Twenty Yearsof Evolutionary Multi-Objecti veOptimization: A Historical View of the Field

CarlosA. CoelloCoelloCINVESTAV-IPN

EvolutionaryComputationGroupDepartamentodeIngenierıaElectrica

Seccion deComputacionAv. InstitutoPolitecnicoNacionalNo. 2508

Col. SanPedroZacatencoMexico,D. F. 07300

ccoello@cs.cinvestav.mx

November11,2005

Abstract

This articleprovidesa generaloverview of thefield now known as“evolution-ary multi-objective optimization”, which refersto the useof evolutionary algo-rithmsto solve problemswith two or more(oftenconflicting)objective functions.Usingasa framework thehistoryof this discipline,we discusssomeof themostrepresentative algorithmsthathave beendevelopedsofar, aswell assomeof theirapplications.Also, we discusssomeof the methodologicalissuesrelatedto theuseof multi-objective evolutionaryalgorithms,aswell assomeof thecurrentandfutureresearchtrendsin thearea.

1 Intr oduction

Optimizationusingmetaheuristicshasbecomeaverypopularresearchtopic in thelastfew years[14]. Optimizationrefersto finding thebestpossiblesolutionto a problemgivena setof limitations (or constraints).Whendealingwith a singleobjective to beoptimized(e.g.,thecostof adesign),weaimto find thebestpossiblesolutionavailable(called“global optimum”),or at leasta goodapproximationof it.

However, whendevising optimizationmodelsfor a problem,it is frequentlythecasethat thereis not one but several objectives that we would like to optimize. Infact, it is normally the casethat theseobjectivesare in conflict with eachother. Forexample,when designinga bridge, we would like to minimize its cost while max-imizing its safety. Theseproblemswith two or more objective functionsare called“multi-objective” andrequiredifferentmathematicalandalgorithmictools thanthose

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adoptedto solve single-objective optimizationproblems. In fact, even the notion of“optimality” changeswhendealingwith multi-objective optimizationproblems.Thegeneralnonlinearsingle-objective optimizationproblemis asmuchan openproblemasthe generalnonlinearmulti-objective optimizationproblem. Therefore,the useofmetaheuristicsis a valid choicewhich, in fact, hasrapidly gainedacceptanceamongresearchersfrom a wide variety of disciplines. From the several metaheuristicscur-rently available, evolutionary algorithms(which are basedon the emulationof themechanismof naturalselection)areamongthe mostpopular[35, 28]. Evolutionaryalgorithmshavebeenpopularin single-objectiveoptimizationand,morerecently, havealsobecomecommonin multi-objective optimization,in which they presentseveraladvantageswith respectto other techniques,aswe will seelater on. In this article,we will provide anoverview of thefield now called“evolutionarymulti-objectiveop-timization”, which refersto theuseof evolutionaryalgorithmsto solvemulti-objectiveoptimizationproblems. The overview will not be comprehensive nor will discussindetail the many approachescurrentlyavailable(moretechnicalsurveys with that sortof informationalreadyexist [5, 85, 88]). Instead,it will beahistoricaltour thatwill tryto illustratetherapiddevelopmentof this field.

2 BasicConcepts

In anattemptfor avoidingtheuseof cumbersomemathematicalterms,wewill provideonly informal definitionsof the main conceptsrequiredto understandthe restof thisarticle. It is assumed,however, thatthereaderis familiarwith thegeneralitiesof evolu-tionaryalgorithms(if morebackgroundonthis topic is required,thereareseveralgoodreferencesthatthereaderis invited to consult,suchas[35, 28, 25]).

First,weneedto defineamulti-objectiveoptimizationproblem(MOP).A MOPis aproblemwhich hastwo or moreobjectivesthatweneedto optimizesimultaneously. Itis importantto mentionthattheremight beconstraintsimposedon theobjectives.It isalsoimportantto emphasizethatit is normallythecasethattheobjectivesof theMOParein conflict with eachother. If this is not thecase,thena singlesolutionexists fortheMOP, andwhat we will discussin the remainderof this articleno longerapplies,becausethe objectivescanbe optimizedoneby one, in sequentialorder to find thissinglesolution.

