artigo nsga-ii

Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.1AnEvolutionaryMany-ObjectiveOptimizationAlgorithmUsingReference-point BasedNon-dominatedSortingApproach,Part I: SolvingProblemswithBoxConstraintsKalyanmoyDeb, Fellow, IEEEandHimanshuJainAbstractHaving developed multi-objective optimization al-gorithms usingevolutionaryoptimizationmethods anddemon-strated their niche on various practical problems involvingmostly two andthree objectives, there is nowa growing needfordevelopingevolutionarymulti-objectiveoptimization(EMO)algorithms for handling many-objective (having four or moreobjectives) optimizationproblems. Inthispaper, werecognizeafewrecent effortsanddiscussanumberof viabledirectionsfordeveloping a potential EMO algorithm for solving many-objectiveoptimizationproblems. Thereafter, wesuggest areference-pointbasedmany-objective NSGA-II (wecall it NSGA-III) that em-phasizespopulationmemberswhicharenon-dominatedyetclosetoaset ofsuppliedreferencepoints. TheproposedNSGA-IIIisappliedtoanumberofmany-objectivetestproblemshavingtwoto15objectives andcomparedwithtwoversions of arecentlysuggested EMOalgorithm(MOEA/D). While each of the twoMOEA/Dmethods works well ondifferent classes of problems,theproposedNSGA-IIIisfoundtoproducesatisfactoryresultsonall problems consideredinthis study. This paper presentsresults on unconstrained problems and the sequel paper considersconstrained and other specialties in handling many-objectiveoptimizationproblems.IndexTermsMany-objectiveoptimization,evolutionarycom-putation, large dimension, NSGA-III, non-dominated sorting,multi-criterionoptimization.I. INTRODUCTIONEvolutionarymulti-objectiveoptimization(EMO) method-ologies have amply shown their niche in nding a set ofwell-convergedandwell-diversiednon-dominatedsolutionsin different two and three-objective optimization problemssincethebeginningof nineties. However, inmost real-worldproblemsinvolvingmultiplestake-holdersandfunctionalities,there often exists manyoptimizationproblems that involvefourormoreobjectives, sometimesdemandingtohave10to15 objectives [1], [2]. Thus, itisnot surprising that handling alargenumber ofobjectiveshadbeenoneofthemainresearchactivities in EMOfor the past fewyears. Many-objectiveK. Deb is with Department of Electrical and Computer Engineering.MichiganState University, 428S. ShawLane, 2120EB, East Lansing, MI48824,USA,e-mail:[email protected](seehttp://www.egr.msu.edu/kdeb).Prof.DebisalsoavisitingprofessoratAaltoUniversitySchoolofBusiness,FinlandandUniversityofSk ovde, Sweden.H. Jain is with Eaton Corporation, Pune 411014, India, email: [email protected] (c) 2012 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtainedfromtheIEEEbysendingarequest [email protected] pose a number of challenges to anyoptimizationalgorithm,includinganEMO.Firstandforemost,thepropor-tionofnon-dominatedsolutionsinarandomlychosenset ofobjective vectors becomes exponentially large with an increaseof number of objectives. Sincethe non-dominatedsolutionsoccupy most of the population slots, any elite-preservingEMOfaces adifcultyinaccommodatingadequatenumberof newsolutions in the population. This slows down thesearchprocessconsiderably[3], [4]. Second, implementationof a diversity-preservation operator (such as the crowdingdistance operator [5] or clusteringoperator [6]) becomes acomputationallyexpensiveoperation. Third, visualizationofa large-dimensional front becomes a difcult task, therebycausing difculties in subsequent decision-making tasks and inevaluatingtheperformanceofanalgorithm.Forthispurpose,the performance metrics (such as hyper-volume measure [7] orother metrics [3], [8]) are either computationally too expensiveormaynotbemeaningful.AnimportantquestiontoaskisthenAreEMOsusefulformany-objectiveoptimizationproblems?. Althoughthe thirddifcultyrelatedtovisualizationandperformancemeasuresmentioned above cannot be avoided, some algorithmic changestotheexistingEMOalgorithmsmaybepossibletoaddressthe rst two concerns. In this paper, we reviewsome ofthe past efforts [9], [10], [11], [12], [13], [14] in devisingmany-objective EMOs and outline some viable directionsfor designingefcient many-objectiveEMOmethodologies.Thereafter, wepropose anew method thatusestheframeworkof NSGA-II procedure[5] but workswithaset of suppliedorpredenedreferencepointsanddemonstrateitsefcacyinsolving two to 15-objective optimization problems. In thispaper, we introduce the framework and restrict to solvingunconstrained problems of various kinds, such as havingnormalized,scaled, convex,concave,disjointed, andfocusingona part of the Pareto-optimal front. Practice mayoffer anumber ofsuch properties to existin aproblem. Therefore, anadequatetest of analgorithmfor theseeventualitiesremainsas an important task. We compare the performance of theproposedNSGA-III withtwoversionsof anexistingmany-objectiveEMO(MOEA/D[10]), asthemethodissomewhatsimilar to the proposed method. Interesting insights aboutworking of both versions of MOEA/Dand NSGA-III arerevealed. The proposed NSGA-III is also evaluated for itsusein afewother interesting multi-objective optimization andCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.2decision-making tasks.Inthesequelofthispaper, wesuggestanextensionof theproposedNSGA-III for handlingmany-objectiveconstrainedoptimizationproblemsandafewotherspecial andchallengingmany-objectiveproblems.Inthe remainder of this paper, we rst discuss the dif-cultiesinsolvingmany-objectiveoptimizationproblemsandthenattempt toanswer the questionposedaboveabout theusefulnessof EMOalgorithmsinhandlingmanyobjectives.Thereafter, in Section III, we present a reviewof some ofthe past studies onmany-objectiveoptimizationincludingarecently proposed method MOEA/D [10]. Then, in Section IV,we outline our proposed NSGA-III procedure in detail. ResultsonnormalizedDTLZtestproblemsupto15objectivesusingNSGA-III andtwoversionsof MOEA/DarepresentednextinSectionV-A. ResultsonscaledversionofDTLZproblemssuggested here are shown next. Thereafter in subsequentsections, NSGA-III procedure is tested on different typesof many-objectiveoptimizationproblems. Finally, NSGA-IIIis applied to two practical problems involving three andnineobjectiveproblems inSectionVII. Conclusions of thisextensivestudyaredrawninSectionVIII.II. MANY-OBJECTIVEPROBLEMSLoosely, many-objectiveproblemsaredenedasproblemshaving four ormore objectives. Twoand three-objective prob-lemsfall intoadifferentclassastheresultingPareto-optimalfront, inmost cases, inits totalitycanbe comprehensivelyvisualized using graphical means. Although a strict upperbound on the number of objectives for a many-objectiveoptimizationproblemisnot soclear, except afewoccasions[15], most practitioners are interestedina maximumof 10to15objectives. Inthissection, rst, wediscussdifcultiesthat anexistingEMOalgorithmmayfaceinhandlingmany-objective problems and investigate if EMOalgorithms areusefulat all inhandlingalargenumberofobjectives.A. DifcultiesinHandlingManyObjectivesIt has beendiscussedelsewhere[4], [16] that thecurrentstate-of-the-art EMOalgorithmsthatworkundertheprincipleofdomination[17]mayfacethefollowingdifculties:1) Alargefractionof populationis non-dominated: It iswell-known[3], [16] that withanincrease innumberof objectives, anincreasinglylarger fractionof aran-domly generated population becomes non-dominated.Sincemost EMOalgorithmsemphasizenon-dominatedsolutions in a population, in handlingmany-objectiveproblems, there is not much roomfor creating newsolutions inageneration. This slows downthesearchprocess andtherefore the overall EMOalgorithmbe-comesinefcient.2) Evaluationof diversitymeasurebecomescomputation-allyexpensive:Todeterminetheextent ofcrowdingofsolutions in a population, the identication of neigh-bors becomes computationally expensive in a large-dimensional space. Anycompromiseor approximationin diversityestimate to make the computations fastermaycauseanunacceptabledistributionof solutionsattheend.3) Recombination operationmaybeinefcient: Inamany-objective problem, if only a handful of solutions aretobefoundinalarge-dimensionalspace, solutionsarelikelytobewidelydistant fromeachother. Insuchapopulation,theeffectofrecombinationoperator(whichis considered as a key search operator in an EMO)becomesquestionable. Twodistant parent solutionsarelikely to produce offspring solutions that are alsodistantfromparents. Thus, special recombination operators(matingrestrictionorotherschemes)maybenecessaryforhandlingmany-objectiveproblemsefciently.4) Representation of trade-off surface is difcult: It isintuitive to realize that to represent a higher dimensionaltrade-offsurface,exponentiallymorepointsareneeded.Thus, alargepopulationsizeisneededtorepresenttheresultingPareto-optimalfront.Thiscausesdifcultyforadecision-makertocomprehendandmakeanadequatedecisiontochooseapreferredsolution.5) Performancemetrics are computationallyexpensivetocompute: Since higher-dimensional sets of points areto be compared against each other to establish theperformanceof onealgorithmagainst another, alargercomputational effortisneeded. Forexample, computinghyper-volumemetricrequiresexponentiallymorecom-putationswiththenumberofobjectives[18], [19].6) Visualization is difcult: Finally, although it is not amatter relatedtooptimizationdirectly, eventuallyvisu-alizationofahigher-dimensional trade-offfrontmaybedifcultformany-objectiveproblems.Therst threedifcultiescanonlybealleviatedbycertainmodications to existing EMOmethodologies. The fourth,fth, andsixthdifcultiesarecommontoall many-objectiveoptimization problems and we do not address them adequatelyhere.B. EMOMethodologiesforHandlingMany-ObjectiveProb-lemsBefore we discuss the possible remedies for three difcultiesmentioned above, here we highlight two different many-objectiveproblemclasses for whichexistingEMOmethod-ologiescanstill beused.First, existing EMO algorithms may still be useful in ndinga preferred subset of solutions (a partial set) from the completePareto-optimalset. Althoughthepreferredsubsetwillstill bemany-dimensional, sincethetargetedsolutionsarefocusedinasmallregiononthePareto-optimalfront,mostoftheabovedifcultieswill bealleviatedbythisprinciple. Anumber ofMCDM-basedEMOmethodologies are alreadydevisedforthispurposeandresultsonaslargeas10-objectiveproblemshaveshowntoperformwell [20], [21], [22], [23].Second, many problems in practice, albeit having manyobjectives, often degenerate to result in a low-dimensionalPareto-optimal front [4], [11], [24], [25]. Insuchproblems,identication of redundant objectives can be integrated with anEMOtondthePareto-optimal frontthatislow-dimensional.