INFORMATICA,2019,Vol.30,No.1,135–152 135 2019 VilniusUniversity DOI:http://dx.doi.org/10.15388/Informatica.2019.201
ABipolarFuzzyExtensionoftheMULTIMOORA Method
DragisaSTANUJKIC1 ∗,DarjanKARABASEVIC2 , EdmundasKazimierasZAVADSKAS3,FlorentinSMARANDACHE4 , WillemK.M.BRAUERS5
1TechnicalFacultyinBor,UniversityofBelgrade,Serbia
2FacultyofAppliedManagement,EconomicsandFinance, UniversityBusinessAcademyinNoviSad,Belgrade,Serbia
3InstituteofSustainableConstruction,LaborofOperationalResearch, FacultyofCivilEngineering,VilniusGediminasTechnical University, Sauletekio11,Vilnius,LT-210233,Lithuania
4DepartmentofMathematics,UniversityofNewMexico, 705GurleyAvenue,Gallup,NM87301,USA
5FacultyofBusinessandEconomics,DepartmentofEconomics, UniversityofAntwerp,Antwerp,Belgium
e-mail:dstanujkic@tfbor.bg.ac.rs,darjan.karabasevic@mef.edu.rs,edmundas.zavadskas@vgtu.lt, fsmarandache@gmail.com,willem.brauers@uantwerpen.be
Received:October2018;accepted:February2019
Abstract. TheaimofthispaperistomakeaproposalforanewextensionoftheMULTIMOORA methodextendedtodealwithbipolarfuzzysets.Bipolarfuzzysetsareproposedasanextension ofclassicalfuzzysetsinordertoenablesolvingaparticularclassofdecision-makingproblems. Unlikeotherextensionsofthefuzzysetoftheory,bipolarfuzzysetsintroduceapositivemembership function,whichdenotesthesatisfactiondegreeoftheelementxtothepropertycorrespondingto thebipolar-valuedfuzzyset,andthenegativemembershipfunction,whichdenotesthedegreeofthe satisfactionoftheelementxtosomeimplicitcounter-propertycorrespondingtothebipolar-valued fuzzyset.Byusingsingle-valuedbipolarfuzzynumbers,theMULTIMOORAmethodcanbemore efficientforsolvingsomespecificproblemswhosesolvingrequiresassessmentandprediction.The suitabilityoftheproposedapproachispresentedthroughanexample.
Keywords: bipolarfuzzyset,single-valuedbipolarfuzzynumber,MULTIMOORA,MCDM.
1.Introduction
Themanagementofverycomplexsystemsisthemostcomplex,andthereforethemostdifficulttaskofthemanagersoftoday’sorganizations.Theeffectivenessofthemanagement andmanagersofanorganizationdependstoalargeextentonthequalityofthedecisions theymakeonadailybasis.
*Correspondingauthor.
D.Stanujkicetal.
Decision-makinganddecisionsarethecoreofmanagerialactivities.Bearinginmind theglobalizationand,therefore,thedynamicsofbusiness doing,alloftheabove-stated havecausedbusinessandthedecision-makingprocesstobecomemoredemanding.Makingqualitydecisionsrequiresanevermoreextensivepreparation,whichalsoinvolvesthe considerationofthedifferentaspectsofadecision,forthe reasonofwhichthedecisionmakingprocessbecomesconsiderablyformalized.Thus,realproblemsandsituationsin reallifearecharacterizedbyalargenumberofmostlyconflictingcriteria,whosestrict optimizationisgenerallyimpossible.
Whenitisnecessarytomakeadecisiononchoosingoneofseveralpotentialsolutions toaproblem,itisdesirabletoapplyoneofthemodelsbasedonmultiple-criteriadecisionmakingmethods(MCDM).Thismostofteninvolvestheprocess ofselectingoneofseveral alternativesolutions,forwhichcertaingoalsareset.WhenMCDMisconcerned,Greco etal. (2010)pointoutthefactthatitisthestudyofthemethodsandproceduresaimed atmakingaproposalforsolutionsintermsofmultiple,oftenconflictingcriteria.Hwang andYoon(1981)statesthatMCDMisbasedonthetwobasicapproaches,i.e.onmultiple attributedecision-making(MADM),whichimpliesachoiceofcoursesinthepresence ofmultiple,andoftenconflictingcriteria,i.e.aselectionofthebestalternativefroma finitesetofpossiblealternatives.UnlikeMADM,inmultipleobjectivedecision-making (MODM),thebestalternativeisthatwhichisformedwithmultiplegoals,basedonthe continuousvariablesofthedecisionwithadditionalconstraints.
