Montgomery, 1995); minimum bias designs (see, e.g., Myers and

Montgomery, 1995; Venter and Haftka, 1997); and other hybrid designs (see, e.g.,

Giovannitti-Jensen and Myers, 1989; Myers and Montgomery, 1995)

Space Filling Experimental Designs: median Latin hypercubes (see, e.g.,

Kalagnanam and Diwekar, 1997; McKay, et al., 1979); minimax and maximin

designs (Johnson, et al., 1990); scrambled nets (Koehler and Owen, 1996);

orthogonal arrays of different strengths (Owen, 1992); Maximum entropy designs

(Currin, et al., 1991; Shewry and Wynn, 1987; Shewry and Wynn, 1988); and

factorial hypercube designs (Salagame and Barton, 1997).

In addition to including a wider array of experimental designs and sampling

strategies in the current testbed of problems, larger problems also should be investigated

because very few engineering problems only have 2-4 variables. However, problems with

larger dimensional design spaces (i.e., more design variables), invoke new complications.

For instance, many of the generators used to create the space filling experimental designs

become computationally expensive in and of themselves for large numbers of factors. For

example, the simulated annealing algorithm for generating maximin Latin hypercube

designs (Morris and Mitchell, 1992; 1995) becomes extremely slow even for four factor

designs with as few as 25 variables as discussed in Section 5.1. Moreover, fractional

factorial based central composite designs are available for problems with five or more

factors. Hence, larger problems require different classes of designs all together.

As for the minimax Latin hypercube design, which is unique to this dissertation, the

genetic algorithm which is employed to generate these designs needs further studying to

develop a better understanding of its workings and to learn the optimal combination of

parameters for their use, namely, population size, number of permissible generations,

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mutation rates, and termination criteria. Also, as it stands right now, the current design

criterion—minimize the maximum distance between sample points and prediction points—

does not yield a unique design for a given sample size and number of design variables.

Developing and implementing an optimization criterion such as that proposed by Mitchell

and Morris (1995) for their maximin Latin hypercube designs could improve the

effectiveness of the minimax Latin hypercubes.

As for kriging, only kriging metamodels which employ an underlying constant for

the global portion of the model have been investigated in this work. In general, f(x) in

Equation 2.14 could be taken as a linear or quadratic model instead of a constant which

may permit more accurate kriging approximations; however, the problem of having a

sufficient number of samples to estimate all of the unknown coefficients in f(x) resurfaces.

A preliminary investigation of such an approach is documented in (Giunta, et al., 1998);

they find that minimal improvement in the accuracy of the kriging approximations is

obtained for their analyses.

Meanwhile, the power of kriging lies in its capability to interpolate accurately a

wide range of linear and non-linear functions. An iterative or sequential strategy which

takes advantage of this may prove useful provided the kriging models can be fit and

validated quickly from one iteration to the next. Consequently, trust region based

approaches which incorporate approximations are being developed by researchers in an

effort to capitalize on the potential of kriging metamodels (see, e.g., Alexandrov, et al.,

1997; Booker, et al., 1996; Booker, et al., 1995; Cox and John, 1995; Dennis and

Torczon, 1996; Osio and Amon, 1996; Schonlau, et al., 1997).

Finally, alternative optimization algorithms for finding the “best” kriging model also

must be investigated for use with larger problems. The simulated annealing algorithm

currently employed to fit the kriging models, see Appendix A, becomes extremely

inefficient for problems with more than eight variables and approximately 180 sample

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points. Moreover, the matrix inversion routines in the current prediction software do not

take full advantage of the properties of the correlation matrix, R, in kriging which is always

symmetric and positive definite. Several matrix decomposition and inversion algorithms

have been developed to take advantage of these properties; however, they have not been

exploited in this work.

8 .4 .2 Future Work in Product Family and Product Platform Design

The concept of scalability and scalable product platforms has provided an excellent

inroads into product family and product platform design, marrying current research efforts

in Decision-Based Design, the Robust Concept Exploration Method, and robust design

with tools from marketing/management science. The end result is the Product Platform

Concept Exploration Method which has been demonstrated by means of two examples: the

design of a family of universal motors and the design of a family of General Aviation

aircraft. While it has been shown that the PPCEM is effective at producing a family of

products based on scaled instantiations of a product platform, verification through

additional applications will help to further verify the method and improve it.

Furthermore, much in the same way that the product platform provides a platform

for leveraging with a product family, the Product Platform Concept Exploration Method

provides a platform for leveraging future work in product family and product platform

design, see Figure 8.2. The different types of systems can be classified on the vertical axis

of a market segmentation grid and different characteristics of product platform design on

the horizontal axis. The use of the PPCEM to design scalable product platforms for a

variety of systems then can be plotted on this market segmentation grid as illustrated in

Figure 8.2 for the two examples in this dissertation. Perhaps through the addition of

different “Processors” to the PPCEM, additional capabilities could be developed within the

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framework of the PPCEM to design modular platforms or facilitate product family redesign

around a common platform, for instance.

ComplexSystems

SimpleSystems

Scalable

UniversalMotor

GAA

Platform


Modular

...

??

...

Figure 8.2 The PPCEM as a Platform for Other Platform Design Methods

Several avenues of future work have also been mentioned during the critical

analysis in Section 8.3. In addition to these potential research areas, additional verification

and extensions of the PPCEM are offered in the following sections as they tie to current

research within the Systems Realization Laboratory in the G. W. Woodruff School of

Mechanical Engineering at the Georgia Institute of Technology. These sections have been

co-written with colleagues who are planning to pursue (or are currently pursuing) the

discussed research. A summary of those providing input for this section and their standing

within the Systems Realization Laboratory are listed in Table 8.2.

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Table 8.2 Contributions to Future Work Discussion

Section is written with input from: Standing§8.4.3 Additional Verification of the

PPCEM and KrigingMetamodels through theConcurrent Design of anEngine Lubrication System

Yao Lin &Kiran Krishnapur M.S. students

§8.4.4 Configuration Design ofCommon AutomotivePlatforms

Zahed Siddique Ph.D. candidate

§8.4.5 Integrated Product and ProcessDesign of Product Familiesand Mass Customized Goods

Gabriel Hernandez Ph.D. student

§8.4.6 Product Family Mappings and“Ideality” Metrics Marc McLean M.S. student

§8.4.7 Modeling the Value of Reuseand Remanufacturing in aProduct Family

Mark McIntosh M.S. student

8 .4 .3 Additional Verification of the PPCEM and Kriging Metamodelsthrough the Concurrent Design of an Engine Lubrication System

The objective in the Ford Engine Design Project is to develop and improve engine

lubrication system models to support advanced concurrent powertrain design and

development (cf., Rangarajan, 1998). As part of this work, robust design specifications

are sought which are capable of satisfying a wide variety of torque and power requirements

for different automobile engines. After developing a better understanding of the engine

lubrication system and its components, potential scaling opportunities within the engine

lubrication systems components can be identified and exploited using the PPCEM to

develop a robust and common platform design for the valves, pistons, bearings, etc. This

platform then can be instantiated quickly using minimal additional analysis for different

classes of vehicles (e.g., automobiles, trucks, and vans) in an effort to maintain better

economies of scale across a wide variety of automobile makes and models.

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In addition to applying the PPCEM to the preliminary design of engine lubrication

components, the use of kriging metamodels for building surrogate approximations of the

associated complex fluid dynamics analyses also can be investigated. Currently second-

order response surfaces are used extensively during the design process; however, the

complex analyses for friction losses, power losses, etc. cannot be modeled well by

response surfaces over a large region of the design space, thus limiting the search for good

solutions. Building accurate global approximations of these analyses using kriging

metamodels may yield additional insight into the complexities of the design space, allowing

better solutions to be identified. The utility of the kriging for partitioning and screening

large systems also can be examined in the context of the engine lubrication system since a

large number of factors (≈ 20) currently are being utilized which would push the limits of

the kriging metamodeling software (i.e., fitting the model, matrix inversion, etc.). Finally,

additional metamodeling techniques such as neural networks (see, e.g., Cheng and

Titterington, 1994; Hajela and Berke, 1992; Rumelhart, et al., 1994; Widrow, et al., 1994)

also can be compared to kriging given the size and complexity of the problem.

8 .4 .4 Configuration Design of Common Automotive Platforms

Balancing the need to customize products for target markets while enabling the

economies of scale of a “world car” is a challenge faced by every automotive manufacturer.

A proliferation of options and model derivatives leads to increased tooling cost and

production line complexity. At first glance, it may appear that automotive platforms are

prime examples for product variety design research. However, in a recent study, Siddique,

et al. (1998) identified significant differences between the variety characteristics of

automotive platforms from some of the examples that other researchers have studied (e.g.,

the Sony Walkman family). For example, the majority of product family design research is

applicable to products that are modular with respect to functions as discussed in Section

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2.2.3. The automotive platform, on the other hand, is not modular because the platform

accomplishes one function as a whole. As a result, many product family design

approaches do not readily apply; however, careful commonization of platforms can still be

used to increase product variety while reducing the number of components between

different models and the product line complexity.

Developing a common platform requires a robust platform that can support all of the

requirements for different car models and also a common assembly process that can

support these variations. For the automotive industry, platform requirements come from

packaging constraints (underhood, passenger, etc.), safety/crash requirements, size of the

vehicle, styling, and other requirements/regulations. Cars in similar classes have similar

types of requirements (except for styling, maybe); as such, the underbody for similar cars

have the potential to be commonized. Toward this end, a method for the configuration

design of common product platforms is to be developed, extending the parameter design

capabilities of the PPCEM for designing scalable product platforms. As discussed in

(Siddique, 1998; Siddique and Rosen, 1998), this includes the following:

• identification of different product family design concepts and investigation of the

applicability of these concepts towards automotive platform design,

• development of a representation scheme for automotive platform commonization,

• development of a scheme to measure commonality for automotive platforms, and

• establishment of configuration design methods for developing common product

platforms.

Using configuration design methods, the underlying common core for different platforms

can be identified along with the required variations. This information then can be used to

increase the commonality of the product platform and determine how to isolate the

variability in specific modules.

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In addition to having a common product platform, a commonized assembly process

also is desired so that the same assembly line can be used to produce all of the (minor)

platform derivatives. Using the same component loading sequences, tooling sequences,

etc. provides some of the requirements when developing a common assembly process (cf.,

Nevins and Whitney, 1989; Whitney, 1993). Other requirements that need to be

considered specifically for automobile platforms include common locators, weld lines,

transfer points, etc. Hence, it is imperative to integrate product and process design of

product families.

8 .4 .5 Integrated Product and Process Design of Product Families andMass Customized Goods

Mass customization, i.e., the manufacture of customized products with the

efficiency and speed of mass produced systems, is increasingly recognized as a source of

competitive advantage and possibly the next world-class manufacturing paradigm.

Although the marketplace is rapidly moving towards mass customization, very little work

has been done on formalizing an integrated product and process development method that

would enable companies to practice mass customization in a systematic and efficacious

manner. For example, the PPCEM provides a method to develop a common product

platform which can be scaled to provide the necessary variety for a product family;

however, its focus is solely on modeling the product itself. Meanwhile, research in

flexible, agile, and/or reconfigurable manufacturing systems (see, e.g., Abair, 1995;

Anderson and Pine, 1997; Chinnaiah, et al., 1998; Dhumal, et al., 1996; Hormozi, 1994;

Richards, 1996) focuses primarily on developing cost effective manufacturing systems to

realize a wide variety of products. Integrating the two fields of research has received little

attention in the context of designing families of products.

Mass customization places an onus on companies to integrate closely their various

activities in order to respond quickly to an environment of continuous change. Thus,

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emphasis should be given to the integration of product design, production system design,

and organization design. The key areas to be addressed include the following:

• Principles of product and process development for mass customized production:

- systematic product and process evolution based on optimization techniques,

parameterization, modularity, standardization, commonality, scalability, and

robustness,

- concurrent design of flexible product and process architectures, and

- optimal design under uncertain demand and customer requirements.

• Structuring design projects for mass customized production:

- identifying appropriate organizational structuring based on the notion of multi-

functional design teams and the particular requirements of mass customized

production, and

- systems to support the required information transfer and group decision

making.

• Design for dis-aggregated production, i.e., decentralized supply chains and

production systems for the growing global economy.

An initial investigation into the concurrent modeling of product and process for

design of make-to-order customized systems is discussed in (Hernandez, et al., 1998)

wherein the integrated product and process design of a family of absorption chillers for a

variety of capacities is presented. In related work, game-theoretic models of product and

process design have been implemented (see, Hernandez, 1998) to facilitate the formulation

and solution of such an approach, providing a foundation for future integration of product

and process design of family of products.

8 .4 .6 Product Family Mappings and “Ideality” Metrics

One of the major obstacles encountered by many industries, particularly in the

telecommunications industry, is (1) that several solution paths exist to satisfy a given set of

customer requirements using available components and (2) when customers ask for new

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functional capabilities, it is difficult to determine how this functionality can be created and

provided seamlessly in the presence of a pre-existing set of components. Therefore,

metrics must be established for the purpose of identifying the most appropriate solution

strategy given the specific design and customer requirements. The NCI and PDI measures

presented in this work provide relative assessments of product commonality and

performance deviation; however, these measures cannot be used in “real-time” by designers

to guide the product platform development process. Therefore, the objective is to survey

further the existing engineering and strategic manufacturing literature to the following:

1. refine the definition of the product family,

2. establish a useful product family model,

3. define useful “real-time” metrics to guide engineering design and improve the

product family architecture,

4. map new functionality and products into the product family, and

5. assess decisions made in the context of the product family.

Establishment of a suitable product family model and corresponding metrics

provides a means of identifying non-ideal substructures within a product (family)

architecture, targeting areas of greatest improvement. Possible metrics include those

relating to the system flexibility, complexity, upgradability, etc., in addition to improving

current metrics for commonality, modularity, etc. for “real-time” use by designers. The

end result will be an efficient process for designing assemble-to-order systems, thereby

replacing the expensive and time consuming process of realizing engineer-to-order

systems.

8 .4 .7 Modeling the Value of Reuse and Remanufacturing in a ProductFamily

As discussed in Section 1.1, there are a number of benefits to developing families

of products, one of which is the ability to reuse and remanufacture components and

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modules from one product to the next (cf., Alexander, 1993; Paula, 1997; Rothwell and

Gardiner, 1990; Sanderson and Uzumeri, 1995). Product reuse is the act of reclaiming

products (or parts of products) from a previous use and remanufacturing them for another

use (where the second use may or may not be the same as the original). Product reuse is

both economically and environmentally desirable due to a number of benefits including:

• previously used products are diverted from landfill or other means of disposal,

• fewer natural resources are consumed in new production, and

• all of the energy, emissions, and financial resources involved in creating the

geometric form of components are reduced.

Assessing the impact of product family development on product reuse can be

accomplished by modeling remanufacturing systems. McIntosh, et al. (1998) develop and

apply a model of an integrated remanufacturing-manufacturing organization to assess the

impact of product design characteristics, product development strategies, and external

factors on the value of reuse and remanufacture over time. The model can be used to

assess the potential value of product remanufacture for an OEM (original equipment

manufacturer) which integrates the reclaiming and reuse of products into its existing

production system. The model allows decision-makers to specify the following:

• product design characteristics of each product model, (e.g., the number of

components and required disassembly sequence),

• product development decisions over time (e.g., the level of product variety, the rate

of product evolution, and the degree of component standardization across product

variety and evolution), and

• external business factors which affect reclaiming and remanufacturing (e.g., the

cost of labor and the retirement distribution of used products over time).

The model then is used to determine which products to reclaim, which components to

recycle and remanufacture, and the resulting costs and benefits of these actions over time.

Thus, it provides an analysis tool to assess the potential value of reuse and remanufacturing

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on the development of product families based on common product platforms, providing

additional cost justification for developing product families.

8.5 CONCLUDING REMARKS

In closing this dissertation, a quote by T.S. Eliot found in the introductory section

of Cressie’s book on Spatial Statistics (Cressie, 1993) is perhaps most fitting.

“We shall not cease from exploration

And the end of all our exploring

Will be to arrive where we started

And know the place for the first time.”

— T.S. Eliot

The PPCEM is not an end in itself; rather, it provides a stepping stone for future research

work in this nascent field of engineering design. For it is only at the end of this

dissertation that the problems and difficulties associated with product family and product

platform design are truly understood and appreciated. And now that we understand them,

either for the first time or in greater depth, new paths can be explored and new methods can

be developed which continue to advance the state-of-the-art in product family and product

platform design. It is the hope of the author that the PPCEM enjoys the same success as

the RCEM, providing a foundation on which future research can be established in the same

manner that this work builds upon the work before it.

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A. APPENDIX A

KRIGING ALGORITHMS AND SOURCE CODE

This appendix is intended to supplement the brief description of kriging and its

equations which is given in Section 2.4.2. In Section A.1, the process for building,

validating, and implementing a kriging model is discussed in detail. In Section A.2 the

kriging source code for mlefinder.f and krigit.f are given in Sections A.2.1 and A.2.2,

respectively, along with a README file which details their execution, see Section A.2.3.

A sample input parameter file, input data file, and corresponding output files are included in

Section A.2.4.

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A.1 BUILDING, VALIDATING, AND IMPLEMENTING A KRIGINGMODEL

Once the appropriate sample data has been obtained, there are three basic steps

required for kriging: (1) building the kriging model, (2) validating the model, and (3) using

the model. The general steps themselves are the same regardless of the metamodeling

technique used (response surfaces, neural nets, etc.); however, special attention is given to

the kriging approach because of its nascency in engineering design applications. Each step

is discussed in turn as it applies to the kriging approach being advocated in this dissertation;

parallels to RS modeling are included when applicable.

A.1.1 Building a Kriging Model

The general approach for building a kriging model is illustrated in Figure A.1. In

order to fit the “best” kriging model, an unconstrained non-linear optimization algorithm is

needed to obtain the maximum likelihood estimates (MLEs) for the θk values used to fit the

“best” kriging as mentioned in the previous section.

UnconstrainedNon-linear

OptimizationAlgorithm

θk

k=1,..., ndv

mlefinder.f

R

detR

invR

mle objectivefunctioin

Sample points(xi,yi), i=1,...,ns

θ*

ˆ σ 2Repeatedfor each

response, yi

ˆ β

Figure A.1 Fitting a Kriging Model

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Once the ns sample points have been obtained, an unconstrained, non-linear

optimizer is invoked to find the MLEs for the θk as illustrated in Figure A.1. The optimizer

guesses a value for the θk which provides the input for the mlefinder.f subroutine used to

find the value of the MLE objective function. Once a set of θk values has been guessed, the

correlation matrix R—using any possible correlation function—is computed along with its

determinant and inverse. The inverse of R is then used to compute ˆ β , and both ˆ β and R-1

are then used to compute ˆ σ 2 . The MLE objective function is computed using the

determinant of R and ˆ σ 2 , and this value is then returned to the optimizer which then selects

a new value for the θk so that the MLE objective function is maximized. This process is

repeated until convergence is achieved, yielding the maximum likelihood estimate, or “best”

guess values, for θk based on the given sample data. This process is then repeated for each

response, yi. Currently, the simulated annealing algorithm from (Goffe, et al., 1994) is

employed to perform the optimization; his algorithm is on-line at the referenced web site.

A.1.2 Validating a Kriging Model

Once the MLEs for each theta have been found, the next step is to validate the

kriging model. Since a kriging model interpolates the data, residual plots and R2 values—

the usual model assessments for response surfaces (Myers and Montgomery, 1995)—are

meaningless because there are no residuals. Therefore, validating the model using

additional data points is essential if they can be afforded. If additional validation points can

be afforded, then the maximum absolute error, average absolute error, and root mean

square error (MSE) for the additional validation points can be calculated to assess model

accuracy. These measures are summarized in Table A.1. In the table, nerror is the number

of random test points used, yi is the actual value from the computer code/simulation, and ˆ y i

is the predicted value from the approximation model.

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Table A.1 Error Measures for Approximation Models

Name Error Measure Eqn. #

max. abs. error max. |y i − ˆ y i | i = 1, ..., nerror [A.1]

avg. abs. error1

n error

y i − ˆ y ii =1

nerror∑ [A.2]

root MSE(y i − ˆ y i )

2

i =1

nerror∑n error

[A.3]

However, sometimes taking additional validation points is not possible due to the

added expense of running additional experiments on the computer code/simulation; thus, an

alternative model assessment which requires no additional points is needed. One such

approach is the leave-one-out cross validation (Mitchell and Morris, 1992). In this

approach, each sample point used to fit the model is removed one at a time, the model is

rebuilt without that sample point, and the difference between the model without the sample

point and actual value at the sample point is computed for all of the sample points. The

cross validation root mean square error (cvrmse) is computed as:

cvrmse = (y i − ˆ y i )

2

i =1

ns∑ns

[A.4]

It is worth noting that the MLEs for the θk are not re-computed for each model; the

initial θk MLEs based on the full sample set are used. Mitchell and Morris (1992) describe

an elegant and efficient approach for performing this cross validation since it can be time

consuming depending on the size of the sample set. In this dissertation, however, a “brute

force” approach—remove each point one at a time, re-compute all of the matrices, etc.—is

used to determine the cvrmse for a given kriging model.

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A.1.3 Using a Kriging Model

Once a kriging model has been built and deemed sufficiently accurate, it is ready to

be used in optimization or for concept exploration. For those familiar with response

surfaces, RS model prediction simply requires substituting the new value of x , the design

variables, into the first-order or second-order response surface equations once the model

has been built and properly validated. Prediction with a kriging model, however, requires

the inversion and multiplication of several matrices; these matrices grow as the number of

sample points increases. Hence, for large problems prediction with the kriging model may

become computationally expensive as well. Regardless, the general approach for using a

kriging model in optimization is shown in Figure A.2.

krigit.f

Sample points(xi,yi), i=1,...,ns

θ*

Repeatedfor each

response, yi

xnew

x*

rT

OptimizationAlgorithm

(e.g., DSIDES)

R invR ˆ β

ˆ y

Goals, bounds, and constraints

ˆ y i

Figure A.2 Optimization Using a Kriging Model

The process of using a kriging model in optimization commences once the

constraints, goals, and variable bounds are fed into a numerical optimization algorithm,

e.g., DSIDES (Decision Support in Designing Engineering Systems (Mistree, et al.,

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1993)) or OptdesX (Parkinson and al., 1998). The optimization algorithm then selects a

value for the design variables: xnew. The sample points, θ* from Figure A.1, and xnew serve

as the input for the dace_eval.f subroutine used for prediction. The krigit.f algorithm is used

to compute R, the inverse of R, and then ˆ β . Meanwhile, rT is also computed, and once rT

and ˆ β are known, ˆ y i can be computed. This process is repeated for each response, and

the vector of predicted values, ˆ y , is returned to the optimizer which adjusts xnew until the

best point, x*, is found. The algorithms krigit.f and mlefinder.f are included in the next

section.

A.2 KRIGING SOURCE CODE

The kriging source code for mlefinder.f and krigit.f are included in Sections A.2.1

and A.2.2, respectively. A README file describing the intricacies of the kriging codes

and file naming conventions is included in Section A.2.3. Finally, sample parameter and

data input files and resulting output files are given in Section A.2.4.

A.2.1 The mlefinder.f Algorithm

************************************************************************ subroutine daceinter(xvector,MLE,krig)** This subroutine acts as an interface between a numerical optimizer by* providing the MLE estimate of the DACE correlation parameter 'theta'* based on the data contained in the input file opened below.** This routine (daceinter) calls the correlation subroutine (correlate)* to evaluate the correlation matrix and DACE model parameters.** Tim Simpson, 25 February 1998 / Tony Giunta, 12 May 1997************************************************************************** Input Variables:* ----------------* xvector = vector of length 'numdv' which contains theta guesses* krig = integer indicating response currently under investigation*

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* Output Variables:* -----------------* MLE = value of the MLE objective function** Parameter Variables:* --------------------* numdv = number of variables* numsamp = number of data samples from which the correlation matrix* and the DACE model parameters are calculated* numnew = number of new points to be predicted* corflag = integer value used to indicate correlation function* 1 -> Exponential correlation function, p=1* 2 -> Gaussian correlation function with p=2* 3 -> Cubic spline correlation function* 4 -> Matern function, once differentiable (exp*linear)* 5 -> Matern function, twice differentiable (exp*quadratic)** Local Variables:* ----------------* DOUBLE PRECISION* ----------------* xmat = numdv x numsamp of sample site locations* cormat = correlation matrix (numsamp x numsamp)* invmat = inverse of the correlation matrix (numsamp x numsamp)* Fvect = matrix (1 x numsamp) of constant terms (all = 1 in* 'correlate')* FRinv = matrix product of 'Fvect' and 'invmat'* yvect = matrix (1 x numsamp) of response values* yfb_vect = matrix (1 x numsamp) resulting from* ('yvect'-'Fvect'*'betahat')* yfbRinv = matrix (1 x numsamp) resulting from* ('yvect' - 'Fvect'*'betahat')*'invmat'* RHSterms = matrix product of 'invmat' and 'yfb_vect'* r_xhat = matrix (1 x numsamp) created by using the vector 'xnew'* in the correlation function* betahat = estimate of the constant term in the DACE model (beta)* sigmahat = estimate of the variance (sigma) term in the data* work = vector of length 'numsamp' used as temporary storage by* the LAPACK subroutine DGEDI** INTEGER* -------* ipvt = vector of length 'numsamp' of pivot locations used in* LAPACK subroutines DGEDI and DGEFA*************************************************************************

integer numdv,numsamp,numresp,krig,corflag,numnew character*16 fprefixCC include parameter settings for numdv, numsamp, numrespC include 'dace.params.h'

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COMMON /filename/ deckfile,fitsfile

double precision xmat(numsamp,numdv), & invmat(numsamp,numsamp),Fvect(numsamp,numdv+1),FRinv(numsamp), & yvect(numsamp),yfb_vect(numsamp),Rinvyfb(numsamp), & betahat, & MLE, work(numsamp), p,detR, & theta(numdv), xvector(numdv), resp(numsamp,numresp)

integer i,j,ipvt(numsamp),iprint character*24 deckfile,fitsfile

CC specify correlation function parameter p if necessaryC if (corflag.eq.2) then p=2.0 else if (corflag.eq.1) then p=1.0 end ifCC open necessary .dek fileC open(21,file=deckfile,status='old')CC initialize the DACE modeling parametersC do 10 i = 1,numdv theta(i) = xvector(i) 10 continueCC read in xmat and response arraysC do 100 i=1,numsamp read (21,*) (xmat(i,j),j=1,numdv),(resp(i,k),k=1,numresp) 100 continue close(21)

CC Assign response to yvect for the response of interest (specifiedC by variable 'krig'); yvect is the response for which model is beingC built using 'theta' parameters; also, initiliaze FvectC do 107 i=1,numsamp yvect(i)=resp(i,krig) Fvect(i,1)=1.0d0 107 continue

CC call subroutine to calculate the inverse of the correlation matrixC

call cormat (xmat,invmat,detR,numsamp,numdv, & work,ipvt,theta,p,corflag,iprint)

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CC call subroutine to compute intermediate kriging equationsC

call compeqn (invmat,Fvect,yvect,FRinv,yfb_vect, & Rinvyfb,betahat,numsamp,numdv)

CC call subroutine to compute sigmahat and mle obj functionC call mleobjfcn (detR,yfb_vect,Rinvyfb,MLE,numsamp)

CC return to simulated annealing algorithm with MLE valueC return end

************************************************************************ subroutine cormat (xmat,invmat,detR,numsamp,numdv, & work,ipvt,theta,p,corflag,iprint)*** This subroutine calculates the correlation matrix and its inverse** Tim Simpson 15 February 1998 / Tony Giunta, 12 May 1997************************************************************************** Needed External Files:* ----------------------* LAPACK subroutines DGEFA and DGEDI (FORTRAN77)** Inputs:* -------* DOUBLE PRECISION:* -----------------* xmat,work,p,theta** INTEGER:* --------* numdv,numsamp,ipvt,corflag** Variables ipvt and work will be changed upon exit.** Outputs:* --------* DOUBLE PRECISION:* -----------------* invmat,detR** Local:* ------

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* DOUBLE PRECISION:* -----------------* det(2) = dummy variable used in DGEFI subroutine* R = value of correlation function between i_th and j_th* sample points************************************************************************CC passed variablesC integer numdv,numsamp,corflag,ipvt(numsamp),iprint

double precision xmat(numsamp,numdv),invmat(numsamp,numsamp), & work(numsamp),p,theta(numdv),detRCC local variablesC integer i,j double precision det(2),RCC calculate terms in the correlation matrixC do 300 i = 1,numsamp do 305 j = i,numsamp if( i .eq. j ) then invmat(i,j) = 1.0d0 elseCC call subroutine to compute spatial correlation function for xmatC call scfxmat(R,xmat,theta,corflag,p,numdv,numsamp,i,j) invmat(i,j) = R invmat(j,i) = invmat(i,j) endif 305 continue 300 continue

if (iprint.eq.1) then do 321 i=1,numsamp 321 write(28,829) (invmat(i,j),j=1,numsamp) 829 format(5(f12.3,1x)) end ifCC calculate the inverse of the correlation matrix using dgefa and dgediC do 307 i=1,numsamp work(i)=0.0d0 ipvt(i)=0 307 continue

call dgefa(invmat, numsamp, numsamp, ipvt, info) if( info .ne. 0 ) then write(*,*)"Error in DGEFA, info = ",info stop endif

331

CC In DGEDI, last flag is: 1 (inverse only), 10 (Det only), 11 (both)C call dgedi(invmat, numsamp, numsamp, ipvt, det, work, 11) detR = det(1) * 10.0d0**det(2)C print *, 'detR=',detR, det(1), det(2)

if (iprint.eq.1) then do 320 i=1,numsamp 320 write(28,828) (invmat(i,j),j=1,numsamp) 828 format(5(f12.3,1x)) end if

return end

************************************************************************* include LAPACK routines used to find inverse of correlation matrix;* available on-line at http://www.netlib.org/************************************************************************

include 'dgefa.f' include 'dgedi.f'

