TODIM Method for Single-Valued Neutrosophic Multiple Attribute Decision Making

TODIMMethodforSingle-ValuedNeutrosophic MultipleAttributeDecisionMaking

Dong-ShengXu 1,CunWei 1,*andGui-WuWei 2,* ID

1 SchoolofScience,SouthwestPetroleumUniversity,Chengdu610500,China;xudongsheng1976@163.com

2 SchoolofBusiness,SichuanNormalUniversity,Chengdu610101,China

* Correspondence:weicun1990@163.com(C.W.);weiguiwu1973@sicnu.edu.cn(G.-W.W.)

Received:20September2017;Accepted:11October2017;Published:16October2017

Abstract: Recently,theTODIMhasbeenusedtosolvemultipleattributedecisionmaking(MADM) problems.Thesingle-valuedneutrosophicsets(SVNSs)areusefultoolstodepicttheuncertaintyof theMADM.Inthispaper,wewillextendtheTODIMmethodtotheMADMwiththesingle-valued neutrosophicnumbers(SVNNs).Firstly,thedefinition,comparison,anddistanceofSVNNsare brieflypresented,andthestepsoftheclassicalTODIMmethodforMADMproblemsareintroduced. Then,theextendedclassicalTODIMmethodisproposedtodealwithMADMproblemswiththe SVNNs,anditssignificantcharacteristicisthatitcanfullyconsiderthedecisionmakers’bounded rationalitywhichisarealactionindecisionmaking.Furthermore,weextendtheproposedmodelto intervalneutrosophicsets(INSs).Finally,anumericalexampleisproposed.

Keywords: multipleattributedecisionmaking(MADM);single-valuedneutrosophicnumbers; intervalneutrosophicnumbers;TODIMmethod;prospecttheory

1.Introduction

Multipleattributedecisionmaking(MADM)isahotresearchareaofthedecisiontheorydomain, whichhashadwideapplicationsinmanyfields,andattractedincreasingattention[1,2].Duetothe fuzzinessanduncertaintyofthealternativesindifferentattributes,attributevaluesindecisionmaking problemsarenotalwaysrepresentedasrealnumbers,andtheycanbedescribedasfuzzynumbers inmoresuitableoccasions,suchasinterval-valuednumbers[3,4],triangularfuzzy variables[5–8], linguisticvariables[9–13]oruncertainlinguisticvariables[14–21],intuitionisticfuzzynumbers (IFSs)[22–27]orinterval-valuedintuitionisticfuzzynumbers(IVIFSs)[28–31],andSVNSs[32]or INSs[33].SinceFuzzyset(FS),whichisaveryusefultooltoprocessfuzzyinformation,was firstlyproposedbyZadeh[34],ithasbeenregardedasanusefultooltosolveMADM[35,36], fuzzylogic[37],andpatternsrecognition[38].Atanassov[22]introducedIFSswiththemembership degreeandnon-membershipdegree,whichwereextendedtoIVIFSs[28].Smarandache[39,40] proposedaneutrosophicset(NS)withtruth-membershipfunction,indeterminacy-membership function,andfalsity-membershipfunction.Furthermore,theconceptsofaSVNS[32]andanINS[33] werepresentedforactualapplications.Ye[41]proposedasimplifiedneutrosophicset(SNS),including theSVNSandINS.Recently,SNSs(INSs,andSVNSs)havebeenutilizedtosolvemanyMADM problems[42–67].

Inordertodepicttheincreasingcomplexityintheactualworld,theDMs’riskattitudes shouldbetakenintoconsiderationtodealwithMADM[68–70].Basedontheprospecttheory, GomesandLima[71]establishedTODIM(anacronyminPortugueseforInteractiveMulti-Criteria DecisionMaking)methodtosolvetheMADMproblemswiththeDMs’psychologicalbehaviors areconsidered.SomescholarshavepaidattentiontodepicttheDMs’attitudinalcharactersinthe MADM[72–74].Also,somescholarsproposedfuzzyTODIMmodels[75,76],intuitionisticfuzzy

Information 2017, 8,125;doi:10.3390/info8040125 www.mdpi.com/journal/information

TODIMmodels[77,78],thePythagoreanfuzzyTODIMapproach[68],themulti-hesitantfuzzy linguisticTODIM approach[79,80],theintervaltype-2fuzzyTODIMmodel[81],theintuitionistic linguisticTODIMmethod[82],andthe2-dimensionuncertainlinguisticTODIMmethod[83].However, thereisnoscholartoinvestigatetheTODIMmodelwithSVNNS.Therefore,itisverynecessarytopay abundantattentiontothisnovelandworthyissue.TheaimofthispaperistoextendtheTODIMidea tosolvetheMADMwiththeSVNNs,tofillupthisvacancy.InSection 2,wegivethebasicconceptsof SVNSsandtheclassicalTODIMmethodforMADMproblems.InSection 3,weproposetheTODIM methodforSVNMADMproblems.InSection 4,weextendtheproposedSVNTODIMmethodto INNs.InSection 5,anillustrativeexampleispointedoutandsomecomparativeanalysisisconducted. WegiveaconclusioninSection 6

2.Preliminaries

SomebasicconceptsanddefinitionsofNSsandSVNSsareintroduced.

