Theweighteddistancemeasurebased methodtoneutrosophicmulti-attribute groupdecisionmaking
ChunfangLiu1,2 ,YueShengLuo1,3 1 CollegeofScience, NortheastForestryUniversity,150040,Harbin,China 2 CollegeofAutomation, HarbinEngineeringUniversity,150001,Harbin,China 3 CollegeofScience, HarbinEngineeringUniversity,150001,Harbin,China liuchunfang1112@163.com
Abstract
Neutrosophicset(NS)isageneralizationoffuzzyset(FS)thatisdesignedforsomepracticalsituationsinwhicheachelementhasdifferent truthmembershipfunction,indeterminacymembershipfunctionandfalsitymembershipfunction.Inthispaper,westudythemulti-attributegroup decisionmaking(MAGDM)problemsunderneutrosophicenvironment withtheincompletelyknownorcompletelyunknownattributeweight. Wefirstdefinethesinglevaluedneutrosophicideal-solution(SVNIS)and theweighteddistancemeasure,andestablishtheprogrammodelstoderivetheattributeweights.Thenwegiveapracticalapplicationinthe frameworkofSVNS,theresultshowsthatourmethodisreasonableand effectiveindealingwithdecisionmaking(DM)problems.Furthermore, weextendthemethodtointervalvaluedneutrosophicset(IVNS).
Keywords: multi-attributegroupdecisionmaking(MAGDM),the weighteddistancemeasurebasedmethod;neutrosophicset(NS).
1Introduction
FuzzysetwasintroducedbyZadeh,whichhasbeenwidelyusedinmanyaspects [1,2].OnthebasisofZadeh’swork,severalhigh-orderfuzzysetshavebeen proposedasanextensionoffuzzysets,includinginterval-valuedfuzzyset,type-2 fuzzyset,type-nfuzzyset,softset,roughset,intuitionisticfuzzyset,intervalvaluedintuitionisticfuzzyset,hesitantfuzzysetandneutrosophicset(NS) [2,3,4,5,6].Sofar,theproposedhigh-orderfuzzysetshavebeensuccessfully utilizedindealingwithdifferentuncertainproblems,suchasdecisionmaking [7],patternrecognition[8],etc.
Asageneralizationoffuzzyset,theNSwasproposedbySmarandache[5] notonlytodealwiththedecisioninformationwhichisoftenincomplete,indeterminateandinconsistentbutalsoincludethetruthmembershipdegree,the
falsitymembershipdegreeandtheindeterminacymembershipdegree.Forsimplicityandpracticalapplication,WangproposedthesinglevaluedNS(SVNS) andtheintervalvaluedNS(IVNS)whicharetheinstancesofNSandgavesome operationsonthesesets[8,9].Sinceitsappearance,manyfruitfulresultshave beenappeared[23,24].Ononehand,manyresearchershaveproposedsome aggregationoperatorsofSVNSandINSandappliedthemtoMADMproblems [10,11,12,25,26].Ontheotherhand,someresearchershavealsoproposed entropyandsimilaritymeasureoftheSVNSandIVNSandappliedthemto MADMandpatternrecognition[13,14].Theaboveproblemsthatrelatedto theattributeweightsarecompletelyknown.However,withthedevelopmentof theinformationsocietyandinternettechnology,thesocio-economicenvironment getsmorecomplexinmanydecisionareas,suchascapitalinvestmentdecision making,medicaldiagnosis,personnelexamination,etc.Onlyonedecisionmaker cannotdealwiththecomplexproblems.Accordingly,itisnecessarytogather multipledecisionmakerswithdifferentknowledgestructuresandexperiences toconductagroupdecisionmaking.Insomecircumstances,itisdifficultfor thedecisionmakerstogivetheinformationoftheattributeweightscorrectly, whichmakestheattributeweightsincompletelyknownorcompletelyunknown. Howtoderivetheattributeweightsfromthegivenneutrosophicinformation isanimportanttopic.Inintuitionisticfuzzyenvironments,manyresearchers haveproposedsomeprogrammodelstoobtaintheincompletelyknownattribute weightsorthecompletelyunknownattributeweights,suchasXuproposedthe deviation-basedmethod[15],theideal-solutionbasedmethod[16],thegroup consensus-basedmethod[17],Liproposedtheconsistency-basedmethod[18], etc.Undertheneutrosophicenvironment,Sahinproposedthemaximizingdeviationmethod[19].Uptonow,wefoundthatthereisnoresearchoftheweighted distancemeasurebasedmethodtoneutrosophicmulti-attributegroupdecision making.Inthispaper,weinvestigatetheMAGDMproblemswhichtheinformationexpressedbySVNSorIVNS,andtheattributeweightsareincompletely knownorcompletelyunknown.