************************************************************************ subroutine compeqn (invmat,Fvect,yvect,FRinv,yfb_vect, & Rinvyfb,betahat,numsamp,numdv)*** This subroutine calculates the DACE correlation matrix and* corresponding equations needed for model prediction.** Tim Simpson 15 February 1998**----------------------------------------------------------------------** Inputs:* -------* DOUBLE PRECISION:* -----------------* invmat,Fvect,yvect** Outputs:* --------* DOUBLE PRECISION:* -----------------* betahat,FRinv,Rinvyfb,yfb_vect** Local:* ------* DOUBLE PRECISION:

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* -----------------* beta_den,beta_num************************************************************************CC passed variablesC integer numsamp,numdv

double precision betahat,invmat(numsamp,numsamp), & Fvect(numsamp,numdv+1),FRinv(numsamp), & Rinvyfb(numsamp),yvect(numsamp),yfb_vect(numsamp)CC local variablesC double precision beta_den,beta_num integer i,j

CC compute F'RinvC do 310 i=1,numsamp FRinv(i)=0.0d0 do 315 j=1,numsamp FRinv(i)=FRinv(i)+Fvect(j,1)*invmat(j,i) 315 continue 310 continue

CC compute betahat = (F'Rinv*yvect)/(F'Rinv*F)C beta_den=0.0d0 beta_num=0.0d0 do 320 i=1,numsamp beta_den=beta_den+FRinv(i)*Fvect(i,1) beta_num=beta_num+FRinv(i)*yvect(i) 320 continue betahat = beta_num / beta_den

CC compute y-f'betahatC do 330 i = 1,numsamp yfb_vect(i) = yvect(i) - betahat*Fvect(i,1) 330 continue

CC compute Rinv*(y-f'beta)C do 340 i=1,numsamp Rinvyfb(i)=0.0d0 do 345 j=1,numsamp Rinvyfb(i)=Rinvyfb(i)+invmat(i,j)*yfb_vect(j) 345 continue 340 continue

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return end

C********************************************************************C subroutine mleobjfcn(detR,yfb_vect,Rinvyfb,MLE,numsamp)CC author: Tim Simpson date: 16 February 1998*** Inputs:* -------* DOUBLE PRECISION:* -----------------* detR,yfb_vect,Rinvyfb** INTEGER:* --------* numsamp*** Outputs:* --------* DOUBLE PRECISION:* -----------------* MLE** Local:* ------* DOUBLE PRECISION:* -----------------* sigmahat = estimated value of*C********************************************************************CC passed variablesC double precision detR,yfb_vect(numsamp),Rinvyfb(numsamp),MLE integer numsampCC local variablesC double precision sigmahat integer i

CC compute sigma_hat = (yfb_vect*Ringyfb)/numsampC sigmahat=0.0d0 do 10 i=1,numsamp sigmahat=sigmahat+yfb_vect(i)*Rinvyfb(i) 10 continue sigmahat=ABS(sigmahat)/numsamp

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CC compute MLE objective functionC

C added first part of if loop to control MLE when near a singularityC if ((detR.lt.0.0).or.(sigmahat.lt.0.0)) then MLE = -5000. else MLE = -0.5d0*(numsamp*log(sigmahat) + log(detR)) end if

C write(6,83) sigmahat,detR,MLEC 83 format('sigmahat=',f12.4,2x,'detR=',f10.7,'obj fcn=',f12.5)

return end

C********************************************************************C subroutine scfxmat(R,xmat,theta,corflag,p,numdv,numsamp,i,j)CC author: tim simpson date: 2/11/98CC subroutine to compute correlation function for correlationC matrix; NOT to compute r_xhat.CCC Output:C -------C R = value of correlation function between two sample points,C given thetaCC Input:C ------C xmat = matrix of sample pointsC theta = array of theta valuesC i,j = i_th and j_th elements of correlation matrix for whichC correlation function is being computedC corflag = integer flag specifying correlation functionCC All variables except R are unchanged upon exitingCC********************************************************************CC passed variablesC integer i,j,corflag,numdv,numsamp double precision R,xmat(numsamp,numdv),theta(numdv),pCC local variablesC double precision sum,thetadist,dist integer k

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if ((corflag.eq.2).or.(corflag.eq.1)) then sum=0.0d0 do 120 k = 1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) sum = sum + theta(k)*((dist)**p) 120 continue R = exp( -1.0d0*sum )

else if (corflag.eq.3) then sum=1.0d0 do 130 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) thetadist=dist*theta(k) if (thetadist.lt.0.5) then sum=sum*(1.0-6.0*(thetadist**2)+6.0*(thetadist**3)) else if (thetadist.ge.1.0) then sum=sum*0.0 else sum=sum*(2.0*(1.0-thetadist)**3) end if 130 continueC print *, theta(1),thetadist,sum R = sum

else if (corflag.eq.4) then sum=1.0 do 140 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) sum = sum*(exp(-theta(k)*dist)*(1.+theta(k)*dist)) 140 continue R = sum

else if (corflag.eq.5) then sum=1.0 do 150 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) sum = sum*(exp(-theta(k)*dist)* & (1.+theta(k)*dist+(theta(K)**2*dist**2)/3.0)) 150 continue R = sum

end if

return end

A.2.2 The krigit.f Algorithm

************************************************************************* program krigit** This program invokes the subroutines (correlate) and (dace_eval)

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* to evaluate the correlation matrix and predict at the value of y* at an untried x, given a correlation function and values for theta** Tim Simpson 25 February 1998 / Tony Giunta, 12 May 1997*************************************************************************** Variables:* ----------* xnew = vector of length 'numdv' where the DACE model is to be* evaluated* theta = supplied value of the correlation parameter 'theta'* ynew = vector of length 'numnew' of the DACE model response value(s)* corflag = flag variable used to specify correlation function:* 1 -> Exponential correlation function, p=1* 2 -> Gaussian correlation function, p=2* 3 -> Cubic spline correlation function* 4 -> Matern function, once differentiable (exp*linear)* 5 -> Matern function, twice differentiable (exp*quadratic)* krig = counter indicating # of response which is currently being* modeled (1 < KRIG < numresp)** Parameter Variables (to be specified by user in krig.params.h):* ----------------------------------------------------* numdv = number of variables* numsamp = number of data samples from which the correlation matrix* and the DACE model parameters are calculated* numnew = number of sample site where the unknown response value* will be calculated* numresp = number of responses for which kriging models have been* constructed** Local Variables:* ----------------* DOUBLE PRECISION* ----------------* xmat = numdv x numsamp of sample site locations* invmat = inverse of the correlation matrix (numsamp x numsamp)* Fvect = matrix (1 x numsamp) of constant terms* (all = 1 in 'correlate')* FRinv = matrix product of 'Fvect' and 'invmat'* yvect = matrix (1 x numsamp) of response values* yfb_vect = matrix (1 x numsamp) resulting from* ('yvect' - 'Fvect'*'betahat')* Rinvyfb = matrix product of 'invmat' and 'yfb_vect'* r_xhat = matrix (1 x numsamp) created by using the vector 'xnew'* in the correlation function* betahat = estimate of the constant term in the DACE model* work = vector of length 'numsamp' used as temporary storage by* the LAPACK subroutine DGEDI** INTEGER* -------* ipvt = vector of length 'numsamp' of pivot locations used in* LAPACK subroutines DGEDI and DGEFA

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** Notes:* ------* A. All points X must be scaled [0,1]* B. Three files are needed for prediction:* 1. '*.dek' -> data file containing sample points* used to fit the model* 2. '*.fits' -> file containing theta parameters for* kriging model* 3. '*.pts' -> file containing new points at which* to predict y_hat(X)* C. Predicted y_hat are written to file 'wei.4_160.out'* D. User must specify modeling parameters in 'krig.params.h', for* instance if you want to predict 100 new points, then numnew* is changed to 100 in 'krig.params.h' file* E. Subroutines 'dgedi.f' and 'dgefa.f' must be in same directory* when compiling this code* F. Files 'krig.params.h' and 'krig.files.h' must also be in same* directory when compiling this code* G. Program must be recompiled any time changes are made to* either 'krig.files.h' or 'krig.params.h'*************************************************************************

integer numdv,numsamp,numnew,numresp,corflag character*16 fprefixCC integer parameters are specified in dace.params.h fileC include 'dace.params.h'

double precision xmat(numsamp,numdv),r_xhat(numsamp),betahat, & invmat(numsamp,numsamp),Fvect(numsamp,numdv+1),FRinv(numsamp), & Rinvyfb(numsamp),yvect(numsamp),yfb_vect(numsamp), & xnew(numnew,numdv),ynew(numnew),work(numsamp),dummy2, & Farray(numnew,numresp),thetaray(numresp,numdv), & theta(numdv),resp(numsamp,numresp),p,detR integer krig,i,j,k,ipvt(numsamp),dummy,lenstr,scfstr character*16 ftitle character*20 deckfile,fitsfile,newptfile,outfile,scfname

scfname='expgaucubma1ma2'

CC specify correlation function; make sure .fits file has correct thetasC if (corflag.eq.2) then p=2.0 else if (corflag.eq.1) then p=1.0 end ifCC open necessary .dek, .fit, .npt, and .out filesC call getlen(fprefix,lenstr)

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ftitle=fprefix

deckfile=ftitle(1:lenstr) // '.dek' newptfile=ftitle(1:lenstr) // '.npt' scfstr=(3*(corflag-1))+1 fitsfile=ftitle(1:lenstr) // '.' // scfname(scfstr:scfstr+2) & // '.fit' outfile=ftitle(1:lenstr) // '.' // scfname(scfstr:scfstr+2) & // '.out'

open(21,file=deckfile,status='old') open(22,file=fitsfile,status='old') open(23,file=newptfile,status='old') open(27,file=outfile,status='unknown')

print * print *, deckfile,fitsfile,newptfile,outfile print *, numnew,numdv,numsamp,numrespCC initialize xmat, response, and theta arrays,C print * write(6,*) 'Reading in sample data...' do 10 i=1,numsamp 10 read (21,*) (xmat(i,j),j=1,numdv),(resp(i,k),k=1,numresp) close(21)

print * write(6,*) 'Reading in theta parameters...' do 20 i=1,numresp read(22,*) dummy,(thetaray(i,j),j=1,numdv),dummy2 write(6,1000) dummy,(thetaray(i,j),j=1,numdv) 1000 format(i2,8f9.5) 20 continue close(22)

print * write(6,*) 'Reading in new points at which to predict y_hat...' do 30 i=1,numnew read(23,*) (xnew(i,j),j=1,numdv) write(6,78) i,(xnew(i,j),j=1,numdv) 78 format(i3,2x,8(f6.4,1x)) 30 continue close(23)CC step through each response, predict new y_hat at each new pointC do 40 krig=1,numresp

write(6,1001) 1001 format(/,60('-'),/) write(6,*) 'Predicing response #',krig,' using theta paramters:'CC Assign response to yvect for the response of interest (specifiedC by variable 'krig'); yvect is the response for which model is being

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C built using 'theta' parameters specified by 'krig'C do 50 j=1,numdv theta(j)=thetaray(krig,j) 50 continue write(6,1002) (theta(j),j=1,numdv) 1002 format(8f9.5) print * write(6,*) 'Predicted values for response #',krig do 60 i=1,numsamp yvect(i)=resp(i,krig) Fvect(i,1)=1.0d0 60 continueCC call subroutine to calculate the inverse of the correlation matrixC

call cormat (xmat,invmat,detR,numsamp,numdv, & work,ipvt,theta,p,corflag)

CC call subroutine to compute intermediate kriging equationsC

call compeqn (invmat,Fvect,yvect,FRinv,yfb_vect, & Rinvyfb,betahat,numsamp,numdv)

CC call subroutine to predict new point given correlation matrix andC the correlation parametersC

call dace_eval(xnew,xmat,r_xhat,betahat,Rinvyfb,numsamp, & numdv,numnew,theta,ynew,p,corflag)

CC store predicted values for current responseC do 80 i=1,numnew Farray(i,krig)=ynew(i) 80 continueCC compute predicted values for next response, i.e., increment 'krig'C 40 continueCC write predicted values to specified .out fileC do 90 i=1,numnew write(27,79) (Farray(i,krig),krig=1,numresp) 79 format(10(f13.5,1x)) 90 continue close(27)

print *

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write(6,*) 'All response values written to specified .out file'

stop end

************************************************************************ subroutine getlen(string,lenstr)*** This subroutine is used to determine the actual length of the* filename prefix specified by the user in 'krig.params.h'.** With this known, the .dek, .fit, .npt, and .out suffixes are* concatenated onto the prefix, and the files are opened.** Author: Tim Simpson, 2/15/98** From: Koffman and Friedman, Fortran (5th ed.), Addison-Wesley,* New York, pp. 537-538.************************************************************************* character*1 blank character*16 string parameter (blank=' ') integer next do 10 next = LEN(string), 1, -1 if (string(next:next).ne.blank) then lenstr=next return end if 10 continue lenstr=0 if (lenstr.eq.0) then write(6,*) 'You have not specified a file name prefix' stop end if return end

************************************************************************ subroutine cormat (xmat,invmat,detR,numsamp,numdv, & work,ipvt,theta,p,corflag)*** This subroutine calculates the correlation matrix and its inverse** Tim Simpson 15 February 1998 / Tony Giunta, 12 May 1997************************************************************************** Needed External Files:* ----------------------

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* LAPACK subroutines DGEFA and DGEDI (FORTRAN77)** Inputs:* -------* DOUBLE PRECISION:* -----------------* xmat,work,p,theta** INTEGER:* --------* numdv,numsamp,ipvt,corflag** Variables ipvt and work will be changed upon exit.** Outputs:* --------* DOUBLE PRECISION:* -----------------* invmat,detR (used in mle*.f but not dace*.f -> done to maintain* consistency between cormat routine for two codes)** Local:* ------* DOUBLE PRECISION:* -----------------* det(2) = dummy variable used in DGEFI subroutine* R = value of correlation function between i_th and j_th* sample points************************************************************************CC passed variablesC integer numdv,numsamp,corflag,ipvt(numsamp)

double precision xmat(numsamp,numdv),invmat(numsamp,numsamp), & work(numsamp),p,theta(numdv),detRCC local variablesC integer i,j double precision det(2),RCC calculate terms in the correlation matrixC do 300 i = 1,numsamp do 305 j = i,numsamp if( i .eq. j ) then invmat(i,j) = 1.0d0 elseCC call subroutine to compute spatial correlation function for xmatC call scfxmat(R,xmat,theta,corflag,p,numdv,numsamp,i,j) invmat(i,j) = R

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invmat(j,i) = invmat(i,j) endif 305 continue 300 continueCC calculate the inverse of the correlation matrix using dgefa and dgefiC do 307 i=1,numsamp work(i)=0.0d0 ipvt(i)=0 307 continue

call dgefa(invmat, numsamp, numsamp, ipvt, info) if( info .ne. 0 ) then write(*,*)"Error in DGEFA, info = ",info stop endifCC In DGEDI, last flag is: 1 (inverse only), 10 (Det only), 11 (both)C call dgedi(invmat, numsamp, numsamp, ipvt, det, work, 11) detR=det(1)*10.0d0**det(2)

return end

************************************************************************* include LINPACK routines used to find inverse of correlation matrix************************************************************************

include 'dgefa.f' include 'dgedi.f'

************************************************************************ subroutine compeqn (invmat,Fvect,yvect,FRinv,yfb_vect, & Rinvyfb,betahat,numsamp,numdv)*** This subroutine calculates the DACE correlation matrix and* corresponding equations needed for model prediction.** Tim Simpson 15 February 1998**----------------------------------------------------------------------** Inputs:* -------* DOUBLE PRECISION:* -----------------* invmat,Fvect,yvect*

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* Outputs:* --------* DOUBLE PRECISION:* -----------------* betahat,FRinv,Rinvyfb,yfb_vect** Local:* ------* DOUBLE PRECISION:* -----------------************************************************************************CC passed variablesC integer numsamp,numdv

double precision betahat,invmat(numsamp,numsamp), & Fvect(numsamp,numdv+1),FRinv(numsamp), & Rinvyfb(numsamp),yvect(numsamp),yfb_vect(numsamp)CC local variablesC double precision beta_den,beta_num

integer i,j

CC compute F'RinvC do 310 i=1,numsamp FRinv(i)=0.0d0 do 315 j=1,numsamp FRinv(i)=FRinv(i)+Fvect(j,1)*invmat(j,i) 315 continue 310 continueCC compute betahat = (F'Rinv*yvect)/(F'Rinv*F)C beta_den=0.0d0 beta_num=0.0d0 do 320 i=1,numsamp beta_den=beta_den+FRinv(i)*Fvect(i,1) beta_num=beta_num+FRinv(i)*yvect(i) 320 continue betahat = beta_num / beta_den print *, 'betahat=',betahatCC compute y-f'betahatC do 330 i = 1,numsamp yfb_vect(i) = yvect(i) - betahat*Fvect(i,1) 330 continueCC compute Rinv*(y-f'beta)

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C do 340 i=1,numsamp Rinvyfb(i)=0.0d0 do 345 j=1,numsamp Rinvyfb(i)=Rinvyfb(i)+invmat(i,j)*yfb_vect(j) 345 continue 340 continue

return end

************************************************************************ subroutine dace_eval(xnew,xmat,r_xhat,betahat,Rinvyfb, & numsamp,numdv,numnew,theta,ynew,p,corflag)*** Use DACE interpolating model to predict response values at unsampled* locations** Tony Giunta, 12 May 1997/Tim Simpson 12 February 1998************************************************************************** Inputs:* -------* xnew* xmat* r_xhat* betahat* Rinvyfb* numsamp* numdv* theta** Outputs:* --------* ynew** Local Variables:* ----------------* sum = temporary variable used for calculating the terms in the* vector 'r_xhat'* yeval = scalar value resulting from matrix multiplication of* 'r_xhat' * 'Rinvyfb'************************************************************************** double precision xnew(numnew,numdv),r_xhat(numsamp), & xmat(numsamp,numdv),Rinvyfb(numsamp),betahat, & theta(numdv),ynew(numnew),yeval,p,R integer i,j,k,numdv,numsamp,numnew,corflagC

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C calculate the vector r(x)C do 200 i = 1,numnew do 210 j = 1,numsamp call scfxnew(R,xnew,xmat,theta,corflag,p,numdv,numsamp, & numnew,i,j)CC equate r_xhat to correlation between ith xnew point and jth sample pointC r_xhat(j)=R 210 continueCC calculate the estimate of Y, i.e., Y_hat(x)C yeval = 0.0d0 do 220 k=1,numsamp 220 yeval=yeval+r_xhat(k)*Rinvyfb(k) ynew(i) = yeval + betahat print '(f20.5)', ynew(i)

CC repeat for next xnewC 200 continue

return end

C********************************************************************C subroutine scfxmat(R,xmat,theta,corflag,p,numdv,numsamp,i,j)CC Author: Tim Simpson Date: 2/11/98CC subroutine to compute spatial correlation function (scf) forC correlation matrix; NOT to compute scf for r_xhat.CC Output:C -------C R = value of correlation function between two sample points,C given thetaCC Input:C ------C xmat = matrix of sample pointsC theta = array of theta valuesC i,j = i_th and j_th elements of correlation matrix for whichC correlation function is being computedC corflag = integer flag specifying correlation function:CC All variables except R are unchanged upon exitingCC********************************************************************CC passed variables

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C integer i,j,corflag,numdv,numsamp double precision R,xmat(numsamp,numdv),theta(numdv),pCC local variablesC double precision sum,thetadist,dist integer k

if ((corflag.eq.2).or.(corflag.eq.1)) then sum=0.0d0 do 120 k = 1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) sum = sum + theta(k)*((dist)**p) 120 continue R = exp( -1.0d0*sum )

else if (corflag.eq.3) then sum=1.0d0 do 130 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) thetadist=dist*theta(k) if (thetadist.lt.0.5) then sum=sum*(1.0-6.0*(thetadist**2)+6.0*(thetadist**3)) else if (thetadist.ge.1.0) then sum=sum*0.0 else sum=sum*(2.0*(1.0-thetadist)**3) end if 130 continue R = sum

else if (corflag.eq.4) then sum=1.0 do 140 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) sum = sum*(exp(-theta(k)*dist)*(1.+theta(k)*dist)) 140 continue R = sum

else if (corflag.eq.5) then sum=1.0 do 150 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) thetadist=dist*theta(k) sum = sum*(exp(-thetadist)* & (1.+thetadist+(thetadist**2)/3.0)) 150 continue R = sum

end if

return end

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C********************************************************************C subroutine scfxnew(R,xnew,xmat,theta,corflag,p,numdv,numsamp, & numnew,i,j)CC Author: Tim Simpson Date: 2/13/98CC subroutine to compute spatial correlation function (scf) betweenC ith xnew point and each sample point; NOT to compute scf for xmat.CC Output:C -------C R = value of correlation function for r_xhatCC Input:C ------C xnew = matrix of new points (numnew,numdv)C xmat = matrix of sample points (numsamp,numdv)C theta = array of theta valuesC i,j = i_th xnew point j_th sample point for whichC correlation function is being computedC corflag = integer flag specifying correlation function (see scfxmat)CC All variables except R are unchanged upon exitingCC********************************************************************CC passed variablesC integer i,j,corflag,numdv,numsamp double precision R,xnew(numnew,numdv), & xmat(numsamp,numdv),theta(numdv),pCC local variablesC double precision sum,thetadist,dist integer k

if ((corflag.eq.2).or.(corflag.eq.1)) then sum=0.0d0 do 400 k = 1,numdv dist = ABS(xnew(i,k)-xmat(j,k)) sum = sum + theta(k)*((dist)**p) 400 continue R = exp( -1.0d0*sum ) else if (corflag.eq.3) then sum=1.0d0 do 410 k=1,numdv dist = ABS(xnew(i,k)-xmat(j,k)) thetadist=theta(k)*dist if (thetadist.lt.0.5) then sum=sum*(1.0-6.0*(thetadist**2)+6.0*(thetadist**3)) else if (thetadist.ge.1.0) then sum=sum*0.0 else

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sum=sum*(2.0*(1.0-thetadist)**3) end if 410 continue R = sum else if (corflag.eq.4) then sum=1.0 do 420 k=1,numdv dist = ABS(xnew(i,k)-xmat(j,k)) sum = sum*(exp(-theta(k)*dist)*(1.+theta(k)*dist)) 420 continue R = sum else if (corflag.eq.5) then sum=1.0 do 430 k=1,numdv dist = ABS(xnew(i,k)-xmat(j,k)) thetadist=theta(k)*dist sum = sum*(exp(-thetadist)* & (1.+thetadist+(thetadist**2)/3.0)) 430 continue R = sum end if

return end

A.2.3 Kriging Algorithm README File

This is the README file for 'kriger.v1'

Author: Tim Simpson Date: June 8, 1998

***************** SOURCE CODES *****************The F77 files needed to build, run, and cross-validate a krigingmodel are the following. All of these codes should be contained inthe same file as the sample data files. For compiling, links aremade internally to the codes; therefore, only need to compile mainroutine in order to compile entire code.

Codes used to build a kriging model: simman.f, mlefinder.f, dgefa.f, dgedi.f----------------------------------------------------------------------------simman.f = simulated annealing algorithm used to find MLE values for theta parameters used to fit a kriging model to sample datamlefinder.f = contains subroutines which estimate the value of the MLE objective functiondgedi.f = LAPACK subroutine used to invert correlation matrixdgefa.f = LAPACK subroutine used to invert correlation matrix

TO COMPILE: f77 -o simmle simman.f -> makes executable 'simmle'

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Codes used to predict with a kriging model: krigit.f, dgefa.f, dgedi.f----------------------------------------------------------------------------krigit.f = used to predict new values of response once MLE values for the theta parameters are found

TO COMPILE: f77 -o krigit krigit.f -> makes executable 'krigit'

******************** PARAMETER FILES ********************Two parameter files are necessary to run these codes. A description ofthe parameters in each file are included in the respecctive '*.h' file.

dace.params.h = contains parameters needed to run simman.f, mlefinder.f, and krigit.f

mle.opt_params.h = contains parameters needed for simulated annealing algorithm, only used by simman.f

*************************** FILE NAMING CONVENTION ***************************The following file naming convention is used in all of the codes.

'fprefix' = is the file name prefix for ALL the files; it is specified in 'dace.params.h'; it should be less than 8 characters long and does not include a final period (the period added to the end of 'fprefix' comes with the file name extension

Input Files Needed:===================

fprefix.dek = input file for all codes; contains sample points for which----------- the kriging model is being built; format is:

x1_1 x2_1 ... x_numdv_1 y1_1 ... y_numresp_1 x1_2 x2_2 ... x_numdv_2 y1_2 ... y_numresp_2 . . . . . . . . . . . . . . . . . . . . .x1_numsamp x2_numsamp ... x_numdv_numsamp y1_numsamp ... y_numsamp_numresp

where 'numsamp' is the number of samples, 'numdv' is the number of designvariables, an d 'numresp' is the number of responses all specified in thefile 'dace.params.h'

fprefix.npt = input for 'krigit'; contains new points at which to predict----------- values of responses; format is:

x1_of_new_pt#1 x2_of_new_pt#1 ... xnumdv_of_new_pt#1

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x1_of_new_pt#2 x2_of_new_pt#2 ... xnumdv_of_new_pt#2 . . . . . . . . . . . .x1_of_new_pt#numnew x2_of_new_pt#numnew ... xnumdv_of_new_pt#numnew

where 'numnew' is the number of new points as specified in 'dace.params.h'

Output Files Generated:=======================

fprefix.*.fit = output from 'simmle'; contains the MLE values for theta------------- and also the corresponding MLE objective function value; format is:

1 MLE_theta_x1 MLE_theta_x2 ... MLE_theta_x_numdv MLE_obj_fcn_value 2 MLE_theta_x1 MLE_theta_x2 ... MLE_theta_x_numdv MLE_obj_fcn_value . . . . . . . . . . . . . . . . . .numresp MLE_theta_x1 MLE_theta_x2 ... MLE_theta_x_numdv MLE_obj_fcn_value

where 'numresp' is the number of responses and 'numdv' is the number ofdesign variables as specified in 'dace.params.h'

The '*' in 'fprefix.*.fit' is a three letter abbreviation for thecorrelation function used to fit the model:

exp = exponential correlation function gau = gaussian correlation function cub = cubic correlation function ma1 = linear matern function (linear times exponential) ma2 = quadratic matern function (quadratic times exponential)

This three letter abbreviation is automatically derived from 'corflag'

fprefix.*.out = output from 'krigit'; contains the predicted values of------------- the response at each new point; format is:

y1_new_pt1 y2_new_pt1 ... y_numresp_new_pt1 . . . . . . . . . . . .y1_new_pt_numnew y2_new_pt_numnew ... y_numresp_new_pt_numnew

where 'numresp' is the number of responses and 'numnew' is the number ofpoints at which y is predicted; both are specified in 'dace.params.h'

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****************** EXAMPLE FILES ******************

An example two-bar truss problem is included to test to see if thecode is working properly. The problem has 2 design variables and3 responses; the file prefix is '2bar'.

The sample data file is '2bar.dek' and the 'dace.params.h' file hasbeen renamed '2bar.h' for ease of association with the example. Thenew point file is '2bar.npt' and contains 11 points at which topredict new values of the responses.

To make sure the code is working properly:

1. copy '2bar.h' to 'dace.params.h' : %user>cp 2bar.h dace.params.h

2. complie 'simmle' : %user>f77 -o simmle simman.f

3. execute 'simmle' to find MLEs : %user>simmle

4. after 'simmle' is complete, check contents of '2bar.gau.fit' and compare to '2bar.gau.fit%'. Here, the correlation function used in '2bar.h' is corflag=2, the Gaussian correlation function

5. compile 'krigit' : %user>f77 -o krigit krigit.f

6. execute 'krigit' to predict new y : %user>krigit

7. after 'krigit' is complete, check contents of '2bar.out' and compare to '2bar.gau.out%'.

*********************************** SOME FINAL NOTES AND POINTERS ***********************************

1. All x (design variables) in *.dek must be scaled to [0,1].

2. As with the 2bar example, it helps to maintain a '*.h' file for each problem and then just copy that file to 'dace.params.h' when needed, rather than recreat 'dace.params.h' each time.

3. The optimization parameters in the 'mle.opt_params.h' file are rather conservatively set in the enclosed example. It is left up to the user to find settings for EPS, RT, NS, and NT which yield sufficiently good results yet do not require long optimization runs. The current settings work well for problems with as many as 10 variables and 121 sample points. Please see the comments in the introduction to the 'simman.f' file to learn more about these parameters.

