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AMultiObjectiveGeneticAlgorithmforRobustDesignOptimizationMianLiGraduateResearchAssistantDepartmentofMechanicalEngineeringUniversityofMarylandCollegePark,MD20742Tel.13014054919Emailmli6umd.eduShapourAzarmProfessorDepartmentofMechanicalEngineeringUniversityofMarylandCollegePark,MD20742Tel.13014055250Emailazarmumd.eduVikrantAuteFacultyResearchAssistantDepartmentofMechanicalEngineeringUniversityofMarylandCollegePark,MD20742Tel.13014058726Emailvikrantumd.eduABSTRACTRealworldmultiobjectiveengineeringdesignoptimizationproblemsoftenhaveparameterswithuncontrollablevariations.Theaimofsolvingsuchproblemsistoobtainsolutionsthatintermsofobjectivesandfeasibilityareasgoodaspossibleandatthesametimeareleastsensitivetotheparametervariations.Suchsolutionsaresaidtoberobustoptimumsolutions.Inordertoinvestigatethetradeoffbetweentheperformanceandrobustnessofoptimumsolutions,wepresentanewRobustMultiObjectiveGeneticAlgorithmRMOGAthatoptimizestwoobjectivesafitnessvalueandarobustnessindex.Thefitnessvalueservesasameasureofperformanceofdesignsolutionswithrespecttomultipleobjectivesandfeasibilityoftheoriginaloptimizationproblem.Therobustnessindex,whichisbasedonanongradientbasedparametersensitivityestimationapproach,isameasurethatquantitativelyevaluatestherobustnessofdesignsolutions.RMOGAdoesnotrequireapresumedprobabilitydistributionofuncontrollableparametersandalsodoesnotutilizethegradientinformationoftheseparameters.Threedistancemetricsareusedtoobtaintherobustnessindexandrobustsolutions.Toillustrateitsapplication,RMOGAisappliedtotwowellstudiedengineeringdesignproblemsfromtheliterature.CategoriesandSubjectDescriptorsG.1.6Optimization–NonlinearProgrammingGeneralTermsDesign,algorithmsKeywordsMultiobjectivegeneticalgorithms,robustdesignoptimization,performanceandrobustnesstradeoff1.INTRODUCTIONTherearemanyengineeringoptimizationproblemsintherealworldthathaveparameterswithuncontrollablevariationsduetonoiseoruncertainty.Thesevariationscansignificantlydegradetheperformanceofoptimumsolutionsandcanevenchangethefeasibilityofobtainedsolutions.Theimplicationsofsuchvariationsaremoreseriousinengineeringdesignproblemsthatoftenhaveaboundedfeasibledomainand/orwheretheoptimumsolutionslieontheboundaryofthefeasibledomain.Manymethodsandapproacheshavebeenproposedintheliteraturetoobtainrobustdesignsolutionsthatis,feasibledesignalternativesthatareoptimumintheirobjectivesandwhoseobjectiveperformanceorfeasibilityorbothisinsensitivetotheparametervariations.Generally,thoseapproachescanbeclassifiedintotwotypesstochasticapproachesanddeterministicapproaches.Stochasticapproachesusetheprobabilityinformationofthevariableparameters,i.e.,theirexpectedvalueandvariance,tominimizethesensitivityofsolutionse.g.,Parkinsonetal.1,YuandIshii2,JungandLee3forobjectiverobustoptimizationandDuandChen4,Chenetal.5,Tu,ChoiandPark6,7,Younetal.8andRay9forfeasibilityrobustoptimization–alsocalledreliabilityoptimization.Also,JinandSendhoff10proposedanevolutionaryapproachtodealwiththetradeoffbetweenperformanceandrobustnessusingvarianceinformation.Themainshortcomingofstochasticapproachesisthattheprobabilitydistributionfortheuncontrollableparametersisknownorpresumed.However,itisdifficultorevenimpossibletoobtainsuchinformationbeforehandinrealworldengineeringdesignproblems.Deterministicapproaches,ontheotherhand,obtainrobustoptimumdesignsolutionsusinggradientinformationoftheparameterse.g.,Ballingetal.11,Sundaresanetal.12,13,ZhuandTing14,LeeandPark15,SuandRenaud16,MessacandYahaya17orusinganongradientbasedparametersensitivityestimationGunawanandAzarm1821.TheaimoftheGunawanandAzarmsapproach1821istoobtainoptimumsolutionswhichessentiallysatisfyanadditionalrobustnessconstraintthatisprescribedbythedecisionmakerDM.Inthispaper,wepresentanewdeterministicapproachtoinvestigatethetradeoffbetweentheperformanceandrobustnessPermissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitationonthefirstpage.Tocopyotherwise,orrepublish,topostonserversortoredistributetolists,requirespriorspecificpermissionand/orafee.GECCO05,June2529,2005,Washington,DC,USA.Copyright2005ACM1595930108/05/00065.00.771ofoptimumsolutions,basedonaMultiObjectiveGeneticAlgorithmMOGA.Thegoalofourapproachistosimultaneouslyoptimizeiameasureoftheoptimumsolutionsperformance,i.e.,thefitnessvalue,thataccountsforobjectiveandconstraintvaluesintheoriginaloptimizationproblemdefinedinSection2,andiiameasureoftheoptimumsolutionsrobustness,therobustnessindex,originallyproposedbyGunawanandAzarm1821,extendedinthispaperwiththeuseoftwoadditionaldistancenorms.Thisapproachisadeterministicmethodusingnongradientbasedparametersensitivityestimation,whichcanbeappliedtooptimizationproblemshavingobjectiveand/orconstraintfunctionsthatarenondifferentiablewithrespecttotheparameters.AnyMOGAintheliteraturecanbeappliedtothisapproach.InGunawanandAzarmsapproach1821,theauthorstriedtoobtainoptimumsolutionsthatareinsensitivetotheparametervariation.Inotherwords,therobustnessrequirementwasconsideredasaconstraintintheirapproach.Onthecontrarywetreatrobustnessasoneofourobjectivesandformanewtwoobjectiverobustoptimizationproblemregardlessofhowmanyobjectivestheoriginalproblemhas,toinvestigatetherelationbetweentheperformanceandrobustnessofsolutions.TheRMOGAhereaimsatsimultaneouslymaximizingperformanceandmaximizingrobustness.TheorganizationoftherestofthispaperisasfollowsInSection2,wepresenttheoriginaloptimizationproblemandexplainsomedefinitionsandterminologies.Basedonabriefdescriptionoftheobjectiveandfeasibilityrobustoptimizationapproach,wepresentournewapproachinSection3.Section4demonstratestheapplicationofourapproachtotwotestproblemsfollowedbyadiscussionoftheresults.ThepaperconcludeswithasummaryinSection5.2.PROBLEMDEFINITIONInthissection,weformallydefinetheproblemandexplainseveraldefinitionsandterminologiesusedinthispaper.Ageneralformulationofmultiobjectiveoptimizationproblemisshownin1.upperlowerjmxxxxJjpxgMmpxf≤≤≤,,10,s.t.,,1,minKK1f1,,fMtaretheobjectivefunctionstreferstoatransposeoftherowvector,xx1,,xNtisthedesignvariablevectorcontrolledinanoptimizationrun,pp1,pGtistheuncontrollableparametervector.Notethatdesignvariablesthatthemselveshaveuncontrollablevariationsareincludedinbothxandp.xlowerandxupperarethelowerandupperboundsofx,respectively.TheproblemhasJinequalityconstraints,gj,j1,,J.Wepresumethatallconstraintscanberepresentedasinequalityfunctions.Inthispaper,wecalltheoptimizationproblemshownin1astheoriginalproblem.SincetherearetradeoffsamongsttheMobjectives,usuallytheoriginalproblemhasmorethanoneoptimumsolution.ThesetoftheseoptimumsolutionsiscalledtheParetoset,asdiscussedinMiettinen22andinDeb23.Inthefollowing,webrieflydescribesometerminologiesthatareusedinthispaper.Nominalparametervaluep0p0,1,,p0,Gtistheparametervectorvalues,pp0,usedtooptimizetheproblemin1.Theparametersvariationis∆p∆p1,,∆pGt.NominalParetosolutionsaretheParetosolutionsoftheoptimizationproblemin1whenpp0.Letx0beadesignsolutionwhoserobustnesswewanttoanalyze.fx0,p0f1x0,p0,,fMx0,p0arethenominalvaluesfortheobjectivefunctions,andgx0,p0g1x0,p0,,gJx0,p0arethenominalvaluesfortheconstraintfunctions.ToleranceRegionisahyperrectangularregionin∆pspaceformedbyasetof∆pvalueswithrespecttowhichthedecisionmakerwantstherobustoptimumsolutiontobeinsensitive.Thisregionisusuallyboundedbyupperlowerppp∆≤∆≤∆,wheretheknown∆plowerand∆pupperarethelowerandupperboundsof∆p,respectively,∆plower∆p1lower,,∆pGlowertand∆pupper∆p1upper,,∆pGuppert.Forsimplicity,thetoleranceregionisassumedtobesymmetric,i.e.,∆pilower∆pT,i∆piupper,.,,10,GipiTK∀∆Sincetherecanbemorethanoneuncontrollableparameterwithdifferentmagnitudes,wenormalizethetoleranceregiontoformahypersquare.Parametervariationspace∆pspaceAGdimensionalspaceinwhichtheaxesaretheparametervariation∆pvalues.AcceptablePerformanceVariationRegionAPVRistheregionformedintheobjectivefunctionspacearoundthepointfx0,p0,whichrepresentsthemaximumacceptableperformancevariationchosenbytheDM,i.e.,∆f0∆f0,1,,∆f0,Mt,whereMifi,,1,0,0K∀≥∆.SeeFigure1afordetails.Fitnessvaluefvisavaluethatmeasuresasolutionsperformanceinacombinedobjectiveandconstraintsense.ThefitnessvalueorrankobtainedfromaMOGA,e.g.,NSGA23,canbeusedasthefitnessvalueinourapproach.Robustnessindexηisaratiothatcalculatestheradiusoftheworst–casesensitivityregiondefinedinSec.3.2withrespectto∆povertheradiusoftheexteriorhypersphereofnormalizedtoleranceregion18.Itisusedasarobustnessmeasureinourmethod.WewilldiscussitfurtherinSection3.DistancenormLpforanNdimensionalvectorxisthevectornormpxforp1,2andinfinity∞definedaspNipipxx11∑.Theinfinitynormisiixxmax≡∞.3.ROBUSTMOGAInthissection,wefirstbrieflydiscussthesensitivityestimationproposedbyGunawanandAzarm1821,basedonwhich,themeasuresforrobustnessandperformanceofsolutionsaredefined.WethenpresentourtwoobjectiveRMOGAapproach.3.1ParameterSensitivityEstimationWefirstdiscusstheapproachformultiobjectiverobustoptimization,followedbytheapproachforfeasibilityrobustoptimizationandthenthecombinedapproach.GivenanAPVRforasolutionx0,thereisasetof∆psuchthatthevariationinobjectivefunctionsvaluesduetothe∆parestillwithintherangesof∆f0,iforalli1,...,M.Thissetof∆pformsahyperregionaroundtheoriginin∆pspace,whichiscalledtheSensitivityRegionSR.Theregionisboundedasshownin2,,where,,1,0000,0pxfppxffMiffiiiii−∆∆∀∆≤∆K2Figure1showstheAPVRanditscorrespondingSRforasolutionx0inatwoparameterandtwoobjectivecase.Graphically,the772pointsinside,outside,andontheboundaryoftheAPVRcorrespondtothepointsinside,outside,andontheboundaryoftheSRFigure1b,respectively.Af2Bfx0,p0∆f0,1∆f0,2CAPVRfaBCA∆p2∆p1SensitivityRegionβαbFigure1aTheAPVRandbtheSREssentially,theSRrepresentstheamountofparametervariationsthatasolutionx0canabsorbbeforeitviolatestheAPVR.WecanusethesizeoftheSRasameasureforthesensitivityofadesignthelargertheSRforadesign,themorerobustthatdesignis.However,ingeneral,theshapeoftheSRcanbeasymmetric,whichmeansadesigncanbeverysensitiveormuchlessrobustinacertaindirectionof∆psuchasdirectionβinFigure1b,butmuchlesssensitiveorveryrobustinotherdirectionssuchasdirectionαinFigure1b.Toovercomethisasymmetry,aWorstCaseSensitivityRegionWCSRisusedtoestimatetheSRofadesign.TheWCSRisasymmetrichyperspherethatapproximatestheSR.Graphically,theWCSRisthesmallesthyperspherethattouchestheSRattheclosestpointtotheorigin,asshowninFigure2foratwoparametercase.SincetheWCSRissymmetric,theradiusoftheWCSR,Rf,insteadofthesizeoftheWCSR,couldbeusedasourrobustnessmeasure.Itmeasurestheoverallrobustnessofadesign.TheradiusoftheWCSRforx0canbecalculatedbysolvingasingleobjectiveoptimizationproblemshownin3.,,where01maxs.t.min0000,0,,11pxfppxffffppRiiiiiMiqGjqjfp−∆∆−∆∆∆∆∆∑K3Inthisproblem,thedesignvariablesarethe∆ps,theobjectivefunctionistheradiusoftheWCSR.Theequalityconstraintfunctionmeansthattheresultantvectorof∆fitouchestheboundaryoftheAPVR,whichmeansthat∆pisontheboundaryofthesensitivityregion.DetaileddiscussionofthisWCSRestimationapproachisgivenelsewhere18,19.Asimilarapproachcanbeusedforthefeasibilityrobustoptimization.Foradesignx0,all∆ppointswhosecorrespondingconstraintfunctionvaluegjx0,p0∆p≤0,j1,,J,formtheFeasibilitySensitivityRegionFSR,whichmeansthe∆pinsidetheFSRwillnotchangethefeasibilityofdesignx0.TheFeasibilityWCSRorFWCSRistheworstcaseestimateoftheFSRsimilartotheWCSRandRgistheradiusofthenormalizedFWCSR.Rgcanbecalculatedby4.0,maxs.t.min00,,11∆∆∆∆∑ppxgppRjJjqGjqjgpK4SincetheSRandtheFSRaredefinedinthesame∆pspaceandareofthesamescale,RminRf,RgimpliesthatwearelookingfortheradiusofworstcaseestimateoftheintersectionoftheSRandtheFSRforadesignsolution,asshowninFigure2.RfWCSR∆p2∆p1SRRgFSRFWCSRFigure2TheintersectionofSRandtheFSRTheradiusRcanbecalculatedbysolvingtheoptimizationproblemshownin5.JjpxgppxgppxgpxggffppRjjjjjJjiiMiqGjqjp,...,1,,,,where01},max,maxmax{s.t.min0000000,,1,0,,11−∆∆∆−∆∆∆∆∆∆∑KK5Forexample,inthecaseshowninFigure2,RminRf,RgRf.3.2RobustnessIndexTheradiusRrepresentsasolutionsrobustnessonanordinalscaleanddoesnothaveaphysicalassociationwiththedesignsolutionitself.AssuchitcanbedifficultfortheDMtodothetradeoffanalysisbetweentheperformanceandrobustness,i.e.,givenR,s/hecannotdecidewhetheradesignsolutionisrobustornot.Toovercomethisdifficulty,weusetheradiusoftheexteriorhypersphereofthenormalizedtoleranceregion,RE,asareferencefortherobustnessrequirementFigure3.WedefinetherobustnessindexERRηandusethisrobustnessindexasoneofthetwoobjectivesinourRMOGA.Ristheoptimumsolutioncalculatedin5.SinceREistheradiusoftheexteriorcircleofthenormalizedtoleranceregion,if1≥ERRη,thenthedesignx0isrobust.NormalizedtoleranceregionRE∆p2∆p1Figure3Theexteriorcircleofthenormalizedtoleranceregion7733.3FitnessValueRecallthatourgoalinthispaperistosimultaneouslymaximizetheperformanceandrobustnessofadesign.Therobustnessindexservesasameasureofrobustnessofthedesignsolutions.Henceweneedanothermeasurefortheoverallperformanceofdesignsolutions.Inmultiobjectiveoptimizationproblem,MOGAisagoodtooltoobtainParetooptimumsolutions.MostMOGAsassignafitnessvalueoraranktoeachalternativesolutioninthepopulationtorepresentitsrelativegoodness,accountingforbothobjectivevaluesandconstraints.SothefitnessvalueorrankorderingobtainedfromanyMOGAapproach,e.g.therankvaluefromNSGA23,canbeusedastheperformancemeasureinourapproachthesmallerthefitnessvalue,thebettertheperformanceofthesolution.Formoredetailsonhowtoobtainthisfitnessvaluethereaderisreferredto23.NotethatdifferentMOGAapproachesmaygeneratedifferentsolutionsinourapproach.However,ourgoalhereisnottodevelopanewgeneticalgorithmordistinguishbetweendifferentMOGAs.3.4RMOGAApproachGiventhetwomeasuresforperformanceandrobustnessofadesignsolution,asdiscussedbefore,wecanformulateourproblemthathastwoobjectivesoneistheperformanceandtheotheristherobustnessforadesignsolution.Theformulationofourrobustmultiobjectiveoptimizationproblemisshownin6ExGMvxRRggfffηmax,,,,,min11KK6Herethefitnessvaluefvisafunctionoftheobjectivesandconstraintsthatarecalculatedin1.Therobustnessindexηiscalculatedfrom5.AnoptimizationapproachFigure4,withanouterinnerstructure,isutilizedtosolvetheproblemshownin6.Theoutersubproblemi.e.,theuppersubprobleminFigure4istosimultaneouslyminimizethefitnessvaluefvandmaximizetherobustnessindexη.Weusetheinnersubproblemi.e.,thelowersubprobleminFigure4tocalculatetheradiusRrecall5withrespectto∆p.upperlowerExGMvxxxxRRggfff≤≤ηmax,,,,,min11KK01},max,maxmax{s.t.min0,,1,0,,11−∆∆∆∆∆∆∑pxggffppRjjJjiiMiqGjqjpKKRx0Figure4OuterinneroptimizationstructureofRMOGAWestartwithanxvalue,calledx0,intheoutersubproblemandsendittotheinnersubproblemincludingthenominalvaluesforfmx0,p0andgjx0,p0.Thenominalvaluesarefixedintheinnersubproblem.WethenoptimizetheradiusRasafunctionof∆pforthenominaldesignx0,andtheoptimalvalueRissentbacktotheoutersubproblem.ThisisrepeatedforalldesignalternativesunderconsiderationseeFigure4.WenowdiscussthefitnessassignmentprocedureusedintheimplementationofRMOGA.Forconciseness,theMOGAdetailsarenotdiscussedhereandthefocusisontheRMOGA.TheGeneticAlgorithmGArequiresascalarfitnessvalueforallthecandidatesolutions.ThemajorstepsinthefitnessassignmentprocedureareasfollowsStep1Evaluatetheobjectivesandconstraintsoftheoriginalproblem1.Step2Calculatetherobustnessindexηforeachofthecandidatesolutions.Step3Performanondominatedsortonthepopulationbasedontheobjectivevaluesoftheoriginalproblem.Considerthisrankandtheconstraintviolationtoassignafitnessvaluefvtothecandidatesolution.Step4Performanondominatedsortingonthepopulationbasedonrobustindexηandfitnessvaluefvastheobjectives.Thisisessentiallyatwoobjectivenondominatedsorting.Step5AssignafitnessbasedonthenondominatedrankfromStep4andcontinuetheGAiterationsuntilconvergenceisreached.3.5DistanceNormIn5wecouldusethreedifferentqvaluesq1,2and∞.DifferentLqnormswillaffectthevalueoftherobustnessindexofadesignsolution.AsshowninFigure5,thedistancefrompointA,BandCtotheorigincorrespondtotheradiussolutionfrom5inL1,L2andL∞norm.Therobustnessmeasureshouldbespecifiedinaparticulardistancenorm.BCA∆p2∆p1SensitivityRegionL2normL1normL∞normFigure5TheeffectofdifferentLpnorms4.TESTRESULTSInthissection,wewilldemonstrateanapplicationoftheproposedapproachtotwotestproblems.4.1TestProblem14.1.1ProblemDescriptionThefirsttestproblemthatweusetodemonstratetheRMOGA774
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