MostMOPs,however, donotlendthemselvesto asinglesolutionandhave,instead,a setof solutions.Suchsolutionsarereally “trade-offs” or goodcompromisesamongtheobjectives.In orderto generatethesetrade-off solutions,anold notionof optimalityis normally adopted.This notion of optimality wasoriginally introducedby FrancisYsidro Edgeworth in 1881[23] andlatergeneralizedby Vilfredo Paretoin 1896[69].It is called Edgeworth-Pareto optimumor, simply, Pareto optimum. In words, thisdefinitionsaysthatasolutionto aMOPis Paretooptimalif thereexistsnootherfeasiblesolutionwhichwoulddecreasesomecriterionwithoutcausingasimultaneousincreasein at leastoneothercriterion. It shouldnot be difficult to realizethat the useof thisconceptalmostalwaysgivesnot asinglesolutionbut asetof them,which is calledtheParetooptimalset. Thevectorsof thedecisionvariablescorrespondingto thesolutionsincludedin theParetooptimal setarecallednondominated. Theplot of theobjective

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functionswhosenondominatedvectorsarein theParetooptimalsetis calledtheParetofront.

3 Why Evolutionary Algorithms?

TheOperationsResearchcommunityhasdevelopedapproachesto solve MOPssincethe1950s.Currently, awidevarietyof mathematicalprogrammingtechniquesto solveMOPsareavailablein thespecializedliterature(seefor example[59, 24]). However,mathematicalprogrammingtechniqueshave certainlimitationswhentacklingMOPs.For example,many of themaresusceptibleto the shapeof the Paretofront andmaynot work whentheParetofront is concaveor disconnected.Othersrequiredifferentia-bility of theobjective functionsandtheconstraints.Also, mostof themonly generatea singlesolutionfrom eachrun. Thus,severalruns(usingdifferentstartingpoints)arerequiredin orderto generateseveralelementsof the Paretooptimal set[59]. In con-trast,evolutionaryalgorithmsdealsimultaneouslywith a setof possiblesolutions(theso-calledpopulation)which allows us to find several membersof the Paretooptimalsetin a singlerunof thealgorithm.Additionally, evolutionaryalgorithmsarelesssus-ceptibleto the shapeor continuity of the Paretofront (e.g.,they caneasilydealwithdiscontinuousandconcaveParetofronts).

4 The Origins of the Field

Thefirst hint regardingthepossibilityof usingevolutionaryalgorithmsto solveaMOPappearsin a PhDthesisfrom 1967[74] in which, however, no actualmulti-objectiveevolutionaryalgorithm(MOEA) wasdeveloped(themulti-objective problemwasre-statedasa single-objective problemandsolved with a geneticalgorithm). Althoughthereis ararelymentionedattemptto useageneticalgorithmto solveamulti-objectiveoptimizationproblemfrom 1983 (see[43]), David Schaffer is normally consideredto be the first to have designeda MOEA during the mid-1980s[76, 77]. Schaffer’sapproach,calledVectorEvaluatedGeneticAlgorithm (VEGA) consistsof asimplege-netic algorithmwith a modifiedselectionmechanism.At eachgeneration,a numberof sub-populationsweregeneratedby performingproportionalselectionaccordingtoeachobjectivefunctionin turn. Thesesub-populationswould thenbeshuffledtogetherto obtaina new population,on which theGA would applythecrossoverandmutationoperatorsin the usualway. VEGA hada numberof problems,from which the mainonehadto dowith its inability to retainsolutionswith acceptableperformance,perhapsaboveaverage,but not outstandingfor any of theobjective functions.Thesesolutionswere perhapsgood candidatesfor becomingnondominatedsolutions,but could notsurviveundertheselectionschemeof this approach.

5 The First Generation: Emphasison Simplicity

After VEGA, researchersadoptedfor severalyearsothernaive approaches.Themostpopularwerethelinearaggregatingfunctions,whichconsistsin addingall theobjective

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functionsinto asinglevaluewhich is directly adoptedasthefitnessof anevolutionaryalgorithm[26]. Nonlinearaggregatingfunctionswerealsopopular[6]. Lexicographicorderingwasanotherinterestingchoice.In this case,a singleobjective (which is con-sideredthe most important)is chosenandoptimizedwithout consideringany of theothers.Then,the secondobjective is optimizedbut without decreasingthequality ofthesolutionobtainedfor thefirst objective. Thisprocessis repeatedfor all theremain-ing objectives[33].

