Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information A novel method for single-valued neutrosophic multi-criteria decision making with incomplete weight information Zhiming Zhang1,2 and Chong Wu21College of Mathematics and Computer Science, Hebei University, Baoding, 071002, China. E-mail: zhimingzhang@ymail.com 2School of Management, Harbin Institute of Technology, Harbin, 150001, China. E-mail: wuchong@hit.edu.cnAbstract. A single-valued neutrosophic set (SVNS) and an interval neutrosophic set (INS) are two instances of a neutrosoph-icset,whichcanefficientlydealwithuncertain,impre-cise,incomplete,andinconsistentinformation.Inthis paper,wedevelopanovelmethodforsolvingsingle-valuedneutrosophicmulti-criteriadecisionmakingwith incompleteweightinformation,inwhichthecriterion values are given in the form of single-valued neutrosoph-icsets(SVNSs),andtheinformationaboutcriterion weightsisincompletelyknownorcompletelyunknown. Thedevelopedmethodconsistsoftwostages.Thefirst stage is to use the maximizing deviation method to estab-lishanoptimizationmodel,whichderivestheoptimal weightsofcriteriaundersingle-valuedneutrosophicen-vironments.Afterobtainingtheweightsofcriteria through the above stage, the second stage is to develop a single-valuedneutrosophicTOPSIS(SVNTOPSIS) method to determine a solution with the shortest distance tothesingle-valuedneutrosophicpositiveidealsolution (SVNPIS)andthegreatestdistancefromthesingle-valuedneutrosophicnegativeidealsolution(SVNNIS). Moreover,abestglobalsupplierselectionproblemis usedtodemonstratethevalidityandapplicabilityofthe developedmethod.Finally,the extendedresultsininter-val neutrosophic situations are pointed out and a compar-ison analysis with the other methods is given to illustrate the advantages of the developed methods. Keywords: neutrosophic set, single-valued neutrosophic set (SVNS), interval neutrosophic set (INS), multi-criteria decisionmak-ing (MCDM), maximizing deviation method; TOPSIS. 1. IntroductionNeutrosophy,originallyintroducedbySmarandache[12],isabranchofphilosophywhichstudiestheorigin, natureandscopeofneutralities,aswellastheirinterac-tionswithdifferentideationalspectra[12].Asapowerful general formal framework, neutrosophic set [12] generaliz-estheconceptoftheclassicset,fuzzyset[24],interval-valued fuzzy set [14,25], vague set [4], intuitionistic fuzzy set [1], interval-valued intuitionistic fuzzy set [2], paracon-sistent set [12], dialetheist set [12], paradoxist set [12], and tautological set [12]. In the neutrosophic set, indeterminacy isquantifiedexplicitlyandtruth-membership,indetermi-nacy-membership, and falsity-membership are independent, which is a very important assumption in many applications such as information fusion in which the data are combined fromdifferentsensors[12].Recently,neutrosophicsets have been successfully applied to image processing [3,5,6]. Theneutrosophicsetgeneralizestheabovementioned setsfromphilosophicalpointofview.Fromscientificor engineeringpointofview,theneutrosophicsetandset-theoreticoperatorsneedtobespecified.Otherwise,itwill be difficult to apply in the real applications [16,17]. There-fore, Wang et al. [17] defined a single valued neutrosophic set(SVNS),andthenprovidedthesettheoreticoperators andvariouspropertiesofsinglevaluedneutrosophicsets (SVNSs).Furthermore,Wangetal. [16] proposedtheset-theoreticoperatorsonaninstanceofneutrosophicset calledintervalneutrosophicset(INS).Asingle-valued neutrosophicset(SVNS)andanintervalneutrosophicset (INS)aretwoinstancesofaneutrosophicset,whichgive us an additional possibility to represent uncertainty, impre-cise,incomplete,andinconsistentinformationwhichexist in real world. Single