Managerial DecisionModeling 6e

Cliff RagsdaleVirginia Polytechnic Institute

and State University

In memory of thosewho were killed and injured

in the noble pursuit of educationhere at Virginia Tech on April 16, 2007

SOUTH-WESTERNCENGAGE Learning-

A u s t r a l i a • B r a z i l • J a p a n • K o r e a • M e x i c o • S i n g a p o r e • S p a i n • U n i t e d K i n g d o m • U n i t e d S t a t e s

Ill

Introduction to Modeling and Decision Analysis 1Introduction 1The Modeling Approach to Decision Making 3Characteristics and Benefits of Modeling 3Mathematical Models 4Categories of Mathematical Models 6The Problem-Solving Process 7Anchoring and Framing Effects 9Good Decisions vs. Good Outcomes 11Summary 12References 12The World of Management Science 12Questions and Problems 14

Introduction to Optimization and Linear Programming 17Introduction 17Applications of Mathematical Optimization 17Characteristics of Optimization Problems 18Expressing Optimization Problems Mathematically 19Decisions 19 Constraints 19 Objective 20

Mathematical Programming Techniques 20An Example LP Problem 21Formulating LP Models 21Steps in Formulating an LP Model 21

Summary of the LP Model for the Example Problem 23The General Form of an LP Model 23Solving LP Problems: An Intuitive Approach 24Solving LP Problems: A Graphical Approach 25Plotting the First Constraint 26 Plotting the Second Constraint 26 Plotting the ThirdConstraint 27 The Feasible Region 28 Plotting the Objective Function 29 Finding theOptimal Solution Using Level Curves 30 Finding the Optimal Solution by Enumeratingthe Corner Points 32 Summary of Graphical Solution to LP Problems 32Understanding How Things Change 33Special Conditions in LP Models 34Alternate Optimal Solutions 34 Redundant Constraints 35 Unbounded Solutions 37Infeasibility 38

Summary 39

Contents xi

References 39Questions and Problems 39

3. Modeling and Solving LP Problems in a Spreadsheet 46Introduction 46Spreadsheet Solvers 46Solving LP Problems in a Spreadsheet 47The Steps in Implementing an LP Model in a Spreadsheet 47A Spreadsheet Model for the Blue Ridge Hot Tubs Problem 49Organizing the Data 50 Representing the Decision Variables 50 Representing theObjective Function 50 Representing the Constraints 51 Representing the Boundson the Decision Variables 51

How Solver Views the Model 52Using Risk Solver Platform 54Defining the Objective Cell 55 Defining the Variable Cells 56 Defining theConstraint Cells 57 Defining the Nonnegativity Conditions 60 Reviewing theModel 61 .Other Options 62 Solving the Problem 63Using Excel's Built-in Solver 64Goals and Guidelines for Spreadsheet Design 64Make vs. Buy Decisions 67Defining the Decision Variables 68 Defining the Objective Function 68 Defining theConstraints 68 Implementing the Model 69 Solving the Problem 70 Analyzing theSolution 71

An Investment Problem 72Defining the Decision Variables 73' Defining the Objective Function 73 Defining theConstraints 73 Implementing the Model 74 Solving the Problem 75 Analyzing theSolution 76

A Transportation Problem 76Defining the Decision Variables 77 Defining the Objective Function 78 Defining theConstraints 78 Implementing the Model 79 Heuristic Solution for the Model 80Solving the Problem 81 Analyzing the Solution 82

A Blending Problem 83Defining the Decision Variables 83 Defining the Objective Function 83 Defining theConstraints 84 Some Observations About Constraints, Reporting, and Scaling 84Re-scaling the Model 85 Implementing the Model 86 Solving the Problem 87Analyzing the Solution 87

A Production and Inventory Planning Problem 89Defining the Decision Variables 90 Defining the Objective Function 90 Defining theConstraints 90 Implementing the Model 91 Solving the Problem 93 Analyzing theSolution 94

