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Design For Variety to Achieve Responsive Design Customization
Based on the Modeling of Design Flexibility
Jilin Zhang
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
Department of Mechanical and Industrial Engineering University of Toronto
S Copyright by Jilin Zhang 2001
Y O W M VornrlYnrier
dur li* N#ndiYlrnci
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Design For Variety to Achieve Responsive Design Customization
Based on the Modeling of Design Flexibility
Jilin Zhang Degree of Master of Appüed Science, 2001
Department of Mechanical and Industrial Engineering University of Toronto
ABSTRACT
Design For Variety (DFV) is an engineering approach in general to accommodate
rapid, unpredictable market changes in an engineered product by means of rapid product
customization. Treated as a systematic approach, it is developed in this thesis for
achieving responsive design customization by generating a family of designs scaled fiom
a commonly shared design platform.
In this methodology, model-based design platfoms are formalized and formulated
fiom a core design to create a set of custom designs. Besides, this thesis also contributes
on three aspects including (1) a fomal approach for design flexibility modeling in the
absence/presence of uncertainty, (2) a suite of DFV models and their computational
approaches for obtaining various design platfoms, and (3) a systematic approach for
achieving responsive design customization.
Finally, a simple yet illustrative design example is utilized to test the validity and
utility of the methodology developed, dong with the discussions of the design results.
1 would like to thank my supmriaor Dr. Li Chen for his full support on my thesis
work. Without his patience, encouragement, guidance, this work would not have been
successfbl . I woutd also like to express my appreciation to my colleagues in Design and
Manufactunng Integration Laboratory for their helps, most especially to Simon Li for his
assistance on using the Matlab software and preparing thesis presentation.
Additionally, my acknowledgments extend to the Department of Mechanical and
Industrial Engineering at the University of Toronto for the financial support through
University Open Fellowship. Also, this work was partially supported by Material &
Manufachiring Ontario ( M M ) through the Enabling Program (DE # 61 g!).
This thesis is dedicated to my wife, Yanfeng Xu. The completion of the thesis
would not have been possible without her understanding and endless support.
iii
TABLE-OF CONTENTS
. . ................................................................................. ABSTRACT ..il
............................................................... ACKNOWLEDGMENTS iii
................................................................... TABLE OF CONTENTS iv
........................................................................... LIST OF TABLES vii ... ......................................................................... LIST OF FIGURES viil
...................................................... NOMENCLATURE AND ACRONYM ix
CHAPTER 2
CHAPTER 3
................................................... INTRODUCTION i
............................................. Motivation and Background 1 ......................................................... Literature Review 2
.................................... 1.2.1 Robust Engineering Design 3 ........................ 1.2.2 Enginee~g Design Ushg Fuzzy Sets 5
.......................................... 1.2.3 Product Family Design 8 ............................................. Objective and ûrganization 10
............................. ROBUST DESIGN MODELING 14
.............................................................. Introduction 14 ....................................... Pararnetric Design Frarnework 14 .......................................... 2.2.1 Design Formalization 14
.......................................... 2.2.2 Design Formulation 17 ................................ Uncertainty Modeling and Treatment 21
.......................................... 2.3.1 Source of Uncertainty 21 ....................................... 2.3.2 Propagation of Variation 23
............................................. Robust Design Formulation 24 ................................. 2.4.1 Design Feasibility Robustness 25 ................................ 2.4.2 Design Sensitivity Robustness 26
....................................... 2.4.3 Total Design Robustness 27
.................... DESIGN FLEXIBILITY MODELING 29
............................................................... Introduction 29 ................................... Fiexibility Modeling and Treatment 30
.............................. 3.2.1 Flexibility in Design Parameten 31 ...................... 3.2.1.1 Parameter Induced Flexibility 31
................. 3.2.1.2 Parameter Transmitted Flexibility 33
CHAPTER 4
CHAPTER 5
CHAPTER 6
....................... 3.2.2 Flexibility in Functional Requirememts 39 ............... Incorporation of Flexibi1ii)i hto Design Fomiulation 40
.......................... 3.3.1 Design Parameter Rooted Flexibility 41 .................. 3.3.2 Functional Requirernent Rwted Flexibility 46
DESIGN FLEXIBILITY MODELING UNDER ....................................................... UNCETGINTY 49
................................................................ Introduction 49 ..................................................... Sequential Approach -50
.......................... 4.2.1 Design Parameter Rooted Flexibility 51 4.2.2 FunctionalRequirernentRootedFlexibility .................. 55 4.2.3 Incorporation of Flexibility into Uncertainty-present Design
....................................................... Formulation 56 .............. 4.2.3.1 Design Parameter Rooted Flexibility - 5 7
4.2.3.2 Functional Requirement Rooted Flexibility ....... 60 .................................................. Sirnultamous Approach 61
.......................... 4.3.1 Design Parameter Rooted Flexibility 62 .................. 4.3.2 Functional Repuirement Rooted Flexibility 68
4.3.3 Incorporation of Flexibility into Uncertainty-present Design ...................................................... Formulation 70
............... 4.3.3.1 Design Parameter Rooted Flexibility 70 4.3.3.2 Functional Requirement Rooted Flexibility ....... 75
Cornparison Between Sequentid and Simultaneous Approaches .. 76
...................................... DESIGN FOR VARIETY -79
................................................................ Introduction 79 ............................................... Design For Variety (DFV) 80
................................ 5.2.1 DFV with Preference Coupling 81 ............................. 5.2.2 DFV with Prefemce Uncouplhg 83
........................ Design For Variety (DFV) under Uncertainty 83 ............................................. 5 .3.1 Sequential Approach 84
..................... 5.3.1.1 DFV with Prefemice Coupling 84 .................. 5.3.1.2 D N with Preference Uncoupling 85
......................................... 5.3.2 Simultaneous Approach 86 ..................... 5.3.2.1 D N with Reference Couplhg 87
.................. 5.3.2.2 DFV with Preference Uncoupling 88 ..................................... Responsive Design Customization 89
................................. 5.4.1 Model-based Design Platform 89 ............................... 5.4.2 Platform-based Custom Design -92
......................................................... CASE STUDY 99
................................................................ Introduction 99 ...................................................... Problern Statement -99
............................................................ 6.3 Scalable Design 1 O0 -2:- a2 . . - 2 - ............... 6.3.1 Results and Discussion of ~ e s i g n For Variety 101
6.3.2 Resuits and Discussion of Design For Variety under ........................................................ Uncertainty 103
................................. . 6.3 .2 i Sequential Appmach 104 .............................. 6.3.2.2 Sirnultanmus Approach 106
............................................................ 6.4 Custom Design 108
.................... CHAPTER 7 RETROSPECT AND CONCLUSIONS i l 9
......................................................... 7.1 Thesis Overview 119 .............................................. 7.2 Surnmary of Contributions 121
................................... 7.3 Recomrnendations and Future Work 121
.................................................................. LIST OF REFERENCES 124
APPENDIX A FUNDAMENTALS OF FUZZY SET THEORY ........... 133
............................................................... Introduction. 133 ............................................ Fuzzy Membership Function 134
.................................................. Basic Fuzzy Operations 135 ......................................................... Fuzzy Probability 136 . . . ....................................................... Fuzzy Optirninition 137
APPENDIX B FORMULATIONS A . MODELS IN WELDED BEAM .................................................................. DESIGN 142
.................................................... 1 Basic Formulation 142 ........................... 2 Design Fomulations in Different Cases 143
- - LIST OF TABLES
Table 2.1 Value of k vs. desired probability
Table 4.1 Cornparison of sequential approach and simultaneous approach
Table 5.1 A Guideline to Selection of Desip Platforms
Table 6.1 Data of welded beam design
Table 6.2 Results of ordinary design opthization and total robust design
Table 6.3 Results of Case 1 based on Design Platfonn "DPI"
Table 6.4 Results of Case 2 based on Design Platfonn "DP2" (with the fixed L)
Table 6.5 Design results of Case 3 based on Design Platform "DP3"
Table 6.6 Results of Case 1 based on Design Platform "DPUCI"
Table 6.7 Results of Case 2 based on Design Plat fonn "DPUCZ" (with the fixed L)
Table 6.8 Results of Case 3 based on Design Platfonn YûPUC3"
Table 6.9 Threshold Values for Design Constraints
Table 6.10 Results of Case 1 based on Desip Platform "DPUAI"
Table 6.1 1 Results of Case 2 based on Design Platfonn "DPUA2" (with the fixed L)
Table 6.12 Results of Case 3 baseâ on Design Platform "DPUA3"
Table 6.13 Custom design parameten based on Design Platform "DPUAI "
Table 6.14 Custom design parameters based on Design Platfonn "DPUA2"
Table 6.15 Custom hctional requirements Uased on Design Platfom "DPUA3"
vii
LIST OF- FIGURES
Figure 3.1 Flexibility characteristic hct ion of design parameter
Figure 3.2 Flexibility characteristic fiuiction of a wish design attribute
Figure 3.3 (a) Violation of the i" must design attribute
Figure 3.3 (b) Flexibility characteristic function of i" must design attribute
Figure 3.4 Flexibility characteristic function of flexible functional requirement
Figure 4.1 Flexibility characteristic hct ion of an uncertainty-present wish attribute
Figure 4.2 Flexibility characteristic function of m uncertainty-present must attribute
Figure 4.3 Flexibility characteristic function of the i h ranged functional
Figure 4.4 incorporation of uncertainty into flexibility modeling for a wish attribute
Figure 4.5 Incorporation of uncertainty into flexibility modeling for the i' must attribute
Figure 4.6 Incorporation of uncertainty into flexibility modeling for the ranged functional
requirement of the i' must attribute
Figure 5.1 Design procedure flowchart for the case of preference uncoupling subject to
given custom design panuneters
Figure 5.2 Design procedure flowchart for the case of preference coupling subject to
given custom design parameters
Figure 5.3 Design procedure flowchart subject to &en custom functional requirements
Figure 6.1. A welded beam
scalar weighting factor of wish attribute
bounding value
general design attribute
design target
constant associated with probabi lity
vector of design parameters
low bound of flexible design panuneter
upper bound of flexible design parameter
the most desirable design parameter
the custom value specified on a design parameter
general scalar weighting factor associateà with the in objective fùnction
vector of design variables
low bound of the k" design variable
upper bound of the k* design variable
the k* design variable
objective function
objective function with uncertainty
design constraint
design constraint with uncertainty
spreads of flexible hmctional nquirement
transmitted variation W. r. t. wish attribute
spread of transmitted flexibility W. r. t. wish attnbute at level of preference, a
spread of design flexibility W. r. t. wish attnbute corresponding to a = O
----= a- - AF?) spread of transmitted flexibility W. r. & an uncertainty-present wish attribute at
level of preference, a d~(') spread of design flexibility W. r. t. an uncertainty-present wish attribute
corresponding to a = O
4g (*) transmitted variation W. r. t. must attribute
4gjU) spread of transmitted flexibility W. r. t. the iH must attribute at the level of
preference, a
4 spread of design flexibility W. r. t. the i lh must atttibute corresponding to
a=O
AG,'") spread of transmitted flexibility W. r. t. the i' uncertainty-present must
attribute at the level of preference, a
AG,!') spread of design flexibility W. r. t. the i' uncertainty-present must attribute
corresponding to n = O
A P ~ to Ierancing uncertainty of the h " design parameter
4'"' flexibility of a design parameter at the level of design preference, a
4 tolerancing uncertainty of a design variable
E [*] expectation of its entity
Pr { a ) probability of its entity
Var [-] design variation caused by design uncertainty
a aggregated degree of design preference
E an infinitesimal number
9 (9 probability distribution h c t i o n of its entity
K a scalar parameter prescribed by designer
Â. overall preference
4 3 expected preference
A- best overall design preference
--- .. -- - Â, threshold to feasible vgiues of the if* must attribute
&in minimum overall design p re fmce aggregated
p, ( 0 ) flexibility characteristic h c t i o n W. r. t. wish attribute
pa(*) flexibility characteristic function W. r. t. must attribute
ph ( 0 ) fl exibility characteristic function W. r. t. design parameter
pF ( ) flexibility characteristic fwiction W. r. t. wish attribute with uncertainty
RA*) flexibility characteristic fimction W. r. t. must attribute with uncertainty
standard deviation of its entity
DFV design for variety
FR huictional requirement
DP design platforni
DPUC design platform with uncertainty in conservative approach
DPUA design platform with uncertainty in aggressive approach
Introduction
1.1 Motivation and Background
Nowadays the demands for engineered products shift rapidly in the market. To
compete globally and stay cornpetitive, manufacturers must react to the frequent,
unpredictable market changes arising from, for example, introduction of new product
models and technologies, fluctuations in customer needs, changes in parts for existing
products, changes in govenunental regulations (safety and environment), and changes in
process technology. This requires a new engineering approach for market-driven, rapid
product customization in a responsive manner.
One way of treating rapid product customization is to expedite the design
customization process, i.e. responsive design customization, by shortening the cycle of
design computing. This strategy lies in the following observation. A computational
design process can be divided into two phases: (1) design selection and (2) design
automation. ùi Phase 1, design entities such as design parameters and fùnctional
requirements are selected manually. In Phase 2, a single or multiple design configurations
are generated aufoniaficutfy on a computer where the decision-making is governed by, for
exarnple, an optirnization algorithm. However, according to psychological study, hurnan
cannot make consistent decisions when sirnultaneoudy considenng more than 7k2 factors.
This implies that a designer, in Phase 1, cunnot select properly design parameters simply
by mind as the total number of design parameters exceeds a certain limit, thereby
necessary to resort to the aid of computer. Furthennore, due to the over-the-wall nature of
the process, there exists a gap in the transition of two design phases between the manual
design selection and the automutic design generation, impeding the enhancement of
design computing speed. To overcome human's limitation and also to bridge the gap
mentioned above, it is intended to provide a linkage by merging Phase 1 and Phase 2 into = -
one unified phase in the spirit of concurrent design. As a result, design selection would be
computenzed, thmugh the modeling of design flexibility, with incorporation of design
automation into one unified phase in design computing, so as to achieve concurrent
design.
On the other hand, it is hoped to formalize and formulate a model-based platfom
for scalable desip on which a farnily of design variants could be generated yet
customized by scaling those flexible design parameters (or huictional requirements).
which consequent ly contributes to supporthg responsive custom design. To this end,
Design For Variety (DFV) approaches to achieve responsive design customization will be
explored and exarnined. DFV is an engineering approach in general to accommodate
rapid, unpredictable market changes in an engineered product by means of rapid product
customization. In this thesis, DFV is treated as a systematic approach for achieving
responsive design customization by generating a family of designs scaled from a
commonly shared design platfom based on engineering models. Hercin, the terni, design
platfom, refers to a central base of engineering models on which a set of desip variants
cm be derived and configureci h m a core design in a cost-effective1 y and time-efficient
fashion. It is therefore the aim of D W to formalize and fortnulate a model-based design
platforni based on which a set of custom designs with different scales can be created from
a core design with ease and efficiency.
In the above context, this thesis presents the methodology of DFV for achieving
responsive design customization through the modeling of design flexibility. The presence
of flexibility in design makes it possible to sale a design uù hoc to achieve a variety of
custorn designs with an enhanced efficiency. Two sources of design flexibility will be
taken into account, and therefore, exarnined and justified. The first one is concemed with
flexible design parameters, and the second one with /lerible fûnctional requirements.
Accorcüngly, appropriate modeling approaches will be suggested to manipulate the
flexibility of design rooted in the above two sources. As well, the propagation of
flexibility in design will be addresseci and theu induced effects on design will be revealed.
Robust engineering design deals with tolerancing uncertainty while fuzzy set
theory helps handle irnprecision in engineering design. They both provide a basis for the
methodology development in this thesis, and therefore, will be reviewed. On the other
hand, since product farnily design is a target application of this study, and thus will be
reviewed as well in this following section.
1.2.1 Robust Engineering Design
Robust engineering design is an approach by which a designer c m synthesize a
product that is less sensitive to engineering uncertainties in design variables and
parameters. It ensures the product to achieve high quaiity with low variability in the
expected performance. Traditionally, the designer is accustomed to prescribing closer
tolerance or choosing a higher factor of safkty to make a design robust. Unfortunately,
this attempt may fail to obtain high robustness in product performance because of
inappropriateness in treating the uncertainties. That is, design variations are reduced b y
passively imposing on tighter tolerance rather thm controlling the effects of variations on
the resulting performance characteristic. Now, the use of robust design schemes enables
to minimize variations in design regardless of the presence of exterior disturbances and
noise.
Taguchi (1987) first advocated the concept of "design-in-quality" through his
"O ff-line quality control" philosophy. He promoted Taguchi method in a systematic way,
by which the optimal values of an economical design with low variability cm be found.
In his appmach, ternis such as "signal-to-noise ( S N ) ratio" are coined to account for
different design case scenarios in the presence of extemal disturbances and noise. He
fiuther classified the design factors as two categories: control factors that affect primarily
the SM ratios but not the mean, and signal factors that contribute mainly to the mean
response of the performance characteristic. The application of Taguchi method follows a
two-step procedure as the template for design: reduce variations in perfomance by
maximizing the SN, then adjust the mean of performance on the desireci target (Roy,
1990).
Many researchers have dedicated to either improvhg Taguchi method or - - + - -- -
extending its application scope. Otto and Antonsson (1993) extended Taguchi method to
allow for formulating possibility and necessity requirements in product design. Chang et
al. (1994) presentbd a robust decision-making procedure based on the framework of
Taguchi method, enabling to accommodate "conceptual noise" resulting fiom
inconsistent judgrnents made by different individuals in a design team. Bras and Mistree
(1 994) used an optimization mode1 evolving from goal programming, called compromise
Decision Support Pmblem (DSP), for robust design by integrating Taguchi robust
philosophy with axiomatic design principles in a unified fashion. Chen et al. (1 9%)
introduced a variant of Taguchi methoâ to robust design through combining the Response
Surface Methodology (RSM) and the compromise Decision Support Problem (DSP),
allowing to resolve robust design problems with no closed-fom solutions. Ku et al. (1998)
elaborated a Taguchi-based design approach for handling unconstrained, constrained, and
multiobjective optimization problems, by which the optimal design generated is
inherently robust. Gold and Knshnamurty (1997) addressed tradeoffs in robust design
under multiple design criteria, in which Taguchi orthogonal arrays were used as a
template to lay out utility analysis. Gadallah and Elnarghy (1993) employed the Taguchi
pararnetric design procedure for designing a c m valve system, subject to three types of
geometric tolerances, based on dynamic simulation and analysis.
Although Taguchi method was initiated h m design experimentation, it has been
shown that Taguchi-orienteci design problems c m be expressed and fonnulated
equivalently by appropnate optimization models. As such, robust product design has
been investigated also by the methods of optimization. Sandgren et al. (1985) discussed
robust design for a class of welded structures based on the framework of optimization to
minirnize design sensitivity transmitted by uncontrollable parameters. Parkinson and his
associates have examined in depth "robust optimal design", as can be found in Parkinson
et al. (1993). Emch and Parkinson (1994). Lewis and Parkinson (1 994), and Parkinson
(1995). In their approaches, the notion of transmitted variation was introduced to account
for fluctuations in design constraint hct ions resulting h m variations in design
variables and parameters, based on which feasibility mbustness can be achieved for
design. Sundaresan et al. (1993) presented an optimization pmedure for robust design, in
which "sensitivity index" was used to define the metric of robustness to characterize - -
variations in design variables and constraints. Ramakrishnan and Rao (1996) proposed an
optimizatioa-based robust design method in which the mbust design problems were
fonnalized in terrns of a general loss hction.
In addition to the above efforts, Chen (1999) suggested a coordination-based
robust design approach to achieve "desiping-in-quality" in engineered product design,
by which both variability and functionality in design c m be guided during the design
iterations. This work shows how to control the computational design process for high-
robustness product performance. An adaptive robust design approach to capture and
handle evolutionary statistical uncertainties in the design fomulation by Bayesian
statistics was also developed (Chen et al., 1999). This method enables the designer to
adjust (or update) the numenc of empirical design parameters in use according to the
design progression. Recently, Chen and Ramani (2000) introduced the notion of
conceptual robustness to ensure design quality in the early-stage process development,
when subjected to conceptual disturbances in design. Accordingly, a conceptual design
methodology was proposed fonnally for achieving conceptual robustness through the
pwsuit of both conceptual performance precision and conceptual performance accuracy
in design.
While the robust design rnethods are used to tackle design problems with
tolerancing uncertainty, this thesis argues that its design strategy can also be extended to
handle design problems with consideration of design flexibility. In this study, extending
robust design strategy to treat design flexibility will be of concem.
1.2.2 Engineering Design Using Fuzzy Sets
Since introduced to academia by Zadeh (1963) firstly, the fuzzy sets theory has
grown rapidly because many researchers have dedicated their efforts to developing
various facets of it. For example, p t contributions can be found in the papers written
by Dubois and Rade (1980), Kaufhann and Cupta (1991), and KLir and Yuan (1995).
As dehed in Klir and Yuan (1995), hizy set is a set with boudaries that is not
precise, and the membership in fuuy set is not a matter of afEnnation or denial, but
rather a matter of a degree. It is u s d ----- 2 c- - - . -
hgmentary, not Mly reliable, vague,
boundaries and cannot be well defined in
to describe the events that
contradictory, deficient or
the way of absolute truth or
6
are incomplete,
have no clear
falseness. These
events are defined as fuuy set because they can not be grouped as members that certain1 y
belong in a set or nonmembers that certainly does not belong in a set. They are not
necessarily either true or fdse, but they may be true or false to some degree, the degree to
which they are members of the set. Comrnonly, the membenhip in a fuuy set can be
expressed as degrees of tnith of the associated propositions by numben in the closed
unite interval [O, 11. The extreme values in this interval, O and 1, then respectively
represent the total denial and affirmation of the membership in a &en fuzzy set as well
as the falsity and tmth of associated proposition. On the other hand, the cnsp set is a
special case of hizzy set, while the degrees of membership in a fuzzy set are O and 1.
The study of engineering design using fuzzy sets emerged in the late 80s and
rapidly grew in 90s. Wood and Antonsson (1989,1990) explored the application of fuzzy
mathematics for semi-automated mechanical design, and incorporated design impression
into preliminary design computing. They also discussed the differentiation between Fuuy
and stochastic uncertainties appearing in the design process (Wood et al., 1990). Based
on the above works, they M e r developed "Method of Irnprecision" for representing and
manipulating fuzzy uncertainties in engineering design, and then compared it with other
techniques such as utility theory (Otto and Antonsson, 19936 (Law and Antonsson,
1994). They also extended this method to incorporate the manufachinng aspect of design
parameters through tuning parameters (Otto and Antonsson, 1993b). A summary of their
work was presented in Antonsson and Otto (1995).
In the meantirne, Rao (1987% 1987b) introduced the notion of hiuiness to the
area of mechanical and structural design, by which the computational methods were
developed for implementation of fuzzy design optimizatian. Since then, Rao and his
group have addressed the general issues of design optimization in the presence of fuvy
uncertainties to general engineering systern, as reported in, for example, Dhingra et al
(1990), Dhingra et al (1992), Dhingra and Rao (1991, 1995), Rao and Chen (1 W6), etc.
In their approaches, it is argued that practical design problems often embody not only the
feature of incompleteness, but also the aspects of uncertainty, inexactness or vagueness.
As such, the design objectives may w t able to be described pncisely because the utility - -
functions may not definable hgmentarily or the phenornena of the design problem may
only be stated in an ambiguous way. As well, the design constraints may not be able to be
stated with certainty because doubt may arise about the exactness of permissible
bounding values or the comtness of judgments on the constraints.