So,alltheproblemsoftodayare,ingeneral,multi-criterial,primarilybecauseproblemsaremainlyrelatedtotheachievementoftheobjectives relatedtoalargernumberof,usuallyconflicting,criteria,whichisagreatapproximationtorealtasksin decision-makingprocesses(Das etal.,2012;Zavadskas etal.,2014).TheincreasingapplicationoftheMCDMmethodtosolvingvariousproblemshas causedanexceptional growthofmulti-criteriadecision-makingasanimportantfieldofoperationalresearch, especiallysince1980(Aouni etal.,2018;Masri, etal.,2018;Wallenius etal.,2008; Dyer etal.,1992).
WithinMADM,someofthemethodsthathavebeenproposedare: WeightedSum Model(WSM)(Fishburn,1967);SimpleAdditiveWeighting(SAW)method(MacCrimon,1968),EliminationEtChoixTraduisantlaREalité(ELECTRE)method(Roy, 1968),DEcision-MAkingTrialandEvaluationLaboratory(DEMATEL)method(Gabus andFontela,1972),CompromiseProgramming(CP)method(Zeleny,1973),Simple MultiAttributeRatingTechnique(SMART)(Edwards,1977), AnalyticHierarchyProcess(AHP)method(Saaty,1978),TechniqueforOrderofPreferencebySimilarityto IdealSolution(TOPSIS)method(HwangandYoon,1981),PreferenceRankingOrganizationMethodforEnrichmentEvaluations(PROMETHEE)method(Brans,1982),MeasuringAttractivenessbyaCategoricalBasedEvaluationTechnique(MACBETH)(Banae CostaandVansnick,1994),ComplexProportionalAssessmentofalternatives(COPRAS) method(Zavadskas etal.,1994),AnalyticNetworkProcess(ANP)method(Saaty,1996), VlseKriterijumskaOptimizacijaikompromisnoResenje(VIKOR)(Opricovic,1998), Multi-ObjectiveOptimizationonbasisofRatioAnalysis(MOORA)method(Brauersand Zavadskas,2006),AdditiveRatioASsessment(ARAS)method (ZavadskasandTurskis,
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ABipolarFuzzyExtensionoftheMULTIMOORAMethod 137 2010),Multi-ObjectiveOptimizationbyRatioAnalysisplustheFullMultiplicativeForm (MULTIMOORA)method(BrauersandZavadskas,2010a),andso on.Whilewithin theMODMmethodsthathavebeenproposedcanbestated:Dataenvelopmentanalysis (DEA)method(Charnes etal.,1978),LinearProgramming(LP)andNonlinearProgramming(NP)(LuenbergerandYe,1984),Multi-ObjectiveProgramming(MOP)technique (Charnes etal.,1989),Multi-ObjectiveLinearProgramming(EckerandKouada,1978), andsoon.
TheMULTIMOORAmethod(BrauersandZavadskas,2010b)isanimportantMCDM methodthathasbeenappliedsofartosolvethemostdiverseproblemsinthefieldofeconomics,management,etc.Basically,theMULTIMOORAmethod consistsofthewellknownMOORAmethod(BrauersandZavadskas,2006)andthemethodofmulti-object optimization(theFullMultiplicativeFormofMultipleObjectsmethod).Thus,Brauers andZavadskas(2010a)proposedtheupdatingoftheMOORAmethodbyaddingamultiobjectoptimizationmethodwhichinvolvesmaximizingandminimizingusefulmultiplicativefunctions(Lazauskas etal.,2015).
Asnotedabove,theMULTIMOORAmethodwasappliedinorderto solveavarietyof problems,suchas:usingMULTIMOORAforrankingandselectingthebestperformance appraisalmethod(Maghsoodi etal.,2018),projectcriticalpathselection(Dorfeshan etal., 2018),theselectionoftheoptimalminingmethod(Liang etal.,2018),pharmacological therapyselection(Eghbali-Zarch etal.,2018),ICThardwareselection(AdaliandIşik, 2017),industrialrobotselection(Karande etal.,2016),aCNCmachinetoolevaluation (Sahu etal.,2016),personnelselection(Karabasevic etal.,2015;Baležentis etal.,2012), theeconomy(BaležentisandZeng,2013;BrauersandZavadskas,2011a,2010b;Brauers andGinevičius,2010),andsoon.