Therestofthepaperisorganizedasfollows.InSection2,werecallthe conceptofNS,SVNS,INSandtheirdistancemeasures.InSection3,wegive theweighteddistancemeasurebasedmethodtosinglevaluedneutrosophicset (SVNS).Furthermore,weextendthemethodtointervalvaluedneutrosophic set(IVNS).Finally,aconclusionisgiveninSection4.
Forsimilarityandpracticalapplication,WangproposedtheSVNSandIVNSwhicharethesubclassesofNSandpreservealltheoperationsonNS.Inthe followingpart,werecallSVNSandIVNSandtheirdistancemeasure,respectively.
Definition2.2 [8]Assume X beauniverseofdiscoursewithagenericelement in X denotedby x.Asinglevaluedneutrosophicset(SVNS) A on X isdefined byatruthmembershipfunction TA(x),anindeterminacymembershipfunction IA(x)andafalsitymembershipfunction FA(x). TA(x), IA(x)and FA(x)are definedby
TA(x): X → [0, 1] IA(x): X → [0, 1] FA(x): X → [0, 1]
where TA(x), IA(x)and FA(x)aresubsetsof[0, 1],andsatisfy0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3.
Forsimilarity,weutilize A = {TA(x),IA(x),FA(x)} todenoteaSVNS A inthefollowingpart.If X hasonlyoneelement,forconvenience,wecall A a singlevaluedneutrosophicnumber(SVNN)anddenotedby A = {TA,IA,FA}
Definition2.3 [20]Let A1 = {T1,I1,F1}, A2 = {T2,I2,F2} betwoSVNNs, thenormalizedHammingdistancemeasurebetween A1 and A2 isdefinedby d(A1,A2)= 1 3 (|T1 T2| + |I1 I2| + |F1 F2|)(1)
Definition2.4 [9]Assume X beauniverseofdiscoursewithagenericelement in X denotedby x,andint[0, 1]bethesetofallclosedsubsetsof[0, 1].An intervalvaluedneutrosophicset(IVNS) A on X isdefinedbyatruthmembershipfunction TA(x),anindeterminacymembershipfunction IA(x)andafalsity membershipfunction FA(x). TA(x), IA(x)and FA(x)aredefinedby
TA(x): X → int[0, 1] IA(x): X → int[0, 1] FA(x): X → int[0, 1]
withthecondition0 ≤ SupTA(x)+SupIA(x)+SupFA(x) ≤ 3.
Herewedenote TA(x)=[TA (x),T + A (x)], IA(x)=[IA (x),I + A (x)], FA(x)= [FA (x),F + A (x)].Forconvenience,wecall A anintervalvaluedneutrosophic number(IVNN)anddenotedby A = {[TA ,T + A ], [IA ,I + A ], [FA ,F + A ]}
Definition2.5 [14]Let A1 = {[T1 ,T + 1 ], [I1 ,I + 1 ], [F1 ,F + 1 ]}, A2 = {[T2 ,T + 2 ], [I2 ,I + 2 ], [F2 ,F + 2 ]} betwoIVNNs,thenormalizedHamming distancemeasurebetween A1 and A2 isdefinedby d(A1,A2)= 1 6 (|T1 T2 | + |T + 1 T + 2 | + |I1 I2 | +|I + 1 I + 2 | + |F1 F2 | + |F + 1 F + 2 |) (2)
3Theweighteddistancemeasurebasedmethod toneutrosophicset
3.1Theweighteddistancemeasurebasedmethodtosinglevaluedneutrosophicset
Let X = {X1,X2,...,Xm} beasetofalternatives, C = {C1,C2,...,Cn} bea setofattributesand w = {w1,w2,...,wn} betheweightvectoroftheattribute with wj ∈ [0, 1]and n j=1 wj =1.Supposethatthereare s decisionmakers D = {D1,D2,...,Ds},whosecorrespondingweightedvectoris λ = {λ1,λ2,...,λs} .Let Ak =(rk ij )m×n (k =1, 2,...,s)besinglevaluedneutrosophicdecision matrix,where rk ij = {T k ij ,I k ij ,F k ij } isthevalueoftheattribute,expressedby SVNNs.
InMADMenvironments,theidealpointisusedtohelptheidentificationof thebestalternativeinthedecisionset.Althoughtheidealpointdoesnotexist inrealworld,itdoesprovideaneffectivewaytoevaluatethebestalternative. NowwesupposetheidealSVNNas α∗ j = {t∗,i∗,f ∗} = {1, 0, 0}.Basedon theidealSVNN,wedefinethesinglevaluedneutrosophicpositiveideal-solution (SVNPIS).