A Multiperiod Cash Flow Problem 95Defining the Decision Variables 96 Defining the Objective Function 96 Defining theConstraints 96 Implementing the Model 98 Solving the Problem 100 Analyzing theSolution 101 Modifying The Taco-Viva Problem to Account for Risk (Optional) 102Implementing the Risk Constraints 104 Solving the Problem 105 Analyzing theSolution 105

xii Contents

Data Envelopment Analysis 106Defining the Decision Variables 107 Defining the Objective 107 Defining the Constraints 107Implementing the Model 108 Solving the Problem 110 Analyzing the Solution 113

Summary 114References 115The World of Management Science 116Questions and Problems 116

4. Sensitivity Analysis and the Simplex Method 136

Introduction 136The Purpose of Sensitivity Analysis 136Approaches to Sensitivity Analysis 137An Example Problem 137The Answer Report 138The Sensitivity Report 140Changes in the Objective Function Coefficients 140 A Note About Constancy 142Alternate Optimal Solutions 143 Changes in the RHS Values 143 Shadow Prices forNonbinding Constraints 144 A Note About Shadow Prices 144 Shadow Prices and theValue of Additional Resources 146 Other Uses of Shadow Prices 146 The Meaning ofthe Reduced Costs 147 Analyzing Changes in Constraint Coefficients 149 SimultaneousChanges in Objective Function Coefficients 150 A Warning About Degeneracy 151

The Limits Report 151Ad Hoc Sensitivity Analysis 152Creating Spider Tables and Plots 152 Creating a Solver Table 156 Comments 159

Robust Optimization 159The Simplex Method 162Creating Equality Constraints Using Slack Variables 163 Basic Feasible Solutions 163Finding the Best Solution 166Summary 166

References 166The World of Management Science 167Questions and Problems 168

5. Network Modeling 178Introduction 178The Transshipment Problem 178Characteristics of Network Flow Problems 178 The Decision Variables forNetwork Flow Problems 180 The Objective Function for Network Flow Problems 180The Constraints for Network Flow Problems 181 Implementing the Model in aSpreadsheet 182 Analyzing the Solution 183

The Shortest Path Problem 184An LP Model for the Example Problem 186 The Spreadsheet Model and Solution 187Network Flow Models and Integer Solutions 187

The Equipment Replacement Problem 190The Spreadsheet Model and Solution 190

Contents xiii

Transportation/Assignment Problems 193Generalized Network Flow Problems 194Formulating an LP Model for the Recycling Problem 196 Implementing the Model 197Analyzing the Solution 198 Generalized Network Flow Problems and Feasibility 199

Maximal Flow Problems 202An Example of a Maximal Flow Problem 202 The Spreadsheet Model and Solution 204

Special Modeling Considerations 206Minimal Spanning Tree Problems 209An Algorithm for the Minimal Spanning Tree Problem 210 Solving the Example Problem 210

Summary 212References 212The World of Management Science 212Questions and Problems 213

6. Integer Linear Programming 230

Introduction 230Integrality Conditions 230Relaxation 231Solving the Relaxed Problem 231Bounds 233Rounding 234Stopping Rules 237Solving ILP Problems Using Solver 237Other ILP Problems 241An Employee Scheduling Problem 241Defining the Decision Variables 242 Defining the Objective Function 242 Defining theConstraints 242 A Note About the Constraints 243 Implementing the Model 243Solving the Model 245 Analyzing the Solution 246Binary Variables 246A Capital Budgeting Problem 246Defining the Decision Variables 247 Defining the Objective Function 247 Defining theConstraints 247 Setting Up the Binary Variables 247 Implementing the Model 248Solving the Model 249 Comparing the Optimal Solution to a Heuristic Solution 249

Binary Variables and Logical Conditions 250The Fixed-Charge Problem 251Defining the Decision Variables 252 Defining the Objective Function 252 Defining theConstraints 253 Determining Values for "Big M" 254 Implementing the Model 254Solving the Model 255 Analyzing the Solution 256 A Comment on IF() Functions 257

Minimum Order/Purchase Size 258Quantity Discounts 259Formulating the Model 259 The Missing Constraints 260