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.3Sincetheultimate front isaslowastwoorthree-dimensional,existingEMOmethodologies shouldworkwell inhandlingsuchproblems. Aprevious studyofNSGA-IIwithaprincipalcomponent analysis (PCA) basedprocedure[4] was abletosolve as large as 50-objective problems having a two-objectivePareto-optimalfront.C. TwoIdeasforaMany-ObjectiveEMOKeepinginmindtherst threedifcultiesassociatedwithadomination-basedEMOprocedure, twodifferent strategiescanbeconsideredtoalleviatethedifculties:1) Use of a special domination principle: The rst difcultymentionedabovecanbealleviatedbyusinga specialdominationprinciplethat will adaptivelydiscretizethePareto-optimal front andnda well-distributedset ofpoints. For example, theuseof -dominationprinciple[26], [27]willmakeall pointswithindistancefromaset ofPareto-optimalpoints-dominatedandhencetheprocesswill generateanitenumberofPareto-optimalpointsastarget.Suchaconsiderationwillalsoalleviatetheseconddifcultyofdiversitypreservation.Thethirddifcultycanbetakencareof bytheuseof amatingrestrictionscheme or a special recombinationschemeinwhichnear-parentsolutionsareemphasized(suchasSBXwithalargedistributionindex[28]).Otherspecialdomination principles [29], [30] can also be used for thispurpose. AguirreandTanaka[31] andSatoet al. [32]suggested the use of a subset of objectives for dominancecheck and using adifferent combination of objectives inevery generation. The use of xed cone-domination [33],[34] or variable cone-domination [35] principles can alsobe tried. Thesestudies weremade in thecontext of low-dimensional problems and their success in solving many-objectiveoptimizationisyet tobeestablished.2) Use of a predened multiple targeted search: It hasbeen increasinglyclear that it is too muchto expectfromasinglepopulation-basedoptimizationalgorithmtohaveconvergenceof its populationnear thePareto-optimal front and simultaneously its distributed uni-formlyaroundthe entire front in a large-dimensionalproblem. Onewaytohandlesuchmany-objectiveopti-mization problems would betoaidthediversity mainte-nanceissuebysomeexternalmeans.Thisprinciplecandirectlyaddresstheseconddifcultymentionedabove.InsteadofsearchingtheentiresearchspaceforPareto-optimalsolutions, multiplepredenedtargetedsearchescanbe set byanalgorithm. Since optimal points arefound corresponding toeachofthetargeted searchtask,the rstdifculty of dealing with alarge non-dominatedset is alsoalleviated. The recombinationissue canbeaddressed by using a mating restriction scheme in whichtwosolutionsfromneighboringtargetsareparticipatedintherecombination operation. Ourproposed algorithm(NSGA-III) is based on this principle and thus wediscussthis aspectin somewhat more detail. Wesuggesttwo different waysto implement the predened multipletargetedsearchprinciple:a) Aset of predened search directions spanningthe entire Pareto-optimal front can be speciedbeforehand and multiple searches can be performedalongeachdirection. Since the searchdirectionsarewidelydistributed,theobtainedoptimalpointsare also likely to be widely distributed on thePareto-optimal front in most problems. RecentlyproposedMOEA/Dprocedure[10] usesthiscon-cept.b) Insteadofmultiplesearchdirections,multiplepre-denedreferencepointscanbespeciedfor thispurpose. Thereafter, pointscorrespondingtoeachreferencepoint canbeemphasizedtondset ofwidelydistributedset of Pareto-optimal points. Afewsuchimplementationswereproposedrecently[36], [37], [38], [14], and this paper suggestsanotherapproachextendingandperfectingtheal-gorithmproposedintherst reference[36].III. EXISTINGMANY-OBJECTIVEOPTIMIZATIONALGORITHMSGarza-Fabre, Pulido and Coello [16] suggested three single-objective measures by using differences in individual objectivevalues between two competing parents and showed that in 5 to50-objectiveDTLZ1, DTLZ3andDTLZ6problemsthecon-vergence propertycangetenhanced compared toanumberofexistingmethodsincludingtheusualPareto-dominancebasedEMO approaches. Purshouse and Fleming [39] clearly showedthatdiversity preservation andachieving convergence near thePareto-optimal front are two contradictory goals and usualgeneticoperatorsarenotadequatetoattainbothgoalssimul-taneously, particularlyfor many-objectiveproblems. Anotherstudy[40] extends NSGA-II byaddingdiversity-controllingoperators to solve six to 20-objective DTLZ2 problems.K oppenandYoshida [41] claimedthat NSGA-II procedureinitsoriginalityisnot suitablefor many-objectiveoptimiza-tionproblems andsuggesteda number of metrics that canpotentiallyreplaceNSGA-IIscrowdingdistanceoperatorforbetter performance. Based onsimulationstudies ontwo to15-objective DTLZ2, DTLZ3 and DTLZ6 problems, theysuggestedtouseasubstituteassignment distancemeasureasthe best strategy. Hadka and Reed [42] suggested an ensemble-basedEMOprocedurewhichuses a suitable recombinationoperatoradaptivelychosenfromaset ofeightto10differentpre-denedoperatorsbasedontheir generation-wisesuccessrateinaproblem. It alsouses -dominanceconcept andanadaptivepopulationsizingapproachthat isreportedtosolveup to eight-objective test problems successfully. Bader andZitzler [43] have suggesteda fast procedurefor computingsample-based hyper-volume and devised an algorithm to nd asetoftrade-off solutionsformaximizingthehyper-volume. Agrowingliteratureonapproximatehyper-volumecomputation[44], [18], [45] may make such an approach practical forsolvingmany-objectiveproblems.Theabovestudiesanalyzeandextendpreviously-suggestedevolutionarymulti-objectiveoptimizationalgorithmsfortheirsuitabilitytosolvingmany-objective problems.Inmostcases,Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.4theresultsarepromisingandthesuggestedalgorithmsmustbetestedonothermorechallengingproblemsthantheusualnormalized test problems such as DTLZ problems. Theymust alsobetriedonreal-worldproblems. Inthefollowingparagraphs,wedescribearecentlyproposedalgorithmwhichts well with our description of a many-objective optimizationalgorithmgiveninSectionII-Candcloselymatcheswithourproposedalgorithm.MOEA/D[10] uses apredenedset of weight vectors tomaintainadiverseset oftrade-offsolutions. Foreachweightvector,theresultingproblemiscalledasub-problem. Tostartwith, every population member (with sizesameasthenumberof weight vectors) is associated with a weight vector randomly.Thereafter, two solutions fromneighboring weight vectors(denedthroughanichingparameter (T)) arematedandanoffspringsolutioniscreated. Theoffspringisthenassociatedwith one or more weight vectors based on a performancemetric. Twometricsaresuggestedinthestudy. Apenalizeddistancemeasureofapointfromtheidealpointisformedbyweightedsum(weightisanotheralgorithmicparameter)ofperpendicular distance(d2) fromthereferencedirectionanddistance(d1)alongthereferencedirection:PBI(x, w) = d1 + d2. (1)We call this procedure here as MOEA/D-PBI. The secondapproachsuggestedis the use of Tchebycheff metric usingautopianpoint zandtheweightvectorw:TCH(x, w, z) =Mmaxi=1wi|fi(x) zi|. (2)Inreportedsimulations[10], theideal point wasusedas zandzero-weight scenarioishandledbyusingasmallnumber.We call this procedureas MOEA/D-TCHhere. Anexternalpopulationmaintains the non-dominatedsolutions. The rsttwodifcultiesmentionedearlier arenegotiatedbyusinganexplicit set of weight vectors to nd points and the thirddifcultyis alleviatedbyusingamatingrestrictionscheme.Simulation results were shown for two and three-objective testproblems only and it was concludedthat MOEA/D-PBI isbetter for three-objectiveproblemsthanMOEA/D-TCHandthe performance of MOEA/D-TCH improved with an objectivenormalizationprocessusingpopulationminimumandmaxi-mumobjectivevalues. Bothversionsof MOEA/Drequiretosetaniching parameter (T). Basedon somesimulation resultsontwoandthree-objectiveproblems, authors suggestedtheuseof alargefractionofpopulationsizeasT. Additionally,MOEA/D-PBIrequiresanappropriatesettingofanadditionalparameter penalty parameter , for which authors havesuggestedavalueof5.Alater study by the developers of MOEA/Dsuggestedthe use of differential evolution (DE) to replace geneticrecombinationandmutationoperators. Also, further modi-cations were done indeningneighborhoodof a particularsolution and in replacing parents in a given neighborhoodbythe correspondingoffspringsolutions [46]. We call thismethod as MOEA/D-DEhere. Results on a set of mostlytwoandathree-objectivelinkedproblems[47]showedbetterperformance with MOEA/D-DE compared to other algorithms.Asmentioned, MOEA/Disapromisingapproachfor many-objectiveoptimizationasit addressessomeofthedifcultiesmentioned above well, but the above-mentioned MOEA/Dstudies did not quite explore their suitability to a large numberof objectives. In this paper, we apply them to problems havingupto15objectives andevaluate their applicabilitytotrulymany-objectiveoptimizationproblemsandreveal interestingpropertiesofthesealgorithms.Another recent study [14] follows our description of amany-objective optimization procedure. The study extendsthe NSGA-II procedure to suggest a hybridNSGA-II (HNalgorithm) for handling three and four-objective problems.Combined population members are projected on a hyper-planeandaclusteringoperationis performedonthehyper-plane toselect a desirednumber of clusters whichis user-dened. Thereafter, basedonthediversityof thepopulation,either alocal searchoperationonarandomcluster memberis used to move the solution closer to the Pareto-optimalfront or adiversityenhancement operator is usedtochoosepopulationmembersfromall clusters. Sincenotargetedanddistributedsearchisused, theapproachismoregenericthanMOEA/Dortheproceduresuggestedinthispaper. However,the efciency of HN algorithm for problems having more thanfourobjectivesisyet tobeinvestigatedtosuggest itsuseformany-objectiveproblems, in general. We nowdescribe ourproposedalgorithm.IV. PROPOSEDALGORITHM:NSGA-IIIThe basic framework of the proposed many-objectiveNSGA-II (or NSGA-III) remains similar to the originalNSGA-II algorithm[5] withsignicant changesinitsselec-tionmechanism. But unlikeinNSGA-II, themaintenanceofdiversity among population membersinNSGA-IIIisaidedbysupplyingandadaptivelyupdatinga number of well-spreadreference points. For completeness, we rst present a briefdescriptionoftheoriginalNSGA-IIalgorithm.Let us consider t-th generation of NSGA-II algorithm.Suppose the parent populationat this generationis PtanditssizeisN, whiletheoffspringpopulationcreatedfromPtis Qthaving Nmembers. The rst step is to choose