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Salvatore Greco

Matthias Ehrgott

José Rui Figueira Editors

Multiple Criteria Decision Analysis

State of the Art Surveys

Second Edition

Volume233

SeriesEditor

CamilleC.Price

StephenF.AustinStateUniversity,TX,USA

AssociateSeriesEditor JoeZhu

WorcesterPolytechnicInstitute,MA,USA

FoundingSeriesEditor

FrederickS.Hillier

StanfordUniversity,CA,USA

Moreinformationaboutthisseriesat http://www.springer.com/series/6161

SalvatoreGreco•MatthiasEhrgott JoséRuiFigueira

Editors MultipleCriteriaDecision Analysis StateoftheArtSurveys

Volume1and2

SecondEdition

Editors

SalvatoreGreco

DepartmentofEconomicsandBusiness

UniversityofCatania Catania,Italy

PortsmouthBusinessSchool

CentreofOperationsResearch andLogistics(CORL)

UniversityofPortsmouth Portsmouth,UK

JoséRuiFigueira

CEG-IST,InstitutoSuperiorTécnico UniversidadedeLisboa Lisboa,Portugal

MatthiasEhrgott

DepartmentofManagementScience

LancasterUniversity Lancaster,UK

ISSN0884-8289ISSN2214-7934(electronic)

InternationalSeriesinOperationsResearch&ManagementScience ISBN978-1-4939-3093-7ISBN978-1-4939-3094-4(eBook) DOI10.1007/978-1-4939-3094-4

LibraryofCongressControlNumber:2015957403

SpringerNewYorkHeidelbergDordrechtLondon

©SpringerScience+BusinessMedia,LLC2005

©SpringerScience+BusinessMediaNewYork2016

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VolumeI

PartITheHistoryandCurrentStateofMCDA

1AnEarlyHistoryofMultipleCriteriaDecisionMaking .............3

MuratKöksalan,JyrkiWallenius,andStanleyZionts

2ParadigmsandChallenges ..............................................19 BernardRoy

PartIIFoundationsofMCDA

3PreferenceModelling .....................................................43

StefanoMoretti,MeltemÖztürk,andAlexisTsoukiàs

4ConjointMeasurementToolsforMCDM ..............................97 DenisBouyssouandMarcPirlot

PartIIIOutrankingMethods

5ELECTREMethods ......................................................155

JoséRuiFigueira,VincentMousseau,andBernardRoy

6PROMETHEEMethods .................................................187

Jean-PierreBransandYvesDeSmet

7OtherOutrankingApproaches .........................................221

Jean-M.MartelandBenedettoMatarazzo

PartIVMultiattributeUtilityandValueTheories

8MultiattributeUtilityTheory(MAUT) .................................285

JamesS.Dyer

9UTAMethods .............................................................315 YannisSiskos,EvangelosGrigoroudis, andNikolaosF.Matsatsinis

10TheAnalyticHierarchyandAnalyticNetworkProcesses fortheMeasurementofIntangibleCriteriaandfor Decision-Making ..........................................................363 ThomasL.Saaty

11OntheMathematicalFoundationsofMACBETH ...................421 CarlosA.BanaeCosta,Jean-MarieDeCorte, andJean-ClaudeVansnick

PartVNon-classicalMCDAApproaches

12DealingwithUncertaintiesinMCDA ..................................467 TheodorJ.StewartandIanDurbach

13DecisionRuleApproach .................................................497 SalvatoreGreco,Benedetto Matarazzo,andRomanSłowi ´ nski

14FuzzyMeasuresandIntegralsinMCDA ..............................553 MichelGrabischandChristopheLabreuche

15VerbalDecisionAnalysis .................................................605 HelenMoshkovich,AlexanderMechitov,andDavidOlson

16AReviewofFuzzySetsinDecisionSciences: Achievements,LimitationsandPerspectives ..........................637 DidierDuboisandPatricePerny

VolumeII

PartVIMultiobjectiveOptimization

17VectorandSetOptimization ............................................695 GabrieleEichfelderandJohannesJahn

18ContinuousMultiobjectiveProgramming .............................739 MargaretM.Wiecek,MatthiasEhrgott,andAlexanderEngau

19ExactMethodsforMulti-ObjectiveCombinatorial Optimisation ..............................................................817 MatthiasEhrgott,XavierGandibleux,andAnthonyPrzybylski

20FuzzyMulti-CriteriaOptimization:Possibilistic andFuzzy/StochasticApproaches ......................................851 MasahiroInuiguchi,KosukeKato,andHidekiKatagiri

21AReviewofGoalProgramming ........................................903 DylanJonesandMehrdadTamiz

22InteractiveNonlinearMultiobjectiveOptimizationMethods .......927 KaisaMiettinen,JussiHakanen,andDmitryPodkopaev

23MCDAandMultiobjectiveEvolutionaryAlgorithms ................977 JuergenBranke

PartVIIApplications

24MulticriteriaDecision Aid/AnalysisinFinance 1011 JaapSpronk,RalphE.Steuer,andConstantinZopounidis

25Multi-ObjectiveOptimizationandMulti-CriteriaAnalysis ModelsandMethodsforProblemsintheEnergySector 1067 CarlosHenggelerAntunesandCarlaOliveiraHenriques

26MulticriteriaAnalysisinTelecommunicationNetwork PlanningandDesign:ASurvey 1167 JoãoClímaco,JoséCraveirinha,andRitaGirão-Silva

27MultipleCriteriaDecisionAnalysisandSustainable Development 1235 GiuseppeMunda

28MulticriteriaPortfolioDecisionAnalysisforProjectSelection .....1269 AlecMorton,JeffreyM.Keisler,andAhtiSalo

PartVIIIMCDMSoftware

29MultipleCriteriaDecisionAnalysisSoftware

H.RolandWeistrofferandYanLi

ListofFigures

Fig.3.1Graphicalrepresentationof R

Fig.3.2Matrixrepresentationof R

Fig.3.3Graphicalrepresentationofthesemiorder

Fig.4.1Comparingthelengthoftworods ................................101

Fig.4.2Comparingthelengthofcompositerods

Fig.4.3Usingstandardsequences .........................................105

Fig.4.4Buildingastandardsequenceon X2

Fig.4.5Buildingastandardsequenceon X

Fig.4.6Thegrid ............................................................119

Fig.4.7Theentiregrid .....................................................120

Fig.4.8TheThomsencondition ...........................................121

Fig.4.9RestrictedSolvabilityon X1

Fig.4.10Valuefunctionwhen Xi isdiscrete

Fig.4.11Valuefunctionwhen Xi iscontinuous

Fig.5.1InferringparametervaluesforELECTRETRI

Fig.6.1Preferencefunction ................................................194

Fig.6.2Valuedoutrankinggraph

Fig.6.3ThePROMETHEEoutrankingflows.(a)The C .a/ outrankingflow.(b)The .a/ outrankingflow ................198

Fig.6.4Proﬁleofanalternative ...........................................200

Fig.6.5ProjectionontheGAIAplane ....................................203

Fig.6.6AlternativesandcriteriaintheGAIAplane

Fig.6.7PROMETHEEIIranking.PROMETHEEdecision axisandstick ......................................................205

Fig.6.8PilotingthePROMETHEEdecisionstick ........................206

Fig.6.9“HumanBrain” ....................................................207

Fig.6.10Twotypesofdecisionproblems.(a)Softproblem (S1).(b)Hardproblem(S2) ......................................208