2.1.NSsandSVNSs

Definition1 [39,40]. Let X beaspaceofpoints(objects)withagenericelementinfixset X,denoted by x.NSs A in X ischaracterizedbyatruth-membershipfunction TA (x),anindeterminacy-membership IA (x) andafalsity-membershipfunction FA (x),where TA (x) : X → ] 0,1+ [, IA (x) : X → ] 0,1+ [ andFA (x) : X → ] 0,1+ [ and 0 ≤ supTA (x) + supIA (x) + supFA (x) ≤ 3+ TheNSswasdifficulttoapplytorealapplications.Wang[32]developtheSNSs.

Definition2 [32]. LetXbeaspaceofpoints(objects);aSVNSsAinXischaracterizedasthefollowing: A = {(x, TA (x), IA (x), FA (x))|x ∈ X } (1) wherethetruth-membershipfunction TA (x),indeterminacy-membership IA (x) andfalsity-membershipfunction FA (x), TA (x) : X → [0,1], IA (x) : X → [0,1] and FA (x) : X → [0,1] ,withthecondition 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3. Forconvenience,aSVNNcanbeexpressedtobe A = (TA, IA, FA ), TA ∈ [0,1], IA ∈ [0,1], FA ∈ [0,1], and 0 ≤ TA + IA + FA ≤ 3.

Definition3 [50]. LetA = (TA, IA, FA ) beaSVNN,ascorefunctionS(A) isdefined: S(A) = (2 + TA IA FA ) 3 , S(A) ∈ [0,1].(2)

Definition4 [50]. LetA = (TA, IA, FA ) beaSVNN,anaccuracyfunctionH(A) ofaSVNNisdefined: H(A) = TA FA, H(A) ∈ [ 1,1].(3)

toevaluatethedegreeofaccuracyoftheSVNN A = (TA, IA, FA ),where H(A) ∈ [ 1,1] .Thelargerthevalue ofH(A) is,thehigherthedegreeofaccuracyoftheSVNNA.

Zhangetal.[50]gaveanorderrelationbetweentwoSVNNs,whichisdefinedasfollows:

Definition5 [50]. Let A = (TA, IA, FA ) and B = (TB, IB, FB ) betwoSVNNs,if S(A) < S(B),then A < B; ifS(A) = S(B),then

(1) ifH(A) = H(B),thenA = B; (2) ifH(A) < H(B),thenA < B.

Definition6 [32]. LetAandBbetwoSVNNs,thebasicoperationsofSVNNsare:

(1) A ⊕ B = (TA + TB TA TB, IA IB, FA FB );

(2) A ⊗ B = (TA TB, IA + IB IA IB, FA + FB FA FB ); (3) λ A = 1 (1 TA )λ , (IA )λ , (FA )λ , λ > 0; (4) (A)λ = (TA )λ , (IA )λ ,1 (1 FA )λ , λ > 0.

Definition7 [42]. LetAandBbetwoSVNNs,thenthenormalizedHammingdistancebetweenAandBis: d(A, B) = 1 3 (|TA TB | + |IA IB | + |FA FB |) (4)

2.2.TheTODIMApproach

TheTODIMapproach[71],developedtoconsidertheDM’spsychologicalbehavior,caneffectively solvetheMADMproblems.Basedontheprospecttheory,thisapproachdepictsthedominanceof eachalternativeoverothersbyconstructingafunctionofmulti-attributevalues[69].

Let G = {G1, G2, , Gn } betheattributes, w = (w1, w2, , wn ) betheweightof Gj, 0 ≤ wj ≤ 1, and n ∑ j=1 wj = 1 A = {A1, A2, ··· , Am } arealternatives.Let A = aij m×n beadecisionmatrix, where aij isgivenforthealternative Ai underthe Gj, i = 1,2, ··· , m,and j = 1,2, ··· , n.Weset wjr = wj /wr (j, r = 1,2, , n) arerelativeweightof Gj to Gr,and wr = max wj |j = 1,2, , n , and0 ≤ wjr ≤ 1.





