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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