@ E-mail: yehjun aliyun.com48 Neutrosophic Sets and Systems, Vol. 2, 2014 48 Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making Jun Ye1 and Qiansheng Zhang21 Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang 312000, P.R. China. 2 School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, P.R. China. E-mail: zhqiansh01@gdufs.edu.cn Abstract.Similaritymeasuresplayanimportantroleindata mining,patternrecognition,decisionmaking,machinelearning, imageprocessetc.Then,singlevaluedneutrosophicsets (SVNSs)candescribeandhandletheindeterminateand inconsistentinformation,whichfuzzysetsandintuitionistic fuzzysetscannotdescribeanddealwith.Therefore,thepaper proposes new similarity meas-ures between SVNSs based on the minimumandmaxi-mumoperators.Thenamultipleattribute decision-makingmethodbasedontheweightedsimilarity measure of SVNSs is established in which attribute values for al-ternativesarerepresentedbytheformofsinglevalued neutrosophicvalues(SVNVs)andtheattributeweightsandthe weightsofthethreeindependentelements(i.e.,truth-membershipdegree,indeterminacy-membership degree, and falsity-membership degree) in a SVNV are consideredinthedecision-makingmethod.Inthe decisionmaking,weutilizethesingle-valued neutrosophicweightedsimilaritymeasurebetweenthe idealalternativeandanalternativetorankthe alternatives corresponding to the measure values and to selectthemostdesirableone(s).Finally,twopractical examplesareprovidedtodemonstratetheapplications andeffectivenessofthesinglevaluedneutrosophic multiple attribute decision-making method. Keywords: Neutrosophic set, single valued neutrosophic set, similarity measure, decision making. 1 Introduction Sincefuzzysets[1],intuitionisticfuzzysets(IFSs) [2],interval-valuedintuitionisticfuzzysets(IVIFSs)[3] wereintroduced,theyhavebeenwidelyappliedindata mining,patternrecognition,informationretrieval, decisionmaking,machinelearning,image process and so on.Althoughtheyareverysuccessfulintheirrespective domains, fuzzy sets, IFSs, and IVIFSs cannot describe and dealwiththeindeterminateandinconsistentinformation that exists in real world. To handle uncertainty, imprecise, incomplete,andinconsistentinformation,Smarandache [4]proposedtheconceptofaneutrosophicset.The neutrosophicsetisapowerfulgeneralformalframework whichgeneralizestheconceptsoftheclassicset,fuzzy set,IFS,IVIFSetc.[4].Intheneutrosophicset,truth-membership,indeterminacy-membership,andfalsity-membershiparerepresentedindependently.However,the neutrosophicsetgeneralizestheabovementionedsets from philosophical point ofview anditsfunctions TA(x), IA(x)andFA(x)arerealstandardornonstandardsubsets of]0,1+[,i.e.,TA(x):X]0,1+[,IA(x):X]0, 1+[, and FA(x): X ]0, 1+[. Thus, it is difficult to apply inrealscientificandengineeringareas.Therefore,Wang etal.[5,6]introducedasinglevaluedneutrosophicset (SVNS) and an interval neutrosophic set (INS), which are the subclass of aneutrosophic set. They can describe and handle indeterminate information and inconsistent information,whichfuzzysets,IFSs,andIVIFSs cannot describe and deal with. Recently, Ye [7-9] proposedthecorrelationcoefficientsofSVNSs andthecross-entropymeasureofSVNSsand appliedthemtosinglevaluedneutrosophic decision-makingproblems.Then,Ye[10] introducedsimilaritymeasuresbasedonthe distancesbetweenINSsandappliedthemto multicriteriadecision-makingproblemswith intervalneutrosophicinformation.ChiandLiu [11]proposedanextendedTOPSISmethodfor themultipleattributedecisionmakingproblems withintervalneutrosophicinformation. Furthermore,Ye[12]introducedtheconceptof