thebest Nmembers fromthe combined parent and offspringpopulation Rt=PtQt(of size 2N), thus allowing topreserveelitemembersof theparent population. Toachievethis, rst, thecombinedpopulationRtissortedaccordingtodifferent non-domination levels (F1, F2and so on). Then, eachnon-dominationlevel isselectedoneat atimetoconstruct anewpopulationSt, startingfromF1, until thesizeof Stisequalto Norforthersttimeexceeds N.Letussaythelastlevel includedisthel-thlevel. Thus, all solutionsfromlevel(l+ 1) onwards arerejectedfromthe combinedpopulationRt. Inmost situations, thelast acceptedlevel (l-thlevel) isonlyacceptedpartially. Insuchacase, onlythosesolutionsthat will maximize the diversity of the l-th front are chosen. InNSGA-II,thisisachievedthroughacomputationallyefcientyet approximateniche-preservationoperatorwhichcomputesthe crowding distance for every last level member as thesummation of objective-wise normalized distance between twoCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.5neighboringsolutions. Thereafter, thesolutionshavinglargercrowdingdistance values are chosen. Here, we replace thecrowding distance operator with the following approaches(subsectionsIV-AtoIV-E).A. ClassicationofPopulationintoNon-dominatedLevelsTheaboveprocedureof identifyingnon-dominatedfrontsusing the usual domination principle [17] is also used inNSGA-III. Allpopulation members from non-dominated frontlevel 1 to level l are rst included in St. If |St| = N, no furtheroperationsareneededandthenext generationisstartedwithPt+1=St. For |St| >N, members fromone to(l1)frontsarealreadyselected, that is, Pt+1=l1i=1Fi, andtheremaining (K= N |Pt+1|)populationmembersarechosenfromthelast front Fl. We describetheremainingselectionprocessinthefollowingsubsections.B. Determinationof ReferencePointsonaHyper-PlaneAs indicated before, NSGA-III uses a predened set ofreference points to ensure diversity in obtained solutions.The chosen reference points can either be predened in astructuredmanner or suppliedpreferentiallybytheuser. Weshall present results of both methods in the results section later.Intheabsenceofanypreferenceinformation,anypredenedstructured placement of reference points can be adopted, but inthispaper weuse Dasand Denniss [48] systematicapproach1that placespointsonanormalizedhyper-planea(M1)-dimensional unit simplexwhichis equallyinclinedtoallobjectiveaxesandhasanintercept ofoneoneachaxis. Ifpdivisions are considered along each objective, the total numberof referencepoints(H) inanM-objectiveproblemisgivenby:H=_M+ p 1p_. (3)For example, in a three-objective problem(M=3), thereference points are created on a triangle with apex at (1, 0, 0),(0, 1, 0)and(0, 0, 1). Iffourdivisions(p = 4)arechosenforeachobjectiveaxis,H=_3+414_or15referencepointswillbe created. For clarity, these referencepoints are showninFigure1.Intheproposed NSGA-III,inadditiontoemphasiz-ingnon-dominatedsolutions, we alsoemphasizepopulationmembers which are in some sense associated with eachof these referencepoints. Since the above-createdreferencepointsarewidelydistributedontheentirenormalizedhyper-plane, the obtained solutions are also likely to be widelydistributed on or near the Pareto-optimal front. In the case of auser-supplied setof preferred reference points, ideally the usercanmarkHpointsonthenormalizedhyper-plane orindicateany H, M-dimensional vectorsforthepurpose. Theproposedalgorithmislikelytondnear Pareto-optimal solutionscor-respondingtothesuppliedreferencepoints, therebyallowingthis methodto be used more fromthe point of viewof acombinedapplicationofdecision-makingandmany-objectiveoptimization. TheprocedureispresentedinAlgorithm1.1Anyother structureddistributionwithorwithout abiasingonsomepartofthePareto-optimal front canbeusedaswell.hyperplaneNormalizedlineReferencepointReferenceIdeal point 11f1f3 1f2Fig. 1. 15referencepointsareshownonanormalizedreferenceplaneforathree-objectiveproblemwithp=4.Algorithm1GenerationtofNSGA-IIIprocedureInput: HstructuredreferencepointsZsor suppliedaspira-tionpointsZa, parentpopulationPtOutput: Pt+11: St= ,i = 12: Qt=Recombination+Mutation(Pt)3: Rt= Pt Qt4: (F1, F2, . . .) =Non-dominated-sort(Rt)5: repeat6: St= St Fiandi = i + 17: until|St| N8: Last fronttobeincluded:Fl= Fi9: if|St| = Nthen10: Pt+1= St, break11: else12: Pt+1= l1j=1Fj13: PointstobechosenfromFl:K= N|Pt+1|14: Normalize objectives and create reference set Zr:Normalize(fn, St, Zr, Zs, Za)15: AssociateeachmembersofStwithareferencepoint:[(s), d(s)] =Associate(St, Zr) %(s):closestreferencepoint,d: distancebetweensand(s)16: Computenichecount ofreferencepointjZr: j=

sSt/Fl(((s) = j)?1 : 0)17: Choose Kmembersoneatatimefrom FltoconstructPt+1: Niching(K, j, , d, Zr, Fl, Pt+1)18: endifC. AdaptiveNormalizationof PopulationMembersFirst, the ideal point of the populationStis determinedbyidentifyingtheminimumvalue(zmini), for eachobjectivefunctioni=1, 2, . . . , Mint=0Sandbyconstructingtheideal point z=(zmin1, zmin2, . . . , zminM). Eachobjectivevalueof Stis thentranslatedbysubtractingobjectivefibyzmini,sothattheidealpointoftranslatedStbecomesazerovector.Wedenotethistranslatedobjectiveasfi(x) = fi(x) zmini.Thereafter, theextremepointineachobjectiveaxisisidenti-edbyndingthesolution(x St)thatmakesthefollowingachievement scalarizing function minimum with weight vectorCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.6wbeingtheaxisdirection:ASF(x, w) =Mmaxi=1fi(x)/wi, for x St. (4)For wi= 0,wereplaceitwithasmallnumber 106.For i-thtranslatedobjectivedirection fi,thiswillresultinanextremeobjectivevector, zi,max. TheseMextremevectors arethenusedtoconstituteaM-dimensional linear hyper-plane. Theintercept aiof thei-thobjectiveaxis andthe linear hyper-planecanthenbecomputed(seeFigure2)andtheobjectivefunctionscanbenormalizedasfollows:fni(x) =fi(x)ai zmini=fi(x) zminiai zmini, fori = 1, 2, . . . , M.(5)Notethat theintercept oneachnormalizedobjectiveaxis isnowat fni=1 and a hyper-plane constructed with theseinterceptpointswill make

Mi=1fni= 1.3,max1,maxzz3212aa13az2,maxf 0 1 2ff 1 2 0 0 1 2 3Fig.2. Procedureofcomputinginterceptsandthenformingthehyper-planefromextremepointsareshownforathree-objectiveproblem.Inthecaseofstructuredreferencepoints(Hofthem), theoriginal referencepoints calculatedusingDas andDenniss[48] approachalreadylieonthisnormalizedhyper-plane. Inthe case of preferred reference points by the user, the referencepoints are simply mapped on to the above-constructed normal-ized hyper-plane usingequation5. Since the normalizationprocedureandthecreationofthehyper-planeisdoneateachgenerationusingextremepointseverfoundfromthestart ofthesimulation, theproposedNSGA-III procedureadaptivelymaintains a diversityinthespace spannedbythe membersof Stat everygeneration. This enables NSGA-III tosolveproblems having a Pareto-optimal front whose objective valuesmaybedifferentlyscaled. TheprocedureisalsodescribedinAlgorithm2.Algorithm2Normalize(fn, St, Zr, Zs/Za)procedureInput: St, Zs(structuredpoints)orZa(suppliedpoints)Output: fn, Zr(reference points on normalized hyper-plane)1: forj= 1toMdo2: Computeideal point:zminj= minsSt fj(s)3: Translateobjectives:fj(s) = fj(s) zminjs St4: Compute extreme points: zj,max= s :argminsStASF(s, wj), where wj= (, . . . , )T = 106, andwjj= 15: endfor6: Computeinterceptsajforj= 1, . . . , M7: Normalizeobjectives(fn)usingEquation58: ifZaisgiventhen9: Mapeach(aspiration)pointonnormalizedhyper-planeusingEquation5andsavethepointsinthesetZr10: else11: Zr= Zs12: endifD. AssociationOperationAfter normalizingeachobjectiveadaptivelybasedontheextent of members of St in the objective space, next we need toassociateeachpopulation memberwithareferencepoint.Forthispurpose, wedeneareferencelinecorresponding toeachreferencepoint onthe hyper-planebyjoiningthe referencepoint withthe origin. Then, we calculate the perpendiculardistanceof eachpopulationmember of Stfromeachof thereferencelines. Thereferencepoint whosereferenceline isclosest toa populationmember inthenormalizedobjectivespaceis consideredassociatedwiththe populationmember.ThisisillustratedinFigure3. Theprocedureispresentedin00.511.500.511.500.511.5f1f2f3Fig. 3. Association of population members with reference points is illustrated.Algorithm3.E. Niche-PreservationOperationIt is worth noting that a reference point may have oneor morepopulationmembers associatedwithit or neednothaveanypopulationmemberassociatedwithit. WecounttheCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.7Algorithm3Associate(St, Zr)procedureInput: Zr, StOutput: (s St), d(s St)1: foreachreferencepoint z Zrdo2: Computereferencelinew =z3: endfor4: foreachs Stdo5: foreachw Zrdo6: Computed(s, w) =s wTs/w7: endfor8: Assign(s) =w :argminwZr d(s, w)9: Assignd(s) = d(s, (s))10: endfornumber of populationmembersfromPt+1=St/Flthat areassociatedwitheachreferencepoint.Letusdenotethisnichecount as jforthe j-threference point. Wenow deviseanewniche-preservingoperationas follows. First, weidentifythereferencepoint set Jmin= {j:argminjj}havingminimumj. Incaseofmultiplesuchreferencepoints, one(j Jmin)ischosenat random.If j= 0 (meaning that there is no associated Pt+1membertothereferencepointj), therecanbetwoscenarioswithjinset Fl. First, there exists one or more members infrontFlthat arealreadyassociatedwiththereferencepointj. Inthiscase, theonehavingtheshortest perpendicular distancefrom thereference lineisadded to Pt+1.Thecount jisthenincrementedbyone. Second, thefrontFldoesnot haveanymember associated with the reference point j. In this case,thereferencepointisexcludedfromfurtherconsiderationforthecurrentgeneration.Intheevent of j1(meaningthat alreadyonemem-ber associated with the reference point exists in St/Fl), arandomly2chosen member, if exists, fromfront Flthat isassociatedwiththereferencepointj is addedtoPt+1. Thecount jisthenincrementedbyone. After nichecountsareupdated, the procedure is repeated for a total of Ktimes to llall vacant population slots of Pt+1. The procedure is presentedinAlgorithm4.F. GeneticOperationstoCreateOffspringPopulationAfter Pt+1is formed, it is then used to create a newoffspring population Qt+1 by applying usual genetic operators.In NSGA-III, we have already performed a careful elitist selec-tionof solutionsandattemptedtomaintaindiversityamongsolutions by emphasizingsolutions closest to the referenceline of eachreference point. Also, as we shall describe inSection V, for a computationally fast procedure, we haveset Nalmost equal toH, therebygivingequal importanceto each population member. For all these reasons, we donot employanyexplicit