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A.2.4 Sample Parameter and Data Input Files and Kriging Output

2bar.dek--------1.0000 0.5000 25.011 0.72046 0.205820.9286 0.7500 26.392 0.63087 0.225080.8571 0.2500 20.395 0.65797 0.020510.7857 0.8750 24.968 0.45596 0.162480.7143 0.3750 19.221 0.50157 -0.030380.6429 0.6250 20.066 0.32731 -0.001340.5714 0.1250 15.300 0.32361 -0.338220.5000 0.9375 19.901 -0.19855 -0.063830.4286 0.4375 15.086 -0.14293 -0.313120.3571 0.6875 15.439 -0.64681 -0.312130.2857 0.1875 11.508 -0.73043 -0.753290.2143 0.8125 13.531 -2.04917 -0.527600.1429 0.3125 9.9140 -2.26733 -1.002840.0714 0.5625 9.7780 -4.18023 -1.04129

dace.params.h-------------

c**********************************************************c *c Parameter input file for dace_v6+, xval_v4+ *c Author: Tim Simpson *c Date: 2/23/98 *c *c**********************************************************cc specify parameter values for dace modeling softwarec

parameter ( numdv=6,numsamp=64,numresp=5,numnew=1000, & fprefix='brake.oa64',mleflag=0 )

cc numdv = # design variablesc numsamp = # samples in data setc numresp = # response models to be builtc numnew = # new points at which to predict y_hatcc fprefix = prefix of titles of files to opened/usedcc linflag = indicates order of model (0=0_th, 1=1_st or linear)c mleflag = indicates whether or not MLE for theta should bec found (0=no, 1=yes)c corflag = indicates which correlation function is used

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2bar.gau.fit------------ 1 0.04879 0.03442 31.34387 2 6.03023 0.06015 9.08923 3 1.13740 0.34200 36.40723

2bar.npt----------0.0 0.10.1 0.40.2 0.90.3 0.00.4 0.50.5 0.60.6 1.00.7 0.20.8 0.80.9 0.31.0 0.7

2bar.gau.out------------ 7.06591 -4.93252 -1.60996 9.59596 -3.19662 -1.06274 13.72457 -2.38054 -0.52032 10.87004 -0.56786 -0.91234 15.00633 -0.29634 -0.32415 17.37792 0.03354 -0.14981 22.42622 0.01010 0.04215 17.66755 0.49679 -0.14165 24.50410 0.50823 0.15698 21.47997 0.69081 0.07588 27.15561 0.66212 0.25021

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B. APPENDIX B

EXPERIMENTAL DESIGN DESCRIPTIONS

Detailed descriptions of the experimental designs used in the kriging/DOE study in

Chapter 5 are presented in this appendix. As introduced in Section 2.4.3, two types of

classical experimental designs are considered in this work: central composite designs and

Box-Behnken designs. They are described in Sections B.1.1 and B.1.2, respectively. In

addition to these two types of classical designs, nine types of space filling designs are

studied in Chapter 5, testing Hypothesis 3 and the utility of space filling designs for

deterministic computer experiments. Of these nine space filling designs, eight are

described in this appendix in Sections B.2.1 through B.2.8. These designs include the

following: random Latin hypercubes, random orthogonal arrays, IMSE optimal Latin

hypercubes, maximin Latin hypercubes, orthogonal-array based Latin hypercubes, uniform

designs, orthogonal Latin hypercubes, and Hammersley sequence designs. The ninth

space filling experimental design—the minimax Latin hypercube which is unique to this

dissertation—is described separately in Appendix C.

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B.1 CLASSICAL EXPERIMENTAL DESIGNS

B.1 .1 Central Composite Designs

As shown in Figure B.1, central composite design consists of a (fractional) factorial

design—the cube portion of the design with corners at ±1—augmented by additional “star”

and “center” points which allow the estimation of a second-order polynomial equation.

Note that the “star” points reside at a distance ±α from the origin. Central composite

designs (CCDs) require 2k + 2k + 1 points for fitting a second-order polynomial with

(k+2)(k+1)/2 coefficients where k is the number of factors being considered. The standard

CCD typically uses α > 1; values for α are determined based on the number of factors in

the experiment (Myers and Montgomery, 1995). However, variations on the standard

CCD do exist. If α = 1, then the design is called a face-centered CCD or CCF for short; if

α = 1/α, then the design is said to be an inscribed central composite or CCI for short.

Run # X1 X2 X31 -1 -1 -12 -1 -1 13 -1 1 -14 -1 1 15 1 -1 -16 1 -1 17 1 1 -18 1 1 19 -α 0 010 +α 0 011 0 -α 012 0 +α 013 0 0 -α14 0 0 +α15 0 0 0

X1

X2

X3Star pts

Center ptFactorial pts

Figure B.1 Central Composite Design for 3 Factors

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Central composite designs have many desirable features (e.g., rotatable,

orthogonal, blockable, etc.) for fitting non-deterministic data with random error (cf.,Myers

and Montgomery, 1995); however, as Sacks, et al. (1989) point out, “because

deterministic computer experiments lack random error...the classical notions of

experimental blocking, replication, and randomization are irrelevant.” Regardless, central

composite designs are employed in this work as a basis for comparison since they are

quickly becoming, or maybe already are, the standard approach for fitting second-order

response surface models in engineering computer applications (cf., Simpson, et al., 1997).

Specifically, CCF designs, a scaled CCD (where α = 1 and the factorial points lie at ±1/α),

a CCI, and a combination CCF+CCD and a combination CCF+CCI are employed in this

work. The reader is again referred to (Simpson, et al., 1997) for a review of several

studies comparing CCDs and other types of classical experimental designs.

B.1 .2 Box-Behnken Designs

Box-Behnken (BB) designs (Box and Behnken, 1960; Myers and Montgomery, 1995) are

a family of efficient three-level designs for fitting second-order response surfaces that are

formed by combining 2k factorial and incomplete block designs. They are an important

alternative to central composite designs because they require only three levels of each

factor; however, Myers and Montgomery (1995) warn that BB designs should not be used

when accurate predictions at the extremes (i.e., the corners of the hypercube) are important.

They recommend using a face-centered CCD in such cases. An inspection of the three

factor Box-Behnken design shown in Figure B.2 illustrates the lack of points near the

corners of the cube and thus the poor prediction capability of the Box-Behnken designs in

this region.

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Run# X1 X2 X31 -1 -1 02 -1 1 03 1 -1 04 1 1 05 0 -1 -16 0 -1 17 0 1 -18 0 1 19 -1 0 -1

10 1 0 -111 -1 0 112 1 0 113 0 0 0

X1

X2

X3

Figure B.2 Box-Behnken Design for 3 Factors

Two factor Box-Behnken designs do not exist. Designs for four to six factors are

constructed using the design matrices shown in Figure B.3. In this work, only one center

point is used in each design since replication of a deterministic computer experiment is

wasted effort. The resulting designs for 4, 5, and 6 factors requires 25, 41, and 49 points,

respectively.

4 Factors 5 Factors 6 FactorsX1 X2 X3 X4 X1 X2 X3 X4 X5 X1 X2 X3 X4 X5 X6±1 ±1 0 0 ±1 ±1 0 0 0 ±1 ±1 0 ±1 0 00 0 ±1 ±1 0 0 ±1 ±1 0 0 ±1 ±1 0 ±1 0

±1 0 0 ±1 0 ±1 0 0 ±1 0 0 ±1 ±1 0 ±10 ±1 ±1 0 ±1 0 ±1 0 0 ±1 0 0 ±1 ±1 0

±1 0 ±1 0 0 0 0 ±1 ±1 0 ±1 0 0 ±1 ±10 ±1 0 ±1 0 ±1 ±1 0 0 ±1 0 ±1 0 0 ±10 0 0 0 ±1 0 0 ±1 0 0 0 0 0 0 0

0 0 ±1 0 ±1±1 0 0 0 ±10 ±1 0 ±1 00 0 0 0 0

Figure B.3 Box-Behnken Design Matrices for 4, 5, and 6 Factors(from Box and Behnken, 1960)

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The procedure for creating Box-Behnken designs is described in (e.g., Box and

Behnken, 1960; Myers and Montgomery, 1995). The software package JMP® (SAS,

1995) can also be used to quickly generate these designs. As a final note, Lucas (1976)

compares Box-Behnken designs to a wide variety of experimental designs used in

conjunction with non-deterministic experiments.

B.2 SPACE FILLING EXPERIMENTAL DESIGNS

Of the numerous space filling designs which exist, random Latin hypercubes,

random orthogonal arrays, IMSE optimal Latin hypercubes, maximin Latin hypercubes,

orthogonal-array based Latin hypercubes, uniform designs, orthogonal Latin hypercubes,

and Hammersley sampling sequence designs are utilized in this dissertation. These designs

have been selected based on availability of code and ease of generation. Each of these

space filling experimental designs is explained in the following sections.

B.2 .1 Random Latin Hypercubes

Perhaps the earliest space filling experimental design intended for use with

computer experiments (Monte Carlo simulation in particular) is a Latin hypercube (McKay,

et al., 1979). A Latin hypercube is a matrix of n rows and k columns where n is the

number of levels being examined and k is the number of design variables. Each column

contains the levels 1, 2, ..., n, randomly permuted, and the k columns are matched at

random to form the Latin hypercube. Three two dimensional Latin hypercubes with n = 9

are shown in Figure B.4. Notice how the points are randomly scattered throughout the

design space; several of the modified Latin hypercube designs discussed later in this section

attempt to control the scattering.

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X1

X2

X1

X2

X1

X2

Figure B.4 Three Nine Point Latin Hypercube Designs for 2 Factors

By their nature, Latin hypercubes are quite easy to generate because they require

only a random permutation of n levels in each column of the design matrix. The big

advantage of Latin hypercube designs is that they ensure stratified sampling, i.e., each of

the input variables is sampled at n levels. Thus, when a Latin hypercube is projected or

collapsed into a single dimension, n distinct levels are obtained. This is extremely

beneficial for deterministic computer experiments since the Latin hypercube points do not

overlap, minimizing any information loss.

B.2 .2 Random Orthogonal Arrays

An orthogonal array (OA) is a matrix of n rows and k columns with every element

being one of q symbols: 0, ..., q-1 (Owen, 1992). An orthogonal array has an associated

strength t depending on the number of combinations of l levels appearing in any of the r

columns of the OA. As an example, an OA with strength r = 2 and l = 3 levels is shown in

Figure B.5. As Booker, et al. (1995) describe, all possible combinations, taken any two at

a time, of the three levels 1, 2, and 3, are as follows: 1 1, 1 2, 1 3, 2 1, 2 2, 2 3, 3 1, 3 2,

and 3 3. If any two columns are chosen in the OA, each of these combinations appears

only once in the design as a row. The same happens for any pair of columns; hence, the

OA has strength r = 2.

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Run # X1 X2 X31 1 1 12 1 2 23 1 3 34 2 1 25 2 2 36 2 3 17 3 1 38 3 2 19 3 3 2

X1

X2

X3

Figure B.5 Orthogonal Array of Strength 2 for 3 Factors

Notice that if the OA design in Figure B.5 is projected into any two dimensions, the

points form a 3x3 grid of nine points. This turns out to be a nice feature of these types of

designs; as Barton (1994) points out, “Orthogonal arrays...are an attractive class of sparse

designs because they provide balanced (full factorial) designs for any projection into r

factors.” Barton also states, however, that this type of balance can lead to problems with

kriging models; when these designs are projected into one dimension, points do overlap

which can lead to ill-conditioning of the correlation matrix. Meanwhile, the classical

experimental designs, in general, do not exhibit good projection capabilities, yielding

several overlapping points when projected into two dimensions as can be ascertained from

Figure B.1 and Figure B.2. However, these designs do not project well into single

dimensions as Latin hypercube designs do. Owen’s algorithm for generating these designs

was obtained from his web site.

B.2 .3 IMSE Optimal Latin Hypercubes

Park (1994) combines Latin hypercube designs and integrated mean square error

(IMSE) optimal designs (Sacks, et al., 1989) to generate a hybrid set of designs which he

refers to as optimal Latin hypercubes or olhd for short. These designs are well spread out

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over the design space because of the IMSE criterion, do not have any replicated points, and

are often symmetric or nearly so. They also maintain the good projection capabilities which

are characteristic of Latin hypercubes. A 7 point olhd and an 8 point olhd for two factors

are shown in Figure B.6. Notice how well the points are spread out in the design space;

yet, they do not extend all the way to the edge of the design space.

X1

X2

X1

X2

(a) 7 point design (b) 8 point design

Figure B.6 Optimal Latin Hypercube Designs for 2 Factors

Park has developed a two-stage (exchange and Newton-type) algorithm which finds

a olhd for a given number of factors and runs; the reader is referred to (Park, 1994) for

details about the algorithm, and he has provided a copy of it for use in this research. The

algorithm works well for problems with a small number of factors and a small number of

runs. The algorithm can also be used to generate optimal Latin hypercube designs which

maximize the entropy of a design following the work in (Shewry and Wynn, 1987). Only

designs which minimize IMSE over the design space are considered in this work.

B.2 .4 Maximin Latin Hypercubes

Morris and Mitchell (1995) develop a maximin Latin hypercube design (Mmlhd) for

computer experiment applications in an effort to compromise between the maximin distance

criterion used in (Johnson, et al., 1990) and the good projective capabilities of Latin

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hypercubes. As the name implies, the maximin distance criterion is used to maximize the

minimum distance (either Euclidean or rectangular) between any two sample points, thus

spreading the points out as much as possible in the design space. As an example, a two

factor Mmlhd with 7 points and 8 points are shown in Figure B.7.

X1

X2

X1

X2

(a) 7 point design (b) 8 point design

Figure B.7 Maximin Latin Hypercube Designs for 2 Factors

Morris and Mitchell use a simulated annealing search algorithm to construct these

designs. They also have generated a catalog of these designs (Morris and Mitchell, 1992)

for both Euclidean and rectangular distances for n (number of points) between 3 and 12 and

k (number of factors) between 2 and 5 and designs for k=2(n-20), k=n(n-9), and k=n/2(n-

14). For designs not listed in the catalog, their algorithm has been obtained for use in this

research. It is worth noting that for problems with large n and large k, the simulated

annealing algorithm used to construct these designs is quite slow, thus limiting the use of

these types of designs to relatively small problems.

B.2 .5 Orthogonal-Array Based Latin Hypercubes

Tang (1993) uses Latin hypercubes to construct strength r orthogonal arrays which

he refers to as U designs when used for designing experiments. His designs are quite

similar to those of Owen (1992) discussed earlier; however, the proposed sampling

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schemes were developed independently of each another. The strength r OA-based Latin

hypercubes stratify each r-dimensional projection—particularly good for one dimensional

projections when fitting a model—providing designs which are better suited for computer

experiments and numerical integration. An example six point U design and Latin

hypercube design are shown in Figure B.8.

X1

X2

X1

X2

(a) 6 point U design (b) 6 point Latin hypercube

Figure B.8 Six Point U Design and Latin Hypercube Design

As Tang points out in his paper, notice in Figure B.8 how the points are more

uniformly scattered in the U design than the Latin hypercube design in the two dimensional

region. Supposed that the area inside the dashed boxes represents the region of interest.

The U design is the more preferable design because each dashed box contains exactly one

design point which is not so for the Latin hypercube. The algorithm for creating these

designs can be found in (Tang, 1993); a copy of it has been provided by Tang for use in

this dissertation.

B.2 .6 Uniform Designs

Uniform designs are a class of designs based on statistical applications of number-

theoretic methods (Fang and Wang, 1994). These designs are denoted as UDn(qt) where

UD signifies uniform design, n is the number of experiments, q is the number of levels of

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each factor, and t is the number of columns in the design, i.e., the number of factors. All

of the uniform designs considered in this work have q = n; moreover, n is strictly an odd

number.

A uniform design is obtained from a generating vector (n; h1, . . . , h t) of a good

lattice point set where 1 = h1 < h2 < ... < ht < n which has greatest common divisor (n, hi)

= 1, i = 1, ..., t. Specifically, the UD is formed from the terms qki which are defined:

qki = khi (mod n), k = 1, ..., n; i = 1, ..., t [B.1]

where 0 < qki ≤ n. Generating vectors (n; h1, ..., h t) for small n are listed in the appendix

of (Fang and Wang, 1994); the criterion for choosing the generating vectors is the mean

square error criterion. The generating vectors utilized in this work are summarized in Table

B.1.

Table B.1 Uniform Design Generating Vectors for Small Sample Sizes

2 Factors, h1=1 3 Factors, h1=1 4 Factors, h1=1n h2 n h2 h3 n h2 h3 h4

7 3 13 3 9 21 2 10 179 4 15 2 7 23 2 5 1011 7 17 3 9 25 4 6 913 5 19 3 9 27 5 17 25

In general, UD are easy to create once the generating vector is known. The first

column of the design is the levels 1, ..., n. Subsequent columns are based on the qki from

Equation 3.17 which requires estimating the multiplication modulo n of khi. Example 7, 9,

and 11 point uniform designs for 2 factors are shown in Figure B.9.

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X1

X2

X1

X2

X1

X2

(a) 7 point designs (b) 9 point designs (c) 11 point designs

Figure B.9 Uniform Designs for 2 Factors with 7, 9, and 11 Points

B.2 .7 Orthogonal Latin Hypercubes

Ye (1997) has created a class of Latin hypercubes which preserves the

orthogonality among columns; hence, these designs are referred to as orthogonal Latin

hypercubes (OLH). These designs retain the orthogonality of traditional experimental

designs (e.g., CCDs) while attempting to maintain a good spread of points throughout the

design space. The orthogonality guarantees that the quadratic and interaction effects are

uncorrelated with the estimates of linear effects. The reader is referred to (Ye, 1997) for a

detailed description for creating orthogonal Latin hypercubes; he has provided a copy of his

algorithm for use in this dissertation. It is worth noting that the orthogonality of these

designs does not depend on the numerical values of the levels; hence, new OLH designs

can be computed by permuting the numerical values or reversing the signs of the columns.

Three permutations of a 9 point OLH for 2 factors are shown in Figure B.10.

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X1

X2

X1

X2

X1

X2

Figure B.10 Nine Point Orthogonal Latin Hypercube Designs for 2 Factors

The OLH designs are constructed by purely algebraic means without the aid of

computers; however, Ye does propose an algorithm to search for optimal OLH designs

based on different selection criteria (minimum entropy, and maximin spacing based on

either Euclidean of rectangular distance). Ye’s algorithm and an example application of

OLH designs to an injection molding cooling process with six variables are given in (Ye,

1997). In this dissertation, optimal OLH designs are not considered, only random

permutations of the algebraically-generated OLH designs as illustrated in Figure B.10.

B.2 .8 Hammersley Sequence

Latin hypercube techniques are designed for uniformity along a single dimension

where subsequent columns are randomly paired for placement on a k-dimensional cube.

Hammersley sequence sampling (HSS) provides a low-discrepancy experimental design

for placing n points in a k-dimensional hypercube (Kalagnanam and Diwekar, 1997),

providing better uniformity properties over the k-dimensional space than Latin hypercubes.

Example 7, 9, and 11 point HSS designs are illustrated in Figure B.11.

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X1

X2

X1

X2

X1

X2

(a) 7 point design (b) 9 point design (c) 11 point design

Figure B.11 HSS Designs for 2 Factors with 7, 9, and 11 Points

In (Kalagnanam and Diwekar, 1997), it is shown by means of an example that HSS

designs require significantly fewer samples to converge to the variance of a derived

distribution than Latin hypercube designs and Monte Carlo (random sampling) methods,

thus verifying the good uniformity properties of these types of designs in k-dimensions.

The reader is referred to (Kalagnanam and Diwekar, 1997) for a definition of Hammersley

points and an explicit procedure for generating them. The algorithm in (Iman and

Shortencarier, 1984) has been modified by Diwekar (see, Diwekar, 1995) to generate HSS

designs efficiently; Diwekar has provided a copy of the algorithm for use in this research.

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C. APPENDIX C

A MINIMAX LATIN HYPERCUBE DESIGNGENERATOR USING A GENETIC ALGORITHM

The details of the minimax Latin hypercube design which are unique to this

dissertation are presented in this appendix. In Section C.1 the question of “Why a minimax

Latin hypercube design?” is addressed. The genetic algorithm which is used to generate

these designs is then presented in Section C.2. Convergence studies of the genetic

algorithm are included in Section C.3.

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C.1 WHY A MINIMAX LATIN HYPERCUBE DESIGN?

From an intuitive stand point, since prediction with a kriging model relies on the

spatial correlation between data points, a design which minimizes the maximum distance

between the sample points and any point in the design space should yield an accurate

predictor. This type of design is referred to as a minimax design (Johnson, et al., 1990).

Thus, the novel space filling design advocated in this work is the combination of the

minimax criterion with a Latin hypercube design. Why combine the two? Alone, the

minimax criterion does not ensure good stratification of the design space, i.e., when the

sample points are projected into 1-dimension, many of the points may overlap. Because a

Latin hypercube ensures good stratification of the design space when projected into 1-

dimension, the minimax criterion combined with a Latin hypercube design provides a good

compromise between minimizing the maximum distance between the sample points and any

point in the design space and the good projection properties of the Latin hypercube. Morris

and Mitchell (1995) used a similar argument when combining the maximin criterion with

Latin hypercubes to create maximin Latin hypercube designs; their designs performed better

than Latin hypercube designs and maximin designs in comparative studies.

How are these novel designs created? The algorithm provided by Morris and

Mitchell (1995) could not be modified easily to accommodate the minimax criterion as

opposed to the maximin criteria. Moreover, their use of a simulated annealing algorithm to

generate maximin Latin hypercube designs is slow at best. Thus, an entirely different

approach for generating minimax Latin hypercube designs was conceived.

Consider that a Latin hypercube is a (n x k) matrix which has n runs with n levels

for k factors. Since each column of the Latin hypercube contains a random permutation of

n numbers, there are n! possible permutations for each of k columns, yielding a total of

(n!)k possible combinations. For large n and k, an exhaustive search of all possible

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combinations to find the minimax design is too time consuming if not impossible. Hence,

an efficient method to search for good designs is needed. Toward this end, a genetic

algorithm (see, e.g., Goldberg, 1989) based design generator is developed to create

minimax Latin hypercube designs. Genetic algorithms have been shown to work well for

large combinatorial problems, and genetic algorithms have been successfully utilized to

create D-optimal designs (see, e.g., Giunta, et al., 1994; Narducci, 1995) and minimum

bias designs (Venter and Haftka, 1997). Moreover, the discrete nature of the Latin

hypercubes appears to be well suited for the genetic algorithm formulation. A description

of the algorithm is provided in the next section.

C.2 A GENETIC ALGORITHM FOR GENERATING MINIMAX LATINHYPERCUBE DESIGNS

The genetic algorithm for creating minimax Latin hypercube designs developed in

this dissertation is shown in Figure C.1. To begin, the user specifies the maximum

number of generations, maxngen, to be created and n and k, the number of levels (and

therefore rows) in each hypercube and the number of factors (i.e., columns), respectively.

In Step 1, an initial population of size npopsize of random Latin hypercubes with n rows

and k columns is created, and the number of generations, ngen, is set to 1.

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1. Generate a population of size npopsize of random Latin hypercube designs, lhds;

initialize number of generations, ngen, to 1

2. For each lhd in lhds, compute the distances, dist, between all the samples and each

of the 2k corners.

3. Rank order the designs based on smallest maximum distance, maxdist, of each lhd.

4. Iterate Steps a-h until ngen is equal to maxngen, the maximum ngen

a. increment ngen to ngen + 1

b. select top 1/2 of population based on rank ordering

c. mate lhd imate={2, ..., npopsize/2} with best design, generate two new lhd:

i. child 1 = best design with randomly chosen column from imate

ii. child 2 = imate with randomly chosen column from best design

d. child 1 is mutated if prob = rand*normalized(maxdist(imate)) > P1

e. child 2 is mutated if prob = rand*normalized(maxdist(imate)) > P2

f. best design is kept in population

g. final design in new generation is best design with a random mutation

h. repeat steps 2 and 3

5. Report best design found after maxngen populations have been created

Figure C.1 Genetic Algorithm for Creating Minimax Latin Hypercubes

Step 2 involves computing the distances between all the sample points in a given

Latin hypercube and each of the 2k corners of the design space. By definition, the minimax

distance criterion serves to minimize the maximum distance between the sample points and

all possible prediction points. However, as the design space under investigation is a

hypercube (as opposed to spherical or irregularly shaped), the prediction point that is

furthest away is always one of the corners, see Figure C.2. Therefore, only n•2k distance

calculations need to be made as opposed to computing distances over the entire design

space. In this work, the squared Euclidean distance is used when computing distances;

however, rectangular distance also could be used.

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X1

X2

Equidistant points lie on

circles of expanding radii

Sample point

In [1,n] hypercube,opposite corner is furthest new point from sample point

k

Figure C.2 Expanding Concentric Circles of Equidistant Points

Once all of the distances have been computed, the Latin hypercubes are rank

ordered in Step 3 based on the maximum distance for each design where the smaller the

maximum distance, the better the design. The top half of the population then is selected for

mating in Step 4, provided the maximum number of generations, maxngen, has not yet

been created. The best design in each population is mated with each of the designs in the

top half of the population, excluding itself. The best design is carried over into the next

generation—reminiscent of a survival of the fittest strategy (Goldberg, 1989)—and a

random mutation of the best design is also included in the new population.

During mating, the usual genetic algorithm process of crossover (see, e.g.,

Goldberg, 1989) is modified because Latin hypercubes must maintain n distinct levels in

each column. So rather than switch individual “genes” within two mating Latin

hypercubes, one column is chosen randomly in each mating design, and the resulting

columns are switched. The first child is the best design with the randomly chosen column

from the mating design, and the second child is the mating design, imate, with the

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randomly chosen column form the best design. After mating is done, mutation may occur.

Child 1 undergoes mutation if the probability P = (random number)(normalized maximum

distance of imate) is greater than P1; Child 2 undergoes mutation if the probability P =

(random number)(normalized maximum distance of imate) is greater than P2. When

computing the probability, normalization of the maximum distance is taken as:

normalized maxdist = maxdist − maxdistmin

maxdist max − maxdist min

[C.1]

where maxdistmax and maxdistmin are for the entire population before mating. In this

manner, the larger (worse) the maximum distance of the parent design, the greater the

probability that its child (Child 2) will undergo a mutation in hopes of further reducing its

maximum distance after mating. Meanwhile, since the normalized maximum distance of

the best design is zero, the probability of mutating Child 1 is also based on the normalized

distance of the mating parent, but the probability of mutation, P1, is higher than P2 because

this design has more in common with the best design and therefore need not mutate as

often.

Once a new population has been created, Steps 2 and 3 are repeated, computing the

distances between the sample points and the corners and rank ordering the designs based

on smaller maximum distance being better. The best 1/2 of the population is then mated

and mutated again as the minimax distance Latin hypercube is sought. A pictorial

illustration of the mating and mutation process is shown in Figure C.3.

In Figure C.3, an initial population of six Latin hypercube designs are generated

randomly. In this example, each Latin hypercube has five sample points for two factors as

illustrated in the 2-D grids of sample points. Once these designs are created, the distance

between each point and each corner is computed, and the maximum distance is recorded.

Multiple points can have the same maximum distance; hence, the number of points having

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this maximum distance is also recorded. In the example shown in Figure C.3, LHDs 4, 5,

and 6 each have two points which are at the maximum distance of 25.

5

6x

x

INITIALPOPULATION

MAXIMUMDISTANCE RANK

NEWPOPULATION

LHD 24 12 45 23 51 3

LHD 41 44 22 53 15 3

LHD 61 52 43 35 24 1

LHD 33 32 14 41 25 5

LHD 12 53 41 25 34 1

➀

➁ ➃

➂

Pt 5, Crnr 2Pt 1, Crnr 3Pt 3, Crnr 4

= 25}

Pt 1, Crnr 3 = 32

Pt 1, Crnr 2Pt 3, Crnr 2 = 25}

= 25}Pt 1, Crnr 3Pt 3, Crnr 3

4

1

LHD 51 44 52 25 33 1

= 25}Pt 2, Crnr 1Pt 1, Crnr 3

MATING &MUTATION

LHD 2’4 12 45 23 51 3

LHD 6’4 15 42 23 51 3

LHD 4’4 42 25 53 11 3

LHD 3’1 14 42 23 55 3

LHD 1’4 12 55 23 41 3

25

25

32

32

25

NEW MAXDISTANCE

x

Mutate col 1 row 2 & 3

col 2

Mutate: col 2row 2 & 4

col 1

col 2

col 1

col 2

col 2no change

25= 32Pt 5, Crnr 1 col 1

col 1

2

3

LHD 5’4 42 55 23 31 1

indicates max distancesample pt ➀ corner numberKEY: x deleted from population

Figure C.3 Genetic Algorithm-Based Minimax Latin Hypercube Generator

Once the distances have been computed, the designs are rank ordered based on

maximum distance with smaller maximum distances preferred over larger ones. Once the

rank ordering is completed, the top half of the population is selected for mating. The best

design, that which is ranked first, then is mated with each of the remaining designs in the

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top half of the population as previously described. The columns which are switched and

the mutations which occur in the example in Figure C.3 are listed alongside each arrow.

For instance, LHD 5’ is a combination of column 2 from LHD 2 and column 2 from LHD

5; no mutation occurs for this particular design. The maximum distance of the resulting

new designs then are computed, and the process is repeated until the specified number of

generations has been created. The best design at the end of this process is a minimax Latin

hypercube design. Convergence studies of the minimum maximum distance are offered in

the next section.

C.3 EXAMPLE MINIMAX LATIN HYPERCUBE DESIGNS ANDCONVERGENCE STUDIES

For the purposes of demonstration, six minimax Latin hypercubes are generated

from varying population sizes to investigate the convergence of the genetic algorithm, see

Table C.1. Population sizes of 20, 40, and 60 Latin hypercubes are investigated for two 2,

3, and 4 variable designs ranging from 9 points to 41 points as shown in the table. In the

current algorithm, the number of generations is used to dictate termination of the algorithm.

For the 2 and 3 variable designs, 50 generations are created before the genetic algorithm is

terminated; in the 4 variable problem, 100 generations are permited before termination. The

resulting minimax distance—the square of the Euclidean distance—between any sample

point and any point in the design space is listed for each combination of number of

variables, sample size, population size, and number of generations. The number of Latin

hypercubes which have this minimax distance also are listed. In general, as the sample size

increases, the number of Latin hypercubes with this minimax distance increase.

Convergence plots for each sample size (e.g., two factors with 14 samples) are shown in

Figure C.4 through Figure C.9.