Fig.6.11ConﬂictbetweenDM’s

Fig.6.12OverviewPROMETHEEGDSSprocedure

Fig.6.13MainfunctionalitiesofD-Sight

Fig.6.14D-Sight:geo-localizationofthealternatives, PROMETHEEIdiamond,comparisonsofproﬁles

Fig.7.1Setoffeasibleweights

Fig.7.2ORESTEﬂowchart ...............................................233

Fig.7.3Outrankinggraph ..................................................237

Fig.7.4Geometricalinterpretationofbasicpreferencesindices

Fig.7.5Indifferenceareas:rectangular

Fig.7.6Indifferenceareas:rhomboidal

Fig.7.7Indifferenceareas:elliptical

Fig.7.8Aggregatedsemiorderstructure

Fig.7.9Aggregatedpseudo-orderstructure

Fig.7.10Partialproﬁleofaction ah

Fig.7.11Partialproﬁlesandpartialbrokenlinesof ar , as , at

Fig.7.12Partialfrequenciesof ar , as , at

Fig.7.13Someexamplesofcompensatoryfunctions

Fig.7.14Determinationofarelationbetweenthetwo alternatives a,b 2 A onthebasisofthevaluesofglobalindices

Fig.7.15Partialpreorder ....................................................279

Fig.8.1Choicebetweentwolotteries .....................................298

Fig.8.2Additiveindependencecriterionforrisk

Fig.8.3Piecewiselinearapproximationof v1 . /

Fig.8.4Piecewiselinearapproximationof

Fig.9.1Theaggregationanddisaggregationparadigmsin MCDA[57] ........................................................318

Fig.9.2Thedisaggregation-aggregationapproach[127]. (a)Thevaluesystemapproach;(b)theoutranking relationapproach;(c)thedisaggregation-aggregation approach;(d)themultiobjectiveoptimizationapproach ........319

Fig.9.3Thenormalizedmarginalvaluefunction .........................321

Fig.9.4Post-optimalityanalysis[56] .....................................323

Fig.9.5Ordinalregressioncurve(rankingversusglobalvalue) .........325

Fig.9.6Robustnessanalysisinpreferencedisaggregation approaches[125] ..................................................328

Fig.9.7Normalizedmarginalvaluefunctions ............................332

Fig.9.8Anon-monotonicpartialutilityfunction[22] ...................335

Fig.9.9Distributionalevaluationandmarginalvaluefunction ..........338

Fig.9.10Distributionoftheactions A1 and A2 on u.g/ [56] ...............340

Fig.9.11Simpliﬁeddecisionsupportprocessbasedon disaggregationapproach[57] .....................................348

Fig.9.12MethodologicalﬂowchartofMARKEX[89] ....................351

Fig.10.1Comparisonsaccordingtovolume ...............................376

Fig.10.2Tochoosethebesthospiceplan,oneconstructsa hierarchymodelingthebeneﬁtstothepatient,to theinstitution,andtosociety.Thisisthebeneﬁts hierarchyoftwoseparatehierarchies .............................378

Fig.10.3Tochoosethebesthospiceplan,oneconstructsa hierarchymodelingthecommunity,institutional, andsocietalcosts.Thisisthecostshierarchyoftwo separatehierarchies ................................................379

Fig.10.4Employeeevaluationhierarchy

Fig.10.5Hierarchiesforratingbeneﬁts,costs,opportunities, andrisks ...........................................................395

Fig.10.6PrioritizingthestrategiccriteriatobeusedinratingtheBOCR 396

Fig.10.7Howahierarchycomparestoanetwork

Fig.10.8Thesupermatrixofanetworkanddetailofa componentinit ....................................................399

Fig.10.9Thesupermatrixofahierarchywiththeresultinglimit matrixcorrespondingtohierarchicalcomposition

Fig.10.10(a)Schoolchoicehierarchycomposition.(b) Supermatrixofschoolchoicehierarchygivessame resultsashierarchiccomposition .................................402

Fig.10.11Theclustersandnodesofamodeltoestimatethe relativemarketshareofWalmart,KmartandTarget ............405

Fig.10.12Theclustersandnodesofamodeltoestimatethe relativemarketshareoffootware .................................410

Fig.10.13Hierarchyforratingbeneﬁts,opportunities,costsandrisks ....416

Fig.10.14Arrow’sfourconditions ...........................................418

Fig.11.1Exampleofsub-typebinconsistency .............................428

Fig.11.2Exampleofincompatibilitybetween(*)and(**) ...............435

Fig.11.3Procedureforallcasesofinconsistency ..........................441

Fig.11.4Suggestionofchangetoresolveinconsistency ..................443

Fig.11.5MatrixofjudgementsandbasicMACBETHscale ..............444

Fig.11.6RepresentationsoftheMACBETHscale ........................445

Fig.11.7ConsistentmatrixofMACBETHqualitative judgementswithnohesitation ....................................447

Fig.11.8FirstattempttoobtainthebasicMACBETHscale ..............448

Fig.11.9SecondattempttoobtainthebasicMACBETHscale ...........449

Fig.11.10“Greatest”closedintervalsincludedinthefreeand dependentintervals ................................................451

Fig.13.1Decisiontreerepresentingknowledgeincludedfrom Table13.1 ..........................................................523

Fig.13.2Thehierarchyofattributesandcriteriaforacar classiﬁcationproblem .............................................532

Fig.14.1Thetwovaluesthat xk ! @U @xi .x/ cantake ........................568

Fig.14.2Differentcasesofinteraction:complementarycriteria (a),substitutivecriteria(b),independentcriteria(c) ............570

Fig.16.1Interval-weightedaveragevs.intervalconvexsum ..............673

Fig.17.1(a)Minimalelement x andmaximalelement y ofaset A.(b)Stronglyminimalelement y ofaset A ....................701

Fig.17.2(a)Weaklyminimalelement y ofaset A.(b)Properly minimalelement y ofaset A ......................................701

Fig.17.3Minimalandmaximalelementsof T D f .S / ....................705

Fig.17.4Section Ay ofaset A ...............................................706

Fig.17.5(a)Arbitraryspins.(b)Parallelandanti-parallel alignedspins .......................................................715

Fig.17.6Spinprecession ....................................................715

Fig.17.7Aso-calledsagittalT1MP-RAGEimagetakenupby the3teslasystemMAGNETOMSkyraproducedby SiemensAG.WithkindpermissionofSiemensAG Healthcaresector ..................................................716

Fig.17.8Qualitativeillustrationoftheimagepointsofminimal solutionsandtheimagepointofthestandard excitationpulse ...................................................717

Fig.17.9Theelement y 2 A isaminimalelementof A w.r.t. theorderingmap D whereas y isnotanondominated elementof A w.r.t.theorderingmap D becauseof y 2fy0 gC D .y0 / nf0Y g,cf.[21,23] .............................720

Fig.17.10Illustrationoftwosets A and B with A 4s B,and a 2 max A and b 2 max B with a b and b a ................730

Fig.17.11Illustrationoftwosets A; B 2 M with A 4m B ..................731

Fig.17.12Illustrationoftwosets A; B 2 M with A 4mc B .................731

Fig.17.13Illustrationofthesets A1 , A2 , A3 , A5 and A6 inExample18 ....734

Fig.19.1FeasiblesetandEdgeworth-Paretohull ..........................819

Fig.19.2(a)Individualandlexicographicminima.(b)(Weakly) non-dominatedpoints ............................................819