Despiteall theseearlyefforts, thedirectincorporationof theconceptof Paretoop-timality into an evolutionaryalgorithmwasfirst hintedby David E. Goldberg in hisseminalbook on geneticalgorithms[35]. While criticizing Schaffer’s VEGA, Gold-berg suggestedthe useof nondominatedrankingandselectionto move a populationtowardstheParetofront in amulti-objectiveoptimizationproblem.Thebasicideais tofind thesetof solutionsin thepopulationthatareParetonondominatedby the restofthepopulation.Thesesolutionsarethenassignedthehighestrankandeliminatedfromfurthercontention.Anothersetof Paretonondominatedsolutionsis determinedfromtheremainingpopulationandareassignedthenext highestrank. This processcontin-uesuntil all thepopulationis suitablyranked.Goldberg alsosuggestedtheuseof somekind of niching techniqueto keepthe GA from converging to a single point on thefront. A nichingmechanismsuchasfitnesssharing[36] would allow theevolutionaryalgorithmto maintainindividualsall alongthe nondominatedfrontier. Goldberg didnot provide anactualimplementationof his procedure,but practicallyall theMOEAsdevelopedafterthepublicationof hisbookwereinfluencedby his ideas.Fromthesev-eralMOEAsdevelopedfrom 1989to 1998,themostrepresentativearethefollowing:

1. NondominatedSorting GeneticAlgorithm (NSGA): This algorithmwaspro-posedby SrinivasandDeb[80], andwasthefirst to bepublishedin aspecializedjournal (EvolutionaryComputation). The NSGA is basedon several layersofclassificationsof the individualsassuggestedby Goldberg [35]. Beforeselec-tion is performed,the populationis ranked on the basisof nondomination:allnondominatedindividualsareclassifiedinto onecategory(with adummyfitnessvalue,whichis proportionalto thepopulationsize,to provideanequalreproduc-tive potentialfor theseindividuals).To maintainthediversityof thepopulation,theseclassifiedindividualsare sharedwith their dummyfitnessvalues. Thenthis group of classifiedindividuals is ignoredand anotherlayer of nondomi-natedindividualsis considered.Theprocesscontinuesuntil all individualsin thepopulationareclassified.Sinceindividualsin thefirst front have themaximumfitnessvalue,they alwaysget morecopiesthanthe restof the population.Thealgorithmof the NSGA is not very efficient, becauseParetorankinghasto berepeatedoveranoveragain.Evidently, it is possibleto achievethesamegoalina moreefficientway.

2. Niched-ParetoGeneticAlgorithm (NPGA):Proposedin [40]. TheNPGAusesatournamentselectionschemebasedonParetodominance.Thebasicideaof thealgorithmis quite clever: Two individualsarerandomlychosenandcomparedagainstasubsetfrom theentirepopulation(typically, around10%of thepopula-tion). If oneof themis dominated(by theindividualsrandomlychosenfrom the

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Figure1: MasahiroTanaka.

population)andtheotheris not, thenthenondominatedindividualwins. All theothersituationsareconsidereda tie (i.e., bothcompetitorsareeitherdominatedor nondominated).Whenthereis a tie, the resultof the tournamentis decidedthroughfitnesssharing.

3. Multi-Objecti ve Genetic Algorithm (MOGA): Proposedin [29]. In MOGA,the rank of a certainindividual correspondsto the numberof chromosomesinthecurrentpopulationby which it is dominated.All nondominatedindividualsareassignedthehighestpossiblefitnessvalue(all of themget thesamefitness,suchthat they canbe sampledat the samerate),while dominatedonesarepe-nalizedaccordingto thepopulationdensityof thecorrespondingregionto whichthey belong(i.e., fitnesssharingis usedto verify how crowded is the regionsurroundingeachindividual).

During thefirst generation,few peopleperformedcomparativestudiesamongdif-ferent MOEAs. However, thosewho comparedthe threeprevious MOEAs unani-mouslyagreedon thesuperiorityof MOGA, followedby theNPGAandby theNSGA(in a distantthird place)[84]. This periodwascharacterizedby the simplicity of thealgorithmsproposedandby the lack of methodologyto validatethem. No standardtestfunctionswereavailableandcomparisonswerenormallydonevisually (noperfor-mancemeasureswereavailable).