selectionoperationwithNSGA-III.The populationQt+1is constructed by applying the usualcrossover and mutation operators by randomly picking parentsfromPt+1. However, tocreate offspringsolutions closer to2The point closest to the reference point or using any other diversitypreservingcriterioncanalsobeused.Algorithm4Niching(K, j, , d, Zr, Fl, Pt+1)procedureInput: K, j,(s St), d(s St), Zr,FlOutput: Pt+11: k = 12: whilek Kdo3: Jmin= {j: argminjZr j}4:j= random(Jmin)5: Ij= {s : (s) = j, s Fl}6: ifIj= then7: ifj= 0then8: Pt+1= Pt+1 _s :argminsIjd(s)_9: else10: Pt+1= Pt+1 random(Ij)11: endif12: j= j + 1,Fl= Fl\s13: k = k + 114: else15: Zr= Zr/{j}16: endif17: endwhileparent solutions(totakecareofthirddifcultymentionedinSectionII-A), we suggest usinga relativelylarger value ofdistributionindexintheSBXoperator.G. Computational Complexity of One Generation of NSGA-IIIThe non-dominatedsorting (line 4 in Algorithm1) of apopulationof size2NhavingM-dimensional objectivevec-tors require O(N logM2N) computations [49]. Identicationof ideal point in line 2 of Algorithm2 requires a totalof O(MN) computations. Translationof objectives (line 3)requires O(MN) computations. However, identication ofextremepoints(line4) requireO(M2N) computations. De-termination of intercepts (line 6) requires one matrix inversionof size MM, requiring O(M3) operations. Thereafter,normalization of a maximumof 2N population members(line7) requireO(N) computations. Line8of Algorithm2requires O(MH) computations. All operations in Algorithm 3in associating a maximumof 2N population members toHreferencepoints wouldrequireO(MNH) computations.Thereafter, in the niching procedure in Algorithm 4, line 3 willrequireO(H) comparisons. Assumingthat L=|Fl|, line5requires O(L)checks.Line8intheworstcaserequires O(L)computations. Other operations have smaller complexity. How-ever, theabove computations intheniching algorithmneedtobeperformed amaximum of Ltimes,thereby requiring largerof O(L2)or O(LH)computations. Intheworstcasescenario(St=F1, that is, therst non-dominatedfront exceedsthepopulation size), L 2N. In all our simulations, we have usedN Hand N> M. Considering all the above considerationsandcomputations, theoverall worst-case complexityof onegenerationof NSGA-III is O(N2logM2N) or O(N2M),whicheverislarger.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.8H. Parameter-LessPropertyofNSGA-IIILikeinNSGA-II, NSGA-IIIalgorithmdoesnot requiretosetany new parameter other thanthe usualgenetic parameterssuchasthepopulationsize, terminationparameter, crossoverand mutation probabilities and their associated parameters.The number of reference points His not an algorithmicparameter, asthisisentirelyat thedisposal of theuser. Thepopulation size N is dependent on H, as N H. Thelocationofthereferencepointsissimilarlydependentonthepreference information thattheuserisinterestedtoachieveintheobtainedsolutions.We shall nowpresent simulationresults of NSGA-III forvarious many-objective scenarios and compare its performancewithMOEA/Dandclassical methods.V. RESULTSInthissection,weprovide thesimulation resultsofNSGA-IIIonthreeto15-objectiveoptimizationproblems. Sinceourmethod has a framework similar to MOEA/D in that both typesof algorithmsrequireaset of user-suppliedreferencepointsor weight vectors, we compare our proposed method withdifferent versions of MOEA/D(codes fromMOEA/Dweb-site[50] are used). The original MOEA/Dstudy proposed twoprocedures(MOEA/D-PBI andMOEA/D-TCH), but didnotsolvefourormoreobjectiveproblems. Here, alongwithouralgorithm, weinvestigatetheperformanceoftheseMOEA/Dalgorithmsonthreeto15-objectiveproblems.As a performance metric, we have chosen the inversegenerational distance (IGD) metric [51], [47] which as asingle metric which can provide a combined informationabouttheconvergence anddiversityoftheobtainedsolutions.SincereferencepointsorreferencedirectionsaresuppliedinNSGA-III andMOEA/Dalgorithms, respectively, andsinceinthissectionweshowtheworkingofthesemethodsontestproblems for which the exact Pareto-optimal surface is known,we canexactlylocate the targetedPareto-optimal points inthe normalizedobjective space. We compute these targetedpoints andcall themaset Z. For anyalgorithm, weobtainthenalnon-dominated pointsintheobjectivespaceandcallthemthe set A. Now, we compute the IGDmetric as theaverage Euclidean distance of points in setZ with their nearestmembersofall pointsinsetA:IGD(A, Z) =1|Z||Z|

i=1|A|minj=1d(zi, aj), (6)where d(zi, aj) = ziaj

2.Itisneedlesstowritethatasetwithasmaller IGDvalueisbetter. If nosolutionassociatedwithareferencepointisfound, theIGDmetricvaluefortheset will be large. For each case, 20 different runs from differentinitial populationsareperformedandbest, medianandworstIGDperformancevaluesarereported. Forall algorithms, thepopulationmembers fromthenal generationarepresentedandusedforcomputingtheperformancemeasure.Table I shows the number of chosen reference points (H) fordifferent sizes of a problem. The population size Nfor NSGA-IIIissetasthesmallestmultipleoffour3higherthanH. ForMOEA/Dprocedures, weuseapopulationsize, N=H, assuggestedbytheirdevelopers. Forthree-objectiveproblems,TABLEINUMBEROFREFERENCEPOINTS/DIRECTIONS ANDCORRESPONDINGPOPULATION SIZESUSEDIN NSGA-III ANDMOEA/DALGORITHMS.No. of Ref. pts./ NSGA-III MOEA/Dobjectives Ref. dirn. popsize popsize(M) (H) (N) (N)3 91 92 915 210 212 2108 156 156 15610 275 276 27515 135 136 135wehaveusedp=12inorder toobtainH=_31+1212_or91reference points (refer toequation3). For ve-objectiveproblems, wehaveusedp=6, sothat H=210referencepoints are obtained. Note that aslong as p Mis not chosen,nointermediate point will be createdbyDas andDennisssystematicapproach. Foreight-objective problems, evenifweusep = 8(tohaveat least oneintermediatereferencepoint),it requires5,040referencepoints. Toavoidsuchasituation,weusetwolayersofreferencepointswithsmallvaluesofp.Ontheboundarylayer, weusep= 3, sothat 120pointsarecreated. Ontheinsidelayer, weusep = 2, sothat 36pointsarecreated. All H=156pointsarethenusedas referencepoints foreight-objective problems. WeillustratethisscenarioinFigure 4for a three-objectiveproblemusingp =2forboundary layer and p = 1 for the inside layer. For 10-objectiveFig. 4. Theconcept for two-layered reference points (withsixpoints ontheboundarylayer (p=2)andthreepointsontheinsidelayer (p=1)) isshownforathree-objective problem, but areimplementedforeight ormoreobjectivesinthesimulationshere.problemsaswell, weusep = 3andp = 2forboundaryandinsidelayers, respectively, therebyrequiringatotal of H=220 + 55or 275referencepoints. Similarly, for15-objectiveproblems, weusep=2andp=1for boundaryandinsidelayers, respectively, therebyrequiringH=120 + 15or 135referencepoints.3SincenotournamentselectionisusedinNSGA-III,afactoroftwowouldbe adequate as well, but as we re-introduce tournament selection in theconstrainedNSGA-IIIinthesequel paper[52], wekeeppopulationsizeasamultipleoffourhereaswell tohaveauniedalgorithm.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.9TableIIpresentsotherNSGA-IIIandMOEA/Dparametersusedinthisstudy. MOEA/Drequiresadditional parametersTABLEIIPARAMETERVALUESUSEDIN NSGA-IIIANDTWOVERSIONSOFMOEA/D. nISTHENUMBEROFVARIABLES.Parameters NSGA-III MOEA/DSBXprobability[28], pc 1 1Polynomial mutationprob. [3], pm 1/n 1/nc[28] 30 20m[28] 20 20whichwehave setaccording tothesuggestions given bytheirdevelopers. The neighborhoodsize T is set as 20for bothapproachesandadditionallythepenaltyparameter for theMOEA/D-PBIapproachisset as5.A. NormalizedTest ProblemsTo startwith, weuse three to 15-objective DTLZ1, DTLZ2,DTLZ3andDTLZ4problems[53]. Thenumberof variablesare (M +k 1),where Misnumber ofobjectivesand k = 5for DTLZ1, while k = 10 for DTLZ2, DTLZ3 and DTLZ4. Inall these problems, the corresponding Pareto-optimal fronts lieinfi [0, 0.5]forDTLZ1problem, or[0, 1]forotherDTLZproblems. Since they have an identical range of values for eachobjective, wecall theseproblemsnormalizedtest problemsin this study. Table III indicates the maximumnumber ofgenerationsusedforeachtest problem.Figure 5 shows NSGA-III obtained front for the three-objectiveDTLZ1problem. This particular runis associatedwith the medianvalue of IGDperformance metric. All 91points are well distributedonthe entire Pareto-optimal set.Results withtheMOEA/D-PBI areshowninFigure6. It isclear that MOEA/D-PBI is also able to nd a good distributionof points similar to that of NSGA-III. However, Figure 7 showsthatMOEA/D-TCHisunable tondauniform distribution ofpoints. Suchadistributionwas alsoreportedintheoriginalMOEA/Dstudy[10].Table III shows that for the DTLZ1 problemNSGA-IIIperformsslightlybetterintermsoftheIGDmetric, followedby MOEA/D-PBI. For the ve-objective DTLZ1 problem,MOEA/D performs better than NSGA-III, but in 8, 10, and 15-objective problems, NSGA-III performs better. MOEA/D-TCHconsistentlydoes not performwell inall higher dimensionsoftheproblem. Thisobservationissimilartothat concludedintheoriginal MOEA/Dstudy[10] basedontwoandthree-objectiveproblems.For DTLZ2problems, the performanceof MOEA/D-PBIis consistently better than NSGA-III, however NSGA-IIIperforms better thanMOEA/D-TCHapproach. Figures 8, 9and10showthedistributionof obtainedpoints for NSGA-III, MOEA/D-PBI, andMOEA/D-TCHalgorithmsonthree-objectiveDTLZ2problem, respectively. Figuresshowthattheperformances of NSGA-III and MOEA/D-PBI are comparabletoeachother, whereastheperformanceof MOEA/D-TCHispoor.Figure 11 shows the variation of the IGDmetric valuewith function evaluations for NSGA-III and MOEA/D-PBIapproachesfor theeight-objectiveDTLZ2problem. AverageIGDmetricvaluefor20runsareplotted. ItisclearthatbothapproachesareabletoreducetheIGDvaluewithelapseoffunctionevaluations.SimilarobservationismadefortheDTLZ3problem. Thisproblemintroduces a number of local Pareto-optimal frontsthat provide a stiff challenge for algorithms tocome closetotheglobal Pareto-optimalfront. WhiletheperformanceofNSGA-III andMOEA/D-PBI aresimilar, withaslight edgefor MOEA/D-PBI, the performance of MOEA/D-TCH is poor.However, insomeruns, MOEA/D-PBIisunabletoget closetothePareto-optimal front, asevident fromalargevalueofIGDmetricvalue. Figure12showsvariationof theaverageIGDmetricof 20runswithfunctionevaluationsfor NSGA-IIIandMOEA/D-PBIapproaches.NSGA-IIImanagestondabetter IGDvaluethanMOEA/D-PBI approachafter about80,000functionevaluations.ProblemDTLZ4hasabiaseddensityofpointsawayfromfM=0, however the Pareto-optimal front is identical tothat inDTLZ2. ThedifferenceintheperformancesbetweenNSGA-III and MOEA/D-PBI is clear