Fig.19.3(a)Extremenon-dominatedpointfor T D .1;1/.(b) Supportednon-dominatedpointintherelativeinterior ofafacefor T D .2;1/ ..........................................821

Fig.19.4(a)Alowerboundset.(b)Anupperboundsetdeﬁned byfeasiblepoints ..................................................825

Fig.19.5(a)Theweightedsumscalarisation.(b)The "-constraintscalarisation ..........................................830

Fig.19.6TheChebychevscalarisation .....................................830

Fig.19.7(a)The "-constraintscalarisation.(b)Theelastic constraintscalarisation ............................................833

Fig.19.8(a)Lexicographicallyoptimalpoints.(b)Theﬁrst weightedsumproblem ............................................835

Fig.19.9Phase1ofthetwophasemethod .................................836

Fig.19.10(a)Thetriangleswherenon-supportednon-dominated pointsmaybelocated.(b)Rankingnon-supported non-dominatedpoints .............................................837

Fig.19.11(a)Thenodecanbefathomedbydominance.(b)The nodecanbefathomedbydominanceassuming Y Zp .........843

Fig.20.1L-Rfuzzynumber c D .cL ; cR ;˛;ˇ/LR ..........................855

Fig.20.2Possibilityandnecessitymeasures ...............................856

Fig.20.3Example1 .........................................................860

Fig.20.4Problem(20.25) ...................................................862

Fig.20.5Problem(20.25)withtheupdatedobjectivefunction

Fig.20.6Anexampleofanecessarilyefﬁcientsolution ...................872

Fig.20.7Anexampleofanon-necessarilyefﬁcientsolution ..............873

Fig.20.8Exampleofatreegeneratedbytheimplicit enumerationalgorithm ............................................880

Fig.20.9Exampleofatreegeneratedbytheextendedimplicit enumeration ........................................................881

Fig.20.10Exampleofdiscretefuzzyrandomvariables

Fig.20.11Exampleofthemembershipfunction Cljsl

Fig.20.12Exampleofthemembershipfunction Clsl x

Fig.20.13Exampleofthemembershipfunctionofafuzzygoal ...........888

Fig.20.14Degreeofpossibility ˘Clsl x . Q Gl / ..................................889

Fig.23.1Basicloopofanevolutionaryalgorithm .........................979

Fig.23.2Non-dominatedsortingofsolutionsasinNSGA-II .............981

Fig.23.3Examplefor(marginal)Hypervolume ...........................982

Fig.23.4Influenceofscalingonthedistributionofsolutions alongtheParetofrontasgeneratedbyMOEAs.On theleftfigure(a),thefrontisplottedwitha1:1ratio. Ontherightfigure(b),they-axishasbeenscaledbya factorof100 .......................................................985

Fig.23.5PartoftheParetooptimalfrontthatremainsoptimal withagivenreferencepoint r andthepreference relationfrom[38].Theleftpanel(a)showsa reachablereferencepoint,whiletherightpanel (b)showsanunreachableone ....................................988

Fig.23.6EffectofthemodiﬁeddominanceschemeusedbyG-MOEA ..991

Fig.23.7Marginalcontributioncalculatedaccordingto expectedutilityresultinaconcentrationofthe individualsinkneeareas ..........................................994

Fig.23.8Resultingdistributionofindividualswiththemarginal expectedutilityapproachandalinearlydecreasing probabilitydistributionfor ......................................994

Fig.23.9Examplefordominatedregionintheapproach from[31].Maximizationofobjectivesisassumed. The curve representsallsolutionsequivalentto B accordingtotheapproximatedvaluefunction.All solutionswithanestimatedvaluebetterthan B (above the curve)dominateallsolutionswithanestimated valueworsethan B (belowthe curve).The greyareas indicatetheareasdominatedbysolutions A and C ,respectively 998

Fig.23.10Visualizationofthepreferenceconein2D,assuming quasiconcaveutilityfunctionandmaximizationofobjectives .1000

Fig.23.11Solutions(blackpoints)andterritories(squares)with differentsizesasusedin[54].Regionswithsmaller territorieswillmaintainahigherdensityofsolutions

Fig.24.1Theneo-classicalviewontheobjectiveoftheﬁrm .............

Fig.24.2SituationsleadingtoMCDAintheﬁrm ..........................

Fig.24.3Abird’s-eyeviewoftheframework

Fig.24.4Feasibleregions Z of(MC-Un)and(MC-B)forthe sameeightsecurities .............................................. 1035

Fig.24.5Unboundedbullet-shapedfeasibleregion Z createdby securitiesA,BandC .............................................. 1036

Fig.24.6Nondominatedfrontiersasafunctionofchangesin thevalueofupperboundparameter ........................... 1039

Fig.24.7Anellipsoidalfeasibleregionprojectedonto two-dimensionalrisk-returnspace ................................ 1043

Fig.25.1Criteriaconsideredinenergydecision-makingstudies .......... 1132

Fig.25.2Technicalcriteria. M miscellaneous, EE energy efﬁciency, SD sitingdecisions, EP energyprojects, EPP energyplansandpolicies, PGT powergeneration technologies ....................................................... 1136

Fig.25.3Economiccriteria .................................................. 1138

Fig.25.4Environmentalcriteria ............................................. 1142

Fig.25.5Socialcriteria ...................................................... 1144

Fig.25.6MCDAmethodsusedinenergydecision-makingstudies ....... 1145

Fig.25.7MCDAmethodsusedineachtypeofenergy application(numberofpapers) ................................... 1145

Fig.25.8Uncertaintyhandlingtechniquesusedwithdifferent MCDAmethods ................................................... 1150

Fig.26.1Priorityregionsandexamplein[16] ............................. 1188

Fig.27.1Asystemicvisionofsustainabilityissues ........................ 1238

Fig.27.2TheidealproblemstructuringinSMCE ......................... 1257

Fig.28.1Bubblechartfortheﬂatfurnishingexample ..................... 1278

Fig.28.2Coreindexdisplayfortheﬂatfurnishingexample .............. 1279

Fig.28.3Paretofrontdisplayfortheﬂatfurnishingexample ............. 1280

ListofTables

Table3.1Principalt-norms,t-conormsandnegations

Table3.2Various "-representationswith " D 1 ...........................71

Table4.1Evaluationofthefiveofficesonthefiveattributes .............107

Table6.1Evaluationtable ..................................................190

Table6.2Weightsofrelativeimportance ..................................193

Table6.3Typesofgeneralizedcriteria(P.d /:preferencefunction)

Table6.4Singlecriterionnetﬂows ........................................202

Table7.1Rankevaluationofalternatives(impactmatrix)

Table7.2Theconcordance/discordanceindices ..........................225

Table7.3Concordancematrix .............................................227

Table7.4Rankevaluationofalternatives(impactmatrix)

Table7.5Regimematrix

Table7.6Position-matrix

Table7.7City-blockdistance

Table7.8Preferencematrixforacriterionwithordinalevaluation

Table7.9Preferencematrixforacriterion(Max)with evaluationonaquantitativescale

Table7.10Preferenceimportancetablefor gj , a; b

Table7.11Combinedpreferenceswithweightsimportance

Table7.12Evaluationofalternatives

Table7.13Criteria g1 and g3 (ordinalscales) ...............................236

Table7.14Criterion g2 (ordinalscale)

Table7.15Criterion g4 (intervalscaleMIN)

Table7.16Preferencestructureofweights .................................237

Table7.17Pairwisecomparisonbetween a1 and a4 ........................237

Table7.18AxiomaticsystemofMAPPACbasicindices

Table7.19Basicpreferencesindices ........................................250

Table7.20Tableofobservedstochasticdominances

Table7.21Explicableconcordancesindices

Table9.1CriteriavaluesandrankingoftheDM

Table9.2Marginalvaluefunctions(initialsolution)

Table9.3Linearprogrammingformulation(post-optimalityanalysis)

Table9.4Post-optimalityanalysisandﬁnalsolution

Table9.5Marginalvaluefunctions(ﬁnalsolution)

Table9.6LPsizeofUTAmodels ..........................................336

Table9.7IndicativeapplicationsoftheUTAmethods

Table10.1Thefundamentalscaleofabsolutenumbers

Table10.2WhichdrinkisconsumedmoreintheU.S.?An exampleofestimationusingjudgments ........................375

Table10.3Theentriesinthismatrixrespondtothequestion: whichcriterionismoreimportantwithrespectto choosingthebesthospicealternativeandhowstrongly?