376

Table C.1 Summary of Minimax Latin Hypercube Convergence Study

number ofvariables

number ofsamples

pop.size

number ofgenerations

minimaxdistance

# lhd withminimax dist

2 9 20 50 89 32 9 40 50 89 112 9 60 50 89 132 14 20 50 244 82 14 40 50 244 162 14 60 50 244 203 15 20 50 417 63 15 40 50 417 93 15 60 50 404 123 23 20 50 1026 53 23 40 50 1014 13 23 60 50 1014 94 25 20 100 1538 44 25 40 100 1502 64 25 60 100 1495 124 41 20 100 4550 44 41 40 100 4550 84 41 60 100 4500 13

General observations regarding the convergence of the genetic algorithm in Figure

C.4 through Figure C.9 include the following:

• The larger the population size, the quicker the convergence to a minimax design.

• For the two and three variable designs, 50 generations is more than sufficient to

achieve a minimax design; however, considerably more generations are needed in

the 4 variable designs to achieve low minimax distances.

• The larger the sample size and design space, the more important the population size.

In the two variable case, convergence to a minimax design occurs regradless of the

population size; however, in the three and four variable cases, a population size of

60 is necessary to achieve the lowest minimax distance.

There are several other factors which can affect the performance of the genetic algorithm

which are not studied (e.g., different mutation probabilities, random seeds, alternative

mating routines, etc.); these factors are discussed in Section 8.3 as possible future work.

377

88

90

92

94

96

98

100

0 10 20 30 40 50Iterations

Max

imum

Dis

tanc

e

popsize=20popsize=40popsize=60

Figure C.4 Convergence of 2 Factor mMlhd with 9 Sample Points

240

245

250

255

260

265

270

0 10 20 30 40 50Iterations

Max

imum

Dis

tanc

e



378

400

405

410

415

420

425

430

435

440

445

0 10 20 30 40 50Iterations

Max

imum

Dis

tanc

epopsize=20popsize=40popsize=60


1010

1020

1030

1040

1050

1060

1070

0 10 20 30 40 50Iterations

Max

imum

Dis

tanc

e



379

1450

1500

1550

1600

1650

1700

1750

0 20 40 60 80 100Iterations

Max

imum

Dis

tanc

epopsize=20popsize=40popsize=60


4500

4550

4600

4650

4700

4750

4800

4850

4900

4950

5000

0 20 40 60 80 100Iterations

Max

imum

Dis

tanc

e



380

D. APPENDIX D

KRIGING TESTBED PROBLEMS

Kriging/DOE Testbed

In Chapter 5 six engineering test problems—consisting of 2 two variable problems,

2 three variable problems, and 2 four variable problems—are introduced to test the utility of

kriging and space filling experimental designs for building metamodels of deterministic

computer experiments. The two variable problems are the design of a two-bar truss,

Section D.1, and the design of a symmetric three bar truss, Section D.2; the three variable

problems are the design of a helical compression spring, Section D.3, and a two-member

frame, Section D.4; and the four variables problems are the design of a welded beam,

Section D.5, and a pressure vessel, Section D.6. As explained in Section 5.1.1, building

kriging approximations of these analyses is overkill to say the least; however, they are

taken to be representative of typical analyses encountered in mechanical engineering design,

allowing Hypotheses 2 and 3 to be tested and verified. Each example is described in turn

along with its corresponding constraints, bounds, and objective function.

381

D.1 DESIGN OF A TWO-BAR TRUSS

The symmetric two-bar truss shown in Figure D.1 has been studied by several

researchers (see, e.g., Balling and Clark, 1992; Schmit, 1981; Sobieszczanski-Sobieski, et

al., 1982). In this example, a single load case of 2P = 66,000 lbs is considered. The

distance between the supports is 2B = 60 in. The two bars are identical, having an annular

cross section with wall thickness T = 0.1 in. The material properties are Young’s modulus

E = 30E6 psi, density ρ = 0.3 lbs/in3 and yield stress σy = 60,000 psi. There are two

design variables—mean tube diameter (D) and height (H) of the truss—which have the

following ranges of interest:

0.5 in. ≤ D ≤ 5 in.5 in. ≤ H ≤ 50 in.

t

D

Section C-C’

H2P

B B

C

C’

Figure D.1 Two-Bar Truss

The objective is to minimize the weight of truss system which is:

W(x) = 2ρπDT(B2 + H2)1/2 [D.1]

382

subject to the following behavioral constraints:

g1(x) = σe - σ = π 2E(D2 + T2)

8(B2 + H2 )−

P(B2 + H2)1/2

πTDH ≥ 0 [D.2]

g2(x) = σy - σ = σy - P(B2 + H2)1/2

πTDH ≥ 0 [D.3]

The first constraint prevents failure due to Euler buckling, the second due to yield. The

resulting optimum value from (Schmit, 1981) for W(x) is 19.8 lbs, occurring at:

D* = 2.47 in.H* = 30.15 in.

Hence, two constraints—Equations D.1 and D.2—and one objective function—

Equation D.3—are to be replaced by kriging models. These three equations are to be

approximated over the region of interest specified by the bounds on the two design

variables given at the beginning of this section. The next example is another two variable,

structural optimization problem.

D.2 DESIGN OF A SYMMETRIC THREE-BAR TRUSS

The second example also comes from (Schmit, 1981); it is the design of a

symmetric three-bar planar truss, see Figure D.2. Two loads are considered: P1 = 20,000

lbs acting at 45° to the x axis and P2 = 20,000 lbs acting at an angle of 135° to the axis.

Pertinent design parameters include the following: N = 10 in., β1 = 135°, β1 = 90°, β1 = 45°,

density ρ = 0.1 lbs/in3 and Young’s modulus E = 10E6 psi.

383

N

x

P1P2

αααα

αα

A1 A2 A3

Figure D.2 Symmetric Three-Bar Truss

The cross sectional areas of the bars are the design variables. Due to symmetry,

there are only two design variables: A1 = A3 and A2; the ranges of interest are:

0.5 in2 ≤ A1 ≤ 1.2 in2

0.0 in2 ≤ A2 ≤ 4.0 in2

The objective is to find A1 and A2 to minimize the weight of the truss subject to stress

constraints. The weight of the structure calculated as:

W(x) = ρN(2 2A1 + A2) [D.4]

Stress constraints guard against both tensile and compressive failure and are

formulated as:

-15,000 ≤ σij(x) ≤ 20,000 i = 1, 2, 3; j = 1, 2

where σij is the stress in the ith member due to the jth load. Because of symmetry, only three

of the six inequality constraints need to be considered. The corresponding expressions for

these are as follows.

384

g1(x) = 20,000 - σ11 = 20,000 - 1

A1

−A2

2A1A2 + 2A12

≥ 0 [D.5]

g1(x) = 20,000 - σ21 = 20,000 - 20,000 2A1

2A1A2 + 2A12 ≥ 0 [D.6]

g1(x) = 15,000 + σ31 = 15,000 - −20,000A 2

2A 1A2 + 2A12 ≥ 0 [D.7]

The optimum value W(x) given in (Schmit, 1981) is 2.64 lbs, occurring at:

A1* = A1* = 0.788 in2

A2* = 0.41 in2

In summary, there are three constraints—Equations D.5-D.7—and one objective—

Equation D.4— which are to be replaced by kriging models for the design space defined by

the bounds on A1 and A2. This concludes the two variables examples considered in this

dissertation; the first three variable problem, the design of a helical compression spring, is

detailed in the next section.

D.3 DESIGN OF A HELICAL COMPRESSION SPRING

A problem which is perhaps more representative of traditional mechanical

engineering design is the design of a helical compression spring (Kannan and Kramer,

1994; Sandgren, 1990; Siddall, 1982), see Figure D.2. There are three design variables—

number of active coils (N), mean coil diameter (D), and wire diameter (d)—with the

following ranges of interest:

3 ≤ N ≤ 301.0 in. ≤ D ≤ 6.0 in.0.2 in. ≤ d ≤ 0.5 in.

The spring is to be manufactured from music wire spring steel ASTM A228; the

maximum outside diameter, Dmax, must be less than 3 in. The allowable stress is S =

189,000 psi, and the shear modulus is G = 1.15E8. The maximum working load is Fmax =

385

1000 lbs, the preload compressive force is Fp = 300 lbs, and the maximum deflection under

preload is δpm = 6 in. The combined deflection from preload to maximum load is δw = 1.25

in. The maximum free length is lmax = 14 in. The spring has an end coefficient equal to 1,

and it is assumed to be guided; therefore, buckling need not be considered.

Figure D.3 Helical Compress Spring (from Siddall, 1982)

The objective is to minimize the volume of material in the spring for a static loading

condition. The volume of material in the spring is given as:

V(x) = π2Dd2(N + 2)/4 [D.8]

The constraints for the design are as follows.

386

D.3.1 Shear Stress

Following the problem formulations given in (Kannan and Kramer, 1994;

Sandgren, 1990; Siddall, 1982), the design shear stress must be less than the allowable

maximum shear stress or:

g1(x) = S - 8CfFmaxD/(πd3) ≥ 0 [D.9]

where:

Cf = (4C - 1)/(4C - 4) + 0.615/C

C = D/d

In reality, the spring correction factor, Cf, given by the preceding equation is for

springs subject to dynamic loading and is only important when fatigue is a concern. In this

example, the spring is assumed to be under a constant load, and the correction factor which

should be used for a spring in static loading is (Shigley and Mischke, 1989):

Cf = 1 + 0.5/C

However, in order to maintain consistency with previous problem formulations and

solutions which are also in error, (see, Kannan and Kramer, 1994; Sandgren, 1990;

Siddall, 1982), the fatigue correction factor is used in this example since it is only for

demonstrative purposes.

D.3.2 Free Length

The spring free length must be less than the specified value:

g2(x) = lmax - lf ≥ 0 [D.10]

where:

387

lf = δ + 1.05(N + 2)d

δ = Fmax/K

K = Gd4/(8ND3)

This assumes that the spring length is 1.05 times the solid length under Fmax.

D.3.3 Preload Deflection

The deflection under preload must not exceed specified deflection δpm:

g3(x) = δpm - δ ≥ 0 [D.11]

D.3.4 Combined Deflections

The combined deflections must be consistent with the length

lf - δp - (Fmax - Fload)/K - 1.05(N + 2)d ≥ 0

where:

δp = Fp/K

As Siddall (1982) points out, this constraint function will always be zero at convergence;

therefore, it is not included in the problem formulation.

D.3.5 Deflection Requirement

The deflection from preload to maximum load must be equal to that specified:

g4(x) = (Fmax - Fload)/K - δw ≥ 0 [D.11]

388

D.3.6 Geometric Constraints

There are two additional geometric constraints which also must be satisfied. The

outside diameter of the coil must be less than the maximum specified:

g5(x) = Dmax - D - d ≥ 0 [D.12]

The inner coil diameter must be at least three times the wire diameter for proper winding:

g6(x) = C - 3 ≥ 0 [D.13]

This spring problem is typically solved using a mixed discrete/continuous solver.

The continuous solution which is given in (Sandgren, 1990) is V(x) = 2.6353 which

occurs at:

N* = 9.192D* = 1.2052 in.d* = 0.2814 in.

Hence, seven constraints—Equations D.9-D.13—and one objective function—

Equation D.8—are to be approximated using kriging models. The design region of interest

is defined by the bounds on the design variables given at the beginning of this section. The

design of a two-member frame presented in the next section is the other three variable test

problem being considered in this dissertation.

D.4 DESIGN OF A TWO-MEMBER FRAME

This example is from (Arora, 1989) and is the design of a two-member frame

subjected to the out-of-plane loads shown as in Figure D.4. There are three design

variables—frame width (d), height (h), and wall thickness (t)—with the following ranges

of interest:

2.5 in. ≤ d ≤ 10 in.2.5 in. ≤ h ≤ 10 in.

389

0.1 in. ≤ t ≤ 1.0 in.

ht

d

(1)

(2)

(3)LL

z

P

x y

U1

U2 U3

Figure D.4 Two-Member Frame

The objective is to minimize the volume of the frame subject to stress constraints

and size limitations. The volume of the structure is given by:

V(x) = 2L(2dt + 2ht - 4t2) [D.14]

where the length, L, of each member is 100 in.

The two members in the frame are subject to both bending and torsional stresses,

and the combined stress only needs to be imposed at nodes (1) and (2) since the frame is

symmetric and the two members of the frame are identical. The stresses are calculated

using the finite element method where the nodal displacements are defined as U1 = vertical

displacement at node (2), U2 = rotation about line (3)-(2) and U3 = rotation about line (1)-

(2). Following (Arora, 1989), the equilibrium equation for the finite element model is:

390

EI

L3

24 −6L 6L

−6L (4L2 +GJ

EIL2) 0

6L 0 (4L2 + GJEI

L2)

U1

U2

U3

=P

0

0

where E = 3.0E7 psi, G = 1.154E7 psi and the load at node (2), P = -10,000 lbs. Values

for I, J, and A are given by:

I = (1/12)[dh3 - (d - 2t)(h - 2t)3]

J = 2t[(d - t)2(h-t)2]/(d + h - 2t)

A = (d - t)(h - t)

Once the displacements U1, U2 and U3 are calculated for a given design, the torque,

T, and bending moments at nodes (1), M1, and (2), M2, for member (1)-(2) are:

T = -GJU3/L

M1 = 2EI(-3U1 + U2L)/L2

M2 = 2EI(-3U1 + 2U2L)/L2

The corresponding torsional shear and bending stress are then computed using:

τ = T/(2At)

σ1 = M1h/(2I)

σ2 = M2h/(2I)

The effective stress, σe, at nodes (1) and (2) is determined using von Mises yield criterion

and must be less than the allowable stress which is taken as 40,000 psi.

g1(x) = σe,1 = (σ12 + 3τ2)1/2 ≤ 40,000 [D.15]

391

g2(x) = σe,2 = (σ22 + 3τ2)1/2 ≤ 40,000 [D.16]

The resulting optimum value from (Arora, 1989) is V(x) = 703.916 in3, occurring at:

d* = 7.798 inh* = 10.00 int* = 0.10 in

In summary, two constraints—Equations D.15 and D.16—and one objective—

Equations D.14—are to be replaced with kriging approximations over the design space

defined by the bounds on the design variables listed at the beginning of this section. This

is the last of the three variable problems considered in this dissertation; the first four

variable test problem is introduced in the next section.

D.5 DESIGN OF A WELDED BEAM

This example is taken (Ragsdell and Phillips, 1976) and has been solved by several

researchers studying design optimization (see, e.g., Eggert and Mayne, 1993; Kannan and

Kramer, 1994; Sandgren, 1989). The objective is to minimize the total system cost of the

welded beam structure shown in Figure D.5 subject to five constraints which define

feasibility. The length, L, of the bar is 14 in., and the force acting on the bar is taken as

6000 lbs. The bar “A” is made out of 1010 steel. There are four design variables—weld

height (h), weld length (l), bar thickness (t), and width (b)—with the following ranges of

interest:

0.125 in. ≤ h ≤ 2.0 in. 2.0 in. ≤ t ≤ 10.0 in.2.0 in. ≤ l ≤ 10.0 in. 0.125 in. ≤ b ≤ 2.0 in.

392

B

A

L

F

h

F

b

l t

Figure D.5 Welded Beam

The objective function is a combination of set-up cost, welding labor cost and

material cost and is given by:

F(x) = (1 + c3)h2l + c4tb(L + l) [D.17]

where c3 = (0.37)(0.283)($/in.) and c4 = (0.17)(0.283)($/in.). The constraints for the

problem include the maximum shear stress in the weld, the maximum normal stress in the

beam, the bar buckling load, the bar end deflection, and a geometric constraint which

ensures that the thickness of the weld, h, is less than the bar width, b, assuming a 45°

weldment. The necessary equations for the remaining constraints are as follows:

D.5.1 Weld Stress

The shear stress in the weld is given by:

τ(x) = [(τ’)2 + 2τ’τ’’cosθ + (τ’’)2]1/2 [D.18]

where:

cosθ = l/2R

R = {(l2/4) + [(t + h)/2]2}1/2

τ’ = F/ 2hl and τ’’ = MR/J

393

M = F[L + (l/2)]

J = 2{0.707hl[(l2/12) + (t + h)/2)2]}

The shear stress in the weld must be less than the design shear stress in the weld, τd, which

is taken to be 13,600 psi.

D.5.2 Bar Bending Stress

The maximum bending stress in the bar is given by:

σ(x) = 6FL/(bt2) [D.19]

which must be less than the design normal stress for the beam material, σd, which is taken

as 30,000 psi.

D.5.3 Bar Buckling Load

For narrow rectangular bars, a good approximation to the buckling load is:

Pc(x) =4.013 EIα

L2 [1− (t

2L)

EI

α] [D.20]

where I = (1/12)tb2, and α = (1/3)Gtb3. Young’s modulus, E, is equal to 30E6 psi; the

shear modulus, G, is equal to 12E6 psi. The value for Pc(x) must be greater than F in

order for the bar not to buckle under the applied load.

D.5.4 Bar Deflection

Assuming that the bar is a cantilever beam of length L, the deflection in the beam is:

DEFL(x) = 4FL3/(Et3b) [D.21]

The deflection of the beam must less than the maximum permissible deflection which is

taken as 0.25 in.

394

The optimum value F(x) given in (Ragsdell and Phillips, 1976) is $2.386,

occurring at:

h* = 0.2455 in. t* = 8.2730 in.l* = 6.1960 in. b* = 0.2455 in.

Hence, there are four constraints—Equations D.18-D.21—and one objective

function—Equation D.17—which are replaced by kriging models and approximated over

the design space defined by the variable bounds given at the beginning of this section. In

the next section, the design of a pressure vessel is introduced as the second four variable

problem for investigation.

D.6 DESIGN OF A PRESSURE VESSEL

The final four variable example is the design of a pressure vessel (Li and Chou,

1994; Sandgren, 1990). The cylindrical pressure vessel is shown in Figure D.6. The shell

is made in two halves of rolled steel plate which are joined by two longitudinal welds.

Available rolling equipment limits the length of the shell to 20 ft. The end caps are

hemispherical, forged, and welded to the shell. All welds are single-welded butt joints

with a backing strip. The material is carbon steel ASME SA 203 grade B. The pressure

vessel should store 750 ft3 of compressed air at a pressure of 3,000 psi. There are four

design variables—radius (R) and length (L) of the cylindrical shell, shell thickness (Ts),

and spherical head thickness (Th)—which have the following ranges of interest:

25 in. ≤ R ≤ 150 in. 1.0 in. ≤ Ts ≤ 1.375 in.25 in. ≤ L ≤ 240 in. 0.625 in. ≤ Th ≤ 1.0 in.

395

R

TsTh

R

L

Figure D.6 Pressure Vessel

The design objective is to minimize total system cost which is a combination of

welding, material, and forming costs. The total system cost is given by:

F(x) = 0.6224T sRL + 1.7781ThR2 + 3.1661T s

2L + 19.84Ts2R [D.22]

Meanwhile, the constraints which limit the minimal wall thicknesses Ts and Th are from the

ASME boiler and pressure vessel codes and are given as:

g1(x) = Ts - 0.0193R ≥ 0 [D.23]

g2(x) = Th - 0.00954R ≥ 0 [D.24]

The constraint for the minimum tank volume is written as:

g3(x) = πR2L + (4/3)πR3 - 1.296E6 ≥ 0 [D.25]

The fourth constraint limiting the length of the cylinder already has been accounted for in

the bounds for the design variable L.

396

This problem is typically solved using a mixed discrete/continuous solver;

however, only the continuous solution which is given by Sandgren (1990) is of interest.

The optimum, continuous solution is F(x) = 7867.0, occurring at:

R* = 47.7008 in. Ts* = 1.1 in.L* = 117.701 in. Th* = 0.6 in.

Thus, three constraints—Equations D.23-D.27—and one objective function—

Equation D.22—are replaced by kriging approximations over the design region of interest

within the bounds of the design variables.

397

E. APPENDIX E

SUPPLEMENTAL INFORMATION FORKRIGING/DOE STUDY

This appendix contains supplemental information for the kriging/DOE study

performed in Chapter 5. Specifically, the process used to cull the data and remove potential

outliers is explained in Section E.1. Through this process, the data is reduced from a total

of 7905 models to 7578 models as discussed in Section 5.1.4. The analysis of variance

(ANOVA) of the resulting data set is given in Section E.2; this information supplements the

discussion in Section 5.2.1. Finally, interaction plots for DOE and NSAMP for each

problem are given in Section E.3 as a supplement to the work in Section 5.4.

398

E.1 CULLING THE DATA TO REMOVE POTENTIAL OUTLIERS

The process employed in this dissertation to cull the data and remove potential

outliers involves three steps:

1. cull the data first based on RMSE.RANGE because it is the most important measure

of model accuracy, see Section E.1.1;

2. cull the data next based on MAX.RANGE because it is the next most important

measure of model accuracy, see Section E.1.2; and

3. cull the data for the final time based on CVRMSE.RANGE because it is the least

important measure of model accuracy, see Section E.1.3.

The results of each step are described in each of the following sections.

E.1.1 Step 1: Culling the Data Based on RMSE.RANGE

To begin the culling process to remove potential outliers, a plot of the distribution

of RMSE.RANGE for each model for each pair of problems is shown in Figure E.1.

Because the majority of the data is scattered near the origin, the points with unusually large

RMSE.RANGE values are considered potential outliers and are thus removed from the data

set. As can be seen in the figure, the RMSE.RANGE for one of the two variable models is

almost 600, and there are two models which have RMSE.RANGE values over 400 in the 3

variable problems.

In each pair of problems, the data is culled until all of the data points cluster

together and there are few, if any, points which reside by themselves. Points to the right of

the dashed lines in Figure E.1 (i.e., where the hump of the distribution returns to the x-

axis) are removed from the data set. In this manner, the number of models in the two,

three, and four variable problems is reduced from 1785 to 1745, 3150 to 3095, and 2970

to 2951, respectively, leaving a total of 7791 models.

399

0.0

0.02

0.04

0 100 200 300 400 500 600

rmse.range

Den

sity

0.00.020.040.06

0 100 200 300 400

rmse.range

Den

sity

0.0

0.4

0 10 20 30

rmse.range

Den

sity

4 Variable Problems

3 Variable Problems

2 Variable Problems

potential outliers removed from data set

Figure E.1 Original Distribution of RMSE.RANGE of Data

The resulting distribution of RMSE.RANGE of the models of the culled data set is

shown in Figure E.2. Notice in the figure how the points are better distributed with few

extreme points.

400

0

612

0.0 0.1 0.2 0.3 0.4

rmse.range

Den

sity

0

10

0.0 0.1 0.2 0.3 0.4

rmse.range

Den

sity

0

10

20

0.0 0.2 0.4 0.6

rmse.range

Den

sity

4 Variable Problems

3 Variable Problems

2 Variable Problems

Figure E.2 Resulting Distribution of RMSE.RANGE after Culling

The next step is to cull the data further based on MAX.RANGE as described in the

next section.

E.1.2 Step 2: Culling the Data Based on MAX.RANGE

After culling the data based on RMSE.RANGE, potential outliers based on

MAX.RANGE are removed from the data set. The distribution of MAX.RANGE after

culling the data based on RMSE.RANGE is shown in Figure E.3. There are few potential

outliers, e.g., the group of points in the four variable problems with MAX.RANGE values

near 14. The dashed lines indicate the approximate cut-off point used to cull the data, with

all points to the right of the dashed line removed because they are potential outliers.

401

0.0

1.0

2.0

0 1 2 3 4

max.range

Den

sity

0.01.02.0

0 1 2 3

max.range

Den

sity

0.00.40.8

0 5 10 15

max.range

Den

sity

4 Variable Problems

3 Variable Problems

2 Variable Problems

potential outliers

Figure E.3 Distribution of MAX.RANGE of Culled Data Set

The resulting distribution of MAX.RANGE after all of the potential outliers are

removed is illustrated in Figure E.4. Notice that the remaining points are clustered together

nicely with relatively few extreme points or groups of points. Twenty-eight models are

removed from the two variable problems, thirty-three models are removed from the three

variable problems, and seventy-one models are removed from the four variable problems,

leaving a total of 7659 models.

402

0.01.0

2.0

0.0 0.5 1.0 1.5 2.0 2.5

max.range

Den

sity

0.01.02.03.0

0.0 0.5 1.0 1.5 2.0 2.5

max.range

Den

sity

0.01.02.0

0 1 2 3 4 5 6

max.range

Den

sity

4 Variable Problems

3 Variable Problems

2 Variable Problems

Figure E.4 Resulting Distribution of MAX.RANGE after Culling

The final step is to cull the data based on CVRMSE.RANGE; this is done in the

next section.

E.1.3 Step 3: Culling the Data Based on CVRMSE.RANGE

After culling the data based on RMSE.RANGE and MAX.RANGE, the resulting

distribution of CVRMSE.RANGE of the models of the culled data set is shown in Figure

E.5. Based on CVRMSE.RANGE, a few potential outliers still remain, and these are

removed from the data set. The dashed lines show the point at which the data is culled with

all the models to the right of the line removed from the data set. This reduces the number

403

of models in the two, three, and four variable problems from 1717 to 1690, 3062 to 3049,

and 2880 to 2839, leaving 7578 models in the final data set.

0

24

0.0 0.5 1.0 1.5 2.0 2.5

cvrmse.range

Den

sity

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4 Variable Problems

3 Variable Problems

2 Variable Problems

potential outliers

Figure E.5 Distribution of CVRMSE.RANGE of Culled Data Set

The resulting distributions of CVRMSE.RANGE of the models after culling are

illustrated in Figure E.6. Since this is the last step in the culling process to remove any

potential outliers, this distribution represents the final distribution of the values of

CVRMSE.RANGE which are analyzed in the data set. Notice that the data is fairly well

clustered with no extreme points. The final distributions for RMSE.RANGE and

MAX.RANGE are shown in Figure E.7 and Figure E.8, respectively.

404

0246

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048

12

0.0 0.2 0.4 0.6 0.8 1.0 1.2

cvrmse.range

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sity

4 Variable Problems

3 Variable Problems

2 Variable Problems

Figure E.6 Distribution of CVRMSE.RANGE of Final Culled Data Set

The final distribution of RMSE.RANGE of the models in the culled data set after all

of the potential outliers have been removed is shown in Figure E.7. The data look good

even through there is a large group of models in the four variable problems which are

slightly removed from the rest of the data. However, these models are not removed from

the data set because there is such a large grouping. The scatter in the two and three variable

problems appears to be okay with only a few extreme points. Regardless, the distributions

in Figure E.7 represent the final distribution of RMSE.RANGE values of the models which

are analyzed.

405

05

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4 Variable Problems

3 Variable Problems

2 Variable Problems

Figure E.7 Distribution of RMSE.RANGE of Final Culled Data Set

The final distribution of MAX.RANGE of the models in the culled data set is

shown in Figure E.8. The data are scattered nicely and well clustered. As such, there are

no more potential outliers which need be removed. These distributions represent the final

distributions of MAX.RANGE of the 7578 models in the data set which are to be analyzed.

406

0.01.02.0

0.0 0.5 1.0 1.5 2.0 2.5

max.range

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max.range

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0 1 2 3 4 5 6

max.range

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4 Variable Problems

3 Variable Problems

2 Variable Problems

Figure E.8 Distribution of MAX.RANGE of Final Culled Data Set

Now that potential outliers have been removed from the data set, it is ready to

analyzed. The analysis of variance is discussed in the next section. This information

supplements the discussion in Section 4.2.1.

E.2 ANALYSIS OF VARIANCE RESULTS

The analysis of variance (ANOVA) results for the culled data set are summarized as

follows. The software package S-Plus4 (MathSoft, 1997) is employed to perform the

necessary analyses; Pr(F) values of 10% or less are considered significant. A discussion

of the ramifications of these results is given in Section 4.2.1, see Table 4.7.