(6) andtheparameter θ showstheattenuationfactorofthelosses.If bij btj > 0,then φj (Ai, At ) representsagain;if bij btj < 0,then φj (Ai, At ) signifiesaloss. Step3. Derivingtheoveralldominancevalueof Ai bytheEquation(7): φ(Ai ) =

m ∑ t=1 δ(Ai, At ) max i

m ∑ t=1 δ(Ai, At ) min i

m ∑ t=1 δ(Ai, At ) min i

m ∑ t=1 δ(Ai, At ) , i = 1,2, , m.(7)

Step4. Rankingallalternativesandselectingthemostdesirablealternativeinaccordancewith φ(Ai ).Thealternativewithminimumvalueistheworst.Inversely,themaximumvalueis thebestone.

3.TODIMMethodforSVNMADMProblems

Let A = {A1, A2, , Am } bealternatives,and G = {G1, G2, , Gn } beattributes.Let w = (w1, w2, , wn ) betheweightofattributes,where wj ∈ [0,1], n ∑ j=1 wj = 1.Supposethat R = rij m×n = Tij, Iij, Fij m×n beaSVNmatrix,where rij = Tij, Iij, Fij ,whichisanattributevalue,givenbyan expert,forthealternative Ai under Gj, Tij ∈ [0,1], Iij ∈ [0,1], Fij ∈ [0,1], 0 ≤ Tij + Iij + Fij ≤ 3, i = 1,2, ··· , m, j = 1,2, ··· , n

TosolvetheMADMproblemwithsingle-valuedneutrosophicinformation,wetrytopresent asingle-valuedneutrosophicTODIMmodelbasedontheprospecttheoryandcandepicttheDMs’ behaviorsunderrisk.

Firstly,wecalculatetherelativeweightofeachattribute Gj as: wjr = wj /wr, j, r = 1,2, , n.(8) where wj istheweightoftheattributeof Gj, wr = max wj |j = 1,2, ··· , n ,and0 ≤ wjr ≤ 1. BasedontheEquation(8),wecanderivethedominancedegreeof Ai overeachalternative At withrespecttotheattribute Gj: φj (Ai, At )

wjr d rij, rtj / n ∑ j=1 wjr, ifrij > rtj 0, ifrij = rtj 1 θ n ∑ j=1 wjr d rij, rtj /wjr, ifrij < rtj

(9) d rij, rtj = 1 3 Tij Ttj + Iij Itj + Fij Ftj .(10) wheretheparameter θ showstheattenuationfactorofthelosses,and d rij, rtj istomeasurethe distancesbetweentheSVNNs rij and rtj byDefinition7.If rij > rtj,then φj (Ai, At ) representsagain; if rij < rtj,then φj (Ai, At ) signifiesaloss.

. φj (Am, A1) φj (Am, A2) ··· 0 

j (A2, A1) 0 ··· φj (A2, Am ) . .

 , j = 1,2, , n (11) OnthebasisofEquation(11),theoveralldominancedegree δ(Ai, At ) ofthe Ai overeach At can becalculated: δ(Ai, At )

      o φj (A1, A2) ··· φj (A1, Am ) φ      

Then,theoverallvalueofeach Ai canbecalculatedEquation(14):

δ(Ai ) =

m ∑ t=1 δ(Ai, At ) max i

m ∑ t=1 δ(Ai, At ) min i

m ∑ t=1 δ(Ai, At ) min i

m ∑ t=1 δ(Ai, At ) , i = 1,2, ··· , m.(14)

Alsothegreatertheoverallvalue δ(Ai ),thebetterthealternative Ai. Ingeneral,single-valuedneutrosophicTODIMmodelincludesthecomputingsteps: (Procedureone)

Step1. Identifyingthesingle-valuedneutrosophicmatrix R = rij m×n = Tij, Iij, Fij m×n inthe MADM,where rij isaSVNN.

Step2. Calculatingtherelativeweightof Gj byusingEquation(8).

Step3. Calculatingthedominancedegree φj (Ai, At ) of Ai overeachalternative At underattribute Gj byEquation(9).

Step4. Calculatingtheoveralldominancedegree δ(Ai, At ) of Ai overeachalternative At byusing Equation(12).

Step5. Derivingtheoverallvalue δ(Ai ) ofeachalternative Ai usingEquation(14). Step6. Determiningtheorderofthealternativesinaccordancewith δ(Ai )(i = 1,2, , m).

4.TODIMMethodforIntervalNeutrosophicMADMProblems

Furthermore,Wangetal.[33]definedINSs.