Other contributions to this research subject can also be found in the literature.
Thurson and Carnahan (1992) proposed a procedure to evaluate multiple fuzzy attributes
in the prelirninary design stage using utility analysis. Lucas et al (1994) extended the
construct of compromise Decision Support Pmblem to enable the design of a hierarchical
engineering system involving information. In Chen et al. (1996a), linguistic
variables have been successfully applied to facilitate the description and selection of
initial process conditions for cases in which subjective uncertainties are present, by which
they developed a modeling and design approach to guide early-stage polymer process
development (Chen et al., 1996b). Recently, based on these efforts, Chen and Ramani
(2000) elaborated a formal, subjective design framework for conceptual design of
polymeric processes with multiple parameters. in addition, Chen et al. (1998) presented a
f u y kinetostatic approach to analyze the dynamic behavior of high-speed planar
mechanisms when involving hiuiness with design parameters.
In fuzzy finite element analysis, the representative works include the following.
Shimuzu and Hiroaki (1993) used hizzy sets as a basis to automatically generate the finite
element mesh. The fuzzy set theory was utiüzed in their method to mathematically mode1
the hurnan thought process. Rao and Sawyer (1995) proposed a fuzzy finite element
rnethod (FEM) for static analysis of enpeering systems using an optimization-based
scheme for the numencai solution of systems of fuvy linear equations. The hizzy
treatment of system parameten, goometry and applied loads was considered and
Unplemented in their approach. Chen and Rao (1997) exarnined the fuuy finite element
method with application to vibration analysis of imprecisely defined engineering systems
in which imprecision d vagueness are present in system parameters, which can be
applied to the early stages of the design process. Later on, they further developed a
numerical technique for solving a general system of fuuy linear equations in engineering
anaiysis with fuzy input parameters (Rao and Chen, 1998).
a -- While fuzzy set theory has been widely apptied to deal with imprecision in
engineering design, little effort has been devoted to an engineering problem involving
both tolerancing uncertainty and design hprecision. Otto and Antonsson ( 1 994)
formalized to define a '%estw of set design parameters for design problerns under multiple
forms of uncertainty. Shih and Wang-sawidjaja (1996) established a mixed &y-
probabilistic programming approach for multiobjective optimization with random
variables. Ragheb and Tsoukalas (1988) developed a coupled probability-possibility
method for monitoring performance of devices in the presence of randomness and
fuuiness. Ayyub and Lai (1996) discussed various types of uncertainty present in
structural systems, also presented uncertainty measues in reliability assessment of ship
structures. Rao et al. (1998) explorad a unified scheme that enables conventional finite
element analysis to combine both stochastic and fuzzy foms of uncertainty to the design
process. Recently, Chen (2000) provided a comprehensive discussion with its approaches
for treating a variety of uncertainties in decision-based design, in which hybrid-
uncertainty based modeling and design approaches were addressed.
in this thesis, fuvy set theory will be used as a mathematical bais for design
flexibility modeling. In particular, the definition of membership hc t ion will be extended
to the detemination of flexibility characteristic huiction.
1 J.3 Product Family Design
Product family design (PFD) is an emerging research area being developed
undeway. Many issues in this area are eithet still open or unexplored as yet. The aim of
PFD is to achieve rapid customization of market-driven products subject to wpredictable
market demands. The research efforts in this subject can be classified into a number of
groups according to motivation, représentation/methodology and design implernentation.
The motivation behind much of this research is the need for companies to be able
to provide increased product variety to the customer without increased complexity for the
manufacturer. Gonzalez-Zugasti et al. (1999) provided a deeper assessment of the value
of product farnily design by considering the benefits, cost, investment, and the
uncertainty associated with each family design.
Current design literature includes a variety of design representations and
methodologies involving the development of product families. Siddique and Rosen
(1999) identified two key problems that are most often addressed. The first is concemed
with platform commonization, which is aimed at finding a suitable common platform for
a given product family. The second is concerned with platform supported product variety,
which involves the development of a product farnily h m given a platfom. The use of
platform-based product architecture was explored by Umeda et al. (1999) as a way to
provide upgradability to a product over time. Here the various products in a family are
not simply related products but rather the successive generations models of a product.
The approach descnbed, used a Function-Behaviour-State model. Zarnirowski and Otto
(1999) introduced function and variety heuristics to help identify potential common
platform components. Martin and Ishii (1997, 2000) extended their previous work on
commonality meûics to detail two indices that help designers identify a platform and
complete architecture that requires less design effort for follow-up products. Fujita et al.
(1999) described general design issues involvecl in commonization in an effort to move
toward a computational methodology.
There are also a number of model-based approaches to the representation and
design of different kinds of product platforni-based architectures. Finch (1999) used set
based models, while Siddique and Rosen (1999) used a graph grammar approach. In this
approach, graphs are used to represent the con huiction and structures, Le., platform, and
grammars to descnbe the relationships between the platform and the possible variants.
Siddique and Rosen (2000) used a hyperarc-based assembly representation to aid in the
àevelopment of a product family h m a &en platfonn using configuration reasoning.
For design implementation, Simpson et al. (1999) used a Decision Support
Pmblem formulation for designing a family of products based on scalable platforms.
They introduced the Product Platform Concept and Exploration Method, which form the
basis for the work by Messac et al. (2000) who included a physical programrning aspect,
and Nayak et al. (2000) who described a variation-based methodology. McAdam et al.
(1998) and Stone et al. (1999) described a quantitative hctional model, which allows
designers to capture product functionality and customer need information during the
development of a product fmi l y architecture.
In spite of various approaches taken by the researchers, still no cornprehensive - - .
theory or methodology exists that-iould provide the guidance to a designer for PFD.
Furthemore, the curreat focus has been placed only on the need for a fixed integrated
platform. Improved modularity throughout the design, with an assurance that cross
product modules do not inteifere with one another, provides a more flexible design. It
should be noted that Dahmus et al. (2000) and Gonzalez-Zugasti and Otto (2000),
described an approach for the benefits of achieving a flexible platform. The approach,
however, relies heavily on single product design techniques and fails to consider the
system level (i.e., product farnily) interactions during the design process. To this end.
most recently, Chen et al. (2001) proposeâ a modularization-based reconfiguration design
method for PFD based on a flexible product architecture platform.
In this thesis, it is argued that there may exist an alternate approach for achieving
the airn of PFD for rapid product customization, that is. the methodology of DFV. In
particular, DFV is an engineering approach to accommodate rapid, lmpredictable market
changes in an engineered product through quick design customization to the products.
This argument lies in the fact that, to expedite product customization, the efficiency of
design computing and production operations should be both enhanced. To date, most
emphasis has been given to the production aspect of enhancement, while the design
aspect of enhancement, especially in view of design computing, has been overlooked. It
is therefore the focus of this thesis to develop a methodology for rapid product
customization through the enhancement of design computing by DFV.
13 Objective and Organization
As described earlier, the traditional design computing process is characterized as
two phases: (1) design selection and (2) design automation. In this context, Phase 1 is
concemed with the munual selection of design entities (e.g. design parameters and
bctional requirements), serving as an input to launch design computing. Thereafler,
design computing is executed in Phase 2 where the design decision on a feasible
configuration(s) is made automaticulij through, for example, an optimization algorithm
d n g on a computer. ln general, however, there exists human's limitation and also a
gap between the above two phases, as described in Section 1.1. It is therefore important
to provide a linkage for bridging the gap between these two design phases by integrating
them Ulto a unified one with a view to concurrent design. More importantly, to meet the
needs of respondve design customization, the aim of this thesis is to develop the
methodology of DFV in two aspects:
1. an approach for modeling design flexibility to account for the fieedom space of
design selection through computerization, and
2. an approach for achieving responsive design customization through the exploration of
DFV upon the availability of jkible design space.
In particular, research efforts will be devoted to
B computerizing the design selection process so that the choice of a design parameter(s)
and a functional requirement(s) cm be descnbed computationally and manipulated in
a cornputer-aided manner;
P characterizing the fieedom space of design selection through design fiexibility
modeling so that the relationship betweenflexibility and preference can be conveyed
in the context of design computing;
P incorporating design flexibility into design fomulation to attain scaleable design
through the strategy of DFV so that a linkage h m the modeling of design flexibility
to the obtaining of design variety can be bridged;
P formalizing a model-based design platform for achieving scaluble design so that a
farnily of design variants fiom a core design can be generated by scaling those
flexible design parameters or hc t iond requirements; and
B exploring DFV approaches to achieve responsive design custornization so that custom
design can be carrieci out with ease and efficiency on the fonnalized design platforni
to cope with typical application scenarios in design customization.
Centered on the above goals, the remainder of this thesis is organized according to
the following arrangements:
- "
Chapter 2 - describes parametric design framework and robust design modeling. First,
the h ~ e w o r k of fomalizing a design problem is introduced based on which design
modeling is described in the context of optimization fornalism. Next, uncertainty
modeling and treatment in paramerric design is discussed with coverage of the source
of uncertainty and the propagation of variation. Last, robust design modeling is
addressed, and three robust design models are presented, which are derived nom an
underlying optimization model.
B Chapter 3 - discusses approaches for design flexibility modeling as well as design
formulations with incorporation of design flexibility. First, the notion of flexibility in
the context of parametric design is defined, and then design fiexibility modeling is
addressed through the introduction of flexibility characteristic fùnction. Discussion is
thus divided into two streams: (1) flexibility in design parameters and (2) flexibility
in functional requirements. Accordingly, design fomulations that incorporate design
fiexibility are denved rnathematically, dong with their computational models for
implementation.
P Chapter 4 - addresses desip flexibility modeling in the presence of uncertainty and
appmaches to formulate their computational design models. Attention is given to the
discussion of two appmaches, sequential and simultaneow. The former treats design
flexibility and uncertainty in a sequential fashion, while the latter handles them both
simultaneoudy. As a result, the cornsponding desip models to account for different
design scenarios are derived, fonnulated and justifiecl, along with discussions of their
computational implementation.
B Chapter 5 - presents the DFV approaches and their computational models. These
appmaches support the transition that transforms design flexibility to ultimately
achieve design variety. The computational models provide a basis for fomalizing a
model-based design platfonn not only to attain scalable design but also to facilitate
custom design. As a result, a model-based design platfom is obtained so that a set of
design variants (or derivatives) h m a core design can be created efficiently. A
systematic approach for achieving responsive design customization is presented to
charactenze typicaî design customization scenarios in practice.
. - L-- B L - Chapter 6 - applies the methodology - - developed - - - and their resulting design models for
a simple design example. In particular, a welded-beam design problem is investigated
with consideration of both flexibiüty and uncertainty. Beyond illustration, the use of
this design study is also for validating the methodology developed and testing
applicability and utility of the design models derived. Various application scenmios
are discussed against each design model.
P Chapter 7 - s u ~ n i z e s this thesis and comments on fùture work. In particular,
research contributions will be highüghted, and the thesis will be reviewed and
concluded dong with recommendations and discussions of fùture work.
Robust Design Modeling
2.1 Introduction
As mentioned previously, robust design is a process by which a designer can
synthesize a product that is less sensitive to design uncertainty. Uncertainty in design is
due to physical limitations such as manufachuing tolerances associated with a
manufacturing process for a part / product under design. In dealing with the uncertainty,
robust design is used to achieve high quality but low variability in the expected
performance through the design. n i e robust design methodology has been well
investigated and well documentecl in the archival literature.
in this study, the robust design strategy is used and extended to deal with design
flexibility, which will be discussed later on. It is therefore necessary to address robust
design modeling in the context of pararneûic design. Especially, topics will cover design
feasibility robustness, design sensitivity robustness, and total design robustness.
In the remaining sections of this chapter, parametric design h e w o r k , including
design fomialization and formulation, will be introduced first. Then, discussion will be
extended to uncertainty modeling and treatment in engineering design, under which the
source of uncertainty will be discussed and propagation of variations will be descnbed.
Finally, robust design formulation will be presented and elaborated.
2.2 Parametric Design Framework
2.2.1 Design Formalization
In engineering design, a parametric design problem c m be formalized by
elaborating five necessary design entities.
- . , &A--.- - - The h t entity is concemecl with design components, which can be furthet broken
down into: design variables and design parameters. Their definitions can be given by
Definition 1 (Design Componmt)
Design component is a factor t h nee& to be confgwed in a design problern.
B Design variable is defined as a desijp component that will be detennined during the
design process. A set of design variables foms the vector of design variables, denoted by
P Design parameter is defined as a design component that is pre-determined pnor to the
execution of a design process. A set of design parameten foms the vector cf design
parameters, denoted by
The second entity is concemeâ with attributes, which c m be m e r classified into
wish attributes and must attributes.
Deflnition 2 (Design Attribute)
Design attribute refers to any property or characteristic d a design, denoted as Ai .
B Wisk amibute is defined as a design attribute whose hctional requirement (FR)
(desired) would be achieved whenever possible, thereby allowing for compromise in
design.
Musi atîribute is defined as a design attribute whose FR (specified) must be achieved
under al1 circurnstances, thereby not allowing for any compromise in design.
In the definition above, the FR cornsponds to the third entity, which is defined
bellow ,
Definition 3 (Functional Requirement) -----a---- E
FR refers to the composition of a design attribute, a permissible value or range of values
associated with the attribute.
The fourth one is concerned with design functions, whose definition is given as
follows.
Definition 4 (Design Function)
Design fknction refers to a finctional mapping thut relates a state of the design
components to a design uttribute, c.g. f,. (X, P) A,.
It is important to note that a design hc t ion can be of any form such as a closed-
form equation or a set of heuristic rules that serves as a bridge for reasoning fiom ( X , P)
to a design attribute, Say A,. . A design fùnction characterizes one huiciional behavior of
an artifact.
The last entity is concemed with design availability associated with the choice of
design variability.
Definition 5 (Design Availability)
Design availabiiity refers to un admissible spuce in which each value is physicaliy
available to the choice of a design variable.
It is noteworthy that design availability is not the same as the notion of a feasible
design space. The latter lies in the space formed with consideration of the behavioral
bounds of design fhctions, in addition to the physical limits of design variables. By
contrast, design availubili@ delimits the space resulting firom the physical limits to a
suitable choice of design variables only.
To formalize a parametric design problern, the above design entities must be first
determineci and then incorporated into design formulation through appropriate modeling.
in this study, optimization forrnalism is useâ as a modeling basis for the derivation of - -
design formulations, which will be fwther discussed in the following section.
2.2.2 Design Formulation
Optimization modeling approaches are employed here to fornulate the parametric
design problems formalized using the five design entities above. in such handling, the
wish design attribute is characterized by modeüng it as an objective hnction, and the
mwt design attribute is ûeated as a design constraint though a constraintfùnction. Each
constraint function in tum also characterizes the expression of a specijied FR.
h general, there exist three application scenarios for treating a wish attribute
fùnction, denoted w (X, P) . They are:
1) Smaller-the-Better: i.e., the smaller the value of w ( X , P) is, the better the design
will be, as can be modeled in the fom of
2) Larger-the-Better: i.e., the larger the value of w (X, P) is, the better the design
will be, as can be modeled in the form of
max w ( X , P ) (2.2a)
which can be fùrther converted into the form of
min -w(X,P) (2.2b)
3) Center-the-Better: i.e., the closs the value of w(X, P) is to its desired target
( A), the better the design will be, as can be modeled according to
min [w(X,P)-A]'
- - -
Since the expressions above al1 involve the "min" operation, a simplifjmg formulation
below is thus applied to uni@ the rnodeling of wish attribute though "minimization", Le.
min f(X,P) (2.4)
where f ( X , P) represents a generai objective fiction for characterization of a wish
design attribute; X and P are the vectors of design variable and parameter, respectively.
To M e r differentiate the must design attnbute in different types of FR, three
modeling cases are taken into account, i.e. at-rnost, at-least and no-bias, as expressed
below respectively:
m j ( X , P ) < bj
m j ( X , P ) > bj
m ( X , P ) = bj
where mj (a) signifies the j" must attribute hction; bj denotes the bounding value
prescnbed for the j" must attribute. Note that m j and bj both contribute to constituting
the jrh FR. According to optimization formalism, Eqs. (2.5) and (2.6) belong to the class
of inequality constraint, and Eq. (2.7) the class of equality constraint.
For simplicity, an inequality constraint function, denoted g,(e), is used to
characterize and uni@ Eqs. (2.5) through (2.7) using
The above treatment lies in the fact that Eqs. (2.6) and (2.7) both can be transfomed into
the fom of inequality constraint through
where 8 is an infinitesimal number for computation convergence checking.
There are alternate ways of integrating wish and must attributes to formulate a
design model, depending on what design intent will be. To summarize, two typical design
cases are often encountmd in practice. The first case is concemed with "one-wish many-
must", in which a design problern is fomalized by a single wish attribute and multiple
m u t attributes plus other four design entities. The second case is concerned with "many-
wish many-must", in which the fomalization of a design problem involves both multiple
wish and multiple must attributes in addition to other four design entities.
To account for the above design cases, optimization theory is used as an
underlying mathematical means to formulate the design models. Optimization formalism
has been well developed and is applied to facilitate computerization of design modeling
and to support automation of design computing. In particular, single-objective
optimization formalism is used to formulate "one-wish many-must" design paradigrn and
multisbjective optimization fonnalism to formulate "many-wish many-must" design
paradi grn .
"One-wish maymust" formalism
Find X
min f (X Pl
s.t. gi (X, P) bj i=1,2 ,...... m
where X is a q -dimensional vector of design variables; P is an h -dimensional vector
of design panuneters; bi denotes a bounding value; xi andx; are lower and upper
bounds of the design variables, respectively. In the above formulation, Eqs. (2.1 1) and
(2.12) characterize the FRs associated with wish and must attributes by a corresponding
objective function and an appropriate set of constraint huictions. Eq. (2.1 3) c haracterizes ----A- ---- . - -
design availability pertaining to the design variables by their side constraints. Any
available mathematical programming software can be employed for the above problem
solving.
Find X
min {fi (x9 PX f2 P)9*b-*0*9 f N ( X 9 (2.14)
set. gi (X, P) 1 bi i=1 ,2 ,...... m (2.15)
1 x, 4 x, 4 x: k = 1,2 ,......, q (2.16)
where N is the total number of objective functions to be minimized. ii is important to
notice that f 's may be mutually conflicting and numerically non-cornmensurate. In this
formulation, each objective fùnction represents a wish attribute, and each constraint
fùnction indicates a must attribute. The overall design goal is to seek a solution that best
compromises al1 objective functions. There are many tradeoff approaches in the literature
for finding the best compromise design h m the Pareto solution set. Nevertheless, a
common solution strategy is to convert the multi-objective optimization problem into an
appropriate single-objective optimization problem. For example, the weighted-surn
method formulates the multiple objectives into a single objective function according to
with
where wi is a scalar weighting factor associated with the ilh objective huiction. Thus,
the resulting design problem tums out to be a single minimization problem, i.e. min
F ( X , P). In this methocl, the wish design attributes are conveyed by the objective
hmctions, and are aggregated through "additive operation". The weighting factors are -L-- -- A- L - --
used to refiect the importance of each wish attribut;. By this beatment, any commercially
available single-obj ective optimization solver can be applied for the remaining prob lem
solving.
What needs to be pointed out is that the above design models do not account for
uncertainty present in a design problem. Furthemore, since a "many-wish" design
problem can be transformecl somehow into an equivalent "one-wish" problem, the design
fomulation given in Eqs. (2.1 1) through (2.13) will be used as an underlying mode1 for
the exploration of new design models in this study. Any result from "one-wish" design
problem is extensible to treat "many-wish" design problem.
2.3 Uncertainty Modeling and Treatment
It is the aim of this section to incorporate the probabilistic fom of uncertainty into
the design fomulatioa(s) presented earlia on. Herein, design uncertainty refers to
tolerancing variations in design due to the presence of uncontrollable disturbances and
noise (e.g. environmental or operational conditions) in the associated manufac turing
approach. Upon design uncertainty, the actual design may deviate from its desired
optimum, leading to either expensive products in production or defective products in
service.
in this section, the source of uncertainty will be discussed, and propagation of
variations in design will be examineci.
2.3.1 Source of Uncertainty
To physically realize a design configuration, an appropnate manufactunng means
must be selected eventually. However, any physical process has limitation(s) to certain
extent on its capabilities, incurring quality variations in the products to be produced. To
convey the capability concem of a manufaturing process, tolerancing information must
be specified in the context of design and manufacturing.
It has been revealed that the tolerance induced variation is inherently probabilistic - - - - - - - -
(or stochastic), which is considerad as the major source of uncertainty affecting design
performance. Due to the association of manufacturing tolerances with design components
(variabledparameters). uncertainty inevitably exists in the context of design even though
it is undesirable. It would even cause the cirift of functional performance away from its
optimal design, which will be further addressecl later on.
In parameüic design, tolemcing information is usually specified on design
variables and design parameters (Le. design components). The size of each tolerance
associateâ with a design component depends on the manufacturing approach selected. In
view of manufactunng cost, a tighta design tolerance requires a more expensive
manufacturing approach insofar as a better quality production system (or line) is
concerned. In order to manipulate the tolerancing uncertainty in parametric design, a
quantitative description is necessary through the application of probability theory. As
such, the probabilistic nature of uncertainty due to tolerancing can be characterized and
quantifieâ.
Given tolerances, Ax, and 4,, associated with the kn design variable and the
h" design parameter. Here are mathematical expressions held as follows:
which relate each tolerancing size (Le. dr, and 4,) to its corresponding standard
deviation (i.e. a,, op*) through the choice of K . DifZerent choices on the value of K
signify different levels of desired probability that detemines to what extent the design
should be feasible in view of likelihood of chance. Table 2 (see below) lists typical values
of K with respect to the comsponding levels of desired probability. The expressions
above, Eqs. (2.18) and (2.19), also contribute to characterization and quantification of the
underlying (or mot) variations in design, resulhing h m the tolerancing uncertainty of
- --. A A- - = d-qign componen@. Note that t ~ , q d O,, designate the standard deviations for the kth
design variable and the hd design puameter, respectively.
Table 2.1. Value of K vs. desired probability
Value of K
1
2
3
4
2.3.2 Propagation of Variation
Desired Robability
%
84.13
97.725
99.865
99,9968
The behavior of a design is characterized by design attributes through their design
hctions, each being composed of desip components. Thus, the underlying variations
due to tolerances embodded in the design components will propagate through the design
fùnctions to cause functional variations (or performance cirift) in design attributes. In this
circurnstance, the solution of nominal design obtained by ordinary optimization may not
be optimally feasible any more. In more detail, wish attribute (or objective function) may
deviate fiom its optimal point, while the must attributes (or constraint Functions) may
shifi their values to violate their design constraints. To handle the above problem, it is
necessary to have an eirective appmach(es) so that the fùnctional level of design
variations can be estimateci quantitatively and efficiently.
There are two altemate approximation approaches able to quanti@ the propagated
design variations at the functional (or attribute) level (Chen and Ramani, 2000), i.e.
Aggressive Design Approach
and
Conservative Design Approach
where 4 and 4, denote the huictional variations, caused by tolerancing uncertainties
of design variables (dr, ) and desip parameters ( dp,), with respect to the wish attribute
(objective huiction) and the must attributes (constraint fùnctions). in principle,
derivations of the above fomulae are grounded on the expansion of Taylor senes in
which the lth -order linear approximation is assumed (see Chen and Ramani, 2000;
Bjorke, 1989; Cox, 1986).
In particular, the aggressive approach is denved using "partial denvative rule" in
the same way as to denve the fiinctional level of standard deviation (see Rao, 1996;
Chen, 2000). In the conservative approach, the use of "absolute" operation eliminates the
negative signs possibly present in the expressions. This treatment assumes that al1 design
variations occur simuhaneously m the worst possible combinations of the design
components. By contrast, the aggressive approach hypothesizes that desig variations
occur randomfy to the design components following a certain probabilistic distribution.