Thereis, however, an importantresult during this period that is normally disre-garded.MasahiroTanaka(seeFigure1) [82] developedthefirst schemeto incorporateuser’s preferencesinto a MOEA. This is a very importanttopic, sincein real-worldapplicationsit is normallythecasethattheuserdoesnotneedtheentireParetooptimalset,but only a small portionof it (or perhapsonly a singlesolution). Thus,it is nor-mally desirablethat theusercandefinecertainpreferencesthatcannarrow thesearchandthat canmagnify certainportionsof the Paretofront. For many years,however,few researchersin this areapaidattentionto this issue(seefor example[12]).

Anotherimportanteventduringthefirst generation,wasthepublicationof thefirstsurvey of the field. FonsecaandFlemingpublishedsuchsurvey in the journal Evo-lutionary Computationin 1995[30]. CarlosM. Fonseca(seeFigure2) alsoproposedthefirst performancemeasurethatdid not requirethetrueParetofront of theproblembeforehand(see[31]), andwasalsothefirst to suggesta way of modifying thePareto

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Figure2: CarlosM. Fonseca.

Figure3: EckartZitzler.

dominancerelationshipin orderto handleconstraints[32]. Themainlessonlearntfromthefirst generationwasthata successfulMOEA hadto combinea goodmechanismtoselectnondominatedindividuals(perhaps,but not necessarily, basedon the conceptof Paretooptimality) combinedwith a goodmechanismto maintaindiversity (fitnesssharingwasa choice,but not the only one). The questionwas: canwe designmoreefficientalgorithmswhile keepingat leasttheeffectivenessachievedby first generationMOEAs?

6 The SecondGeneration: Emphasison Efficiency

Fromtheauthor’sperspective,asecondgenerationof MOEAsstartedwhenelitismbe-camea standardmechanism.Althoughthereweresomeearlystudiesthatconsideredthe notion of elitism in a MOEA (seefor example[42]), mostauthorscredit EckartZitzler (seeFigure3) with theformal introductionof this conceptin a MOEA, mainlybecausehis StrengthPareto EvolutionaryAlgorithm (SPEA)waspublishedin a spe-cialized journal (the IEEE Transactionson EvolutionaryComputation), [92] whichmadeit a landmarkin the field. Needlessto say, after the publicationof this paper,mostresearchersin thefield startedto incorporateexternalpopulationsin theirMOEAsandthe useof this mechanism(or an alternative form of elitism) becamea commonpractice. In fact, the useof elitism is a theoreticalrequirementin orderto guarantee

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convergenceof aMOEA andthereforeits importance[75].In thecontext of multi-objectiveoptimization,elitism usually(althoughnot neces-

sarily) refersto theuseof anexternalpopulation(alsocalledsecondarypopulation)toretainthe nondominatedindividualsfoundalongthe evolutionaryprocess.The mainmotivationfor this mechanismis thefact thata solutionthatis nondominatedwith re-spectto its currentpopulationis not necessarilynondominatedwith respectto all thepopulationsthat areproducedby an evolutionaryalgorithm. Thus,what we needisa way of guaranteeingthat the solutionsthat we will reportto the userarenondomi-natedwith respectto every othersolutionthatour algorithmhasproduced.Therefore,the most intuitive way of doing this is by storingin an externalmemory(or archive)all the nondominatedsolutionsfound. If a solution that wishesto enterthe archiveis dominatedby its contents,thenit is not allowed to enter. Conversely, if a solutiondominatesanyonestoredin thefile, thedominatedsolutionmustbedeleted.Theuseof thisexternalfile raisesseveralquestions:

� How doesthe externalfile interactwith the main population?In otherwords,do we selectindividualsfrom theunionof themainpopulationandtheexternalfile?,or doweselectonly from themainpopulation,ignoringthecontentsof theexternalfile?

� Whatdo we do whentheexternalfile is full (assumingthat the capacityof theexternal file is bounded)?Sincememorycapabilitiesarealways limited, thisissuedeservesspecialattention.