Definition8 [33]. Let X beaspaceofpoints(objects)withagenericelementinfixset X,anINSs A in X is characterizedasfollows: A = x, TA (x), IA (x), FA (x) |x ∈ X (15)

wheretruth-membershipfunction TA (x),indeterminacy-membership IA (x) andfalsity-membershipfunction FA (x) areintervalvalues, TA (x) ⊆ [0,1], IA (x) ⊆ [0,1] and FA (x) ⊆ [0,1],and 0 ≤ sup TA (x) + sup IA (x) + sup FA (x) ≤ 3. Anintervalneutrosophicnumber(INN)canbeexpressedas A = TA, IA, FA = TL A, TR A , I L A, I R A , FL A, FR A ,where TL A, TR A ⊆ [0,1], I L A, I R A ⊆ [0,1], FL A, FR A ⊆ [0,1], and 0 ≤ TR A + I R A + FR A ≤ 3.

Definition9 [84]. Let A = TL A, TR A , I L A, I R A , FL A, FR A beanINN,ascorefunction S ofanINNcanbe representedasfollows: S A = 2 + TL A I L A FL A + 2 + TR A I R A FR A 6 , S A ∈ [0,1].(16)

Definition10 [84]. Let A = TL A, TR A , I L A, I R A , FL A, FR A beanINN,anaccuracyfunction H A isdefined: H A = TL A + TR A FL A + FR A 2 , H A ∈ [ 1,1].(17)

Tang[84]definedanorderrelationbetweentwoINNs.

Definition11 [84]. Let A = TL A, TR A , I L A, I R A , FL A, FR A and B = TL B , TR B , I L B , I R B , FL B , FR B betwoINNs, S A = 2+TL A I L A FL A + 2+TR A I R A FR A 6 and S B = 2+TL B I L B FL B + 2+TR B I R B FR B 6 bethe scores,and H A = TL A +TR A FL A +FR A 2 and H B = TL B +TR B FL B +FR B 2 betheaccuracyfunction,thenif S A < S B ,then A < B;ifS A = S B ,then

(1) ifH A = H B ,then A = B; (2) ifH A < H B , A < B

Definition12 [33,61]. Let A1 = TL 1 , TR 1 , I L 1 , I R 1 , FL 1 , FR 1 and A2 = TL 2 , TR 2 , I L 2 , I R 2 , FL 2 , FR 2 betwoINNs,andsomebasicoperationsonthemaredefinedasfollows:

(1) A1 ⊕ A2 = TL 1 + TL 1 TL 1 TL 1 , TR 1 + TR 1 TR 1 TR 1 , I L 1 I L 2 , I R 1 I R 2 , FL 1 FL 2 , FR 1 FR 2 ; (2) A1 ⊗ A2 = TL 1 TL 2 , TR 1 TR 2 , I L 1 + I L 1 I L 1 I L 1 , I R 1 + I R 1 I R 1 I R 1 , FL 1 + FL 1 FL 1 FL 1 , FR 1 + FR 1 FR 1 FR 1 ; (3) λ A1 = 1 1 TL 1 λ ,1 1 TR 1 λ , I L 1 λ , I R 1 λ , FL 1 λ , FR 1 λ , λ > 0; (4) A1 λ = TL 1 λ , TR 1 λ , I L 1 λ , I R 1 λ , 1 1 FL 1 λ ,1 1 FR 1 λ , λ > 0.

Definition13 [84]. Let A1 = TL 1 , TR 1 , I L 1 , I R 1 , FL 1 , FR 1 and A2 = TL 2 , TR 2 , I L 2 , I R 2 , FL 2 , FR 2 betwoINNs,thenthenormalizedHammingdistancebetween A1 = TL 1 , TR 1 , I L 1 , I R 1 , FL 1 , FR 1 and A2 = TL 2 , TR 2 , I L 2 , I R 2 , FL 2 , FR 2 isdefinedasfollows: d A1, A2 = 1 6 TL 1 TL 2 + TR 1 TR 2 + I L 1 I L 2 + I R 1 I R 2 + FL 1 FL 2 + FR 1 FR 2 (18)

Let A, G and w bepresentedasinSection 3.Supposethat R = rij m×n = TL ij , TR ij , I L ij , I R ij , FL ij , FR ij m×n istheintervalneutrosophicdecisionmatrix,where TL ij , TR ij , I L ij , I R ij , FL ij , FR ij istruth-membershipfunction,indeterminacy-membershipfunctionand falsity-membershipfunction, TL ij , TR ij ⊆ [0,1], I L ij , I R ij ⊆ [0,1], FL ij , FR ij ⊆ [0,1], 0 ≤ TR ij + I R ij + FR ij ≤ 3, i = 1,2, , m, j = 1,2, , n