valued neutrosophic sets andinterval neutrosophic sets are different from intuitionistic fuzzy sets andinterval-valuedintuitionisticfuzzysets.Intuitionistic fuzzysetsandinterval-valuedintuitionisticfuzzysetscan only handle incomplete information, but cannot handle the indeterminateinformationandinconsistentinformation which exist commonly in real situations. The connectors in the intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set are defined with respect to membership and non-membershiponly(hencetheindeterminacyiswhatisleft from 1), while in the single valued neutrosophic set and in-terval neutrosophic set, they can be defined with respect to anyofthem(norestriction).Forexample[17],whenwe ask the opinion of an expert about certain statement, he or shemaysaythatthepossibilityinwhichthestatementis true is 0.6 and the statement is false is 0.5 and the degree in which he or she is not sure is 0.2. This situation can be ex-pressedasasinglevaluedneutrosophicset0.6, 0.2, 0.5 , whichisbeyondthescopeoftheintuitionisticfuzzyset. 35Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information For another example [16], suppose that an expert may say that the possibility that the statement is true is between 0.5 and 0.6, andthestatementisfalseisbetween0.7 and0.9, and the degree that he or she is not sure is between 0.1 and 0.3.Thissituationcanbeexpressedasanintervalneutro-sophicset| | | | | | 0.5, 0.6 , 0.1, 0.3 , 0.7, 0.9 ,whichisbeyondthe scope of the interval-valued intuitionistic fuzzy set. Due to their abilities to easily reflect the ambiguous na-tureofsubjectivejudgments,singlevaluedneutrosophic sets(SVNSs)andintervalneutrosophicsets(INSs)are suitableforcapturingimprecise,uncertain,andincon-sistentinformationinthemulti-criteriadecisionanalysis [20,21,22,23].Mostrecently,somemethods[20,21,22,23] have been developed for solving the multi-criteria decision making (MCDM) problems with single-valued neutrosoph-icorintervalneutrosophicinformation.Forexample,Ye [20] developed a multi-criteria decision making method us-ingthecorrelationcoefficientundersingle-valuedneutro-sophicenvironments.Ye[21]definedthesinglevalued neutrosophiccrossentropy,basedonwhich,amulti-criteriadecisionmakingmethodisestablishedinwhich criteriavaluesforalternativesaresinglevaluedneutro-sophicsets(SVNSs).Ye[23]proposedasimplifiedneu-trosophic weighted arithmetic average operator and a sim-plifiedneutrosophicweightedgeometricaverageoperator, andthenutilizedtwoaggregationoperatorstodevelopa methodformulti-criteriadecisionmakingproblemsunder simplified neutrosophic environments. Ye [22] defined the similaritymeasuresbetweenintervalneutrosophicsets (INSs),andthenutilizedthesimilaritymeasuresbetween eachalternativeandtheidealalternativetorankthealter-natives and to determine the best one. However, it is noted thattheaforementionedmethodsneedtheinformation aboutcriterionweightstobeexactlyknown.Whenusing thesemethods,theassociatedweightingvectorismoreor lessdeterminedsubjectivelyandthedecisioninformation itself is not taken into consideration sufficiently. In fact, in the process of multi-criteria decision making (MCDM), we often encounter the situations in which the criterion values taketheformofsinglevaluedneutrosophicsets(SVNSs) orintervalneutrosophicsets(INSs),andtheinformation aboutattributeweightsisincompletelyknownorcom-pletelyunknownbecauseoftimepressure,lackof knowledge or data, and the experts limited expertise about theproblemdomain[18].Consideringthattheexisting methods are inappropriate for dealing with such situations, in this paper, we develop a novel method for single valued neutrosophic or interval neutrosophic MCDM with incom-plete