VIKOR Method for Interval Neutrosophic Multiple Attribute Group Decision-Making

VIKORMethodforIntervalNeutrosophicMultiple AttributeGroupDecision-Making

Yu-HanHuang 1,*,Gui-WuWei 2,* ID andCunWei 3

1 CollegeofMathematicsandSoftwareScience,SichuanNormalUniversity,Chengdu610068,China

2 SchoolofBusiness,SichuanNormalUniversity,Chengdu610101,China

3 SchoolofScience,SouthwestPetroleumUniversity,Chengdu610500,China;weicun1990@163.com

* Correspondence:hyh85004267@163.com(Y.-H.H.);weiguiwu1973@sicnu.edu.cn(G.-W.W.)

Received:21October2017;Accepted:8November2017;Published:10November2017

Abstract: Inthispaper,wewillextendtheVIKOR(VIsekriterijumskaoptimizacijaiKOmpromisno Resenje)methodtomultipleattributegroupdecision-making(MAGDM)withintervalneutrosophic numbers(INNs).Firstly,thebasicconceptsofINNsarebrieflypresented.Themethodfirst aggregatesallindividualdecision-makers’assessmentinformationbasedonanintervalneutrosophic weightedaveraging(INWA)operator,andthenemploystheextendedclassicalVIKORmethod tosolveMAGDMproblemswithINNs.Thevalidityandstabilityofthismethodareverifiedby exampleanalysisandsensitivityanalysis,anditssuperiorityisillustratedbyacomparisonwiththe existingmethods.

Keywords: MAGDM;INNs;VIKORmethod

1.Introduction

Multipleattributegroupdecision-making(MAGDM),whichhasbeenincreasinglyinvestigated andconsideredbyallkindsofresearchersandscholars,isoneofthemostinfluentialpartsofdecision theory.Itaimstoprovideacomprehensivesolutionbyevaluatingandrankingalternativesbased onconflictingattributeswithrespecttodecision-makers’(DMs)preferences,andhaswidelybeen utilizedinengineering,economics,andmanagement.SeveraltraditionalMAGDMmethodshavebeen developedbyscholarsinliterature,suchastheTOPSIS(TechniqueforOrderPreferencebySimilarity toanIdealSolution)method[1,2],theVIKOR(VIsekriterijumskaoptimizacijaiKOmpromisno Resenje)method[3–5],thePROMETHEE(PreferenceRankingOrganizationMethodforEnrichment Evaluations)method[6],theELECTRE(ELiminationEtChoixTraduisantlaRealité)method[7],the GRA(GreyRelationalAnalysis)method[8–10],andtheMULTIMOORA(MultiobjectiveOptimization byRatioAnalysisplusFullMultiplicativeForm)method[11,12].

Duetothefuzzinessanduncertaintyofthealternativesindifferentattributes,attributevaluesin MAGDMarenotalwaysrepresentedasrealnumbers,andtheycanbedescribedasfuzzynumbers inmoresuitableoccasions[13–15].Sincefuzzyset(FS)wasfirstdefinedbyZadeh[16],ishasbeen usedasabettertooltosolveMAGDM[17,18].Smarandache[19,20]proposedaneutrosophicset(NS). Furthermore,theconceptsofsingle-valuedneutrosophicsets(SVNSs)[21]andintervalneutrosophic sets(INSs)[22]werepresentedforactualapplications.Ye[23]proposedasimplifiedneutrosophic set(SNS).BroumiandSmarandache[24]definedthecorrelationcoefficientofINS.Zhangetal.[25] gavethecorrelationcoefficientofintervalneutrosophicnumbers(INNs)inMAGDM.Zhangetal.[26] gaveanoutrankingapproachforINNMAGDM.Tianetal.[27]definedacross-entropyinINN MAGDM.Zhangetal.[28]proposedsomeINNaggregating.SomeotherINNoperatorsareproposed inReferences[29–32].Ye[33]proposedtwosimilaritymeasuresbetweenINNs.TheSVNSandINS havereceivedmoreandmoreattentionsincetheirappearance[34–42].