Table10.4Theentriesinthismatrixrespondtothequestion: whichsubcriterionyieldsthegreaterbeneﬁtwith respecttoinstitutionalbeneﬁtsandhowstrongly?

Table10.5Theentriesinthismatrixrespondtothequestion: whichmodelyieldsthegreaterbeneﬁtwithrespect todirectcareandhowstrongly?

Table10.6Theentriesinthismatrixrespondtothequestion: whichcriterionisagreaterdeterminantofcostwith respecttothecaremethodandhowstrongly?

Table10.7Theentriesinthismatrixrespondtothequestion: whichcriterionincursgreaterinstitutionalcostsand howstrongly? ....................................................382

Table10.8Theentriesinthismatrixrespondtothequestion: whichmodelincursgreatercostwithrespectto institutionalcostsforrecruitingstaffandhowstrongly? ......382

Table10.9Synthesis(P=Priorities,M=Model) ..........................383

Table10.10Rankingintensities ..............................................386

Table10.11Rankingalternatives .............................................387

Table10.12Randomindex ....................................................390

Table10.13Calculatingreturnsarithmetically ..............................391

Table10.14Normalizedcriteriaweightsandnormalized alternativeweightsfrommeasurementsinthesame scale(additivesynthesis) ........................................392

Table10.15Priorityratingsforthemerits:beneﬁts,costs, opportunities,andrisks ..........................................396

Table10.16FourmethodsofsynthesizingBOCRusingtheidealmode ...397

Table10.17Thesupermatrix .................................................403

Table10.18Thelimitsupermatrix ............................................403

Table10.19Theunweightedsupermatrix ....................................406

Table10.20Theclustermatrix ...............................................408

Table10.21Theweightedsupermatrix .......................................408

Table10.22Thesynthesizedresultsforthealternatives .....................409

Table10.23Footwearactualstatisticsandmodelresultsalong withthecompatibilityindex ....................................413

Table10.24Priorityratingsforthemerits:beneﬁts,opportunities, costsandrisks ....................................................417

Table10.25Overallsynthesesofthealternatives ............................417

Table12.1Descriptionofconsequencesforthesimpleexample ..........489

Table13.1Datatablepresentingexamplesofcomprehensive evaluationsofstudents ..........................................501

Table13.2QualityofclassiﬁcationandShapleyvaluefor classiﬁcation Cl andsetofcriteria P ............................512

Table13.3Evaluationsofnewstudents .....................................521

Table13.4Evaluationsofnewstudents .....................................522

Table13.5Informationtableoftheillustrativeexample ...................525

Table13.6Studentswithintervalevaluations ..............................526

Table13.7Exampleofmissingvaluesintheevaluationofstudents ......529

Table13.8Substitutionofmissingvaluesintheevaluationofstudents ...530

Table13.9Decisiontablewithreferenceobjects ...........................539

Table13.10Afragmentof SPCT ..............................................540

Table13.11Rankingofwarehousesforsalebydecisionrules andtheNetFlowScoreprocedure ..............................541

Table15.1Criteriaforapplicantevaluation ................................612

Table15.2Comparisonofhypotheticalalternatives .......................614

Table15.3Anexampleofajointordinalscale .............................615

Table15.4RanksforJSQV ..................................................618

Table15.5EffectivenessofSTEP-ZAPROS ...............................620

Table15.6Presentationofa“tryad”tothedecisionmaker ................622

Table19.1ComplexityresultsforMOCOproblems

Table19.2Propertiesofpopularscalarisationmethods ....................831

Table19.3Algorithmsbasedonscalarisation ..............................834

Table19.4Two-phasealgorithms ...........................................841

Table19.5Multi-objectivebranchandboundalgorithms ..................845

Table21.1DistancemetricsusedinMCDMdistance-basedtechniques ..913

Table23.1Comparisonofsomeselectedapproachesto incorporatepartialuserpreferences ............................. 1003

Table24.1ApplicationsofMCDAapproachesinbankruptcy andcreditriskassessment ....................................... 1052

Table24.2ApplicationsofMCDAapproachesinportfolio selectionandmanagement ...................................... 1053

Table24.3ApplicationsofMCDAapproachesinthe assessmentofcorporateperformance 1054

Table24.4ApplicationsofMCDAapproachesininvestmentappraisal ..1055

Table24.5ApplicationsofMCDAapproachesinotherﬁnancial decision-makingproblems 1055

Table25.1Categoriesofplanningproblemsinpowersystems accordingtotheorganizationallevelandtimeframe 1071

Table25.2Studiesgroupedinpowergenerationcomparisonproblems ..1106

Table25.3Studiesgroupedinenergyplansandpoliciesproblems 1112

Table25.4Studiesgroupedinenergyprojectproblems 1122

Table25.5Studiesgroupedinsitingdecisionproblems ................... 1124

Table25.6Studiesgroupedinenergyefﬁciencyproblems ................ 1126

Table25.7Studiesgroupedinenergymiscellaneousproblems ........... 1127

Table27.1Impactmatrixforthefourchosencitiesaccordingto theselectedindicators ........................................... 1249

Table27.2Normalisedimpactmatrix ....................................... 1250

Table27.3Outrankingmatrixofthefourcitiesaccordingtothe nineindicators ................................................... 1250

Table27.4Weightedoutrankingmatrix ..................................... 1252

Table29.1MADAandMOOsoftware

Table29.2Multiplecriteriaevaluationmethods

Table29.3Softwarebymethodimplemented

Table29.4Softwarewithgroup

Table29.5Softwareplatforms ..............................................