407

ANOVA FOR 2 VARIABLE PROBLEMS

RMSE.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 12 0.790807 0.0659006 72.3791 0.0000000 nsamp 7 0.231039 0.0330056 36.2503 0.0000000 eqn 6 3.236058 0.5393430 592.3647 0.0000000 corfcn 4 0.109507 0.0273766 30.0680 0.0000000 doe:nsamp 31 0.414692 0.0133772 14.6923 0.0000000 doe:eqn 72 0.729166 0.0101273 11.1229 0.0000000 doe:corfcn 48 0.137153 0.0028574 3.1383 0.0000000 nsamp:eqn 42 0.225910 0.0053788 5.9076 0.0000000nsamp:corfcn 28 0.017491 0.0006247 0.6861 0.8900391 eqn:corfcn 24 0.076081 0.0031700 3.4817 0.0000000 Residuals 1415 1.288346 0.0009105

MAX.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 12 39.2740 3.27283 132.978 0.0000000 nsamp 7 13.2717 1.89596 77.035 0.0000000 eqn 6 200.1000 33.35000 1355.043 0.0000000 corfcn 4 1.9708 0.49271 20.019 0.0000000 doe:nsamp 31 7.8845 0.25434 10.334 0.0000000 doe:eqn 72 53.5926 0.74434 30.243 0.0000000 doe:corfcn 48 2.8798 0.05999 2.438 0.0000003 nsamp:eqn 42 11.5988 0.27616 11.221 0.0000000nsamp:corfcn 28 0.1757 0.00627 0.255 0.9999754 eqn:corfcn 24 1.0262 0.04276 1.737 0.0150119 Residuals 1415 34.8256 0.02461

CVRMSE.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 12 0.49719 0.041432 7.922 0.0000000 nsamp 7 0.25892 0.036989 7.073 0.0000000 eqn 6 13.91795 2.319658 443.548 0.0000000 corfcn 4 38.94373 9.735933 1861.635 0.0000000 doe:nsamp 31 0.16915 0.005457 1.043 0.4020019 doe:eqn 72 0.81794 0.011360 2.172 0.0000001 doe:corfcn 48 1.13349 0.023614 4.515 0.0000000 nsamp:eqn 42 0.37403 0.008905 1.703 0.0036142nsamp:corfcn 28 3.51757 0.125628 24.022 0.0000000 eqn:corfcn 24 13.38914 0.557881 106.674 0.0000000 Residuals 1415 7.40013 0.005230

408


RMSE.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 14 0.396151 0.0282965 39.6287 0.0000000 corfcn 4 0.468482 0.1171205 164.0250 0.0000000 nsamp 9 0.517310 0.0574789 80.4980 0.0000000 eqn 9 4.548163 0.5053515 707.7348 0.0000000 doe:corfcn 56 0.200964 0.0035886 5.0258 0.0000000 doe:nsamp 39 0.281790 0.0072254 10.1190 0.0000000 doe:eqn 126 0.927631 0.0073622 10.3105 0.0000000corfcn:nsamp 36 0.025811 0.0007170 1.0041 0.4624071 corfcn:eqn 36 0.121429 0.0033730 4.7238 0.0000000 nsamp:eqn 81 0.576383 0.0071158 9.9656 0.0000000 Residuals 2641 1.885782 0.0007140

MAX.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 14 63.6899 4.54928 95.4241 0.0000000 corfcn 4 16.9380 4.23449 88.8212 0.0000000 nsamp 9 18.6596 2.07328 43.4885 0.0000000 eqn 9 356.9195 39.65772 831.8472 0.0000000 doe:corfcn 56 6.1304 0.10947 2.2962 0.0000002 doe:nsamp 39 37.2497 0.95512 20.0343 0.0000000 doe:eqn 126 81.4085 0.64610 13.5524 0.0000000corfcn:nsamp 36 1.7057 0.04738 0.9938 0.4796303 corfcn:eqn 36 3.4569 0.09603 2.0142 0.0003410 nsamp:eqn 81 32.5882 0.40232 8.4390 0.0000000 Residuals 2641 125.9078 0.04767

CVRMSE.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 14 0.33509 0.02393 5.706 0.0000000 corfcn 4 49.62834 12.40709 2957.611 0.0000000 nsamp 9 0.39570 0.04397 10.481 0.0000000 eqn 9 12.65203 1.40578 335.111 0.0000000 doe:corfcn 56 3.03595 0.05421 12.923 0.0000000 doe:nsamp 39 0.35440 0.00909 2.166 0.0000416 doe:eqn 126 1.86217 0.01478 3.523 0.0000000corfcn:nsamp 36 0.47594 0.01322 3.152 0.0000000 corfcn:eqn 36 18.17634 0.50490 120.358 0.0000000 nsamp:eqn 81 0.38010 0.00469 1.119 0.2222467 Residuals 2641 11.07891 0.00419

409


RMSE.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 14 0.92494 0.066067 51.370 0.0000000 nsamp 13 0.28108 0.021622 16.812 0.0000000 eqn 8 13.86528 1.733160 1347.593 0.0000000 corfcn 4 0.08302 0.020756 16.139 0.0000000 doe:nsamp 38 0.20198 0.005315 4.133 0.0000000 doe:eqn 111 6.72750 0.060608 47.125 0.0000000 doe:corfcn 56 0.54880 0.009800 7.620 0.0000000 nsamp:eqn 104 0.97659 0.009390 7.301 0.0000000nsamp:corfcn 52 0.01214 0.000233 0.182 1.0000000 eqn:corfcn 32 0.22220 0.006944 5.399 0.0000000 Residuals 2406 3.09439 0.001286

MAX.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 14 252.069 18.0049 120.648 0.0000000 nsamp 13 41.469 3.1899 21.375 0.0000000 eqn 8 2551.345 318.9182 2137.007 0.0000000 corfcn 4 4.116 1.0290 6.895 0.0000162 doe:nsamp 38 61.576 1.6204 10.858 0.0000000 doe:eqn 111 1007.096 9.0729 60.796 0.0000000 doe:corfcn 56 5.537 0.0989 0.662 0.9748813 nsamp:eqn 104 161.063 1.5487 10.377 0.0000000nsamp:corfcn 52 0.524 0.0101 0.068 1.0000000 eqn:corfcn 32 5.331 0.1666 1.116 0.2994217 Residuals 2406 359.062 0.1492

CVRMSE.RANGE Df Sum of Sq Mean Sq F Value Pr(F) doe 14 0.97608 0.06972 17.785 0.0000000 nsamp 13 0.18734 0.01441 3.676 0.0000082 eqn 8 12.16207 1.52026 387.799 0.0000000 corfcn 4 42.77639 10.69410 2727.929 0.0000000 doe:nsamp 38 0.21336 0.00561 1.432 0.0426066 doe:eqn 111 4.12844 0.03719 9.487 0.0000000 doe:corfcn 56 4.24417 0.07579 19.333 0.0000000 nsamp:eqn 104 0.47239 0.00454 1.159 0.1343197nsamp:corfcn 52 0.08734 0.00168 0.428 0.9998909 eqn:corfcn 32 17.00269 0.53133 135.537 0.0000000 Residuals 2406 9.43206 0.00392

410

E.3 INTERACTION OF DOE AND NSAMP

In addition to looking at how well individual sample sizes in a design affect the

accuracy of the resulting kriging model as is done in Section 4.4, it also is interesting to

look at how the number of sample points in a design affects the accuracy of the model to try

to determine the relationship between sample size and model accuracy. As the number of

sample points increases, the approximation should become more accurate, but how many

points is enough for designs with variable sample sizes? This is examined in this section,

particularly in Figure E.9 - Figure E.11.

Of the designs utilized in this dissertation, only five designs allow variable sample

sizes: Hammersley sampling sequence (hamss) designs, minimax Latin hypercube (mnmxl)

designs, maximin Latin hypercube (mxmnl) designs, optimal Latin hypercube designs

(oplhd), random Latin hypercube designs (rnlhd), and uniform designs (unifd). A plot of

the average effect of each type of DOE across the sample range for each pair of problems is

given in Figure E.9 - Figure E.11. Recall that the following sample sizes are used for each

pair of problems (refer to Table 5.2 - Table 5.4):

• NSAMP = 7-14 for the two variable problems,

• NSAMP = 14-25 for the three variable problems, and

• NSAMP = 20-33 for the four variable problems.

411

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ange

0.04

0.06

0.08

0.10

0.12

0.14

7 8 9 10 11 12 13 14

DOEunifdoplhdrnlhdhamssmxmnlmnmxl

Figure E.9 Effect of DOE and Sample Size on RMSE.RANGE for the TwoVariable Problems

In Figure E.9, the general observed trend is that the accuracy of the resulting

kriging model (as measured by RMSE.RANGE) improves as the sample size increases

regardless of the type of design. This is of no surprise since additional sample points

should yield a more accurate model because there is more information on which to base the

approximation. The same trends can observed in Figure E.10 and Figure E.11 for the three

and four variable problems, respectively. However, it appears that a wide enough spread

of sample sizes has not been examined to draw any conclusions regarding the minimum

number of samples which should be taken in order to ensure minimum prediction error.

412

0.04

0.06

0.08

0.10

13 14 15 16 17 18 19 21 23 25

mnmxlrnlhdhamssunifdoplhd

DOE

NSAMP

mea

n of

rm

se.r

ange

Figure E.10 Effect of DOE and Sample Size on RMSE.RANGE for theThree Variable Problems

NSAMP

mea

n of

rm

se.r

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0.02

0.04

0.06

0.08

20 21 22 23 24 25 26 27 28 29 31 33

DOEunifdhamssoplhdrnlhdmnmxl

Figure E.11 Effect of DOE and Sample Size on RMSE.RANGE for theFour Variable Problems

413

F. APPENDIX F

SUPPLEMENTAL INFORMATION FOR GAAEXAMPLE PROBLEM

This appendix contains detailed supplemental information for the GAA example

problem in Chapter 7. Section F.1 contains a brief description of the design variables used

in the GAA example. The sample points, data, and MLE theta parameters for the kriging

metamodels are given in Section F.2. Analysis of variance of the sample data is offered in

Section F.3; it is used to generate the Pareto plots in Section 7.6.3. Additional design

scenarios for the exercising the PPCEM product platform compromise DSP are presented

and discussed in Section F.4, and convergence histories for the individually designed

benchmark aircraft for Scenarios 1-3 are plotted in Section F.5. Information for the GAA

product variety tradeoff study discussed in Section 7.6.3 is included in Section F.6.

Justification of the design variable weights used in the computation of NCI are explained in

Section F.6.1 while Sections F.6.2 and F.6.3 contain all of the results—instantiations of

each aircraft, responses, and corresponding deviation function values—of the product

variety tradeoff study for Scenarios 2 and 3, respectively. Finally, sample DSIDES files

for the PPCEM platform and the individual aircraft are included in Sections F.7 and F.8.

414

F.1 GAA DESIGN VARIABLE DESCRIPTION

The intent in this section is to provide the reader with a brief description of the

design variables considered in the GAA example—cruise speed, aspect ratio, propeller

diameter, wing loading, engine activity factor, and seat width—and why they are important

in aircraft design. Where applicable, typical ranges of the design variables for current GAA

are provide to justify the design variable bounds employed in this work.

Cruise speed (CSPD): General Aviation aircraft are typically subsonic aircraft with

cruise speed in the range of 0 < M < 0.6 where M is the mach number. Based on

the GAA Design Competition guidelines (NASA and FAA, 1994) the aircraft

should be capable of maintaining a cruise speed, CSPD, between 150 and 300 kts.

Therefore, the cruise speed Mach number varies from Mach 0.24 to Mach 0.48 in

order to find the “best” velocity at which to fly the cruise portion of the mission.

Aspect ratio (AR): Aspect ratio of a wing is defined as the span of the wing squared

divided by its area; span is the distance from wing tip to wing tip measured

perpendicular to the plane (Raymer, 1992). Raymer states that the Wright brothers

were the first to investigate the effect of aspect ratio on flight performance,

observing that a long, skinny wing (high aspect ratio) has less drag than a short, fat

wing (low aspect ratio). In general, wings with a high aspect ratio tend to have

lower induced drag and therefore larger values of (L/D)max. Moreover, a high

aspect ratio wing tends to have high lift-curve slopes. High lift curve slopes have

two consequences:

1. at low speeds, the approach attitude is conducive to good runway visibility from

the cockpit, and

2. the ride through turbulence is rougher.

Also, high aspect ratio wings usually weigh more than low aspect ratio wings. In

general, General aviation aircraft have aspect ratios ranging from 7.6 (single

engine) to 9.2 (twin turboprop). In this investigation, aspect ratios ranging from 7-

11 are investigated.

415

Propeller diameter (DPRP): Propeller diameter is self-explanatory. Generally

speaking, the larger the propeller diameter, the more efficient the propeller will be;

however, the limitation on the length of propeller diameter is the propeller tip speed

which should be kept below sonic speed (Raymer, 1992). From a survey of

modern GAA, the propeller diameter varies from 5.5 ft to 6.8 ft. Raymer suggests

to limit the tipspeed to be less than 850 ft/sec for wooden propellers; this translates

into a maximum diameter of 5.96 ft. A range of 5 ft to 5.96 ft is examined in this

problem.

Wing loading (WL): Wing loading is the ratio of the aircraft weight to the area of

the wing and affects stall speed, climb rate, takeoff and landing distances, and turn

performance; wing loading also has a very strong effect upon the takeoff gross

weight of an aircraft (Raymer, 1992). The larger the wing, the lower the wing

loading. Raymer notes that the larger wing helps reduce takeoff/landing field

lengths; however, the additional drag and empty weight due to the larger wing will

increase takeoff gross weight in order to perform the mission. A larger wing also is

liable to affect operating costs because more fuel is consumed to provide the

necessary lift to achieve the mission. Representative wing loadings of General

Aviation aircraft vary between 17 (single engine) and 26 (twin engine); the range

utilized in the GAA example is 17 to 25.

For the GAA flight envelope, i.e., cruise speeds of Mach 0.2 to Mach 0.6, piston

and turboprop engines are the most widely used engines. In addition, a piston engine has a

low fuel consumption compared to a turboprop engine. Although, the engine weight of a

piston engine (1.1 - 1.75 lb/take-off hp) is heavier than a turboprop (.35 - .55 lb/take-off

hp), the acquisition cost of the piston engine ($25 - $ 50/hp) is much cheaper than the

turboprop engine ($60 - $100/hp). General Aviation aircraft pilots are primarily concerned

about the direct operating cost (DOC) of the aircraft. In general practice, piston engines are

very competitive for GAA which require less than 500 hp compared to turboprop engines

which are more suitable for a powerful GAA engine of more than 500 hp. For this work,

the engine selected for the GAA design is a piston, reciprocate fuel injection, engine

equipped with 350 hp.

416

Engine activity factor (AF): To help size the engine, the engine activity factor is

used; it is a measure of the amount of power being absorbed by the propeller

(Raymer, 1992). Raymer states that engine activity factors range from about 90 -

200. For example, a typical light-aircraft, such as a general aviation aircraft, has an

activity factor of 100. A range of 85-110 is examined in this study where 110 is the

value of the baseline design.

Seat width (WS): The seat width is used to size the width of the cabin; in particular:

cabin width = 2 x (seat width) + (aisle width) + 12 in [A.1]

where aisle width is taken as a fixed parameter and 12 in. is the total thickness of

the fuselage walls. As seat width increases, cabin width increases and the surface

area of the fuselage increases causing an increased drag. Seat widths for an

economy sized aircraft vary between 17 in and 22 in, whereas seat widths in a small

aircraft vary between 16 in and 18 in (Raymer, 1992). In this study, the baseline

value of 20 in is used as the upper bound for seat width while the lower bound is

taken as 14 in for comfort reasons.

These six variables constitute the control factors for the General Aviation aircraft

family. The bounds on the design variables define the design space in which the kriging

models are constructed. The kriging model construction is documented in the next section.

F.2 GAA KRIGING METAMODELS: SAMPLE POINTS, RESPONSEVALUES, AND FITTED MLE PARAMETERS

Table F.1 contains the 64 sample points from the randomized orthogonal array used

in this study, and are scaled to fit the design space based on the bounds listed in the

beginning of Section 7.2. GASP is invoked using these 64 points for each aircraft; the

response values are listed in Table F.2 for the one passenger (two seater) aircraft, in Table

F.3 for the three passenger (four seater) aircraft, and in Table F.4 for the five passenger

(six seater) aircraft.

417

Kriging metamodels are constructed for the mean, µ, and standard deviation, σ, of

each response, yielding a total of 18 models for the nine GAA responses. For the GAA

example problem, it is assumed that the demand for each aircraft is uniform; therefore, the

scale factor—the number of passengers—is assumed to be uniformly distributed and so are

the corresponding responses. In light of this, the mean and standard deviation of each

response are computed from the GASP response values as follows.

• Mean: µy j,i=

y j,i,1 + y j,i,3 + y j,i,5

3, j = {noise, wemp, ..., ldmax}, i = {1, 2, ..., n} [F.1]

• Std. Dev.: σy j,i=

y j,i,1 − y j,i,5

12, j = {noise, wemp, ..., ldmax}, i = {1, 2, ..., n} [F.2]

So for example, the mean and deviation of the direct operating cost, DOC, for the 3rd

experimental design is estimated as:

µyDOC,3=

yDOC,3,1 + yDOC,3,3 + yDOC,3,5

3=

59.289 + 60.593 + 61.322

3= 60.4013

σyDOC,3=

yDOC,3,1 − yDOC,3,5

12=

59.289 − 61.322

12= 0.587

To fit the kriging models to the data, the sample points listed in Table F.1 are scaled

to be between [0,1] and the simulated annealing algorithm and mlefinder.f algorithm given

in Section A.2.1 are invoked to find the maximum likelihood estimates for each θk in the

Gaussian correlation function. The resulting MLE values for the theta parameters, θk, used

to fit each kriging model are listed in Table F.5. These parameters produce the “best”

kriging model for the sample data and are used within the GAA Compromise DSP and

DSIDES to facilitate the design of a family of GAA using the PPCEM.

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Table F.1 64 Point Orthogonal Array Used to Build Kriging Metamodelsfor GAA Example Problem

Run # CSPD AR DPRP WL AF WS Run # CSPD AR DPRP WL AF WS1 0 0 0 0 2 4 33 2 0 5 5 3 02 0 2 7 1 3 3 34 2 2 4 4 2 63 0 3 2 6 4 5 35 2 3 6 3 0 74 0 1 3 7 0 2 36 2 1 1 2 4 15 0 7 5 4 1 1 37 2 7 0 1 7 26 0 5 4 5 7 7 38 2 5 7 0 1 57 0 4 6 2 5 6 39 2 4 2 7 6 38 0 6 1 3 6 0 40 2 6 3 6 5 49 1 0 7 6 0 1 41 3 0 4 3 4 210 1 2 0 7 4 7 42 3 2 5 2 0 511 1 3 3 0 3 6 43 3 3 1 5 2 312 1 1 2 1 2 0 44 3 1 6 4 3 413 1 7 4 2 6 4 45 3 7 7 7 5 014 1 5 5 3 5 3 46 3 5 0 6 6 615 1 4 1 4 7 5 47 3 4 3 1 1 716 1 6 6 5 1 2 48 3 6 2 0 7 117 4 0 2 4 5 7 49 6 0 6 1 6 518 4 2 3 5 6 1 50 6 2 1 0 5 219 4 3 0 2 1 0 51 6 3 5 7 7 420 4 1 7 3 7 6 52 6 1 4 6 1 321 4 7 6 0 4 3 53 6 7 2 5 0 622 4 5 1 1 0 4 54 6 5 3 4 4 023 4 4 5 6 2 2 55 6 4 0 3 3 124 4 6 4 7 3 5 56 6 6 7 2 2 725 5 0 3 2 7 3 57 7 0 1 7 1 626 5 2 2 3 1 4 58 7 2 6 6 7 027 5 3 7 4 6 2 59 7 3 4 1 5 128 5 1 0 5 5 5 60 7 1 5 0 6 729 5 7 1 6 3 7 61 7 7 3 3 2 530 5 5 6 7 2 1 62 7 5 2 2 3 231 5 4 4 0 0 0 63 7 4 7 5 4 432 5 6 5 1 4 6 64 7 6 0 4 0 3

419

Table F.2 Response Values for 1 Passenger GAA for 64 Point OA

PAX NOISE WEMP DOC ROUGH WFUEL PURCH RANGE VCRMX LDMAX1 72.557 1884.54 63.124 2.028 447.596 41403.1 2983 184.628 16.0731 73.834 1894.91 60.672 2.236 438.478 42613.8 2823 198.503 17.0911 73.001 1889.44 59.289 1.972 450.704 42457.9 2164 199.883 16.4841 73.192 1816.08 57.814 1.905 523.536 41264.1 2167 205.805 15.9111 73.484 1878.83 57.027 2.318 464.365 42609 2377 205.608 19.121 73.301 1955.33 60.303 2.075 384.01 43843.4 2119 198.371 17.0991 73.638 1953 61.177 2.217 382.699 43714.4 2602 196.692 17.4031 72.766 1865.21 57.705 2.299 478.156 42000.3 2725 200.446 19.0191 73.893 1807.07 65.091 1.948 529.699 41357.2 2252 208.777 15.7951 72.624 1906.7 70.292 1.803 431.643 42363.3 1965 193.086 15.6781 73.107 1952.68 72.466 2.236 384.506 43203.4 2983 190.378 17.4921 72.97 1835.46 68.823 2.177 502.377 41130.1 3318 196.206 17.1131 73.272 1943.61 68.558 2.346 397.792 43572.4 2649 198.129 19.1561 73.468 1904.32 67.194 2.258 436.036 42976.2 2513 201.539 18.3421 72.78 1913.85 69.355 2.084 427.426 42759.6 2458 196.092 17.4061 73.67 1879.05 64.918 2.228 463.882 42727.6 2181 206.98 18.61 72.994 1912.96 83.018 1.869 419.247 42581.9 2543 193.104 15.5871 73.149 1834.45 82.796 2.058 505.538 41693.3 2421 204.48 17.4461 72.571 1842.57 95.928 2.17 499.555 41057 3100 193.176 18.5411 73.846 1926.6 83.071 2.026 403.906 43285.7 2444 196.817 16.4291 73.614 1953.73 83.177 2.49 386.029 43744.4 2946 196.706 20.2561 72.758 1924.92 87.896 2.252 416.307 42549.1 3157 190.261 19.0871 73.507 1858.01 83.004 2.105 483.301 42326.9 2139 207.308 18.1191 73.326 1913.58 83.365 2.059 428.406 43270.5 1903 203.93 18.2461 73.129 1868.49 82.837 2.042 466.84 41942 2963 196.348 16.5841 72.988 1885.51 83.241 2.068 452.136 42107 2790 195.578 17.3971 73.847 1875.17 83.152 2.155 461.134 42643.5 2273 204.065 18.0241 72.603 1878.8 80.39 1.877 456.447 41852.1 2371 193.308 16.4051 72.797 1957.43 82.199 2.04 384.397 43591.6 1999 195.558 18.521 73.688 1849.04 83.168 2.127 492.969 42384 1945 210.852 18.6811 73.283 1879.81 82.696 2.403 461.839 42090.5 3366 196.674 19.4231 73.449 1980.06 83.443 2.336 358.95 44141.9 2804 194.624 19.2671 73.517 1801.96 78.834 1.991 537.274 41161.6 2482 206.886 16.2181 73.318 1912.87 83.034 2.017 425.578 42840.1 2483 197.619 16.521 73.663 1947.52 83.17 2.106 390.13 43541.3 2538 197.102 16.9851 72.775 1840.03 82.578 2.099 499.966 41220.9 3155 195.949 17.0611 72.529 1917.35 80.334 2.363 427.24 42607.3 3098 192.691 19.8981 73.82 1961.03 83.205 2.386 376.73 43757 2820 195.237 18.7721 73.001 1863.4 80.381 2.009 481.741 42234.9 2029 204.399 17.4931 73.146 1902.26 81.269 2.113 443.218 42984.5 2074 203.254 18.2971 73.315 1841.93 82.827 2.029 494.783 41588 2858 200.196 16.3531 73.485 1910.72 82.979 2.135 426.787 42748.5 2905 196.903 17.1491 72.807 1862.12 80.866 2.026 477.791 41802.7 2390 198.855 17.4791 73.676 1876.65 83.17 2.026 458.394 42417.1 2470 201.639 16.4681 73.86 1854.67 83.365 2.209 487.486 42671.1 1792 212.367 19.3921 72.592 1923.47 80.04 1.985 416.958 42885.9 2066 195.418 17.6551 73.119 1969.26 83.205 2.212 368.491 43554.8 2991 190.894 17.951 72.92 1911.33 83.139 2.446 431.433 42674.6 3357 194.807 19.941 73.646 1913.87 82.961 2.076 418.014 42854.3 2643 194.644 16.5711 72.745 1886.34 82.812 2.225 452.166 41924.3 3213 191.292 18.2961 73.506 1883.04 83.151 1.981 454.951 42843.3 1930 205.321 17.3711 73.346 1845.59 82.848 1.941 491.884 41826 2264 203.683 16.6981 72.992 1946.02 83.284 2.119 396.058 43394.9 2188 196.527 19.0651 73.135 1855.94 82.96 2.247 488.335 42080.8 2519 203.967 19.4281 72.571 1860.63 88.447 2.163 480.849 41581.7 2786 195.383 18.9931 73.829 1991.8 83.512 2.278 344.412 44500.9 2458 195.942 19.0271 72.854 1874.45 80.302 1.738 459.146 41829.4 2055 194.868 15.61 73.681 1827.34 83.056 2.059 510.184 41944.7 2119 208.918 17.7471 73.278 1881.32 82.916 2.298 458.798 42314.1 3140 198.097 18.8621 73.453 1964.89 83.26 2.144 367.103 43625.5 2625 190.495 17.1111 73.125 1950.06 83.457 2.258 391.466 43564.4 2533 196.797 19.7441 72.959 1896.25 83.161 2.287 446.543 42462.4 2925 197.121 19.5721 73.854 1906.05 83.356 2.119 430.593 43237.6 2085 203.73 18.2541 72.588 1894.72 94.83 2.132 447.424 42112.4 2463 193.412 19.53

420


PAX NOISE WEMP DOC ROUGH WFUEL PURCH RANGE VCRMX LDMAX3 72.557 1918.88 64.063 2.014 412.124 41922.6 2822 181.831 15.7543 73.834 1928.1 61.734 2.226 404.586 43125.8 2704 195.66 16.7563 73 1925.67 60.593 1.958 413.277 43024.6 2108 196.887 16.1193 73.192 1849.37 59.014 1.895 489.025 41740.6 2111 202.305 15.5523 73.483 1910.54 58.096 2.312 431.878 43105 2328 202.889 18.7083 73.3 1993.62 61.557 2.063 344.707 44474.5 2072 195.605 16.7283 73.638 1990.4 62.354 2.208 344.484 44336 2556 194.113 17.0493 72.766 1895.28 58.879 2.295 447.241 42447.7 2661 197.602 18.6223 73.892 1838.43 66.418 1.939 497.408 41828.8 2206 205.824 15.4543 72.623 1946.14 72.437 1.785 390.773 42978 1907 189.914 15.333 73.107 1990.4 74.287 2.227 345.985 43820.7 2828 187.784 17.1463 72.97 1860.74 70.579 2.172 476.278 41462.4 3174 193.144 16.7473 73.271 1978.84 70.129 2.339 361.849 44132.3 2597 195.285 18.7593 73.468 1938.46 68.667 2.253 401.155 43526.4 2464 198.914 17.9583 72.779 1949.21 71.166 2.074 391.06 43312 2397 193.19 17.0333 73.67 1912.1 66.321 2.222 430.041 43231.6 2137 203.902 18.1923 72.993 1952.37 83.139 1.854 378.457 43215.9 2482 190.248 15.2673 73.149 1863.17 82.996 2.053 474.828 42092.9 2365 201.257 17.063 72.571 1872.74 95.278 2.16 466.836 41493.6 3020 190.278 18.1573 73.846 1964.34 83.115 2.012 364.829 43903.6 2401 194.125 16.0933 73.619 1986.98 83.418 2.485 351.476 44266.9 2839 193.94 19.8563 72.758 1960.12 90.09 2.24 378.344 43101.4 3079 187.519 18.7033 73.506 1888.49 83.148 2.099 451.216 42770.1 2092 204.137 17.7153 73.326 1948.95 83.382 2.048 391.932 43821.9 1857 200.84 17.8343 73.13 1900.19 82.997 2.031 432.777 42421.6 2828 193.559 16.2533 72.988 1919.09 83.139 2.054 416.669 42603.4 2722 192.414 17.0353 73.847 1904.74 83.175 2.15 429.793 43066.5 2233 200.904 17.643 72.603 1915.58 82.05 1.862 418.25 42419.1 2307 190.3 16.0543 72.804 1995.43 82.24 2.031 345.096 44186.8 1944 192.387 18.1093 73.688 1880.24 83.291 2.122 460.288 42845.4 1903 207.68 18.2563 73.282 1909.54 82.83 2.397 431.04 42537.9 3223 193.995 19.0463 73.448 2017.81 83.547 2.325 318.929 44757.7 2752 191.832 18.8813 73.517 1831.63 80.707 1.979 506.32 41587.6 2431 203.781 15.8793 73.318 1951 83.256 2.004 385.524 43458.8 2427 194.826 16.173 73.663 1986.32 83.407 2.095 349.085 44164.7 2487 194.094 16.6313 72.775 1868.5 82.325 2.092 469.152 41609.5 3077 192.754 16.6963 72.529 1949.89 80.762 2.355 392.093 43104.4 3025 189.898 19.4973 73.82 1997.35 83.436 2.377 338.826 44343.6 2692 192.487 18.4043 73.001 1895.96 82.577 2.001 447.565 42715.1 1977 201.172 17.13 73.145 1937.58 83.201 2.103 406.199 43523.3 2024 200.012 17.8873 73.315 1872.96 82.792 2.016 461.365 42056.1 2798 197.414 16.0133 73.484 1946.57 83.101 2.124 388.517 43312 2830 194.004 16.7953 72.807 1894.54 80.836 2.016 444.05 42278.5 2327 195.714 17.0993 73.676 1912.21 83.312 2.015 421.139 42973.4 2421 198.686 16.1213 73.859 1885.3 83.514 2.207 455.039 43121.8 1758 209.18 18.9523 72.592 1960.38 80.721 1.972 378.712 43450.6 2009 192.191 17.2673 73.119 2007.83 83.126 2.196 327.766 44183.8 2830 188.152 17.5943 72.92 1942.18 83.328 2.439 399.149 43141.9 3239 192.065 19.5553 73.646 1950.27 83.107 2.064 379.303 43436.4 2520 191.91 16.2433 72.745 1918.93 82.698 2.216 417.39 42417.8 3057 188.503 17.9363 73.506 1918.74 83.281 1.972 418.152 43391.7 1888 202.094 16.9833 73.346 1879.75 82.826 1.926 456.426 42345.3 2210 200.605 16.3363 72.991 1983.83 83.029 2.106 357.035 44007.8 2129 193.699 18.6463 73.135 1886.05 83.082 2.243 455.805 42523.7 2462 200.983 19.0163 72.57 1892.09 91.357 2.155 447.189 42041.9 2714 192.375 18.5973 73.829 2030.56 83.605 2.267 303.987 45153 2414 193.317 18.6353 72.854 1912.5 81.33 1.721 419.482 42407.4 1996 191.586 15.2613 73.681 1852.94 83.162 2.056 483.12 42293.3 2078 205.855 17.3633 73.277 1912.28 83.053 2.291 426.385 42786.4 3031 195.386 18.4913 73.453 2002.79 83.47 2.132 326.876 44236.6 2495 187.706 16.7783 73.125 1986.87 83.672 2.25 352.966 44155.4 2475 193.938 19.3273 72.958 1928.88 83.146 2.279 411.42 42951.6 2858 194.113 19.1733 73.854 1939.11 83.368 2.111 396.482 43739.5 2046 200.636 17.8593 72.588 1929.07 94.311 2.121 411.6 42620 2394 190.201 19.111