Mathematically, it c m be proved that (4). d (4), and (dg,). 5 (dgj ) =, implying that
the use of Eqs. (2.22) and (2.23) are, by cornparison, conservative to the use of Eqs.
(2.20) and (2.21).
2.4 Robust Design Formulation
The effect of tolerancing uncertainty on a design is Uuee-fold. (1) It may cause an - - - - - -
optimal (feasible) design to be infeeasiIe in view of a rnust design attribute (or constraint
hction). (2) It may lead to the dniff of an optimized design away from its target
performance level in view of a wish design attribute (or objective hction). (3) The
above cases may both occur at the same time. To attenuate the effect of design variations,
robust design modeling is the best means to control design variations by incorporating
them into the context of desip formulations. In accordance with the above three-fold
effect, three robust design moâeling approaches for achieving "designing-in-quality" will
be addressed in this section, Le., desip feasibility robustness, design sensitivity
robustness, and total design robustness.
As discussed in Section 2.2.2, most parametric design problems c m be somehow
generalized and treated as a single-objective optimization problem. In this regard, the
formulation of single-objective optimization given in Eqs. (2.1 1) to (2.1 3), w i 11 be
referenceâ as an underlying model for the derivation of robust design models. The
resulting design models can be viewed as an extension (or variant) to the underlying
design model.
2.4.1 Design Feasibility Robustness
A feasible design refers to the one that satisfies al1 design constraints (or meet al1
FRs) under consideration. However, in the presence of tolerancing uncertainties, design
variation in a must attribute (or constraint function) may cause an optimal (feasible)
desip to be inferniMe, meaning that one or some design constraints wouM be vioiated.
To account for this situation, the notion of design feasibility robustness is introduced so
that a jkasible yet optimal design is still attainable even when subjected to design
variations. This solution concept is r e f e d to as robust optimal design (Parkinson et al,
1993).
To safeguard a design against undesirable disturbances, design formulation should
account for the handling of design variation by which the optimal design generated can
sustain the side-effect due to design variations. Under this thnist, a abust design model
for achieving feasibility robustness is given below:
in the above formulation, the probabilistic form of tolerancing uncertainty has been
incorporatecl into the design constraints, Le. Eqs. (2.25) and (2.26), where 4, is the
prescribed level of probability that quantifies how reliable the design constraint(s) is to
sustain the side-effect of design variations. For design computing, Eqs. (2.25) and (2.26)
are M e r sirnplified according to the following approximations.
where 4, denotes the functional level of variation in a must design attribute, whose
arnount can be estimated using Eq. (2.21) or Eq. (2.23). Note that, for the convenience of
later discussions, the constraint huictions (Le. must attributes) are classified into (1) those
having design parameters (P) in their expressions (i.e. g,(X,P)); and (2) those not
ha* âesign parameters ( P ) in their expressions (Le. g,(X) ).
Design feasibility robustness contributes to design controllability for achieving
desired performance accuracy in design, as commented in Chen (1999) and Chen and
Ramani (2000). This robust design scheme mures the obtaining of a feasible yet optimal
design regardless of the existence of design variations. However, the original feasible
design space is shnuik with the amount of 4, as a result of design variations (see Eqs.
(2.28) and (2.29)).
2.4.2 Design Sensitivity Robustness
- - -
When exposed to design variations, an optimized design may shifr away fiom its
optimal point even though the design may still remain feasible. The drift of target
petformance level in an optimized design may lead to a much worse design as compared
to its originally optimized one. To this end, the notion of design sensitivity robustness is
explored in such a way that the amount of design variation in the target performance
characteristic is minimized while maintaining the design to be feasible, even though
subject to design tolerances.
To reduce the amount of such variation to the ieast extent, a different robust
design mode1 should be explored by which the loss of target performance huiction (i.e.
wish attribute) can be dictated toward its minimum. The design formulation for achieving
sensitivity robustness is presented below:
Find X
min dfw, p) (2.30)
s.t. Pr[g,(X.P) s bJ 2 i = 1,2, .....+, m (2.3 1)
Wg1 (XI bj I 2 P' j = 1,2 ,... ..., n (2.32)
where 4 denotes the variation of (target) performance characteristic in the wish design
attribute, whose quantity can be estimated using Eq. (2.20) or Eq. (2.22).
Design sensitivity robustness contributes to design stability for achieving desired
performance precision in design, as commented in Chen (1999) and Chen and Ramani
(2000). This robust design scheme warrants the resulting design performance to be least
sensitive to the side-effect of design variations. It is important to note that the original
feasible design space is shrunk again with the amount of 4,. in order to dictate the
generation of feasible designs when undergoing variations in design.
2.43 Total Design Robustness
. - -----:A- - To achieve "desip-inquality" by mbwt design, both feasibility robustness and - - - - - - -
sensitivity robustness should be taken into account in the design formulation. Therefore,
their individual models should be brought together to fom a uni fied robust design model.
The resulting model is refend to as "total design robustness" (Chen, 2000), which is
formulated as follows:
where a and b both are scalar parameters used to make the values of f and Af
cornmensurate.
In this formulation, the unified wish attribute consists of f and 4. The use of
f ( X , P ) is intended to guarantee an optimal design to converge with feasibility
robustness. Meanwhile, the inclusion of 4(X,P) is purporteci to ensure the resulting
(optimal) design as stable as possible with sensitivity robustness. Furthemore, each pair
of settings of (a$), selected by the designer, corresponds to a specific trade off
settlement (or agreement) between f ( X , P) and qf (X, P) . Compared with the design
models discussed in the last two sections, the solution obtained using this design mode1
represents a more stable but less feasible design as compared to the one obtained solely
by design feasibility robustness, or a less stable yet more feasible design as compared to
the one obtained solely by design sensitivity robustness.
Design Flexibility Modeling
3.1 Introduction
Robust design discussed in the last chapter is a systematic approach by which a
designer can synthesize the design lay out to minimize the design variations while
maximizing the design performance. Although it is used to deal with only the design
problems subjected to tolerancing uncertainties, the design philosophy itself can be used
as a strategy with extension to tackle the design problems subjected to design flexibility.
This chapter will discuss the way of handlùig design flexibility based on robust design
strategy.
In this thesis, design flexibility refm ta the M o m capacity of a selection
insofar as the nominal value of a design parameter or the bounding value of a design
constraint (or FR) is concemed. Metrically, design flexibility is quantified as a physical
range within which any point represents a feasible choice (with preference rating) and
therefore, cm be flexibfy selected against one's preference. For example, in material
selection, a designer may consider the feasible choice of Young's modulus to be around
2ûû GPa, or between 199 GPa and 201 GPa. In this case, the physical range from the b w
limit to the upper limit indicates the choice of fieedom on selecting the value of Yong's
modulus and hence, may be specified as the flexibility in design that characterizes the
fieedom capacity in the selection of Young's modulus. This example shows only the
design parameter induced flexibility, which will be fûrther discussed in the remainder of
this chapter. The provision of flexibility in the early stages of design is useful because it
allows the designer to obtain design altematives with more choices of M o m .
In general, design flexibility hinges closely on designeris preference. To be more
precise, the spread of flexibility in design is inversely proportional to the degree of
preference - - - - in design selection. For example, - if - the designer exact& knows about his
choice, thea a spec$c choice will be made on a design. Since the designer has made only
one choice on the design, that choice must nflect his favorite or preference over others.
In this case, the room of flexibility is zen, while the degree of preference is unity. On the
other hand, if the designer doesn't show his particular preference over several design
alternatives, then each alternative could be selected jierxibly. That is to Say, the room of
flexibility is large while the degree of prefmnce tends to be low. In this circurnstance, no
choice is particularly favorite. To conclude, the larger the room (or spread) of design
flexibility. the lower the degree of design preference, and vice versa.
In Section 3.2, flexibility modeling and treatment will be discussed, the notion of
flexibility characteristic function will be introduced and explored. Section 3.3 presents
the design formulations that incorporate design flexibility according to three design
scenarios: flexible wish attribute, non-flexible wish attribute, and flexible FR. These
explorations will be used as a basis for the development of DFV, which will be
elaborated in the following chapters. It is important to note that discussions in the
remaining chapters are al1 centered on "one-wish many-must" design paradigm for its
generality and extensibility in design applications, as justified in Chapter 2.
3.2 Fiexibility Modeling and Treatment
In parametric design stage, designers usually need to select the proper values of
design parameters and/or bounding values of design constraints (FRs) before the
execution of design computing (for fidmg féasible design sohtions). As discussed in
Section 1.1, the computational design process may be divided into two phases. In Phase
1, the value of a design parameter (or FR) is selected rnanualiy according to designer's
judgrnent (or pnor experience) based on, for example, the existing design information.
Ofientimes, a trial-and-emr approach is used to resolve some doubt and imprecision
involved in such design selection. However, even for the same design parameter (or FR),
different designers may have different decision choices to account for different design
intents (or desires). It is evident that the degree of preference varies with the level of
fîexibility inverse-proportionally h f a r as design selection is concemed. That is, the
more preference given to a vaiue for a design panuneter, the l a s flexibility for selection - - -- - - - -
of the other values for the parameta. The provision of flexibility modeling in the early
design stages is so necessary that it enables designers with roorn to adjust a design
without resorting to any change of the design platfonn when incorporating other issues
(e.g., manufacturability) in the latter design stages. Flexibility modeling makes it possible
to combine Phase 1 with Phase 2 (Le. design automation). As such, design concurrency
can be enhanced in the way that decisions on design parameters and design variables are
both selected with the aid of computer. In this pmcess, the pararnetnc configuration of a
design is generated on a computer where the decision making is governed auiomatically
by an optirnization aigorithm. Nevertheless, the niche of design automation requires the
development of computerized design models, which is the focus of this chapter.
To develop the computerized design models, flexibility modeling and treatment is
the key which will be emphasized in this section. First, discussion will be given to two
design cases: flexibility in design parameters and flexibility in FRs. Next, design
flexibility charactenstic Auictions will be developed based on the notion and form of a
fuzzy membership function, the lata design being overviewed in Appendix A.
3.2.1 Flexibility in Design Parameters
Discussions herein are divided into two parts: parameter induced flexibility and
parameter transmitted flexibility. The former is concemed with the flexibility rooted in
the numeric selection for a design parameter. The later is to deal with the flexibility in a
design a(tribute that is iianrimitted due to flexible design panmeters through a design
function. In either case, an appropriate fom of flexibility characteristic function is
de fined to facili tate flexibility modeling.
3.2.1.1 Parameter Induceâ Flexibiiity
As discussed earlier, parameter induced flexibility is mted in Phase 1 of a
computational design pmcess through design selection. In such selection, designers may
select design parameters in a certain range subjectively according to their preferences, so
as to accommodate theu desire, instinct or experience. Essentially, there is a relationship - -
between designer' prefaence and flexibility of selection, which cm be charactenzed
mathematically by a hction, temed "flexibility characteristic fûnction".
Consider a flexible design parameter whose value falls in the range, [ pu 1, to
designate the span (or m m ) of flexibility. Each point within this range represents a
feasible choice on the patametric value(s), which in turn is associated with preference
meûics assigned over the interval [0,1]. The degree of design preference at the most
desirabie point, p,, is assigned "unity"; the degree of preference at two boundary points,
pl and p u , are assigneci "zeros", implying that any point beyond the boundary points is
not preferred at all. The points locating between the two boundary ones are assigned with
preference metrics h m O to 1 appropriately. To characterize this relationship, a
flexibility characteristic fhction with a linear variation is defined mathematically as
analogous to the form of hizzy membenhip fwiction (see Appendix A):
where p , (p ) is called "flexibility characteristic huiction" for the hfh flexible design
parameter ( h = 1,2, ......, r ) that is defined against preference metrics over the interval
[0,1]; p l , pu and p, correspond to the low bound, upper bound and the most desirable
point of the hlh flexible desip parameter. Figure 3.1 illustrates a graphical representation
of the flexibility characteristic hc t ion for a flexible desip parameter, where 4'")
indicates a room of flexibility at the level of design preference, a.
4 ( a ) i i I <-.---..-- ..-................. ".".. >; i 1 1 spread of design flexibility
>[ I i
Figure 3.1 Flexibility characteristic function of design parameter
in general, if one shows more preference on a design, he or she has less choice of
fieedom, i.e. less room of flexibility, in the design selection. This essence is exactly
captured and conveyed through the expression of a flexibility characteristic function. In
Figure 3.1, the spread of flexibility (4) refers to afeasible range within which al1 points
can be flexibly selected pertaining to a certain degree of preferences (a). Note that the
flexibility characteristic function (see Figure 3.1) couples design selection with one's
preference. This treatment provides a mathematicai basis for relating the extent (or room)
of flexibility (4) in the choice of parametric values to one's preference (a), resulting in
4p'"'. That is to Say, the spread of flexibility delimits a feasible choice of "design
parameter" against "preference". In addition, it also provides possibility for supponing
rapid design custornization according to "preference", which will be M e r discussed
later on.
3.23.2 Parameter Transmitted Flexibility
Due to the presence of flexibility in design parameters, an attribute function
becornes flexible proviâing that it involves any flexible design parameter. The flexibility
of a design amibute is propagated due to a flexible design parameter@) through a
correspondhg design hction, i.e. attribute hction. The amount of such propagation
may be quantified through the expansion of lth- order Taylor series, which will be - L w - - - - - - -- -
addressed in this section. Discussions are divided into two parts according to the
classification of desip atûibutes: wish attribute and must attribute. As a result, two
attribute-level flexibility characteristic functions will be derived to facilitate flexibility
modeling in the context of design computing.
Wish Attribute
Consider a wish amibute function involving flenble design parameters (P), Say
f (X, P) . Based on the methods discusseâ in Chapter 2 for quantifjmg propagation of
variations, the conservative approach, which is derived from the 1st-order Taylor
expansion, will be extended to estimate the spread of transmitted flexibility in a wish
attribute. This conservative method accounts for wotst-case design analysis. where the
approximation of a linear variation is assumed.
The spread of the transmitted flexibility in a wish attribute due to the jlexible
design parameten may be determined appmximaiely with reference to Eq. (2.22):
where dfta) is the spread of transmitted flexibility of a wish design attribute at the
prescribed level of preference, a, which is approximated as a linear variation with
respect to 44@; 4:") indicates the spread of design flexibility of the h" design
parameter, which is a linear variation against the prescribed level of preference, a; X,
and 4, are the vectors of design variable and design parameter, respectively, pertaining
to their nominal values that are obtained, for example, h m a standard design
optimization (without the consideration of design flexibility).
The introduction of design flexibility leads to variations in design attribute
funaions (for both "wish" and "must"). The resulting side-effect is similar to that of
tolerancing uncertainty in robust design. In Chapter 2, toleracing uncertainty is quantified
by the span of tolerance with characterization of probabilistic metrics. By analogy, design - - .AL -- a---- " -. - L - - - -
flexibility herein is measured by the spread of flexibility with characterization of
preference ranks. In this spirit, the flexibility characteristic function of a wish attribute,
denoted p, ( X , P) , is dehed in the following fonn:
whose graphic representation is displayed in Figure 3.2 for illustration.
Figure 3.2 Flexibility characteristic function of a wish design attribute
where f,, denotes the nominal minimum of the objective function of a wish design
attribute obtained by solving an ordinary optimization problem (without considenng
flexibility in design); 4@"' signifies the spread of design flexibility of a wish design
attribute correspondhg to a = O quantifid using Eq. (3.2). It can be mathematically
proved that the linear shape of ,uf(X, P) is inherited fiun the linearity of p J p )
( h = 1,2 ,....... r ) through flexibility propagation (Le. 4'"').
The existence of transmitted flexibility ~ d f ( ~ - ) ) may deteriorate the nominal
design (f,) that is optimized without considering design flexibility. Due to the presence
of flexibility in design, a variation with the amount of 4'"' is introduced such that the
original design ( f&) may shift h m its nominal optimum to end up with a likely
unacceptable design, which is truly undesirabIe. To characterize one's desire to such a
design variation, the flexibility characteristic huiction, shown in Figure 3.2, exactly
captures and conveys the trends in the change of f,. In this figure, the degree of design
preference (a) reduces inverse-proprtionally to the spnad of the transmitted flexibility
(4'"') in design. In specific, it is wished that the design should maintain its nominal
optimum as possible as it could, thereby f, corresponding to a = 1.0 (i.e. most
desirable). When undergoing design variations, it is anticipated that the maximum
amount of flexibility induced variation to sustain the design still acceptable should be
govemed within the m m of @O), therefore f, + 4'') corresponding to a = O (i.e.
least desirable). Note that the above description targets a generalized design minimization
problem, h m which any conclusion is equally applicable (or extensible) to other types
of design problem (see Chapter 2).
Coupled with design preference, the flexibility characteristic fiuiction of a wish
attribute provides a meaningful way to bridge a gap between "flexibility" and
"preference" in design modeling. This treatment enables the design progression to be
dictated toward optimizing the objective function of a wish attribute according to design
preference.
Consider a must atüibute fiinction involving flexible design parameters (P), Say
g,(X,P) . To estimate the spread of transmitted flexibility of a must attribute, the
consemative approach for quantifymg propagation of variations in Chapter 2 is extended
in the same way as to treat "wish attribute", acconiing to
where 4:"' is the spread of transmitteà flexibility of the i" mut design attribute at the
prescribed level of preference, a, which is approximated as a linear variation with
respect to 4,. Likewise, the same argument as held in dealing with "wish athibute" is
applied here except that the spread of transmitteà flexibility of a must attribute may incur
such an infeasible design that was o*iginally feasible if without considering design
flexibility.
When incorporating "flexibility" into design, the effect of flexibility induced
variations on the feasibility of design solutions should be taken into consideration. In
particular, the flexibility induced variation in a must attribute may shifl the boundary of
the design constraint to result in a sntaller design domain even though it's still feasible. in
the worst case scenario, a f e d l e design originally selected will tum out to be infeasible
because of a shift of the design boundary and the resulting shrinkage of the feasible
design space. To account for such side-effect, robust design strategy is extended here to
facilitate the control of flexibility induced design variations. In robust design,
probabilistic metrics is used to describe tolerancing uncertainty according to the span of
tolerance specified. Similarly, in the handling of flexibility, preference rating is applied to
characterize the spread of flexibility in design.
In this context, a flexibility characteristic fùnction is accordingly introduced to
assist in handling the transmitted flexibility in a must attribute:
Figure 3.3 (a) illustrates the case in which the i" design coastraint is violated as a result
of the introduction of transmitted flexibility to the i" must attribute. Figure 3.3 @) shows
---A . a graphical representation of the flexibility characteristic hction correspondhg to the
i" must attribute. Note that the linearity of p, (X, P) is due to an inheritance fiom the
linearity associated with ph (p) ( h = 1,2, ..... .,r ), which can be proved with mathematical
rigorousness.
Figure 3.3 (a)
Violation of the i" must design attribute
Figure 3.3 (b)
Flexibility characteristic function
of in must design attnbute
As seen in Figure 3.3 (b), the spread of transmitted flexibility of a must attribute
4, varies inverse-proportionally to the degree of design preference (a). To compensate
for the flexibility induced variation (&!"'), a variant fom of design constraint is
introduced to the i" must attribute such that g, (X, P) +dg!") s b, . Since the variation
on a design constraint (i.e. Ag,) is not desired, 4, + O is most desirable in design while
Ag, + 4 j 0 ) (i.e. maximum) is least desirable in design. In this sense, Figure 3.3 (b)
exactly describes such design intent through proper assignments of preference ranking. In
two extreme locations, the degree of preference is set 'iuiity" (Le. a = 1 ) at " b, - dg:' "
conesponding to the most conservative (or least aggressive) boundary point; on the
contrary, the degree of preference is set "zero" (Le. a = O ) at " bi " corresponding to the
m m uggressive (or -La- --- -- -.
preference thus holds
3.2.2 Flexibility in
leaîr conservutive) boundary point. A gradient of degree of
for the other boundary points locating between the two extreme
Funetional Requiremeats
In this study, the FRs are consideredflexible providing that their bounding values
can be jlexibly selected. Assume that the spread of flexibility of the i" FR is Ab,.
Accordingly, b, +db, is used to express the general f o m of a flexible bounding value.
Thus, the resul ting constraint, g, (X) S [b, , b, + db, ] ', corresponds to a range of FRs for
the i" must amibute that should be satisfied in design by order of desired preference.
Obviously, if a design solution satisfies the iM design constraint with its bounding value
b, , it also satisfies the same design constraint with a wider bounding value, b, + Ab,. To
characterize the variation of one's desire to the change of a bounding value, a flexibility
characteristic hct ion is established for describing aflexible FR in the form of design
constraint, as illustrated in Fig. 3.4:
Figure 3.4 Flexibility characteristic hct ion of flexible fiinctional requirement
'since the design parameters (P) in this use are fircd, for simplicity, the design amibute (with al1 its
related f o m ) is expressed as a function of X only. This adoption is extended from here forward once
dealing with Jtexible FRs.
.-
-=+. . - . ..."-.-.A- -- .A a - . In Figure 3.4, bi
satisfying g,(X) 5 4 are
acts as a threshold - - -.
the most desirable;
40
value such that those design solutions - -
dbi serves as a leeway adding to bi to
characterize those des ip solutions, feasible in [b,, bi + 4 . 1 , that are acceptable but not
most desirable. To further differentiate the acceptable designs h m the most desirable
ones upon the design constraint(s), p,(X) shows the variation of preference metrics
against different levels of the spread of flexibility of the i" FR.
Mathernatically, this flexibility characteristic function can be expressed as
follows:
g, ( X ) > b, + Abi
bi s g, (X) S bi + Ab,
gj (XI < bi
As seen, the fieedom of design selection is proportional to design preference.
Outside the boundary points (bi and bi +Ab,) in Figure 3.4, the level of preference is set
either 1 or O. Between the boundary points, a range of FR is characterized by a linear
variation of design preference in the form of dgj". The bigger the degree of design
preference (a), the smaller the room of desip flexibility (4gja) ).
3.3 Incorporation of Flexibility into Design Formulation
A Bexibility characteristic function is dehed based on the mathematical
expression of hizy membership fiiriction. Similady, the flexibility-incorporateci design
fomulations may be derived mathematicdly with reference to the formulation of fuzy
design optirnization (see Appendix A). By aggngating individual design preferences over
al1 FRs ihrough their flexibility characteristic bctions, an overall design preference can
be detennined and thus, the design solution process is dictated toward maximizing the
overall design preference. In this section, discussions will be held in view of two cases:
--a=----- -- - - (1) . design - parameter induced flexibility and (2) fictional requirement induced -- -
flexibility.
It is important to note that the aew design models derivexi in this section c m be
viewed as an extension to the underlying design models presented in Chapter 2.
3.3.1 Design Parameter Rooted Flexibiiity
Under this circumstance, design flexibility is rooted in the design parameters that
can be flexibly selected. Two sets of design formulation will be addressed according to
whether the wish attribute in the formulation is flexible or not.
Flexible Wish Attribute
If a flexible design parameter is involved in the objective function of a wish
attribute, the propagation of flexibility leads to a flexible wish attribute that can be
chanicterized by an appropriate flexibility characteristic fiinction. This argument is also
applicable to justify a PexibIe must attribute. Consider a design problem involving a
flexible wish attribute f ( X , P), and a number of flexible must attributes
g, ( X , P) Vi E [l, m] (for the presence of" P "), plus a set of non-flexible must attributes
(for the absence of " P ") g, (X) Vj E [1, n] . In this case, the feasible design space,
denoted D, is govemed by an intersecting region due to the decision domains of not only
gr. g,, ph but f as weU, Le.
where F , G,. , G j , Ph represent the decision domains with respect to f , g,, g, , ph .