fromthis problem.BothMOEA/Dalgorithms arenot able tondanadequatedistributionofpoints, whereasNSGA-IIIalgorithmperformsasit didinother problems. Figures13, 14and15showtheobtaineddistributionofpointsonthethree-objectiveDTLZ4problem. The algorithms are unable to nd near f3= 0 Pareto-optimal points, whereas the NSGA-III is able tonda setof well-distributedpointsontheentirePareto-optimal front.These plots are made using the median performed run ineachcase. TableIIIclearlyshowthat theIGDmetricvaluesfor NSGA-III algorithmare better than those of MOEA/Dalgorithms. Figure 16 shows the value path plot of all obtainedsolutionsforthe10-objectiveDTLZ4problembyNSGA-III.Aspread ofsolutionsover fi [0, 1]forall10objectives andatrade-off amongthemareclear fromtheplot. Incontrast,Figure 17shows the value pathplot for the same problemobtainedusingMOEA/D-PBI. ThegureclearlyshowsthatMOEA/D-PBI isnot abletondawidelydistributedset ofpoints for the 10-objective DTLZ4 problem. Points havinglarger values of objectives f1to f7are not found by MOEA/D-PBIapproach.The right-most column of Table III presents the performanceof the recently proposedMOEA/D-DEapproach[46]. Theapproachusesdifferentialevolution(DE)insteadofSBXandpolynomial mutation operators. Twoadditional parameters areintroducedintheMOEA/D-DEprocedure.First, amaximumbound(nr)onthenumberofweightvectorsthat achildcanbeassociatedtoisintroducedandisset asnr=2. Second,for choosingamatingpartner of aparent, aprobability()isset for choosinganeighboringpartnerandtheprobability(1 ) for choosinganyother populationmember. Authorssuggestedtouse=0.9. Weusethesamevaluesof theseparametersinour study. TableIII indicatesthat this versionofMOEA/DdoesnotperformwellonthenormalizedDTLZproblems, however it performsbetter thanMOEA/D-PBI onDTLZ4problem. Duetoits poor performanceingeneral intheseproblems, wedonotapplyit anyfurther.In the above runs with MOEA/D, we have used SBXrecombination parameter index c= 20 (as indicated inCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.10TABLEIIIBEST,MEDIANANDWORST IGD VALUESOBTAINEDFOR NSGA-III ANDTWOVERSIONSOFMOEA/DON M-OBJECTIVE DTLZ1, DTLZ2, DTLZ3ANDDTLZ4PROBLEMS. BESTPERFORMANCEISSHOWNINBOLD.Problem M MaxGen NSGA-III MOEA/D-PBI MOEA/D-TCH MOEA/D-DEDTLZ1 4.880 1044.095 1043.296 1025.470 1033 400 1.308 1031.495 1033.321 1021.778 1024.880 1034.743 1033.359 1023.394 1015.116 1043.179 1041.124 1012.149 1025 600 9.799 1046.372 1041.129 1012.489 1021.979 1031.635 1031.137 1013.432 1022.044 1033.914 1031.729 1013.849 1028 750 3.979 1036.106 1031.953 1014.145 1028.721 1038.537 1032.094 1014.815 1022.215 1033.872 1032.072 1014.253 10210 1000 3.462 1035.073 1032.147 1014.648 1026.869 1036.130 1032.400 1014.908 1022.649 1031.236 1023.237 1018.048 10215 1500 5.063 1031.431 1023.438 1018.745 1021.123 1021.692 1023.634 1011.008 101DTLZ2 1.262 1035.432 1047.499 1023.849 1023 250 1.357 1036.406 1047.574 1024.562 1022.114 1038.006 1047.657 1026.069 1024.254 1031.219 1032.935 1011.595 1015 350 4.982 1031.437 1032.945 1011.820 1015.862 1031.727 1032.953 1011.935 1011.371 1023.097 1035.989 1013.003 1018 500 1.571 1023.763 1036.301 1013.194 1011.811 1025.198 1036.606 1013.481 1011.350 1022.474 1037.002 1012.629 10110 750 1.528 1022.778 1037.266 1012.873 1011.697 1023.235 1037.704 1013.337 1011.360 1025.254 1031.000 3.131 10115 1000 1.726 1026.005 1031.084 3.770 1012.114 1029.409 1031.120 4.908 101DTLZ3 9.751 1049.773 1047.602 1025.610 1023 1000 4.007 1033.426 1037.658 1021.439 1016.665 1039.113 1037.764 1028.887 1013.086 1031.129 1032.938 1011.544 1015 1000 5.960 1032.213 1032.948 1012.115 1011.196 1026.147 1032.956 1018.152 1011.244 1026.459 1036.062 1012.607 1018 1000 2.375 1021.948 1026.399 1013.321 1019.649 1021.123 6.808 1013.9238.849 1032.791 1037.174 1012.549 10110 1500 1.188 1024.319 1037.398 1012.789 1012.083 1021.010 8.047 1012.998 1011.401 1024.360 1031.029 2.202 10115 2000 2.145 1021.664 1021.073 3.219 1014.195 1021.260 1.148 4.681 101DTLZ4 2.915 1042.929 1012.168 1013.276 1023 600 5.970 1044.280 1013.724 1016.049 1024.286 1015.234 1014.421 1013.468 1019.849 1041.080 1013.846 1011.090 1015 1000 1.255 1035.787 1015.527 1011.479 1011.721 1037.348 1017.491 1014.116 1015.079 1035.298 1016.676 1012.333 1018 1250 7.054 1038.816 1019.036 1013.333 1016.051 1019.723 1011.035 7.443 1015.694 1033.966 1017.734 1012.102 10110 2000 6.337 1039.203 1019.310 1012.885 1011.076 1011.077 1.039 6.422 1017.110 1035.890 1011.056 4.500 10115 3000 3.431 1011.133 1.162 6.282 1011.073 1.249 1.220 8.477 101Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.1100.250.500.250.500.10.20.30.40.50.6f2 f1f300.250.500.250.500.10.20.30.40.50.6f1f2f3Fig. 5. Obtained solutions by NSGA-III forDTLZ1.00.250.500.250.500.10.20.30.40.50.6f2f1f300.250.500.250.500.10.20.30.40.50.6f1f2f3Fig. 6. ObtainedsolutionsbyMOEA/D-PBI forDTLZ1.00.250.500.250.500.10.20.30.40.50.6f2f1f300.250.500.250.500.20.40.6f1f2f3Fig. 7. ObtainedsolutionsbyMOEA/D-TCHforDTLZ1.00.5100.20.40.60.8100.20.40.60.811.2f2 f1f300.5100.5100.51f1f2f3Fig. 8. Obtained solutions by NSGA-III forDTLZ2.00.5100.20.40.60.8100.20.40.60.811.2f2f1f300.5100.5100.51f1f2f3Fig. 9. ObtainedsolutionsbyMOEA/D-PBI forDTLZ2.00.5100.5100.20.40.60.811.2f2 f1f300.5100.5100.51f1f2f3Fig. 10. Obtained solutionsby MOEA/D-TCH forDTLZ2.0 2 4 6 8x 10410-310-210-1100Function EvaluationsIGD NSGA-IIIMOEA/D-PBIFig.11. VariationofIGDmetricvaluewithNSGA-IIIandMOEA/D-PBIforDTLZ2.0 5 10 15x 10410-210-1100101102103IGDFunction Evaluations NSGA-IIIMOEA/D-PBIFig.12. VariationofIGDmetricvaluewithNSGA-IIIandMOEA/D-PBIforDTLZ3.TableII)mainlybecausethisvaluewaschosen intheoriginalMOEA/Dstudy [10]. Next, we investigate MOEA/D-PBIsperformancewithc= 30(whichwasusedwithNSGA-III).TableIVshows that theperformanceof MOEA/Ddoesnotchangemuchwiththeabovechangeincvalue.Next, we applyNSGA-III andMOEA/D-PBI approachestotwo WFGtest problems. The parameter settings are thesameasbefore. Figures18and19showtheobtainedpointsonWFG6problem. The convergenceis slightlybetter withNSGA-III approach. Similarly, Figures 20 and 21 showasimilarperformancecomparisonoftheabovetwoapproachesfor WFG7 problem. Table V shows the performances ofthe two approaches up to 15-objective WFG6 and WFG7problems.Basedonthe results onthree to15-objectivenormalizedDTLZandWFGtest problems, it canbeconcludedthat (i)MOEA/D-PBI performs consistently better than MOEA/D-TCHapproach, (ii) MOEA/D-PBI performs best in someproblems, whereas the proposed NSGA-III approach performsbest in some other problems, and (iii) in a non-uniformlydistributedPareto-optimalfront(likeintheDTLZ4problem),both MOEA/D approaches failtomaintain agood distributionofpoints, whileNSGA-IIIperformswell.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.1200.5100.20.40.60.8100.20.40.60.811.2f2 f1f300.5100.5100.51f1f2f3Fig. 13. Obtained solutions by NSGA-III forDTLZ4.00.5100.20.40.60.8100.20.40.60.811.2f2f1f300.5100.5100.51f1f2f3Fig.14. ObtainedsolutionsbyMOEA/D-PBIforDTLZ4.00.5100.20.40.60.8100.20.40.60.811.2f2f1f300.5100.5100.51f1f2f3Fig. 15. Obtained solutionsby MOEA/D-TCH forDTLZ4.1 2 3 4 5 6 7 8 9 1000.20.40.60.81Objective ValueObjective No.Fig. 16. NSGA-III Solutions are shownusing10-objective value pathformat forDTLZ4.1 2 3 4 5 6 7 8 9 1000.20.40.60.81Objective ValueObjective No.Fig.17. MOEA/D-PBISolutionsareshown using10-objectivevaluepathformat forDTLZ4.00.5100.5100.20.40.60.811.2f2 f1f300.5100.5100.51f1f2f3Fig. 18. NSGA-III solutions areshown on three-objectiveWFG6 prob-lem.00.5100.5100.20.40.60.811.2f2f1f300.5100.5100.51f1f2f3Fig. 19. MOEA/D-PBI solutionsare shown on three-objective WFG6problem.00.5100.5100.20.40.60.811.2f2f1f300.5100.5100.51f1f2f3Fig. 20. NSGA-III solutions areshown on three-objectiveWFG7 prob-lem.00.5100.5100.20.40.60.811.2f2 f1f300.5100.5100.51f1f2f3Fig. 21. MOEA/D-PBI solutionsare shown on three-objective WFG7problem.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.13TABLEIVIGDVALUESOFMOEA/D-PBI APPROACHWITH c=30.Function M MaxGen NSGA-III MOEA/D-PBIDTLZ1 4.880 1044.047 1043 400 1.308 1031.797 1034.880 1035.699 1035.116 1042.849 1045 600 9.799 1045.717 1041.979 1031.647 1032.044 1034.053 1038 750 3.979 1036.839 1038.721 1031.127 1022.215 1034.057 10310 1000 3.462 1035.251 1036.869 1036.300 1032.649 1031.301 10215 1500 5.063 1031.653 1021.123 1022.695 102DTLZ2 1.262 1034.535 1043 250 1.357 1035.778 1042.114 1038.049 1044.254 1031.096 1035 350 4.982 1031.208 1035.862 1031.370 1031.371 1022.793 1038 500 1.571 1023.615 1031.811 1025.231 1031.350 1022.286 10310 750 1.528 1022.433 1031.697 1022.910 1031.360 1025.292 10315 1000 1.726 1025.952 1032.114 1028.413 103DTLZ3 9.751 1041.001 1033 1000 4.007 1035.080 1036.665 1031.154 1023.086 1037.191 1045 1000 5.960 1032.344 1031.196 1026.118 1031.244 1026.285 1038 1000 2.375 1022.032 1029.649 1021.1338.849 1032.848 10310 1500 1.188 1026.110 1032.083 1024.834 1011.401 1024.647 10315 2000 2.145 1021.110 1024.195 1021.271DTLZ4 2.915 1041.092 1013 600 5.970 1044.277 1014.286 1015.235 1019.849 1042.342 1015 1000 1.255 1034.384 1011.721 1037.347 1015.079 1033.821 1018 1250 7.054 1038.015 1016.051 1019.686 1015.694 1035.083 10110 2000 6.337 1038.321 1011.076 1011.0247.110 1039.406 10115 3000 3.431 1011.1571.073 1.219TABLEVIGD VALUESFOR NSGA-III ANDMOEA/D-PBI APPROACHESONTHREETO15-OBJECTIVE WFGPROBLEMS.Problem M MaxGen NSGA-III MOEA/D-PBIWFG6 4.828 1031.015 1023 400 1.224 1023.522 1025.486 1021.066 1015.065 1038.335 1035 750 1.965 1024.230 1024.474 1021.058 1011.009 1021.757 1028 1500 2.922 1025.551 1027.098 1021.156 1011.060 1029.924 10310 2000 2.491 1024.179 1026.129 1021.195 1011.368 1021.513 10215 3000 2.877 1026.782 1026.970 1021.637 101WFG7 2.789 1031.033 1023 400 3.692 1031.358 1024.787 1031.926 1028.249 1038.780 1035 750 9.111 1031.101 1021.050 1021.313 1022.452 1021.355 1028 1500 2.911 1021.573 1026.198 1022.626 1023.228 1021.041 10210 2000 4.292 1021.218 1029.071 1021.490 1023.457 1027.552 10315 3000 5.450 1021.063 1028.826 1022.065 102TABLEVIBEST,MEDIANANDWORST IGDANDCONVERGENCEMETRICVALUESOBTAINEDFOR NSGA-III ANDCLASSICAL GENERATIVEMETHODSFORTHREE-OBJECTIVE DTLZ1ANDDTLZ2PROBLEMS.Prob.FE NSGA-III GenerativeMethodIGD GD IGD GDDTLZ136,4004.8801044.8801046.4001021.7021011.3081036.5261048.0801021.8081014.8801037.4501041.0831011.848101DTLZ222,7501.2621031.2641031.1131039.6781051.3571031.2701036.5971031.0191042.1141031.2741039.5511031.082104Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.14B. Classical GenerativeMethodFor solvingmany-objectiveoptimizationproblems, were-quireaset of referencepointseither suppliedbytheuseror