TocopewiththeMADMwithINNs,wedevelopintervalneutrosophicTODIMmodel. Firstly,wecalculatetherelativeweightofeachattribute Gj as: wjr = wj /wr, j, r = 1,2, , n (19) where wj istheweightoftheattributeof Gj, wr = max wj |j = 1,2, ··· , n ,and0 ≤ wjr ≤ 1. BasedontheEquation(20),wecanderivethedominancedegreeof Ai overeachalternative At withrespecttotheattribute Gj: φj (Ai, At ) =

wjr d rij, rtj / n ∑ j=1 wjr, if rij > rtj 0, if rij = rtj 1 θ n ∑ j=1 wjr d rij, rtj /wjr, if rij < rtj

(20)

d rij, rtj = 1 6 TL ij TL tj + TR ij TR tj + I L ij I L tj + I R ij I R tj + FL ij FL tj + FR ij FR tj .(21) wheretheparameter θ showstheattenuationfactorofthelosses,and d rij, rtj istomeasurethe distancesbetweentheINNs rij and rtj byDefinition13.If rij > rtj,then φj (Ai, At ) representsagain;if rij < rtj,then φj (Ai, At ) signifiesaloss.

Forindicatingfunctions φj (Ai, At ) clearly,adominancedegreematrix φj = φj (Ai, At ) m×m under Gj isexpressedas: A1 A2 Am φj = φj (Ai, At ) m

o φj (A1, A2) φj (A1, Am ) φj (A2, A1) 0 φj (A2, Am ) . . . φj (Am, A1) φj (Am, A2) 0

      , j = 1,2, , n (22)

OnthebasisofEquation(22),theoveralldominancedegree δ(Ai, At ) ofthe Ai overeach At can becalculated: δ(Ai, At ) = n ∑ j=1 φj (Ai, At ), (i, t = 1,2, ··· , m) (23)

Thus,theoveralldominancedegreematrix δ = [δ(Ai, At )]m×m canbederivedbyEquation(23): A1 A2 Am δ = [δ(Ai, At )]m×m =

A1 A2 . Am

      o δ(A1, A2) δ(A1, Am ) δ(A2, A1) 0 δ(A2, Am ) . . . δ(Am, A1) δ(Am, A2) 0

      (24)

Then,theoverallvalueofeach Ai canbecalculatedEquation(25): δ(Ai ) =

m ∑ t=1 δ(Ai, At ) max i

m ∑ t=1 δ(Ai, At ) min i

m ∑ t=1 δ(Ai, At ) min i

m ∑ t=1 δ(Ai, At ) , i = 1,2, , m.(25)

Alsothegreatertheoverallvalue δ(Ai ),thebetterthealternative Ai Ingeneral,intervalneutrosophicTODIMmodelincludesthecomputingsteps: (Proceduretwo)

Step1. Identifyingtheintervalneutrosophicmatrix R = rij m×n = TL ij , TR ij , IL ij , IR ij , FL ij , FR ij m×n intheMADM,where rij isanINN.

Step2. Calculatingtherelativeweightof Gj byusingEquation(19).

Step3. Calculatingthedominancedegree φj (Ai, At ) of Ai overeachalternative At underattribute Gj byEquation(20).

Step4. Calculatingtheoveralldominancedegree δ(Ai, At ) of Ai overeachalternative At byusing Equation(23).

Step5. Derivingtheoverallvalue δ(Ai ) ofeachalternative Ai usingEquation(25). Step6. Determiningtheorderofthealternativesinaccordancewith δ(Ai )(i = 1,2, , m)

5.NumericalExampleandComparativeAnalysis

5.1.NumericalExample1

Inthispart,anumericalexampleisgiventoshowpotentialevaluationofemerging technologycommercializationwithSVNNs.Fivepossibleemergingtechnologyenterprises(ETEs) Ai (i = 1,2,3,4,5) aretobeevaluatedandselected.Fourattributesareselectedtoevaluatethe fivepossibleETEs: 1 G1 istheemploymentcreation; 2 G2 isthedevelopmentofscienceand technology; 3 G3 isthetechnicaladvancement;and 4 G4 istheindustrializationinfrastructure.The fiveETEs Ai (i = 1,2,3,4,5) aretobeevaluatedbyusingtheSVNNsundertheabovefourattributes (whoseweightingvector ω = (0.2,0.1,0.3,0.4)T ),aslistedinthefollowingmatrix.