Definition3.1 Let α∗ j = {1, 0, 0} (j =1, 2,...,n)be n idealSVNNs,thena SVNPISisdefinedby A∗ = {α ∗ 1 ,α ∗ 2 ,...,α ∗ n}
Definition3.2 Let Ak i = {rk i1,rk i2,...,rk in} (i =1, 2,...,m)bethe i thalternativeofthe kthdecisionmakers(k =1, 2,...,s), A∗ = {α∗ 1 ,α∗ 2 ,...,α∗ n} bethe SVNPIS,thentheweightedHammingdistancemeasure(WHDM)between Ai and A∗ isdefinedby d(Ai,A∗)= s k=1 λk
n j=1 wj d(r k ij ,α ∗ j ) (3)
3.1.1Incompletelyknownattributeweights
Inthedecisionmakingprocess,theincompleteinformationoftheattribute weightprovidedbythedecisionmakerscanusuallybeconstructedusingseveral basicrankingforms[21].Let H bethesetofinformationabouttheincompletely knownattributeweights,whichmaybeconstructedinthefollowingforms[22], for i = j:
(a)Aweakranking:{wi ≥ wj };
(b)Astrictranking: {wi wj ≥ δi(> 0)};
(c)Arankingwithmultiples: {wi ≥ δiwj }, 0 ≤ δi ≤ 1;
(d)Anintervalform: {δi ≤ wi ≤ δi + εi}, 0 ≤ δi ≤ δi + εi;
(e)Arankingofdifferences: {wi wj ≥ wk wl} ,for j = k = l. Wenowestablishthefollowingsingle-objectiveprogrammingmodelbasedon theweighteddistancemeasuremethod:
(M1) Minf (w)= s k=1 λk m i=1 n j=1 wj d(rk ij ,α∗ j ) s.t.wj ∈ H, n j=1 wj =1,wj ≥ 0,j =1, 2,...,n.
where λk istheweightofthedecisionmaker Dk(k =1, 2,...,s)and d(r k ij ,α ∗ j )= 1 3 (|T k ij 1| + I k ij + F k ij )(4) d(rk ij ,α∗ j )representstheweighteddistancemeasurebetweentheattributevalue rk ij andtheSVNPIS α∗ j .Thedesirableweightvector w =(w1,w2,...,wn) shouldmakethesumofalltheweighteddistancemeasure(3)small.Sowe constructthismodeltomaketheoveralldistancesmall. Bysolvingthemodel(M1)withMatlabsoftware,wegettheoptimalsolution w∗ =(w∗ 1 ,w∗ 2 ,...,w∗ n),whichisconsideredastheweightoftheattributes C1,C2,...,Cn.Thenweutilize d(Ai,A∗)torankallthealternatives.The smallertheweighteddistancemeasure,thebetterthealternative.
3.1.2Completelyunknownattributeweights
Iftheinformationabouttheattributeweightiscompletelyunknown,weestablishthefollowingprogrammingmodel: (M2) Minf (w)= s k=1 λk m i=1 n j=1 wj d(rk ij ,α∗ j ) s.t. n j=1 w2 j =1,wj ≥ 0,j =1, 2,...,n.
Tosolvethismodel,weconstructtheLagrangefunctionasfollows: L(w,λ)= s k=1 λk
m i=1
(6)
n j=1 wj d(r k ij ,α ∗ j )+ λ 2 ( n j=1 w 2 j 1)(5) where λ istheLagrangemultiplier. Differentiating(5)withrespectto wj (j =1, 2,...,n)and λ,settingthesepartial derivativesequaltozero,thefollowingsetoftheequationsareobtained: ∂L ∂wj = s k=1 λk m i=1 d(rk ij ,α∗ j )+ wj λ =0 ∂L ∂λ = n j=1 w2 j =1
BysolvingEq.(6),weobtaintheweight wj andnormalizeitwith w∗ j = wj n j=1 wj , thenweget w ∗ j = s k=1 λk m i=1 d(rk ij ,α∗ j ) n j=1 s k=1 λk m i=1 d(rk ij ,α∗ j )
(7)
wegettheoptimalsolution w∗ =(w∗ 1 ,w∗ 2 ,...,w∗ n),whichisconsideredasthe weightoftheattributes C1,C2,...,Cn.Later,wecalculatethedistancemeasure (3)andgetthemostdesirableone.