421


PAX NOISE WEMP DOC ROUGH WFUEL PURCH RANGE VCRMX LDMAX5 72.556 1944.19 64.584 1.987 386.305 42380.4 2737 181.049 15.6595 73.833 1950.23 62.147 2.2 382.171 43533.8 2642 194.847 16.6565 72.999 1954.13 61.322 1.928 384.146 43517.5 2082 195.301 15.955 73.191 1872.07 59.556 1.868 465.777 42127.5 2089 200.992 15.4215 73.483 1927.39 58.497 2.286 414.76 43401.8 2312 201.983 18.5755 73.3 2024.38 62.331 2.032 313.32 45037.5 2048 194.222 16.5375 73.637 2018.16 62.986 2.179 316.304 44838.8 2534 192.824 16.8925 72.765 1909.15 59.223 2.269 433.14 42679.8 2645 196.727 18.5225 73.892 1859.15 66.916 1.91 476.335 42179.9 2191 204.558 15.3515 72.622 1976.38 73.757 1.757 359.64 43487.5 1876 188.102 15.1445 73.106 2017.68 75.158 2.199 318.293 44308.5 2737 186.542 17.0065 72.969 1881.11 70.826 2.143 455.724 41870.1 3116 193.144 16.7265 73.271 2001.28 70.852 2.31 339.084 44541 2575 194.285 18.65 73.467 1959.61 69.272 2.222 379.7 43901.1 2446 197.821 17.825 72.778 1976.84 72.083 2.044 362.901 43801.8 2371 191.854 16.8765 73.669 1931.83 66.893 2.193 410.013 43588.8 2120 202.98 18.0475 72.992 1987.8 83.501 1.825 342.41 43894.6 2464 189.404 15.1645 73.148 1886.04 83.097 2.024 451.468 42509.8 2350 200.386 16.9635 72.571 1886.59 95.214 2.136 452.629 41745.9 3007 189.863 18.0935 73.846 1992.26 83.262 1.987 336.432 44406 2354 192.836 15.9425 73.618 2008.64 83.602 2.459 329.356 44681.7 2774 193.296 19.7415 72.757 1982.31 95.498 2.216 355.279 43478.4 3051 186.253 18.5635 73.506 1912.23 83.238 2.068 427.021 43196 2075 203.004 17.5785 73.325 1977.67 83.588 2.018 362.565 44323.4 1835 199.172 17.6365 73.129 1924.53 83.06 2.005 407.622 42857.9 2752 192.563 16.1525 72.987 1945.15 83.26 2.028 389.855 43085.4 2696 191.571 16.9035 73.846 1928.08 83.274 2.12 405.969 43498.4 2220 200.029 17.5285 72.602 1942.54 83.132 1.835 390.551 42889.4 2277 189.011 15.8975 72.803 2026.78 82.485 1.998 312.929 44754.8 1914 190.957 17.8925 73.687 1900.91 83.295 2.09 439.452 43207.2 1890 206.508 18.1255 73.282 1920.44 82.879 2.376 419.891 42735.6 3185 193.581 18.995 73.448 2043.11 83.609 2.296 292.726 45209.4 2678 190.542 18.7125 73.516 1850.06 81.226 1.956 487.528 41926.8 2418 203.16 15.8025 73.317 1979.25 83.392 1.976 356.248 43953.3 2400 193.322 16.0065 73.662 2016.96 83.537 2.065 317.717 44726.7 2462 192.789 16.465 72.774 1889.78 82.366 2.064 447.291 42012.7 3050 192.348 16.6345 72.528 1969.72 81.084 2.33 371.431 43472 3005 189.253 19.3855 73.819 2021.3 83.546 2.35 314.079 44794 2620 191.643 18.265 73 1921.68 83.144 1.97 420.987 43164.6 1956 199.789 16.9435 73.144 1962.73 83.337 2.072 380.188 43963.3 2002 198.605 17.7095 73.314 1895.96 82.943 1.991 437.611 42476.3 2780 196.602 15.9185 73.484 1973.45 83.317 2.098 360.755 43792.8 2743 192.762 16.6565 72.806 1919.67 80.959 1.984 418.363 42712.1 2304 194.378 16.9615 73.675 1937.35 83.391 1.988 395.361 43430.3 2400 197.631 15.9835 73.859 1900.99 83.447 2.175 439.079 43394 1747 208.227 18.8195 72.59 1990 81.4 1.942 348.302 43968.7 1981 190.637 17.0735 73.118 2036.91 83.249 2.168 297.777 44710.7 2729 186.886 17.4295 72.919 1959.5 83.367 2.412 381.461 43472.6 3180 191.643 19.4795 73.646 1975.94 83.376 2.043 352.808 43907.8 2449 190.953 16.1245 72.744 1935.9 82.804 2.195 399.789 42725.3 2999 187.905 17.8625 73.505 1944.4 83.407 1.944 391.938 43844.2 1869 200.711 16.8175 73.345 1903.64 83.103 1.902 431.991 42758.4 2189 199.293 16.1985 72.991 2011.36 83.124 2.077 328.837 44496.6 2101 192.293 18.4455 73.134 1899.64 83.148 2.216 441.72 42777.1 2450 200.537 18.9255 72.57 1908.13 91.649 2.128 430.605 42315.5 2697 191.516 18.4995 73.828 2058.89 83.845 2.241 274.964 45660.3 2393 191.836 18.4535 72.852 1940.67 82.161 1.697 390.35 42876.2 1964 189.836 15.0865 73.681 1875.46 83.195 2.026 460.258 42706.5 2067 204.98 17.2725 73.277 1926.63 83.196 2.268 411.705 43043.4 2984 194.765 18.4215 73.452 2032.34 83.614 2.106 296.421 44802.2 2411 186.909 16.6385 73.124 2010.5 83.861 2.224 328.725 44580.1 2450 192.805 19.1495 72.957 1946.13 83.191 2.253 393.548 43254.4 2839 193.254 19.0585 73.853 1966.44 83.519 2.082 368.695 44229.5 2029 199.277 17.7015 72.587 1949.46 94.147 2.094 390.73 42967.5 2369 189.069 18.956

422

Table F.5 MLE Values for Kriging Metamodels Parameters

Model# Response θCSPD θAR θDPRP θWL θAF θWS MLE Obj Fcn

1 µNOISE 0.0015 0.0135 4.3221 0.0081 0.0101 0.0010 318.022

2 µWEMP 0.0063 0.0361 0.0182 0.0699 0.0972 0.3787 -58.576

3 µDOC 32.1074 0.0303 2.4160 0.0254 0.0011 0.0314 -66.260

4 µROUGH 0.0010 0.1154 0.3397 0.0461 0.0158 0.0034 304.616

5 µWFUEL 0.0042 0.0433 0.0364 0.2373 0.0160 1.7512 -87.657

6 µPURCH 0.0010 0.0266 0.5815 0.0166 0.0741 0.0462 -260.478

7 µRANGE 0.0189 0.6119 0.4512 4.4036 0.0326 0.2096 -269.953

8 µVCRMX 0.0021 0.0248 1.1897 0.0611 0.0443 0.0644 45.676

9 µLDMAX 0.0554 0.1724 0.0013 0.5892 0.0077 0.0361 177.236

10 σNOISE 0.0010 1.0361 32.7718 0.2702 3.8878 3.9316 539.114

11 σWEMP 0.0044 0.7930 0.0327 0.8117 0.0010 4.6024 59.282

12 σDOC 18.7774 29.7456 5.5055 0.0010 1.7481 0.0010 69.821

13 σROUGH 0.4232 0.2684 0.4734 0.6765 11.1107 2.9553 462.787

14 σWFUEL 0.2518 0.1051 0.2409 0.7221 0.0032 7.4614 44.767

15 σPURCH 0.0131 2.7522 0.0440 0.6500 0.0010 3.5610 -151.602

16 σRANGE 0.0010 0.4205 1.9821 9.7039 3.8147 0.2138 -143.105

17 σVCRMX 0.2443 0.0181 3.8395 3.5390 0.0627 2.8240 178.677

18 σLDMAX 0.0114 1.5707 0.0070 0.1971 0.0065 3.3875 361.215

The MLE values for the θk listed in Table F.5 are used in conjunction with the

sample points listed in Table F.1 (scaled to [0,1]), the µ and σ of each response (computed

from the data in Table F.2 through Table F.4), and the Gaussian correlation function,

Equation 2.17, to estimate µ and σ response values at untried points in the design space.

The krigit.f algorithm given in Section A.2.2 is employed to make these predictions.

F.3 ANALYSIS OF VARIANCE OF GAA RESPONSE MEANS

The analysis of variance of the means of the GAA responses are listed in Table F.6.

This information is used to generate the Pareto plots for the GAA response means give in

Figure 7.7. Looking at the information in Table F.6, it is noted that for empty weight

(WEMP), fuel weight (WFUEL), and purchase price (PURCH), all of the design variables

are significant, i.e., Pr(F) > 0.01. For maximum flight range (RANGE), all but cruise

423

speed are significant , whereas the opposite is true for direct operating cost (DOC); cruise

speed is the only variable with a significant effect on DOC. For maximum cruise speed

(VCRMX), all but engine activity factor are significant. Finally, cruise speed, aspect ratio,

wing loading, and seat width all have significant effects on maximum lift to drag ratio

(LDMAX). However, insignificant variables are not removed from any of the metamodels.

Table F.6 Main Effects ANOVA Results for GAA Response Means

WEMP DOC

Df Sum of Sq Mean Sq F Value Pr(F) Df Sum of Sq Mean Sq F Value Pr(F)

CSPD 7 334.45 47.78 17.984 0.0000 CSPD 7 4522.135 646.0194 91.50201 0.0000 AR 7 25940.78 3705.83 1394.927 0.0000 AR 7 44.150 6.3071 0.89334 0.5292 DPRP 7 1249.92 178.56 67.213 0.0000 DPRP 7 113.404 16.2005 2.29464 0.0665 WL 7 19038.63 2719.80 1023.774 0.0000 WL 7 61.516 8.7880 1.24473 0.3236 AF 7 581.71 83.10 31.281 0.0000 AF 7 32.378 4.6255 0.65515 0.7065 WS 7 93613.86 13373.41 5033.946 0.0000 WS 7 32.472 4.6389 0.65705 0.7050Resid 21 55.79 2.66 Resid 21 148.263 7.0602

WFUEL PURCH


CSPD 7 534.0 76.28 25.748 0.0000 CSPD 7 131673 18810 12.195 0.0000 AR 7 19764.3 2823.47 953.023 0.0000 AR 7 12976562 1853795 1201.819 0.0000 DPRP 7 1891.0 270.14 91.183 0.0000 DPRP 7 7363576 1051939 681.975 0.0000 WL 7 20985.0 2997.86 1011.885 0.0000 WL 7 1309407 187058 121.270 0.0000 AF 7 673.1 96.16 32.456 0.0000 AF 7 873248 124750 80.876 0.0000 WS 7 101120.4 14445.77 4875.974 0.0000 WS 7 24701497 3528785 2287.719 0.0000Resid 21 62.2 2.96 Resid 21 32392 1542

RANGE VCRMX


CSPD 7 114956 16422 3.2275 0.0174 CSPD 7 9.5692 1.3670 3.6772 0.0095 AR 7 131815 18831 3.7009 0.0092 AR 7 24.2662 3.4666 9.3249 0.0000 DPRP 7 593925 84846 16.6752 0.0000 DPRP 7 615.1011 87.8716 236.3669 0.0000 WL 7 7971133 1138733 223.7989 0.0000 WL 7 816.5116 116.6445 313.7635 0.0000 AF 7 177899 25414 4.9947 0.0019 AF 7 4.3497 0.6214 1.6715 0.1706 WS 7 617934 88276 17.3492 0.0000 WS 7 499.8083 71.4012 192.0630 0.0000Resid 21 106852 5088 Resid 21 7.8069 0.3718

LDMAX KEY:

Df Sum of Sq Mean Sq F Value Pr(F) Df = degree of freedom

CSPD 7 6.98133 0.99733 310.093 0.0000 CSPD = Cruise speed AR 7 71.52682 10.21812 3177.036 0.0000 AR = Aspect ratio DPRP 7 0.02329 0.00333 1.034 0.4373 DPRP = Propeller diameter WL 7 9.33662 1.33380 414.709 0.0000 WL = Wing loading AF 7 0.02003 0.00286 0.889 0.5319 AF = Engine activity factor WS 7 8.93490 1.27641 396.865 0.0000 WS = Seat widthResid 21 0.06754 0.00322 Resid = residuals

424

F.4 ADDITIONAL DESIGN SCENARIOS FOR STUDY GAA PRODUCTPLATFORM USING Cdk FORMULATION

In all, eight different design scenarios are considered to investigate several

performance and economic tradeoffs within the product family, see Table F.7. Only the

first three scenarios are discussed in Section 7.5 in regards to the PPCEM product

platform. These remaining scenarios are described as follows.

Group I - Tradeoff studies: includes Scenarios 1, 2 and 3. In these scenarios, the

tradeoffs of achieving a set of conflicting goals are examined. An overall tradeoff

study with all goals weighted equally is performed in Scenario 1. Scenario 2 is an

economic tradeoff study with WEMP, PURCH, and DOC given first priority. A

performance tradeoff study with WFUEL, VCRMX, RANGE, and LDMAX

ranked highest is Scenario 3. The deviation function formulations for Scenarios 1-

3 are summarized in Table F.7. In the table, PLEV# represents the priority level.

Group II - Economic tradeoff scenarios: includes Scenarios 4 and 5. In these

two scenarios, the economic goals are given first priority over performance

concerns. Two scenarios are considered, one with manufacturing considerations

(WEMP and PURCH) as the main driver, Scenario 4, and one with direct operating

cost (DOC) as the main driver, Scenario 5. See Table F.7 for the corresponding

deviation function formulations.

Group III - Performance tradeoff scenarios: includes Scenario 6, 7, and 8. In

these scenarios performance considerations are given first priority; economics ranks

second. Scenario 6 is based on speed considerations with VCRMX and WFUEL

ranked first, Scenario 7 is based on aerodynamic considerations with LDMAX

ranked first, and Scenario 8 is based on the flight distance with RANGE ranked

first. The corresponding deviation function formulations are listed in Table F.7.

425

Table F.7 GAA Product Platform Design Scenarios

Deviation FunctionScenario PLEV1 PLEV2 PLEV3 PLEV4

Group I: TradeoffStudies

d1- drives fuel weight Cdk to 1

d2- drives empty weight Cdk to 1


- + d2- + d3

-

+ d4- + d5

- + d6-

+ d7-)/7

d3- drives direct operating cost Cdk to 1

d4- drives purchase price Cdk to 1

d5- drives max. lift/drag Cdk to 1

2. Economic tradeoff (d2- + d3

- +d4

-)/3(d1

- + d5- + d6

-

+ d7-)/4

d6- drives max. speed Cdk to 1

d7- drives max. range Cdk to 1

3. Performance tradeoff (d1- + d5

- + d6-

+ d7-)/4

(d2- + d3

- +d4

-)/3Group II: Economic

Tradeoff Studies4. Economic driver:

manufacturing (d2- + d4

-)/2 d3- (d1

- + d5- + d6

-

+ d7-)/4

5. Economic driver:operator cost d3

- (d2- + d4

-)/2 (d1- + d5

- + d6-

+ d7-)/4

Group III: PerformanceTradeoff Studies

6. Performance driver:speed and fuel (d1

- + d6-)/2 (d5

- + d7-)/2 (d2

- + d3- +

d4-)/3

7. Performance driver:aerodynamics d5

- d7- (d1

- + d6-)/2 (d2

- + d3- +

d4-)/3

8. Performance driver:distance d7

- d5- (d1

- + d6-)/2 (d2

- + d3- +

d4-)/3

These design scenarios are used in conjunction with the GAA product platform

compromise DSP given in Figure 7.7 in Section 7.4. As with Scenarios 1-3 discussed in

Section 7.5, three starting points are used when solving the compromise DSP for each

scenario: the lower, middle, and upper bounds of the design variables; the resulting

design(s) with the lowest deviation function is taken as the best solution. The Cdk

solutions are given in Section F.4.1; the instantiated aircraft are listed in Section F.4.2.

F.4 .1 PPCEM Cdk Values for Design Scenarios 1-8

The resulting Cdk solutions for these 8 scenarios are summarized and discussed in

Table F.8. The initial Cdk designs are included as Scenario 0 in each graph for the sake of

426

comparison; they are designated by a black square. The white circle represents the Cdk

values obtained in Scenario 1, the overall tradeoff study; the black diamonds indicate Cdk

values obtained for the economically oriented design scenarios (Scenarios 2, 4, and 5); and

the white triangles indicate Cdk values obtained for the performance oriented design

scenarios (Scenarios 3, 6, 7, and 8). A horizontal dashed line at Cdk = 1 is included in each

plot to designate the target value for each Cdk; points above the line indicate that the target

has been met in that particular scenario. Although it is not noted, all designs are feasible

based on the constraints specified in the GAA product platform compromise DSP.

Table F.8 PPCEM Solutions - Goal Cdk Values

Fuel Weight (WFUEL) Cdk

• The target for WFUEL is always met,

regardless of the design scenario.

• Considerable improvement over the initial

Cdk of -0.7 is obtained in all scenarios.

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6 7 8

Scenario

Cd

k -

WF

UE

L

Aircraft Empty Weight (WEMP) Cdk

• The target for WEMP is almost always

satisfied regardless of the design scenario.

• When placed at the first priority level in

Scenario 4, the Cdk target of 1 is achieved.

• Considerable improvement over the initial

Cdk (which is only slight greater than zero)

is also obtained in all scenarios.

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6 7 8

Scenario

Cd

k -

WE

MP

427

Direct Operating Cost (WEMP) Cdk

• The target for DOC is never met; the best

Cdk value that can be achieved is 0.5 when

DOC is given the highest priority in

Scenario 5.

• The initial Cdk is quite poor (-670 which is

well off the chart); so, considerable

improvement is made despite not being able

to achieve the target of 1.

-6.0-5.0-4.0-3.0-2.0-1.00.01.02.0

0 1 2 3 4 5 6 7 8

Scenario

Cd

k -

DO

C

Purchase Price (PURCH) Cdk

• The target for PURCH is met when given

the highest priority (Scenario 4) and by

happenstance in Scenario 8.

• All design scenarios yield improvement

over the baseline Cdk . -3.0

-2.0

-1.0

0.0

1.0

2.0

0 1 2 3 4 5 6 7 8

Scenario

Cd

k -

PU

RC

H

Max. Lift/Drag Ratio (LDMAX) Cdk

• The target for LDMAX is only met when

given the highest priority (Scenario 7).

• In general, LDMAX is worse than the

original Cdk; unless given a high priority,

the achievement of the LDMAX target is

compromised to improve the other goals. -5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

0 1 2 3 4 5 6 7 8

Scenario

Cd

k -

LD

MA

X

428

Max. Cruise Speed (VCRMX) Cdk

• The target for VCRMX is only met when

given the highest priority (Scenario 6).

• Six out of the eight design scenarios yield

improvement over the initial Cdk value.

• In Scenarios 4 and 8, priority is placed on

WFUEL and RANGE, which greatly

compromises max cruise speed as might be

expected (fly slower yet further).

-7.0-6.0-5.0-4.0-3.0-2.0-1.00.01.02.0

0 1 2 3 4 5 6 7 8

Scenario

Cd

k -

VC

RM

X

Max. Flight Range (RANGE) Cdk

• The target for RANGE is achieved in

Scenario 8 when placed at the highest

priority level and when WFUEL is given

the highest priority in Scenario 4 (which

causes VCRMX to decrease, thereby

enabling the plane to fly further).

• In all scenarios, however, improvement is

made over the initial Cdk value of -4 for the

baseline design.

-5.0-4.0-3.0-2.0-1.00.01.02.03.0

0 1 2 3 4 5 6 7 8

Scenario

Cd

k -

RA

NG

E

The end result of having examined all these design scenarios is that significant

tradeoffs are occurring in Scenarios 1, 2, and 3 where the economic and performance Cdk

goals are equally weighted at different priority levels. Only when a particular Cdk is given

first priority (i.e., placed at PLEV1) in the GAA product platform compromise DSP can the

target of 1 be achieved. Any other time, the results indicate the best possible compromise

which can be obtained for a particular design scenario.

429

F.4 .2 PPCEM Aircraft Instantiations

The resulting PPCEM platform designs for Scenarios 1-8 (and the baseline design)

are summarized in Table F.9; the resulting instantiations of each aircraft are listed in Table

F.10. Note that these instantiations are directly from GASP based on the PPCEM platform

specifications given in Table F.9; they are not based on kriging metamodel predictions.

Also, all solutions listed in the table are feasible, and the values for ride roughness and

takeoff noise have not been included.

Table F.9 PPCEM Platform Specifications for Scenarios 1-8

Design CSPD AR DPRP WL AF WSScenario [Mach] [ft] [lb/ft2] [in]

0 0.310 7.88 6.30 20.50 110.00 20.001 0.244 8.00 5.13 22.45 89.60 18.602 0.242 8.09 5.19 22.63 89.40 18.723 0.291 7.62 5.55 22.48 85.63 18.704 0.244 8.11 5.04 21.97 85.00 18.775 0.241 8.21 5.30 23.06 87.62 18.736 0.240 8.31 5.90 24.00 88.94 18.647 0.283 9.93 5.00 24.30 85.00 18.148 0.264 8.49 5.00 22.21 85.00 18.21

430

Table F.10 PPCEM Instantiations in GASP from Scenarios 1-8

ResponseDesign No. of wfuel wemp doc purch ldmax vcrmx range Dev.Fcn.

Scenario Seats [lbs] [lbs] [$/hr] [$] [kts] [nm] PLEV1 PLEV2 PLEV3 PLEV42 364.39 1961.95 83.46 43908.9 16.49 193.88 2237.0

0 4 324.00 2000.43 83.40 44556.1 16.16 191.39 2136.06 293.66 2030.03 83.53 45099.0 16.01 190.15 2067.02 449.43 1887.15 61.98 41817.0 15.89 190.83 2491.0 0.024

1 4 413.80 1921.71 63.31 42374.5 15.61 188.47 2436.0 0.0386 388.49 1946.59 63.85 42827.0 15.53 187.61 2420.0 0.0512 447.25 1889.73 61.60 41959.4 15.91 192.24 2446.0 0.017 0.031

2 4 411.34 1924.56 62.85 42502.8 15.63 189.51 2393.0 0.020 0.0516 385.69 1949.77 63.38 42989.1 15.55 189.07 2377.0 0.019 0.0732 450.92 1886.97 77.43 42150.1 15.79 195.53 2499.0 0.024 0.106

3 4 415.34 1921.70 79.13 42727.0 15.52 193.32 2451.0 0.045 0.1126 389.84 1946.82 79.76 43190.3 15.44 192.46 2437.0 0.067 0.1112 442.52 1893.71 62.41 41685.2 16.03 186.82 2569.0 0.008 0.040 0.031

4 4 406.67 1928.43 63.20 42229.0 15.76 184.25 2509.0 0.003 0.053 0.0426 381.37 1953.28 63.61 42702.3 15.68 183.82 2492.0 0.000 0.060 0.0632 448.84 1888.86 60.96 42077.4 15.93 194.39 2353.0 0.016 0.013 0.037

5 4 412.80 1923.83 62.13 42659.0 15.65 192.20 2303.0 0.036 0.008 0.0576 386.92 1949.25 62.64 43126.4 15.56 191.33 2287.0 0.044 0.001 0.0802 440.78 1896.38 59.62 42879.9 16.12 203.32 2092.0 0.000 0.029 0.015

6 4 402.41 1933.72 60.77 43481.9 15.77 200.43 2049.0 0.003 0.059 0.0166 374.03 1961.53 61.43 43968.4 15.60 198.87 2027.0 0.037 0.062 0.0152 447.55 1896.55 72.01 42180.9 17.61 194.45 2072.0 0.000 0.171 0.014 0.076

7 4 411.12 1931.83 76.37 42694.1 17.21 191.00 2010.0 0.000 0.196 0.036 0.0976 382.46 1959.79 80.21 43170.0 17.03 189.41 1982.0 0.000 0.207 0.073 0.1142 450.38 1888.00 69.26 41603.1 16.51 187.43 2503.0 0.000 0.029 0.032 0.056

8 4 420.42 1916.93 71.48 42019.1 16.21 184.65 2441.0 0.024 0.047 0.064 0.0646 391.41 1945.51 72.33 42565.7 16.15 184.21 2427.0 0.029 0.050 0.099 0.069

F.5 CONVERGENCE HISTORIES OF INDIVIDUALLY DESIGNEDBENCHMARK TWO, FOUR, AND SIX SEATER AIRCRAFT

The convergence plots for the two, four, and six seater GAA have been included in

this section to provide the full details of the model convergence history for the individually

designed aircraft. Unlike the convergence plots for the PPCEM Cdk solutions, the

iterations have not been out carried to match the longest run; thus, some of the convergence

lines appear to stop abruptly.

431

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25Iterations

PLEV

1

Hi

Low

Mid

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 5 10 15Iterations

Hi

Low

Mid

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15Iterations

Hi

Low

Mid

(a) 2 Seater - PLEV1 (b) 4 Seater - PLEV1 (c) 6 Seater - PLEV1

Figure F.1 Convergence of Individual Benchmark Aircraft - Scenario 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 5 10 15Iterations

PLEV

1

Hi

Low

Mid

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 5 10 15 20Iterations

Hi

Low

Mid

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20

Iterations

Hi

Low

Mid


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 5 10 15

Iterations

PLEV

2

Hi

Low

Mid0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20

Iterations

Hi

Low

Mid

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18


Hi

Low

Mid

(d) 2 Seater - PLEV2 (e) 4 Seater - PLEV2 (f) 6 Seater - PLEV2


432

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09


PLEV

1Hi

Low

Mid

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 5 10 15Iterations

Hi

Low

Mid

0

0.02

0.04

0.06

0.08

0.1

0.12

0 5 10 15Iterations

Hi

Low

Mid


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18


PLEV

2

Hi

Low

Mid

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 5 10 15Iterations

Hi

Low

Mid

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15Iterations

Hi

Low

Mid

(d) 2 Seater - PLEV2 (e) 4 Seater - PLEV2 (f) 6 Seater - PLEV2


F.6 GAA PRODUCT VARIETY TRADEOFF STUDY INFORMATION

Supplemental information for the GAA product variety tradeoff study is included in

this section. In particular, the rank ordering of the design variables for computing the non-

commonality index (NCI) are listed in Section F.3.1. In Section F.3.2, the detailed results

of the product variety tradeoff study for Scenario 2 are listed.

F.6 .1 Rank Ordering of Design Variables

The justification for the rank ordering of the design variables in the GAA product

variety case study are listed in Table F.11. The viewpoints for these decisions are listed in

the table. The resulting preferences are then quantified in Table F.12 and the total and

relative importance of each design variable are computed.

433

Table F.11 Viewpoints for Pairwise Comparisons for GAA DesignVariables

AF < AR* It is easier to derate the engine than it is to allow aspect ratio to vary.AF > CSPD Cruise speed is the cheapest variable to allow to vary.AF > DPRP Propeller diameter is the second cheapest variable to allow to vary.AF < WL It is easier to derate the engine than to allow wing loading to vary.AF < WS Seat width is the most costly variable to vary.AF > dummy All variables are preferred to the dummy.AR > CSPD Cruise speed is the cheapest variable to vary.AR > DPRP Propeller diameter is the second cheapest variable to allow to vary.AR > WL It is more costly to allow aspect ratio to vary than wing loading.AR < WS Seat width is the most costly variable to vary.AR > dummy All variables are preferred to the dummy.CSPD < DPRP Cruise speed is the cheapest variable to vary.CSPD < WL “ “ “ “ “ “ “ “CSPD < WS “ “ “ “ “ “ “ “CSPD > dummy All variables are preferred to the dummy.DPRP < WL Propeller diameter is the second cheapest variable to allow to vary.DPRP < WS Seat width is the most costly variable to allow to vary.DPRP > dummy All variables are preferred to the dummy.WL < WS Seat width is the most costly variable to vary.WL > dummy All variables are preferred to the dummy.WS > dummy All variables are preferred to the dummy.

* The symbol > indicates preference.

Table F.12 Summary of Pairwise Comparisons and Resulting RelativeImportance of GAA Design Variables

Decision: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Total† RelativeAF 0 1 1 0 0 1 3 0.1429AR 1 1 1 1 0 1 5 0.2381

CSPD 0 0 0 0 0 1 1 0.0476DPRP 0 0 1 0 0 1 2 0.0952WL 1 0 1 1 0 1 4 0.1905WS 1 1 1 1 1 1 6 0.2857

dummy 0 0 0 0 0 0 — —21

† Larger numbers indicate preference.

The relative importances listed in Table F.12 are used to compute the non-

commonality index (NCI) for each family of aircraft in the product variety tradeoff study in

434

Section 7.6.2. The results for the product variety tradeoff study for Scenario 2 are listed in

the next section. The results for Scenario 3 are listed in F.3.3.

F.6 .2 Product Variety Tradeoff Study Results for Scenario 2

The PPCEM solutions based on the GAA Compromise DSP employing the Cdk

formulations are summarized in Table F.13.

Table F.13 PPCEM Platform Designs for Scenario 2

PPCEM Designs CSPD AR DPRP WL AF WS

2 Seater ñ ñ ñ ñ ñ ñ

4 Seater 0.242 8.085 5.20 22.63 89.40 18.726 Seater ò ò ò ò ò ò

Vary: WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2

2 Seater 447.25 1889.73 61.60 41959.4 15.91 192.24 2446.0 0.017 0.0314 Seater 411.34 1924.56 62.85 42502.8 15.63 189.51 2393.0 0.020 0.0516 Seater 385.69 1949.77 63.38 42989.1 15.55 189.07 2377.0 0.019 0.073

The following tables list the actual goal values for each aircraft (platform instantiation)

found by allowing one, two, and three variables to vary at a time between aircraft while

holding the remaining variables fixed at the PPCEM platform design values.