Introduction

JoséRuiFigueira,SalvatoreGreco,andMatthiasEhrgott

1TenYearsofSuccessofMultipleCriteriaDecisionAnalysis andReasonsforThisNewEdition

After10yearswepresentanupdatedrevisionofthecollectionofstate-of-theartsurveysonMultipleCriteriaDecisionAnalysis(MCDA).Thisisagood occasiontobrieﬂycommentonthelatestadvancesinthedomain.Webelieve thatinthelast10yearswehaveseengreatprogressofMCDA,frombotha theoreticalpointofviewandareal-lifeapplicationpointofview.Wehaveseen theconsolidationofthemain“traditional”methodologiessuchasmultipleattribute utilitytheory, outrankingmethods,interactivemultiobjectiveoptimization,aswell asthegrowingsuccessofnewapproachessuchasEvolutionaryMultiobjective Optimization(EMO).Thespectrumofapplicationshasbeenconstantlyexpanding withparticularemphasisonverycomplexproblemssuchasindustrialdesignorgrid optimization.Takingintoaccountthisevolutionofthedomain,wepartlymodiﬁed thestructureandthecontentofthebookgivingspacetonewmethodologies(e.g., EMOormulti-criteriaportfoliodecisionanalysisforprojectselection)orsplitting chaptersintoseveralnewones(e.g.,thechapteronmultiobjectiveprogrammingof thepreviouseditionthathasnowbeensubstitutedbythreechapters,oneonvector andsetoptimization,oneoncontinuousmultiobjective,andone onmultiobjective combinatorialoptimization).Moreover,allauthors,sometimeswiththehelpof

J.R.Figueira

CEG-IST,InstitutoSuperiorTécnico,UniversidadedeLisboa,A.RoviscoPais,1,1049-001 Lisboa,Portugal

S.Greco

UniversityofCatania,Catania,Italy

UniversityofPortsmouth,Portsmouth,UK

M.Ehrgott

DepartmentofManagementScience,LancasterUniversity,Bailrigg,LancasterLA14YX,UK

anewcolleague,haveupdatedthecontentsoftheircontributionsincorporating thenoveltiesofthelast10years.Ofcourse,manysophisticatedtechnicaldetails thatappearintheneweditionofthebook willsoonerorlaterbedestinedto besupersededbytheincessantevolutionofresearchandapplications.Wethink, however,thatthebasicprinciplesasstatedbytheexpertswhopreparedthedifferent chaptersinthebookwillremainreferencepointsfortheyearstocome.Moreover, webelievethatthespiritwithwhichexpertsonMCDAareworkingtoday,inthese sorichandfruitfulyears,willremainforeverinthisbook.Thisspiritisstrongly relatedwiththespiritwithwhich,inthelate1960sandearly1970softhelast century,the“pioneers”(manyofwhoareamongthemanyauthorsofthechapters inthisbook)outlinedthebasicprinciples ofMCDAwiththegenuineaimtogivea satisfactoryanswertoconcreterealworldproblemsforwhichtheclassicalmethods ofoperationsresearchwerenotabletoﬁndadequateanswers.Thereforethebasic principlesofthepresentedmethodologiesandtheirrelationshipswiththeMCDA spiritarethingsthatwerecommendthereadertolookforineachchapter.After thesewordsabouttheintuitionthatguidedtherevisionofthisbook,letusenter“in mediasres”,comingbacktotheintroductionoftheﬁrsteditionthatwasofcourse alsoupdated.

2HumanReﬂectionAboutDecision

Decision-makinghasinspiredreﬂectionsofmanythinkerssinceancienttimes.The greatphilosophersAristotle,Plato,andThomasAquinas,tomentiononlyafew, discussedthecapacityofhumanstodecideandinsomemannersclaimedthatthis possibilityiswhatdistinguisheshumansfro manimals.Toillustratesomeimportant aspectsofdecision-making,letusbrieﬂyquotetwoimportantthinkers,Ignatiusof Loyola(1491–1556)andBenjaminFranklin(1706–1790).

Toconsider,reckoningup,howmanyadvantagesandutilitiesfollowformefromholding theproposedofﬁceorbeneﬁce[ ],and,toconsiderlikewise,onthecontrary,the disadvantagesanddangerswhichthereareinhavingit.Doingthesameinthesecondpart, thatis,lookingattheadvantagesandutilitiestherearein nothavingit,andlikewise,on thecontrary,thedisadvantagesanddangersinnothavingthesame.[ ]AfterIhavethus discussedandreckoneduponallsidesaboutthethingproposed,tolookwherereasonmore inclines:andso,accordingtothegreaterinclinationofreason,[ ],deliberationshouldbe madeonthethingproposed.

Thisfragmentfromthe“SpiritualExercises”ofSt.[14]hasbeentakenfroma paperbyFortempsandSlowinski[12].

London,Sept19,l772

DearSir,

Intheaffairofsomuchimportancetoyou,whereinyouaskmyadvice,Icannot,forwant ofsufﬁcientpremises,adviseyouwhattodetermine,butifyoupleaseIwilltellyouhow. [::: ],mywayistodividehalfasheetofpaperbyalineintotwocolumns;writingoverthe onePro,andovertheotherCon.[::: ]WhenIhavethusgotthemalltogetherinoneview, Iendeavortoestimatetheirrespectiveweights;andwhereIﬁndtwo,oneoneachside,that

seemequal,Istrikethembothout.IfIfindareasonproequaltosometworeasonscon, Istrikeoutthethree.IfIjudgesometworeasonscon,equaltothreereasonspro,Istrike outthefive;andthusproceedingIfindatlengthwherethebalancelies;andif,afteraday ortwooffurtherconsideration,nothingnewthatisofimportanceoccursoneitherside,I cometoadeterminationaccordingly.[ ]Ihavefoundgreatadvantagefromthiskindof equation,andwhatmightbecalledmoralorprudentialalgebra.Wishingsincerelythatyou maydetermineforthebest,Iamever,mydearfriend,yoursmostaffectionately.

B.Franklin

ThisletterfromBenjaminFranklintoJosephPrestlyhasbeentakenfromapaper byMacCrimmon[17].

Whatisinterestingintheabovetwoquotationsisthefactthatdecisionisstrongly relatedtothecomparisonofdifferentpointsofview,someinfavorandsomeagainst acertaindecision.Thismeansthatdecisionisintrinsicallyrelatedtoapluralityof pointsofview,whichcanroughlybedefinedascriteria.Contrarytothisverynatural observation,formanyyearstheonlywaytostateadecisionproblemwasconsidered tobethedefinitionofasinglecriterion,whichamalgamatesthemultidimensional aspectsofthedecisionsituationintoa singlescaleofmeasure.Forexample,even todaytextbooksofoperationsresearchsuggesttodealwithadecisionproblem asfollows:Tofirstdefineanobjectivefunction,i.e.,asinglepointofviewlike acomprehensiveprofitindex(oracomprehensivecostindex)representingthe preferability(ordis-preferability)oftheconsideredactionsandthentomaximize (minimize)thisobjective.Thisisaveryreductive,andinsomesensealsounnatural, waytolookatadecisionproblem.Thus,foratleast40years,anewwaytolook atdecisionproblemshasmoreandmoregainedtheattentionofresearchersand practitioners.ThisistheapproachconsideredbyLoyolaandFranklin,i.e.,the approachofexplicitlytakingintoaccounttheprosandtheconsofapluralityof pointsofview,inotherwordsthedomainofmultiplecriteriadecisionanalysis. Therefore,MCDAintuitioniscloselyrelatedtothewayhumanshavealways beenmakingdecisions.Consequently,despitethediversityofMCDAapproaches, methodsandtechniques,thebasicingredientsofMCDAareverysimple:Afinite orinfinitesetofactions(alternatives,solutions,coursesofaction, ),atleast twocriteria,and,obviously,atleastonedecision-maker(DM).Giventhesebasic elements,MCDAisanactivitywhichhelpsmakingdecisionsmainlyintermsof choosing,ranking,orsortingtheactions.

Ofcourse,notonlyphilosophersreasonedaboutdecision.ManyimportanttechnicalaspectsofMCDAarelinkedtoclassicworksineconomics,inparticular, welfareeconomics,utilitytheory,andvoting-orientedsocialchoicetheory(see [28]).Aggregatingtheopinionorthepreferencesofvotersorindividualsofa communityintocollectiveorsocialpreferencesisquitesimilaraproblemto

devisingcomprehensivepreferencesofadecision-makerfromasetofconﬂicting criteriainMCDA[7].