� Do we imposeadditionalcriteria for a nondominatedsolutionto be allowed toenterthe file ratherthanjust usingParetodominance(e.g.,usethe distributionof solutionsasanadditionalcriterion)?

Theseandsomeotherissuesrelatedto externalarchives(alsocalled“elite” archives)have beenstudiedboth from an empiricalandfrom a theoreticalperspective (seeforexample[49, 27]). Besidesthe useof an externalfile, elitism canalsobe introducedthroughtheuseof a( ����� )-selectionin whichparentscompetewith theirchildrenandthosewhich arenondominated(and possiblycomply with someadditionalcriterionsuchasproviding a betterdistribution of solutions)areselectedfor thefollowing gen-eration.Many MOEAshavebeenproposedduringthesecondgeneration(whichwearestill living today). However, mostresearcherswill agreethat few of theseapproacheshave beenadoptedasa referenceor have beenusedby others. The authorconsidersthatthemostrepresentativeMOEAsof thesecondgenerationarethefollowing:

1. Strength Pareto Evolutionary Algorithm (SPEA):This algorithmwasintro-ducedin [91, 92]. Thisapproachwasconceivedasawayof integratingdifferentMOEAs. SPEAusesan archive containingnondominatedsolutionspreviouslyfound (the so-calledexternal nondominatedset). At eachgeneration,nondom-inatedindividualsarecopiedto the externalnondominatedset. For eachindi-vidual in this externalset,a strengthvalueis computed.This strengthis similarto the rankingvalueof MOGA [29], sinceit is proportionalto the numberofsolutionsto which a certainindividual dominates.In SPEA,thefitnessof each

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Figure4: JoshuaD. Knowles.

memberof thecurrentpopulationis computedaccordingto thestrengthsof allexternalnondominatedsolutionsthatdominateit. Thefitnessassignmentprocessof SPEAconsidersbothclosenessto thetrueParetofront andevendistributionof solutionsat the sametime. Thus,insteadof usingnichesbasedon distance,Paretodominanceis usedto ensurethat the solutionsareproperlydistributedalongtheParetofront. Althoughthis approachdoesnot requirea nicheradius,its effectivenessreliesonthesizeof theexternalnondominatedset.In fact,sincetheexternalnondominatedsetparticipatesin theselectionprocessof SPEA,if itssizegrows too large, it might reducetheselectionpressure,thusslowing downthesearch.Becauseof this, theauthorsdecidedto adopta techniquethatprunesthe contentsof the externalnondominatedsetso that its size remainsbelow acertainthreshold.

2. Strength Pareto Evolutionary Algorithm 2 (SPEA2):SPEA2hasthreemaindifferenceswith respectto its predecessor[89]: (1) it incorporatesafine-grainedfitnessassignmentstrategy whichtakesinto accountfor eachindividualthenum-berof individualsthatdominateit andthenumberof individualsby which it isdominated;(2) it usesa nearestneighbordensityestimationtechniquewhichguidesthesearchmoreefficiently, and(3) it hasanenhancedarchive truncationmethodthatguaranteesthepreservationof boundarysolutions.

3. ParetoAr chivedEvolution Strategy (PAES):Thisalgorithmwasintroducedin[52]. PAESconsistsof a(1+1)evolutionstrategy (i.e.,asingleparentthatgener-atesa singleoffspring) in combinationwith a historicalarchive that recordsthenondominatedsolutionspreviously found. This archive is usedasa referencesetagainstwhich eachmutatedindividual is beingcompared.Sucha historicalarchive is the elitist mechanismadoptedin PAES. However, an interestingas-pectof this algorithmis theprocedureusedto maintaindiversitywhich consistsof acrowdingprocedurethatdividesobjectivespacein arecursivemanner. Eachsolutionis placedin a certaingrid locationbasedon thevaluesof its objectives(which areusedasits “coordinates”or “geographicallocation”). A mapof suchgrid is maintained,indicatingthenumberof solutionsthatresidein eachgrid lo-cation.Sincetheprocedureis adaptive,noextraparametersarerequired(exceptfor thenumberof divisionsof theobjective space).This adaptive grid (or vari-

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ationsof it) hasbeenadoptedby severalmodernMOEAs (e.g.,[10]), andis thecontribution by which JoshuaD. Knowles (seeFigure4) is morewell-known,althoughhe hasmadeseveral othersignificantcontributions to the field (e.g.,[49, 50, 51]).