3.1.3Ilustrativeexample
Example1.Herewechoosethedecisionmakingproblemadaptedfrom[19].An automotivecompanyisdesiredtoselectthemostappropriatesupplierforoneof thekeyelementsinitsmanufacturingprocess.Afterpre-evaluation,foursuppliershaveremainedasalternativesforfurtherevaluation.Inordertoevaluate alternativesuppliers,acommitteecomposedoffourdecisionmakershasbeen
formed.Thecommitteeselectsfourattributestoevaluatethealternatives: (1)C1:productquality,(2)C2:relationshipcloseness,(3)C3:deliveryperformance,(4)C4:price.Supposethattherearefourdecision-makers,denotedby d1,d2,d3,d4,whosecorrespondingweightvectoris λ =(0 25, 0 25, 0 25, 0 25). Thefourpossiblealternativesaretobeevaluatedunderthesefourattributes andareintheformofSVNNsforeachdecision-maker,asshowninthefollowing singlevaluedneutrosophicdecisionmatrix:
{0 4, 0 2, 0 3}{0 4, 0 2, 0 3}{0 2, 0 2, 0 5}{0 7, 0 2, 0 3} {0 6, 0 1, 0 2}{0 6, 0 1, 0 2}{0 5, 0 2, 0 3}{0 5, 0 1, 0 2} {0.3, 0.2, 0.3}{0.5, 0.2, 0.3}{0.1, 0.5, 0.2}{0.1, 0.4, 0.5} {0 7, 0 2, 0 1}{0 6, 0 1, 0 2}{0 4, 0 3, 0 2}{0 4, 0 5, 0 1}
D2 =
{0 1, 0 3, 0 5}{0 5, 0 1, 0 5}{0 3, 0 1, 0 6}{0 4, 0 1, 0 4} {0 2, 0 5, 0 4}{0 3, 0 4, 0 3}{0 2, 0 3, 0 1}{0 2, 0 3, 0 5} {0 5, 0 2, 0 6}{0 2, 0 4, 0 3}{0 5, 0 2, 0 5}{0 1, 0 5, 0 3} {0.2, 0.4, 0.2}{0.1, 0.1, 0.3}{0.1, 0.5, 0.4}{0.5, 0.3, 0.1}
D3 =
{0 3, 0 2, 0 1}{0 3, 0 1, 0 3}{0 1, 0 4, 0 5}{0 2, 0 3, 0 5} {0 6, 0 1, 0 4}{0 6, 0 4, 0 2}{0 5, 0 4, 0 1}{0 5, 0 2, 0 4} {0 3, 0 3, 0 6}{0 4, 0 2, 0 4}{0 2, 0 3, 0 2}{0 3, 0 5, 0 1} {0.3, 0.6, 0.1}{0.5, 0.3, 0.2}{0.3, 0.3, 0.6}{0.4, 0.3, 0.2}
{0 2, 0 2, 0 3}{0 3, 0 2, 0 3}{0 2, 0 3, 0 5}{0 4, 0 2, 0 5} {0 4, 0 1, 0 2}{0 6, 0 3, 0 5}{0 1, 0 2, 0 2}{0 5, 0 1, 0 2} {0.3, 0.5, 0.1}{0.2, 0.2, 0.3}{0.5, 0.4, 0.3}{0.5, 0.3, 0.2} {0.3, 0.1, 0.1}{0.2, 0.1, 0.4}{0.2, 0.3, 0.2}{0.3, 0.1, 0.6}
D4 =
Case1.Incompletelyknownattributeweights Supposetheincompletelyknowninformationoftheattributeweightisgiven asfollows: H = {0 18 ≤ w1 ≤ 0 2, 0 15 ≤ w2 ≤ 0 25, 0 30 ≤ w3 ≤ 0 35, 0 3 ≤ w4 ≤ 0 4, 4 j=1 wj =1}
Step1. Bymodel(M1),weestablishthefollowingmodel: Minf (w)=1 5833w1 +1 5038w2 +1 825w3 +1 625w4 s.t.w ∈ H
Step2. BysolvingthismodelwithMatlabsoftware,wegettheweight vector: w1 =0 18,w2 =0 22,w3 =0 30,w4 =0 30 Step3. Usethedistancemeasure(3),wehave d(A1,A∗)=0 4365,d(A2,A∗)=0 3618,d(A3,A∗)=0 4502,d(A4,A∗)=0 4033
Step4. Rankthealternatives. Since d(A3,A∗)isthebiggest,and d(A2,A∗)isthesmallest,werankthealternativesasfollows: A2 A4 A1 A3, 6
where indicatestherelationshipsuperiororpreferredto,and A2 isthebest alternative.
Case2.Completelyunknownattributeweights
Step1.Bymodel(M2),weestablishthefollowingmodel: Minf (w)=1 5833w1 +1 5038w2 +1 825w3 +1 625w4 s.t. 4 j=1 w2 j =1,wj ≥ 0,j =1, 2, 3, 4 Step2. UseEq.(7)toobtaintheweightvectorofattributes: w ∗ 1 =0 18,w ∗ 2 =0 22,w ∗ 3 =0 30,w ∗ 4 =0 30 Step3. Usethedistancemeasure(4)and(5),wehave d(A1,A∗)=0 4352,d(A2,A∗)=0 3613,d(A3,A∗)=0 4482,d(A4,A∗)=0 3984 Step4. Rankthealternatives.Since d(A3,A∗)isthebiggest,and d(A2,A∗) isthesmallest,werankthealternativesasfollows: A2 A4 A1 A3, where indicatestherelationshipsuperiororpreferredto,and A2 isthebest alternative.