The above set operation can be fiuther expresseci b y their flexibility c harac teristic
funcrions mathematically, Le.
where " A " stands for an aggregating operator comsponding to the set intersecting
operator " I "; pd (X) is defied as
To account for those non-flexible must attributes due to the absence offlexible design
parameters (P ), Eq. (3.8) would fall into either scenario:
Clearly, the design scenario expressed in Eq. (3.1 1) is not desired at al1 because its
aggregated design prefemice will nim out to be zero. Therefore, the design scenario
reflected in Eq. (3.10) is dictated such that gj (X) < bj . Note that p, (X, P) indicates an
overall design preference resulting h m aggregating the preferences associated with
individual design attributes and desip parameters. To evaluate the overall design
preference, a generalized intersecting operation in fuzzy set theory is adopted through
"min" aggregating operator for quantifjing p, (X, P) , accoràing to
Note that p# ( X ) is omitted in Eq. (3.12) by dictating g, (X) <; bj so that p, (X)
aiways mains unity (Le. p, (X) 4, j = 1 ,......, n ). Thus, to achieve a design with the
overall prrferaice to an extreme extent, the solution seeking process is directed toward
maximizing p, (X , P) such that
where X ' denotes the resulting design configuration eventually obtained, and P '
indicates the parametnc design configuration selected simuitaneouîfy with X' . Owing to
the existence of flexibility in both the wish and must attributes, Eq. (3.13) couples f
with the aggregated design preference, p, . The denvations above are based on the max-min aggregating approach in fuzzy
set theory. Since the lowest prefeience is picked out to quanti@ the overall design
preference, it is implied that the design aspect with the worst satisfaction is targeted for
enhancement, thereby accounting for the worstcase design scenario. With the max-min
aggregating operation, the overall design preference hinges on the weakest design aspect
having the least degree of design preference. As such, the design progress through
iterations is dictated toward the maximization of the overall design preference.
For design automation, a computational model is needed so that design computing
cm be executed on a cornputer. To this end, a design model suitable for computenzation
of Eq. (3.13) is developed with reference to ''fuzzy optimization" formulation:
Find [X,P,A]
max iZ
s.t. A I p, (X, P)
........ A.<pa(X,P) i=1,2 m
........ A 5 P ~ P ) h =1,2 r
........ g j ( X ) S b j j =1,2 n
X: s X, a X; k =1,2 ,......, q
where R is a scalar variable (0 S L i 1 ) to quanti@ the overall design preference.
Non-flexible Wish Attribute
A wish attribute is considereâ non-fleible if its objective îùnction does not
contain flexible design parameter. In this context, consider a design problern similar to
the one discussed above, with the expectation that the wish attribute of concem is non-
flexible. In this case, the feasible design space @) is defined as a comrnon region
fonnulated by intersecting the decision domains across must attributes ( G, and Gj ) and
design parameters ( P, ), i.e.
Likewise, the above set operation can M e r be mathematically conveyed by their
flexibility characteristic functions, Le.
Since p, (X) is either zero or one (as argued earlier), Eq. (3.21) can be fùrther
simpli fied into
by holding gj (X) a bj such that /rd (X) 4.
At this point, as justifieci eariier, the ''min" operator is applied to aggregate
p, (X, P) with ph ( p ) , thereby obtaining p, (X, P) quantitatively according to
--2---- - * -
In the expression above, it can be seen that the wish attribute (f ) is not
fomulated into the aggregate design preference p,, due to the non-flexibility of f (X) . This case reveals that the aggregated design preference is not coupled with the wish
attribute. For this sake, the design process is dedicated toward optimizing (or minimizing)
f (X) subject to the prescribed level of p, , Le.
where D is characterized by p, according to
Note that a is the level of design preference prescribed by a designer with O S a I 1 . Based on Eqs. (3.24) and (3.25), a computerized design model for this design case
can be m e r elaborated below:
Find [X ,P ]
min f ( X )
........ s.t. a 6 ,u,(X, P) i =1,2 m
........ a PJP) h =1,2 r
........ g j ( W Sbj j =1,2 n
.... x: xk x; k = 1,2,... q
This design model is dedicated to dealing with a design problem having no
(flexible) design parameter involved in the wish attribute. The solution procedure
discussed above is based on the a t u t approach of fuzy set theory. This treatment
allows for the incorporation of flexibility into the context of design formulation. By
varying the value of a with assignment to the aggregated design preference, multiple - = -
design solutions can be found to result in a set of design alternatives.
For cornparison, one remarkabie différence between the two design models newly
developed is that the level of aggregated preference p, in the first one is detennined
through maximization during design computing while, in the second one, being pre-
detennined before design execution. Furthemore, the second model also allows for the
generation of altemate design solutions by the adjusmient of a over the interval from O
to 1.
3.3.2 Funetional Requirement Rooted Flexibility
Under this circumstance, design flexibility is rooted in the FRs whose bounding
values can be fiexibly selected. This design case can be treated as an extension to the
design problem in the second model above for which the wish atttibute is considered non-
jZexible. Only those must attributes whose FRs cm be jlexibiy selected are considered
flexible. In this discussion, it is assumed that al1 design parameters remain constant (Le.
non- flexible).
As discussed earlier in Section 3.2.2, the flexibility of FRs results from the
possibility of flexible choices on theù bounding values. Consider a design problem in
which the set of must attributes is classified into two subsets, depending on whether the
bounding values of their FRs can bepexibfy selected. Say
where dbi is the leeway prescribed for characterization of the spread of flexibility for the
bounding values, bi . Following the same denvation procedure as described earlier, the
feasible design space (D) is delimiteci as an overlapping region, Le.
By dictating gj(X) S bj (Le. non-flexible attributes), it yields that pU (X) = 1 for
j = 1 ,......, n . Thus Eq. (3.35) is simplifieâ into
subject to which the decision process is toward seeking X' such that f ( X ) is optimized
(or minimized), according to
f (x') = min XCD f ( X )
where D is quantified by p, using the a -cut approach from fuzzy set theory.
Accordingly, a computerized design mode1 can be finally derived with reference to Eq.
(3.26) through (3.30):
is -the flexibility cmgcteristic function of a must
of the FR cm be flexibly selectd; Eq. (3.40) denotes
must attribute for which the FR is notflex'ble.
This design model is used to tackle the design problems whose
48
attribute whose
the non-flexible
FRs are flexible
but design parameters arefied. What needs to be detemineci in the design process is
only the nominal values of design variables. Similar to the second design model
presented earlier, the a- cut approach is also used as a basis for design solutions. By
assigning different values of a (O S a S 1 ) to the aggregated design preference ( p, ),
this design model allows to generate a variety of altemate design solutions.
Design Flexibility Modeling Under Uncertainty
4.1 Introduction
The robust design approaches discussed in Chapter 2 are used to treat design
variations caused by tolerancing uncertaiw, while the design modeling approaches
explored in Chapter 3 are used to handle design variations induced by designflexb&y.
In reality, however, a design problem may involve not on& design uncertuinty but also
design jlexibdity, both likely present in the context of design computing. This chapter is
therefore intended to address such a design case (as mentioned above). In particular, new
design modeling approaches will be m e r explored in order to incorporate both design
flexibility and design uncertainty into the context of design formulation.
To cope with design flexibility modeling in the presence of uncertainty, two
strategic approaches will be developed in this chapter specifically, that is, sequential and
simultaneous approaches, both of which can be regarded as an extension to the
approach(es) aven in Chapter 2. The squential approach treats tolerancing uncertainty
and design flexibility in sequence through the use of "superposition", that is, tolerancing
uncertainty (associated with design components) is first incorporated into the context of
design formulation using the robust design appmaches h m Chapter 2; design flexibility
(associated with design parameters or FR'S) is then takm into account using the
flexibility modeling approach(es) h m Chapter 3. In conhast, the simultaneous approach,
by its narne, handles tolerancing uncertainty and design flexibility ut the same tinte in
design modeling and formulation, by extending the notion of mathematical expectation of
a membership fùnction h m hiuy set theory.
The remahder of Chapter 4 is arranged as follows. Section 4.3 will elaborate on
the sequential approach, and Section 4.4 will detail the simultaneous approach. The
discussion of each approach will be deployed h m the modeling of flexibility (rooted in
design parameters and FR'S, respectively) to the formulation of design models.
4.2 Sequential Approach
Both tolerancing uncertainty and design flexibili ty incur variations in design,
which can be manipulateci sequentially using "sequential approach" in a hvo-step
procedure by "superposition":
Step 1. Incorporate tolerancing uncertainty into design models using the robust design
approaches given in Chapter 2.
Step 2. Add design flexibility to the context of design formulation using the flexibility
modeling approach(es) presented in Chapter 3.
To account for the presence of tolerancing uncertainty, the nominal design
functions in this approach are adapted with refaence to robust design formulations in
Chapter 2. As a result, a variant is introduced to both wish and must attributes that
associates the nominal design (attribute) hct ions ( f , g, ) with their transmitted
variations (@,dg i ), according to
where F (9) and Gi ( 0 ) are the modified fonn of desip fùnctions, ( f , gi ), that incorporate
uncertainty induced design variations in the wish amibute and the i' must attribute,
respectively; 4 (e) and dg, (e) are the correspondhg amount of transmitted variations in
the wish and must attributes (Le. f , g, ), which can be estimated using the formulae given
in Eqs. (2.20) to (2.23) (set Chapter 2). Note that the use of Eq. (4.1) is a result of --a -- - -- - - L -
refmncing Eq. (2.34) while Eq. (4.2) refemces the form appearing in the left side of Eq.
(2.28). The modifieâ form of a design fûnction above consists of a nominal terni of the
attribute function ( f , g, ) and a terni of its variation (4, Agi), the latter resulting fkom
the presence of tolerancing uncertainty. Such treatment is borrowed fiom robust design
approaches insofar as "one-wish many must" fonnalism is concemed in the context of
desip (see Chapter 2 for M e r details). The same justification as argued in Chapter 2
cm be exactly extended to hold Eqs. (4.1) and (4.2). Hence, in the remaining discussion
of the sequential approach, F (e) and G, (9) are used to replace f ( 0 ) and g, (*) in the
context of flexibility modeling with uncertainty.
4.2.1 Design Parameter Rooted Flexibility
This section considers such a design scenario that design flexibility roots fiom the
prescnce of flexible desip parameters, subject to the tolerancing uncertainty associated
with design components (Le. variables and parameters). Discussions will be held on the
transmitted flexibility to both wish and must attributes through the propagation of their
design fùnctions. Specifically, their flexibility characteristic fûnctions will be formulated
and described.
Wish Attribute
In the presence of uncertainty, a variation term (4) is added to combine with its
nominal design fûnction ( f ). Consequently, it yields a variant (F ), as fonnulated in Eq.
(4.1), to replace the original fimction fom of the wish amibute in the context of
flexibility modeling. This implies that the derivation previously presented in Chapter 3 is
exactly applicable h m .
At the above point, the spread of transmitted flexibility of a wish attribute under
uncertainty can be esthateci, with nference to Eq. (3.2), using
where f (*) is replaceci with F (O); LW(@ denotes the spread of transmitted flexibility of
an uncertainty-present wish attribute against a given preference level (a) of flexible
design parameters (P ). Note that F (m) has incorporated the uncertainty induced design
variation into the formula, as such, X, (and Po ) may use the total robust design solution
obt ained fiom solving Eqs. (2.34) through (2.37). 6
To associate design preference with flexibility modeling under uncertainty, the
flexibility characteristic function of an uncertainty-present wish attribute can be
expresseci with teference to Eq. (3.3), Le.
where F, stands for the uncertainty-present minimum of Eq. (4.1) by solving the total
robust design problem fonnulated in Section 2.4.3 (see Chapter 2); LW(') represents the
spread of design flexibility of an uncertainty-present wish attribute corresponding to
a = O while is given in Eq. (4.3) for quantification. Figure 4.1 shows its graphical
representation, simlar to Fig. 3.2. Since p, (.) is an extension to pf ') by incorporating
design uncertainty? the same interpretation as used to Fig. 3.2 cm be exactly employed
here to justify the configuration of Fig. 4.1, with the replacement of gi by Gi .
Due to the linear variation in pJp), it can be mathematically proved that
,uF (X, P) also varies linearly as shown in Fig. 4.1. It is important to note that p,(-)
characterizes the variation of the s p d of ttansmitted flexibility of an uncertainty-
present wish attribute against the distribution of design preference. The higher the degree
of design prefetence (a), the smaller the spread of design flexibility ( M("' ).
Figure 4.1 Flexibility characteristic bc t ion of an uncertainty-present wish amibute
Must Attribute
To account for the presmce of uncertainty, the must attribute function, g, (X, P) ,
is adapted to G,(X, P) by adding the variation term, dg,(X, P ) , into g, (X, P)
according to Eq. (4.2). Thus, the resulting constraint becomes G, (X, P) b, (i.e.
g, (X, P) + 4, (X, P) S b, ) as an adaptation to g, (X, P ) 6 6, (see Chapter 2).
Correspondingly, the spread of transmitted flexibility, due to the existence of flexible
design parameters (P ), of an uncertainty-present must attribute can be estimated, with
reference to Eq. (3.4), using
where AG,'") designates the spread of transmitted flexibility of the i" uncertainty-present
must attribute at the prescribed level of preference (a ). Note that the selection of
(X, , Po ) may result h m solving the total robust design problem formuiated in Chapter
2.
- L. - - - + .- - To
flexibility
expressed with reference to Eq. (3 .9 , i.e.
associate design preference with flexibility moâeling under uncertainty, the - -
characteristic fiinction of the i' uncertainty-present must attribute can be
where AG:') is the spread of design flexibility of the i' uncertainty-present must
attribute corresponding to a = O . Figure 4.2 gives its graphical representation, similar to
Fig. 3.3 (b).
Figure 4.2 Flexibility characteristic hct ion of an uncertainty-present must attribute
Since kt(*) is an extension to ,ugi (*) by incorporating design uncertainty, the same
interpretation as used to Fig. (3.36) can be equally extended here to justify the
configuration of Fig. 4.3, in which g, is replaceci by G,. kt(*) shows a linear fom of
variation due to the presence of linearity in &(P), as argued earlier in Chapter 3, that
propagates through AG:"). It is important to note that kt(*) characterizes the variation of
the spread of transmitted flexibility of the i' uncertainty-present must attribute against
-- the distribution of design prefennce. The higher the degree of design preference (a ), the
smailer the spread of design flexibility (AG:)).
4.2.2 Functional Requirement Rooted Flexibility
In this scenario, the selection on the nominal values of design parameters is fued
while the choice on the bounding values (b, ) of fùnctional requirements is madejlexible.
In the absence of uncertainty, the above design concem can be formulated as such a
design constraint that g , ( X ) < [b,, b, + db,] to express a range of FRs for the i" must
attribute. Using this as a badine, the expression of a range of FRs in the presence of
uncertainty can be M e r derived by replacing g, by G, in the sequential approach, i.e.,
G, (X) S [b, , b, + Ab, ] , where G, is fonnulated in Eq. (4.2).
In this context, the correspondhg flexibility characteristic function to describe a
flexible FR in the fonn of design constraint can be written, with reference to Eq. (3.6), as
G , ( X ) > b , +db,
bf 9 G, (X) S b, + Ab,
G, (XI < 4 whose graphical representation is given in Fig. 4.3, similar to Fig. 3.4.
Figure 4.3 Flexibility characteristic Wtion of the i ranged functional
requirement subject to uncertainty
where Ab,. is the spread of flexibility for the i" must attribute. The explanation of Fig.
4.3 is analogous to that of Fig. 3.4. The higher the degree of design preference (a ), the
smaller the room of design flexibility ( AG,!") ).
4.23 Incorporation of Flexibility into Uncertainty-present Design
Formulation
Referencing the fomulation of fuzzy design optimization (see Appendix A) and
robust design models (see Chapter 2). design flexibility and uncertainty can be both
incorporated into design formulation based on the flexibility characteristic functions
given above. Also, similar to the discussion held in Chapter 3, the solution process will
be grounded on aggregating the preference of design attributes using the flexibility
characteristic functions established earlier on. Mer executing the solution process, an
overall design preference will be maximized by searching for an optimal design solution
(for both variables and parameters) over such a feasible design space that accounts for
the presence of not only design unceriainty but also design flexibility. The design models
to be derivecî in this section are classified, consistently with Chapter 3, according to (1)
design parameter mted flexibility, and (2) functional requirement rooted flexibility. It is
important to note that these new design models are considereâ as an extension to those
denved in Chapter 3 that also incorporate design uncertainty into the context of design
formulation.
4.23.1 Design Parameter Rooted Flexibility
The discussion of this design case is m e r divided into two parts according to
whether the wish attribute present in a design problem isfkxible or not.
Flexible Wlsh Attribute
This scenario deals with a design pmblem in which both flexibility and
uncertainty are present. Design flexibility arises from the presence of flenble design
parameter(s) in the objective huiction of a wish attribute and the constraint functions of
must attributes. Design uncertainty results h m the association of tolerances with design
components (variables and/or parameters), incurring the detenoration of design
performance in the objective hct ion of a wish attnbute and the constraint bctions of
must amibutes. To handle such a design problem, the squential approach adopts the
strategy of superposition that incorporates "uncertainty" into design functions ( f , gi )
first and then adds ''flexibility" into the modified design functions (F, G,). This
treatment allows applying the results of Chapter 3 directly to derive flexibility-
incorporated, uncertainty-present design models.
With this view, the design mode1 to be derived here can be viewed as a
counterpart to that derived in Section 3.3.1 (under the same heading). This means that the
same derivation and arguments as given in Section 3.3.1 are exactly shared here in
deriving a new design model to account for the uncertainty-present design problem.
Nevertheless, it should be noticed that the attnbute functions used in Chapter 3 such as
f (*) and g, (9) are replaced with the uncertainty-present attribute functions in a modified
fom given in Eqs. (4.1) and (4.2). Le. F (*) and G, (m), in the context of denvation of the
new design model. That is to Say, al1 fomulae necessary to denve the new design model
exactly foilow what have been presented in Section 3.3.1, Le. Eqs. (3.7) through (3.13).
with the exception that al1 f 's and g, 's are replaced by F (B ) and G,. (-),
correspondingly.
As a result, the new design model can be fonnulated for computationai
implementation, according to
Find [X, P,A]
where p, (X, P) and p, (X, P) denote the flexibility characteristic bctions of wish
and must design attributes given in Eqs. (4.4) and (4.7); 4g,(X) denotes the fùnctional
level of uncertainty induceâ variation in the j' (nonflexible) must attribute, which can
be quantified using Eq. (2.21) or Eq. (2.23) (without including the tenn due to " P '3. To summarize, this design model shows the foilowing salient features. (1) The
overall design preference ( A ) is coupled with individual design preference for the wish
and must attributes, and therefore, is evaluated during the design process. (2) The design
process is dictated toward the maximization of the overall design preference ( A ) through
the "max-min" aggngating operation. (3) The flexibility of design is characterized by
those flexibility characteristic fùnctions in connection with design preference. (4) The
best overall design preference (Le. A,) found using this design model provides a basis
for m e r achieving a family of designs over the preference range [O, A-] through
DFV (to be discussed in detail in the next chapter).
Non-flexible Wish Attribute
This scenario is complementary to the preceding one or rather, is considered as a
special case of the design problem discussed earlier on. That is, only uncertainty is
present in the objective fbnction of a wish attribute with no flexible design parameten.
Notwithstanding this, some must attributes may still involve both Pexibiliity and
uncertainty in theu constraint functions.
Since the wish attribute prescrits non-flexibility, then is no room of flexibility in
its objective function. This dictates the objextive huiction to achieve its optimum at
which the corresponding preferrnce Ievel always reaches unity (100%). This is because,
as ergued in Chapter3, the m m of fiexibility varies inverse-proportionally to the level of - - & - - L - -
design preference. By using the uncertainty-present attribute functions (F, Gi ) to
replace their correspondhg nominal ones ( f , g, ), the procedure of deriving the cunent
design model exactly follows that given in Section 3.3.1, i.e. Eq. (3.20) through (3 .Z), so
do the arguments in justifjing the context of modeling and formulation of the design
problem.
As a result, the new design formulation is given as follows:
Find [X,P]
min F(X ) (4.14)
....... s.t. a l p G i ( X , P ) i = 1,2 rn (4.1 5)
........ a P h (P) h =1,2 r (4.16)
........ G j ( X ) = g j ( X ) + ~ , ( X ) ~ o j = 1,2 n (4.1 7)
I x, i x, S x; k = l,2 ......., q (4.18)
where F (e) is given in Eq. (4.1) for quantification and evaluation; the same notation as
used earlier applies to the symbols in the above formulation. This design model is viewed
as a counterpart to that presented in Eqs. (3.26) to (3.30).
To summarize, the following characteristics are associated with this design model.
(1) The (overall) design preference (a) is govemed by that of the individual must
attributes solely and thus, should be prescribed pnor to the design process. (2) The design
process is âictated toward the optimization of the wish attnbute F ( X ) , subject to the
prescribed level of preference (a ) pertaining to the flexible must attributes plus other
(non-flexible) design constraints. (3) The a t u t approach of fuvy set theory is used for
the treatment of prefmnce. (4) It allows for the creation of scalable design or a family of
designs upon the adjustment of a over the iatmal [O, 11.
4.23.2 hinetionil Requirement Rooted FlexibUity
- - In this design case, the design problem solving involves the concems of both
- -
flexibility and unceruiinty where the source of flexibility is not due to flexible design
parameters but flexible FRs. To be pncise, the selection of bounding values of FRs is
treated flexible while the desip parameters stayfied during the design process. Note
that the source of uncertainty roots h m the sarne as discussed earlier.
Since the flexibility is present sole& in the bounding values of some FRs, only the
design constraints containhg flexible bounding values are considered flexible (while the
desip parameters (P) stay fmd in this case). At this point, the design objective
(pertaining to a wish attribute) is always treated non-flexible. The design model for the
present problem solving is viewed as a couterpart to that derived in Section 3.3.2. As
such, the procedure of deriving the present desip model exactly complies with thiit given
in Section 3.3.2, except that the original atûibute hurctions (f ,gi) are replaced by the
modified ones ( F , Gi ) using Eqs. (4.1) and (4.2) in order to account for the presence of
uncertainty in the cumnt desip problem. As such, the formulation and related arguments
held at Eqs. (3.3 1) through (3.35) can be extended directly to the present design model.
As a result, the design formulation for the current problem solving can be
expressed below:
Find X
min F ( X ) (4.19)
s.t. a S p , ( X ) i = 1,2 ,....., m (4.20)
G j ( X ) = g,(X)+dg,(X)SO j = 1, 2 ,......, n (4.2 1 )
xi Sx, 6x,Y k =1,2 ,......, q (4.22)
Note that, since the design parameten are fued, the related fùnctions in the formulation
above, for simplicity, are expressed in ternis of X only.
To summarize, the important features of this design model are revealed below. (1)
The (overall) design prefermce (a ) is govemed by that of theflexible design constraints
oniy and thus, should be specified before design computing. (2) The design process is
dictated toward the optimization of the uncertainty-present design objective F ( X ) ,
subject to the prescribed level of preference (a ) for the flexible design constraints
together with other non-flexible design constraints. (3) The a s u t approach of fuzzy set
theory is used for the treatment of preference. (4) It allows for the creation of scalable
design or a family of designs upon the tuning of a ova the interval [O, 1 1.