simplifiedneutrosophicsetsandpresented simplifiedneutrosophicweightedaggregation operators,andthenheappliedthemto multicriteriadecision-makingproblemswith simplifiedneutrosophicinformation.Majumdar andSamanta[13]introducedseveralsimilarity measuresbetweenSVNSsbasedondistances,a matchingfunction,membershipgrades,andthen proposedanentropymeasureforaSVNS. BroumiandSmarandache[14]definedthe distance between neutrosophic sets on the basis of theHausdorffdistanceandsomesimilarity Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making Neutrosophic Sets and Systems, Vol. 2, 2014 49 49 measuresbasedonthedistances,settheoreticapproach, andmatchingfunctiontocalculatethesimilaritydegree between neutrosophic sets. Becausetheconceptofsimilarityisfundamentally importantinalmosteveryscientificfieldandSVNSscan describeandhandletheindeterminateandinconsistent information,thispaperproposesnewsimilaritymeasures betweenSVNSsbasedontheminimumandmaximum operatorsandestablishesamultipleattributedecision-making method based on the weighted similarity measure of SVNSs under single valued neutrosophic environment. Todoso,therestofthearticleisorganizedasfollows. Section2introducessomebasicconceptsofSVNSs. Section3proposesnewsimilaritymeasuresbetween SVNSsbasedontheminimumandmaximumoperators andinvestigatestheirproperties.InSection4,asingle valuedneutrosophicdecision-makingapproachis proposedbasedontheweightedsimilaritymeasureof SVNSs. In Section 5, two practical examples are given to demonstratetheapplicationsandtheeffectivenessofthe proposeddecision-makingapproach.Conclusionsand further research are contained in Section 6. 2 Some basic concepts of SVNSs Smarandache[4]originallyintroducedtheconceptof aneutrosophicsetfromphilosophicalpointofview, which generalizes that of fuzzy set, IFS, and IVIFS etc.. Definition 1 [4]. Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A inXischaracterizedbyatruth-membershipfunction TA(x),anindeterminacy-membershipfunctionIA(x)anda falsity-membershipfunctionFA(x).ThefunctionsTA(x), IA(x) and FA(x) are real standard or nonstandard subsets of ]0,1+[.ThatisTA(x):X]0,1+[,IA(x):X]0,1+[, andFA(x):X]0,1+[.Thus,thereisnorestrictionon the sum of TA(x), IA(x) and FA(x), so 0 sup TA(x) + sup IA(x) + sup FA(x) 3+. Obviously, it is difficult to apply in real scientific and engineeringareas.Hence,Wangetal.[6]introducedthe definition of a SVNS. Definition 2 [6]. Let X be a universal set. A SVNS A in X ischaracterizedbyatruth-membershipfunctionTA(x),an indeterminacy-membershipfunctionIA(x), andafalsity-membershipfunctionFA(x).Then,aSVNSAcanbe denoted by { } X x x F x I x T x AA A Ae = | ) ( ), ( ), ( , , whereTA(x),IA(x),FA(x)e[0,1]foreachpointxinX. Therefore,thesumofTA(x),IA(x)andFA(x)satisfiesthe condition 0 TA(x) + IA(x) + FA(x) 3. Definition 3 [6]. The complement of a SVNS A is denotedbyAcandisdefinedasTAc(x)=FA(x), IAc(x)=1IA(x),FAc(x)=TA(x)foranyxinX. Then, it can be denoted by { } X x x T x I x F x AA A Ace = | ) ( ), ( 1 ), ( , . Definition4[6].ASVNSAiscontainedinthe other SVNS B, A B, if and only if TA(x) TB(x), IA(x) IB(x), FA(x) FB(x) for any x in X. Definition 5 [6]. Two SVNSs A andB are equal, i.e., A = B, if and only if A B and B A.3 Similarity measures of SVNSs Thissectionproposesseveralsimilarity measuresofSVNSsbasedontheminimumand maximumoperatorsandinvestigatestheir properties.Ingeneral,asimilaritymeasurebetweentwo SVNSs A and B is a function definedas S: N(X)2 [0, 1] which satisfies the following properties: (SP1) 0 s S(A, B) s 1; (SP2) S(A, B) = 1 if A = B; (SP3) S(A, B) = S(B, A); (SP4) S(A, C) s S(A, B) and S(A, C) s S(B, C) if A _ B _ C for a SVNS C. LettwoSVNSsAandBinauniverseof discourseX={xl,x2,,xn}be { } X x x F x I x T x Ai i A i A i A ie = | ) ( ), ( ), ( , and { } X x x F x I x T x Bi i B i B i B ie = | ) ( ), ( ), ( ,,where TA(xi),IA(xi),FA(xi),TB(xi),IB(xi),FB(xi)e[0,1] foreveryxieX.Basedontheminimumand maximumoperators,wepresentthefollowing similarity measure between A and B: ( )( )( )( )( )( )||.|+