A Contract Award Problem 260Formulating the Model: The Objective Function and Transportation Constraints 261Implementing the Transportation Constraints 262 Formulating the Model: The SideConstraints 263 Implementing the Side Constraints 264 Solving the Model 266Analyzing the Solution 266

xiv Contents

The Branch-and-Bound Algorithm (Optional) 266Branching 268 Bounding 269 Branching Again 270 Bounding Again 272Summary of B&B Example 272

Summary 274References 274The World of Management Science 274Questions and Problems 275

7. Goal Programming and Multiple Objective Optimization 292

Introduction 292Goal Programming 292A Goal Programming Example 293Defining the Decision Variables 294 Defining the Goals 294 Defining the GoalConstraints 294 Defining the Hard Constraints 295 GP Objective Functions 296Defining the Objective 297 Implementing the Model 298 Solving the Model 299Analyzing the Solution 299 Revising the Model 300 Trade-offs: The Nature of GP 301

Comments about Goal Programming 303Multiple Objective Optimization 303An MOLP Example 305Defining the Decision Variables 305 Defining the Objectives 306 Defining theConstraints 306 Implementing the Model 306 Determining Target Values for theObjectives 307 Summarizing the Target Solutions 309 Determining a GP Objective 310The MINIMAX Objective 312 Implementing the Revised Model 313 Solving theModel 314

Comments on MOLP 316Summary 317References 317The World of Management Science 317Questions and Problems 318

8. Nonlinear Programming and Evolutionary Optimization 331Introduction 331The Nature of NLP Problems 331Solution Strategies for NLP Problems 333Local vs. Global Optimal Solutions 334Economic Order Quantity Models 336Implementing the Model 339 Solving the Model 339 Analyzing the Solution 341Comments on the EOQ Model 341Location Problems 342Defining the Decision Variables 343 Defining the Objective 343 Defining theConstraints 344 Implementing the Model 344 Solving the Model and Analyzing theSolution 345 Another Solution to the Problem 346 Some Comments About the Solutionto Location Problems 346

Contents xv

Nonlinear Network Flow Problem 347Defining the Decision Variables 348 Defining the Objective 348 Defining theConstraints 349 Implementing the Model 349 Solving the Model and Analyzing theSolution 352

Project Selection Problems 352Defining the Decision Variables 353 Defining the Objective Function 353 Defining theConstraints 354 Implementing the Model 354 Solving the Model 356

Optimizing Existing Financial Spreadsheet Models 356Implementing the Model 358 Optimizing the Spreadsheet Model 359 Analyzing theSolution 360 Comments on Optimizing Existing Spreadsheets 360

The Portfolio Selection Problem 360Defining the Decision Variables 362 Defining the Objective 362 Defining theConstraints 363 Implementing the Model 363 Analyzing the Solution 365 HandlingConflicting Objectives in Portfolio Problems 367

Sensitivity Analysis 369Lagrange Multipliers 371 Reduced Gradients 372

Solver Options for Solving NLPs 372Evolutionary Algorithms 373Forming Fair Teams 375A Spreadsheet Model for the Problem 375 Solving the Model 377 Analyzing theSolution 377

The Traveling Salesperson Problem 378A Spreadsheet Model for the Problem 379 Solving the Model 381 Analyzing theSolution 381

Summary 382References 383The World of Management Science 383Questions and Problems 384

9. Regression Analysis 400Introduction 400An Example 400Regression Models 402Simple Linear Regression Analysis 403Defining "Best Fit" 404Solving the Problem Using Solver 405Solving the Problem Using the Regression Tool 407Evaluating the Fit 409The R2 Statistic 411Making Predictions 413The Standard Error 413 Prediction Intervals for New Values of Y 414Confidence Intervals for Mean Values of Y 416 Extrapolation 416

Statistical Tests for Population Parameters 417Analysis of Variance 417 Assumptions for the Statistical Tests 418

xvi Contents

Introduction to Multiple Regression 420A Multiple Regression Example 422

Selecting the Model 423Models with One Independent Variable 423 Models with Two IndependentVariables 424 Inflating R2 427 The Adjusted-R2 Statistic 427 The Best Modelwith Two Independent Variables 428 Multicollinearity 428 The Model with ThreeIndependent Variables 428