Information 2017, 8,144;doi:10.3390/info8040144 www.mdpi.com/journal/information

Opricovic[3]proposedtheVIKORmethodforaMAGDMproblemwithconflicting attributes [43–45].SomescholarsproposedfuzzyVIKORmodels[46],intuitionisticfuzzyVIKOR models[47–49],thelinguisticVIKORmethod[50],theintervaltype-2fuzzyVIKORmodel[51],the hesitantfuzzylinguisticVIKORmethod[52],thedualhesitantfuzzyVIKORmethod[53],thelinguistic intuitionisticfuzzy[54],andthesingle-valuedneutrosophicnumber(SVNN)VIKORmethod[38]. However,therehasnotyetbeenanacademicinvestigationoftheVIKORmethodforMAGDM problemswithINNs.Therefore,itisnecessarytopaygreatattentiontothisnovelandworthyresearch issue.ThepurposeofourpaperistousetheVIKORideatosolveMAGDMwithINNs,tofillthis vacancyofknowledge.InSection 2,wegivethedefinitionofINNs.WeproposetheVIKORmethod forINNMAGDM.InSection 3,anexampleisprovided,andthecomparativeanalysisisproposedin Section 4.WefinishwithourconclusionsinSection 5

2.Preliminaries

TheconceptsofSVNSsandINSsareintroduced.

SVNSsandINSs

NSs[19,20]arenoteasytoapplytorealapplications.Wangetal.[21]developedSNSs. Furthermore,Wangetal.[22]definedINSs.

Definition1[21]. LetXbeaspaceofpoints(objects),aSVNSsAinXischaracterizedasfollowing: A = {(x, ξ A (x), ψA (x), ζ A (x))|x ∈ X } (1)

wherethetruth-membershipfunction ξ A (x),indeterminacy-membership ψA (x) andfalsity-membershipfunction ζ A (x), ξ A (x) → [0,1], ψA (x) → [0,1] and ζ A (x) → [0,1] ,withthecondition 0 ≤ ξ A (x) + ψA (x) + ζ A (x) ≤ 3.

Definition2[22]. Let X beaspaceofpoints(objects)withagenericelementinfixedset X,denotedby x,where anINS AinXischaracterizedasfollows:

A = x, ξ A (x), ψA (x), ζ A (x) |x ∈ X (2) wheretruth-membershipfunction ξ A (x),indeterminacy-membership ψA (x),andfalsity-membershipfunction ζ A (x) areintervalvalues, ξ A (x) ⊆ [0,1], ψA (x) ⊆ [0,1] and ζ A (x) ⊆ [0,1],and 0 ≤ sup ξ A (x) + sup ψA (x) + sup ζ A (x) ≤ 3 AnINNcanbeexpressedas A = ξ A, ψA, ζ A = ξ L A, ξ R A , ψL A, ψR A , ζ L A, ζ R A ,where ξ L A, ξ R A ⊆ [0,1], ψL A, ψR A ⊆ [0,1], ζ L A, ζ R A ⊆ [0,1],and 0 ≤ ξ R A + ψR A + ζ R A ≤ 3 Definition3[45]. Let A = ξ L A, ξ R A , ψL A, ψR A , ζ L A, ζ R A beanINN,thenascorefunction,SF,is: SF A = 2 + ξ L A ψL A ζ L A + 2 + ξ R A ψR A ζ R A 6 , SF A ∈ [0,1] (3) Definition4[45]. Let A = ξ L A, ξ R A , ψL A, ψR A , ζ L A, ζ R A beanINN,thenanaccuracyfunction, AF A , isdefinedas: AF A = ξ L A + ξ R A ζ L A + ζ R A 2 , AF A ∈ [ 1,1] (4)