\|+ ==) ( ), ( max) ( ), ( min) ( ), ( max) ( ), ( min) ( ), ( max) ( ), ( min31) , (11i B i Ai B i Ai B i Ai B i Ani i B i Ai B i Ax F x Fx F x Fx I x Ix I x Ix T x Tx T x TnB A S. (1) Thesimilaritymeasurehasthefollowing proposition. Proposition1.LetAandBbetwoSVNSsina universeofdiscourseX={x1,x2,,xn}.The singlevaluedneutrosophicsimilaritymeasure S1(A, B) should satisfy the following properties: Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making =1/3,Eq.(2) reduces 50 Neutrosophic Sets and Systems, Vol. 2, 2014 50 (SP1) 0 s S1(A, B) s 1; (SP2) S1(A, B) = 1 if A = B; (SP3) S1(A, B) = S1(B, A); (SP4) S1(A, C) s S1(A, B) and S1(A, C) s S1(B, C) if A _ B _ C for a SVNS C. Proof.ItiseasytoremarkthatS1(A,B)satisfiesthe properties (SP1)-(SP3). Thus, we must prove the property (SP4). LetA_B_C,then,TA(xi)sTB(xi)sTC(xi),IA(xi)> IB(xi) > IC(xi), and FA(xi) > FB(xi) > FC(xi) for every xi e X. Accordingtotheseinequalities,wehavethefollowing similarity measures: =||.|

\|+ + =ni i Ai Bi Ai Bi Bi Ax Fx Fx Ix Ix Tx TnB A S11) () () () () () (31) , (, =||.|

\|+ + =ni i Ai Ci Ai Ci Ci Ax Fx Fx Ix Ix Tx TnC A S11) () () () () () (31) , (=||.|

\|+ + =ni i Bi Ci Bi Ci Ci Bx Fx Fx Ix Ix Tx TnC B S11) () () () () () (31) , (. Since there are ) () (i Bi Ax Tx T>) () (i ci Ax Tx T, ) () (i Ai Bx Ix I> ) () (i Ai Cx Ix I, and ) () (i Ai Bx Fx F> ) () (i Ai Cx Fx F, we can obtain that S1(A, C) s S1(A, B). Similarly,thereare ) () (i Ci Bx Tx T> ) () (i Ci Ax Tx T, ) () (i Bi Cx Ix I> ) () (i Ai Cx Ix I,and ) () (i Bi Cx Fx F> ) () (i Ai Cx Fx F.Then,wecanobtain that S1(A, C) s S1(B, C). Thus S1(A, B) satisfies the property (SP4). Therefore, we finish the proof.If the important differences are considered in the three independentelements(i.e.,truth-membership, indeterminacy-membership,andfalsity-membership)ina SVNS,weneedtotaketheweightsofthethree independenttermsinEq.(1)intoaccount.Therefore,we develop another similarity measure between SVNSs: ( )( )( )( )( )( )||.|+

\|+ = =) ( ), ( max) ( ), ( min) ( ), ( max) ( ), ( min) ( ), ( max) ( ), ( min 1) , (12i B i Ai B i Ai B i Ai B i Ani i B i Ai B i Ax F x Fx F x Fx I x Ix I x Ix T x Tx T x TnB A S |o, (2) whereo,|,aretheweightsofthethree independentelements(i.e.,truth-membership, indeterminacy-membership,andfalsity-membership)inaSVNSando+|+=1. Especially,wheno=|= to Eq. (1). Then,thesimilaritymeasureofS2(A,B)also has the following proposition: Proposition2.LetAandBbetwoSVNSsina universeofdiscourseX={x1,x2,,xn}.The singlevaluedneutrosophicsimilaritymeasure S2(A, B) should satisfy the following properties: (SP1) 0 s S2(A, B) s 1; (SP2) S2(A, B) = 1 if A = B; (SP3) S2(A, B) = S2(B, A); (SP4) S2(A, C) s S2(A, B) and S2(A, C) s S2(B, C) if A _ B _ C for a SVNS C.By the similar proof method in Proposition 1, wecanprovethatthesimilaritymeasureofS2(A, B) alsosatisfiestheproperties(SP1)-(SP4)(omitted). Furthermore,iftheimportantdifferencesare consideredintheelementsinauniverseof discourse X = {xl, x2, , xn}, we need to take the weightofeachelementxi(i=1,2,,n)into account.Therefore,wedevelopaweighted similarity measure between SVNSs. Let wi be the weight for each element xi (i = 1, 2,, n), wi e [0, 1], and 11==niiw, and then we have the following weighted similarity measure: ( )( )( )( )( )( )||.|+