However,mostdecisionsmadeintherealworldaremadeinanenvironmentinwhich goalsandconstraintscannotbepreciselyexpressedduetotheircomplexity;therefore,a problemcannotbedisplayedexactlyincrispnumbers(BellmanandZadeh,1970).For suchproblems,characterizedbyuncertaintyandindeterminacy,itismoreappropriateto usevaluesexpressedinintervalsinsteadofconcrete(crisp)values.Inthiscase,theexisting,ordinaryMCDMmethodsareexpandedbyusingtheextensionsbasedonfuzzysets (Zadeh,1965),intuitionisticfuzzysets(Atanassov,1986),andneutrosophicsets(Smarandache,1999).Accordingly,inordertoallowamuchwideruse oftheMULTIMOORA method,someextensionsoftheMULTIMOORAmethodhavebeenproposed,someof whichareasfollows:Brauers etal. (2011)proposedafuzzyextensionoftheMULTIMOORAmethod;BaležentisandZeng(2013)proposedanIVFNextensionoftheMULTIMOORAmethod;Baležentis etal. (2014)alsoproposedanIFNextensionoftheMULTIMOORAmethod;Stanujkic etal. (2015)proposedanextensionoftheMULTIMOORA methodbasedontheuseofinterval-valuedtriangularfuzzy numbers;Zavadskas etal. (2015)proposedanIVIF-basedextensionoftheMULTIMOORAmethod;Hafezalkotob etal. (2016)proposedanextensionoftheMULTIMOORAmethodbased ontheuseof intervalnumbers;Stanujkic etal. (2017a)proposedaneutrosophicextensionoftheMULTIMOORAmethod,andsoon.
Inadditiontotheaforementionedextensionsofthefuzzysettheory,Zhang(1994) introducedtheconceptofbipolarfuzzysetsandproposedtheusageofthetwomembership
D.Stanujkicetal.
functionsthatrepresentmembershiptoasetandmembership toacomplementaryset,thus providinganefficientapproachtosolvingawidelylargernumberofcomplexdecisionmakingproblems.
Despiteanadvantagethatcanbeachievedbyusingbipolarfuzzylogic,theyaresignificantlylessusedforsolvingMCDMproblemscomparedtootherfuzzylogicextensions. Thefollowingexamplescanbementionedassomeofthereally rareusagesofBFSfor solvingMCDMproblems:Alghamdi etal. (2018)andAkramandArshad(2018)proposedbipolarfuzzyextensionsofTOPSISandELECTREImethods;whileHan etal. (2018)provideacomprehensivemathematicalapproachbasedontheTOPSISmethod forimprovingtheaccuracyofbipolardisorderclinicaldiagnosis.
Itisalsoimportanttonotethatthesearecurrentresearches.Inaddition,thebipolarlogichasbeenconsiderablyusedintheneutrosophicset theory,whereUluçay etal. (2018),Pramanik etal. (018)andTian etal. (2016)canbecitedassomeofthecurrent researches.
Therefore,inordertoenableawideruseoftheMULTIMOORAmethodforsolving evenawiderrangeofproblems,abipolarextensionoftheMULTIMOORAmethodis proposedinthispaper.Accordingly,thepaperisstructuredasfollows:inSection1,the introductoryconsiderationsarepresented.InSection2,somebasicdefinitionsregardingbipolarfuzzysetsaregiven.InSection3,theordinaryMULTIMOORAmethodis presented,whereasinSection4,anextensionoftheMULTIMOORAmethodbasedon single-valuedbipolarfuzzynumbersisproposed.InSection5,anumericalexampleis demonstrated,andfinally,theconclusionsaregivenattheendofthepaper.
2.TheBasicElementsofaBipolarFuzzySet
Definition 1(Fuzzyset,Zadeh,1965).Let X beanonemptyset,withagenericelement in X denotedby x.Then,afuzzyset A in X isasetoforderedpairs:
A = x,µA(x) x ∈ X , (1)
wherethemembershipfunction µA(x) denotesthedegreeofthemembershipoftheelement x totheset A,and µA(x) ∈[0, 1]
Definition 2(Bipolarfuzzyset,Lee,2000).Let X beanonemptyset.Then,abipolar fuzzyset(BFS)isdefinedas:
A = x,µ+ A (x),νA (x) x ∈ X , (2) where:thepositivemembershipfunction µ+ A (x) denotesthesatisfactiondegreeofthe element x tothepropertycorrespondingtothebipolar-valuedfuzzyset,andthenegativemembershipfunction νA (x) denotesthedegreeofthesatisfactionoftheelement x tosomeimplicitcounter-propertycorrespondingtothebipolar-valuedfuzzyset,respectively; µ+ A (x) : X →[0, 1] and νA (x) : X →[−1, 0]
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ABipolarFuzzyExtensionoftheMULTIMOORAMethod 139
Definition 3.Asingle-valuedbipolarfuzzynumber(SVBFN) a = a+,a isaspecialbipolarfuzzysetontherealnumberset R,whosepositivemembershipandnegative membershipfunctionareasfollows:
µ+(x) = 1 x = a+ , 0 otherwise, (3)
ν (x) = 1 x = a , 0 otherwise, (4) respectively.