Making Predictions 430Binary Independent Variables 430Statistical Tests for the Population Parameters 431Polynomial Regression 432Expressing Nonlinear Relationships Using Linear Models 433 Summary of NonlinearRegression 436

Summary 437References 438The World of Management Science 438Questions and Problems 439

10. Discriminant Analysis 448Introduction 448The Two-Group DA Problem 449Group Locations and Centroids 449 Calculating Discriminant Scores 451The Classification Rule 454Refining the Cut-off Value 455 Classification Accuracy 456 Classifying NewEmployees 457

The /(-Group DA Problem 459Multiple Discriminant Analysis 460 Distance Measures 462 MDA Classification 463

Summary 466

References 467

The World of Management Science 467Questions and Problems 468

11. Time Series Forecasting 473Introduction 473

Time Series Methods 474

Measuring Accuracy 474Stationary Models 475

Moving Averages 476Forecasting with the Moving Average Model 478

Weighted Moving Averages 480Forecasting with the Weighted Moving Average Model 481

Exponential Smoothing 482Forecasting with the Exponential Smoothing Model 484

Contents xvii

Seasonality 486Stationary Data with Additive Seasonal Effects 487Forecasting with the Model 490

Stationary Data with Multiplicative Seasonal Effects 491Forecasting with the Model 494

Trend Models 495An Example 495

Double Moving Average 495Forecasting with the Model 498

Double Exponential Smoothing (Holt's Method) 499Forecasting with Holt's Method 502

Holt-Winter's Method for Additive Seasonal Effects 502Forecasting with Holt-Winter's Additive Method 506

Holt-Winter's Method for Multiplicative Seasonal Effects 506Forecasting with Holt-Winter's Multiplicative Method 510

Modeling Time Series Trends Using Regression 510Linear Trend Model 510Forecasting with the Linear Trend Model 512

Quadratic Trend Model 513Forecasting with the Quadratic Trend Model 515

Modeling Seasonality with Regression Models 516

Adjusting Trend Predictions with Seasonal Indices 516Computing Seasonal Indices 516 Forecasting with Seasonal Indices 519 Refining theSeasonal Indices 519

Seasonal Regression Models -522The Seasonal Model 522 Forecasting with the Seasonal Regression Model 525

Combining Forecasts 525Summary 526References 526The World of Management Science 527Questions and Problems 527

12. Introduction to Simulation Using Risk Solver Platform 537Introduction 537Random Variables and Risk 537Why Analyze Risk? 538Methods of Risk Analysis 538Best-Case/Worst-Case Analysis 539 What-If Analysis 540 Simulation 540

A Corporate Health Insurance Example 541A Critique of the Base Case Model 543

Spreadsheet Simulation Using Risk Solver Platform 543Starting Risk Solver Platform 544

xviii Contents

Random Number Generators 544Discrete vs. Continuous Random Variables 546

Preparing the Model for Simulation 547Alternate RNG Entry 550

Running the Simulation 551Selecting the Output Cells to Track 551 Selecting the Number of Replications 552Selecting What Gets Displayed on the Worksheet 554 Running The Simulation 554

Data Analysis 555The Best Case and the Worst Case 555 Viewing the Distribution of the Output Cells 556Viewing the Cumulative Distribution of the Output Cells 557 Obtaining OtherCumulative Probabilities 558 Sensitivity Analysis 559

The Uncertainty of Sampling 559Constructing a Confidence Interval for the True Population Mean 560 Constructing aConfidence Interval for a Population Proportion 561 Sample Sizes and ConfidenceInterval Widths 562

Interactive Simulation 562The Benefits of Simulation 564Additional Uses of Simulation 565A Reservation Management Example 565Implementing the Model 566 Details for Multiple Simulations 567 Running theSimulations 569 Data Analysis 569