\|+ ==) ( ), ( max) ( ), ( min) ( ), ( max) ( ), ( min) ( ), ( max) ( ), ( min) , (13i B i Ai B i Ai B i Ai B i Ani i B i Ai B i Aix F x Fx F x Fx I x Ix I x Ix T x Tx T x Tw B A S |o. (3) Similarly, the weighted similarity measure of S3(A, B) also has the following proposition: Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making , then Eq. (3) reduces to Eq. (2).Neutrosophic Sets and Systems, Vol. 2, 2014 51 51 Proposition 3. Let A and B be two SVNSs in a universe of discourseX={x1,x2,,xn}.Then,thesinglevalued neutrosophicsimilaritymeasureS3(A,B)shouldsatisfy the following properties: (SP1) 0 s S3(A, B) s 1; (SP2) S3(A, B) = 1 if A = B; (SP3) S3(A, B) = S3(B, A); (SP4) S3(A, C) s S3(A, B) and S3(A, C) s S3(B, C) if A _ B _ C for a SVNS C. SimilartotheproofmethodinProposition1,wecan prove that the weighted similarity measure of S3(A, B) also satisfies the properties (SP1)(SP4) (omitted). If w = (1/n, 1/n,, 1/n)T ForExample,Assumethatwehavethefollowing three SVNSs in a universe of discourse X = {xl, x2}: A = {, }, B = {, }, C = {, }. Then,thereareA_B_C,withTA(xi)sTB(xi)s TC(xi),IA(xi)>IB(xi)>IC(xi),andFA(xi)>FB(xi)>FC(xi) for each xi in X = {x1, x2}. By using Eq. (1), the similarity measures between the SVNSs are as follows: S1(A,B)=0.6996,S1(B,C)=0.601,andS1(A,C)= 0.4206. Thus,thereareS1(A,C)sS1(A,B)andS1(A,C)s S1(B, C). If the weight values of the three independent elements (i.e.,truth-membershipdegree,indeterminacy-membershipdegree,andfalsity-membershipdegree)ina SVNS are o = 0.25, | = 0.35, and = 0.4, by applying Eq. (2)thesimilaritymeasuresbetweentheSVNSsareas follows: S2(A,B)=0.6991,S2(B,C)=0.5916,andS2(A,C)= 0.4143. Then,thereareS2(A,C)sS2(A,B)andS2(A,C)s S2(B, C). Assume that the weight vector of the two attributes is w=(0.4,0.6)Tandtheweightvaluesofthethree independentelements(i.e.,truth-membershipdegree, indeterminacy-membershipdegree,andfalsity-membershipdegree)inaSVNSareo=0.25,|=0.35, and=0.4.ByapplyingEq.(3),theweightedsimilarity measures between the SVNSs are as follows: S3(A,B)=0.7051,S3(B,C)=0.4181,andS3(A,C)= 0.5912. Hence, there are S3(A, C) s S3(A, B) and S3(A, C) s S3(B, C).4 Decisions making method using the weighted similarity measure of SVNSs In this section, we propose a multiple attribute decision-makingmethodbasedontheweighted similaritymeasuresbetweenSVNSsundersingle valued neutrosophic environment.Let A = {A1, A2,, Am} be a set of alternatives andC={C1,C2,,Cn}beasetofattributes. AssumethattheweightoftheattributeCj(j=1, 2,,n)iswjwithwje[0,1], 11= =njjxandthe weightsofthethreeindependentelements(i.e., truth-membership,indeterminacy-membership, and falsity-membership) in a SVNS are o, |, and ando+|+=1,whichareenteredbythe decision-maker.Inthiscase,thecharacteristicof the alternative Ai (i = 1, 2,, m) on an attribute Cj (j = 1, 2,, n) is represented by a SVNS form: } | ) ( ), ( ), ( , { C C C F C I C T C Aj j A j A j A j ii i ie ) ( =, where ) (j AC Fi, ) (j AC Ii, ) (j AC Fie [0, 1] and 0 s ) (j AC Ti + ) (j AC Ii + ) (j AC Fi s 3 for Cj e C, j = 1, 2, , n, and i = 1, 2, , m. For convenience, the three elements ) (j AC Ti, ) (j AC Ii, ) (j AC FiintheSVNSaredenotedbya single valued neutrosophic value (SVNV) aij = (tij, iij,fij)(i=1,2,,m;j=1,2,,n),whichis usuallyderivedfromtheevaluationofan alternative Ai with respect to an attribute Cj by the expert or decision maker. Hence, we can establish asinglevaluedneutrosophicdecisionmatrixD= (aij)mn: |||||.|