G1 G2 G3 G4

(0.5,0.8,0.1)(0.6,0.3,0.3)(0.3,0.6,0.1)(0.5,0.7,0.2) (0.7,0.2,0.1)(0.7,0.2,0.2)(0.7,0.2,0.4)(0.8,0.2,0.1) (0.6,0.7,0.2)(0.5,0.7,0.3)(0.5,0.3,0.1)(0.6,0.3,0.2) (0.8,0.1,0.3)(0.6,0.3,0.4)(0.3,0.4,0.2)(0.5,0.6,0.1) (0.6,0.4,0.4)(0.4,0.8,0.1)(0.7,0.6,0.1)(0.5,0.8,0.2)

      

Then,weuse ProcedureOne toselectthebestETE. Firstly,since w4 = max{w1, w2, w3, w4},then G4 isthereferenceattributeandthereference attribute’sweightis wr = 0.4.Then,wecancalculatetherelativeweightsoftheattributes Gj (j = 1,2,3,4) as w1r = 0.50, w2r = 0.25, w3r = 0.75 and w4r = 1.00.Let θ = 2.5,thenthedominance degreematrix φj (Ai, At )(j = 1,2,3,4) withrespectto Gj canbecalculated:

A1 A2 A3 A4 A5

0.0000 0.4619 0.2828 0.5657 0.4619 0.23090.00000.21600.16330.2000 0.1414 0.43200.0000 0.4899 0.3651 0.2828 0.32660.24490.00000.2000 0.2309 0.40000.1826 0.40000.0000

A1 A2 A3 A4 A5

0.0000 0.40000.12910.05770.1732 0.10000.00000.16330.11550.1826 0.5164 0.65320.0000 0.5657 0.4619 0.2309 0.46190.14140.00000.1826 0.6928 0.73030.1155 0.73030.0000

0.0000 0.4422 0.2981 0.2309 0.2667 0.33170.0000 0.32660.28280.2646 0.22360.24490.00000.20000.2236 0.1732 0.3771 0.26670.0000 0.3528 0.2000 0.3528 0.29810.26460.0000

0.0000 0.3464 0.2582 0.16330.1155 0.34640.00000.23090.30550.3651 0.2582 0.23090.00000.25820.2828 0.1633 0.3055 0.25820.00000.2000 0.1155 0.3651 0.2828 0.20000.0000

Theoveralldominancedegree δ(Ai, At ) ofthecandidate Ai overeachcandidate At canbederived byEquation(13):



   

A1 A2 A3 A4 A5 

0.0000 1.6505 0.7100 0.9022 0.4399 1.00900.00000.28360.86711.01234 0.1068 1.07120.0000 0.5974 0.3206 0.3884 1.4711 0.13860.00000.2298 0.3774 1.8482 0.2828 1.06570.0000

  

   

Finally,wegetorderofETEsby δ(Ai )(i = 1,2,3,4,5): A2 A4 A3 A5 A1,andthusthe mostdesirableETEis A2.

5.2.ComparativeAnalysis1

Inwhatfollows,wecompareourproposedmethodwithotherexistingmethodsincludingthe SVNWAoperatorandSVNWGoperatorproposedbySahin[85]asfollows:

Definition14 [85]. Let Aj = Tj, Ij, Fj (j = 1,2, , n) beacollectionofSVNNs, w = (w1, w2, , wn )T betheweightofAj (j = 1,2, , n),andwj > 0, n ∑ j=1 wj = 1 .Then

ri = (Ti, Ii, Fi )

= SVNWAw (ri1, ri2, , rin ) = n ⊕ j=1 wjrij = 1 n ∏ j=1 1 Tij wj , n ∏ j=1 Iij wj , n ∏ j=1 Fij wj

ri = (Ti, Ii, Fi ) = SVNWGω (ri1, ri2, ··· , rin ) = n ⊗ j=1 rij wj = n ∏ j=1 Tij wj ,1 n ∏ j=1 1 Iij wj ,1 n ∏ j=1 1 Fij wj

(26)

(27)

Byutilizingthe R,aswellastheSVNWAandSVNWGoperators,theaggregatingvaluesare derivedinTable 1

Table1. TheaggregatingvaluesoftheemergingtechnologyenterprisesbytheSVNWA (SVNWG)operators.

SVNWASVNWG

A1 (0.4591,0.6307,0.1473)(0.4369,0.6718,0.1627) A2 (0.7449,0.2000,0.1625)(0.7384,0.2000,0.2124) A3 (0.5627,0.3868,0.1692)(0.5578,0.4571,0.1822) A4 (0.5497,0.3464,0.1762)(0.4799,0.4381,0.2067) A5 (0.5822,0.6389,0.1741)(0.5610,0.6933,0.2083)

AccordingtotheaggregatingresultsinTable 1,thescorefunctionsarelistedinTable 2.

Table2. Thescorefunctionsoftheemergingtechnologyenterprises.