3.2Theweighteddistancemeasurebasedmethodtointervalvaluedneutrosophicset
Let X = {X1,X2,...,Xm} beasetofalternatives, C = {C1,C2,...,Cn} bea setofattributesand w = {w1,w2,...,wn} betheweightvectoroftheattribute with wj ∈ [0, 1]and n j=1 wj =1.Supposethatthereare s decisionmakers D = {D1,D2,...,Ds},whosecorrespondingweightedvectoris λ = {λ1,λ2,...,λs} .Let Ak =(rk ij )m×n (k =1, 2,...,s)beintervalvaluedneutrosophicdecision matrix,where rk ij = {T k ij ,I k ij ,F k ij } isthevalueoftheattribute,expressedby IVNNs.
NowwesupposetheidealIVNNas β∗ j = {t∗,i∗,f ∗} = {[1, 1], [0, 0], [0, 0]} BasedontheidealIVNN,wedefinetheintervalvaluedneutrosophicpositive ideal-solution(IVNPIS).
Definition3.3 Let β∗ j = {[1, 1], [0, 0], [0, 0]} (j =1, 2,...,n)be n idealIVNNs, thenaIVNPISisdefinedby A∗ = {β∗ 1 ,β∗ 2 ,...,β∗ n}.
n j=1 wj d(r k ij ,β∗ j )(8)
3.2.1Incompletelyknownattributeweights
Wenowestablishthefollowingsingle-objectiveprogrammingmodelbasedon theweighteddistancemeasuremethod:
(M3) Minf (w)= s k=1 λk m i=1 n j=1 wj d(rk ij ,β∗ j ) s.t.wj ∈ H, n j=1 wj =1,wj ≥ 0,j =1, 2,...,n. where λk istheweightofthedecisionmaker Dk (k =1, 2,...,s)and d(r k ij ,β∗ j )= 1 6 (|T k ij 1| + |T +k ij 1| + |I k ij | + |I +k ij | + |F k ij | + |F +k ij |)(9) d(rk ij ,β∗ j )representsthedistancemeasurebetweentheattributevalue rk ij and theIVNPIS β∗ j .Thedesirableweightvector w =(w1,w2,...,wn)shouldmake thesumofalltheweighteddistance(8)small.Soweconstructthismodelto maketheoveralldistancessmall.ThesmallertheWHD,thebetterthealternative.Weuse(8)torankthealternative.
Bysolvingthemodel(M3)withMatlabsoftware,wegettheoptimalsolution w∗ =(w∗ 1 ,w∗ 2 ,...,w∗ n),whichisconsideredastheweightoftheattributes C1,C2,...,Cn.Thenweutilize d(Ai,A∗)torankallthealternatives.The smallerthedistance,thebetterthealternative.
3.2.2Completelyunknownattributeweights
Iftheinformationabouttheattributeweightiscompletelyunknown,weestablishthefollowingprogrammingmodel:
(M4) Minf (w)= s k=1 λk m i=1 n j=1 wj d(rk ij ,β∗ j ) s.t. n j=1 w2 j =1,wj ≥ 0,j =1, 2,...,n.
ByLagrangemultiplemethod,wegetthecompletelyunknownweight wj and normalizeitwith w∗ j = wj n j=1 wj asfollows: w ∗ j = s k=1 λk m i=1 d(rk ij ,β∗ j ) n j=1 s k=1 λk m i=1 d(rk ij ,β∗ j )
(10)
whichisconsideredastheweightoftheattributes Cj Later,wecalculatethe distancemeasure d(Ai,A∗),andthengetthemostdesirableone.