\|=mn m mnna a aa a aa a aD 2 12 22 211 12 11. Inmultipleattributedecisionmaking environments, the concept of ideal point has been usedtohelpidentifythebestalternativeinthe decisionset[7,8].Generally,theevaluation attributescanbecategorizedintotwokinds: benefitattributesandcostattributes.LetHbea collectionofbenefitattributesandLbea collectionofcostattributes.Inthepresented decision-makingmethod,anidealalternativecan be identified by using a maximum operator for the benefit attributes and a minimum operator for the cost attributes to determine the best value of each Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making 52 Neutrosophic Sets and Systems, Vol. 2, 2014 52 attributeamongallalternatives.Therefore,wedefinean idealSVNVforabenefitattributeintheidealalternative A* as ) ( min ), ( min ), ( max , ,* * * *ijiijiijij j j jf i t f i t a = = for jeH; while for a cost attribute, we define an ideal SVNV in the ideal alternative A* as ) ( max ), ( max ), ( min , ,* * * *ijiijiijij j j jf i t f i t a = = for jeL. Thus, by applying Eq. (3), the weighted similarity measure between an alternative Ai and the ideal alternative A* are written as ( )( )( )( )( )( )||.|+

\|+ ==****1***4, max) , min, max) , min, max) , min) , (j ijj ijj ijj ijnj j ijj ijj if ff fi ii it tt tw A A S |o, (4) whichprovidestheglobalevaluationforeachalternative regardingallattributes.Accordingtotheweighted similaritymeasurebetweeneachalternativeandtheideal alternative, the bigger the measure valueS4(Ai, A*) (i = 1, 2,3,4),thebetterthealternativeAi.Hence,theranking orderofallalternativescanbedeterminedandthebest one can be easily selected as well. 5 Practical examples Thissectionprovidestwopracticalexamplesfor multipleattributedecision-makingproblemswithsingle valuedneutrosophicinformationtodemonstratethe applicationsandeffectivenessoftheproposeddecision-making method. Example 1. Let us consider the decision-making problem adaptedfrom[7,8].Thereisaninvestmentcompany, which wants to invest a sum of money in the best option. Thereisapanelwithfourpossiblealternativestoinvest themoney:(1)A1isacarcompany;(2)A2isafood company; (3) A3 is a computer company; (4) A4 is an arms company.Theinvestmentcompanymusttakeadecision accordingtothethreeattributes:(1)C1is therisk;(2)C2 isthegrowth;(3)C3istheenvironmentalimpact,where C1andC2arebenefitattributesandC3isacostattribute. Theweightvectorofthethreeattributesisgivenbyw= (0.35, 0.25, 0.4)T. The four possible alternatives are to be evaluatedundertheabovethreeattributesbytheformof SVNVs.FortheevaluationofanalternativeAi(i=1,2,3,4) withrespecttoanattributeCj(j=1,2,3),itisobtained fromthequestionnaireofadomainexpert.Forexample, when we ask the opinion of an expert about an alternative A1 with respect to an attribute C1, he/she may say that the possibility in which the statement is good is0.4andthestatementispooris0.3andthe degreeinwhichhe/sheisnotsureis0.2.Forthe neutrosophic notation, it can be expressed as a11 = (0.4,0.2,0.3).Thus,whenthefourpossible alternativeswithrespecttotheabovethree attributesareevaluatedbytheexpert,wecan obtainthefollowingsinglevaluedneutrosophic decision matrix D: ((((((

=8 . 0 , 3 . 0 , 6 . 0 2 . 0 , 1 . 0 , 6 . 0 1 . 0 , 0 . 0 , 7 . 08 . 0 , 3 . 0 , 5 . 0 3 . 0 , 2 . 0 , 5 . 0 3 . 0 , 2 . 0 , 3 . 08 . 0 , 2 . 0 , 5 . 0 2 . 0 , 1 . 0 , 6 . 0 2 . 0 , 1 . 0 , 6 . 05 . 0 , 2 . 0 , 8 . 0 3 . 0 , 2 . 0 , 4 . 0 3 . 0 , 2 . 0 , 4 . 0D. Withoutlossofgenerality,lettheweight valuesofthethreeindependentelements(i.e., truth-membershipdegree,indeterminacy-membershipdegree,andfalsity-membership degree)inaSVNV beo= |= = 1/3.Thenwe utilize the developed approach to obtain the most desirable alternative(s). Firstly,fromthesinglevaluedneutrosophic decisionmatrixwecanyieldthefollowingideal alternative: } 8 . 0 , 3 . 0 , 5 . 0 , , 2 . 0 , 1 . 0 , 6 . 0 , , 1 . 0 , 0 . 0 , 7 . 0 , {3 2 1*) ( ) ( ) ( = C C C A. Then, by using Eq. (4) we can obtain the values of the weighted similarity measure S4(Ai, A*) (i =1, 2, 3, 4): S4(A1, A*) = 0.6595, S4(A2, A*) = 0.9805, S4(A3, A*) = 0.7944, and S4(A4, A*) = 0.9828. Thus,therankingorderofthefour alternativesisA4A2A3A1.Therefore,the alternativeA4isthebestchoiceamongthefour alternatives. Fromtheaboveresultswecanseethatthe rankingorderofthealternativesandbestchoice are in agreement with the results (i.e., the ranking orderisA4A2A3A1andthe bestchoiceis A4.) in Yes method [8], but not in agreement with the results (i.e., the ranking order is A2 A4 A3 A1andthebestchoiceisA2.)inYesmethod [7]. The reason is that different measuremethods mayyielddifferentrankingordersofthe alternatives in the decision-making process.Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making Neutrosophic Sets and Systems, Vol. 2, 2014 53 53 Example 2. A multi-criteria decision making problem adoptedfromYe[9]isconcernedwithamanufacturing companywhichwantstoselectthebestglobalsupplier accordingtothecorecompetenciesofsuppliers.Now suppose that there are aset offour suppliersA ={A1, A2, A3, A4} whose core competenciesare evaluated bymeans ofthefourattributes(C1,C2,C3,C4):(1)thelevelof technology innovation (C1), (2) the control ability of flow (C2),(3)theabilityofmanagement(C3),(4)thelevelof service(C4),whichareallbenefitattributes.Then,the weight vector for the four attributes is w = (0.3, 0.25, 0.25, 0.2)T.Thefourpossiblealternativesaretobeevaluated under the above four attributes by the form of SVNVs. FortheevaluationofanalternativeAi(i=1,2,3,4) with respect to an attribute Cj (j =1, 2, 3, 4), by the similar evaluationmethodinExample1itisobtainedfromthe questionnaireofadomainexpert.Forexample,whenwe asktheopinionofanexpertaboutanalternativeA1with respecttoanattributeC1,he/shemaysaythatthe possibilityinwhichthestatementisgoodis0.5andthe statement is poor is 0.3 and the degreein whichhe/sheis notsureis0.1.Fortheneutrosophicnotation,itcanbe expressedasa11=0.5,0.1,0.3 .Thus,whenthefour possiblealternativeswithrespecttotheabovefour attributesareevaluatedbythesimilarmethodfromthe expert,wecanestablishthefollowingsinglevalued neutrosophic decision matrix D:1 . 0 , 2 . 0 , 7 . 0 2 . 0 , 3 . 0 , 4 . 0 5 . 0 , 2 . 0 , 2 . 0 2 . 0 , 1 . 0 , 6 . 02 . 0 , 2 . 0 , 6 . 0 4 . 0 , 0 . 0 , 5 . 0 3 . 0 , 1 . 0 , 5 . 0 1 . 0 , 3 . 0 , 4 . 02 . 0 , 3 . 0 , 5 . 0 1 . 0 , 0 . 0 , 9 . 0 4 . 0 , 2 . 0 , 3 . 0 3 . 0 , 2 . 0 , 4 . 01 . 0 , 2 . 0 , 3 . 0 2 . 0 , 1 . 0 , 7 . 0 4 . 0 , 1 . 0 , 5 . 0 3 . 0 , 1 . 0 , 5 . 0D. Without loss of generality, let the weight values of the threeindependentelements(i.e.,truth-membership degree,indeterminacy-membershipdegree,andfalsity-membership degree) in a SVNV be = = = 1/3. Then theproposeddecision-makingmethodisappliedtosolve this problem for selecting suppliers. Fromthesinglevaluedneutrosophicdecisionmatrix, we can yield the following ideal alternative: } 1 . 0 , 2 . 0 , 7 . 0 , , 1 . 0 , 0 . 0 , 9 . 0 ,, 3 . 0 , 1 . 0 , 5 . 0 , , 1 . 0 , 1 . 0 , 6 . 0 , {4 32 1*C CC C A. ByapplyingEq.