4. NondominatedSorting GeneticAlgorithm II (NSGA-II): This approachwasintroducedin [19, 20] asanimprovedversionof theNSGA[80].1 In theNSGA-II, for eachsolutiononehasto determinehow many solutionsdominateit andthe setof solutionsto which it dominates.The NSGA-II estimatesthe densityof solutionssurroundinga particularsolution in the populationby computingtheaveragedistanceof two pointson eithersideof this point alongeachof theobjectivesof theproblem.Thisvalueis theso-calledcrowdingdistance. Duringselection,the NSGA-II usesa crowded-comparisonoperatorwhich takes intoconsiderationboththenondominationrankof anindividualin thepopulationandits crowdingdistance(i.e.,nondominatedsolutionsarepreferredoverdominatedsolutions,but betweentwo solutionswith thesamenondominationrank,theonethatresidesin thelesscrowdedregion is preferred).TheNSGA-II doesnot useanexternalmemoryastheotherMOEAspreviouslydiscussed.Instead,theelitistmechanismof theNSGA-II consistsof combiningthebestparentswith thebestoffspringobtained(i.e., a ( ����� )-selection).Dueto its clever mechanisms,theNSGA-II is muchmoreefficient(computationallyspeaking)thanits predecessor,andits performanceis so good,that it hasbecomevery popularin the last fewyears,becominga landmarkagainstwhich other multi-objective evolutionaryalgorithmshave to becompared.

Many otheralgorithmshave beenproposedduring the secondgeneration(seeforexample[10, 16, 15]). Also, fitnesssharingis no longertheonly alternative to main-taindiversity, sinceseveralotherapproacheshavebeenproposed.Besidestheadaptivegrid from PAES, researchershave adoptedclusteringtechniques[61], crowding [20],entropy [48], andgeometrically-basedapproaches[86], amongothermechanisms.Ad-ditionally, someresearchershave alsoadoptedmatingrestrictionschemes[62]. Morerecently, the useof relaxed forms of Paretodominancehasbeenadoptedasa mech-anismto encouragemoreexplorationand,therefore,to provide morediversity. Fromthesemechanisms,� -dominancehasbecomeincreasinglypopular, not only becauseof its effectiveness,but alsobecauseof its soundtheoreticalfoundation[56]. Also,new surveys on evolutionarymulti-objective optimizationwerepublishedduring thisperiod,aswell asseveralmonographicbooks[18, 12, 13, 81, 67].

During thesecondgeneration,many otheraspectswereemphasized.Themainonehasbeen,with no doubt,efficiency. Researchersraisedconcernsaboutefficiency bothat analgorithmiclevel andat thedatastructureslevel [45, 51]. Theincreasingnumberof publicationsduringthesecondgeneration(from theendof 1998to date)makesuswonderwhatwill thethird generationhaveto offer.

1Notehowever that thedifferencesbetweentheNSGA-II andtheNSGA aresosignificantthat they areconsideredastwo completelydifferentalgorithmsby severalresearchers.

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Figure5: Kalyanmoy Deb.

7 Methodological Issues

During the secondgeneration,several researchersproposeda varietyof performancemeasuresto allow a quantitative (ratherthanonly qualitative) comparisonof results[87, 31, 92]. Zitzler et al. [87] statedthat,whenassessingperformanceof a MOEA,onewasinterestedin measuringthreethings:

1. Maximizethenumberof elementsof theParetooptimalsetfound.

2. Minimize thedistanceof theParetofront producedby ouralgorithmwith respectto theglobalParetofront (assumingwe know its location).