Definition5[45]. Let A = ξ L A, ξ R A , ψL A, ψR A , ζ L A, ζ R A and B = ξ L B, ξ R B , ψL B, ψR B , ζ L B, ζ R B betwoINNs, SF A = 2+ξ L A ψL A ζ L A + 2+ξ R A ψR A ζ R A 6 and SF B = 2+ξ L B ψL B ζ L B + 2+ξ R B ψR B ζ R B 6 be thescorefunctions,and AF A = ξ L A +ξ R A ζ L A +ζ R A 2 and AF B = ξ L B +ξ R B ζ L B +ζ R B 2 betheaccuracy functions,thenif SF A < SF B ,then A < B;if SF A = SF B ,then(1)if AF A = AF B ,then A = B;(2)ifAF A < AF B ,then A < B. Definition6[22,33]. Let A = ξ L A, ξ R A , ψL A, ψR A , ζ L A, ζ R A and B = ξ L B, ξ R B , ψL B, ψR B , ζ L B, ζ R B betwoINNs,then: (1) A ⊕ B = ξ L A + ξ L B ξ L A ξ L B, ξ R A + ξ R B ξ R A ξ R B , ψL A ψL B, ψR A ψR B , ζ L A ζ L B, ζ R A ζ R B ; (2) A ⊗ B = ξ L A ξ L B, ξ R A ξ R B , ψL A + ψL B ψL A ψL B, ψR A + ψR B ψR A ψR B , ζ L A + ζ L B ζ L A ζ L B, ζ R A + ζ R B ζ R A ζ R B ; (3) λ A = 1 1 ξ L A λ ,1 1 ξ R A λ , ψL A λ , ψR A λ , ζ L A λ , ζ R A λ , λ > 0; (4) A λ = ξ L A λ , ξ R A λ , ψL A λ , ψR A λ , 1 1 ζ L A λ ,1 1 ζ R A λ , λ > 0.

Definition7[45]. Let A and B betwoINNs,thenthenormalizedHammingdistancebetween A and B is definedasfollows: d A, B = 1 6 ξ L A ξ L B + ξ R A ξ R B + ψL A I L B + ψR A ψR B + ζ L A ζ L B + ζ R A ζ R B (5)

3.VIKORMethodforINNMAGDMProblems

Let φ = {φ1, φ2, , φm } bealternativesand ϕ = { ϕ1, ϕ2, , ϕn } beattributes.Let τ = (τ1, τ2, ··· , τn ) betheweightof ϕj, 0 ≤ τj ≤ 1, n ∑ j=1 τj = 1.Let D = {D1, D2, ··· , Dt } betheset ofDMs, σ = (σ1, σ2, , σt ) betheweightingofDMs,with 0 ≤ σk ≤ 1, t ∑ k=1 σk = 1.Suppose that Rk = r(k) ij m×n = ξ L(k) ij , ξ R(k) ij , ψL(k) ij , ψR(k) ij , ζ L(k) ij , ζ R(k) ij m×n istheINNdecisionmatrix ξ L(k) ij , ξ R(k) ij ⊆ [0,1], ψL(k) ij , ψR(k) ij ⊆ [0,1], ζ L(k) ij , ζ R(k) ij ⊆ [0,1], 0 ≤ ξ R(k) ij + ψR(k) ij + ζ R(k) ij ≤ 3, i = 1,2, , m, j = 1,2, , n, k = 1,2, , t. TocopewiththeMAGDMwithINNs,wedeveloptheINNVIKORmodel.

Step1. Utilizethe Rk andtheintervalneutrosophicnumberweightedaveraging (INNWA)operator rij = ξ L ij, ξ R ij , ψL ij, ψR ij , ζ L ij, ζ R ij = INNWAσ r(1) ij , r(2) ij , ··· , r(t) ij i = 1,2, ··· , m, j = 1,2, ··· , n (6) toget R = rij m×n . Step2. Definethepositiveidealsolutions R+ andnegativeidealsolutions R R+ = ξ L+ j , ξ R+ j , ψL+ j , ψR+ j , ζ L+ j , ζ R+ j (7) R = ξ L j , ξ R j , ψL j , ψR j , ζ L j , ζ R j (8)

Forthebenefitattribute:

ξ L+ j , ξ R+ j , ψL+ j , ψR+ j , ζ L+ j , ζ R+ j

= max i ξ L ij,max i ξ R ij , min i ψL ij,min i ψR ij , min i ζ L ij,min i ζ R ij (9)

ξ L j , ξ R j , ψL j , ψR j , ζ L j , ζ R j

= min i ξ L ij,min i ξ R ij , max i ψL ij,max i ψR ij , max i ζ L ij,max i ζ R ij (10)

Forthecostattribute:

ξ L+ j , ξ R+ j , ψL+ j , ψR+ j , ζ L+ j , ζ R+ j

= min i ξ L ij,min i ξ R ij , max i ψL ij,max i ψR ij , max i ζ L ij,max i ζ R ij (11)

ξ L j , ξ R j , ψL j , ψR j , ζ L j , ζ R j

= max i ξ L ij,max i ξ R ij , min i ψL ij,min i ψR ij , min i ζ L ij,min i ζ R ij (12)

      

      

 τ





 (14) where τj isweightof ϕj. Step4. Computethe Θi bythefollowingformula: Θi = θ (Γi Γ∗ i ) Γi Γ∗ i + (1 θ) (Zi Z∗ i ) Zi Z∗ i (15) where Γ∗ i = min i Γi, Γi = max i Γi (16) Z∗ i = min i Zi,Zi = max i Γi (17)

where θ depictsthedecision-makingmechanismcoefficient.If θ > 0.5,itisfor“themaximumgroup utility”;If θ < 0.5,itis“theminimumregret”;anditisbothif θ = 0.5.

4.NumericalExample

4.1.NumericalExample

Inthissection,anumericalexampleisgivenwithINNs.Fivepossibleemergingtechnology enterprises(ETEs) φi (i = 1,2,3,4,5) areselected.Fourattributesareselectedtoevaluatethefive possibleETEs: 1 ϕ1 istheemploymentcreation; 2 ϕ2 isthedevelopmentofscienceandtechnology; 3 ϕ3 isthetechnicaladvancement; 4 ϕ4 istheindustrializationinfrastructure.ThefiveETEsaretobe evaluatedbyusingINNsundertheattributes(τ = (0.2,0.1,0.3,0.4)T )bytheDMs(σ = (0.2,0.5,0.3)T ), aslistedinTables 1–3.

Table1. Thedecisionmatrix R1 ϕ1 ϕ2

φ1 ([0.3,0.4],[0.6,0.7],[0.3,0.5])([0.4,0.5],[0.2,0.3],[0.1,0.2]) φ2 ([0.5,0.7],[0.6,0.8],[0.2,0.4])([0.5,0.6],[0.3,0.5],[0.2,0.3])

φ3 ([0.4,0.5],[0.5,0.6],[0.2,0.3])([0.3,0.4],[0.5,0.6],[0.1,0.2])

φ4 ([0.6,0.7],[0.2,0.3],[0.1,0.2])([0.4,0.5],[0.1,0.2],[0.2,0.3])

φ5 ([0.4,0.5],[0.2,0.3],[0.2,0.3])([0.2,0.3],[0.6,0.7],[0.2,0.3]) ϕ3 ϕ4

φ1 ([0.1,0.2],[0.4,0.5],[0.1,0.2])([0.3,0.4],[0.5,0.6],[0.2,0.3])

φ2 ([0.5,0.7],[0.4,0.6],[0.2,0.3])([0.6,0.7],[0.3,0.4],[0.2,0.3]) φ3 ([0.3,0.4],[0.1,0.2],[0.2,0.3])([0.4,0.5],[0.1,0.2],[0.3,0.4]) φ4 ([0.4,0.5],[0.2,0.3],[0.1,0.2])([0.3,0.4],[0.4,0.5],[0.2,0.3])

φ5 ([0.5,0.6],[0.4,0.5],[0.2,0.3])([0.3,0.4],[0.6,0.7],[0.3,0.4])

Table2. Thedecisionmatrix R2 ϕ1 ϕ2

φ1 ([0.4,0.6],[0.5,0.7],[0.3,0.4])([0.6,0.7],[0.5,0.6],[0.5,0.6]) φ2 ([0.6,0.9],[0.4,0.5],[0.3,0.4])([0.7,0.8],[0.6,0.7],[0.4,0.5]) φ3 ([0.8,0.9],[0.8,0.9],[0.4,0.5])([0.7,0.8],[0.5,0.6],[0.5,0.6]) φ4 ([0.6,0.7],[0.3,0.4],[0.5,0.6])([0.8,0.9],[0.5,0.6],[0.6,0.7]) φ5 ([0.4,0.5],[0.6,0.7],[0.6,0.7])([0.6,0.7],[0.3,0.4],[0.3,0.4]) ϕ3 ϕ4