4.3 Simultaneous Approach
The sequential approach discussed earlier is a rwo-step method, which copes with
design uncertainty and design flexibility in sequence in handling a design problem. In
contnist, the simultaneow appmach treats design uncertainty and design flexibility ut the
same time when dealing with the same design problern, thereby is a single-step method.
Consider a design attribute (either wish or mut), A(X, P) , involving both design
uncertainty and designflexibility. As discussed in Chapter 3, the ruom of flexibility over
the choice of each nominal value of the attribute function is associated with design
preference through a flexibility characteristic furiction, ( X , P . In the presence of
uncertainty, the acnial value of the attribute huiction, as known, varies around its
specified nominal value as the result of the uncertainty induced design variation. This
means that the nominal fonn of a flexibility characteristic hction, p, ( X , P) , is subject
to variation due to the presence of uncertainty.
To characterize the effect of uncertainty, the notion of mathematical expectation
of a membership hction (or hizy probability) in tuuy set theory (see Appendix A) is
extended to treat uncertainty aad flexibility simultaneously in design through the concept
of expectution of flexibility characteristic finetion. The expectation of a flexibility
characteristic fiinction is expressed as E[p, (X, P)] , where M e ) is to characterize "design
flexibility" through preference metrics and E[w] to account for "design uncertainty"
through the averaging of probabilistically distributed values. Mathematically,
E[p, (X , P)] is defineâ according to
where ,&A) = p, (X, P) as one descnbed earlier, and ((A) = #A (X, P) representing the
probability distribution function to characterize design uncertainty involved in A(X, P) . In the simultaneous approach, design flexibility moâeling witb uncertainty is analogous
to that with no uncertainty (discussed in Chapter 3) by simply adding the averaging
operator (E), signifjmg an expectation jknction, ont0 the top of a flexibility characteristic
function (p), thus resulting in sucb a fonn as E [ U a ) ] in the context of design modeling.
Discussions as follows are devoted to (1) design parameter rooted flexibility, (2)
huictional requirement rooted flexibility, and (3) uncertainty-present design formulation
based on the simultaneous approach.
4.3.1 Design Pirameter Rooted Flexibility
The flexibility modeling approach discussed in Chapter 3 is extended to include
design uncertainty based on Eq. (4.24). This design scenario assumes that the flexibility
of design is due to the presence offlexbfe desip parametea.
Wish Attribute
Consider a flexible wish attribute (involving jlexible design parameters) in a
general fom, f (X, P), whose flexibility characteristic hction, denoted pf (X, P) , is
defined as one according to Eq. (3.3). To incorporate uncertainty into Eq. (3.3), the
original fom of Eq. (3.3) is adapted, with reference to Eq. (4.24), as
P - - L - -- 2 - - - -
where p (X, P) = p[ f (X, P)] is the flexibility characteristic h c t i o n for afrexible wish
attribute V); #f (X, P) = O[ f ( X , P)] is the probability density function of an uncerioin
wish attribute 0; 7 = f ( X , P) denotes the nominal (or mean) function of a wish attribute
0; and q f ( X , P) = ml stands for the uncertainty transmitted design variation to a wish
attribute V ) as can be approximated using Eq. (2.20). Note that both 7 and Af are
evaluated with the iterative values of (X, P) and then fixed throughout each design
iteration. In the case of nonnal distribution, 4, ( X , P) can be hirther expressed using
where f denotes the wish attribute whose values are dictated over the interval,
[7 - 4 ( X , P), 7 + 4 ( X , P) 1, which is updated and fixed during each design iteration.
Note that the expression, al = f (X, P)] = mf (X, P) , indicates the transmitted
standard deviation of a wish attribute V) that can be evaluated approximately using
where qk and a, indicate the standard deviations of the k" design variable and the
h th design parameter, respectively.
To M e r reveal E[p,(X, PI], Eq. (4.26) is illustrated graphically through Fig.
4.4, in which p,(X,P) is the same as one in Fig. 3.2. Figure 4.4 displays the essence
interacting between the flexibility characteristic fùnction ( p /) and the probabilistic
distribution hction (+f ). In pariicular, there are three cases in E[p (X, P)] bounded by
points (A, C) that characterize different value regions of E[p,(X, P)] against the
- -- - presence - - of uncertainty. (1) Ebf(+q = O as the uncertainty distribution is located
beyond "C" on its right; (2) O < E[pf(X, P)] 5 1 as the uncertainty distribution is located
between "A" and "C"; (3) E[p, (X, P)] = 1 as the uncertainty distribution falls in the
region beyond point A on its left. To summarize, this can be expressed mathematically
as follows:
Pf (XJ=)
4 1.0
Figure 4.4 Incorporation of uncertainty into flexibility modeling for a wish attribute
As a result, B[p (X, P)] described h u g h Eq. (4.29) can also be viewed as an
extension to pf (X,P) that is formuiated in Eq. (3.3), where design uncertainty is
incorporated with design flexibility through the expectation fbnction (E). The higher
expectation of the flexibility characteristic fùnction indicates the higher uverage design
preference but the smaller merage room of flexibility in the wish attribute.
Consider a must attribute, g, ( X , P) S b, , which undergoes both uncertainty and
flexibility. Correspondingly, the flexibility charactenstic function subject to the presence
of uncertainty can be expressed with reference to Eq. (4.24), yielding
where pJX, P) - p[g, (X, P)] is the flexibility charactenstic function of the i" flexible
must amibute (gi); & (X, P) = dg, (X, P)] is the pmbability density function of the i" -
uncertain must attribute (gi); gi = g, (X, P) denotes the nominal (or mean) function of
the i' must amibute (&); and 4, (X, P) = md stands for the uncertainty transmitted
design variation to the ih must atüibute (g,) as c m be approximated using Eq. (2.2 1).
Note that both and 4, are evaluated with the iterutive values of (A', P) and then
fixed throughout each design iteration. In the case of normal distribution, &. (X, P) can
be fûrther expresseà using
where g, denotes the i* must atûibute whose values are dictated over the interval,
[E - &(X, P), + &(X, P) 1, which is updated and fixed during each design
iteration. Note that the expression, a, - a[g, (X, P)] - a, ( X , P) , indicates the
transmitted standard deviation of the ih must attribute that cm be evaluated
approxirnately using
To incorporate design uncertainty into flexibility modeling, the expectation of
flexibility characteristic funetion is used to characterize the i' must attribute with
re ference to Eq. (4.29), yielding
Figure 4.5 Lncorporation of uncertainty into flexibility modeling for the i' must attribute
To illustrate Eq. (4.33), Eb,'(X,P)l is fkrther revealed through Fig. 4.5 in which
pJX, P) is the same as one in Fig. 3.2.
In Fig. 4.5 (a), four cases cbaracterized using points (A, B, C) are displayed: (1)
E[p, (X, P)] = O as the uncertainty distribution is located beyond "C" on i ts right; (2)
O S E[y,(X, P)] c Azn as the uncertainty distribution is located between "B" and "C";
(3) A:,, S E[p& P)] c 1 as the uncertainty distribution is located between "A" and
"B"; (4) E[p,(X, P)] = 1 as the uncertainty distribution is located beyond "A" dong its
leR (which is the most desired to a design). However, cases 1 and 2 must be eliminated
because they both violate the design constraint according to g i ( X , P) 5 b,. As can be
seen in Fig. (4.9, the marginal point for a design to sustain is located at "B", Le..
bounding point as gi (X, P) = bi . In the presence of uncertainty, this expression should
be adapted, through "expectation", to E(x, P) S bi plus 4gi ( X , P) = KG to
characterize the associated spread of uncertainty. This treatment leads to a new quantity,
A:,, , to characterize the threshold to the feasible values of the i" must attribute such that
E[pn ( X , P)] 2 Azn . Numerically, this threshold value ( At!n ) may be estimated
according to
where X, represents the design solution obtained h m solving the ordinary optimization
problem formulated in Eqs. (2.1 1) to (2.13); Po denotes the collection of P in solving
that ordinary optimization problem without considering flexibility. Figure 4.5 (b) gives
an expended view ta A:,, , which is based on Fig. 4.5 (a).
As a result, E[p,(X, P)] described through Eq. (4.33) cm also be viewed as an
extension to p,. ( X , P) that is fonnulated in Eq. (3.9, where design imcertainty is
incorporated with design flexibility through the expectation bct ion (E). The higher
Pm - expectation of the flexibility characteristic fùnction indicates the higher average design preference but the smaller average m m of flexibility in the must attribute.
4.3.2 Funetional Requirement Rooted Flexibility
The design problem being dealt with in this section is the sarne as one discussed
in Section 4.2.2. Consider a must attribute subject to a range of functional requirements,
i.e., gi (X) 5 [b,, 4 + Ab,], in which the bounding values of some functional requirement
( b j ) are treated flexiMe and the design parameters (P) keep fmed. In this case, the
flexibility modeling subject to the presence of uncertainty is based on the same
formulations as those given in Eqs. (4.30) to (4.32), along with the same argument and
justification. Nevertheless, an adaptation is introduced to the expectation of its flexibility
characteristic function to account for the ranged bounding value, [b,, bi +Abi] ,
according to
where pn(X) = d g , ( X ) ] is the flexibility characteristic function expressed in Eq. (3.6)
due to the flexible fimctional requirement associated with the i' must attribute (gi);
A(mn is defined as a threshold to delimit the feasible values of the i' must attribute (gi)
such that E[p,(X)] 2 A z n . With nference to Eq. (4.34), is estimated in a similar
way according to
which extends the same notation as used in Eq. (4.34). - 2. --z. 2- -- ..--- - . - . -- - - A -
Figure 4.6 illustrates Eq. (4.35) graphically according to different locations of the
uncertainty distribution. Similar to Fig. 4.5 (a), Fig. 4.6 (a) also shows four cases based
on points (A, B, C). (1) E[p, (X)] = O as the uncertainty distribution is located beyond
" C ' o n its nght; (2) O S E[p, (X)] < A!;,, as the uncertainty distribution is located
between "B" and "Cm; (3) A E (X, P l < 1 as the uncertainty distribution is
located between "A" and "B"; (4) E[p, (X)] = 1 as the uncertainty distribution is located
beyond "A" dong its lefi. Obviously, both case 1 and case 2 mu t be eliminated because
they violate the ranged design constraint, that is, g, (X) > b, + Ab,. Therefore, the design
solution should be sought subject to A(& Ei(, (X)] S 1 , where At, is illustrated as a
shaded m a in Fig. 4.6 @).
Figure 4.6 Incorporation of uncertainty into flexibility modeling for the ranged
bctional requirement of the i' must attribute
As a result, E[p, (X)] described through Eq. (4.35) cm also be viewed as an
extension to p, ( X ) that is fomulated in Eq. (3.6), where design uncertainty is
incorporated with design flexibility through the expectation fhction (E). The higher
--TL- - - - expectation of the flexibility characteaistic fiuiction indicates the higher average design
preference but the smallei average m m of flexibility in the fiuictional requirement.
4.33 Incorporation of Flexibility into Uncertainty-present Design
Formulation
To incorporate design flexibility into uncertainty-present design formulation,
three design models will be presented, based on the simultaneous approach, according to
(1 ) design parameter rooted flexibility and (2) hctional requirement rooted flexibi li ty.
The former in turn is M e r divided into two scenarios, i.e.,flexible wish attribute and
n o n - - b l e wish attribute.
4.3.3.1 Design Parameter Rooted Flexibility In accordance with Section 4.3.1, the development of design models is classified
according to whether the wish attribute is flexible or not, subject to the simultaneous
approach for treating uncertainty and flexibility in design.
Flexible Wish Attribute
This design problem is the same as that discussed in the sequential approach,
which involves both uncertainty and flexibility in design. In particular, the wish attribute
is considedflexible providing that it involves anyflexble design parameten. Different
than the one addresseci earlier, an e~rpectation fonn offlexibility churucterisiic function is
used for representing the wish and muri attributes in order to characterize the
simultaneous treatment of uncertainty and flexibility in design. Still, the design model is
denved such that an overall design preference is maximized over the feasible design
space delimited by each individual design amibute (both wish and must). In this view, the
dendg of this model is considemi as an extension to that in Section 3.3.1 (see Chapter
3) by incorporating uncertainty through the expectation operathg on the wish and m u t
attributes.
---x.--.--- - - a
Based on the discussion held in Section 3.3.1, Eq. (3.7) is still applicable in this . . - - - - -
case to define a feasible design space where the design amibutes delimiting the design
space rernain sarne. Nevertheless, the expression of an overd1 design preference in Eq.
(3.8) should be adapted to account for the presence of uncertainty, according to
where E[pJX)] is nothing more than the case in which the must attributes are
considered non-jlexibfe for the absence of flexible design parameters, thus having
for which E[p, (X)] = 1 must be held in order to have a feasible design. Extending this
to simplifjt Eq. (4.37) yields
As before, a general "min'' operator is ernployed for aggregating design preference across
allflexibfe atüibutes and parameters such that p, ( X , P) , an overall design preference, is
quantifid through
Ta obtain a design with the highest overall design preference, Eq. (3.13) is furiher -
employed to dictate the solution seeking process to maximize p, (X, P) . Based on this
denvation, a general fom of the computational design mode1 arrives ultimately at
where E[pf ( X , P)] and E[pd (X, P)] extend the sarne notation as given in Eqs. (4.29)
and (4.33) respectively. Yet, caution is called that the overall design preference A is
defined over the intmtal [ amin , 11 while ilfi,, is detemineâ by integrating Agi,, through
alljlexible must attributes according to
where A::,, (i = 1,2, .... m) is evaluated using Eq. (4.34).
This design mode1 extends the same applicability as that corresponding one
discussed in Section 3.3.1, plus an extra merit on the handling of uncertainty. It provides
a basis for fivther achieving DFV under uncertainty, in which the finding of A,, and
A, is a critical step for eventuaîly a family of design alternatives (to be addressed in
detail in Chapter 5).
Non-flexible Wish Attribute
-%- A- - . - This design problem is the same as that
- & -
which involves both uncertainty and flexibility in
considered non-flexible but some must attributes
73
discussed in the sequential approach,
design. Or rather, the wish attribute is
are considered flexible, subject to the
presence of design uncertainty. Extenàing the expectation fom of jlexibiliry
characteristic function to this pmblem allows for a similar procedure, as referenced to
Section 3.3.1, to denve the present design modei.
The expression for defining a feasible design space is the same as Eq. (3.20) foe
the fact that only the must attributes in this case contribute to delimiting the design space.
With extension to Eq. (3.21), the feasible design space is evaluated by finding an overall
design preference, p, (X, P) , according to
By holding E[pd ( X I ] = 1, the above expression can be M e r simplified as
A s before, a general "min" operator is used to aggregate design preference across al1
flexible must attributes and parameters such that
Since the wish attribute is n o n - / d i e in this design pmblem, there is no preference
coupling between the wish and nusf attributes. As such, the overall design preference in
this case can be prescribed pnor to design execution, thus yielding
P L - - - when - Te a is a scalar parameter designahg - - the level of design preference pre-assigned by a
designer. Accordingly, the solution process is dictatexi to optimizing the uncertainty-
present wish atûibute, quantified as F(X) = af (X) + b 4 ( X ) , subject to Eq. (4.51), i.e.
F, (x' ) = min [af (X) + b4(X)] K P ) t D
where D represents a feasible design domain quantified by Eq. (4.51).
Centered on Eq. (4.52), a corresponding computational design model c m be
denved according to the following formulation:
where pJX,P) is given in Eq. (3 .9 , and the same notation as used before is extended
here. However, it is important to note that the range of a is no longer [O, 11 (as applied
earlier) but [ Ami,, , 11 to account for the treatrnent of uncertainty in the simultaneous
approach for which A,, is treated in the same way as given in Eq. (4.47).
This design model extends the same applicability as that corresponding one
discussed in Section 3.3.1, plus the additional capability of handling uncertainty. It
provides a bais for m e r achieving DFV under uncertainty, especially the achieving of
scalable design over the range of design prefermce [Amin, 11 given flexible design
parameters in some must attributes.
4.33.2 Funetional Requirement Rooted Flexibility
This design problem is the same as that discussed in the sequential approach. That
is, some hctional requirements are considerd flex'ble while design uncertainty is
associateci with both the wish and must attributes. Note that the design parameters keep
f h i in this design problem.
Herein, an approach based on the notion of expectation offlexibiility characteristic
fitnction is presented so that design flexibility can be treated simultaneously with design
uncertainty. Yet, the procedure to derive the current design model is similar to that
discussed in Section 3.3.2, where an adaptation to H-) is introduced with EL&*)] in the
context of design modeling. As such, the solution process is carried out to optimize the
uncertainty-present wish attribute (Le., of (X) + bqf (X) ) thiough Eq. (4.52), subject to
the design space @) that is quantified using p, (X) according to
Accordingly, the computational design model can be fomulated, with reference to Eqs.
(3.36) to (3.39), as
Find [XI
min af ( X ) + b 4 ( X )
where E[pJX)] is given in Eq. (4.35). standing for the expectation of the flexibility
characteristic fùnctioa p,(X) . It is important to note that the value of a should be pre-
determined fiom the interval over [rl,, 11, as opposed to [O, 11, where A- is given in
Eq. (4.47) but A z n should be estirnateci wing Eq. (4.36) in this case.
-= LL - -- - - This design mode1 extends the same - applicability as that corresponding one
discussed in Section 3.3.2, plus the additional capability of handling uncertainty. It
provides a basis for m e r achieving DFV under uncertainty, especially the achieving of
%alable design over the range of design prefaence [Afin, 11 subject to given flexibility
4.4Comparison Between Sequential and Simultaneous Approaches
Sequential and simultaneous approaches represent two alternate resolutions to a
design problem in the presence of both uncertainty and flexibility. However, they also
represent two difleerent ways to treat uncertainty and flexibility in design. At this point,
they should be applied alternative& according to an application scenario. Toward this
end, this section reviews and compares these two approaches in three aspects to guide
their usages in practice.
In view of solution strategy, the sequential approach is based on robust design
theory to treat desip variations induced respectively by uncertainty and flexibility. To
account for the presence of uncertainty and flexibility in design, this approach attempts to
deactivate the induced design variations that possibly cause the violation of a design
constraint to fail a design. To avoid such a defective design, the constraint bounds are
shified toward shrinàuge of the feasible design space with the same amount of the
resultant design variations as to compensate the design constraint(s). Upon the reduced
solution boundary, the resulting desip is sustainable when subjected to the two sources
of design variation. In contrast, the simultaneous approach is based on the notion of Fuzzy
probability âom f b y set theory in an effort to incorporate the handling of uncertainty
into flexibility modeling. In this approach, the uveragz'ng operation through "expectation"
is used to treat the uncertainty-present flexibility modeiuig. However, due to the essence
of "expectation", only the mean aspect of uncertainty modeling is emphasized with
neglect of other aspects such as standard derivation. This may lead to the situation that
the mean quantity satisfies design tequirements but the aciual quaatity falls outside the
feasible design domain. From this point of view, therefore, the solution strategy of
simultaneous - -- approach is relatively aggregative as compared to the one of sequential
approach that is consewative.
In view of modeling approach, the narne of sequential approach reveals the
treatment of uncertainty and flexibility in sequence with a two-step procedure; in
contrast, the name of simultaneous approach refers to the concurrent handling of
uncertainty and flexibility in a single-step procedure. In the sequential approach, the
modeling of uncertainty takes place first with incorporation into the nominal fom of
design attributes, leading to the resulting fonn of uncertclinty-presenf design amibutes;
then, based on the unceriainty-present desip attributes, the modeling of flexibi li ty is
carried out in which the total robust design solution is used as a benchmark (called
nominal design) for constnicting the uncertainry-present flexibili ty characteristic
functions. In the simultaneous approach, on the other hand, the modeling of uncertainty
and flexibility is perfonned together through the expectation (E) operating ont0 the
nominal flexibility characteristic hctions (p) whose constructions are based on the use
of the standard optimum design solution as a benchmark (i.e., nominal design).
In view of design computing, the sequential approach involves the evaluation of
the 2"dsrder derivatives to design attributes when quantifjing the uncerfainty-ptesent
flexibility characteristic fûnctions. Although this is not required in the simultaneous
approach, the numerical integration is demanded to operate instead in order to evaluate
the expectation of flexibility characteristic function for each design attribute.
Furthemore, the simultaneous approach also requires the calculation of Ac!, for al1 must
attributes prior to design computing, which adds an extra computing load to the
implementation of the approach.
To summarize, Table 4.1 tabulates the above discussions that serves as a guideline
ta the practical usages. Either approach should be applied to accommodate an application
scenario in reality.
--A& a
Table - . - 4.1 Cornparison of sequential appmach and simultaneous approach Cornputing Complexity
Evaluation of the 2d- order derivative to an attribute hction a
(1) Evaluation of the integral of an attn'bute hction; (2) Calculation of for must attributes
r
- -A-*- y
' ~o te : Nominal design rcfers to the one obtaintd without consideringjlexibility in design, which is used in the flexibility modeling for a dcsign attri'butc.
Nominal D a i g ~ ' ( x,, Po
Total mbust design ( F a by solving Eqs. (2.34) - (2.37) Ordinary optimum design (fd by solving Eqs. (2.1 1) - (2.1 3)
Modeling Procedure
Tw-step
Single-step
Scpuentid Approacb
SlmulClncous Approieh
Solution Tmtment Strattgy
Comerwative stcategy based on mbust design theor y Aggtessive stmtegy bascd on fiizzy probability h m fiizzy set thcory
Design For Variety to Achieve Responsive Design Customization
5.1 Introduction
In response to rapid market changes in an engineered product, the needs of rapid
product customization become more signifiant than ever. Though meeting such needs
require an investigation into the life cycle of product development, one way of fûlfilling
rapid pmduct customization is via responslw design customization by expediting the
design computing process, as arguecl in Chapter 1. Dnven by this motivation, the aim of
this chapter is to develop model-based DFV approaches to achieve responsive design
customization.
The existence of flexibility in design mots the possibility of achieving design
variety and consequently responsive design customization. The modeling of design
flexibility, discussed in the preceding chapters, grounds the methodology of DFV on a
scalable design space necessary for realizing responsive design customization, that is,
generating a set of desired design variants in a responsive marner to accommodate the
needs of ropid pmduct customization. By incorporatiag design flexibility into design
formulation, the flexibility ernbedded design models make it possible to offer such a
scalable design space in that the fùrther application of DFV approaches enables the
creation of a farnily of d e s i g or the derivation of design variants (or derivatives) from a
core design. in this context, DFV nfers to the methodology of achieving responsive
design customization through the development of a model-based design platfonn by
which, once needed, mtomized design variants can be generated easii'y and responsively.
As argued throughout this thesis, the strategy adopted here for achieving
responsive design customization is based on the notion of concurrent design, that is,
integmting munuof panuneter selection into automatic design generation through the A
modeling of design flexibility into design formulation. Inasmuch as both concurrent and
automatic design cornputhg is concerne4 this treatment attempts to uni@ Phase 1 (i.e.
design selection) with Phase 2 (Le. design automation) through design flexibility
modeling. This consequence shortens the cycle of design computing, thereby expediting
the computationai design process.