3. Maximize the spreadof solutionsfound, so that we canhave a distribution ofvectorsassmoothanduniform aspossible.

This, however, raisedsomeissues.First, it wasrequiredto know beforehandtheexactlocationof thetrueParetofront of aproblemin orderto usea performancemea-sure. This may not be possiblein real-world problemsin which the locationof thetrue Paretofront is unknown. The secondissuewas that it is unlikely that a singleperformancemeasurecanassessthe threethings indicatedby Zitzler et al. [87]. Inotherwords,assessingtheperformanceof a MOEA is, also,aMOP! Onceresearchersstartedto proposeperformancemeasures,criticismsarose.Someresearchersrealized(empirically)thatmany of thenew performancemeasureswerebiased.In otherwords,therewerecasesin which they provided resultsthat did not correspondto what wecouldseefrom thegraphicalrepresentationof theresults(seefor example[84]). Iron-ically, many researcherswentbackto thegraphicalcomparisonswhensuspectedthatsomethingwaswrong with the numericalresultsproducedfrom applyingthe perfor-mancemeasuresavailable. Althoughslowly, researchersstartedto proposeda differ-enttypeof performancemeasuresthatconsiderednot onealgorithmat a time,but two[31, 92]. Theseperformancemeasureswerecalled“binary” (in contrastto thosethatassessperformanceof a singlealgorithmat a time, which werecalled “unary”). In2002,thetruthwasfinally in theopen:Unaryperformancemeasuresarenotcompliantwith Paretodominanceand,therefore,arenot reliable[90, 93]. Not everythingis lost,however, sincebinaryperformancemeasurescanovercomethis limitation [88, 93].

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Concurrentlywith the researchon performancemeasures,otherresearchersweredesigningtestfunctions. The mostremarkablework in this regardis dueto Kalyan-moy Deb (seeFigure5) who proposedin 1999a methodologyto designMOPsthatwaswidely usedduring several years[17]. Later on, an alternative setof test func-tionswasproposed,but this time, dueto their characteristics,no enumerative processwasrequiredto generatetheir true Paretofront [21, 22]. Consideringthat thesetestfunctionsarealsoscalable,their usehasbecomewidespread.So,today, researchersinthefield normallyvalidatetheirMOEAswith problemshaving threeor moreobjectivefunctions,and10 or moredecisionvariables.

8 Applications

MOEAshavebecomeincreasinglypopularin awidevarietyof applicationdomains,asreflectsarecentbookentirelydevotedto this topic [9]. In orderto providearoughideaof thesortof applicationsthatarebeingtackledin thecurrentliterature,wewill classifythe applicationsin threelarge groups: engineering,industrial and scientific. Somespecificareaswithin eachof thesegroupsare indicatednext. We will startwith theengineeringapplications,which are,by far, themostpopularin theliterature,perhapsdueto the fact thatengineeringproblemshave well-studiedmathematicalmodels.Arepresentativesampleof engineeringapplicationsis thefollowing:

� Electricalengineering[73]

� Hydraulicengineering[63]

� Structuralengineering[37]

� Aeronauticalengineering[4]

� RoboticsandControl[70, 60]

A representativesampleof industrialapplicationsis thefollowing:

� Designandmanufacture[47]

� Scheduling[39]

� Management[66]

Finally, wehaveavarietyof scientificapplications:

� Chemistry[57]

� Physics[58]

� Medicine[55]

� Computerscience[34]

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Thestronginterestfor usingMOEAsin somany differentdisciplinesreinforcestheideastatedat thebeginningof thisarticleregardingthemulti-objectivenatureof manyreal-worldproblems.However, someapplicationdomainshavereceivedrelatively littleattentionfrom researchersarerepresentareasof opportunity. For example: cellularautomata[64], patternrecognition[65], datamining [68], bioinformatics[41], andfinancialapplications[78].

9 Curr ent Research Trends

Fromtheauthor’sperspective,researchershaven’t producedanotherbreakthroughthatis so significant(aselitism) asto redirectmostof the researchinto a new direction.Thus,thethird generationis yet to appear. However, thereareseveralinterestingideasthathavecertainlyinfluencedalot of thework beingdonethesedaysandwhichdeservecloserattention.Someexamplesarethefollowing:

� Theuseof relaxedformsof Paretodominancehasbecomepopularasa mecha-nismto regulateconvergenceof aMOEA. Fromthesemechanisms,� -dominanceis, with no doubt,themostpopular[56], but it is not theonly mechanismof thistype (seefor example[54]). � -dominanceallows to control the granularityoftheapproximationof theParetofront obtained.As a consequence,it is possibleto accelerateconvergenceusingthis mechanism(if we aresatisfiedwith a verycoarseapproximationof theParetofront).