SVNWASVNWG

A1 0.56040.5341 A2 0.79420.7753

A3 0.66890.6398 A4 0.67570.6117 A5 0.58980.5531

AccordingtothescorefunctionsshowninTable 2,theorderoftheemergingtechnology enterprisesareinTable 3

Table3. Orderoftheemergingtechnologyenterprises.

Order

SVNWAA2 >A4 >A3 >A5 >A1 SVNWGA2 >A3 >A4 >A5 >A1

Fromtheaboveanalysis,itcanbeseenthattwooperatorshavethesamebestemergingtechnology enterpriseA2 andtwomethods’rankingresultsareslightlydifferent.However,theSVNTODIM approachcanreasonablydepicttheDMs’psychologicalbehaviorsunderrisk,andthus,itmaydeal withtheaboveissueeffectively.Thisverifiesthemethodweproposedisreasonableandeffectivein thispaper.

5.3.NumericalExample2

Ifthefivepossibleemergingtechnologyenterprises Ai (i = 1,2,3,4,5) aretobeevaluatedby usingtheINNSundertheabovefourattributes(whoseweightingvector ω = (0.2,0.1,0.3,0.4)T ), aslistedinthematrix R,then:

([0.5,0.6], [0.8,0.9], [0.1,0.2])([0.6,0.7], [0.3,0.4], [0.3,0.4]) ([0.7,0.9], [0.2,0.3], [0.1,0.2])([0.7,0.8], [0.1,0.2], [0.2,0.3]) ([0.6,0.7], [0.7,0.8], [0.2,0.3])([0.5,0.6], [0.7,0.8], [0.3,0.4]) ([0.8,0.9], [0.1,0.2], [0.3,0.4])([0.6,0.7], [0.3,0.4], [0.4,0.5]) ([0.6,0.7], [0.4,0.5], [0.4,0.5])([0.4,0.5], [0.8,0.9], [0.1,0.2]) ([0.3,0.4], [0.6,0.7], [0.1,0.2])([0.5,0.6], [0.7,0.8], [0.1,0.2]) ([0.7,0.9], [0.2,0.3], [0.4,0.5])([0.8,0.9], [0.2,0.3], [0.1,0.2]) ([0.5,0.6], [0.3,0.4], [0.1,0.2])([0.6,0.7], [0.3,0.4], [0.2,0.3]) ([0.3,0.4], [0.4,0.5], [0.2,0.3])([0.5,0.6], [0.6,0.7], [0.1,0.2]) ([0.7,0.8], [0.6,0.7], [0.1,0.2])([0.5,0.6], [0.8,0.9], [0.2,0.3])

      

Then,weuse ProcedureTwo toselectthebestETE. Firstly,since w4 = max{w1, w2, w3, w4},then G4 isthereferenceattributeandthereference attribute’sweightis wr = 0.4.Then,wecancalculatetherelativeweightsoftheattributes

Gj (j = 1,2,3,4) as: w1r = 0.50, w2r = 0.25, w3r = 0.75 and w4r = 1.00.Let θ = 2.5,thenthedominance degreematrix φj (Ai, At )(j = 1,2,3,4) withrespectto Gj canbecalculated:

A1 A2 A3 A4 A5

0.0000 0.4761 0.2828 0.5657 0.4619 0.23800.00000.22360.15280.2082 0.1414 0.44720.0000 0.4899 0.3651 0.2828 0.30550.24490.00000.2000 0.2309 0.41630.1826 0.40000.0000

0.0000 0.46190.12910.05770.1732 0.11550.00000.17320.12910.1915 0.5164 0.69280.0000 0.5657 0.4619 0.2309 0.51640.14140.00000.1826 0.6928 0.76590.1155 0.73030.0000

0.0000 0.4522 0.2981 0.2309 0.2667 0.33910.00000.25500.29150.2739 0.2236 0.33990.00000.20000.2236 0.1732 0.3887 0.26670.0000 0.3528 0.2000 0.3651 0.29810.26460.0000

A1 A2 A3 A4 A5

0.0000 0.3266 0.2828 0.11550.1633 0.32660.00000.23090.30550.3651 0.2828 0.23090.00000.25820.2828 0.1155 0.3055 0.25820.00000.2000 0.1633 0.3651 0.2828 0.20000.0000

Theoveralldominancedegree δ(Ai, At ) ofthecandidate Ai overeachcandidate At canbederived byEquation(24):

A1 A2 A3 A4 A5

0.0000 1.7168 0.7346 0.75060.0698 1.01920.00000.37270.35130.8305 0.1314 1.03100.0000 0.47260.0445 0.3406 1.5161 0.13860.20000.0298 0.4252 1.9124 0.8654 0.66570.0000