This chapter is dedicated to presenting the details of DFV approaches for
achieving responsive design customization. In particular, the layout of this chapter is
arrangeci as follows. Section 5.2 will discuss the methodology of DFV without
c o n s i d e ~ g design uncertainty, while Section 5.3 will m e r elaborate on DFV in the
presence of design uncertainty according to two modeling appmaches (Le. sequential and
simultaneous). The discussion of Section 5.4 will be devoted to the development of a
model-based design platfonn in the context of scalable design and custom design for
achieving responsive design customization
5.2 Design For Variety @FV)
DFV hinges on the existence of flexibility in design, as descnbed earlier. The
modeling of design flexibility, however, is made possible when in association with
preference modeling. In Chapter 3, such an essence has been revealed that the room of
design flexibility is inversely proportional to the degree of design preference. This
relationship can be caphued and conveyed through the use of an appropriate flexibility
characteristic fitnction. Chapter 3 hes pvided various fleKibility chaiacteristic functions
to descnbe, respectively,~exibfe design panuneters,/ImexlbIe fùnctional requirements, and
(induceâ)jïaibIe design attributes.
Using the flexibility characteristic hurctions as a baseline, a variety of design
models hcorporating flexibility have been developed in Chapter 3 and Chapter 4. These
design models, nevertheless, can be categorized into two renarios according to the
treatment of design preference; that is, those (1) by which the design process is dictated
toward the maximization of an overall design preference (A), and (2) those by which the
design process is dictated toward the optimization of an objective h c t i o n subject to a
given level of preference (a). in scenario 1, design preference is coupled between the - A - . & - - - -
wish amibute (design objective) and the must attributes (design constraints) for the
presence of flexibility in both classes of design attribute. In scenario 2, design preference
is uncoupled between the wish and must attributes for the presence of flexibility solefy in
same must attributes (but not in the wish attribute). With this view, the development of
DFV appmhes is deployed accordhg to prefemice coupling or uncoupling.
5.2.1 DFV with Preference Coupling
To quanti@ design fîexibility, prefmnce modeling through a flexibility
characteristic hction is used to relate the room of flexibility with desip preference
metrics. Thus the handling of design flexibility is simply to deal with design preference
instead. With this view, the feasibility of obtaining various designs (Le. design variety)
based on flexibility modeling is dependent on the availability of various prefermces in
design. Since prefmnce metrics is modeled mathematically over an intemal irom O to 1,
degree of design preference varies within the range of 0% to 100%; narnely, the
maximum availability of various prefemces in design is subject to an extent limit, [O, 11.
This span not only delimits the muximum space of design variety upon the maximum
desip flexibility, but also characterizes an ideal DFV situation in which the most design
variety is attained upon the adjustment of design preference (hm 0%) up to 100%.
In the case of preference couplin& both a whh and m m attributes in a design
problem are assumed flexible. However, design preference over the wish and must
attributes may not be able to reach tmity munimousty. This is because the wish and m m
attributes rnay be mutuaily competing in design. That is to say, an individual preference
on the wish attribute may not be able to achieve 100% shultaneourly with 100%
preference on the must attributes. ûr rather, the ovetall design preference (denoted A) by
aggregating both the wfsi, and maut attributes is most likely lower than 100% but located
between O and 1, i.e. R €[O, 1). The inconsistency of wish and must attributes in view of
design preference requires the hding of the ma« allowable preference for them to reveal
the best aggregated (or overall) design preference (denoted A, ). As such, the space of
design - - variety can be characterized through a range of design preference as [O, A,,].
Accordingly, a variety of designs (X, P) can be readily generated by simply varying A
over [O, A,, ] through the adjustment of A,, , accorâing to
Find [X,P,A]
min
s.t. A s p,(X,P)
A ~ P J X ~ ' ) i = I,2, ....... m
........ W ~ P ) h =1,2 t
........ gr (X I b, j =1,2 A
xi S x, S x i k =1,2 ,......, q
where A,, E [O, Am] is a scalar parameter assigned prior to daign computing; A,, is
obtained by solving the desip problem formulated in Eqs. (3.14) through (3.19). Note
that Eqs. (5.1) to (5.6) represent the DFV model in the case of preference coupling.
Based on the above design model, a two-stage approach is suggested to deal with
DFV in the case of preference coupling:
Stage 1 : Find A, using the design mode1 aven in Eqs. (3.14) through (3.19).
Stage 2: Generate a variety of designs using the design mode1 given above by
adjusting A , over [O, A- ] appropriately.
As a result, a family of various designF can be generated efficiently with variously pre-
assigned values of A,, subject to rl, E [O, A, 1, achieving the transformation fiom
"design flexibility" to "design variety". Since such design alternatives are al1 derived
essentially h m the same design model, such a comrnon formulation base is referred to
as a moùel-based design platfom, which will be fiuther addressecl in Section 5.4. As can
be seen, a variety of designs are produced without any change to the model fundarnentals.
- - -
5.2.2 DFV with Prefereace Uncoupîing
in the case of preference uncoupling, only the m u t attributes in a design problem,
other than the wish attribute, are assurned/lexibe to account for the existence ofjlexible
design pararneter(s) or hctional requirement(s) in the design cons traints related.
Therefore, the span of design preference ranges perlcti'y fiom 0% to 100%, i.e., [O, 11.
This span over which a design can be adjusted to achieve its variety is broader than [O,
A- ] in the case of preference coupling (where A,, < 1). Accordingly, an extent to
design variety also reaches its maximum to match the available room of flexibility
associated with theflexible must attributes. This reveals that a variety of designs upon
given flexibility are attainable through the adjustment of an assigned preference (a) to a
design over [O, 11.
Two design models presented in Chapter 3 have lent themselves to achieving
design variety under two circumstances. In specific, the design model given in Eqs. (3.26)
through (3.30) is dedicated to DFV in the case of preference uncoupling due to the
presence ofjlerble design parameters soleiy in the musf attributes. On the other hand, the
design model given in Eqs. (3.36) to (3.39) contributes to DFV in the case of preference
uncoupling due to the association ofmble FRs with the m u t attributes.
To implement these two models, a single-stage approach to DFV is devised that
simply requires an adjustment of the prescnbed level of preference (a) over [O, 1 ] to
obtain various designs. It is noteworiby that these two design models are also considered
as a model-based design plat,$iom on which a family of design alternatives can be
produced with ease and efficiency (to be fùrther discussed in Section 5.4).
5 3 Design for Variety @FV) under Uncertainty
Under this circumstance, a design problern involves not only design fiexibility but
also design uncertainty. This section devotes attention to DFV under uncertainty;
specifically, discussions are extended to encompass two modeling approaches: sequential - -
approach and simultaneous approach.
5.3.1 Sequential Approach
As addressed in Chapter 4, sequential approach adopts "superposition" with
attempt to mat unceriainty and flexibility in sequence according to the two-step
procedure descnbed in Section 4.2. With such treatment, design flexibility modeling in
the presence of uncertainty is analogous to that in the absence of uncertainty. In this
modeling approach, the uncertainty-present attribute fuicctions (bath wish and niust) are
used to replace those nominal ones in the context of the problem handling, which is the
only exception that makes it différent h m what has been given in Chapter 3 without
considenng uncertainty. As commented in Section 4.2, design flexibility modeling with
uncertainty by the sequential approach can be viewed as an extension to design flexibility
modeiing with no uncertainty.
Likewise, the handhg of desip variety under uncertainty is also vieweâ as an
extension to the handing of design variety with no uncertainty. This means that the results
obtained in the former section can be extended directly to DFV under uncertainty where
the nominal attribute huictions V; gi) in Section 5.2 are replaced with those uncertainty-
present ones (F, Gr) according to Eqs. (4.1) and (4.2). What follows is to address
uncertainty-presmt DFV according to preference coupling or uncoupling in view of
sequential approach.
5.3.1.1 DFV with Preference Coupiing
In the presence of preference coupling, the availability of vurious designs relies
on a range of design preference subject to [O, A, 1, as argued in Section 5.2.1. The
current mode1 of DFV is a cornterpart to that in Section 5.2.1 and therefore, can be
formulated with refe~nce to Eqs. (5.1) to (5.6). In this derivation, it should be noted that
(F, Gi, ,) are used to substitute V; gr,,), as justified earlier. As a nsult, it yields
Find [X, P,A]
min (A-&)*
s.t. ASpF(X,P)
........ A P G ~ (X, i = 1,2 m
........ G j ( X ) = g j ( X ) + A g j ( X ) I O j = 1 , 2 n
........ 1 ~ P J P ) h = 1,2 r
xi 5 x, S xi k =1,2 ,......, q
where II , E [O, A-] is a scalar parameter assigned prior to design computing; A,, is
obtained by solving the design problem fomulated in Eqs. (4.8) through (4.13). As can
be seen, a variety of designs (X, P) can be readily generated by simply varying A over [O,
A- ] through the tuning of an adjustable parameter (Ao 1).
A two-stage approach is suggested accordingly to implement the above DFV
model:
Stage 1:
Stage 2:
Find A,, using the design mode1 given in Eqs. (4.8) through (4.1 3).
Generate a variety of designs using the design model given above by
adj usting i& over [O, A- ] appropriately.
Likewise, since various designs can be generated altematively h m a common model
base, the above DFV mode1 is aiso considered as a moàef-based design pfatjorm (to be
M e r addresseci in Section 5.4). As can be seen, a variety of designs are produced
without any change to the model hdamentals. The design rnodel given above treats
uncertuinty andmbility in design sequentially based on a conse~vutive solution strategy
(see Section 4.4).
5.3.1.2 DFV with Preference Uncoupling
In the absence of preference coupling, the availability of various d e s i , has afill -
range of desip preference over [O, 11, as pointed out in Section 5.2.2. Under such a
circumstance, two desip rnodels derived from the sequmtial approach in Chapter 4 have
lent themselves to DFV under uncertainty to account for two application scenarios.
Specifically, the desip mode1 outlined through Eqs. (4.14) to (4.18) is aimed at DFV
with uncertainty in the case of preference uncoupling due to the presence offlexible
desip parameters solely in the must attributes. On the other hand, the design mode1
outlined through Eqs. (4.19) to (4.22) is aimed at DFV with uncertainty in the case of
preference uncoupling due to the association ofmible FRs with the rnust attributes.
A single-stage approach to DFV is suggested to apply these two design models
for computation implementation. That is, to obtain a family of various designs, it simply
needs to settle, before design execution, the adjustable parameta (a) whose value can be
huted over [O, 11. It is noteworthy that these two desip models are also considered as a
model-bused design plarform on which a family of design alternatives can be produced to
account for both uncertuinty andflm'biliiry in design (to be further discussed in Section
5.4).
5.3.2 Simultaneous Approacb
As addressed in Chapter 4, sirnultaneous approach is centered on the notion of
mathematical expectation of a memôership fhction (or nizzy probability) h m hzzy set
theory (see Appendix A) for treating uncertainty and flexibility simultaneously. The
mathcmatid cxpectation of a mmbtnhip h t i o n is expresscd in such a fom ss E[p
(-11, in which Hm) is to characterize "desip flexibility" through preference metrics and
E[-] to account for "design uncertainly" through the averaging of probabilisticall y
distributeci values. With such treatment, design flexibility modeling in the presence of
uncertainty is analogous to that in the absence of uncertainty by simply adding the
averaging operator (E) ont0 the top of a flexibility characteristic function (p) in the
context of design modeling, as elaborated in Section 4.3. Therefore, design flexibility
moâeling with uncertainty by the simultaaeous approach c m be viewed as an extension
to desip flexibility modeling with no uncertainty.
----= - & - Likewise, the handling of desip variety under uncertainty is also viewed as an
- -
extension to the handing of design variety with no uncertainty. That is, the results
obtained in the previous section can be extended somehow to DFV under uncertainty with
such a treatment that the nominal flexibility characteristic functions p (9) in Section 5.2
are replaced using the f o m of mathematical expectation by E[p (*)le What follows is to
address DFV with uncertainty accorâing to preference coupling or uncoupling in view of
simultaneous approach.
5.3.2.1 DW with Preference Coupling
In the presence of prefeience coupling, the availability of various designs depends
on a range of design preférence, as has been argwd earlier. Nevertheless, the treatment of
uncertainty in simultuneacs appioach ernploys an uggressive solution strategy as
compared to sequential approach. In Chapter 4, it has been revealed that the lowest
degree of design prefaence ( A , ) in the simultaneous approach cannot be zero but
Ad > O, as opposed to A, = O in the sequential approach. This means that the range of
design preference to attain design variety is characterized as [ Â, , A,, ] subject to Anun >
o. Using the DFV model given in Section 5.2.1 as a baseline, the simultaneous
approach leads to an extended DFV model with incorporation of uncertainty into
jkibility modeling, according to
Find [ X , P , A ]
min ( A -&)'
-.Lx. . -& -
where rl, E [Adn , A- ] is a scalar parameter assigmd pnor to design computing; Ad, is
determineci using Eq. (4.47) that M e r teferences Eq. (4.33), and /1,, is obtained by
solving the design problem formulated in Eqs. (4.41) through (4.46). As can be seen, a
variety of desigas (X, P) c m be readily generated by simply varying A. over [ A,,, , A, ]
through the tuning of an adj ustable parameter ( 4, ). A ihree-stage epproach is suggested accordingly to implement the above DFV
model:
Stage 1 : Determine { A$!, , , K K , AL:,,) } throughout allflerible must attributes
using Eq. (4.33) and thus obtain A, according to Eq. (4.47).
Stage 2: Find R, using the design mode1 given in Eqs. (4.41) through (4.46).
Stage 3: Generate a variety of designs using the design mode1 given above by
adjusting rl, over [ A ~ , , , A- ] appropriately.
Likewise, since various designs can be generated alternatively from a common model
base, the above DFV model is also considend as a model-based design plarfonn (to be
M e r addressed in Section 5.4). As cm be seen, a variety of designs are produced
without any change to the model fbndamentals.
5.3.2.2 D W with Preference Uncoupling
In the case of preference uncoupling, the availability of various designs has a
brouder range of design prefereuce [Â, , 11 as compared to [A,,,,, , A, ] in the case of
preference coupling. Under the same circurnstance, however, the range of [ ilmi2,, , 11 in the
sirnultamous approach, due to its adoption of an aggressve solution strategy, is nuwower
than that of [O, 11 in the squential approach. Two design models denved h m the
simultaneous approach in Chapter 4 have lent themselves to DFV under uncertainty to
account for two application scemrios. In particular, the desip model outlined through
Eqs. (4.53) to (4.57) is concerned with DFV under uncertainty in the case of preference --A--- -- > -
uncoupling due to the presence offlex'ble design parameters solefy in the must attributes.
On the other hand, the design mode1 outlined through Eqs. (4.59) to (4.62) is concemed
with DFV under uncertainty in the case of preference uncoupling due to the association
offlexible FRs with the mwt attributes.
A two-stage approach to DFV is suggested to execute these two design models
according to
Stage 1: Determine { A:!, , , K K , As)) ) throughout al1 flexible must attributes
using Eq. (4.33) and thus obtain & according to Eq. (4.47).
Stage 2: Generate a variety of designs, by tuning the adjustable parameter (a) over
[ A , 11 appmpnately, using Eqs. (4.53) to (4.57) in case of flexible
design parameters or Eqs. (4.59) to (4.62) in case of flexible functional
requirements.
It is noteworthy that these two design models are also considered as a model-based
design plarfom on which a family of design alternatives can be produced to account for
both uncertainty andjlexibility in design (to be M e r discussed in the next section).
5.4 Responsive Design Customization
This section elaborates platforni-based approaches for responsive design
customization through DFV. Toward this end, the notion of design platform will be
introclucecl first. Then, centered on this notion, how to formalize a model-based design
platfonn to achieve sculable design will be describeci with diffennt procedws according
to design scenarios. Later, a platform-bas& approach for satisfaction-driven custom
design will be presented to account for typical customization scenarios.
5.4.1 Model-based Design Platform
-- in Uiis thesis, design platfonn is dehed as a central base of engineering models
- -
on which a set of design variants can be derived and configured fiom a core design in a
cost-effectively and time-efficient fashion. n i e DFV models presented in the last two
sections grounds a design platfom on the basis of design flexibility. That is, the
modeling of design flexibility makes it possible to achieve DFV, which in tum
contributes to fomalizing a design platform. Extending this view, the DFV models
derived in Sections 5.2 and 5.3 can be M e r treated as a variety of model-based design
plat forms to account for di fferent design scenarios . Each design platform enables to deliver a scafabfe design and consequently a
family of design variants derived h m a desired core design with different scales. Herein,
the notion of core design refers to such a design that is generated against the &est (overall)
design preference, and thereby representing the most desirable design. in particular, two
cases are encountered according to whether preference coupling or uncoupling is present
in the wish and must attributes. (1) In the case of preference coupling, the core design is
defïned as that corresponding to A- (Le., A = A- ). (2) in the case of preference
uncoupling, the core design is defined as that corresponding to uniiy (i.e., a 400%).
In the case of preference coupling, a design platform is fomalized according to
the following three-step procedure, i .e.
Step 1: Develop an appropriate DFV model according to the related approach described
in Section 5.2 (with no uncertainty) or Section 5.3 (with uncertainty).
Step 2: Determine a core design ( X' , P' ) by maximizing the overall design preference
(A) such that ( X' , P* ) is found at A =A,, (denoted 1' = AmUr ).
Step 3: Generate a f h l y of design variants centered on (X*, P* ) to achieve scalable
design using the same DFV mode1 by scaling A,, over [O, A, ] or [ A,,,, , Am ] in
hnn from A- to O or A, (subject to the DFV model selected).
Lkewise, in the case ofpreference uncoupfing, e design platform is fomalized according
to the following three-step procedure, Le.
Aggressive stratcgy via
simultaneoics approach
Table 5.1 A Guideline to Selection of Design Platforms
Eqs. (5.1 MS.6) 5 DPUCl Eqs. (5.7H5.12)
DPUC2 Eqs. (4.14H4.18) 1
DPUA2 Eqs. (4.53)-(4.57) 5 DPUA3 Eqs. (4.59)-(4.62) I
Sourct of Dc8ign Varkty
Fkible dtsign parameters in boih
wisA and mwt
pammtcrs in mut aîûibutes solely
Flaible fiuictional requiremtnts with
m u t atcributcs Flexible design
paramctnsinborih wish and must
a t î r i i Flexible design
panuntters in mus2 attributes solefy
Flexible functioaal requiremc~lts witb
muFI attnhtes Flaibfe design
parameters in both wish and must
athributes Flexible design
parameters in must attributes solelv
Flexible fiuictional requirements witb
musr attributes 'Note: ( A * , a* ) represent the degm of design preference dictated at the core design; ( 4, a) are adjustable scalar panmeters to characterize scalable design.
Prcference coupüng
Range of -W0
D W ~pproicb
Core -O
Step 1: Develop an appropriate DFV mode1 accordhg to the related approach described A--- -&-
in Section 5.2 (with no uncertainty) or Section 5.3 (with uncertainty).
Step 2: Determine a core design (X0,P') by prescribing the level of preference (a) to
design such that ( Xe, Pm ) is found at a 400% (denoted a' = 100%).
Step 3: Generate a fmily of design variants centered on ( X* , P' ) to achieve scaIabie
design using the same DFV model by scaling a over [O, 11 or [ AM,, , 11 in turn
h m 1 to O or A, (subject to the DFV model selected).
To facilitate the application of a proper design platforrn, Table 5.1 provides a
guideline for the coverage of various (scaiable) design scenarios. It is important to note
that, in Table 5.1, "Con Design" and "Range of Scaiable Design" are both charactenzed
through design prefaence. The design platforms summarized according to Table 5.1
enable design to be scaluble, based on which responsive design customization is made
possible to achieve (to be discussed in the next section). Chapter 6 will illustrate the
usage of these design platforms through the shidy of a design exarnple.
5.4.2 Platform-based Custom Design
The modeling of flexibility into design formulation enables the achieving of
design variety through computerized scalable design based on an appropriate design
plat fo m. In scalable design, a varîety of designs are generated upon a design platfom b y
scaltng a design preference pawneter ( 5 or a) adjustable over a certain range (see
Table 5.1). In such a design pmcess, the design preference parameter (i.e. ilo or a), as
opposed to the (actual) design parameters, is prescribed first, and then a corresponding
design solution (X, P) is generated. This effort makes it possible to further achieve
responsive design customization tbrough computerized m t o m design, b y which
customized design variants cm be generated to fit specific needs to a customer. It is
therefore the aim to propose a platfonn-based approach for satisfaction-driven custom
design. Toward this end, two customization scenarios in design will be discussed as
follows.
----. 2 - --- - .
Scenario 1: Custom Desinn Parameters
In this scenario, assuming that specifications on certain design parameters are
ctcstom-made, it is to seek a custom design based on a proper design platfonn to besr
meet the customer's satisfaction in ternis of design preference.
Given custom values for a set of selected design parameters ( p, , pl , . . . . . ., p, ),
upon the availability of their associated flexibility characteristic functions ( p, , p, , . . . . . ., pm ), the individual prefrrence ranks cm be detennined accordingly with respect to each
custom design panmieter. With this view, the design pmcess starts with the detemination
of degree of preference, according to the mappings prescribed into the flexibility
characteristic fimctions, for each custom design parameter. This ends up with a set of
custom prefemice ratings, denoted (A, , 4, ....., A,,, ). Thereafter, an overall custom
preference rating (Ao) is fiuthm determined by aggregathg al1 the individuai custom
preference ratings according to
where "min" operator is used for the aggregation of custom design preference; 4 is the
aggregated design prefaence corresponding to the custom design parameters selected,
representing the minimum extent of preference to which the custom design must satisQ
as an enforcd requirement. It is important to note that, to ensure feusible design
customization, the resulting value (4) must be validated through a validity check against
the design platform in use. This checkhg is to verify whether do falls within a specifid
mge of scalable design (listed in Table 5.1) pertaining to the design platform in use. By
the validity check, an invalid A, should be rectified through adjusting the custom values
of the selected design parametem.
ûnce rt, is obtained subject to the vaiidity check, what follows in the custom
design is to tailor the design prefemice parsmeter ( A or a) to achieve the desired design
-- L 2s. A-
by dictating the design process, according to the design platfom in use (see Table SA),
such that the resulting overd1 design preference (denotd Ai) matches A. (i.e., Ai -+ for the case of prefmnce coupling or a = A. for the case of preference uncoupling). This
solution process is similar to that of scalable design but A, in custom design is
customized with Eq. (5.19) as opposeâ to being scaled in scalable design. Furthennore,
an additional set of constraints subject to given custom design parameters is irnposed
ont0 the design platfom selected, during the solution seeking process, according to
where p:' denotes the custom value specified on a design parameter @); E is an
infinitesimal number, i.e. E + O. This additional set of constraints dictates those custom
design parameters to stay faed (on their custom values) during the design process.
Subject to this additional set of constraints plus other constraints associated with a DFV
mode1 (listed in Table 5. l), the custom design solution can be sought out upon the design
platfonn selected.
In the case of preference coupling, however, it may not be always guaranteed that
the overall design preference found (Ai) is absolutely consistent with the custom design
preference (4) dictated at the design, Le. (4 - 41 >> 8 . This is perhaps due to the
reason that, for example, (1) the wish and must attributes may be mutually conflicting in
view of k i r design preferetlce OF (2) the c h o k of custom values on some design
parameters may be coupled with design preference. Therefore, the custom design solution
needs to be fùrther validated to ensure that Ai agrees with A, in ternis of 14; -A, ( '. E .
To rectify an invalid custom design solution such that 2; +A,, , it is suggested that the
additional set of constraints on the custom design panmeters, expressed in Eq. (5.20), be
relaxed by having the design process detamine their values as if it were one scalable
design.
--AL
To summarize, Figures 5.1 and 5.2 show the custom design procedure for the case
of preference uncoupling and coupling, respectively. Based on the computerized design
platfomis summarized in Table 5.1, a custom design can be obtained with ease and
efficiency by simply following the design procedure presented in either figure.
SpeciS, custom design parameters (pi, pz, . . . p,) and their values.
r 7
(1) ûetermine mtom prcference ratings for each custom design parameter (Ri , A, ... A), according to their prescribedflexibility characteristic firnctiom (pi, fi, .. . k).