systematicallyconstructedas discussedearlier. One maythen ponder how the proposed NSGA-III method will comparewith a classical generative method in which multiple scalarizedsingle-objectiveoptimizationproblemscanbeformulatedforeach of thepreferred orstructured reference points and solvedindependently.Forthispurpose, weminimizethePBI-metricfor each case. Along the reference direction obtained byjoiningtheidealpoint(z)tothesuppliedreferencepoint(z)dictatedbyaw-vector(aunitvectorw = (z z)/|z z|),thedistance(d1)alongthew-directionandthedistance(d2)perpendiculartothew-directionarecomputedfor anypointx.Aweighted-sum ofthesetwodirectionsisthenminimized,asfollows:Minimizexd1+d2=wTf (x)+f(x)wTf (x)w. (7)Theparameteris set to5for all referencepoints[10]. Inotherwords, theaboveminimizationprocessislikelytondtheintersectionpoint betweenthereferencedirectionwandthe Pareto-optimal surface, if there exists an intersection. Sincetheideal point (z)isrequiredtobeknownforthismethod,weassumeorigintobetheideal vectorforthisproblem.To make a fair comparison, for Hreference points, we allo-cateamaximum of T= FENSGA-III/H(where FENSGA-IIIisthe total number of solutionevaluations neededbyNSGA-III) function evaluations to each optimization process. Forthree-objectiveDTLZ1andDTLZ2problems, FENSGA-III=92400 or 36,800 was required to nd 91 solutions. Therefore,we allocate 36,800/91 or 405 function evaluations to eachoptimizationrunbyfminconroutineof Matlab. Arandomsolutionisusedforinitialization.Figures22and23showthenal solutionsobtainedbythisgenerativemethod.TheDTLZ1 problem hasmultiplelocalfronts andMatlabsfmincon routine could not nd a Pareto-optimal solutioneverytimewiththeallottednumber of functionevaluations.Table VI shows the IGDandGDmetric (average distanceofobtainedpointsfromtheclosest referencepoints, ametricoppositeinsensetoIGDmetric) values. For three-objectiveDTLZ2problem, thegenerative methodisabletondawell-distributedsetofpoints,asshowninthegure.However,thetableindicates that thedistributionof points is not as goodas that obtainedbyNSGA-III withthe number of functionevaluations restricted in this study. The inherent parallel searchofNSGA-IIIconstitutesamoreefcientoptimization.C. ScaledTest ProblemsTo investigate the algorithms performance on problemshaving differently scaled objective values, we consider DTLZ1and DTLZ2 problems again, but nowwe modify themasfollows. Objective fiis multiplied by a factor 10i. To illustrate,objectives f1, f2and f3forthethree-objective scaledDTLZ1problemaremultipliedby100, 101and102, respectively.To handle different scaling of objectives and make thedistances (d1andd2) alongandperpendicular toreferencedirections in MOEA/D-PBI approach, we normalize the objec-tivevaluesusingtheproceduresuggestedforMOEA/D-TCHapproach intheoriginal MOEA/Dstudy [10].Wealsouse thecodefromMOEA/Dweb-site[50] toobtainthepoints. Fig-ures 24, 25, and 26 show the obtained distribution of points us-ing NSGA-III, MOEA/D-PBI and MOEA/D-TCH approaches,respectively. It is clear that bothMOEA/Dapproacheswithobjective normalization are not able to handle the scalinginvolved in the objectives adequately, whereas NSGA-IIIsoperators, normalization of objectives, and adaptive hyper-planeconstructionprocessareabletonegotiatethescalingofthe objectives quite well. In the scaled problems, IGD metric iscomputedbyrst normalizingtheobjectivevaluesusingtheideal andnadir points of theexact Pareto-optimal front andthencomputingtheIGDvalueusingthereferencepointsasbefore. Resulting IGD values areshown inTable VII. SimilarTABLEVIIBEST,MEDIANANDWORST IGDVALUESFORTHREE-OBJECTIVE SCALEDDTLZ1AND DTLZ2PROBLEMSUSING NSGA-IIIAND MOEA/DALGORITHMS. ASCALINGFACTOROF10i, i=1, 2, . . . , M, ISUSED.Prob.MaxGenNSGA-III MOEA/D MOEA/D-PBI -TCHDTLZ14003.8531042.4161028.0471021.2141032.0541012.0511011.1031024.1131012.704101DTLZ22501.3471034.7491022.3501012.0691033.8011015.3081015.2841035.3171015.321101performanceisalsoobservedforthescaledDTLZ2problem,asshowninFigures27, 28, and29andresultsaretabulatedinTableVII.It is interestingtonote that the MOEA/D-PBI algorithmthat workedsowell inthe normalizedtest problems intheprevious subsection did not perform well for the scaled versionofthesameproblems. Practical problemsarefar frombeingnormalizedandtheobjectives areusuallyscaleddifferently.An efcient optimization algorithmmust handle differentscalingofobjectivesaseffectivelyasshownbyNSGA-III.Next, Table VIII shows the IGDvalues of the obtainedsolutions for 5, 8, 10, and15-objectivescaledDTLZ1andDTLZ2 problems. Although difcult tovisualize, asmall IGDvalueineachcaserefers toawell-distributedset of points.Due to the poor performance of MOEA/D algorithms in scaledproblems,wedidnotapplythemfurthertotheabovehigher-dimensionalscaledproblems.D. ConvexDTLZ2ProblemDTLZ1 problemhas a linear Pareto-optimal front andDTLZ2 to DTLZ4 problems have concave Pareto-optimalfronts. ToinvestigatetheperformanceofNSGA-IIIandbothversions of MOEA/Don scalable problems having a con-vexPareto-optimal front, we suggest a newproblembasedon DTLZ2 problem. After the objective values (fi, i =1, 2, . . . , M)arecomputed using theoriginal DTLZ2 problemCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.1500.250.500.250.500.10.20.30.40.50.6f2f1f300.250.500.250.500.10.20.30.40.50.6f1f2f3Fig. 22. Solutions obtainedbyclassical generative methodfor DTLZ1.OnlythepointsobtainedclosetothetruePareto-optimal frontareshown.00.5100.20.40.60.8100.20.40.60.811.2f2f1f300.5100.5100.51f1f2f3Fig. 23. Solutionsobtainedbyclassical generativemethodforDTLZ2.00.250.502.550102030405060f2 f1f300.250.502.5560204060f1f2f3Fig. 24. Obtained solutions by NSGA-III forscaledDTLZ1.00.250.502.550102030405060f2f1f300.250.502.550102030405060f1f2f3Fig.25. ObtainedsolutionsbyMOEA/D-PBIforscaledDTLZ1.00.250.502.550102030405060f2 f1f300.250.502.550102030405060f1f2f3Fig. 26. Obtained solutionsby MOEA/D-TCH forscaledDTLZ1.00.51024681012020406080100120f2 f1f300.510510050100f1f2f3Fig. 27. Obtained solutions by NSGA-III forscaledDTLZ2.00.51024681012020406080100120f2f1f300.510510050100f1f2f3Fig.28. ObtainedsolutionsbyMOEA/D-PBIforscaledDTLZ2.00.20.40.60.810510 020406080100120f2f1f300.510510050100f1f2f3Fig. 29. Obtained solutionsby MOEA/D-TCH forscaledDTLZ2.description, wemapthemasfollows:fi f4i , i = 1, 2, . . . , M1,fM f2M.Thus, thePareto-optimalsurfaceisgivenasfM+M1

i=1_fi= 1. (8)Figure30shows the obtainedpoints ona three-objectiveconvexPareto-optimal front problemwithH=91referencepoints. As the gure shows, the Pareto-optimal surface isalmost at at the edges, but changes sharply in the intermediateregion. Althoughthe reference points are uniformlyplacedon a normalizedhyper-plane makingan equal angle to allobjective axes, thedistribution of points on thePareto-optimalfront is not uniform. Despite such a nature of the front, NSGA-III is able tonda widelydistributedset of points ontheedgesandalsointheintermediateregionofthesurface.Notethat therangeof eachobjectivefor thePareto-optimal set isidentical (within[0,1]), henceMOEA/D-PBI is expectedtoCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.1600.5100.5100.20.40.60.811.2f2 f1f300.5100.5100.20.40.60.811.2f1f2f3Fig. 30. ObtainedsolutionsbyNSGA-IIIfortheconvexPareto-optimal front problem.00.5100.5100.20.40.60.811.2f2 f1f300.5100.5100.20.40.60.811.2f1f2f3Fig.31. ObtainedsolutionsbyMOEA/D-PBIfortheconvexPareto-optimal front problem.00.5100.20.40.60.811.200.20.40.60.811.2f2f1f300.5100.5100.51f1f2f3Fig. 32. Obtained solutionsby MOEA/D-TCH fortheconvexPareto-optimal front problem.TABLEVIIIBEST,MEDIANANDWORST IGD VALUESFORSCALED M-OBJECTIVEDTLZ1ANDDTLZ2PROBLEMS.Problem M Scaling MaxGen NSGA-IIIfactorDTLZ1 3.853 1043 10i400 1.214 1031.103 1021.099 1035 10i600 2.500 1033.921 1024.659 1038 3i750 1.051 1021.167 1013.403 10310 2i1000 5.577 1033.617 1023.450 10315 1.2i1500 6.183 1031.367 102DTLZ2 1.347 1033 10i250 2.069 1035.284 1031.005 1025 10i350 2.564 1028.430 1021.582 1028 3i500 1.788 1022.089 1022.113 10210 3i750 3.334 1022.095 1012.165 10215 2i1000 2.531 1024.450 102performbetter. As showninFigure31, a nicelydistributedsetofpointsarefoundinthemiddlepartofthefront,butthealgorithmfailstondpointsontheboundaryof thePareto-optimalfront.Thishappensduetothevery smallslopeofthesurfaceat theboundaryregion. Theuseofpenaltyparametervalue= 5ndsanon-boundarypoint tohaveabetterPBImetricvaluefor theboundaryreferencedirections. Alargervalue of may allow MOEA/D-PBI to nd the exact boundarypoints, thereby requiring to do a parametric study with .However, theadvantage ofnoadditionalparameter inNSGA-III becomes clear from this problem. MOEA/D-TCH performspoorlyagain, asshowninFigure32.TableIXshowsthebest, medianandworst IGDvaluesof20independentruns(obtainedusingequation6)forall threemethods. Clearly, NSGA-III performs the best by nding asetofpointshavinganorderofmagnitudesmallerIGDvaluesinthree, ve, eight, 10, and15-objectiveversionsoftheconvexDTLZ2problem.TABLEIXBEST,MEDIANANDWORST IGDVALUESFORTHECONVEX DTLZ2PROBLEM.M MaxGen NSGA-III MOEA/D-PBI MOEA/D-TCH2.603 1033.050 1026.835 1023 250 4.404 1033.274 1026.871 1028.055 1033.477 1026.912 1027.950 1031.082 1011.211 1015 750 1.341 1021.103 1011.223 1011.917 1021.117 1011.235 1012.225 1022.079 1012.235 1018 2000 2.986 1022.086 1012.396 1014.234 1022.094 1012.490 1017.389 1022.117 1012.373 10110 4000 9.126 1022.119 1012.472 1011.051 1012.120 1012.514 1012.169 1023.681 1013.851 10115 4500 2.769 1023.682 1013.911 1014.985 1023.683 1013.972 101To demonstrate the distribution of obtained points on ahigher-objective problem, we showthe obtained points ofthemedianperformedrunofthe15-objectiveconvexDTLZ2problemonavaluepathusingNSGA-IIIandMOEA/D-PBIapproaches inFigures33and34,respectively. Itisclearfromthe plots that NSGA-III is able to nd a good spread ofsolutions in the entire range of Pareto-optimal front (fi [0, 1]forall i),butMOEA/D-PBIisunable tondsolutionshavinglargerobjectivevalues.BeforeweevaluatetheproposedNSGA-III approachfur-ther, we highlight a couple of properties of MOEA/D-TCH andMOEA/D-PBI approachesthat becomeclear formtheabovesimulationresults:1) As observedinthree-objectiveresults withMOEA/D-TCHabove (Figure 7, for example), it ends upgen-erating a non-uniformdistribution of points. This isduetothewaytheTchebysheff metricis constructed.Moreover,multipleweightvectorsproduceanidenticalCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.171 2 3 4 5 6 7 8 9 10 11 12 13 14 1500.20.40.60.81Objective ValueObjective No.Fig. 33. NSGA-III Solutions are shownusing15-objective value