DespitetheimportanceofRamonLlull’s(1232–1316)andNicolausCusanus’ (1401–1464)concernsaboutandinterestsinthisverytopic,theoriginsofvoting systemsareoftenattributedtoLeChevalierJean-CharlesdeBorda(1733–1799) andMarieJeanAntoineNicolasdeCaritat(1743–1794),LeMarquisdeCondorcet. However,RamonLlullintroducedthepairwisecomparisonconceptbeforeCondorcet[13],whileNicolausCusanusintroducedthescoringmethodaboutthreeand ahalfcenturiesbeforeBorda[27].Furthermore,itshouldbenotedthataletter fromPlinytheYounger( AD105)toTitusAristoshowsthatheintroduced theternaryapprovalvotingstrategyandwasinterestedinvotingsystemsalong timebeforeRamonLlullandNicolausCusanus[18,Chapter2].Anyway,Borda’s scoringmethod[4]hassomesimilaritieswithcurrentutilityandvaluetheoriesas hasCondorcet’smethod[10]withtheoutrankingapproachofMCDA.Inthesame lineofconcerns,i.e.,theaggregationofindividualpreferencesintocollectiveones, JeremyBentham(1748–1832)introducedtheutilitariancalculustoderivethetotal utilityforthesocietyfromtheaggregationofthepersonalinterestsoftheindividuals ofacommunity[3].InspiredbyBentham’sworks,FrancisYsidroEdgeworth (1845–1926),autilitarianeconomist,wasmainlyconcernedwiththemaximization oftheutilityofthedifferent competingagentsinaneconomy.Edgeworthtriedto findthecompetitiveequilibriumpointsfor thedifferentagents.Heproposedtodraw indifferencecurves(linesofequalutility)foreachagentandthenderivethecontract curve,acurvethatcorrespondstothenotionoftheParetoorefficientset[21].Not longafterward,VilfredoFedericoDamasoPareto(1848–1923)gavethefollowing definitionofophelimity[utility]forthewholecommunity[22].

Wewillsaythatthemembersofacollectivityenjoymaximumophelimityinacertain positionwhenitisimpossibletoﬁndawayofmovingfromthatpositionveryslightly insuchamannerthattheophelimityenjoyedbyeachoftheindividualsofthatcollectivity increasesordecreases.Thatistosay,anysmalldisplacementindepartingfromthatposition necessarilyhastheeffectofincreasingtheophelimitywhichcertainindividualsenjoy,of beingagreeabletosome,anddisagreeabletoothers.

Fromthisdeﬁnitionitiseasytoderivetheconceptofdominance,whichtodayis oneofthefundamentalconceptsinMCDA.

MCDAalsobeneﬁtsfromthebirthanddevelopmentofgametheory.Félix EdouardJustinEmileBorel(1871–1956)andJohnvonNeumann(1903–1957)are consideredthefoundersofgametheory[5, 6, 19, 20].Manyconceptsfromthis disciplinehadastrongimpactonthedevelopmentofMCDA.

Theconceptofefficientpointwasfirstintroducedin1951byTjallingKoopmans (1910–1985)inhispaper“Analysisofproductionasanefficientcombinationof activities”[15].

Apossiblepointinthecommodityspaceiscalledefﬁcientwheneveranincreaseinoneof itscoordinates(thenetoutputofonegood)canbeachievedonlyatthecostofadecrease insomeothercoordinate(thenetoutputofagood).

Inthesameyear(1951)HaroldWilliamKuhn(born1925)andAlbertWilliam Tucker(1905–1995)introducedtheconceptofvectormaximumproblem[16].In the1960s,basicMCDAconceptswereexplicitlyconsideredfortheﬁrsttime.As twoexampleswementionCharnes’andCooper’sworksongoalprogramming[8] andthepropositionofELECTREmethodsbyRoy[23].The1970ssawwhatis conventionallyconsideredthe“ofﬁcial”startingpointofMCDA,theconferenceon “MultipleCriteriaDecisionMaking”organizedin1972byCochraneandZeleny atColumbiaUniversityinSouthCarolina[9].SincethenMCDAhasseena tremendousgrowthwhichcontinuestoday.

4ReasonsforThisCollectionofState-of-the-ArtSurveys

TheideaofMCDAissonaturalandattractivethatthousandsofarticlesand dozensofbookshavebeendevotedtothesubject,withmanyscientiﬁcjournals regularlypublishingarticlesaboutMCDA.Toproposeanewcollectionofstateof-the-artsurveysofMCDAinsorichacontextmayseemarashenterprise. Indeed,someobjectionscometomind.Therearemanyandgoodhandbooksand reviewsonthesubject(togiveanideaconsider[1, 11, 25, 26, 29]).Themainideas arewellestablishedforsomeyearsandonemayquestionthecontributionsthis volumecanprovide.Moreover,theﬁeldissolargeandcomprisesdevelopments soheterogeneousthatitisalmosthopelesstothinkthatanexhaustivevisionofthe researchandpracticeofMCDAcanbegiven.

Wemustconfessthatattheendoftheworkofeditingthisvolumeweagreewith theaboveremarks.However,webelievethatanewandcomprehensivecollection ofstate-of-the-artsurveysonMCDAcanbeveryuseful.Themainreasonswhich, despiteouroriginalresistance,broughtustoproposethisbookarethefollowing:

1.Manyoftheexistinghandbooksandreviewsarenottoorecent.SinceMCDAis aﬁeldwhichisdevelopingveryquicklythisisanimportantreason.

2.EventhoughtheﬁeldofresearchandapplicationofMCDAissolarge,thereare somemaincentralthemesaroundwhichMCDAresearchandapplicationshave beendeveloped.Thereforeourapproachwastotrytopresentthe—atleastinour opinion—mostimportantoftheseideas.

Withreferencetotheﬁrstpoint,wecansaythatweobservedmanytheoretical developmentswhichchangedMCDAoverthelast20years.Wetriedtoconsider thesechangesasmuchaspossibleandinthisperspectivestrongpointsofthebook arethefollowing:

1.Itpresentsthemostup-to-datediscussionsonwell-establishedmethodologies andtheoriessuchasoutranking-basedmethodsandMAUT.

2.Thebookalsocontainssurveysofnew,recentlyemergedﬁeldssuchasconjoint measurement,fuzzypreferences,fuzzyintegrals,roughsets,andothers.

Followingthesepointswedraftedalistoftopicsandaskedwell-known researcherstopresentthem.Weencouragedtheauthorstocooperatewiththeaim topresentdifferentperspectivesiftopicshadsomeoverlap.Weaskedtheauthors topresentacomprehensivepresentationofthemostimportantaspectsoftheﬁeld coveredbytheirchapters,asimpleyetconcisestyleofexposition,andconsiderable spacedevotedtobibliographyandsurveyofrelevantliterature.Wealsorequesteda sufﬁcientlydidacticpresentationandatextthatisusefulforresearchersinMCDA aswellasforpeopleinterestedinreal-lifeapplications.

Theimportanceoftheserequirementsisalsorelatedtothespeciﬁcwaythe MCDAcommunitylooksatitsresearchﬁeld.Itcanbesummarizedintheobservationthatthereisaverystrongandvitallinkbetweentheoreticalandmethodological developmentsontheonehandandrealapplicationsontheotherhand.Thus,the validityoftheoreticalandmethodologicaldevelopmentscanonlybemeasuredin termsoftheprogressgiventoreal-worldpractice.Moreover,interestofMCDA todealwithconcreteproblemsisrelatedtotheconsiderationofasoundtheoretical basiswhichensuresthecorrectapplicationofthemethodologies takenintoaccount.