weight information, in which the criterion values take the form of single valued neutrosophic sets (SVNSs) or in-tervalneutrosophicsets(INSs),andtheinformationabout criterion weights is incompletely known or completely un-known.Thedevelopedmethodiscomposedoftwoparts. First,weestablishanoptimizationmodelbasedonthe maximizing deviation method to objectively determine the optimalcriterionweights.Then,wedevelopanextended TOPSISmethod,whichwecallthesinglevaluedneutro-sophicorintervalneutrosophicTOPSIS,tocalculatethe relative closeness coefficient of each alternative to the sin-glevaluedneutrosophicorintervalneutrosophicpositive ideal solution and to select the optimal one with the maxi-mum relative closeness coefficient. Two illustrative exam-plesandcomparisonanalysiswiththeexistingmethods showthatthedevelopedmethodscannotonlyrelievethe influence of subjectivity of the decision maker but also re-main the original decision information sufficiently. Todoso,theremainderofthispaperissetoutas follows.Section2brieflyrecallssomebasicconceptsof neutrosophicsets,single-valuedneutrosophicsets (SVNSs),andintervalneutrosophicsets(INSs).Section3 developsanovelmethodbasedonthemaximizing deviationmethodandthesingle-valuedneutrosophic TOPSIS(SVNTOPSIS)forsolvingthesingle-valued neutrosophicmulti-criteriadecisionmakingwith incomplete weight information. Section 4 develops a novel method based on the maximizing deviation method and the interval neutrosophic TOPSIS (INTOPSIS) for solving the intervalneutrosophicmulti-criteriadecisionmakingwith incompleteweightinformation.Section5providestwo practicalexamplestoillustratetheeffectivenessand practicalityofthedevelopedmethods.Section6endsthe paper with some concluding remarks. 2 Neutrosophic sets and and SVNSs In this section, we will give a brief overview of neutro-sophicsets[12],single-valuedneutrosophicset(SVNSs) [17], and interval neutrosophic sets (INSs) [16]. 2.1 Neutrosophic sets Neutrosophicsetisa part of neutrosophy, whichstud-iestheorigin,nature,andscopeofneutralities,aswellas their interactions with different ideational spectra [12], and is a powerful general formal framework, which generalizes the above mentioned sets from philosophical point of view. Smarandache[12]definedaneutrosophicsetasfol-lows: Definition2.1 [12].LetXbe a space of points (objects), with a generic element inXdenoted byx . A neutrosoph-icsetAinXischaracterizedbyatruth-membership function( )AT x ,aindeterminacy-membershipfunction ( )AI x ,andafalsity-membershipfunction( )AF x .The functions( )AT x ,( )AI xand( )AF xarerealstandardor nonstandardsubsetsof0 ,1 +( . Thatis ( ) : 0 ,1AT x X +( , ( ) : 0 ,1AI x X +( ,and( ) : 0 ,1AF x X +( .36Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information There is no restriction on the sum of( )AT x ,( )AI xand ( )AF x , so( ) ( ) ( ) 0 sup sup sup 3A A AT x I x F x +s + + s . Definition 2.2 [12]. The complement of a neutrosophic set Aisdenotedby cAandisdefinedas ( ) { } ( ) 1cAAT x T x+= ,( ) { } ( ) 1cAAI x I x+= ,and ( ) { } ( ) 1cAAF x F x+=for everyx inX . Definition2.3[12].AneutrosophicsetA iscontainedin theotherneutrosophicsetB ,A B _ifandonlyif ( ) ( ) inf infA BT x T x s ,( ) ( ) sup supA BT x T x s , ( ) ( ) inf infA BI x I x > ,( ) ( ) sup supA BI x I x > , ( ) ( ) inf infA BF x F x > ,and( ) ( ) sup supA BF x F x >for everyx inX . 2.2 Single-valued neutrosophic sets (SVNSs) A single-valued neutrosophic set (SVNS) is an instance ofaneutrosophicset,whichhasawiderangeofapplica-tions in real scientific and engineering fields. In the follow-ing,wereviewthedefinitionofaSVNSproposedby Wang et al. [17]. Definition2.4[17].LetXbeaspaceofpoints(objects) withgenericelementsinXdenotedbyx .Asingle-valuedneutrosophicset(SVNS)A inXischaracterized bytruth-membershipfunction( )AT x ,indeterminacy-membershipfunction( )AI x ,andfalsity-membership function( )AF x ,where( ) ( ) ( ) | | , , 0,1A A AT x I x F x efor each pointx inX . A SVNSA can be written as ( ) ( ) ( ){ }, , ,A A AA x T x I x F x x X = e(1) Let( ) ( ) ( ){ }, , ,i A i A i A i iA x T x I x F x x X = eand ( ) ( ) ( ){ }, , ,i B i B i B i iB x T x I x F x x X = ebetwosin-gle-valuedneutrosophicsets(SVNSs)in { }1 2, , ,nX x x x = .Thenwedefinethefollowingdis-tances forA andB. (i) The Hamming distance ( )( ) ( ) ( ) ( )( ) ( ) 11,3nA i B i A i B iiA i B iT x T x I x I xd A BF x F x =| | + + | = |\ . (2) (ii) The normalized Hamming distance ( )( ) ( ) ( ) ( )( ) ( ) 11,3nA i B i A i B iiA i B iT x T x I x I xd A BnF x F x =| | + + | = |\ . (3) (iii) The Euclidean distance ( )( ) ( ) ( ) ( )( ) ( )2 2211,3nA i B i A i B iiA i B iT x T x I x I xd A BF x F x=| | + |= |+ \ . (4) (iv) The normalized Euclidean distance ( )( ) ( ) ( ) ( )( ) ( )2 2211,3nA i B i A i B iiA i B iT x T x I x I xd A BnF x F x=| | + |= |+ \ .(5) 2.3 Interval neutrosophic sets (INSs) Definition2.5[16].LetXbeaspaceofpoints(objects) withgenericelementsinX denotedbyx .Aninterval neutrosophicset(INS)AinX ischaracterizedbya truth-membershipfunction( )AT x ,anindeterminacy-membershipfunction( )AI x ,andafalsity-membershipfunction( )AF x .ForeachpointxinX ,wehavethat ( ) ( ) ( ) | | inf ,sup 0,1A A AT x T x T x( = _ , ( ) ( ) ( ) | | inf ,sup 0,1A A AI x I x I x( = _ , ( ) ( ) ( ) | | inf ,sup 0,1A A AF x F x F x( = _ ,and( ) ( ) ( ) 0 sup sup sup 3A A AT x I x F x s + + s . Let( ) ( ) ( ){ }, , ,i i i i iA A AA x T x I x F x x X = eand ( ) ( ) ( ){ }, , ,i i i i i B B BB x T x I x F x x X = ebetwointer-val neutrosophic sets (INSs) in{ }1 2, , ,nX x x x = , where( ) ( ) ( ) inf ,supi i iA A AT x T x T x( = ,( ) ( ) ( ) inf ,supi i iA A AI x I x I x( = ,( ) ( ) ( ) inf ,supi i iA A AF x F x F x( = ,( ) ( ) ( ) inf ,supi i i B B BT x T x T x( = ,( ) ( ) ( ) inf ,supi i i B B BI x I x I x( = ,and( ) ( ) ( ) inf ,supi i i B B BF x F x F x( = .ThenYe[22]defined the following distances forA andB. (i) The Hamming distance ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )1,inf inf sup sup1inf inf sup sup6inf inf sup supi i i i B B A Ani i i i B B A Aii i i i B B A Ad A BT x T x T x T xI x I x I x I xF x F x F x F x==| | + + | | + + | | + \ .(6) (ii) The normalized Hamming distance 37Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )1,inf inf sup sup1inf inf sup sup6inf inf sup supi i i i B B A Ani i i i B B A Aii i i i B B A Ad A BT x T x T x T xI x I x I x I xnF x F x F x F x==| | + + | | + + | | + \ . (7) (iii) The Euclidean distance ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )2 22 212 2,inf inf sup sup1inf inf sup sup6inf inf sup supi i i i B B A Ani i i i B B A Aii i i i B B A Ad A BT x T x T x T xI x I x I x I xF x F x F x F x==| | + + | | + + | | | + \ . (8) (iv) The normalized Euclidean distance ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )2 22 212 2,inf inf sup sup1inf inf sup sup6inf inf sup supi i i i B B A Ani i i i B B A Aii i i i B B A Ad A BT x T x T x T xI x I x I x I xnF x F x F x F x==| | + + | | + + | | | + \ .