435

Table F.14 Product Variety Tradeoff Study Results for Scenario 3,Allowing 1 Design Variable to Vary Between Aircraft

Vary: AF WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2

2 Seater 86.462 448.04 1888.96 61.62 41908.0 15.91 191.81 2448.0 0.016 0.0314 Seater 90.500 410.92 1925.01 62.76 42544.3 15.63 189.95 2394.0 0.020 0.0506 Seater 95.473 383.76 1951.71 63.17 43083.4 15.55 189.51 2375.0 0.018 0.071

Vary: AR WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2



Vary: AF AR WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2

2 Seater 89.402 8.085 447.15 1889.85 61.55 41963.6 15.91 192.24 2447.0 0.016 0.0314 Seater 92.351 8.326 407.23 1929.33 62.58 42665.9 15.82 190.38 2378.0 0.020 0.0466 Seater 90.630 8.505 380.04 1956.52 63.04 43160.7 15.87 189.51 2354.0 0.018 0.066

Vary: AF CSPD WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AF DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


436

Vary: AF WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AF WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR CSPD WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: DPRP WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: DPRP WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: WL WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


437


Vary: AF AR DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 89.402 8.085 5.195 447.15 1889.85 61.55 41963.6 15.91 192.24 2447.0 0.016 0.0314 Seater 89.402 8.085 5.195 411.25 1924.68 62.79 42532.8 15.63 189.95 2394.0 0.020 0.0506 Seater 93.099 8.065 5.220 384.69 1950.74 63.14 43083.5 15.52 189.95 2379.0 0.018 0.071

Vary: AF AR WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 96.128 8.096 23.093 449.14 1888.09 61.24 42038.4 15.84 193.64 2344.0 0.015 0.0414 Seater 97.736 7.712 24.697 430.38 1905.18 62.06 42380.2 14.97 193.46 2017.0 0.014 0.1056 Seater 94.927 7.836 24.728 403.26 1932.16 62.55 42858.9 14.97 192.46 1990.0 0.014 0.128

Vary: AF AR WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 88.726 8.134 18.540 449.97 1887.25 61.42 41933.2 15.97 192.67 2451.0 0.015 0.0294 Seater 93.981 7.872 17.021 444.59 1891.81 61.67 42047.9 15.73 192.67 2460.0 0.010 0.0606 Seater 100.63 7.738 15.299 455.87 1880.92 60.88 41985.6 15.86 194.90 2509.0 0.005 0.099

Vary: AF DPRP WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 98.111 5.309 22.872 445.34 1891.78 61.07 42216.0 15.87 194.90 2386.0 0.016 0.0344 Seater 89.025 5.222 23.116 415.44 1920.75 62.46 42525.7 15.55 191.14 2294.0 0.018 0.0636 Seater 97.700 5.260 24.688 398.24 1938.01 62.16 43074.9 15.18 193.67 1986.0 0.013 0.121

Vary: AF DPRP WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 89.492 5.040 18.482 452.83 1883.95 62.28 41646.4 15.95 189.24 2435.0 0.018 0.0374 Seater 98.601 5.164 18.201 418.94 1917.23 62.42 42473.7 15.72 190.96 2403.0 0.017 0.0526 Seater 89.402 5.195 16.557 430.53 1906.48 61.61 42311.8 15.94 192.46 2456.0 0.009 0.087

Vary: AF WL WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 100.89 22.778 18.457 449.60 1887.56 61.20 42053.9 15.93 193.74 2413.0 0.015 0.0324 Seater 89.387 22.690 16.941 443.99 1893.11 61.51 42043.9 15.92 192.50 2441.0 0.009 0.0596 Seater 86.076 23.531 14.761 473.40 1865.24 60.05 41689.5 16.10 196.35 2329.0 0.000 0.123

Vary: AR DPRP WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 8.085 5.195 22.627 447.15 1889.85 61.55 41963.6 15.91 192.24 2447.0 0.016 0.0314 Seater 7.642 5.244 24.177 429.50 1905.81 62.24 42305.0 15.00 192.46 2116.0 0.015 0.0966 Seater 7.543 5.453 23.512 397.10 1937.45 62.49 43033.4 14.94 193.21 2233.0 0.014 0.099

Vary: AR DPRP WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 8.085 5.195 18.644 448.57 1888.48 61.50 41948.5 15.92 192.46 2449.0 0.016 0.0304 Seater 7.676 5.361 16.568 454.70 1881.64 61.10 41975.6 15.64 194.82 2490.0 0.006 0.0626 Seater 8.134 5.503 17.821 400.83 1935.54 61.71 43070.9 15.75 194.32 2399.0 0.010 0.072

Vary: AR WL WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 8.460 22.996 18.606 447.62 1890.59 61.08 42037.5 16.16 193.26 2351.0 0.014 0.0364 Seater 8.171 23.904 15.921 471.22 1867.30 60.28 41725.9 15.95 196.01 2224.0 0.002 0.0936 Seater 8.172 22.936 15.672 449.42 1888.55 60.80 42057.6 16.11 194.32 2415.0 0.004 0.100

Vary: DPRP WL WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV22 Seater 5.195 22.627 18.724 447.15 1889.85 61.55 41963.6 15.91 192.24 2447.0 0.016 0.0314 Seater 5.270 23.697 16.672 458.04 1879.84 60.64 41976.1 15.79 195.74 2246.0 0.004 0.0856 Seater 5.401 23.867 16.392 442.02 1895.88 60.43 42425.2 15.76 197.45 2213.0 0.002 0.116

438

F.6 .3 Product Variety Tradeoff Study Results for Scenario 3

The PPCEM solutions based on the GAA Compromise DSP employing the Cdk

formulations are summarized in Table F.17.

Table F.17 Initial PPCEM Platform Design for Scenario 3

PPCEM Designs CSPD AR DPRP WL AF WS

2 Seater ñ ñ ñ ñ ñ ñ

4 Seater 0.291 7.617 5.55 22.48 85.63 18.706 Seater ò ò ò ò ò ò

Vary: WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2

2 Seater 450.92 1886.97 77.43 42150.1 15.79 195.53 2499.0 0.024 0.1064 Seater 415.34 1921.70 79.13 42727.0 15.52 193.32 2451.0 0.045 0.1126 Seater 389.84 1946.82 79.76 43190.3 15.44 192.46 2437.0 0.067 0.111

The following tables list the actual goal values for each aircraft (platform instantiation)

found by allowing one, two, and three variables to vary at a time between aircraft while

holding the remaining variables fixed at the PPCEM platform design values.

439


Vary: AF WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Table F.19 Product Variety Tradeoff Study Results for Scenario 3,Allowing 2 Design Variables to Vary Between Aircraft

Vary: AF AR WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AF CSPD WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AF DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


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Vary: AF WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AF WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR CSPD WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: AR WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD DPRP WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: CSPD WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: DPRP WL WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: DPRP WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


Vary: WL WS WFUEL WEMP DOC PURCH LDMAX VCRMX RANGE PLEV1 PLEV2


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Table F.20 Product Variety Tradeoff Study Results for Scenario 3,Allowing 3 Design Variables to Vary Between Aircraft

Vary: AF AR DPRP WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 86.088 7.637 5.543 450.62 1887.34 77.46 42158.0 15.81 195.53 2498.0 0.024 0.1064 Seater 103.172 7.717 5.532 407.38 1929.19 79.01 42978.4 15.60 193.54 2400.0 0.043 0.1136 Seater 85.344 7.944 5.574 385.39 1952.11 79.51 43327.7 15.71 192.89 2413.0 0.062 0.111

Vary: AF AR WL WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 86.803 7.521 22.091 448.27 1889.13 77.87 42151.2 15.77 194.69 2590.0 0.025 0.1094 Seater 85.631 7.617 22.000 411.11 1925.75 79.66 42754.7 15.60 192.40 2557.0 0.037 0.1156 Seater 87.706 7.639 21.962 384.38 1952.03 80.24 43255.2 15.55 191.59 2546.0 0.056 0.114

Vary: AF AR WS WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 91.408 7.701 18.496 451.01 1886.96 77.14 42204.0 15.89 195.95 2484.0 0.024 0.1054 Seater 85.650 8.008 19.256 399.03 1938.74 79.56 43037.4 15.75 192.68 2407.0 0.037 0.1176 Seater 85.248 7.974 18.996 378.80 1958.65 79.98 43419.0 15.67 192.25 2403.0 0.059 0.114

Vary: AF DPRP WL WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 85.631 5.547 22.482 450.92 1886.97 77.43 42150.1 15.79 195.53 2499.0 0.024 0.1064 Seater 90.961 5.563 22.327 411.78 1924.96 79.13 42810.3 15.54 193.11 2468.0 0.041 0.1136 Seater 91.032 5.579 21.518 379.01 1956.87 80.48 43363.1 15.61 191.35 2635.0 0.052 0.116

Vary: AF DPRP WS WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 85.631 5.547 18.701 450.92 1886.97 77.43 42150.1 15.79 195.53 2499.0 0.024 0.1064 Seater 85.858 5.570 19.372 401.75 1934.85 79.70 42949.5 15.40 192.46 2426.0 0.041 0.1176 Seater 85.631 5.547 18.701 389.84 1946.82 79.76 43190.3 15.44 192.46 2437.0 0.067 0.111

Vary: AF WL WS WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 91.408 22.360 18.493 451.18 1886.52 77.29 42189.7 15.84 195.85 2516.0 0.023 0.1064 Seater 97.816 22.355 18.790 407.73 1928.68 79.18 42927.2 15.52 193.24 2441.0 0.041 0.1146 Seater 91.824 21.649 18.222 390.78 1945.57 79.97 43163.7 15.68 191.92 2625.0 0.059 0.112

Vary: AR DPRP WL WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 7.634 5.553 22.467 450.70 1887.22 77.43 42152.7 15.80 195.48 2500.0 0.024 0.1064 Seater 7.815 5.553 22.507 413.02 1924.55 79.02 42793.0 15.67 193.45 2431.0 0.043 0.1126 Seater 7.674 5.666 22.184 384.82 1951.55 79.39 43336.4 15.54 192.89 2476.0 0.058 0.110

Vary: AR DPRP WS WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 7.639 5.541 18.716 450.45 1887.51 77.49 42159.2 15.80 195.53 2499.0 0.024 0.1074 Seater 8.046 5.571 19.257 398.16 1939.66 79.42 43072.9 15.78 192.89 2401.0 0.037 0.1166 Seater 8.020 5.545 18.994 378.31 1959.28 80.00 43419.1 15.72 192.03 2401.0 0.059 0.114

Vary: AR WL WS WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 7.636 22.475 18.709 450.39 1887.56 77.44 42159.4 15.81 195.48 2499.0 0.023 0.1064 Seater 7.617 22.482 18.701 415.34 1921.70 79.13 42727.0 15.52 193.32 2451.0 0.045 0.1126 Seater 7.989 22.380 18.913 379.47 1958.03 79.96 43408.4 15.73 192.25 2426.0 0.057 0.114

Vary: DPRP WL WS WFUEL WEMP DOC PURCH LDMAXVCRMX RANGE PLEV1 PLEV22 Seater 5.597 22.416 18.649 450.48 1887.28 77.20 42199.8 15.81 196.11 2504.0 0.023 0.1054 Seater 5.571 22.212 18.940 408.01 1928.74 79.55 42836.2 15.52 192.68 2499.0 0.036 0.1156 Seater 5.615 21.550 19.024 373.92 1961.89 80.68 43428.3 15.55 191.06 2625.0 0.050 0.118

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F.7 SAMPLE GAA DSIDES FILES FOR PPCEM FAMILY

The following sample DSIDES files are for solving the GAA Compromise DSP

within the PPCEM. These files are explicitly for the Cdk formulation which uses the

kriging metamodels based on the 64 point orthogonal array (OA). These particular files are

for Scenario 1 with a high starting point (i.e., all design variables at their upper bound).

The .dat file is given Section F.4.1, and the . f file is given in Section F.4.2.

F.7 .1 Sample DSIDES File: gasp.oa64.cdk.s1h.dat

PTITLE : Problem Title, User Name and DatePAX=1-5,rb3l GASP; Tim Simpson and Jonathan Maier; June 18, 1998

NUMSYS : Number of system variables: real,integer,boolean 6 0 0

SYSVAR : System variable informationCSPD 1 0.24 0.48 0.48 : cruise speedAR 2 7.0 11.0 11.0 : aspect ratioDPRP 3 5.0 5.96 5.96 : propeller diameterWL 4 19.0 25.0 25.0 : wing loadingAF 5 85.0 110.0 110.0 : engine activity factorWS 6 14.0 20.0 20.0 : seat width

NUMCAG : Number of constraints and goals 0 6 0 0 7 : nlinco,nnlinq,nnlequ,nlingo,nnlgoa

ACHFUN : Achievment function 1 : level 1 7 : level 1, 7 terms (-1,0.1429) (-2,0.1429) (-3,0.1429) (-4,0.1429) (-5,0.1429) (-6,0.1429) (-7,0.1429)

STOPCR : Stopping criteria1 0 50 0.005 0.005 : perfm cal, prt intereslts, Mcyles,sta dev, sta var

NLINCO : Names of nonlinear constraintsnoise 1 : takeoff noisedoc 2 : direct operating costrough 3 : ride roughness coefficientwemp 4 : aircraft empty weightwfuel 5 : fuel weightrange 6 : minimum range

NLINGO: names of nonlinear goalswfulcd 1 : minimize fuel weight - Cdkwempcd 2 : minimize empty weight - Cdkdoccd 3 : minimize direct operating cost - Cdk

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prchcd 4 : minimize purchase price - Cdkldmxcd 5 : maximize lift to drag ratio - Cdkmxspcd 6 : maximize max. speed - Cdkrngecd 7 : maximize range - Cdk

ALPOUT : Output Control 1 1 1 1 0 0 0 0 1 1

USRMOD : User module flags 1 1 0 0

OPTIMP : Optimization parameters -0.01 0.5 0.005 : VIOLIM, REMO, STEP

ADREMO : Using adaptive reduced move8 0.05 : Max. calls, deltaR

ENDPRB :**STOP reading the data file at this point**

F.7 .2 Sample DSIDES File: gasp.oa64.cdk.s1h.f

SUBROUTINE USRINP (NDESV,NINP,NOUT,DESVAR, & numdv,numsamp,numresp,corflag,fprefix,invmat)C*********************************************************************CC Modified call to subroutine; Tim Simpson - June 18, 1998CC Now, this subroutine does the necessary preprocessing to expediteC prediction with a kriging model; the user doesn't need to changeC anything in this routine--just make sure that it is invoked byC USRMOD in the .dat file (i.e., set first user module flag to 1)CC********************************************************************* INTEGER NDESV, NINP, NOUT REAL DESVAR(NDESV)

integer numdv,numsamp,numresp,corflag character*16 fprefix

COMMON /KRIG1/ xmat,betahat,thetaray,Rinvyfb

double precision p, invmat(numsamp,numsamp)

integer scfstr,lenstr,i,j,k,dummy,krig real dummy2 character*16 ftitle character*24 deckfile,fitsfile,scfnameCC using: maxsamp=500, maxdv=25, maxresp=20C double precision xmat(500,15),betahat(20), & thetaray(20,15),theta(15),Rinvyfb(20,500), & Fvect(500,1),FRinv(500),yvect(500),yfb_vect(500), & resp(500,20)

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scfname='expgaucubma1ma2'

if (corflag.eq.2) then p=2.0 else if (corflag.eq.1) then p=1.0 end if

call getlen(fprefix,lenstr) ftitle=fprefix

deckfile=ftitle(1:lenstr) // '.dek' scfstr=(3*(corflag-1))+1 fitsfile=ftitle(1:lenstr) // '.' // scfname(scfstr:scfstr+2) & // '.fit'

print *, '#samp,dv,resp; corflag =',numsamp,numdv,numresp,corflag print *, ' ' print *, 'Files = ', deckfile,fitsfile

open(21,file=deckfile,status='old') open(22,file=fitsfile,status='old')

write(6,*) ' ' write(6,*) 'Reading in sample data...' do 10 i=1,numsamp read (21,*) (xmat(i,j),j=1,numdv),(resp(i,k),k=1,numresp)C write (6,76) (xmat(i,j),j=1,numdv),(resp(i,k),k=1,numresp) 76 format(2f8.4,3(f12.4,1x)) 10 continue close(21)

write(6,*) ' ' write(6,*) 'Reading in theta parameters...' do 20 i=1,numresp read(22,*) dummy,(thetaray(i,j),j=1,numdv),dummy2C write(6,78) dummy,(thetaray(i,j),j=1,numdv) 78 format(i2,8f9.5) 20 continue close(22) print *, ' ' write(6,*) 'Preprocessing kriging computations...' print *, ' 'CC step through responses, computing kriging equationsC do 40 krig=1,numresp do 50 j=1,numdv theta(j)=thetaray(krig,j) 50 continue write(6,*) '...kriging model #', krig,'...'C write(6,1002) (theta(j),j=1,numdv)C 1002 format(8f9.5) do 60 i=1,numsamp

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yvect(i)=resp(i,krig) Fvect(i,1)=1.0d0 60 continue

call cormat (xmat,invmat,numsamp,numdv, & theta,p,corflag) call compeqn (invmat,Fvect,yvect,FRinv,yfb_vect, & Rinvyfb,betahat,numsamp,numdv,krig)CC increment variable krig to look at next responseC 40 continue

RETURN ENDCC subroutine to parse filename specified in ‘dace.params.h’ file.C subroutine getlen(string,lenstr) character*1 blank character*16 string parameter (blank=' ') integer next,lenstr do 10 next = LEN(string), 1, -1 if (string(next:next).ne.blank) then lenstr=next return end if 10 continue lenstr=0 if (lenstr.eq.0) then write(6,*) 'You have not specified a file name prefix' stop end if return endCC subroutine needed for kriging metamodel computationC subroutine cormat (xmat,invmat,numsamp,numdv,theta,p,corflag) integer numdv,numsamp,corflag,ipvt(500) double precision xmat(500,15),invmat(numsamp,numsamp), & work(500),p,theta(15) integer i,j,info double precision det(2),R

do 300 i = 1,numsamp do 305 j = i,numsamp if( i .eq. j ) then invmat(i,j) = 1.0d0 else call scfxmat(R,xmat,theta,corflag,p,numdv,numsamp,i,j) invmat(i,j) = R invmat(j,i) = invmat(i,j) endif

446

305 continue 300 continue

C do 306 i=1,numsampC write(6,82) (invmat(i,j),j=1,numsamp)C 82 format(14(f4.2,1x))C 306 continue

do 307 i=1,numsamp work(i)=0.0d0 ipvt(i)=0 307 continue call dgefa(invmat, numsamp, numsamp, ipvt, info) if( info .ne. 0 ) then write(*,*)"Error in DGEFA, info = ",info stop endif call dgedi(invmat, numsamp, numsamp, ipvt, det, work, 11)

C do 310 i=1,numsampC write(6,84) (invmat(i,j),j=1,numsamp)C 84 format(14(f10.2,1x))C 310 continue

return end************************************************************************* include LINPACK routines used to find inverse of correlation matrix************************************************************************* include 'dgefa.f' include 'dgedi.f'************************************************************************CC subroutine needed for kriging metamodel computationC subroutine compeqn (invmat,Fvect,yvect,FRinv,yfb_vect, & Rinvyfb,betahat,numsamp,numdv,krig)

integer numsamp,numdv double precision betahat(20),invmat(numsamp,numsamp), & Fvect(500,1),FRinv(500), & Rinvyfb(20,500),yvect(500),yfb_vect(500) double precision beta_den,beta_num integer i,j,krigCC compute F'RinvC do 310 i=1,numsamp FRinv(i)=0.0d0 do 315 j=1,numsamp FRinv(i)=FRinv(i)+Fvect(j,1)*invmat(j,i)

447

315 continue 310 continueCC compute betahat = (F'Rinv*yvect)/(F'Rinv*F)C beta_den=0.0d0 beta_num=0.0d0 do 320 i=1,numsamp beta_den=beta_den+FRinv(i)*Fvect(i,1) beta_num=beta_num+FRinv(i)*yvect(i) 320 continue betahat(krig) = beta_num / beta_denCC compute y-f'betahatC do 330 i = 1,numsamp yfb_vect(i) = yvect(i) - betahat(krig)*Fvect(i,1) 330 continueCC compute Rinv*(y-f'beta)C do 340 i=1,numsamp Rinvyfb(krig,i)=0.0d0 do 345 j=1,numsamp Rinvyfb(krig,i)=Rinvyfb(krig,i)+invmat(i,j)*yfb_vect(j) 345 continue 340 continue return end

CC subroutine needed for kriging metamodel computationC subroutine scfxmat(R,xmat,theta,corflag,p,numdv,numsamp,i,j) integer i,j,corflag,numdv,numsamp double precision R,xmat(500,15),theta(15),p double precision sum,thetadist,dist integer k

if ((corflag.eq.2).or.(corflag.eq.1)) then sum=0.0d0 do 120 k = 1,numdv dist = DABS(xmat(i,k)-xmat(j,k)) sum = sum + theta(k)*((dist)**p) 120 continue R = DEXP( -1.0d0*sum ) else if (corflag.eq.3) then sum=1.0d0 do 130 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) thetadist=dist*theta(k) if (thetadist.lt.0.5) then sum=sum*(1.0-6.0*(thetadist**2)+6.0*(thetadist**3)) else if (thetadist.ge.1.0) then sum=sum*0.0

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else sum=sum*(2.0*(1.0-thetadist)**3) end if 130 continue R = sum else if (corflag.eq.4) then sum=1.0 do 140 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) sum = sum*(exp(-theta(k)*dist)*(1.+theta(k)*dist)) 140 continue R = sum else if (corflag.eq.5) then sum=1.0 do 150 k=1,numdv dist = ABS(xmat(i,k)-xmat(j,k)) thetadist=dist*theta(k) sum = sum*(exp(-thetadist)* & (1.+thetadist+(thetadist**2)/3.0)) 150 continue R = sum end if

return endC*********************************************************************C SUBROUTINE USROUT (NDESV, NOUT, DESVAR, LCONDF, LCONSV, LXFEAS)CC Tim Simpson - June 18, 1998CC This subroutine postprocesses the solution by taking the finalC converged values and instantiating each aircraft in GASP. TheC resulting deviation function is computed for each aircraft asC specified by the variable ‘iscen’. After processing, a file calledC ‘gasp.oa64.cdk.s1h.mod’ is created which containts all of theC necessary information for each instantiation. (The output fileC name can be changed by changing variable ‘outfile’.CC********************************************************************* INTEGER NDESV, NOUT REAL DESVAR(NDESV) LOGICAL LCONDF, LCONSV, LXFEASC CHARACTER*80 LINE

INTEGER NUM, IPARAMETER (NUM=7)CHARACTER*10 CHGVAR(NUM)REAL CHGVAL(NUM)INTEGER CHRLEN(NUM), INDEXinteger INLINE,NUMOUT,nscaleparameter(NUMOUT=11,nscale=3)REAL NOISE,WEMP,DOC,ROUGH,WFUEL,PURCH,VCRMX,LDMAX,RANGE

REAL RESPONSE(NUMOUT,nscale),GOALS(7,nscale),

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& dplus(7,nscale),dminus(7,nscale),devfcn(4,nscale), & convio(nscale),CONSTR(6,nscale)

real noisec(nscale),tpspdc(nscale),docc(nscale), & roughc(nscale),wempc(nscale),wfuelc(nscale), & wfuelt(nscale),wempt(nscale),doct(nscale), & ldmaxt(nscale),vcrmxt(nscale),ranget(nscale), & purcht(nscale),rangec(nscale)

integer ipax REAL xlo(6),xhi(6),xnew(6) REAL EMCRU,AR,DPROP,WGS,AF,WS,PAX integer j,i,iscen character*28 outfile

data xlo / 0.24, 7.0, 5.0, 19.0, 85.0, 14.0 / data xhi / 0.48, 11.0, 5.96, 25.0, 110.0, 20.0 /CC define array of constraints for each response for each aircraftC data noisec / 75., 75., 75. / data tpspdc / 850., 850., 850. / data docc / 80., 80., 80. / data roughc / 2., 2., 2. / data wempc / 2200., 2200., 2200. / data wfuelc / 450., 475., 500. / data rangec / 2000., 2000., 2000. /CC define array of goal targets for each response for each aircraftC data wfuelt / 450., 400., 350. / data wempt / 1900., 1950., 2000. / data doct / 60., 60., 60. / data purcht / 41000., 42000., 43000. / data ldmaxt / 17., 17., 17. / data vcrmxt / 200., 200., 200. / data ranget / 2500., 2500., 2500. /

iscen=1 outfile='gasp.oa64.cdk.s1h.mod' open(unit=27,file=outfile,status='unknown')

EMCRU = DESVAR(1) AR = DESVAR(2) DPROP = DESVAR(3) WGS = DESVAR(4) AF = DESVAR(5) WS = DESVAR(6)CC scale the design variables to [0,1] - necessary for krigingC do 10 j=1,6 xnew(j)=(DESVAR(j)-xlo(j))/(xhi(j)-xlo(j)) 10 continue

write(6,*) ' ' write(6,*) outfile

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write(6,*) ' ' write(6,*) 'Instantiate final values in GASP:' write(6,*) ' ' write(6,*) ' INPUTS: Act. Value Scaled Value' write(6,73) 73 format(3(12('-'),1x)) write(6,74) 'Cruise_Spd=',EMCRU,xnew(1) write(6,74) 'Aspct_Rtio=',AR,xnew(2) write(6,74) 'Prop_Diamr=',DPROP,xnew(3) write(6,74) 'Wing_Loadg=',WGS,xnew(4) write(6,74) 'Eng_Act_Fc=',AF,xnew(5) write(6,74) 'Seat_Width=',WS,xnew(6) 74 format(A12,1x,f12.4,1x,f12.4)CC write instantiations to output fileC write(27,*) ' ' write(27,*) outfile write(27,*) ' ' write(27,*) 'Instantiate final values in GASP:' write(27,*) ' ' write(27,*) ' INPUTS: Act. Value Scaled Value' write(27,73) write(27,74) 'Cruise_Spd=',EMCRU,xnew(1) write(27,74) 'Aspct_Rtio=',AR,xnew(2) write(27,74) 'Prop_Diamr=',DPROP,xnew(3) write(27,74) 'Wing_Loadg=',WGS,xnew(4) write(27,74) 'Eng_Act_Fc=',AF,xnew(5) write(27,74) 'Seat_Width=',WS,xnew(6)CC specify number of passengersC do 300 ipax=1,3 PAX = (2.*REAL(ipax))-1C***********************************************************************C USER has to redifine the following block to create GASP input fileC*********************************************************************** CHGVAR(1) = 'EMCRU'

CHRLEN(1) = 5CHGVAL(1) = EMCRU

CHGVAR(2) = 'AR'CHRLEN(2) = 2CHGVAL(2) = AR

CHGVAR(3) = 'DPROP'CHRLEN(3) = 5CHGVAL(3) = DPROPCHGVAR(4) = 'WGS'CHRLEN(4) = 3CHGVAL(4) = WGS

CHGVAR(5) = 'AF' CHRLEN(5) = 2 CHGVAL(5) = AF CHGVAR(6) = 'WS' CHRLEN(6) = 2 CHGVAL(6) = WS

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CHGVAR(7) = 'PAX' CHRLEN(7) = 3 CHGVAL(7) = PAX

C**********************************************************************C user doesn't have to change anything in the following blockC*********************************************************************CC create GASP input file: GASP.INC OPEN(UNIT=100,FILE='GASPINPUT.rb3l') call system('rm GASP.IN') OPEN(UNIT=110,FILE='GASP.IN',STATUS='NEW') DO WHILE ( .TRUE. ) READ(100,70,END=1000) LINE

INDEX = INLINE(NUM, LINE, CHGVAR, CHRLEN, CHGVAL)IF (INDEX .NE. 0) THEN

WRITE(110, 75) CHGVAR(INDEX)(1:CHRLEN(INDEX)), + CHGVAL(INDEX) ELSE WRITE(110,70) LINE

END IF 70 FORMAT ( A80) 75 FORMAT (2X, A, '=', F10.2, ',') ENDDO1000 CONTINUE CLOSE(100) CLOSE(110)CC execute GASP with current values of design variablesC CALL SYSTEM('rm performance') call system('rm GASP.OUT') print *, ' ' print *, 'Executing GASP for Passenger #',PAX CALL SYSTEM('gasp <GASP.IN>GASP.OUT') PRINT *, '' PRINT *, 'GASP IS DONE'CC read GASP output file named performance into response arrayC OPEN(UNIT=120,FILE='performance', STATUS='UNKNOWN') DO 200 I =1, NUMOUT READ (120, *) RESPONSE(I,ipax)200 CONTINUE CLOSE(120)300 continue write(6,*) ' ' write(6,*) ' OUTPUTS: 1 Pax 3 Pax 5 Pax' write(6,77) 77 format(4(12('-'),1x)) write(6,78) 'Noise=', (RESPONSE(2,j),j=1,3) write(6,78) 'Empty_Wgt=', (RESPONSE(3,j),j=1,3) write(6,78) 'Oper_Cost=', (RESPONSE(5,j),j=1,3) write(6,78) 'Ride_Rough=', (RESPONSE(6,j),j=1,3)