φ1 ([0.5,0.6],[0.4,0.5],[0.3,0.4])([0.6,0.7],[0.4,0.5],[0.3,0.4]) φ2 ([0.7,0.8],[0.3,0.4],[0.3,0.4])([0.8,0.9],[0.4,0.5],[0.3,0.4]) φ3 ([0.7,0.8],[0.1,0.2],[0.3,0.4])([0.8,0.9],[0.5,0.6],[0.2,0.3]) φ4 ([0.5,0.6],[0.2,0.3],[0.4,0.5])([0.5,0.6],[0.7,0.9],[0.3,0.4]) φ5 ([0.9,1.0],[0.4,0.5],[0.3,0.4])([0.7,0.8],[0.8,0.9],[0.1,0.2])

Table3. Thedecisionmatrix R3 ϕ1 ϕ2 φ1 ([0.7,0.8],[0.4,0.5],[0.4,0.5])([0.7,0.8],[0.3,0.4],[0.6,0.7]) φ2 ([0.6,0.7],[0.5,0.6],[0.4,0.5])([0.7,0.8],[0.6,0.7],[0.5,0.6]) φ3 ([0.7,0.8],[0.3,0.4],[0.5,0.6])([0.8,0.9],[0.2,0.4],[0.6,0.7]) φ4 ([0.7,0.8],[0.4,0.5],[0.6,0.7])([0.6,0.9],[0.1,0.2],[0.7,0.8]) φ5 ([0.6,0.7],[0.7,0.8],[0.2,0.3])([0.7,0.8],[0.3,0.5],[0.4,0.5]) ϕ3 ϕ4

φ1

([0.6,0.7],[0.3,0.4],[0.4,0.5])([0.5,0.6],[0.4,0.5],[0.4,0.5])

φ2 ([0.8,0.9],[0.2,0.3],[0.7,0.8])([0.6,0.7],[0.3,0.4],[0.4,0.6])

φ3 ([0.8,0.9],[0.2,0.4],[0.4,0.5])([0.9,1.0],[0.1,0.2],[0.5,0.6])

φ4 ([0.6,0.7],[0.1,0.2],[0.5,0.6])([0.6,0.7],[0.3,0.4],[0.4,0.5])

φ5 ([0.7,0.9],[0.3,0.4],[0.40.5])([0.8,0.9],[0.5,0.6],[0.5,0.6])

Then,weusetheproposedmodeltoselectthebestETE.

Step1. Utilize Rk (k = 1,2,3) andtheINNWAoperator,inordertoobtainmatrix R = rij 5×4 by Equation(6)whichislistedinTable 4

Table4. Thedecisionmatrix R

ϕ1 ϕ2

φ1 ([0.4974,0.6477],[0.4850,0.6328],[0.3270,0.4472])([0.6021,0.7058],[0.3571,0.4625],[0.3828,0.5044])

φ2 ([0.5817,0.8268],[0.4638,0.5802],[0.3016,0.4277])([0.6677,0.7703],[0.5223,0.6544],[0.3723,0.4768])

φ3 ([0.7186,0.8301],[0.5426,0.6507],[0.3723,0.4768])([0.6853,0.7976],[0.3798,0.5313],[0.3828,0.5044])

φ4 ([0.6331,0.7344],[0.3016,0.4038],[0.3828,0.5044])([0.6933,0.8620],[0.2236,0.3464],[0.5044,0.6150])

φ5 ([0.4687,0.5710],[0.5044,0.6150],[0.3464,0.4583])([0.5785,0.6853],[0.3446,0.4783],[0.3016,0.4083])

ϕ3

ϕ4

φ1 ([0.4740,0.5785],[0.3669,0.4676],[0.2625,0.3723])([0.5127,0.6243],[0.4183,0.5186],[0.3016,0.4038])

φ2 ([0.7058,0.8238],[0.2814,0.3979],[0.3567,0.4649])([0.7172,0.8268],[0.3464,0.4472],[0.3016,0.4265])

