a compromise programming approach to robust designCHEN W 19991Word文档下载推荐.docx

a compromise programming approach to robust designCHEN W 19991Word文档下载推荐.docx

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MechanicalEngineering（M/C251）

842W.TaylorSt.UniversityofIllinoisatChicagoChicagoIL60607-7022

Phone:

（312）996-6072

Fax:

（312）413-0447

e-mail:

weichen1@uic.edu

ModifiedManuscripttoJMD,Oct.14,98

ABSTRACT

Inrobustdesign,associatedwitheachqualitycharacteristic,thedesignobjectiveofteninvolvesmultipleaspectssuchas“bringingthemeanofperformanceontarget”and“minimizingthevariations”.CurrentwaysofhandlingthesemultipleaspectsusingeithertheTaguchi’ssignal-to-noiseratioortheweighted-summethodarenotadequate.Inthispaper,wesolvebi-objectiverobustdesignproblemsfromautilityperspectivebyfollowingupontherecentdevelopmentsonrelatingutilityfunctionoptimizationtoaCompromiseProgramming（CP）method.Arobustdesignprocedureisdevelopedtoallowadesignertoexpresshis/herpreferencestructureofmultipleaspectsofrobustdesign.TheCPapproach,i.e.,theTchebycheffmethod,isthenusedtodeterminetherobustdesignsolutionwhichisguaranteedtobelongtothesetofefficientsolutions（Paretopoints）.ThequalityutilityatthecandidatesolutionisrepresentedbymeansofaquadraticfunctioninacertainsenseequivalenttotheweightedTchebycheffmetric.Theobtainedutilityfunctioncanbeusedtoexplorethesetofefficientsolutionsinaneighborhoodofthecandidatesolution.Theiterativenatureofourproposedprocedurewillassistdecisionmakinginqualityengineeringandtheapplicationsofrobustdesign.

Keywords:

RobustDesign,MultiobjectiveOptimization,CompromiseProgramming,UtilityFunction,DecisionAnalysis

MainTextWordCount:

5,658;

Characterswithspace:

35,013.

NOMENCLATURE

CP（,w）WeightedTchebycheffApproachf（x）ObjectiveFunction

F（x）VectorofObjectiveFunctionsFCandidateEfficientSolutiong（x）ConstraintFunction

kjPenaltyFactors

KLossFunctionCoefficient

L1ManhattanMetric

L2EuclideanMetric

LTchebycheffMetric

LpLp-metricS/NSignal/NoisewWeights

WSWeighted-sum

WSP（w）Weighted-sumProblem

xVectorofDesignVariables

XDesignSpace

xLLowerBoundforDesignVariablesxUUpperBoundforDesignVariables

x0ParetoSolutionforDesignVariables

x*OptimalSolutionforDesignVariables

xCandidateSolutioninDesignSpace

YRandomVariable;

ObjectiveSpacexDeviationsoftheDesignVariables

*OptimalSolutionof-problem

QualityCharacteristicofS/N

u*UtopiaPointinCP

fMeanoftheObjectiveFunctionf（x）

*OptimalValueoftheMean

f

fStandardDeviationoftheObjectiveFunctionf（x）

*OptimalValueoftheStandardDeviation

1.INTRODUCTION

Inrecentyears,theTaguchirobustdesignmethodhasbeenwidelyusedtodesignqualityintoproductsandprocesses（Phadke,1989）.Usingthismethod,thequalityofaproductisimprovedbyminimizingtheeffectofthecausesofvariationwithouteliminatingthecauses（Taguchi,1993）.Whilethemajorityoftheearlyapplicationsofrobustdesignconsidermanufacturingasthecauseforperformancevariations,recentdevelopmentsindesignmethodologyhaveproducedapproachesthatutilizethesameconcepttoimprovetherobustnessofdesigndecisionswithrespecttothevariationsassociatedwiththedesignprocess（Changetal.,1994;

Chenetal.,1996b）.

AlthoughTaguchi'

srobustdesignprinciplehasbeenwidelyaccepted,themethodsTaguchioffersforrobustdesignhavereceivedmuchcriticism,inparticularthetwo-partorthogonalarrayforexperimentaldesignandthesignal-to-noise-ratio（S/Nratio）usedastherobustoptimizationcriterion（Box,1988;

Nair,1992）.Intheengineeringdesigncommunity,researchersareworkingondevelopingnonlinearprogrammingmethodsthatcanbeusedforavarietyofrobustdesignapplications（OttoandAntonsson,1991;

Parkinsonetal.,1993;

YuandIshii,1994;

andCaganandWilliams,1993）,includingprobabilisticoptimizationmethodsforrobustdesign（Siddall,1984;

EggertandMayne,

1993）.Acomprehensivereviewofexistingrobustoptimizationmethodsisprovidedby

SuandRenaud（1997）,andwillnotberepeatedhere.