3.2.3Ilustrativeexample
Example2.Thedecisionmakingproblemisadaptedfrom[19].SupposeanorganizationplanstoimplementERPsystem.Thefirststepistoformaprojectteam thatconsistsofCIOandtwoseniorrepresentativesfromuserdepartments.By collectingallinformationaboutERPvendorsandsystems,projectteamchoosesfourpotentialERPsystems Ai(i =1, 2, 3, 4)ascandidates.Thecompany employssomeexternalprofessionalorganizations(experts)toaidthisdecision making.Theprojectteamselectsfourattributestoevaluatethealternatives: (1)C1:functionandtechnology,(2)C2:strategicfitness,(3)C3:vendorsability, (4)C4:vendorsreputation.Supposethattherearethreedecision-makers,denotedby D1,D2,D3,whosecorrespondingweightvectoris λ =( 1 3 , 1 3 , 1 3 ).The
fourpossiblealternativesaretobeevaluatedunderthesefourattributesand areintheformofIVNNsforeachdecision-maker,asshowninthefollowing intervalvaluedneutrosophicdecisionmatrix: D1
{{[0 4, 0 5], [0 2, 0 3], [0 3, 0 5]}{[0 3, 0 4], [0 3, 0 6], [0 2, 0 4]}{[0 2, 0 5], [0 2, 0 6], [0 3, 0 5]} {[0 5, 0 6], [0 3, 0 5], [0 2, 0 5]}} {{[0.6, 0.7], [0.1, 0.2], [0.2, 0.3]}{[0.1, 0.3], [0.1, 0.4], [0.2, 0.5]}{[0.4, 0.5], [0.2, 0.5], [0.3, 0.7]} {[0 2, 0 4], [0 1, 0 4], [0 3, 0 3]}} {{[0 3, 0 4], [0 2, 0 3], [0 3, 0 4]}{[0 3, 0 6], [0 2, 0 3], [0 2, 0 5]}{[0 2, 0 7], [0 2, 0 4], [0 3, 0 6]} {[0 2, 0 6], [0 4, 0 7], [0 2, 0 7]}} {{[0 2, 0 6], [0 1, 0 2], [0 1, 0 2]}{[0 2, 0 5], [0 4, 0 5], [0 1, 0 6]}{[0 3, 0 5], [0 1, 0 3], [0 2, 0 2]} {[0.4, 0.4], [0.1, 0.6], [0.1, 0.5]}}
{{[0.4, 0.6], [0.1, 0.3], [0.2, 0.4]}{[0.3, 0.5], [0.1, 0.4], [0.3, 0.4]}{[0.4, 0.5], [0.2, 0.4], [0.1, 0.3]} {[0 3, 0 6], [0 3, 0 6], [0 3, 0 6]}} {{[0 3, 0 5], [0 1, 0 2], [0 2, 0 3]}{[0 3, 0 4], [0 2, 0 2], [0 1, 0 3]}{[0 2, 0 7], [0 3, 0 5], [0 3, 0 6]} {[0 2, 0 5], [0 2, 0 7], [0 1, 0 2]}} {{[0.5, 0.6], [0.2, 0.3], [0.3, 0.4]}{[0.1, 0.4], [0.1, 0.3], [0.3, 0.5]}{[0.5, 0.5], [0.4, 0.6], [0.3, 0.4]} {[0.1, 0.2], [0.1, 0.4], [0.5, 0.6]}} {{[0 3, 0 4], [0 1, 0 2], [0 1, 0 3]}{[0 3, 0 3], [0 1, 0 5], [0 2, 0 4]}{[0 2, 0 3], [0 4, 0 5], [0 5, 0 6]} {[0 3, 0 3], [0 2, 0 3], [0 1, 0 4]}}
{{[0 1, 0 3], [0 2, 0 3], [0 4, 0 5]}{[0 3, 0 3], [0 1, 0 3], [0 3, 0 4]}{[0 2, 0 6], [0 3, 0 5], [0 3, 0 5]} {[0 4, 0 6], [0 3, 0 4], [0 2, 0 3]}} {{[0 3, 0 6], [0 3, 0 5], [0 3, 0 5]}{[0 3, 0 4], [0 3, 0 4], [0 3, 0 5]}{[0 3, 0 5], [0 2, 0 4], [0 1, 0 5]} {[0.1, 0.2], [0.3, 0.5], [0.3, 0.4]}} {{[0 4, 0 5], [0 2, 0 4], [0 2, 0 4]}{[0 2, 0 3], [0 1, 0 1], [0 3, 0 4]}{[0 1, 0 4], [0 2, 0 6], [0 3, 0 6]} {[0 4, 0 5], [0 2, 0 6], [0 1, 0 3]}} {{[0 2, 0 4], [0 3, 0 4], [0 1, 0 3]}{[0 1, 0 4], [0 2, 0 5], [0 1, 0 5]}{[0 3, 0 6], [0 2, 0 4], [0 2, 0 2]} {[0 2, 0 4], [0 3, 0 3], [0 2, 0 6]}}
Case1.Incompletelyknownattributeweights Supposetheincompletelyknowninformationoftheattributeweightisgivenas follows: H = {0 18 ≤ w1 ≤ 0 2, 0 15 ≤ w2 ≤ 0 25, 0 30 ≤ w3 ≤ 0 35, 0 3 ≤ w4 ≤ 0 4, 4 j=1 wj =1}
Step1. Bymodel(M3),weestablishthefollowingmodel: Minf (w)=1 4278w1 +1 7278w2 +1 8278w3 +1 7667w4 s.t.w ∈ H
Step2. BysolvingthismodelwithMatlabsoftware,wegettheweight vector: w1 =0 18,w2 =0 25,w3 =0 20,w4 =0 37 Step3. Usethedistancemeasure(9)and(10),wehave d(A1,A∗)=0 4204,d(A2,A∗)=0 4182,d(A3,A∗)=0 4471,d(A4,A∗)=0 41
Step4. Rankthealternatives. Since d(A3,A∗)isthebiggest,and d(A4,A∗)isthesmallest,werankthealternativesasfollows: A4 A2 A1 A3,
where indicatestherelationshipsuperiororpreferredto,and A4 isthebest alternative.