An Inventory Control Example 570Creating the RNGs 572 Implementing the Model 573 Replicating the Model 575Optimizing the Model 577 Analyzing the Solution 583 Other Measures of Risk 585

A Project Selection Example 586A Spreadsheet Model 587 Solving and Analyzing the Problem with RiskSolver Platform 588 Considering Another Solution 590

A Portfolio Optimization Example 591A Spreadsheet Model 592 Solving the Problem with Risk Solver Platform 594

Summary 597References 597The World of Management Science 598Questions and Problems 598

13. Queuing Theory 613

Introduction 613The Purpose of Queuing Models 613Queuing System Configurations 614Characteristics of Queuing Systems 615Arrival Rate 616 Service Rate 617

Kendall Notation 619Queuing Models 619The M/M/s Model 621An Example 622 The Current Situation 622 Adding a Server 623Economic Analysis 624

Contents xix

The M/M/s Model with Finite Queue Length 624The Current Situation 625 Adding a Server 626

The M/M/s Model with Finite Population 626An Example 627 The Current Situation 628 Adding Servers 629

The M/G/1 Model 630The Current Situation 631 Adding the Automated Dispensing Device 632

The M/D/1 Model 634Simulating Queues and the Steady-state Assumption 634Summary 635References 635The World of Management Science 636Questions and Problems 637

14. Decision Analysis 644

Introduction 644Good Decisions vs. Good Outcomes 644Characteristics of Decision Problems 645An Example 645The Payoff Matrix 646Decision Alternatives 646 States of Nature 647 The Payoff Values 647

Decision Rules 648Nonprobabilistic Methods 648The Maximax Decision Rule 649 The Maximin Decision Rule 650 The Minimax RegretDecision Rule 650

Probabilistic Methods 652Expected Monetary Value 653 Expected Regret 654 Sensitivity Analysis 655

The Expected Value of Perfect Information 657Decision Trees 659Rolling Back a Decision Tree 660

Creating Decision Trees with Risk Solver Platform 661Adding Event Nodes 662 Determining the Payoffs and EMVs 666 . Other Features 666

Multistage Decision Problems 667A Multistage Decision Tree 668 Developing A Risk Profile 669

Sensitivity Analysis 670Tornado Charts 671 Strategy Tables 674 Strategy Charts 676

Using Sample Information in Decision Making 678Conditional Probabilities 679 The Expected Value of Sample Information 680

Computing Conditional Probabilities 681Bayes's Theorem 683

Utility Theory 684Utility Functions 684 Constructing Utility Functions 685 Using Utilities to MakeDecisions 688 The Exponential Utility Function 688 Incorporating Utilities inDecision Trees 689

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Multicriteria Decision Making 690The Multicriteria Scoring Model 691The Analytic Hierarchy Process 695Pairwise Comparisons 695 Normalizing the Comparisons 696 Consistency 697Obtaining Scores for the Remaining Criteria 699 Obtaining Criterion Weights 700Implementing the Scoring Model 700

Summary 701References 701The World of Management Science 702Questions and Problems 703

15. Project Management (Online) 15-1

Introduction 15-1An Example 15-1Creating the Project Network 15-2Start and Finish Points 15-4

CPM: An Overview 15-5The Forward Pass 15-6The Backward Pass 15-8Determining the Critical Path 15-10A Note on Slack 15-11

Project Management Using Spreadsheets 15-12Important Implementation Issue 15-16

Gantt Charts 15-16Project Crashing 15-18An LP Approach to Crashing 15-19 Determining the Earliest Crash CompletionTime 15-20 Implementing the Model 15-22 Solving the Model 15-23 Determining aLeast Costly Crash Schedule 15-24 Crashing as an MOLP 15-25

PERT: An Overview 15-26The Problems with PERT 15-27 Implications 15-29

Simulating Project Networks 15-29An Example 15-30 Generating Random Activity Times 15-30 Implementingthe Model 15-31 Running the Simulation 15-32 Analyzing the Results 15-34

Microsoft Project 15-35Summary 15-37References \5-37The World of Management Science 15-38Questions and Problems 15-38

Cases 715

Index 777