Definition 4.Let a1 = a+ 1 ,a1 and a1 = a+ 2 ,a2 betwoSVBFNs,and λ> 0.Then, thebasicoperationsforthesenumbersaredefinedasshownbelow:
a1 + a2 = a+ 1 + a+ 2 a+ 1 a+ 2 , a1 a2 , (5) a1 · a2 = a+ 1 a+ 2 , a1 a2 a1 a2 , (6)
λa1 = 1 1 a+ 1 λ , a1 λ , (7) a λ 1 = a+ 1 λ , 1 1 a1 λ . (8)
Definition 5.Let a = a+,a beanSVBFN.Then,thescorefunction s(a) isasfollows:
sa = 1 + a+ + a 2 (9)
Definition 6.Let a1 and a2 betwoSVBFNs.Then, a1 >a2 if sa1 >sa2
Definition 7.Let a1 = a+ 1 ,a1 and a1 = a+ 2 ,a2 betwoSVBFNs.TheHamming distancebetween a1 and a2 isasfollows: dH (a1,a2) = 1 2 a+ 1 a+ 2 + a1 a2 (10)
Definition 8.Let aj = a+ j ,aj beacollectionofSVBFNs.Thebipolarweightedaverageoperator(Aw )ofthe n dimensionsisamappingasfollows:
Aw (a1,a2,...,an) = n j =1 wj aj = 1 n j =1 1 a+ j wj , 1 n j =1 1 aj wj , (11)
where: wj istheelement j oftheweightingvector, wj ∈[0, 1] and n j =1 wj = 1
140 D.Stanujkicetal. Definition 9.Let aj = a+ j ,aj beacollectionofSVBFNs.Thebipolarweightedgeometricoperator(Gw )ofthe n dimensionsisamapping Gw : Qn → Q asfollows: Gw (a1,a2,...,an) = n j =1 a wj j = n j =1 a+ j wj , n j =1 aj wj (12)
3.TheMULTIMOORAMethod
ComparedtotheotherMCDMmethods,theMULTIMOORAmethodis characteristic becauseitcombinesthreeapproaches,namely:theRatioSystem(RS)Approach,theReferencePoint(RP)ApproachandtheFullMultiplicativeForm (FMF)Approach,inorder toselectthemostappropriatealternative.
Inaddition,thismethoddoesnotcalculateanddoesnotusetheoverallsignificance forrankingalternativesandselectingthemostappropriateone.Insteadofusinganoverall parameterforrankingalternatives,thefinalrankingorder oftheconsideredalternatives, aswellastheselectionofthemostappropriatealternative,isbasedontheuseofthetheory ofdominance.
ForanMCDMproblemthatincludesthemalternativesthatshouldbeevaluatedon thebasisofthencriteria,thecomputationalprocedureoftheMULTIMOORAcanbe expressedasfollows:
Step1. Constructadecisionmatrixanddeterminetheweightsofcriteria.
Step2. Calculateanormalizeddecisionmatrix,asfollows: rij = xij n i=1 x 2 ij , (13)
where: rij denotesthenormalizedperformanceofthealternative i withrespecttothe criterion j ,and xij denotestheperformanceofthealternative i tothecriterion j
Step3. Calculatetheoverallsignificanceofeachalternative,asfollows: yi = j ∈ max
wj rij j ∈ min
wj rij , (14)
where: yi denotestheoverallimportanceofthealternative i, max and min denotethe setsofthebenefitcostcriteria,respectively.
Step4. Determinethereferencepoint,asfollows: r ∗ = r ∗ 1 ,r ∗ 2 ,...,r ∗ n = max i rij j ∈ max , min i rij j ∈ min . (15)
ABipolarFuzzyExtensionoftheMULTIMOORAMethod 141
Step5. Determinethemaximaldistancebetweeneachalternativeandthereferencepoint, asfollows: d max i = max j wj r ∗ j rij , (16) where: d max i denotesthemaximaldistanceofthealternative i tothereferencepoint. Step6. Determinetheoverallutilityofeachalternative,asfollows: ui = j ∈ max wj rij j ∈ min wj rij , (17) where: ui denotestheoverallutilityofthealternative i Inparticularcase,whenevaluationismadeonlyonthebasis ofbenefitcriteria,Eq.(17) isasfollows: ui = j ∈ max
wj rij (18) Step7. Determinethefinalrankingorderoftheconsideredalternativesandselectthe mostappropriateone.Inthisstep,theconsideredalternativesarerankedbased ontheir: –overallsignificance, –maximaldistancetothereferencepoint,and –overallutility.