Case2.Completelyunknownattributeweights
Step1.Bymodel(M4),weestablishthefollowingmodel: Minf (w)=1 4278w1 +1 7278w2 +1 8278w3 +1 7667w4 s.t. 4 j=1 w2 j =1,wj ≥ 0,j =1, 2, 3, 4.
Step2. UseEq.(10)toobtaintheweightvectorofattributes: w ∗ 1 =0 2115,w ∗ 2 =0 2560,w ∗ 3 =0 2708,w ∗ 4 =0 2617
Step3. Usethedistancemeasure(9)and(10),wehave d(A1,A∗)=0 4200,d(A2,A∗)=0 3776,d(A3,A∗)=0 4421,d(A4,A∗)=0 4054
Step4. Rankthealternatives. Since d(A3,A∗)isthebiggest,and d(A2,A∗)isthesmallest,werankthealternativesasfollows: A2 A4 A1 A3, where indicatestherelationshipsuperiororpreferredto,and A2 isthebest alternative.
3.3Comparativeanalysis
ConsideringtheproposedmethodandthemaximizingdeviationmethodproposedbySahin,thereexsitsomedifferences.InSahinsmethod,theycalculated thedistancemeasureofalltheattributesandassignasmallweighttotheattributewhichhasasimilareffectamongthealternatives,thentheyusedthe weightedaggregationoperatorsandthescorefunctionstorankthealternatives;while,theproposedmethodcalculatesthedistancemeasurebetweenthe attributesandtheidealsolution,andobtaintheweightthatmaketheweighted distancemeasuresmall,wethenusetheweighteddistancemeasuretorankthe alternativeswhichavoidthecomplexcalculationofaggregationoperatorsprocessing.Thetwomethodsarealleffectivetodealwiththeincompletelyknown orcompletelyunknownattributeweightbysolvetheprogrammodels.Theadvantageoftheproposedmethodisthatcalculationissimpleandconvenient, whichcandealwiththeMAGDMproblemeffectively.
4Conclusion
Inthispaper,weinvestigatethemulti-attributegroupdecisionmakingproblemsexpressedwithneutrosophicsetandtheattributeweightsareincompletely knownorcompletelyunknown.Wefirstdefinethesinglevaluedneutrosophic idealsolution(SVNIS),andthenestablishtheoptimalmodelstoderivethe attributeweight.Furthermore,anapproachtoMAGDMwithintheframework ofSVNSisdeveloped,andtheresultshowsthatourapproachisreasonable andeffectiveindealingwithdecisionmakingproblems.Finally,weextendthe methodtoIVNS.
ConflictofInterests
Theauthorsdeclarethatthereisnoconflictofinterestsregardingthepublicationofthispaper.
Acknowledgment
ThisworkwasfinanciallysupportedbytheFundamentalResearchFundsfor theCentralUniversities(2572014BB19)andChinaNaturalScienceFundunder grant(11401084).
References
[1]L.A.Zadeh.Fuzzysets.InformationandControl,vol.8,pp.338-353,1965.
[2]L.A.Zadeh.Inthereaneedforfuzzylogic?.InformationSciences,vol.178, pp.2751-2779,2008.
[3]K.Atanassov.Intuitionisticfuzzysets.Fuzzysetsandsystems,vol.20,pp. 87-96,1986.
[4]V.Torra.Hesitantfuzzysets.InternationalJournalofIntelligentSystems, vol.25,pp.529-539,2010.
[5]F.Smarandache.Neutrosophy.NeutrosophicProbability,Set,andLogic, ProQuestInformationandLearning,AnnArbor,Michigan,USA,1998.
[6]Y.Zhang,P.Li,Y.Wang,P.Ma,X.Su.Multiattributedecisionmaking basedonentropyunderinterval-valuedintuitionisticfuzzyenvironment. MathematicalProblemsinEngineering,vol.69,pp.831-842,2013.