pathformat fortheconvexproblem.1 2 3 4 5 6 7 8 9 10 11 12 13 14 1500.20.40.60.81Objective ValueObjective No.Fig.34. MOEA/D-PBISolutionsareshown using10-objectivevaluepathformat fortheconvexproblem.extremesolution, therebywastingincomputational ef-forts.2) Interestingly,dependingontheparameter, MOEA/D-PBI is potentially capable of generating dominated solu-tions. Thiscertainlyresultsinawasteofcomputationaleffort.Althoughspecial carecanbetakentoreducethechanceofabove, theyareimportantforonetobeawarewhileworkingwithMOEA/D-PBIorMOEA/D-TCHapproaches.VI. FURTHERINVESTIGATIONSOFNSGA-IIIHavingperformedwell onall test problemsinabovesec-tions, wenowinvestigateNSGA-IIIsperformanceincertainspecial typesofmany-objectiveproblemsolvingtasks.A. FindingaPreferredPart ofPareto-frontIn many-objective optimization problems, the user maynot alwaysbeinterestedinndingtheentirePareto-optimalfront. Practically speaking, an user is often interested in apreferredpartofthePareto-optimalfront.Insuchascenario,theuser mayrepresent his/her preferredregionusingafewrepresentativereferencepoints(oraspirationpoints).Theaimin such a many-objective optimization task is to nd thePareto-optimal points that are in some sense closest to thesuppliedreferencepoints.Recall fromSection IV-B that the proposed NSGA-IIIalgorithmcanalsobeappliedwithaset ofHuser-suppliedreference points,insteadofDasandDennissstructuredrefer-encepointsusedsofar. Here,wedemonstratetheworkingofNSGA-IIIforscaledDTLZ1andDTLZ2problemsforafewuser-suppliedpreferenceinformation.Insuchacase, inordertodeterminetherightnormalizedhyper-plane, bydefault,wesupplyMadditional referencepoints, oneat eachobjectiveaxis at anintercept of unity. Thepresenceof theseextremepoints will attempt to keep extreme Pareto-optimal points,therebyformingaproper normalizedhyper-planeneededtohaveappropriatescalingoftheobjectives.For bothscaledDTLZ1andDTLZ2problems, wesupply10referencepoints inthe middleof the normalizedhyper-plane (scenario 1, as shown in Figure 35). As mentioned, threeadditionalreferencepoints (1, 0, 0)T,(0, 1, 0)Tand(0, 0, 1)Tarealsoincludedfor theexecutionof thealgorithm, therebymaking a total of 13 supplied reference points. Figure 37Fig. 35. Reference points usedforScenario1onnormalizedhyper-plane.Fig. 36. Reference points usedforScenario2onnormalizedhyper-plane.showsthepointsobtainedbyNSGA-IIIonthescaledPareto-optimal front of DTLZ1 problem. Figure 38 shows 10 obtainedpoints for the scaled DTLZ2 problem for scenario 1. Next, wesupplyaset of10referencepointsonadifferent part ofthePareto-optimal, as shown as scenario 2 in Figure 36. Figure 38showstheobtainedpointsonthescaledDTLZ2problem. Inboth cases,asetof10 near-Pareto-optimal points are obtainedclosetowherethereferencepointsweresupplied.Best, medianandworstIGDvaluesfortheobtainedpointsfor three and 10-objective scaled DTLZ1 and DTLZ2 problemsare shown in Table X. The IGDvalues are computed bynormalizingtheobtainedobjectivevaluesbytheoreticalidealand nadir points and by projecting 10 supplied referencepointsonthenormalizedPareto-optimal front. Asmall IGDvalueineachcaseensurestheconvergenceanddiversityofthe obtainedsolutions. The success of NSGA-III inndingjust10Pareto-optimal pointson10-objective scaledproblemsindicates its potential use inndinga handful of preferredsolutionsinamany-objectivepracticaloptimizationproblem.B. RandomlygeneratedpreferredreferencepointsThesuppliedpreferredreferencepoints consideredintheprevioussubsectionarestructured. Infact, theywerecreatedusing Das and Denniss [48] structured reference point creationprocedure. Here, we investigate NSGA-IIIs ability to dealwith a set of randomly created reference points. The procedureused here is the same as in previous subsection MadditionalaxispointsareincludedforNSGA-IIItoadaptivelycreateitshyper-plane.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.1800.250.502.550102030405060f2 f1f300.250.502.550102030405060f1f2f3Fig.37. Preferredsolutions(scenario1)obtainedbyNSGA-IIIforthree-objectiveDTLZ1.00.510510020406080100120f2f1f300.510510050100f1f2f3Fig.38. Preferredsolutions(scenario1)obtainedbyNSGA-IIIforthree-objectiveDTLZ2.00.510510020406080100120f2f1f300.510510050100f1f2f3Fig.39. Preferredsolutions(scenario2)obtainedbyNSGA-IIIforthree-objectiveDTLZ2.00.250.502.550102030405060f2f1f300.250.502.550204060f1f2f3Fig. 40. Preferred solutions obtained byNSGA-III for three-objectiveDTLZ1.00.510510020406080100120f2f1f300.510510050100f1f2f3Fig. 41. Preferred solutions obtained byNSGA-III for three-objectiveDTLZ2.For decision-makingpurposes, onlyafewtrade-offpointscanbe analyzed, evenfor a many-objectiveproblem. Here,weinvestigateif theproposedNSGA-IIIisabletondonlyve Pareto-optimal points in a preferred region for threeand10-objectivescaledDTLZ1andDTLZ2problems. Fiverandompointsaresuppliedintheintermediateregionof thenormalized hyper-plane (within zi [0.4, 0.6] for all i).Figures40and41showtheobtainedpointsonscaledthree-objective DTLZ1 and DTLZ2 problems, respectively. Table XIshows the IGDvalues for three and10-objectiveproblems.In each case, a small IGDvalue indicates that NSGA-IIIis adequatetosuccessfullyndthedesiredsolutions. Thisapplicationshows promise in applyingNSGA-III to many-objectiveproblems(10-objectiveorlike)withahandful(veor like) of preferredreferencepointsamatter whichisofgreatimportancetopractitioners.C. SmallpopulationsizePrevioussubsectionshaveshownthat NSGA-IIIcanworkwitha small number of reference points inorder tondafewPareto-optimal pointsinapreferredregion. Inprevioussections, we have used a population size that is almost equal tothe number of reference points. A natural question then arises:CanNSGA-IIIworkwellwithasmallpopulationsize?.Weinvestigatethisaspecthere.Table XII tabulates the number of reference points (H)andcorrespondinglayer-wisepandthemaximumnumberofgenerations consideredfor different many-objectiveDTLZ2problems. We performthis study for three, ve, and 10-objectivescaledDTLZ2problems.Identicalscalingfactorsasthose shown in Table VIII are used here. In each case, the pop-ulation size (N) is always kept as the smallest number multipleoffourandgreaterthanthenumberofreferencepoints(H).Alarger number of generationsis kept for higher objectiveproblems. Thechosenmaximumnumber of generations arealsotabulatedinthetable.For the three-objective DTLZ2 problem, when we areinterested innding 10 well-distributed Pareto-optimal points,wendthat apopulationof size 12is adequatetondall10Pareto-optimal points after 250generations (Figure 42).Forthe10-objective problem, ifwedesire65reference points(with p = 2), we nd that a population of size 68 is adequate tondall65solutionsafter1,000 generations. Thebest,medianandworst IGDvaluesareshowninthetablefor three, veand10-objectiveDTLZ2problems. These results showthatNSGA-III canworkwithasmall populationsizeduetoitsfocusandbalancedemphasisforndingnear Pareto-optimalpointscorrespondingtoasuppliedset ofreferencepoints.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.19TABLEXBEST,MEDIANANDWORST IGD VALUESFORSCALED DTLZ1ANDDTLZ2PROBLEMSWITH 10REFERENCEPOINTS.Function M N MaxGen NSGA-IIIDTLZ1 4.945 1043 28 750 4.076 1031.898 1021.671 103(Scenario1) 10 100 2000 5.792 103(Figure35) 1.212 101DTLZ1 2.742 1043 28 750 2.875 1031.741 1024.194 103(Scenario2) 10 100 2000 7.393 103(Figure36) 5.799 102DTLZ2 6.459 1043 28 250 1.473 1033.883 1032.840 103(Scenario1) 10 100 1500 4.907 103(Figure35) 1.074 102DTLZ2 1.327 1033 28 250 2.706 1038.099 1031.576 102(Scenario2) 10 100 1500 2.548 102(Figure36) 5.893 102TABLEXIBEST,MEDIANANDWORST IGD VALUESFORSCALED DTLZ1ANDDTLZ2PROBLEMSWITHRANDOMLYSUPPLIED REFERENCEPOINTS.Problem M N MaxGen NSGA-IIIDTLZ1 1.621 1033 28 600 9.870 1033.186 1013.244 10310 100 2000 8.227 1031.688 102DTLZ2 5.763 1043 28 250 1.951 1036.766 1031.317 10210 100 1500 2.463 1026.439 102D. NadirPointEstimationIn multi-objective optimizationproblems, the nadir pointis important tobefoundfor various reasons. Anadir pointis constructedfromtheworst objectivevaluesfor theentirePareto-optimal set. Thus, along with the ideal point, nadirpoint can be used to normalize the objective functions fora systematic and reliable execution of many classical andnon-classical optimizationmethods. Inanotherstudy, anadirpoint estimation strategy was proposed based on a bilevelevolutionary approach [54]. In that study, three to 20-objectiveDTLZ1andDTLZ2problemswereconsideredforestimatingthenadirpoint. Sincestudiesintheprevioustwosubsectionshaveshownamplythat NSGA-III canbeusedtondafewPareto-optimal solutionsevenin10-objectiveproblems, hereweapplyNSGA-III tondonlytheextremePareto-optimalpointssothatthenadirpointcanbeestimatedaccurately.Forthispurpose, wesuggestusing Mreference points, oneon each objective axis, located at unity. We apply NSGA-III toTABLEXIIMINIMUMPOPULATION SIZEANDBEST,MEDIANANDWORST IGDVALUESOFTHEOBTAINEDSOLUTIONSFORTHESCALED DTLZ2PROBLEM.M (pB, pI) H MaxGen N NSGA-III1.464 1033 (3,0) 10 250 12 4.206 1031.524 1014.952 1035 (2,1) 20 500 20 8.537 1031.195 1018.641 10310 (2,1) 65 1,000 68 1.155 1021.842 10200.51024681012020406080100120f2f1f300.510510050100f1f2f3Fig.42. ObtainedpointsusingNSGA-IIIforthescaledDTLZ2 witha smallpopulationsizeof12.three to 20-objective DTLZ1 and DTLZ2 problems and recordtheoverall functionevaluationsneededtondthetruenadirpoint with the following termination condition. When the errorvalue(E)computedasbelowE=_M