Asaresultoftheserankings,thethreedifferentrankinglistsareformed,representing therankingsbasedontheRSapproach,theRPapproachandthe FMFapproachofthe MULTIMOORAmethod.
Thefinalrankingofthealternativesisbasedonthedominancetheory,i.e.thealternativewiththehighestnumberofappearancesinthefirstpositionsonallrankinglistsis thebest-rankedalternative.
4.AnExtensionoftheMULTIMOORAMethodBasedonSingle-ValuedBipolar FuzzyNumbers
ForanMCDMprobleminvolvingmalternativesandncriteriaandKdecision-makers, wherebytheperformancesofthealternativesareexpressed byusingSVBFNs,thecalculationprocedureoftheextendedMULTIMOORAmethodcanbeexpressedasfollows: Step1. Evaluatethealternativesinrelationtotheevaluationcriteria,anddothatforeach DM.Inthisstep,eachDMevaluatesthealternativesandformsanevaluationmatrix. Inordertoprovideaneasierevaluation,thefollowingLikertscale,showninTable1, isproposedforevaluatingalternativesinrelationtotheevaluationcriteria.
D.Stanujkicetal.
Table1 Nine-pointLikertscaleforexpressingdegreeofsatisfaction.
SatisfactionlevelNumericalvalue Neutral/withoutattitude0 Extremelylow1 Verylow2 Low3 Mediumlow4 Medium5 Mediumhigh6 High7 Veryhigh8 Extremelyhigh9 Absolute10
However,therespondentsshouldbeintroducedthatthevalueslistedinTable1are onlyapproximativeandthattheycanuseanyvaluefromtheinterval [0, 10] and [−10, 0].
Afterforminginitialdecision-makingmatrix,obtainedresponsesshouldbedividedby 10inordertotransformitintotheallowedinterval [−1, 1].Thisapproachforevaluating alternativesisproposedtoavoidtheuseofvectornormalizationprocedure,usedinthe ordinaryMULTIMOORAmethod.
Step2. Determinetheimportanceoftheevaluationcriteria,anddothatforeachDM.In thisstep,eachDMdeterminestheweightsofthecriteriabyusingoneofseveralexisting methodsfordeterminingtheweightsofcriteria.
Step3. Determinethegroupdecisionmatrix.Inordertotransformindividualintogroup preferences,individualevaluationmatricesaretransformedintogrouponebyapplying Eq.(11).
Step4. Determinethegroupweightsofthecriteria.Inordertotransformindividualinto grouppreferenceswithrespecttotheweightsofcriteria,thegroupweightsofcriteriacan bedeterminedasfollows:
(19)
where: wj denotestheweightofthecriterion j ,and wk j denotestheweightofthecriterion j obtainedfromtheDM k.
Aftercalculatingthegroupevaluationmatrixandthegroup weightsofthecriteria, allthenecessaryprerequisitesforapplyingallthethreeapproachesintegratedinthe MULTIMOORAmethodareobtained.BasedontheapproachproposedbyStanujkic et al. (2017b),theremainderstepsoftheproposedapproachareas follows:
Step5. DeterminethesignificanceoftheevaluatedalternativesbasedontheRSapproach Thisstepcanbeexplainedthroughthefollowingsub-steps:
142
w
=
=
w
j
K k
1
k j
ABipolarFuzzyExtensionoftheMULTIMOORAMethod 143
Step5.1. Determinetheimpactofthebenefitandcostcriteriatotheimportanceofeach alternative,asfollows:
Y + i = 1 n j ∈ max
(1 rij )wj , 1 n j ∈ max
1 ( rij ) wj , (20) Yi = 1 n j ∈ min
(1 rij )wj , 1 n j ∈ min
1 ( rij ) wj , (21)
where: Y + i and Yi denotetheimportanceofthealternative i obtainedonthebasisofthe benefitandcostcriteria,respectively; Y + i and Yi areSVBFNs.
Itisevidentthat Aw operatorisusedtocalculatetheimpactofthebenefitandcost criteria.
Step.5.2. Transform Y + i and Yi intocrispvaluesbyusingtheScoreFunction,asfollows: y+ i = s Y + i , (22) yi = s Yi (23)
Step5.3. Calculatetheoverallimportanceforeachalternative,asfollows: yi = y+ i yi (24)
Step6. DeterminethesignificanceoftheevaluatedalternativesbasedontheRPapproach Thisstepcanbeexplainedthroughthefollowingsub-steps:
Step6.1. Determinethereferencepoint.Thecoordinatesonthebipolarfuzzyreference point r ∗ ={r ∗ 1 ,r ∗ 2 ,...,r ∗ n } canbedeterminedasfollows: r ∗ = max i rij , min i rij j ∈ max , min i rij , max i rij j ∈ min (25) where: r ∗ j denotesthecoordinate j ofthereferencepoint.