[7]H.L.Larsen,R.R.Yager.Aframeworkforfuzzyrecognitiontechnology, IEEETransactionsonSystems.ManandCybernetics-PartC:Applications andReviews,vol.30,pp.65-75,2000.
[8]H.Wang,F.Smarandache,Y.Q.Zhang,R.Sunderraman.Intervalneutrosophicsetsandlogic:theoryandapplicationsincomputing.Hexis, Phoenix,AZ,2005.
[9]H.Wang,F.Smarandache,Y.Q.Zhang,R.Sunderraman.Singlevalued neutrosophicsets.MultispaceMultistruct,no.4,pp.410-413,2010.
[10]P.D.Liu,Y.M.Wang.IntervalneutrosophicprioritizedOWAoperatorand itsapplicationtomultipleattributedecisionmaking,JournalofSystems ScienceandComplextity,DOI:10.1007/s11424-015-4010-7.
[11]S.Broumi,F.Smarandache.Singlevaluedneutrosophictrapezoidlinguistic aggregationoperatorsbasedmulti-attributedecisionmaking.BullPure Appl.Sci,vol.33,no.2,pp.135-155,2014.
[12]C.F.Liu,Y.S.Luo.Correlatedaggregationoperatorsforsimplifiedneutrosophicsetandtheirapplicationinmulti-attributegroupdecisionmaking, JournalofIntelligentandFuzzySystems,vol.30,pp.1755-1761,2016.
[13]Z.S.Xu,M.M.Xia.DistanceandSimilaritymeasuresforhesitantfuzzy sets.InformationSciences,vol.181,pp.2128-2138,2011.
[14]YeJ.Similaritymeasuresbetweenintervalneutrosophicsetsandtheirapplicationinmulticriteriadecisionmaking.Journalofintelligentandfuzzy systems,vol.26,no.1,pp.165-172,2014.
[15]Z.S.Xu.Adeviation-basedapproachtointuitionisticfuzzymultipleattributegroupdecisionmaking.GroupDecision.Neot.,vol.19,pp.57-76, 2010.
[16]Z.S.Xu,Modelsformultipleattributedecisionmakingwithintuitionistic fuzzyinformation.Int.J.Uncertain,FuzzyKnowledge-basedSystems,vol. 15,pp.285-297,2007.
[17]Z.S.Xu,X.Q.Cai,Nonlinearoptimizationmodelsformultipleattribute groupdecisionmakingwithintuitionisticfuzzyinformation.Int.J.Intell.Syst,vol.25,pp.489-513,2010.
[18]D.F.Li,G.H.Chen,Z.G.Huang,LinearprogrammingmethodformultiattributegroupdecisionmakingusingIFsets,InformationScience.vol.180, pp.1591-1609,2010.
[19]RidvanSahin,P.D.Liu.Maximizingdeviationmethodforneutrosophic multipleattributedecisionmakingwithincompleteweightinformation. NeuralComputingandApplications,DOI:10.1007/s00521-015-1995-8.
[20]P.Majumdar,SK.Samanta.Onsimilarityandentropyofneutrosophic sets.JournalofIntelligentFuzzySystems,vol.26,no.3,pp.1245-1252, 2014.
[21]S.H.Kim,S.H.ChoiandK.Kim.Aninteractiveprocedureformultiple attributegroupdecisionmakingwithincompleteinformation:Range-based approach.EuropeanJournalofOperationalResearch,vol.118,pp.139-152, 1999.
[22]K.S.Park.Mathematicalprogrammingmodelsforcharacterizingdominanceandpotentialoptimalitywhenmulticriteriaalternativevaluesand weightsaresimultaneouslyincomplete,IEEETran.Sys.,ManCybernet., A:Syst.Hum.,vol.34,pp.601-614,2004.
[23]S.Broumi,M.Talea,A.Bakali,F.Smarandache.SingleValuedNeutrosophicGraphs,JournalofNewTheory,N10,pp.86-101,2016.
[24]S.Broumi,M.Talea,A.Bakali,F.Smarandache.OnBipolarSingleValued NeutrosophicGraphs,JournalofNewTheory,N11,pp.84-102,2016.
[25]P.D.Liu,Y.C.Chu,Y.W.Li,Y.B.Chen.Somegeneralizedneutrosophic numberHamacheraggregationoperatorsandtheirapplicationtoGroup DecisionMaking,InternationalJournalofFuzzySystems,vol.16,no.2, pp.242-255,2014.
[26]P.D.Liu,Y.M.Wang.MultipleAttributeDecision-MakingMethodBased onSingleValuedNeutrosophicNormalizedWeightedBonferroniMean, NeuralComputingandApplications,vol.25,no.7,pp.2001-2010,2014.