(2) Detemine the overall custom design preference RU according to A, = min {A,, A, .. . A,,,}.
(3) Check validity of against the design platfonn in use, according to a sptcified range of scalable design listed in Table 5.1
Find the custom design dictateâ by =a on the selected design platform h m Table 5.1, subject to one additionai
set of constraints on the custom design parameters, Le. 1 [p-pC"]tsa (h= 1,2, ..., in)
Figure 5.1 Design procedure flowchart for the case ofprefeence uncoupling
subject to giwn custom design parumeters
SpccifL autom design parameters (pi, p, . . . p.) and their values. l
(1) Determine ctrstom preference ratings for cach custom design paramctet (A,, A, .. . A), according to their ptescribcdf7exibifity characteristic finciions u,, f i , . . . fi).
(2) Detemine the overall w t o m design preference A, according to A,, = min (Ai, A, . . . L) ,
(3) Check validity of & against the design platfonn in use, according to a specified range of scalable design listed in Table 5.1
Find the custom design based on the selected design platfonn h m Table 5.1, subject to one additional set of constraints on the custom design parameters, i .e. 1
,-, Yes / Solution validation \ End 4
Solve the design problem again by reluring that additional set to have the computerized design process.
Suggest values for the custom design parameters.
Figure 5.2 Design procedure flowchart for the case ofprefieence coupling
subject to given custom design parameters
----- - Scenario 2: Custom Functional Reuuirements
In this scenario, a s s d g that specifications on certain finciional requirements
are mstom-made, it is to seek a custom design based on a proper design platform to best
meet the customer's satisfaction in terms of design preference.
Given custom bounding values (4 , 4, . . . . . ., b,,, ) for a set of selected functional
requirements, upon the availability of their associated flexibility characteristic hctions
(,us,, pg2, . ....., p,), the individual preference ranks can be determined accordingly
with respect to each custom bounding value. With this view, the design process starts
with the determination of degree of preference, according to the mappings prescribed into
the flexibility characteristic fictions, for each custom fûnctional requirement. This ends
up with a set of custom preference ratings, dmoted (a, ,a2, . . ..., a, ). Thereafter, an
overail custom preference rating ( 4 ) is further detemined by aggregating al1 the
individual custom preference ratings according to
A, = min {a,, a, ,......, a,)
where "min" operator is used for the aggregation of custom design preference; A, is the
aggregated design preference corresponding to the custom fuactional requirements
select& representing the minimum extent of prefercnce to which the custom design must
satisfy as a mandatory requirement. Likewise, to ensure a feasible custom design, a
validity check against the design platform in use should be carried out to validate the
value of A,, , subject to a possible rectification.
Once i&, is obtained subject to the validity check, its value is then used as the
prescnbed level of preference (a) dictated on the relevant design platforms, outlined in
Table 5.1, where a s 4. Figure 5.3 shows the custom design procedure subject to given
custom finctional requirements. Based on the computerized design platfoms, design
customization can be accomplished time-efficiently and cost-effectively with the aid of
computer.
Spccitj. crutom functional requirements and their bounding values (b,, h, . . . 6,). 1
( 1 ) Determine CtLTtom prefcrcnce ratings for each custom design hctional requircment (a,, a, ... G), according to their prescribedflexibility choracteristicftnctions hi, h, . . . 4-1-
(2) Determine the overall custom design preference A, according to a= = min (A,, A, ... A}.
(3) Check validity of A,, against the design platform in use, according to a spccified range of scalable design listed in
Find the custom design dictated at a = Ro on the selected design platfonn h m Table 5.1, subject
to given w t o m firnctiond requirements. 1
Figure 5.3 Design procedure flowchart subject to given custoin finctional
requirements
Case Study
6.1 Introduction
A welded barn design shown in Fig. 6.1 is utilized in this chapter to demonstrate
the proposed methodology of DFV, and also to show how to achieve custom design
besed on the design platfoms developed. This example is simple yet illustrative through
which the methodology will be validated, and applicability and utility of each design
mode1 will be testeà according to diffmnt design cases presumed. The focus of this
stuây is on the illustration of the design methdology but not on the results of the design
example, per se.
In the following sections, the desip problem will be stated first, and then a family
of design alternatives will be generated through the application of the DFV models
proposed earlier on. Thus, the scalable design of achieving DFV cm be tested to validate
these design models developed in the previous chapters, along with the discussions of the
desip results. Finally, design customization will be demonstrated based on the solution
procedures given in Chapter 5; particularly, two design scenarios will be illustrated: (1)
custom desip parameters and (2) custom fûnctional requirements.
6.2 Problem Statement
As illustrated in Fig. 6.1, there are four design variables, which are the depth of
the weld (h) , the length of the weld (1). the height of the beam ( t ) and the thickness of
the beam (b) ; there are three design parameters, which are the length of the beam (L) ,
the applieâ load (F,) and Young's modulus (E) plus the shear modulus (G) that is not
show in the figure. In this design study, t h m are six design functions as fomiulated in
Appendix B. where two design constants (matenal costs c, and c,) are involved along
with the design variables and parameters.
- - - In order to demonstrate the - - applications - of DFV and responsive design
customization, three design cases are considered and examined in this case study. Case 1
is concerneci with the design problem that involvesflexible design parameters in both the
wish and mu t attributes under Scenario 1 given in Table 6.1, where "rigidity" is taken as
a wish attribute. In this case, the other five design hct ions are treated as must attributes,
including the shear stress in the weld (r-) , the bending stress in the beam (s,), the
geornetry limit (e, ) , the welding cost (cm ) and the buckling load on the beam (Fm ) .
Case 2 is concemed with the design problem that involves flexible design parameters
solely in the must attributes subject to Scenario t given in Table 6.1, where "welding
cost" is taken as a wish attribute with t h e m d length of the beam (L). In this case, the
other design functions are treated as must attributes. Case 3 is concemed with the design
problem involvingflexible functional requhnents only. In this case, "rigidity" is taken
as a wish attribute in the formulation under Scenario 2 given in Table 6.1, where d l the
design parameters are assumed fued. B a d on optimization fomialism, al1 these design
problems are fonnulated in Appendix B.
In this design example, DFV with no uncertainiy and DFV with uncertuinty are
studied, respectively, depending on whether or not "tolerance" is associated with the
design components (refcr to Table 6.1). Usually such design uncertainties follow a
nomal distribution, as assurned in this study. Besides, it is also hypothesized that design
fiexibility mots h m either the design parameters or the bctional requirements but not
h m both. The former situation, as shown in Table 6.1, is named Scenario 1, while the
later is narned Scenarîo 2. Table 6.1 lists the design data necessary for this study.
6.3 Scalable Design
DFV will be demonstratecl in this section based on the design platfonns shown in
Table 5.1 and the data given in Table 6.1. DFV without/with uncertainty will be worked
out subject to the cases of preference coupling and uncoupling. In order to do so, both the
ordinary optimum design and the total robust design are detemiined first, and their results
are tabulated in Table 6.2. These results will be utilized not only for calculating the
s p d of transmitted flexibility as well as the thresholds of must amibutes (see Figs. 4.5
(b) and 4.6 (b)), but also for carrying out a comparative study later on.
- - --
6.3.1 Results and Discussion of Design For Variety
Based on the guideline of selecting design platfonns showed in Table 5.1,
'Design Platform" is implemented in this section for DFV without considering
uncertainty. The design platfoms coded as DP 1, DP2 and DP3 in Table 5.1 are employed
for the undertaking of Cases 1,2 and 3, respectively.
Using the two-stage approach suggested for the case of preference coupling (see
Section 5.2.1). the highest degree of design preference ( A,, = A' = 0.8756) plus the core
design is determined first according to Eqs. (3.14) through (3.19). Then design variants
for Case 1 are generated based on DPI (see Table 5.1) by tuning the scalar parameter A,
defined over [O, A,, 1, whose results an outlined in Table 6.3. In contrast, a single-stage
approach for the case of preference uncoupling (see Section 5.2.2) is applied to obtain
design variants for Cases 2 and 3, based respectively on DP2 and DP3, by adjusting the
scalar parameter a over [OJ]. The corresponding results are outlined in Tables 6.4 and
6.5. Note that the results highlighted represent a core design obtained at the highest
degree of preference ( R' or a', see Table 5.1).
As demonstrateci, DFV is achieved through design variants, which are generated
based on the design platfonns @Pl, DP2, DP3) without resorting to any extra design
models. The design processes are highly computerized so that there is no need for the
manual design selection during design iterations. Based on the results in Table 6.3, 6.4
and 6.5, observations are given and discussed as follows. Note that, according to Table
4.1, the ordinary o p h u m soIution given in TabIe 6.2 is used as a nominal design to
initiate the design computing process.
Tendencv of Desinn Performance
B Design performances (rigidity, f,) for case 1 showed in Table 6.3 increase
monotonously as design preferences (A,,) increase. This is because case 1 is a
prefernice couphg problem of design rnaxhization, whose design performances are
directly proportional to their corilesponding -- design pteferences in their flexibility
characteristic function (see Section 3.2.2.2 in Chapter 3). From this point of view, the
bigger the design preference, the better the design performance, e.g., the bigger value
of ngidities.
B Conversely, desip performances (cost, f,, ) for case 2 given in Table 6.4 at lower
degrees of desip prefemces are superior to their higher ones such that design
performances increase as the correspondhg desip prefmences decrease. This is a
preference uncoupling problem of design minimization, whose wish attribute in
design platfom of DP2 is not characterizeâ by flexibility characteristic fiction.
Therefore, its design performances are not necessary to change regularly along with
the changes of their design preferences, such as increase or decrease. However, the
lower degree of design preference defines the bigger feasible desip space with more
fkeedorn of choices in design selection and therefore, gives better results of design
performances.
P Design performances (rigidity) for case 3 outlined in Table 6.5 are getting wone
(smaller) as theù design preferences increase. The reason for this is the same as those
explained above since it is desip preference uncoupling problem.
B In case 1 using design platfonn of DP 1, the best design rigidity ( f- ) shown in Table
6.3 is worse than the one obtained in ordinary optimization approach (see Table 6.2).
To this prefemce coupling design problern, flexibility characteristic functions are
used in design platfom, DPI, to account for transmitted design flexibility occurred in
both wish and must attributes. Because of this (taking flexibility into consideration),
the onginally feasible desip space in ordinary optimization approach is shrunk,
which leads to less M o m of choice in searching best design result. Therefore, the
desip result tums out to be worse but more robust.
B In case 2 using design platform of DP2, the least cost ( fh ) given in Table 6.4 is
better than that obtained in optimization apptoach (see Table 6.2). Here are two
a factors-that a é c t the desip cost: U) the shrinkage of feasible design space caused by
transmitted design flexibility, which leads the design result to be worse (bigger), and
(2) more flexible design components such as flexible design parameters involved in
design, which create more fteedom of choices in desip selection towards searching
optimal results. This compensates design with a better chance to malce the cost less.
These two factors interact each other and yield such a desip result.
b In case 3 using design platform of DP3, the worst rigidity given in Table 6.5 happens
as design preference (a) is assigned one. This result is the same as the one obtained
in ordinary opthkation approach (see Table 6.3) since the design becomes the one
with no flexibility that is exactly the same as the formulation of ordinary optimization
when a=l.
Potential to Welded Beam Familv Design
The results given in Table 6.3,6.4 and 6.5 show that al1 designs may be extended
with contribution to a welded beam family design so that a ranged design with a ranged
solutions can be found based on some conunon, stably fixed design components. For
such a design purpose, the shear modulus (G) in design platfonn of DPI may be fixed as
basis for developing the welded beam family design since its results in Table 6.3 only
changes slightly. Also, design platforms DP2 cm, of course, be applied in a same way
since length of the beam (L) is fixed in Table 6.4. Genedly speaking, al1 design
platforms incorporating flexibility of fiuictional quirements in their deign fomulations
are inherently possible to be extended to welded bearn farnfly design since aii the design
parameters are fixed in these design formuîations. Design platfonn of DP3 is such a case
but with an extra component, height of the beam (t), to be fixed since its design results
showeâ in Table 6.5 do not change at dl.
6.33 Results and Discussion of Design For Variety under Uncertainty
- =-a-- - =-- - - In this section, - -- DFV with uncertainty . - - - GA- -- is implernented - - - - - to obtain design variants
based on the design platforms given in Table 5.1. The results are given and discussed in
two sections, nspectively, according to sequential and simultaneous approaches.
6.3.2.1 Sequential Approach
DFV with uncertainty in sequential approach is demonstrated here to generate
design variants by adjusting scalar parameters (either A,, or a ) . As such the design
platfonns coded as DPUCI, DPUC2 and DPUC3 are tested against Cases 1, 2 and 3.
Their formulations are outlined in Table S. 1.
Using the two-stage approach suggested for the case of preference coupling (see
Section 5.3.1 A), the value of R' (0.7531) plus the cored design is determined fint
according to Eqs. (4.18) through (4.13). Then the design variants for Case 1 are generated
based on DPUCl (see Table 5.1) by adjusting the scalar parameter A, over [O, A-],
whose results are listed in Table 6.6. In contrast, a single-stage approach for the case of
preference uncoupling (see Section 5.3.1.2) is applied to obtain design variants for Cases
2 and 3, based respectively on DPUCZ and DPUC3, by adjusting the scalar parameter a
over [0,1]. The comsponding results are outlined in Tables 6.7 and 6.8. Note that the
results highlighted npresent a core design obtained at the highest degree of preference
(A* or a' , see Table 5.1).
According to the tables, it can be found that rnost of the design results follow
those patterns discussed in Section 6.3.1. In other words, the majority of the observations
mede in the Iast section still hoM me. Besides, additional observations due to the
presmce of uncertainty are given and discussed as follows. Note that, according to Table
4.1, the total robust design solution given in Table 6.2 is used as a nominal design to
initiate the design computing process.
Tendencv of Desinn Performance
The tendencies of design perfomauces showed in Table 6.6 and 6.7 are exactly the
same as those explained in Section 6.3.1, except the rigidities ( f ,, ) show in Table
- 6.81hat stay the same* T h i a r n e a n s t h a t ~ d e s i ~ t g i v e n is the most optimal and
robwt and does not change even though the design prefanice changes.
The tendencies of transrnitted design variations (4) given in Table 6.6.6.7 and 6.8
follow the same comsponding panmis as those of design results ( f,, or f,, ) do.
The reasons for these are still dmown and necessary to cany on for M e r study.
Com~arison Based on Different Desien A~~roaches
P In case 1 using design platform of DPUCI, al1 rigidities except one at design
preference of a =O given in Table 6.6 are better (bigger) than the result of total robust
desip showed in Table 6.2. The result at a =O happens to confom to the one in
Table 6.2. This design observation still ma ins uncertain and needs further study to
reveal its real cause.
* in case 2 using desip platform of DPUCZ, some of the design results (cost, f,, j
given in Table 6.7 is better than the result of total robust design showed in Table 6.2,
while some others are even worse. As mentioned previously, more transmitted
variations (induced by design flexibility besides design uncertainty) involved in must
attributes cause the feasible design space of the initial design (total robust design) to
shrink more, which could lead result to be worse. However, two factors could also
make the design result better: (1) flexible design parameters involved in this design
case gives more &dom of choices on searching best design results, and (2) lower
de- of design preference also increase the W o m of choices in such a search.
These three aspects function togethex in generating such design results.
B In case 3 using design platfonn of DPUC3, the rigidities showed in Table 6.8 are so
stable and the design solutions are so robust that even design flexibility involved in
(caud more design variations) in design they still can not be pushed away from the
optimal result of initial design (total mbust design). They stand still and are the sarne
as that obtained in robust design approach.
> Comparing with the transmitted design variations (4) of robust design given in
Table 6.2, al1 results of tnmsmitted design variations outlined in Table 6.6, 6.7 and
6,8 gxactly follow the same pattern as those of design nsults (either f, or 1,)
discussed previously.
Potential to Welded Beam Familv Design
As argued in last section, Yong's moddus (E) and shear modulus (G) in Table 6.6
for design platfonn of DPUC1 may be fixed as the basis to fom welded beam farnily
design. To desip platform of DPUCZ and DPUC3, the same argument made in last
section is still applicable here with no extra fixed components involved.
6.3.2.2 Simultaneous Approach
In this section, DFV with uncertainty using the simultaneous apprcach is tested to
generate design variants by adjusting scalar parameters (either A, or a ). The design
platfonns in the simultaneous approach coded as DPUAI, DPUA2 and DPUA3 are tested
against Cases 1,2 and 3. Their formulations are outlined in Table 5.1.
As before, the highest degree of design preference (A- =0.7967) plus the core
design is detennined fkst for the case of preference coupling (i.e., Case 1), for which the
h - s t a g e approach suggested in Section 5.3.2.1 is applied. Then, according to the
simultaneous approach, the threshold values for each design constraint, A , , , A$!, ,
A;: , A:: , are cdculated (see Table 6.9), through which their minimum is obtained as
A-= 0.3274. This yields the range of d a b l e desip, [0.3274, 0.79691, over which the
scalabîe parameter (4) is adjusted to generate design variants based on the platform
"DPUAI". The design nsults for Case 1 are given in Table 6.10. In contrast, for the case
of preference uncoupling (i.e., Cases 2 and 3). ody the threshold values for each
conseaint need to be calcuiated (see Table 6.9). Consequently design variants, based
respectively on DPUA2 and DPUA3, are obtained by hiniag the scalar parameter (a )
over the range of [Ah, 11 or [0.3274, 11 according to the two-stage approach suggested
in Section 5.3.2.2. The comsponding design results are listeâ in Tables 6.1 1 and 6.12.
--- =--a- - -- - . - According to the tables, it cm be found - - - - that - most of the design results follow
those patterns discussed in Section 6.3.1 and Section 6.3.2.1 . In other words, the majority
of the observations discussed in the last two sections still hold true. Besides, additional
observations resulting h m the use of sirnultamous appmach for treating uncertainty are
summarized and discussed as follows. Note that, according to Table 4.1, the ordinary
optimum solution given in Table 6.2 is used as a nominal design to initiate the design
computing process.
Tendencv of Desien Performance
A different pattern of tendency of design performance is presented in Table 6.1 1
for case 1 using desip platfom of DPUAI. In this table, rigidities increase as the
comsponding design performances decreape. This is because that lower design
preference with more fieedom of choices in bigger feasible design space plays a major
role in design so that to cause such design results.
Cornnarison Based on Different Design A~nroaches
P In case 1 using design platform of DPUAI, the design rigidity ( f,, ) given in Table
6.10 as A,, = 4 is worse than that of ordinery optimization appmach showed in Table
6.2. This is because the feasible design space of initial design is shnink, which leads
to a worse design result as the transmitted design variations are taken into
consideration in design formulation. However, design rigidities are getting larger as
design preferences decrease and hally better than that of initial design when
A, S0.6 in Table 6.10. This is because the lower design preference gives better
fieedom of choice in searching the bat design results.
> In case 3 using design platfomi of DPUA3, design rigidities given in Table 6.12 are
worse than that of initiai design (ordinary optimization) showed in Table 6.2 because
of the shrinkage of feasible design space, even though they are better as design
prefe~ences (a ) decrease. However, the design solutions are robust since tnuisrnitted -- A..--- - - - -- -- - - - -
design variations are considered in this design formulation.
l+ In case 2 and 3 ushg design platform of DPUA2 and DPUA 3, results given in Table
6.1 1 and 6.12 show that there aie no design solutions as design preference (a ) is
assigned "unity". The nasons for this could be: (1) too large design spreads of design
parameters caused by improper design selections are unfeasibly transmitted into
design attributes. In tum, such transmitted design variations cause feasible design
space of initial desip to shrink t w rnuch and therefore, no optimal design solution
can be found as a =l. (2) Sirnultanaus approach is more aggressive than sequential
approach as discussed in previous chaptets. Therefore, the approach discussed in this
section may involve such risk that design does not have rational solution.
Potential to Welded Beam Familv Desien
Design results given in Table 6.10 show that Yong's modulus (E) and shear
modulus (G) in design platfom of DPUAl can be fixed to form welded beam family
design since their results Vary slightly. Besides, the arguments made in last two sections
are still applicable to design platforms of DPUA2 and DPUA3 in this section based on
the design results showed in Table 6.1 1 and 6.12.
6.4 Custom Design
In the last section, design variants are generated baseâ on design platfoms given
in Table 5.1 by adjusting a scalar parameter (4 or a ) over a range. In such a design
process, the scalar parameter is prescribed first, and then a corresponding solution is
calculated later. Nevertheles, oftentimes design variants may need to be customized to
best satisfy specific needs to a customer, for example, a set of design parameters or
functional requirements may be customized. To demonstrate custom design, two design
scenarios discussed in Section 5.4.2 are tested in this section through implementing three
design platforms, based on the simultaneous approach, to examine the t h e cases
prescribed earlia (see Appendix B). Al1 the other design platfoms can be applied in a , - - - , eh-- - - --- - -- - - -
similar way as demonstrateci in this section.
Custom Design Parameters
In this scenario, two design parameters are customized, which are Young's
modulus (E) and shear modulus (G) with the values of 201.85 (GPa) and 87.5 (GPa),
respectively. Using the flexibility characteristic huictions associated with, their
preference ranks against the given values can be obtained, i.e., A, = 0.7037 and
4 = 0.6558 . Based on (A,, A, } , the overall custom design preference ( A, ) is obtained
accordingly with nfmnce to Eq. (5.19), Le.
A. = min (A.,, 4 }= min (0.7037,0.6558 } = 0.6558 (6.1)
As descnbed in Section 5.4.2, two additional constraints are introduced to accommodate
the custornized Young's modulus and shear modulus and thus imposed ont0 the design
platfom, Say DPUA 1, according to
It has been found that the scaIar parameter (4) on DPUAl is adjustable over
[0.3274,0.7967] in Case 1 (see Table 6.10). Apparently, ilo= 0.6558 falls in the range of
[0.3274, 0.79671 and therefore, is valid. According to the solution procedure outlined in
Fig. 5.2, the custom design results can be obtained and are summarized in Table 6.13.
However, the overall design preference found, 4 4.3442, does not match the custom
design preference, A,, = 0.6558, according to
- F A - - - - - -
This is because, for example, the choice of the custom values may be naive due to the
lack of experience h m the customer. In such a circumstance, the customer must resort to
the cornputer to grnerate an optimized design that best matches the overall custom design
preference (65.58%), by relaxing the additional constraints imposed on the (custom)
design parameters, as descnbed in Section 5.4.2. In this custom design, this means that
the two constraints expresseci in Eqs. (6.2) and (6.3) are removed to have the computer
suggest the best appropriate values for the custom design parameters such that the custom
design preference (Le., 4 = 0.6558) is aîtainable. With such an effort, a newly optimized
design is obtained automatically through a computer, whose results are listed in Table
6.13, where
I& - & l a 0
Consider another scenario based on the design platform "DPUA2", by assuming
two custom values, 203.5406 and 84.9159, specified on Young's modulus (E) and shear
modulus (G). Their aggregated design preference is obtained as A0 = 0.7 by using a
similar approach as earlier. According to Table 5.1, this value falls in the range of
[0.3274, 11 theoretically and therefore, is valid. By adding two additional design
constraints ont0 'PPUAî", Le.,
a faible yet optimized design solution, obtained by dictating a =A,, on the design
platform "DPUA2" (see Table 5.1) cm be found, whose results are listed in Table 6.13.