Step6.2. Determinethemaximumdistancefromeachalternativetoall thecoordinatesof thereferencepoint.Themaximumdistanceofeachalternativetothereferencepointcan bedeterminedasfollows: d max ij = dmax rij ,r ∗ j wj , (26) where d max ij denotesthemaximumdistanceofthealternative i tothecriterion j determinedbyEq.(10).
Step6.3. Determinethemaximumdistanceofeachalternative,asfollows: d max i = max j d max ij , (27) where d max i denotesthemaximumdistanceofthealternative i
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Step7. Determinethesignificanceoftheevaluatedalternativesbasedon theFMF. Thisstepcanbeexplainedthroughthefollowingsub-steps:
Step7.1. Calculatetheutilityobtainedbasedonthebenefit U + i andcost Ui criteria,for eachalternative,asfollows: U + i = n j ∈ max
(rij )wj , n j ∈ max
( rij )wj , (28) Ui = n j ∈ min
(rij )wj , n j ∈ min
where U + i and Ui areSVBFNs.
( rij )wj , (29)
Step7.2. Transform U + i and Ui intocrispvaluesbyusingtheScoreFunction,asfollows: u+ i = s U + i , (30) ui = s Ui . (31)
Step7.3. Determinetheoverallutilityforeachalternative,asfollows: ui = u+ i ui . (32)
Inthecasewhenevaluationismadeonlyonthebasisofbenefit criteria,Eq.(32)isas follows: ui = u+ i . (33)
Step8. Determinethefinalrankingorderofthealternatives.Thefinalrankingorderof thealternativescanbedeterminedasinthecaseoftheordinaryMULTIMOORAmethod, i.e.basedonthedominancetheory(BrauersandZavadskas,2011b).
Inthisstage,thealternativesarerankedbasedontheiroverallimportance,maximum distancetothereferencepointandoverallutility.Asaresultofthat,threerankinglistsare formed.
Basedontheserankinglists,thefinalrankinglistofthealternativesisformedonthe basisofthetheoryofdominance,i.e.thealternativewiththelargestnumberofappearancesonthefirstpositioninthethreerankinglistsisthemostacceptable.
5.ANumericalExample
Inthissection,anumericalexampleofpurchasingrentalspaceisconsideredinorderto explaintheproposedapproachindetail.
ABipolarFuzzyExtensionoftheMULTIMOORAMethod 145
Table2
TheratingsobtainedfromthefirstofthethreeDMs.
C1 C2 C3 C4 C5 a+ a a+ a a+ a a+ a a+ a
A1 7 27 35 17 58 1
A2 4 15 24 24 67 1
A3 7 13 1202 17 2 A4 9 14 1303 16 1
Table3
TheratingsobtainedfromthefirstofthethreeDMs,intheformofSVBFNs.
C1 C2 C3 C4 C5
A1 0 70, 0 20 0 70, 0 30 0 50, 0 10 0 70, 0 50 0 80, 0 10
A2 0 40, 0 10 0 50, 0 20 0 40, 0 20 0 40, 0 60 0 70, 0 10
A3 0 70, 0 10 0 30, 0 10 0 20, 0 00 0 20, 0 10 0 70, 0 20
A4 0 90, 0 10 0 40, 0 10 0 30, 0 00 0 30, 0 10 0 60, 0 10
Table4
TheratingsobtainedfromthesecondofthethreeDMs,intheformofSVBFNs.
C1 C2 C3 C4 C5
A1 0 70, 0 20 0 70, 0 50 0 40, 0 20 0 70, 0 50 0 80, 0 10 A2 0 60, 0 10 0 40, 0 60 0 40, 0 20 0 40, 0 60 0 80, 0 10
A3 0.80, 0.10 0.20, 0.10 0.20, 0.10 0.20, 0.10 0.70, 0.10
A4 0 90, 0 10 0 30, 0 10 0 30, 0 10 0 30, 0 10 0 60, 0 10
Supposethatacompanyisplanningtostartitssalesbusinessinanewlocation,and thereforeislookingforanewsalesbuilding.Aftertheinitialconsiderationoftheavailable alternatives,fouralternativeshavebeenidentifiedassuitable.Forthisreason,ateamof threedecision-makers(DMs)wasformedwiththeaimofevaluatingsuitablealternatives basedonthefollowingcriteria: C1 –Rentalspacequality; C2 –Rentalspaceadequacy; C3 –Locationquality; C4 –Locationdistancefromthecitycentre,and C5 –Rentalprice.
Aspreviouslyreasoned,inthisevaluationtheratingsofthealternativesinrelationto thecriteriaareexpressedbyusingSVBFNs.