Infact,notonlythechaptersofourbookbutratherallMCDAcontributions shouldsatisfytherequirementsstatedoutabovebecausetheyshouldbenottoo “esoteric”andthereforeunderstandableforstudents,theoreticallywellfounded,and applicabletosomeadvantageinreality.

5AGuidedTouroftheBook

Ofcourse,thisbookcanbereadfromtheﬁrsttothelastpage.However,wethink thatthisisnottheonlypossibilityanditmaynotevenbethemostinteresting possibility.Inthefollowingweproposea guidedtourofthebooksuggestingsome referencepointsthatarehopefullyusefulforthereader.

5.1PartI:TheHistoryandCurrentStateofMCDA

ThispartisimportantbecauseMCDAisnot justacollectionoftheories,methodologies,andtechniques,butaspeciﬁcperspectivetodealwithdecisionproblems. Losingthisperspective,eventhemostrigoroustheoreticaldevelopmentsand applicationsofthemostreﬁnedmethodologiesareatriskofbeingmeaningless becausetheymissanadequateconsiderationoftheaimsandoftheroleofMCDA. WesharethisconvictionwithmostMCDAresearchers.

Fromthisperspectiveitisimportanttohaveaclearvisionoftheoriginof themainbasicconceptsofthedomain.Forthisreason,MuratKöksalan,Jyrki Wallenius,andStanleyZiontspresenttheearlyhistoryofMCDAandrelated areasshowinghowmanydevelopmentsin theﬁeldweremadebymajorcontributorstooperationsresearch,managementscience,economics,andotherareas.

ThenBernardRoydiscusses“pre-theoretical”assumptionsofMCDAandgives anoverviewoftheﬁeld.BernardRoy,besidesmakingmanyimportanttheoretical contributions,engagedhimselfinthoroughreﬂectionsonthemeaningandthevalue ofMCDA,proposingsomebasickeyconceptsthatareacceptedthroughoutthe MCDAcommunity.

5.2PartII:FoundationsofMCDA

ThispartofthebookisrelatedtoafundamentalproblemofMCDA,therepresentationofpreferences.Classically,forexampleineconomics,itissupposedthat preferencecanberepresentedbyautilityfunctionassigninganumericalvalueto eachactionsuchthatthemorepreferablean action,thelargeritsnumericalvalue. Moreover,itisveryoftenassumedthatthecomprehensiveevaluationofanaction canbeseenasthesumofitsnumericalvaluesfortheconsideredcriteria.Letus callthistheclassicalmodel.Itisverysimplebutnottoorealistic.Indeed,there isalotofresearchstudyingunderwhich conditionstheclassicalmodelholds. Theseconditionsareveryoftenquitestrictanditisnotreasonabletoassume thattheyaresatisfiedinallreal-worldsituations.Thus,othermodelsrelaxingthe conditionsunderlyingtheclassicalmodelhavebeenproposed.Thisisaveryrich fieldofresearch,whichisfirstofallimportantforthoseinterestedinthetheoretical aspectsofMCDA.However,itisalsoofinteresttoreadersengagedinapplications ofMCDA.Infact,whenweadoptaformalmodelitisnecessarytoknowwhat conditionsaresupposedtobesatisfiedbythepreferencesoftheDM.Inthetwo chaptersofthispart,problemsrelatedto therepresentationsofpreferencesare discussed.

StefanoMoretti,MeltemÖztürk,andAlexisTsoukiàspresentaveryexhaustive reviewofpreferencemodeling,startingfromclassicalresultsbutarrivingatthe frontierofsomechallengingissuesofscientiﬁcactivityrelatedtofuzzylogicand non-classicallogic.

DenisBouyssouandMarcPirlotdiscusstheaxiomaticbasisofthedifferent modelstoaggregatemultiplecriteriapreferences.Webelievethatthischapteris veryimportantforthefutureofMCDA.Initially,theemphasisofMCDAresearch wasonproposalofnewmethods.Butgraduallythenecessitytounderstandthebasic conditionsunderlyingeachmethodanditsspeciﬁcaxiomatizationbecamemoreand moreapparent.ThisistheﬁrstbookonMCDAwithsomuchspacededicatedtothe subjectoffoundationsofMCDA.

5.3PartIII:OutrankingMethods

Inthispartofthebooktheclassofoutranking-basedmultiplecriteriadecision methodsispresented.Givenwhatisknownaboutthedecision-maker’spreferences

andgiventhequalityoftheperformancesoftheactionsandthenatureofthe problem,anoutrankingrelationisabinaryrelation S definedonthesetofpotential actions A suchthat aSb ifthereareenoughargumentstodecidethat a isatleast asgoodas b,whereasthereisnoessentialargumenttorefutethatstatement [24].Methodswhichstrictlyapplythisdefinitionofoutrankingrelationarethe ELECTREmethods.Theyareveryimportantinmanyrespects,notleasthistorically, sinceELECTREIwasthefirstoutrankingmethod[2].

However,withintheclassofoutrankingmethodswegenerallyconsiderall methodswhicharebasedonpairwisecomparisonofactions.Thus,anotherclassof verywell-knownmultiplecriteriamethods,PROMETHEEmethods,isconsidered inthispartofthebook.BesidesELECTREandPROMETHEEmethods,many otherinterestingMCDAmethodsarebasedonthepairwisecomparisonofactions. JoséFigueira,VincentMousseau,andBernardRoypresenttheELECTREmethods; Jean-PierreBransandYvesDeSmetpresentthePROMETHEEmethods;andJeanMarcMartelandBenedettoMatarazzoreviewtherichliteratureofotheroutranking methods.

5.4PartIV:Multi-attributeUtilityandValueTheories

Inthispartofthebookweconsidermultipleattributeutilitytheory(MAUT).This MCDAapproachtriestoassignautilityvaluetoeachaction.Thisutilityisareal numberrepresentingthepreferabilityoftheconsideredaction.Veryoftentheutility isthesumofthemarginalutilitiesthateachcriterionassignstotheconsidered action.Thus,thisapproachveryoftencoincideswithwhatwecalledtheclassical approachbefore.AswenotedincommentingPart I,thisapproachisverysimple atﬁrstglance.Itisoftenappliedinreal life,e.g.,everytimeweaggregatesome indicesbymeansofaweightedsum,weareapplyingthisapproach.Despiteits simplicity,theapproachpresentssometechnicalproblems.Theﬁrstisrelatedtothe axiomaticbasisandtheconstructionofmarginalutilityfunctions(i.e.,theutility functionsrelativetoeachsinglecriterion),bothincaseofdecisionundercertainty anduncertainty.TheseproblemsareconsideredbyJamesDyerinacomprehensive chapteraboutthefundamentalsofthisapproach.

YannisSiskos,VangelisGrigoroudis,andNikolaosMatsatsinispresentthevery well-knownUTAmethods,whichonthebasisofthephilosophyoftheaggregation–disaggregationapproachandusinglinearprogrammingbuildaMAUTmodelthat isasconsistentaspossiblewiththeDM’spreferencesexpressedinactualprevious decisionsorona“trainingsample”.Thephilosophyofaggregation–disaggregation canbesummarizedasfollows:Howisitpossibletoassessthedecision-maker’s preferencemodelleadingtoexactlythesamedecisionastheactualoneoratleast themost“similar”decision?