� Thetransformationof single-objectiveproblemsinto amulti-objectiveform thatsomehow facilitatestheir solution. For example,someresearchershave pro-posedthehandlingof theconstraintsof a problemasobjectives[8], andothershave proposedtheso-called“multi-objectivization” by which a single-objectiveoptimizationproblemis decomposedinto severalsubcomponentsconsideringamulti-objective approach[44, 53]. This procedurehasbeenfoundto behelpfulin removing local optimafrom a problemandhasattracteda lot of attentioninthelastfew years.

� The useof alternative bio-inspiredheuristicsfor multi-objective optimization.The most remarkableexamplesare particle swarm optimization[46] and dif-ferentialevolution [71], whoseusehasbecomeincreasinglypopularin multi-objective optimization(seefor example[1, 11]). However, otherbio-inspiredalgorithmssuchasartificial immunesystemsandantcolony optimizationhavealsobeenusedto solvemulti-objectiveoptimizationproblems[7, 38].

10 Futur e Research Trends

Thereareseveral topicsinvolving challengesthatwill keepbusyto theresearchersinthisareafor thenext few years.Someof themarethefollowing:

� Parametercontrol is certainly a topic that hasbeenonly scarcelyexplored inMOEAs. Is it possibleto designa MOEA that self-adaptsits parameterssuch

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that the userdoesn’t have to fine-tunethemby hand? Someresearchershaveproposeda few self-adaptationandon-line adaptationproceduresfor MOEAs(seefor example[83, 3]), but recently, not muchwork seemsto begoingin thisdirection.

� What is the minimum numberof fitnessfunction evaluationsthat areactuallyrequiredto achievea minimumdesirableperformancewith a MOEA?Recently,someresearchershave proposedthe useof black-boxoptimizationtechniquesnormallyadoptedin engineeringto performanincrediblylow numberof fitnessfunctionevaluationswhile still producingreasonablygoodsolutions(seefor ex-ample[51]). However, this sortof approachis inherentlylimited to problemsoflow dimensionality. So,thequestionis: arethereany otherwaysof reducingthenumberof evaluationswithout sacrificingdimensionality?

� Thedevelopmentof implementationsof MOEAsthatareindependentof theplat-form andprogramminglanguagein which they weredevelopedis an importantsteptowardsa commonplatformthatcanbeusedto validatenew algorithms.Inthis direction,PISA (A PlatformandprogramminglanguageindependentInter-facefor SearchAlgorithms)[2] constitutesanimportantstep,andmorework isexpectedin this direction.

� How to dealwith problemsthat have “many” objectives? Somerecentstudieshave shown that traditionalParetorankingschemesdo not behave well in thepresenceof many objectives(where“many” is normallya numberabove3 or 4)[72].

� Thereareplentyof fundamentalquestionsthatremainunanswered.Forexample:whatarethesourcesof difficulty of amulti-objectiveoptimizationproblemfor aMOEA?Whatarethedimensionalitylimitationsof currentMOEAs?Canweusealternativemechanismsinto anevolutionaryalgorithmto generatenondominatedsolutionswithout relying on Paretoranking(e.g.,adoptingconceptsfrom gametheory[79])?

11 Conclusions

Usingasabasisahistoricalframework,wehaveattemptedtoprovideageneraloverviewof thework thathasbeendonein thelasttwentyyearsin evolutionarymulti-objectiveoptimization.Many detailswereleft outdueto obviousspacelimitations,but themostsignificantachievementsto datewereat leastmentioned.Our discussionincludedal-gorithms,methodsto maintaindiversity, methodologicalissuesandapplications.Someof themostrepresentative currentresearchtrendswerealsodiscussed,andin the lastpartof thearticle,theauthorprovidedhisown insightsregardingthefutureof thefield.We arestill awaiting for the third generationto arrive. As morepapersgetpublishedin this area2, it getsharderevery day to producenew contributionsthataretruly sig-

2TheauthormaintainstheEMOO repository, which currentlyholdsover 2100bibliographicreferences.TheEMOOrepositoryis locatedat: http://delta.cs.cinvestav.mx/˜EMOO

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nificant. So,we wonderwhatsortof changewill make it possibleto shift theresearchtrendsin anentirelynew direction.

Acknowledgements

Theauthoracknowledgessupportfrom CONACyT throughproject42435-Y.

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