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write(6,78) 'Fuel_Wgt=', (RESPONSE(7,j),j=1,3) write(6,78) 'Purch_Price=', (RESPONSE(8,j),j=1,3) write(6,78) 'Max_Range=', (RESPONSE(9,j),j=1,3) write(6,78) 'Max_Speed=', (RESPONSE(10,j),j=1,3) write(6,78) 'L/D_Max=', (RESPONSE(11,j),j=1,3) 78 format(A12,1x,f12.4,1x,f12.4,1x,f12.4) write(6,*) ' 'CC write instantiations to fileC write(27,*) ' ' write(27,*) ' OUTPUTS: 1 Pax 3 Pax 5 Pax' write(27,77) write(27,78) 'Noise=', (RESPONSE(2,j),j=1,3) write(27,78) 'Empty_Wgt=', (RESPONSE(3,j),j=1,3) write(27,78) 'Oper_Cost=', (RESPONSE(5,j),j=1,3) write(27,78) 'Ride_Rough=', (RESPONSE(6,j),j=1,3) write(27,78) 'Fuel_Wgt=', (RESPONSE(7,j),j=1,3) write(27,78) 'Purch_Price=', (RESPONSE(8,j),j=1,3) write(27,78) 'Max_Range=', (RESPONSE(9,j),j=1,3) write(27,78) 'Max_Speed=', (RESPONSE(10,j),j=1,3) write(27,78) 'L/D_Max=', (RESPONSE(11,j),j=1,3) write(27,*) ' 'CC compute deviation function for each aircraftC do 310 j=1,3

NOISE = RESPONSE(2,j) WEMP = RESPONSE(3,j) DOC = RESPONSE(5,j) ROUGH = RESPONSE(6,j) WFUEL = RESPONSE(7,j) PURCH = RESPONSE(8,j) RANGE = RESPONSE(9,j) VCRMX = RESPONSE(10,j) LDMAX = RESPONSE(11,j)

GOALS(1,j) = WFUEL/wfuelt(j)-1.0 GOALS(2,j) = WEMP/wempt(j) - 1.0 GOALS(3,j) = DOC/doct(j) - 1.0 GOALS(4,j) = PURCH/purcht(j) - 1.0 GOALS(5,j) = LDMAX/ldmaxt(j) - 1.0 GOALS(6,j) = VCRMX/vcrmxt(j) - 1.0 GOALS(7,j) = RANGE/ranget(j) - 1.0

do 320 i=1,7 if(GOALS(i,j).ge.0)then dplus(i,j)=ABS(GOALS(i,j)) dminus(i,j)=0.0 else dplus(i,j)=0.0 dminus(i,j)=ABS(GOALS(i,j)) endif 320 continue

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if(iscen.eq.1)then devfcn(1,j)=(dplus(1,j)+dplus(2,j)+dplus(3,j)+dplus(4,j)+ & dminus(5,j)+dminus(6,j)+dminus(7,j))/7. devfcn(2,j)=0.0 devfcn(3,j)=0.0 devfcn(4,j)=0.0

else if(iscen.eq.2) then devfcn(1,j)=(dplus(2,j)+dplus(3,j)+dplus(4,j))/3. devfcn(2,j)=(dplus(1,j)+dminus(5,j)+ & dminus(6,j)+dminus(7,j))/4. devfcn(3,j)=0.0 devfcn(4,j)=0.0

else if(iscen.eq.3) then devfcn(1,j)=(dplus(1,j)+dminus(5,j)+ & dminus(6,j)+dminus(7,j))/4. devfcn(2,j)=(dplus(2,j)+dplus(3,j)+dplus(4,j))/3. devfcn(3,j)=0.0 devfcn(4,j)=0.0

else if(iscen.eq.4) then devfcn(1,j)=(dplus(2,j)+dplus(4,j))/2. devfcn(2,j)=dplus(3,j) devfcn(3,j)=(dplus(1,j)+dminus(5,j)+ & dminus(6,j)+dminus(7,j))/4. devfcn(4,j)=0.0

else if(iscen.eq.5) then devfcn(1,j)=dplus(3,j) devfcn(2,j)=(dplus(2,j)+dplus(4,j))/2. devfcn(3,j)=(dplus(1,j)+dminus(5,j)+ & dminus(6,j)+dminus(7,j))/4. devfcn(4,j)=0.0

else if(iscen.eq.6) then devfcn(1,j)=(dplus(1,j)+dminus(6,j))/2. devfcn(2,j)=(dminus(5,j)+dminus(6,j))/2. devfcn(3,j)=(dplus(2,j)+dplus(3,j)+dplus(4,j))/3. devfcn(4,j)=0.0

else if(iscen.eq.7) then devfcn(1,j)=dminus(5,j) devfcn(2,j)=dminus(7,j) devfcn(3,j)=(dplus(1,j)+dminus(6,j))/2. devfcn(4,j)=(dplus(2,j)+dplus(3,j)+dplus(4,j))/3.

else if(iscen.eq.8) then devfcn(1,j)=dminus(7,j) devfcn(2,j)=dminus(5,j) devfcn(3,j)=(dplus(1,j)+dminus(6,j))/2. devfcn(4,j)=(dplus(2,j)+dplus(3,j)+dplus(4,j))/3.

endif

454

310 continue

write(6,78) 'P_lev_1=', (devfcn(1,j),j=1,3) if(iscen.ge.2) then write(6,78) 'P_lev_2=', (devfcn(2,j),j=1,3) if(iscen.ge.4) then write(6,78) 'P_lev_3=', (devfcn(3,j),j=1,3) if(iscen.ge.7) then write(6,78) 'P_lev_4=', (devfcn(4,j),j=1,3) end if end if end ifCC write instantiations to fileC write(27,78) 'P_lev_1=', (devfcn(1,j),j=1,3) if(iscen.ge.2) then write(27,78) 'P_lev_2=', (devfcn(2,j),j=1,3) if(iscen.ge.4) then write(27,78) 'P_lev_3=', (devfcn(3,j),j=1,3) if(iscen.ge.7) then write(27,78) 'P_lev_4=', (devfcn(4,j),j=1,3) end if end if end ifCC compute constraint violationC do 410 j=1,3

NOISE = RESPONSE(2,j) WEMP = RESPONSE(3,j) DOC = RESPONSE(5,j) ROUGH = RESPONSE(6,j) WFUEL = RESPONSE(7,j) RANGE = RESPONSE(9,j)

CONSTR(1,j) = 1.0 - NOISE/noisec(j) CONSTR(2,j) = 1.0 - DOC/docc(j) CONSTR(3,j) = 1.0 - ROUGH/roughc(j) CONSTR(4,j) = 1.0 - WEMP/wempc(j) CONSTR(5,j) = 1.0 - WFUEL/wfuelc(j) CONSTR(6,j) = RANGE/rangec(j) - 1.0

convio(j)=0.0 do 420 i=1,6 if(CONSTR(i,j).lt.0.0)then convio(j)=convio(j)+ABS(CONSTR(i,j)) end if 420 continue 410 continue

write(6,*) ' ' write(6,78) 'Con_Vio=',(convio(j),j=1,3)

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write(6,*) ' ' write(27,*) ' ' write(27,78) 'Con_Vio=',(convio(j),j=1,3) write(27,*) ' '

close(27)

RETURN ENDC*********************************************************************C user does not need to change the following function blockC********************************************************************** FUNCTION INLINE(NUM, LINE, CHGVAR, CHRLEN, CHGVAL)

INTEGER INLINE, NUM, ICHARACTER*80 LINECHARACTER*10 CHGVAR(NUM)

INTEGER CHRLEN(NUM)REAL CHGVAL(NUM)

DO 10 I = 1, NUMIF (LINE(3:3+CHRLEN(I)) .EQ. CHGVAR(I)(1:CHRLEN(I))

+ //'=') THENINLINE = IGOTO 20END IF

10 CONTINUEINLINE = 0

20 ENDC***********************************************************************CC Subroutine USRSETCC Purpose: Evaluate non-linear constraints and goals.C NOTE - Do not specify the deviation variablesCC-----------------------------------------------------------------------C Arguments Name Type DescriptionC --------- ---- ---- -----------C Input: IPATH int = 1 evaluate constraints and goalsC = 2 evaluate constraints onlyC = 3 evaluate goals onlyC NDESV int number of design variablesC MNLNCG int maximum number of nonlinearC constraints and goalsC NOUT int unit number of output data fileC DESVAR real vector of design variablesCC Output: CONSTR real vector of constraint valuesC GOALS real vector of goal valuesCC Input/Output: noneC-----------------------------------------------------------------------C Common Blocks: noneC

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C Include Files: noneCC Called from: GCALCCC Calls to: noneC-----------------------------------------------------------------------C Development HistoryCC Author: BHARAT PATELC Date: 13 MARCH, 1992.CC Modifications:CC***********************************************************************CC SUBROUTINE USRSET (IPATH, NDESV, MNLNCG, NOUT, DESVAR, & CONSTR, GOALS)CC------------------------------C Passed variables from RUNALPC------------------------------C INTEGER IPATH, NDESV, MNLNCG, NOUTC REAL DESVAR(NDESV) REAL CONSTR(MNLNCG), GOALS(MNLNCG) INTEGER VVNUMCC------------------------------------------------C Variables particular to kriging implementationC------------------------------------------------C COMMON /KRIG1/ xmat,betahat,thetaray,Rinvyfb COMMON /KRIG2/ nmdv,nmsamp,nmresp,corfcn double precision xmat(500,15),betahat(20), & thetaray(20,15),Rinvyfb(20,500) integer nmdv,nmsamp,nmresp,corfcn integer j,krig double precision ynew(20),xnew(15),theta(15), & R,r_xhat(500),yevalCC---------------------------------------------C Local variables particular to c-dsp problemC---------------------------------------------C REAL xlo(6),xhi(6) REAL EMCRU,AR,DPROP,WGS,AF,WS, & NOISEm,WEMPm,DOCm,ROUGHm,WFUELm, & PURCHm,VCRMXm,LDMAXm,RANGEm, & NOISEv,WEMPv,DOCv,ROUGHv,WFUELv, & PURCHv,VCRMXv,LDMAXv,RANGEv, & NOISEcd,NOISEcdu,ROUGHcd,ROUGHcdu, & VCRMXcdl,LDMAXcdl,RANGEcdl, & VCRMXcd,LDMAXcd,RANGEcd,

457

& WFUELcd,WFUELcdu,WFUELcdl, & WEMPcd,WEMPcdu,WEMPcdl, & PURCHcd,PURCHcdu,PURCHcdl, & DOCcd,DOCcdu

data xlo / 0.24, 7.0, 5.0, 19.0, 85.0, 14.0 / data xhi / 0.48, 11.0, 5.96, 25.0, 110.0, 20.0 /

EMCRU = DESVAR(1) AR = DESVAR(2) DPROP = DESVAR(3) WGS = DESVAR(4) AF = DESVAR(5) WS = DESVAR(6)CC scale the design variables to [0,1] - necessary for krigingC do 10 j=1,nmdv xnew(j)=(DESVAR(j)-xlo(j))/(xhi(j)-xlo(j)) 10 continue

VVNUM=VVNUM+1

PRINT *, ' ' PRINT *, ' INPUTS: Act. Value Scaled Value Run#',VVNUM write(6,73) 73 format(3(12('-'),1x)) write(6,74) 'Cruise_Spd=',EMCRU,xnew(1) write(6,74) 'Aspct_Rtio=',AR,xnew(2) write(6,74) 'Prop_Diamr=',DPROP,xnew(3) write(6,74) 'Wing_Loadg=',WGS,xnew(4) write(6,74) 'Eng_Act_Fc=',AF,xnew(5) write(6,74) 'Seat_Width=',WS,xnew(6) 74 format(A12,1x,f12.4,1x,f12.4)

C*********************************************************************CC kriging prediction block; Tim Simpson - June 18, 1998C *user does not have to change anything in this block*CC********************************************************************* do 40 krig=1,nmrespCC transfer theta parameters for current response (designated by krig)C do 50 j=1,nmdv theta(j)=thetaray(krig,j) 50 continueCC compute correlation vector for xnewC do 60 j=1,nmsamp call scfxnew(R,xnew,xmat,theta,corfcn,nmdv,nmsamp,j) r_xhat(j)=R 60 continue

458

CC calculate ynew of current responseC yeval = 0.0d0 do 220 j=1,nmsamp 220 yeval=yeval+r_xhat(j)*Rinvyfb(krig,j) ynew(krig) = yeval + betahat(krig)CC increment variable krig to predict next ynewC 40 continueCC equate predicted ynew vector to appropriate responsesC NOISEm = REAL(ynew(1)) WEMPm = REAL(ynew(2)) DOCm = REAL(ynew(3)) ROUGHm = REAL(ynew(4)) WFUELm = REAL(ynew(5)) PURCHm = REAL(ynew(6)) RANGEm = REAL(ynew(7)) VCRMXm = REAL(ynew(8)) LDMAXm = REAL(ynew(9)) NOISEv = REAL(ynew(10)) WEMPv = REAL(ynew(11)) DOCv = REAL(ynew(12)) ROUGHv = REAL(ynew(13)) WFUELv = REAL(ynew(14)) PURCHv = REAL(ynew(15)) RANGEv = REAL(ynew(16)) VCRMXv = REAL(ynew(17)) LDMAXv = REAL(ynew(18))CC compute Cdk valuesC NOISEcdu=(75.-NOISEm)/(NOISEv*(3.**0.5)) NOISEcd=NOISEcdu

ROUGHcdu=(2.00-ROUGHm)/(ROUGHv*(3.**0.5)) ROUGHcd=ROUGHcdu

DOCcdu=(60.-DOCm)/(DOCv*(3.**0.5)) DOCcd=DOCcdu

WEMPcdl=(WEMPm-1900.)/(WEMPv*(3.**0.5)) WEMPcdu=(2000.-WEMPm)/(WEMPv*(3.**0.5)) WEMPcd=AMIN1(WEMPcdl,WEMPcdu)

WFUELcdl=(WFUELm-350.)/(WFUELv*(3.**0.5)) WFUELcdu=(450.-WFUELm)/(WFUELv*(3.**0.5)) WFUELcd=AMIN1(WFUELcdl,WFUELcdu)

PURCHcdl=(PURCHm-41000.)/(PURCHv*(3.**0.5)) PURCHcdu=(43000.-PURCHm)/(PURCHv*(3.**0.5)) PURCHcd=AMIN1(PURCHcdu,PURCHcdl)

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VCRMXcdl=(VCRMXm-200.0)/(VCRMXv*(3.**0.5)) VCRMXcd=VCRMXcdl

RANGEcdl=(RANGEm-2500.0)/(RANGEv*(3.**0.5)) RANGEcd=RANGEcdl

LDMAXcdl=(LDMAXm-17.0)/(LDMAXv*(3.**0.5)) LDMAXcd=LDMAXcdlCC print out corresponding values of constraint and goal responseC PRINT *, ' ' PRINT *, ' OUTPUTS: Mean Variance Cdk' write(6,77) 77 format(4(12('-'),1x)) write(6,78) 'Noise=', NOISEm, NOISEv, NOISEcd write(6,78) 'Empty_Wgt=', WEMPm, WEMPv, WEMPcd write(6,78) 'Oper_Cost=',DOCm, DOCv, DOCcd write(6,78) 'Ride_Rough=', ROUGHm, ROUGHv, ROUGHcd write(6,78) 'Fuel_Wgt=', WFUELm, WFUELv, WFUELcd write(6,78) 'Purch_Price=', PURCHm, PURCHv, PURCHcd write(6,78) 'Max_Range=', RANGEm, RANGEv, RANGEcd write(6,78) 'Max_Speed=', VCRMXm, VCRMXv, VCRMXcd write(6,78) 'L/D_Max=', LDMAXm, LDMAXv, LDMAXcd 78 format(A12,1x,f12.4,1x,f12.4,1x,f12.5) print *, ' 'CC 3.0 Evaluate non-linear constraintsC IF (IPATH .EQ. 1 .OR. IPATH .EQ. 2) THENCC constraint formulations for restricting appropriate Cdk >= 1C CONSTR(1) = NOISEcd - 1.0 CONSTR(2) = (80.-DOCm)/(DOCv*(3.**0.5)) - 1.0 CONSTR(3) = ROUGHcd - 1.0 CONSTR(4) = (2200.-WEMPm)/(WEMPv*(3.**0.5)) - 1.0 CONSTR(5) = (450.-WFUELm)/(WFUELv*(3.**0.5)) - 1.0 CONSTR(6) = (RANGEm-2000.)/(RANGEv*(3.**0.5)) - 1.0 END IFCC 4.0 Evaluate non-linear goalsC IF (IPATH .EQ. 1 .OR. IPATH .EQ. 3) THENCC specify goals; want all Cdk >= 1.C GOALS(1) = WFUELcd-1.0 GOALS(2) = WEMPcd - 1.0 GOALS(3) = DOCcd - 1.0 GOALS(4) = PURCHcd - 1.0 GOALS(5) = LDMAXcd - 1.0 GOALS(6) = VCRMXcd - 1.0 GOALS(7) = RANGEcd - 1.0

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END IFCC 5.0 Return to calling routineC RETURN ENDCC subroutine needed for kriging metamodel computation to compute correlationC between sample points and new prediction pointC subroutine scfxnew(R,xnew,xmat,theta,corfcn,nmdv,nmsamp,j) integer j,corfcn,nmdv,nmsamp double precision R,xnew(15), & xmat(500,15),theta(15) double precision sum,thetadist,dist,p integer k

if ((corfcn.eq.2).or.(corfcn.eq.1)) then if(corfcn.eq.2)then p=2. else p=1. endif

sum=0.0d0 do 400 k = 1,nmdv dist = ABS(xnew(k)-xmat(j,k)) sum = sum + theta(k)*((dist)**p) 400 continue R = exp( -1.0d0*sum ) else if (corfcn.eq.3) then sum=1.0d0 do 410 k=1,nmdv dist = ABS(xnew(k)-xmat(j,k)) thetadist=theta(k)*dist if (thetadist.lt.0.5) then sum=sum*(1.0-6.0*(thetadist**2)+6.0*(thetadist**3)) else if (thetadist.ge.1.0) then sum=sum*0.0 else sum=sum*(2.0*(1.0-thetadist)**3) end if 410 continue R = sum else if (corfcn.eq.4) then sum=1.0 do 420 k=1,nmdv dist = ABS(xnew(k)-xmat(j,k)) sum = sum*(exp(-theta(k)*dist)*(1.+theta(k)*dist)) 420 continue R = sum else if (corfcn.eq.5) then sum=1.0 do 430 k=1,nmdv

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dist = ABS(xnew(k)-xmat(j,k)) thetadist=theta(k)*dist sum = sum*(exp(-thetadist)* & (1.+thetadist+(thetadist**2)/3.0)) 430 continue R = sum end if

return end

F.8 SAMPLE GAA DSIDES FILES FOR BENCHMARK AIRCRAFT

The following sample DSIDES files are for solving the GAA Compromise DSP for

an individual (benchmark) aircraft. These files are explicitly for linking GASP to DSIDES.

These particular files are the five passenger (six seater) GAA for Scenario 3 with a low

starting point (i.e., all design variables at their upper bound). The .dat file is given

Section F.5.1, and the USRSET subroutine from the . f file is given in Section F.5.2.

F.8 .1 Sample DSIDES File: gasp5s3.dat

PTITLE : Problem Title, User Name and DatePAX=1,rb3l GASP; Tim Simpson and Jonathan Maier; June 16, 1998

NUMSYS : Number of system variables: real,integer,boolean 6 0 0

SYSVAR : System variable informationCSPD 1 0.24 0.48 0.24 : cruise speedAR 2 7.0 11.0 7.0 : aspect ratioDPRP 3 5.0 5.96 5.0 : propeller diameterWL 4 19.0 25.0 19.0 : wing loadingAF 5 85.0 110.0 85.0 : engine activity factorWS 6 14.0 20.0 14.0 : seat width

NUMCAG : Number of constraints and goals 0 5 0 0 7: nlinco, nnlinq, nnlequ, nlingo, nnlgoa

ACHFUN : Achievment function 2 : level 1 4 : level 1, 4 terms (+1,0.25) (-5,0.25) (-6,0.25) (-7,0.25) 2 3 : level 2, 3 terms (+2,0.333) (+3,0.333) (+4, 0.333)

STOPCR : Stopping criteria1 0 25 0.05 0.05 : perfm cal, prt intereslts, Mcyles,sta dev,sta var

462

NLINCO: names of nonlinear constraintsnoise 1 : takeoff noisedoc 2 : direct operating costrough 3 : ride roughness coefficientwemp 4 : aircraft empty weightwfuel 5 : fuel weight

NLINGO: names of nonlinear goalswfuel 1 : minimize fuel weightwemp 2 : minimize empty weightdoc 3 : minimize direct operating costpurch 4 : minimize purchase priceldmax 5 : maximize lift to drag ratiomxspd 6 : maximize max. speedrange 7 : maximize range

ALPOUT : Input/output Control 1 1 1 1 0 0 0 0 1 1

USRMOD : Input/Output flags 1 0 0 0

OPTIMP : Optimization parameters -0.01 0.5 0.005 : VIOLIM, REMO, STEP

ENDPRB :**STOP reading the data file at this point**

F.8 .2 Sample DSIDES File: gasp5s3lo.f

SUBROUTINE USRSET (IPATH, NDESV, MNLNCG, NOUT, DESVAR, & CONSTR, GOALS)CC---------------------------------------C Arguments:C--------------------------------------- INTEGER IPATH, NDESV, MNLNCG, NOUTC REAL DESVAR(NDESV) REAL CONSTR(MNLNCG), GOALS(MNLNCG)CC---------------------------------------C Local variables:C---------------------------------------CC******************************************************************C 1.0 Set the values of the local design variables (optional)C******************************************************************CC User has to define the type of variables in the followingC blockC REAL EMCRU,AR,DPROP,WGS,AF,WS,PAX, & TPSPD,NOISE,WEMP,DOC,ROUGH,WFUEL,PURCH,VCRMX,LDMAX,RANGE

463

INTEGER VVNUMCC******************************************************************C CHARACTER*80 LINE

INTEGER NUM, IPARAMETER (NUM=7)CHARACTER*10 CHGVAR(NUM)REAL CHGVAL(NUM)INTEGER CHRLEN(NUM), INDEXINTEGER INLINEINTEGER NUMOUTinteger nscale

PARAMETER (NUMOUT=11,nscale=3)REAL RESPONSE(NUMOUT)real noisec(nscale),tpspdc(nscale),docc(nscale),

& roughc(nscale),wempc(nscale),wfuelc(nscale), & wfuelt(nscale),wempt(nscale),doct(nscale), & ldmaxt(nscale),vcrmxt(nscale),ranget(nscale), & purcht(nscale) integer ipax VVNUM=VVNUM+1CC define array of constraints for each response for each aircraftC data noisec / 75., 75., 75. / data tpspdc / 850., 850., 850. / data docc / 80., 80., 80. / data roughc / 2., 2., 2. / data wempc / 2200., 2200., 2200. / data wfuelc / 450., 475., 500. /CC define array of goal targets for each response for each aircraftC data wfuelt / 450., 400., 350. / data wempt / 1900., 1950., 2000. / data doct / 60., 60., 60. / data purcht / 41000., 42000., 43000. / data ldmaxt / 17., 17., 17. / data vcrmxt / 200., 200., 200. / data ranget / 2500., 2500., 2500. /C******************************************************************CC Define the design variablesC EMCRU = DESVAR(1) AR = DESVAR(2) DPROP = DESVAR(3) WGS = DESVAR(4) AF = DESVAR(5) WS = DESVAR(6)CC specify number of passengersC PAX = 5.0

464

ipax = INT((PAX+1.)/2.)C***********************************************************************C USER has to redifine the following block to create GASP input fileC*********************************************************************** CHGVAR(1) = 'EMCRU'

CHRLEN(1) = 5CHGVAL(1) = EMCRU

CHGVAR(2) = 'AR'CHRLEN(2) = 2CHGVAL(2) = AR

CHGVAR(3) = 'DPROP'CHRLEN(3) = 5CHGVAL(3) = DPROPCHGVAR(4) = 'WGS'CHRLEN(4) = 3CHGVAL(4) = WGS

CHGVAR(5) = 'AF' CHRLEN(5) = 2 CHGVAL(5) = AF CHGVAR(6) = 'WS' CHRLEN(6) = 2 CHGVAL(6) = WS CHGVAR(7) = 'PAX' CHRLEN(7) = 3 CHGVAL(7) = PAX

C**********************************************************************C user doesn't have to change anything in the following blockC*********************************************************************CC create GASP input file: GASP.INC OPEN(UNIT=100,FILE='GASPINPUT.rb3l') call system('rm GASP.IN') OPEN(UNIT=110,FILE='GASP.IN',STATUS='NEW') DO WHILE ( .TRUE. ) READ(100,70,END=1000) LINE

INDEX = INLINE(NUM, LINE, CHGVAR, CHRLEN, CHGVAL)IF (INDEX .NE. 0) THEN

WRITE(110, 75) CHGVAR(INDEX)(1:CHRLEN(INDEX)), + CHGVAL(INDEX) ELSE WRITE(110,70) LINE

END IF 70 FORMAT ( A80) 75 FORMAT (2X, A, '=', F10.2, ',') ENDDO1000 CONTINUE CLOSE(100) CLOSE(110)CC print out values of design variables at current stepC PRINT *, '' PRINT *, 'INPUTS Run #', VVNUM

465

PRINT *, '' print *, 'Cruise_Spd=',EMCRU PRINT *, 'Aspct_Rtio=',AR PRINT *, 'Prop_Diamr=',DPROP print *, 'Wing_Loadg=',WGS print *, 'Eng_Act_Fc=',AF print *, 'Seat_Width=',WSCC execute GASP with current values of design variablesC CALL SYSTEM('rm performance') call system('rm GASP.OUT') CALL SYSTEM('gasp <GASP.IN>GASP.OUT') PRINT *, '' PRINT *, 'GASP IS DONE'CC read GASP output file named performance into response arrayC OPEN(UNIT=120,FILE='performance', STATUS='UNKNOWN') DO 200 I =1, NUMOUT READ (120, *) RESPONSE(I)200 CONTINUE

CLOSE(120)CC equate response array values to goal and constraint variables;C response(4) in performance file is SFC which is not usedC TPSPD = RESPONSE(1) NOISE = RESPONSE(2) WEMP = RESPONSE(3) DOC = RESPONSE(5) ROUGH = RESPONSE(6) WFUEL = RESPONSE(7) PURCH = RESPONSE(8) RANGE = RESPONSE(9) VCRMX = RESPONSE(10) LDMAX = RESPONSE(11)CC print out corresponding values of constraint and goal responseC PRINT *, '' PRINT *, 'OUTPUTS' PRINT *, '' PRINT *, 'Tipspd=', TPSPD, ' Noise=', NOISE PRINT *, 'Empty Wgt=', WEMP, ' Oper Cost=',DOC PRINT *, 'Ride Rough=', ROUGH, ' Max Range=', RANGE PRINT *, 'Fuel Wgt=', WFUEL, ' Purch Price=', PURCH PRINT *, 'Max Speed=', VCRMX, ' L/D Max=', LDMAXCC*************************************************************************C 3.0 Evaluate non-linear constraintsC**************************************************************************C IF (IPATH .EQ. 1 .OR. IPATH .EQ. 2) THEN

466

CC specify constraints; all constraints are less than constraintsC CONSTR(1) = 1.0 - NOISE/noisec(ipax) CONSTR(2) = 1.0 - DOC/docc(ipax) CONSTR(3) = 1.0 - ROUGH/roughc(ipax) CONSTR(4) = 1.0 - WEMP/wempc(ipax) CONSTR(5) = 1.0 - WFUEL/wfuelc(ipax)

END IFCC**************************************************************************C 4.0 Evaluate non-linear goalsC************************************************************************** IF (IPATH .EQ. 1 .OR. IPATH .EQ. 3) THENCC specify goalsC GOALS(1) = WFUEL/wfuelt(ipax)-1.0 GOALS(2) = WEMP/wempt(ipax) - 1.0 GOALS(3) = DOC/doct(ipax) - 1.0 GOALS(4) = PURCH/purcht(ipax) - 1.0 GOALS(5) = LDMAX/ldmaxt(ipax) - 1.0 GOALS(6) = VCRMX/vcrmxt(ipax) - 1.0 GOALS(7) = RANGE/ranget(ipax) - 1.0

END IFCC 5.0 Return to calling routineC RETURN ENDC*********************************************************************C user does not need to change the following function blockC********************************************************************** FUNCTION INLINE(NUM, LINE, CHGVAR, CHRLEN, CHGVAL)

INTEGER INLINE, NUM, ICHARACTER*80 LINECHARACTER*10 CHGVAR(NUM)

INTEGER CHRLEN(NUM)REAL CHGVAL(NUM)DO 10 I = 1, NUM

IF (LINE(3:3+CHRLEN(I)) .EQ. CHGVAR(I)(1:CHRLEN(I)) + //'=') THEN

INLINE = IGOTO 20END IF

10 CONTINUEINLINE = 0

20 END

467

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VITA

Timothy W. Simpson was born in Arlington, Massachusetts on February 25, 1972. He

grew up in Corning, New York and Wilmington, North Carolina before moving to

Danville, Kentucky where he attended Boyle County High School. After high school, he

attended Cornell University where he graduated in 1994 with distinction, earning his

Bachelor of Science degree in Mechanical Engineering. He earned his Master of Science

degree in Mechanical Engineering in 1995 from the Georgia Institute of Technology. In

1997, he had a summer research appointment at the Institute for Computer Applications in

Science and Engineering at NASA Langley in Hampton, Virginia. His graduate work has

been funded by a Graduate Research Fellowship from the National Science Foundation, a

Presidential Fellowship from the Georgia Institute of Technology, and a Woodruff

Teaching Fellowship from the G. W. Woodruff School of Mechanical Engineering at the

Georgia Institute of Technology. Upon graduation, Timothy has accepted a tenure-track,

joint appointment as an Assistant Professor in the Department of Mechanical & Nuclear

Engineering and the Department of Industrial & Manufacturing Engineering at the

Pennsylvania State University at University Park in State College, Pennsylvania.

a concept exploration method for product family design

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