Definition15 [50]. Let Aj = TL j , TR j , I L j , I R j , FL j , FR j (j = 1,2, , n) beacollectionofINNs, w = (w1, w2, , wn )T betheweightofAj (j = 1,2, , n),andwj > 0, n ∑ j=1 wj = 1 .Then ri = TL i , TR i , I L i , I R i , FL i , FR i = INWAw (ri1, ri2, , rin ) = n ⊕ j=1 wjrij =  

1 n ∏ j=1 1 TL ij wj ,1 n ∏ j=1 1 TR ij wj , n ∏ j=1 I L ij wj , n ∏ j=1 I R ij wj , n ∏ j=1 FL ij wj , n ∏ j=1 FR ij wj



(28) ri = TL i , TR i , I L i , I R i , FL i , FR i = INWGw (ri1, ri2, , rin ) = n ⊗ j=1 rij wj

(29) Byutilizingthedecisionmatrix R,andtheINWAandINWGoperators,theaggregatingvalues areinTable 4

Table4. TheaggregatingvaluesoftheemergingtechnologyenterprisesbytheINWAand INWGoperators.

INWA

A1 ([0.4591,0.5611],[0.6307,0.7342],[0.1116,0.2144])

A2 ([0.7449,0.8928],[0.1866,0.2881],[0.1625,0.2742])

A3 ([0.5627,0.6634],[0.3868,0.4925],[0.1692,0.2734])

A4 ([0.5497,0.6674],[0.3464,0.4657],[0.1762,0.2844]) A5 ([0.5822,0.6863],[0.6389,0.7421],[0.1741,0.2825])

INWG

A1 ([0.4369,0.5395],[0.6718,0.7805],[0.1223,0.2227])

A2 ([0.7384,0.8895],[0.1905,0.2906],[0.2124,0.3144])

A3 ([0.5578,0.6581],[0.4571,0.5685],[0.1822,0.2825])

A4 ([0.4799,0.5851],[0.4381,0.5440],[0.2067,0.3077])

A5 ([0.5610,0.6624],[0.6933,0.8082],[0.2083,0.3097])

AccordingtotheaggregatingvaluesinTable 4,thescorefunctionsareinTable 5.

Table5. Thescorefunctionsoftheemergingtechnologyenterprises.

INWAINWG

A1 0.55490.5298

A2 0.78770.7700

A3 0.65070.6209

A4 0.65740.5948 A5 0.57180.5340

AccordingtothescorefunctionsshowninTable 5,theorderoftheemergingtechnology enterprisesareinTable 6.

Table6. Orderoftheemergingtechnologyenterprises.

Ordering

INWAA2 >A4 >A3 >A5 >A1

INWGA2 >A3 >A4 >A5 >A1

Fromtheaboveanalysis,itcanbeseenthattwooperatorshavethesamebestemerging technologyenterpriseA2 andtwomethods’rankingresultsareslightlydifferent.However,the intervalneutrosophicTODIMapproachcanreasonablydepicttheDMs’psychologicalbehaviorsunder risk,andthus,itmaydealwiththeaboveissueeffectively.Thisverifiesthemethodweproposedis reasonableandeffective.

6.Conclusions

Inthispaper,wewillextendtheTODIMmethodtotheMADMwiththesingle-valued neutrosophicnumbers(SVNNs).Firstly,thedefinition,comparisonanddistanceofSVNNsare brieflypresented,andthestepsoftheclassicalTODIMmethodforMADMproblemsareintroduced. Then,theextended classicalTODIMmethodisproposedtodealwithMADMproblemswiththe SVNNs,anditssignificantcharacteristicisthatitcanfullyconsiderthedecisionmakers’bounded rationalitywhichisarealactionindecisionmaking.Furthermore,weextendtheproposed modeltointervalneutrosophicsets(INSs).Finally,anumericalexampleisproposedtoverifythe developedapproach.

Inthefuture,theapplicationoftheproposedmodelsandmethodsofSVNSsandINSsneeds tobeexploredinthedecisionmaking[86–99],riskanalysisandmanyotheruncertainandfuzzy environment[100–112].

Acknowledgments: TheworkwassupportedbytheNationalNaturalScienceFoundationofChinaunder GrantNo.71571128andtheHumanitiesandSocialSciencesFoundationofMinistryofEducationofthe People’sRepublic ofChina(17XJA630003)andtheConstructionPlanofScientificResearchInnovationTeamfor CollegesandUniversitiesinSichuanProvince(15TD0004).

AuthorContributions: Dong-ShengXu,CunWeiandGui-WuWeiconceivedandworkedtogethertoachieve thiswork,Gui-WuWeiwrotethepaper,CunWeimadecontributiontothecasestudy.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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