Oneissuethatwefindhasnotbeenadequatelyaddressedinthepreviousinvestigationsisthemultipleaspectsoftheobjectiveinrobustdesign.Itwasillustratedbyoneoftheauthors（Chenetal.,1996b）thatassociatedwitheachquality（performance）characteristic,therobustdesignobjectivecouldbegeneralizedintotwoaspects,namely,

“optimizingthemeanofperformance”and“minimizingthevariationofperformance”.A

briefmathematicalbackgroundthatsupportstheabovestatementisprovidedinSection

2.1.Throughourpreviousapplications（Chenetal.,1996aandChenetal.,1997）,weobservethattheperformancevariationisoftenminimizedatthecostofsacrificingthebestperformance,andthereforethetradeoffbetweentheaforementionedtwoaspectscannotbeavoided.Intheliterature,thoughthemultipleaspectsoftheobjectiveinrobustdesignisacknowledged（Sundaresanetal.,1993）,singlerobustdesignobjectivefunctionisoftenutilized.RamakrishnanandRao,1991,formulatetherobustdesignproblemasanonlinearoptimizationproblemwithTaguchi'

slossfunctionastheobjective.Sundaresanetal.（1993）employasingleobjectivefunctionthatutilizesweightingfactorsfortargetperformanceandvariancerepresentedbytheSensitivityIndex（SI）.BrasandMistree（1995）andChenetal.（1996b）introducethecompromiseDecisionSupportProblem（DSP）（Mistreeetal.,1993）,agoalprogrammingapproach,tomodelthemultipleaspectsofrobustdesignobjectiveasseparategoals.Weassertthattheuseofweightedsumsofobjectivesisaverysimplisticapproachtomultiobjectiveoptimizationproblems.AcloserlookatthedrawbacksofminimizingweightedsumsofobjectivesinmulticriteriaoptimizationisprovidedbyDasandDennis（1997）.Morerigorousmethodsneedtobeconsideredforrepresentingthepreferencestructureofmultipleobjectivesinrobustdesign.

Formodelingdesigner’spreferencestructure,oneofthecommonlyusedmethodsisbasedontheutilitytheory（vonNeumannandMorgenstern,1947;

KeeneyandRaifa,

1976;

Hazelrigg,1996;

Thurston,1991）.Underthenotionofutilitytheory,theultimateoverallworthofadesignisrepresentedbyamultiattributeutilityfunctionwhichincorporatesconsiderationofattributesthatcannotbedirectlyconvertedtoacommon

metric.Ideally,whenthepreferenceofthemultipleaspectsoftheobjectiveinrobustdesigncouldbecapturedbythemultiattributeutilityanalysis,robustdesigncouldbesolvedasasingleobjectiveoptimizationproblem.However,onedifficultyassociatedwithusingtheutilityfunctionapproachisthat,inpractice,itisoftenimpossibletoobtainareliablemathematicalrepresentationofthedecision-maker’sactualutilityfunction.Intheliterature,approachesthattakedifferentparadigmsforsolvingmulticriteriaoptimizationproblemsareproposed.Forinstance,Messac（1996）developsthemethodofphysicalprogrammingwhicheliminatestheneedforweightsettingorutilityfunctionbuildinginmulticriteriaoptimization.Inthiswork,weproposetouseCompromiseProgramming（CP）（Yu,1973andZeleny,1973）toaddressthemultipleaspectsofrobustdesign.

CPisoneoftheapproachesthattakeaparadigmdifferentfromtheutilitytheory.ThebasicideainCPistheidentificationofanidealsolutionasapointwhereeachattributeunderconsiderationachievesitsoptimumvalueandseekasolutionthatisascloseaspossibletotheidealpoint（Zeleny’saxiomofchoice）.ThoughtheweightsrepresentingrelativeimportanceareusedasthepreferencestructurewhenapplyingCP,ithasbeenmathematicallyproventhatCPissuperiortotheweighted-sum（WS）methodinlocatingtheefficientsolutions,orthesocalledParetopoints（Steuer,1986）.However,therearefewapplicationsofCPtomechanicalengineeringdesignproblems.MiuraandChargin（1996）developavariationofCPandapplyittooptimalstructuraldesign.AthanandPapalambros（1996）donotrefertoCPbutproposetominimizethesumoftheexponentiallyweightedobjectivefunctionsandillustratetheirapproachalsoonsomestructuraldesignproblems.

ThoughutilitytheoryandCPareconsideredverydifferentparadigmsandmethodologiestomeasurepreferencesaswellastodeterminedecisionmaker’soptimaon

theefficientfrontier,researchershaveillustratedalinkagebetweenthetwoapproaches（BallesteroandRomero,1991）.OneoftheauthorsestablishedarelationshipbetweenaCPapproachandaquadraticweighted-sumsscalarizationofmultiobjectiveproblems（TindandWiecek,1997）.Inthispaper,weapplyCP,specificallytheTchebycheffmethod,tomultiobjectiverobustdesignproblemsfromautilityperspectivebyfollowingupontherecentdevelopments.Aninteractiverobustdesignprocedureisdevelopedtosupportdecisionmakinginrobustdesignapplications.

2.TECHNOLOGICALBASISOFOURAPPROACH

2.1MultipleQualityAspectsofRobustDesign

ThequalitylossfunctionisusedbyTaguchiasametricforrobustoptimization.Therelationshipbetweenqualitylossandtheamountofdeviationfromthetargetvalueisexpressedbythelossfunctionsfordiffe