(4),theweightedsimilaritymeasure valuesbetweenanalternativeAi(i=1,2,3,4)andthe ideal alternative A* are as follows:S4(A1, A*) = 0.7491, S4(A2, A*) = 0.7433,S4(A3, A*) = 0.7605, and S4(A4, A*) = 0.6871.According to the measure values, the ranking order of thefoursuppliersisA3A1A2A4.Hence,thebest supplier is A3. From the results we can see that the ranking order of the alternatives andbest choice arein agreement with the results in Yes method [9]. Fromtheabovetwoexamples,wecansee thattheproposedsinglevaluedneutrosophic multipleattributedecision-makingmethodis moresuitableforrealscientificandengineering applicationsbecauseitcanhandlenotonly incomplete information but also the indeterminate informationandinconsistentinformationwhich existcommonlyinrealsituations.Especially,in theproposeddecision-makingmethodwe considertheimportantdifferencesinthethree independentelements(i.e.,truth-membership degree,indeterminacy-membershipdegree,and falsity-membershipdegree)inaSVNVandcan adjust theweightvaluesofthethreeindependent elements.Thus,theproposedsinglevalued neutrosophicdecision-makingmethodismore flexibleandpracticalthantheexistingdecision-making methods [7-9]. The technique proposed in thispaperextendstheexistingdecision-making methodsandprovidesanewwayfordecision-makers. 6 Conclusion Thispaperhasdevelopedthreesimilarity measures between SVNSs based on the minimum andmaximumoperatorsandinvestigatedtheir properties. Then the proposed weighted similarity measureofSVNSshasbeenappliedtomultiple attributedecision-makingproblemsundersingle valuedneutrosophicenvironment.Theproposed methoddiffersfrompreviousapproachesfor singlevaluedneutrosophicmultipleattribute decisionmakingnotonlydueto thefactthatthe proposedmethodusetheweightedsimilarity measureofSVNSs,butalsoduetoconsidering theweightsofthetruth-membership,indeter-minacy-membership,andfalsity-membershipin SVNSs,whichmakesithavemoreflexibleand practicalthanexistingdecisionmakingmethods [7-9]inrealdecisionmakingproblems.Through theweightedsimilaritymeasurebetweeneach alternative and the ideal alternative, we can obtain therankingorderofallalternativesandthebest alternative.Finally,twopracticalexamples demonstratedtheapplicationsandeffectiveness ofthedecision-makingapproachundersingle valuedneutrosophicenvironments.Theproposed decision-making method can effectively deal with decision-makingproblemswiththeincomplete, indeterminate, and inconsistent information which existcommonlyinrealsituations.Furthermore, bythesimilarmethodwecaneasilyextendthe proposedweightedsimilaritymeasureofSVNSs anditsdecision-makingmethodtothatofINSs. Inthefuture,weshallinvestigatesimilarity measuresbetweenSVNSsandbetweenINSsin the applications of other domains, such as pattern recognition,clusteringanalysis,imageprocess, and medical diagnosis. Jun Ye, Qiansheng Zhang, Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making 54 Neutrosophic Sets and Systems, Vol. 2, 2014 54 ACKNOWLEDGEMENTS ThisworkwassupportedbytheNationalSocial ScienceFundofChina(13CGL130),theGuangdong ProvinceNaturalScienceFoundation(S2013010013050), theHumanitiesandSocialSciencesResearchYouth FoundationofMinistryofEducationofChina (12YJCZH281),andtheNationalStatisticalScience Research Planning Project (2012LY159). References [1]L.A. Zadeh. Fuzzy Sets. Information and Control, 8 (1965), 338-353. [2]K.Atanassov.Intuitionisticfuzzysets.FuzzySetsand Systems, 20 (1986), 87-96. [3]K.AtanassovandG.Gargov.Intervalvaluedintuitionistic fuzzy sets. Fuzzy Sets and Systems, 31 (1989), 343-349. 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