ThomasSaatypresentsaverywell-knownmethodologytobuildutilityfunctions, theAHP(AnalyticHierarchyProcess),anditsmorerecentextension,theANP (AnalyticNetworkProcess).AHPisatheoryofmeasurementthatusespairwise

comparisonsalongwithexpertjudgments todealwiththemeasurementofqualitativeorintangiblecriteria.TheANPisageneraltheoryofrelativemeasurementused toderivecompositepriorityratioscalesfromindividualratioscalesthatrepresent relativemeasurementsoftheinﬂuenceofelementsthatinteractwithrespectto controlcriteria.TheANPcapturestheoutcomeofdependenceandfeedbackwithin andbetweenclustersofelements.ThereforeAHPwithitsdependenceassumptions onclustersandelementsisaspecialcaseoftheANP.

CarlosBanaeCosta,Jean-ClaudeVansnick,andJean-MarieDeCortepresent anotherMCDAmethodologybasedontheadditiveutilitymodel.ThismethodologyisMACBETH(MeasuringAttractivenessbyaCategoricalBasedEvaluation Technique).ItisanMCDAapproachthat requiresonlyqualitativejudgmentsabout differencesofvaluesofattractivenessofoneactionoveranotheractiontohelpan individualoragrouptoquantifytherelativepreferabilityofdifferentactions.In simplewords,theMACBETHapproachtriestoanswerthefollowingquestions: Howcanwebuildanintervalscaleofpreferencesonasetofactionswithoutforcing evaluatorstoproducedirectnumericalrepresentationsoftheirpreferences?How canwecoherentlyaggregatethesequalitativeevaluationsusinganadditiveutility model?

5.5PartV:Non-classicalMCDAApproaches

ManyapproacheshavebeenproposedinMCDAbesidesoutrankingmethodsand multi-attributeutilitytheory.Inthispartofthebookwetrytocollectinformation aboutsomeofthemostinterestingproposals.First,thequestionofuncertaintyin MCDAisconsidered.TheoStewartandIanDurbachdiscussriskanduncertainty inMCDA.Itisnecessarytodistinguishbetweeninternaluncertainties(relatedto decision-makervaluesandjudgments)andexternaluncertainties(relatedtoimperfectknowledgeconcerningconsequences ofactions).Thelatter,correspondingto themostacceptedinterpretationofuncertaintyinthespecializedliterature,has beenconsideredinthechapter.Fourbroadapproachesfordealingwithexternal uncertaintiesarediscussed.Thesearemulti-attributeutilitytheoryandsome extensions;stochasticdominanceconcepts,primarilyinthecontextofpairwise comparisonsofalternatives;theuseofsurrogateriskmeasuressuchasadditional decisioncriteria;andtheintegrationofMCDAandscenarioplanning.

SalvatoreGreco,Benedetto Matarazzo,andRomanSłowi ´ nskipresentthedecisionruleapproachtoMCDA.Thisapproachrepresentsthepreferencesintermsof “if :::,then :::”decisionrulessuchas,forexample,“ifthemaximumspeedofcar x isatleast175km/handitspriceisatmost$12000,thencar x iscomprehensively atleastmedium”.Thisapproachisrelatedtoroughsettheoryandtoartiﬁcial intelligence.Itsmainadvantagesarethefollowing.TheDMgivesinformationin theformofexamplesofdecisions,whichrequiresrelativelylowcognitiveeffortand whichisquitenatural.Thedecisionmodelisalsoexpressedinaverynaturalway bydecisionrules.Thispermitsanabsolutetransparencyofthemethodologyforthe

DM.Anotherinterestingfeatureofthedecisionruleapproachisitsﬂexibility,since anydecisionmodelcanbeexpressedintermsofdecisionrulesand,evenbetter, thedecisionrulemodelcanbemuchmoregeneralthanallotherexistingdecision modelsusedinMCDA.

MichelGrabischandChristopheLabreuchepresentthefuzzyintegralapproach thatisknowninMCDAforthelasttwodecades.Inverysimplewordsthis methodologypermitsaﬂexiblemodelingoftheimportanceofcriteria.Indeed,fuzzy integralsarebasedonacapacitywhichassignsanimportancetoeachsubsetof criteriaandnotonlytoeachsinglecriterion.Thus,theimportanceofagivensetof criteriaisnotnecessarilyequaltothesumoftheimportanceofthecriteriafromthe consideredsubset.Consequently,iftheimportanceofthewholesubsetofcriteriais smallerthanthesumoftheimportancesofitsindividualcriteria,thenweobserve aredundancybetweencriteria,whichinsomewayrepresentsoverlappingpoints ofview.Ontheotherhand,iftheimportanceofthewholesubsetofcriteriais largerthanthesumoftheimportancesofitsmembers,thenweobserveasynergy betweencriteria,theevaluationsofwhichreinforceoneanother.Onthebasisofthe importanceofcriteriameasuredbymeansofacapacity,thecriteriaareaggregated bymeansofspeciﬁcfuzzyintegrals,themostimportantofwhicharetheChoquet integral(forcardinalevaluations)andtheSugenointegral(forordinalevaluations).

HelenMoshkovich,AlexanderMechitov,andDavidOlsonpresenttheverbal decisionmethodsMCDA.Thisisaclassofmethodsoriginatedfromtheworkofone oftheMCDApioneers,thelateOlegLarichev.Theideaofverbaldecisionanalysis istobuildadecisionmodelusingmostlyqualitativeinformationexpressedinterms ofalanguagethatisnaturalfortheDM.Moreover,measurementofcriteriaand preferenceelicitationshouldbepsychologicallyvalid.Themethods,besidesbeing mathematicallysound,shouldchecktheDM’sconsistencyandprovidetransparent recommendations.

Mostreal-worlddecisionproblemstakeplaceinacomplexenvironmentwhere conﬂictingsystemsoflogic,uncertain,andimpreciseknowledge,andpossibly vaguepreferenceshavetobeconsidered.Tofacesuchcomplexity,preference modelingrequirestheuseofspeciﬁctools,techniques,andconceptswhichallow theavailableinformationtoberepresentedwiththeappropriategranularity.Inthis perspective,fuzzysettheoryhasreceivedalotofattentioninMCDAforalongtime. DidierDuboisandPatricePernytrytoprovideatentativeassessmentoftheroleof fuzzysetsindecisionanalysis,takingacriticalstandpointonthestate-of-the-art,in ordertohighlighttheactualachievementsandtryingtobetterassesswhatisoften considereddebatablebydecisionscientistsobservingthefuzzydecisionanalysis literature.

5.6PartVI:MultiobjectiveOptimization

Theclassicalformulationofanoperations researchmodelisbasedonthemaximizationorminimizationofanobjectivefunctionsubjecttosomeconstraints.Avery

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2206 Journeay Jno, Fireman June 19

6417 Jackson J, Shawsheen Aug 22

8291

Johnson G P Sept 9

8858 James F A Sept 15

9392 Johnson M Sept 20

10218 Joseph F Oct 2

602 Keefe Jno, Housatonic Apr 18

698 Kultz A T, Ward Apr 23

1546 Kelley Jas, Underwriter June 1

3850 Kinney J, Water Witch July 24

7375 Lodi Jno Aug 31

2813 Lindersmith E, Montg’y July 3

4201

Lawton Jas, Ladona July 30