φ3 ([0.6853,0.7976],[0.1231,0.2462],[0.3016,0.4038])([0.7976,1.0000],[0.2236,0.3464],[0.2855,0.3912])

φ4 ([0.5150,0.6163],[0.1625,0.2656],[0.3241,0.4397])([0.4998,0.6021],[0.4854,0.6274],[0.3016,0.4038])

φ5 ([0.8082,1.0000],[0.3669,0.4676],[0.3016,0.4038])([0.6853,0.7976],[0.6559,0.7579],[0.2019,0.3194])

 ([0.7186,0.8301], [0.3016,0.4038], [0.3016,0.4277]), ([0.6933,0.8620], [0.2236,0.3464], [0.3016,0.4038]), ([0.8082,1.0000], [0.1231,0.2462], [0.2625,0.3723]), ([0.7976,1.1000], [0.2236,0.3464], [0.2019,0.3194])



 R = 



   



Step4. Computethe Θi (let θ = 0.5)byEquation(15).

Θ1 = 0.8974, Θ2 = 0.3772, Θ3 = 0.0000, Θ4 = 0.8477, Θ5 = 0.7006

Step5. TheorderofETEsisdeterminedby Θi (i = 1,2,3,4,5): φ3 φ2 φ5 φ4 φ1,and thusthemostdesirableETEis φ3

4.2.ComparativeAnalysis

Inwhatfollows,wecomparewiththeintervalneutrosophicnumberweightedaveraging (INNWA)operatorandintervalneutrosophicnumberweightedgeometric(INNWG)operator[28], INNsimilarity[33],andINNVIKOR[55].TheresultsareshowninTable 5

Fromtheaboveanalysis,itcanbeseenthatthefivemethodshavethesamebestemerging technologyenterprise φ3,andtherankingresultsofMethod1andMethod2areslightlydifferent.The proposedINNVIKORmethodcanreasonablyfocusaMAGDMproblemwithINNs.Atthesametime, comparedwithMethod5basedontheINNVIKORmethodinReference[55],ourproposedmethod avoidstheintervalnumbers’comparison.

Table5. Theordersbyutilizingfivemethods.

MethodsRankingOrdersBestAlternatives

Method1withINNWAoperatorin[28] φ3 φ5 φ2 φ4 φ1 φ3

Method2withINNWGoperatorin[28] φ3 φ2 φ5 φ4 φ1 φ3

Method3basedonsimilarityin[33] φ3 φ2 φ5 φ4 φ1 φ3

Method4basedonsimilarityin[33] φ3 φ2 φ5 φ4 φ1 φ3

Method5basedonINNVIKORin[55] φ3 φ2 φ5 φ4 φ1 φ3

Theproposedmethod φ3 φ2 φ5 φ4 φ1 φ3

5.Conclusions

TheVIKORmethodforaMAGDMpresentssomeconflictingattributes.WeextendedtheVIKOR methodtoMAGDMwithINNs.Firstly,thebasicconceptsofINNswerebrieflypresented.Themethod firstaggregatesallindividualdecision-makers’assessmentinformationbasedonanINNWAoperator, andthenemploystheextendedclassicalVIKORmethodforMAGDMproblemswithINNs.The validityandstabilityofthismethodwereverifiedbyexampleanalysisandcomparativeanalysis,and itssuperioritywasillustratedbyacomparisonwiththeexistingmethods.Inthefuture,manyother methodsofINSsneedtobeexploredinforMAGDM,riskanalysis,andmanyotheruncertainand fuzzyenvironments[56–78].

Acknowledgments: TheworkwassupportedbytheNationalNaturalScienceFoundationofChinaunderGrant No.71571128andtheHumanitiesandSocialSciencesFoundationofMinistryofEducationofthePeople’s RepublicofChina(17YJA630115)andtheConstructionPlanofScientificResearchInnovationTeamforColleges andUniversitiesinSichuanProvince(15TD0004).

AuthorContributions: Yu-HanHuang,Gui-WuWeiandCunWeiconceivedandworkedtogethertoachievethis work,Yu-HanHuangcompiledthecomputingprogrambyMatlabandanalyzedthedata,Gui-WuWeiwrotethe paper,CunWeimadecontributiontothecasestudy. ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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