As can be seen, this design solution perfectly meets the custom requirements on Young's
modulus and shear modulus. Nevertheless, it is still possible that, for an inexperienced
customer, some infkasible values may be specified on the custom design parameten. In
this situation, the added eonstraints must be r e l d to have the computer suggest an
optimum design (including the proper values for design parameters) based on the design --- -2 -=,2 2- - &. -- - platform.
Custom Functional Reauirements
In this section, the scenario of custom hmctional requirements is considered and
implemented on the design platform 'PPUA3". Assume two custom bounding values
given to the shear stress ( 2, ) and the welding cost ( c, ), Le., r , = 96.2(MPa) and
r , = 3.1($). Likewise, their comesponding preference ranks are determinecl as 0.7407
and 0.6667, respectively, and therefore, their aggregated preference is 0.6667 based on
Eq. (5.19). The validity check is based on the given range of [0.3274, 11 theoretically.
Obviously, this preference value falls in this range. Based on the computing procedure
illustrated in Fig. 5.3, the custom design resu1ts are obtained and tabulate6 in Table 6.15.
Generally, based on the computenzed design platforms, design customization cm
be accomplished tirne-efficiently and cost-effectively with the aid of cornputer. This is
because (1) additional design models need not to be derived, (2) a custom design solution
may be obtained by interpolation of the existing design solutions within the sarne family
of design alternatives.
Figure 6.1. A welded beam
Table 6.1. Data of welded beam design
Design Componcn t
Tolcrance Initial mw Spmd of '
DFV with DFV wih Feasible Value Symbol value ~icxiiiiity no Range
Unccrtahty Uncertainty (or value)
h (mm) 0.254 [3.1 75,50.8] 10.0 - --
1 (Jm 0.254 [2.54,254.0] 150.0 t 0.0254 [2.54,254.0] 150.0 b(=) 1 0.0254 [2.54,50.8] 10.0 L (mm) 355.6 30 5.9267 [345,375] 355.6
E(GPa) 206.84 25 3.4473 [ 1 90,2 151 206.84
G(GP3 82.737 10 1.379 [80,90] 82.737 L (mm) 355.6 5.9267 355.6
F, (KN) 26.689 0.4448 26.689
E(GPa) 206.84 3.4473 206.84
G (GP,) 82.737 1.379 82.737
( ) 4.5
?, (MPJ 93.769
Sm (MPJ 206.843
&,(mm) 0.0
cm ($1 3.0
F - ( W 0.0
v, (~d) 4.5 0.45 [4.5,4.95]
7, (MPJ 93.769 9.3769 E93.769,
- -- W.----
103.1459]
S, (MPJ 206.843 20.6843 L206.843,
'
227.52731
&,,(mm) 0.0 0.5 [O.O,O.S]
c, ($1 3 .O 0.3 [3 .0,3.3]
F-(KN) 0.0 0.5 [0.0,0.5] I
C, ($Id 6.74 1 e-5
Table 6.2 Results of ordinary design optimization and total robust design
Note: In Cases 1 and 2, maximization is applied with f,, = "rigidity" whik in Case 3, rninimization is
adopted with f*, = "welding cost*'.
Table 6.3 Results of Case 1 based on Design Platfom 'PP 1 "
ûesip Approach & design case
I 1 Case 2 1 6.1461 1 81.148 1 2W.8427 1 6.4û14 1 1.9291 1 0.1280 1
9.3631
6.2070
4.8376
adinsry Optimization
L
cas^ 1 & 3
case 2
C~SC 1 & 3
1
4.1039
77.2226
143.7752
t
254.0000
2 10.6020
176.1 1 16
b
9.3631
6.2070
9.0869
f -orfdn
6.61 12
1.8617
2.1386
4
1
0.0395
Table 6.4 Results of Case 2 based on Design Platfonn "DP2" (with the fixed L)
229.1958 6.5698 355.6 24.6823 203.9264 84.91 59
222.6291 6.4469 355.6 24.0134 2 IO. 1040 85.6422
Table 6.5 Design results of Case 3 based on Design Platform 'DP3"
115
Table 6.6 Results of Case 1 based on Design Platforni "DPUC 1" ( a =0.0 1 8 1, b =0.98 19)
Note: a and b stand for scalars in the cxprcssion of an uncertainty-prcscnt wish attribute for f,, and , rtspec tivcly.
Table 6.7 Results of Cape 2 baseâ on Design Platforni "DPUC2" (with the fixed L)
fmin 1~ a I
= 1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
h
15.4428
13,2249
10.0272
7.6820
6.6276
6.5223
6.4089
6.2864
6.1539
6.0 101
5.8534
1
59.2051
59.0374
69.0202
80.0090
84.6269
80.8902
77.5356
74.5219
71.8172
69.3987
67.2515
t
159.6891
176.9548
195.2084
2 14.4659 t
222.2413
2 15.79 18
209.829 1
204.3214
199.2467
194.5928
190.3587
b
17.6448
13.2249
10.0272
7.6820
6.6276
6.5223
6.4089
6.2864
6.1539
6.010 1
5.8534
L
355.6
355.6
355.6
355.6
355.6
355.6
355.6
355.6
355.6
355.6
355.6
E
206.8400
205.1560
203.4720
201.7880 ,
2 IO. 1040
2 lO.9îOo
2 1 1.7360
2 l2.5520
213.3680
214.1840
215.0000
FO
26.6890
26.0201
25.3512
24.6823
24.0 134
23.3445
22.6756
22.0067
21.3378
20.6689
20.0000
G
82.7370
82.7370
82.7370
82.7354
85.6422
86.3685
87.0948
87.821 1
88.5474
89.2737
90.0000
Table 6.9 Threshold Values for Design Constraints
Taôle 6.10 Results of Case 1 bassd- on Design Platfom 'DPUA 1 "
Case
Case1&3
Case2
1,
0.3274
0.0226
(4) Am,
J
0.0 176
0.0056
(1) 1-
0,0094
0,0125
(2) Amin
0.0102
0.0226
(3) 1,
0.3274
0.0103
Table 6.1 1 Results of Case 2 bacad on Design Platfonn ''DPUA2" (with the fixed L) ---.---A - .- - - - A . - - ( q,9- 33665F -
No prrrcdcrl solution at the given initial design
66.8646 1 190.2739) 11.2007 1 355.6 126.0201 1205.l560(82.737013.l83~ )O. 1551
Table 6.12 Results of Case 3 based on Design Platfonn "DPUAJ" (a=0.0181, b=û.9819)
1 a m = l 1 No practicrl solution at the given initial design
Retrospect and Conclusions
7.1 Thesis Overview
Beginning with the inaoduction of this work, this thesis has been presented
through six chapters and will be ended up with this last chapter for conclusions. in
Chapter 1, the motivation behind this research was introâuced and the background
materials were reviewed through the s w e y made in (1) robust engineering design, (2)
engineering design using f u z y sets and (3) product farnily design. Insofar this work is
concemeâ, the first one provides a basis for the handing of flexibiiitpinduced design
variations, the second one a basis for the modeling of design flexibility and the 1st one a
basis for hiture applications. Centered on the motivation and background, the research
objectives were defined specifically, dong with the organization of the thesis in a
chapter-b y-chapter marner.
In Chapter 2, the parametric design fhmework together with the robust design
modeling approaches has been discussed, which makes it possible to elaborate on the
methodology of DFV later on. In more detail, the framework of design fomdization to
facilitate engineering design modeling was introduced, the issue of uncertainty modeling
and treatment was adàressed, and the robust design modeling approaches were presented
dong with three robust design models for achieving feasibility robustness, sensitivity
mbustness and total mbustness in design, respectively. These represent the fundarnentals
for developing DFV approaches and their computational design models with contribution
to eventually achieve respomive design customization, which have been presented
graâually in the next three chapters.
Chapters 3 and 4 have laid emphasis on the modeling of design flexibility as well
as the denving of desip formulations to account for the presence of both design
flexibility and design preference in a design problem. In Chapter 3, design flexibility
modeling - - in the absence of design uncertainty - - - was examined, - while Chapter 4 explored
the approaches for cophg with design flexibility mdeling in the presence of design
uncertainty* In both chaptem, ernphasis has been given to the discussion of different
forms of flexibility characteristic fiinction according to two streams: (1) flexibility in
design parameters and (2) flexibility in huictional requirements. Particularly, two
approaches for the handling of both flexibility and uncertainty in design were presented
in Chapter 4 - sequential approach and sirnulfaneuus approach - along with their
comparisons. By their names, the former ûeats design flexibility and design uncertainty
sequentiah'y with a comervative solution strategy, while the latter handles them
sirnulfaneously with an aggressive one. B a d on the modeling of design flexibility, a
variety of preferencediven design models to account for various design scenarios have
been derived. These design models l a d themselves to be applicable for automated
design implementations on a computer.
Chapter 5 has been focused on the development of DFV approaches and their
computational models baseâ on the modeling of design flexibility h m Chapters 3 and 4.
These approaches support the transition that transfonns design flexibility to ultimately
achieve design variety on the model-based design platfonns formalizeà in Chapter 5. In
particular, three design platfoms for the case of preference coupling have been
developed and six ones for the case of preference uncoupling have been suggested
developed, al1 serving as an engineering mdel base for attaining different scalable
desips in diffemt design scenarios. Basad on the fhmework of scalable design, it
enables the M e r obtaining of custom desip with ease and efficiency in the way that a
set of design variants (or deriuatiues) h m a cure design can be created responsivefy wi th
no additional effort needed to derive any new model. A systematic approach for
achieving responsive design customization has been presented conclusively to cope with
typical design customization scenarios in practice.
Finally, in Chapter 6, the newly developed approaches were illustrated and tested
using a welded bearn design example. Various design cases were demonstrated through
the study of this design example upon certain assumptions. Design variants for each
model were obtained in a responsive way, and their results were presented and discussed
accordhg to each design case given. It has been showeâ that the methodology of DFV is
contributive gac-hieving a family of d-~ign &matives *th ease and efficiency, without L.-- - -- LL-
any needs to derive any new model.
7.2 Summary of Contributions
In this thesis, thm major contributions have been made, as highlighted below
according to the presentation flow of the thesis.
1) A formal approach for design flexibiflis, modeling in the absencelpresence of
uncertainty ha9 been presenteâ such that the gap in computerization between the
manual design selection and the design generation is bndged and consequently
concurrent design computing is achieved. This contribution not only shortens design-
computing cycle to support rapid proâuct customization but also relieves human
difficulty in facing the case of desip selection subject to more than 712 factors.
2) The methdology of DFV consisting of nine computational models (and design
platfoms) has been developed such that a variety of scdable design to account for
diffemt design scenarios is attainable with eese and efficiency and consequently
design variety is achieved through different design platfoms. This contribution
enables a designer to obtain a family of design alternatives (variants or derivatives)
varying fiam its con design ia a time-efficient and cost-effective manner.
3) A syskmatic appmach for achieving respomive design customization has been
suggested such that a custom design is generated, based on an appropriate design
platform, in a responsive manner with no extra effort in deriving a new design model.
This contribution makes mpid product ct~ptontizorion possible through responsive
design customization, so as to accommodate unpredictable market changes to an
engineered product.
7.3 Recommendations and Future Work
- -A- -L - - A a - This thesis represents a trial attempt made at the modeling of design flexibility
and the handling of responsive design customization based on the methodology of DFV.
Thus, for sirnplicity, the development of this pioneering work has been based on certain
assumptions as outlined in the following.
The distribution of uncertainty in a design attribute was assumed to comply with
normality, while the shape of flexibility characteristic fùnction of a flerble design
parameter is assumai with a Ihear variation. In a more general case, different
uncertainty distributions may be encountereâ dong with the consideration of a non-
linear fom of flexibility chmctenstic function.
Al1 design moâels were based on the handling of a single wish atûibute, although it
has been claimed in Chapter 2 that the case of multi-wish attribute can always be
converted somehow into an equivalent one-wish attribute scenario. However, in
reality, this treatment still seems simplistic from a case-to-case point view. A more
general mode1 in the presence of multiple wish attributes appears more promising.
Some of the newly developed design models involve the evaluation of either the 2nd-
order derivatives or an integral, which inherently incurs an increase of complexity in
desip computing and thereby encumbering the enhancement of design efficiency.
This requires extra efforts to be devoted to more efficient numencal algorithms and
techniques.
The modeling of design flexibility implicitly assumes that the physical range
specifieû on aflexible desip parameter is feasible. This requires that the designer is
kmowledgeable with prior experietlce on the current desip problem. Ln practice,
however, this may not be true in the situation that, for example, a new design problem
is encountered. Even for an experienced designer, rather than a novice, the selection
of a flexible range may not dways be pmperly feasible. As such, it would result in
improper or misleadhg design solutions.
The above assumptions point out possible directions on the funw research. In this
sense, it is recommended that these assumptions be relaxed for generality of the present
methodology. On the other hand, f i d e r efforts should be dedicated to the following = -
works:
1) Develop a more comprehensive flexibility modeling approach to take both flexible
design parameters and flexrexrbIe fhctional requirements into consideration by
incorporating them both sirnultaneously into the context of modeling, rather than
treating them separately in the cumnt approach.
2) Expand the scope of applicability of the present DFV methodology by extending it to
product farnily design in which it is expected to generate a farnily of products, instead
of a farnily of designs.
Yet, it is important to note tâat the above prornising works are al1 'oased on the current
work, which serves as a basis for the future developments.
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Fundamentals of Fuzzy Set Theory
Introduction
In the world some things an so clear that they can be defined in such a way as to
dichotomize the individuals in some given universe of discourse into two groups,
mernbers that certainly belong in a set or nonmembers that certainly does not belong in a
set (George J. Klir, Bo Yuan, 1995). A sharp, unambipous distinction exists between the
members and nonmembers of the set. However, some other things, especially the ones
expressed in nanual lmguage, do not exhibit this characteristic. They are incomplete,
hgmentary, not filly reliable, vague, contradictory, deficient or have no clear
boundaries and cannot be well defined in the way of absolute huth or falseness. In hiuy
set theory the ones that cm explicitly be described by memben or nonmembers are
narned crisp set, but the ones that can not clearly be depicted are designated as hvzy set.
Actually, that elements in a set are members of the set is not necessarily either true or
false, but they may be hue or false to some degree, the degree to which elements are
members of the set. From this point of Mew, it can be said that fuzzy set is 'a set with
boundaries that are not precise, and the membership in a fiizzy set is not a matter of
affirmation or denial, but rather a matter of a degrre.' (George J. mir, Bo Yuan, 1995).
Commonly, the membership in a fuzy set CM be expressed as degrees of tnith of the
associated propositions by numbers in the closed unite interval [O, 11. The extreme values
in this interval, O and 1, then respectively represent the total denial and affirmation of the
mernbership in a given hizzy set as well as the falsity and tmth of associated proposition
(George J. Klir, Bo Yuan, 1995). From this sense, the cnsp set is a special case of fuuy
set, while the d e g i of membership in a fuzzy set an O and 1.
Fuzy set theory has been well developed, and more attention recently is given to
its application in engineering design. As mathematical tools, the approaches showed in
fuzy set theory, such as fhzy membership hc t ion and expectation of fuuy --A- - -
probability, are adopted in the discussion of flexibility modeling. For convenience of the
thesis study, they are provided as follows in temis of fuzy membership fhction, basic
fuzy operations, fuzzy probability and hizy optimization.
Fuuy Membenhip Function
Regarding above introduction, the characteristic of a crisp set, a special case of
hizy set, may be expressed in the following fom while the individual object x is a
member or nonrnember, e.g. elemmt of a set A or non-element of a set A :
where x is an individual object in the universe of discome, or universai set X, A is a
set that x may belong. When x belongs A , the degree of membership fiinction p, ( x ) is
1, otherwise it is O.
A hiuy set cm also be defined mathematically by assigning to each possible
individual in the universe of discourse a value within a M t e interval of specified range
representing its degree of membership in the fuuy set. Larger values denote higher
degrees of set membership. Such a huiction is called a membership huiction, and the set
defined by it a fuzy set. Each fuzy set is compleiely and uniquely defined by one
particular membership hinction. When A is a fuzzy set and x is a relevant object in a set
X , the membership fiinction of the Aizy set is denoted by p, (x) ; that is
ûne of important concepts in hiuy set theory is the a-cut. Given a fuzzy set A
dehed on X aud any number a E [O, 11, then the a -cut is the crisp sets
nipt is, the cnsp set A, contains al1 the elements of the universal set X whose
mmbership grades in A are p a t e r tban or equal to the specified value of a.
Shapes of hizzy membership functions are context dependent issues, so they may
Vary depending on very different fuzzy sets. For sirnplicity, linear membership functions
are usually used.
Basic Fuzzy Operations
Fuzzy sets may be manipulated using various fuzzy operations such as Standard,
Algebraic, Bounded and Drastic operations, but in fuvy optimization only Standard
operations are employeâ and therefon briefly reviewed here for use on design flexibility
modeling in the study. The basic standard operations are k z y complement, intersection
and union, which are expressed by
where x is defined on X and x É X by definition. Given two fuuy sets A and B , an . - aggregated fuzzy set may be obtained by applying the equations (AS) and (A6).
Computationally, the equations, (A4). (AS) and (A6) can be expressed respectively as
following :
where equation (A8) (Ag) am binary operations. Similarly operations of more than two
h i u y sets can be expressed by applying the binary operation as follows:
Clearly, the fuuy complement, intersection, and union are not d l the operations
involved in hiuy set theory. However, among the great varieties of fuzzy operations, the
standad o p d o n s possess certain pmperties that give them a special significance.
Although they do not cover al1 operations by which hiuy sets cm be aggregated, but
they cover al1 aggregating operations associated and employed in this thesis for fuzzy
optimization.
It is known that a fuzzy event may be associated with both a possibility
distribution and a probability distribution accorduig to probabiIity/possibiJity consistency
principle (Zadeh, L. A., 1978). This principle indicates that a high degree of possibility
does not irnply a higher degree of probability, but an event is not probable if it is not
possible. In fact, there is a connection between those two distributions and in f u z y set
theory, it is described by fuzy probability.
A fuw set A , characterized by possibility distribution bction, p, (X) , which
mathematically equates to its mcmbership nuiction, may comprise many elementary
events x which associate with probability distribution p, (X) . Their comection in hiuy
probability is mathematically described by expectation of fuzzy membership hinction,
which is given by
where A denotes a fiizzy set on the universe X; x stands for an element of X. In last
equation it is assumed that the integral of probability on the entire universe of discourse
must qua1 unity, Le.,
In this case, the expectation of membership huiction may be simplified as follow:
Instead of a fuzzy set, if A is a crisp set, then the expectation of fûzzy
membership fiuiction will m e r be simplified to probability density function which is
given by
It can be seen h m equation (A15) that crisp set with no design fuzzy
membership hction involved is a special case of fuzy set in huzy probability theory.
Traditiondly, the objective bctions md constraints of optimization are assumed
king defined precisely. This ignores the fact that they may not be in the situation to be
described in such a way because sometimes the utility fiinctions are not definable or
phenomcaa of design problems may only be stateâ in an ambiguous way. On the other -- --- - - --- - L - -< - -
hand the doubts about the exact selections of design parameters and functional
reqWrements may also anse in an early design stage. Generally, design problems may
contain fuuy infornation since so much design uncertain factors involved in reality.
In iùzzy environment, the notion of conventional optimization is modified. Fuzzy
objective bctions are chanicterized by their membership hctions and so are the
constraints. The optimal solutions, overall satisfactions, are semhed and selected in the
intersection of fuzzy objective functions and constraints. Therefore, the feasible domain
for fûzzy optimal solution is defined not only by fiizzv constraints but also by fuzzy
objective bction. Assume a set of fuuy constraints, gi (X) Vi E [1, ml, plus a set of
cnsp constraints g, (X) Vj E [l , n] . In this case, the feasible design space, denoted D, is
govemed by an intersecting region due to the decision domains of not only ai, g j but
hizzy objective, f , as well, i.e.
where F , G, , G represent the decision domains with respect to f , gi , g j .
The above set operation can be M e r expressed by their fuzzy membership
funciions mathematically, i.e.
where "A" stands for an aggregating operator comsponding to the set intersecting
operator " 1 "; p, ( X ) is defied as
- -
In hizy optimization, the degree of membership bctions for this crisp constraint,
pd ( X ) , must be one since gj ( X ) O is the only case concemed. Therefore, Eq. (A 17)
can be M e r expressad as
which can be fiutha simplified by
To evaluate the overall satisfaction, a generalized intersecting operation in hizzy
set theory is adopted through "min" aggregating operator for quantifjhng p,(X) ,
accordhg to
Thus, to achieve a design optimization, the solution seeking process is directed toward
maximizing p, (X) such that
where X' denotes the resulting design configuration eventually obtained.
B a r d on the above discussion, the design formulation of fuzzv optimization is
given by
Find X
max A
where A. is the overall satisfaction maximized; X denotes a vector of design variables;
,uf (X) and ,ugi (X) indicate the rnembership function of design objective and
constrahts; the equation (A26) stands for the jU crisp constraint, a special case of fuvy
one.
The last design mode1 discussed is used to account for the design problems with
the design objective charactenzed by membenhip function since it involves fuzzy
infonnation. in rcality, there may be the case of design objective with no design fuuy
infonnation involved, this means that the design objective function, f (X) , would not be
able to be characterized by the fiizzy rnembership function. To this case, Eq. (A16) is
modified by
To achieve a design optimization by andogy to discussion made above, the solution
seeking pmess for this case is directeci toward maximizing f ( X ) such that
subject to
- - Actually, this is known as a -1evel cuts method of solution accorâing to the paper
written by S. S. Rao (1987). Base on this solution process, the design problem can be
formulated by
Find X
min f (X) s.t. a S ,u,(X) i = 1,2 ........ m
........ gJX) 0 j =1,2 n
xi S xk x i k = 1,2 ........ q
where a is the degree of design satisfaction selected by designer over [O, 11. Each
number of a defines the constraint interval of equation (A32) as
This a -1evel cuts method transfers the fuzzy design problems into a crisp one and
has design problem solved by using the conventional optimization approach after
assigning a certain value to a in the design model.
Formulations and Models in Welded Beam Design
1. Basic Formulation: Design variables:
X ' ( x 1 9 ~ 2 , ~ 3 > ~ 4 ) ' =(h91,t9 b)'
Design parameters:
P = (L, F,, E, G)Z
Design hctions
c(X, P) = c,+, + c,x,x, (L + x,)
Rigidi ty
Welding cost
Shear stress
Bending
Bounding value parameters:
2. Design Formulations in Different Cases
Case 1: Design variables:
Flexible design parameters:
P = (L,F, ,E,G)Z Boundhg value panmeters:
E,,
F m
Shear stress Welding cost
Bending Geometry Buckling
Rigidity (C 1-3)
Case 2: Design variables:
X= (XI, XI, ~ 3 9 ~ 4 ) ~ = (h, k t , b)r
Flexible design parameter:
P = (F,,E,G)'
Bounding value parameters:
Objective
Constraints
Shear stress (C 1-4)
Bending (C 1-5)
Geometry (C 1-6)
Welding cost (C 1-7)
Buckling (C 1-8)
Shear stress Weldeâ bearn cost Bending Geometry Buckling
Welding cost (C2-3)
Shear stress v - 4 )
Bending (C2-5)
Geomeüy (C2-6)
Rigidiîy (CS-7)
Buckling (C2-8)
Note that one design parameter is fixed and given by
Case 3: Design variables:
X = ( X , , X ~ , X ~ , X , ) ~ = ( & ~ , t , b ) ~
Flexible bounding value parameters:
Shear stress Welding cost Bending Geometry Buckling
Rigidity
Shear stress
Bending
Geome try
Welding cost
Buckling
Note that al1 functional requirements are flexible, but al1 design parameters are
fixed and given by