i=1_znadizestiznadizi_2, (9)is less than a threshold (0.01, used here), the run is terminated.Here, z, znadand zestare the ideal point, the true nadirpoint, and the estimated nadir point by NSGA-III, respectively.TableXIII presents theresults andcomparesthemwiththeevaluations neededtondthe nadir point withanidenticaltermination condition in the previous study [54]. In most cases,NSGA-III requires smaller number of function evaluationsthantheprevious procedure tolocatethenadirpointreliably.Figure 43 shows the value path plot of obtained pointsfor estimating the nadir point for the 10-objective DTLZ2problems. Understandably,thepointsneartheextremeofthePareto-optimal frontarefound.Fromthisplot,thenadirpointoftheproblemisestimatedtobe(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T.VII. TWOPROBLEMSFROMPRACTICEAfter solvinga number of test problems, we nowapplyNSGA-III to a couple of engineering design optimizationproblems. The rst problemhas three objectives and thesecondonehasnineobjectives.Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.20TABLEXIIINADIRPOINTESTIMATION BYNSGA-III INTHREETO20-OBJECTIVEDTLZ1ANDDTLZ2PROBLEMS.Problem M NSGA-III NSGA-II[54]DTLZ1 3 7, 400 18, 80016, 660 26, 50072, 180 45, 7005 25, 000 35, 30062, 900 58, 400152, 040 107, 10010 58, 640 239, 800259, 760 274, 200483, 880 358, 00015 356, 460 750, 000589, 140 1, 100, 0001, 200, 000 1, 800, 00020 274, 800 2, 600, 000956, 960 3, 500, 0001, 600, 000 4, 000, 000DTLZ2 3 1, 368 4, 1001, 920 4, 9002, 532 5, 5005 4, 320 9, 4006, 900 11, 40013, 000 14, 20010 31, 400 77, 60052, 840 92, 80076, 080 128, 00015 153, 540 346, 500213, 060 429, 000316, 860 810, 00020 309, 440 1, 089, 000589, 520 1, 415, 000734, 400 2, 102, 0001 2 3 4 5 6 7 8 9 1000.20.40.60.81Objective ValueObjective No.Fig. 43. Avaluepathplot forthe10-objectiveDTLZ2problemshowsthatNSGA-IIIisabletondtheextremeobjectivevalues(zeroandone)ineachobjective.A. Crash-worthinessDesignof VehiclesforCompleteTrade-offFrontThisproblemaimsat structural optimizationofthefrontalstructure of vehicle for crash-worthiness [55]. Thickness ofve reinforced members around the frontal structure arechosenas designvariables, while mass of vehicle, deceler-ationduringthe full frontal crash(whichis proportional tobiomechanical injuries causedtotheoccupants) and toeboardintrusioninthe offset-frontal crash(whichaccounts for thestructural integrity of the vehicle) are taken as objectives.Mathematical formulation for the three objectives can be foundintheoriginalstudy[55].All objectivesaretobeminimized. For this problem, wechoose p =16 so that there are H =_31+1616_or 153structured reference points. The reference points are initializedon the entire normalizedhyper-plane in the three-objectivespace. NSGA-III is applied with 156 population size andrunfor 200generations. Other parameters are the same asbefore. Figure 44 shows the obtained points with an opencircle. The non-uniformscalingof objectives is clear fromFig. 44. 60solutions arefoundontheentire front onthethree-objectivecrush-worthinessproblem.thegure. Although153referencepointswereused, only60ofthemhaverepresentativesolutionsonthenalfront.Other93referencedonot correspondtoanyPareto-optimal point,therebyindicatingpossibleholesinthePareto-optimalfrontcorrespond to remaining reference points. The ideal point (z)andthe nadir point (znad) are estimatedfromthe obtainedfront.Toinvestigatethenatureof thetruePareto-optimal front,next,wecreate7,381 referencepointszref,i(i = 1, . . . , 7381)withp=120. Usingtheideal andnadir pointsobtainedasabove, we normalize the objectives and then solve the achieve-mentscalarizationfunction(ASF)(equation4)correspondingtoeachweight vector wi=zref,i. EachASFis minimizedusingMatlabs fminconroutine. Resulting7,381points arecollectedanddominatedpointsareremoved.Wenoticedthatonly4,450points are non-dominated. These non-dominatedpoints are shown in Figure 44 with a dot. It is clear thattheNSGA-IIIobtainedsetof60pointsarewidely-distributedontheentirefront.Interestingly,4,450trade-offpointsrevealthecomplexnatureofPareto-optimalfront(havingholesandvaryingdensityof points) for thispractical problemandtheabilityofNSGA-IIItondafewwidely-distributed pointsontheentirefront.Toinvestigatethe closeness of NSGA-III points withtheclassicallyoptimizedsolutions, wecomputetheconvergencemetric(averagedistanceofNSGA-IIIpointsfromtheclosestfmincon-optimized points) and the minimum, median andmaximumconvergence metric values are found be 0.0019,0.0022, and 0.0026, respectively. Sincethese valuesare small,they indicate that obtained NSGA-III solutions are close to theclassically optimized front. This problem demonstrates the useof NSGA-III inndingarepresentativeset of pointsontheentirePareto-optimal frontonproblemshavingpracticaldif-culties(suchasdifferencesinscalingofobjectives, potentialholesinthePareto-optimalfront, etc.).Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.21B. CarCabDesignwithPreferenceInformationNext, we consider a vehicle performance optimization prob-lemhaving11decisionvariablesinvolvingdimensionsofthecar body and bounds on natural frequencies. The probleminvolvesnineobjectivesonroominessofthecar, fuelecon-omy, acceleration time, and road noise at different speeds.Mathematicalformulationcanbefoundelsewhere[56].In this problem, we demonstrate NSGA-IIIs ability touse preference informationin ndinga fewpreferrednon-dominated solutions on a part of the Pareto-optimal front,insteadof onthe completefront. We applyNSGA-III withonly 10 reference points in the intermediate part of the Pareto-optimal front. For this purpose, we rst create 10 randompointsontheentireunit hyper-planeandthenshrink themwithin25%aroundthecentroidof theunit hyper-plane. AssuggestedinSectionVI-A, nineextremereferencepointsarealso included for a better normalization purpose. Apopulationof size 100 is used and NSGA-III isrun for 4,000 generations.20independentrunsaremadewithdifferentsetsofreferencepoints andresultingsolutions arecomparedwiththeclassi-cal achievement scalarizing functionapproachimplementedwithMatlabs fminconroutine. Tohavegoodrepresentativesolutionsinthechosenintermediatepart, fminconroutineisrun for20,000 random reference points initializedinthesameintermediate range one at a time. Adomination check ofnal solutionsmake11,280of themnon-dominatedtoeachother. Eachset of 10solutions fromNSGA-III runsis thencomparedagainstthese11,280fminconsolutionsandtheGDmetric value is computed. The solutions fromthe specicNSGA-IIIruncorrespondingtothemedianGDmetricvalueis shown against the silhouette of 11,280 fmincon solutions onavaluepathplot inFigure45tomakeavisual comparison.Thetrade-off informationandspreadof obtainedNSGA-IIIsolutionsareclear fromtheplot. Whilethelargenumberoffminconsolutionsprovidearangeofobjectivevaluesforthechosenrangeofreferencepoints, onlyoneout of10NSGA-IIIsolutionshastwoobjectivevalues(f4andf9)outsidetherange. Other nine solutions quite nicely represent relevant partof thePareto-optimal front. WithNSGA-IIIs abilitytondjust 10preferredtrade-offsolutionsonanine-objectivenon-dominatedfrontshouldremainasasignicantsteptowardsaconvenientpost-optimalmulti-criteriondecision-makingtask.Fig. 45. Avalue pathplot for thenine-objective car cabdesignproblemshowsthat NSGA-IIIisabletondasfewas10preferredsolutionswithinthesilhouetteofsolutionsfoundbyclassical means.To quantify the convergence of NSGA-III solutions, thebest, median and worst GD metric values against 11,280 fmin-consolutionsonthenormalized objective spacearecomputedas follows: 0.0245, 0.0408, and 0.0566. These values aresmall enoughtoindicatetheclosenessofobtainedNSGA-IIIsolutions from the fmincon solutions. The ability of NSGA-IIItondahandful of trade-off pointsinapreferredpart of anine-objective practicaloptimizationproblemhavingdifferentscalingof objectivesshowsapromiseof itsuseinpracticalmany-objectiveoptimizationproblems.VIII. CONCLUSIONSIn this paper, we have suggested a reference point based ap-proach toan earlier-proposed NSGA-II framework for solvingmany-objectiveoptimizationproblems.TheproposedNSGA-IIIapproach hasbeen appliedtothreeto15-objective existingand new test problems and to three and nine-objective practicalproblems. Thetest problemsinvolvefrontsthat haveconvex,concave, disjointed, differently scaled, biased density of pointsacross the front, andmultimodalityinvolvingmultiplelocalfronts where an optimization algorithmcan get stuck to.Inall suchproblems, the proposedNSGA-III approachhasbeen able to successfully nd a well-converged and well-diversiedset of points repeatedlyover multiple runs. Theperformancescalingto15-objectivesisachievedmainlyduetotheaidindiversitypreservation bysupplying asetofwell-distributedreferencepoints. Inhigher-dimensionalproblems,EMOalgorithms face withan increasinglydifcult task ofmaintainingdiversityaswell asinconvergingtothePareto-optimal front. The supplyof a set of reference points andNSGA-IIIs efcient niching methodology in nding a Pareto-optimal solutionassociatedtoeachreferencepoint hasmadethediversitypreservationofobtainedsolutionsinaslargeas15-objectivespossible.The performance of NSGA-III has been compared withseveral versionsof arecentlyproposedMOEA/Dprocedure.Althoughdifferent MOEA/Ds haveshowntheir workingondifferent problems, no single version is able to solve allproblemsefciently. Havingsolvedall problemswell bytheproposedNSGA-III procedure, thereisanotheradvantageofit that is worthmentioninghere. UnlikeMOEA/Dversions,NSGA-III procedure does not require any additional parametertobeset.Furthermore. NSGA-III has beentested for its abilitytosolveanumberofdifferenttypesofmany-objectiveproblemsolving tasks. It has been demonstrated that NSGA-III isable to work with a small number of user-supplied struc-turedor randomlyassignedreferencepoints, therebymakingthe method suitable for a many-objective preference-basedoptimization-cum-decision-making approach.Ithasalsobeenshown that the NSGA-III procedure can be used to quickly es-timate the nadir point in many-objective optimization problemscompared to another EMO-based method proposed earlier. Fora practical application, only a fewhandful trade-off pointsare required for decision-making in multi- and many-objectiveoptimizationproblems. It hasbeenshownthat NSGA-IIIcanbe usedtondonlya fewpoints (20inve-objectiveandCopyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected] article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.2265ina10-objectiveproblem) withasmall populationsize,therebyreducingthecomputationalefforts.TheproposedNSGA-III approachwithasupplyof a setof reference points has a similar implicit principle to thehyper-gridbasedarchivingtechniques, suchasadaptivegridalgorithm[57], [58] and -MOEA[59], [60]. Grid basedmethods require exponentiallymore grids to be consideredwithanincreaseinobjectives,whereas -domination concept,althoughreducetheburdenof ndinganappropriatediver-sityinsolutionssomewhat, still poses enoughdifculties inndingwell-distributedPareto-optimal solutions for alarge-dimensional problem. However, it remains as aninterestingfuturestudytocompareNSGA-IIIsperformancewithan-dominationbasedalgorithmfor many-objectiveoptimizationproblems. NSGA-IIIs performance has been found to be muchbetter thana classical generatingmethodinmany-objectiveproblems.However,amorethoroughstudyusingpopulation-based aggregation methods [61], [62] in which multiple searchdirectionsareusedsimultaneouslyisworthpursuing.Results onmanydifferent problems withboxconstraintsusedinthis paper have clearlyshownpromise for NSGA-IIIs further application in other challenging probl

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