TheratingsobtainedfromthefirstofthethreeDMsareshowninTable2,asthepoints oftheLikertscale,whereasinTable3,theyareshownintheformofSVBFNs.
TheratingsobtainedfromthesecondandthethirdofthethreeDMsareaccountedfor inTable4andTable5.
Thegroupdecisionmatrix,calculatedbyapplyingEq.(11), ispresentedinTable6.
D.Stanujkicetal.
Table5
TheratingsobtainedfromthethirdofthethreeDMs,intheformofSVBFNs.
C1 C2 C3 C4 C5
A1 0 60, 0 10 0 90, 0 20 1 00, 0 00 1 00, 0 00 0 80, 0 10
A2 0 40, 0 60 0 40, 0 60 1 00, 0 40 1 00, 0 00 0 80, 0 10
A3 0 20, 0 10 0 90, 0 40 0 80, 0 30 0 70, 0 10 0 70, 0 10
A4 0 30, 0 10 1 00, 0 30 0 80, 0 20 0 80, 0 10 0 60, 0 10
Table6
Thegroupdecision-makingmatrix.
C1 C2 C3 C4 C5
A1 0 67, 0 16 0 79, 0 32 1 00, 0 00 1 00, 0 00 0 80, 0 10
A2 0 47, 0 18 0 43, 0 42 1 00, 0 26 1 00, 0 00 0 77, 0 10
A3 0 64, 0 10 0 61, 0 16 0 49, 0 00 0 41, 0 10 0 70, 0 13
A4 0 81, 0 10 1 00, 0 15 0 53, 0 00 0 53, 0 10 0 60, 0 10
Table7
Theweightsofthecriteriaobtainedfromthefirstofthethree DMs.
sj kj qj wj
C1 110.19 C2 1.20.801.250.23 C3 0.91.101.140.21 C4 0.71.300.870.16 C5 1.20.801.090.20 5.005.351.00
Table8
Thegroupcriteriaweights.
w1 j w2 j w3 j wj C1 0.190.170.190.18 C2 0.230.240.230.24 C3 0.210.220.210.21 C4 0.160.170.160.16 C5 0.200.210.200.21 1.00
TheweightsobtainedfromthefirstofthethreeDMsbyapplyingthePIPRECIA method(Stanujkic etal.,2017b)areaccountedforinTable7,whilethegroupweights ofthecriteria,calculatedbyapplyingEq.(19),areshowninTable8.
OnthebasisoftheratingsfromTable6andtheweightsfromTable8,theoverallsignificance,themaximumdistancetothereferencepointandtheoverallutilityarecalculated foreachalternativeinthenextstep.
Theoverallsignificances,accountedforinTable9,arecalculatedbyapplying Eqs.(20)–(24).
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Table9
Theoverallsignificancesoftheconsideredalternatives.
Y + i Yi y+ i yi yi Rank
A1 1 00, 0 11 1 00, 0 02 0.940.99-0.053
A2 1 00, 0 20 1 00, 0 02 0.900.99-0.094 A3 0 42, 0 06 0 30, 0 05 0.680.630.052
A4 1 00, 0 06 0 29, 0 04 0.970.620.351
Table10 Thereferencepoints.
max min r+ r r+ r r∗ 1.00 0.200.29 0.02
Table11
Theratingsofthealternativesobtainedbasedonthereferencepointapproach.
C1 C2 C3 C4 C5 d max i Rank
A1 0.080.160.130.290.100.084 A2 0.170.280.000.290.090.001 A3 0.130.330.380.050.060.053 A4 0.040.140.360.110.000.001
Table12
Theoverallutilityoftheconsideredalternatives.
U + i Ui u+ i ui ui Rank
A1 1 00, 0 11 1 00, 0 02 0.940.99 0.053
A2 1 00, 0 20 1 00, 0 02 0.900.99 0.094 A3 0 42, 0 06 0 30, 0 05 0.680.630.052 A4 1 00, 0 06 0 29, 0 04 0.970.620.351
Afterthat,thereferencepointshowninTable10isdeterminedbyapplyingEq.(25).
ThemaximumdistancestothereferencepointaccountedforinTable11aredeterminedbyapplyingEq.(26)andEq.(27).
TheoverallutilityshowninTable12iscalculatedbyapplyingEqs.(28)–(32).
Finally,onthebasisoftherankingordersshowninTables9, 11and12,themost appropriatealternativeisdeterminedbymeansofthetheoryofdominance,asisshown inTable13.
AscanbeseenfromTable12,themostappropriatealternativeisthealternative denotedas A4
ABipolarFuzzyExtensionoftheMULTIMOORAMethod 147
D.Stanujkicetal.
Table13
Thefinalrankingorderoftheconsideredalternatives.
RSRPFMP Finalrank
A1 3433 A2 4144 A3 2322 A4 1111
6.Conclusions
Thebipolarfuzzysetsintroducedtwomembershipfunctions,namelythemembership functiontoasetandthemembershipfunctiontoacomplementaryset.
Ontheotherhand,theMULTIMOORAmethodisanefficientandalreadyproven multiple-criteriadecision-makingmethod,whichhasbeen usedforsolvinganumberof differentdecision-makingproblemssofar.
Therefore,anextensionoftheMULTIMOORAmethodenablingtheuseofsinglevaluedbipolarfuzzynumbersisproposedinthisarticle.Theusabilityandefficiencyof theproposedextensionissuccessfullydemonstratedonthe exampleoftheproblemofthe bestlocationselection.
Intheliterature,numerousextensionsoftheMULTIMOORAmethodshavebeenproposedwiththeaimtoadaptitfortheuseofgreysystemtheory,fuzzysettheory,aswell asvariousextensionsoffuzzysettheory.Someextensionsthatenabletheuseofneutrosophicsetsarealsoproposed.Thementionedextensionsaim toexploitthespecificities ofparticularsetsforsolvingcertaintypesofdecision-makingproblems,andthusenable moreefficientdecisionmaking.
Becauseofthespecificitythatbipolarfuzzysetsprovide,theproposedexpanded MULTIMOORAmethodcanbeexpectedtobeacceptableforsolvingaparticularclass ofcomplexdecision-makingproblems.
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D.Stanujkic isanassociateprofessorofinformationtechnologyattheTechnicalFaculty inBor,UniversityofBelgrade.HehasreceivedhisMScdegreeininformationscienceand PhDinorganizationalsciencesfromtheFacultyofOrganizationalSciences,University ofBelgrade.Hiscurrentresearchisfocusedondecision-makingtheory,expertsystems andintelligentdecisionsupportsystems.
D.Karabasevic isanassistantprofessorattheFacultyofAppliedManagement,EconomicsandFinance,UniversityBusinessAcademyinNoviSad.Heobtainedhisdegrees atallthelevelsofstudies(BScappl.ineconomics,BScineconomics,academicspecializationinthemanagementofbusinessinformationsystemsandPhDinmanagementand business)attheFacultyofManagementinZajecar,JohnNaisbittUniversity,Belgrade. Hiscurrentresearchisfocusedonthehumanresourcemanagement,managementand decision-makingtheory.
E.K.Zavadskas isaprofessoroftheDepartmentofConstructionManagement andReal Estate,directorofInstituteofSustainableConstruction,andChiefResearchFellowof LaboratoryofOperationalResearchatVilniusGediminasTechnicalUniversity,Vilnius, Lithuania.HehasaPhDinbuildingstructures(1973)andDrSc(1987)inbuildingtechnologyandmanagement.HeisamemberoftheLithuanianandseveralforeignAcademies ofSciences.HeisdoctorhonoriscausaatPoznan,SaintPetersburg,andKievuniversities.Heistheeditorinchiefandamemberofeditorialboardsofanumberofresearch journals.Heisanauthorandcoauthorofmorethan400papers andanumberofmonographs.Researchinterestsare:buildingtechnologyandmanagement,decision-making theory,automationindesignanddecisionsupportsystems.
D.Stanujkicetal.
F.Smarandache isaprofessorofmathematicsattheUniversityofNewMexico,USA. Hehaspublishedmanypapersandbooksonneutrosophicsetandlogicandtheirapplicationsandhaspresentedtomanyinternationalconferences. HegothisMScinmathematics andcomputersciencefromtheUniversityofCraiova,Romania,PhDfromtheStateUniversityofKishinev,andpost-doctoralinappliedmathematicsfromOkayamaUniversity ofSciences,Japan.
W.K.M.Brauers,doctorhonoriscausaVilniusGediminasTechnicalUniversity,was graduatedas:PhDineconomics(un.ofLeuven),MasterofArts(ineconomics)of ColumbiaUniversity(NewYork),masterineconomics,masterinmanagementandfinancialsciences,masterinpoliticalanddiplomaticsciencesandbachelorinphilosophy (allintheUniversityofLeuven).HeisprofessorordinariusattheFacultyofApplied EconomicsoftheUniversityofAntwerp,honoraryprofessor attheUniversityofLeuven, theBelgianWarCollege,theSchoolofMilitaryAdministratorsandtheAntwerpBusinessSchool.Hisscientificpublicationsconsistofeighteenbooksandseveralhundredsof articlesandreportsinEnglish,DutchandFrench.
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