(9) 3Anovelmethodforsingle-valuedneutrosophic multi-criteriadecisionmakingwithincomplete weight information 3.1 Problem description Theaimofmulti-criteriadecisionmaking(MCDM) problems is to find the most desirable alternative(s) from a set of feasible alternatives according to a number of criteria or attributes. In general, the multi-criteria decision making problem includes uncertain, imprecise, incomplete, and in-consistentinformation,whichcanberepresentedby SVNSs. Inthissection, wewillpresentamethod forhan-dling the MCDM problem under single-valued neutrosoph-icenvironments.First,aMCDMproblemwithsingle-valuedneutrosophicinformationcanbeoutlinedas:let { }1 2, , ,mA A A A =beasetofmalternativesand { }1 2, , ,nC c c c =beacollectionofncriteria,whose weightvectoris( )1 2, , ,Tnw ww w = ,with | | 0,1jw e , 1, 2, , j n = ,and 11njjw==.Inthiscase,thecharacter-isticofthealternative iA( 1, 2, , i m = )withrespectto all the criteria is represented by the following SVNS: ( ) ( ) ( ){ }, , ,i i ii j A j A j A j jA c T c I c F c c C = ewhere ( ) ( ) ( ) | | , , 0,1i i iA j A j A jT c I c F c e ,and ( ) ( ) ( )0 3i i iA j A j A jT c I c F c s + + s ( 1, 2, , i m = ,1, 2, , j n = ). Here, ( )iA jT cindicatesthedegreetowhichthealter-native iAsatisfiesthecriterion jc , ( )iA jI cindicatesthe indeterminacydegreetowhichthealternative iAsatisfies or does not satisfy the criterion jc , and ( )iA jF cindicates the degree to which the alternative iAdoes not satisfy the criterion jc .Forthesakeofsimplicity,acriterionvalue ( ) ( ) ( ), , ,i i ij A j A j A jc T c I c F cin iAisdenotedbyasin-gle-valuedneutrosophicvalue(SVNV), ,ij ij ij ija T I F =( 1, 2, , i m = , 1, 2, , j n = ),whichisusuallyderived fromtheevaluationofanalternative iAwithrespecttoa criterion jCbymeansofascorelawanddataprocessing inpractice[19,22].All ija( 1, 2, , i m = , 1, 2, , j n = ) constituteasinglevaluedneutrosophicdecisionmatrix ( ) ( ), ,ij ij ij ijmnmnA a T I F= =(see Table 1): Table 1: Single valued neutrosophic decision matrixA. 3.2Obtainingtheoptimalweightsofcriteriaby the maximizing deviation method DuetothefactthatmanypracticalMCDMproblems are complex and uncertain and human thinking is inherent-lysubjective,theinformationaboutcriterionweightsis usually incomplete. For convenience, letAbe a set of the knownweightinformation[8,9,10,11],whereAcanbe constructed by the following forms, fori j = : Form 1. A weak ranking: { }i jw w > ; 1 1cjcnc1A11 11 11, , T I F1 1 1, ,j j jT I F1 1 1, ,n n nT I FiA1 1 1, ,i i iT I F , ,ij ij ijT I F , ,in in inT I FmA1 1 1, ,m m mT I F , ,mj mj mjT I F , ,mn mn mnT I F38Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information Form 2. A strict ranking: { }i j iw w o >( 0io > ); Form 3. A ranking of differences: { }i j k lw w w w > , forj k l = = ; Form4.Arankingwithmultiples: { }i i jw w o >( 0 1io s s ); Form5.Anintervalform:{ }i i i iw o o c s s +( 0 1i i io o c s s + s ). Wang[15]proposedthemaximizingdeviationmethod forestimatingthecriterionweightsinMCDMproblems withnumericalinformation.AccordingtoWang[15],if theperformancevaluesofallthealternativeshavesmall differences under a criterion, it shows that such a criterion plays a less important role in choosing the best alternative andshould be assigned a smallerweight. Onthecontrary, if a criterion makes the performance values of all the alter-nativeshaveobviousdifferences,thensuchacriterion playsamuchimportantroleinchoosingthebestalterna-tiveandshouldbeassignedalargerweight.Especially,if allavailablealternativesscoreaboutequallywithrespect to a given criterion, then such a criterion will be judged un-important by most decision makers and should be assigned averysmallweight.Wang[15]suggeststhatzeroweight should be assigned to the criterion of this kind. Here,basedonthemaximizingdeviationmethod,we constructanoptimizationmodeltodeterminetheoptimal relative weights of criteria under single valued neutrosoph-ic environments. For the criterion jc C e , the deviation of thealternative iAtoalltheotheralternativescanbede-fined as below: ( )2 2 21 1,3m mij qj ij qj ij qjij ij qjq qT T I I F FD d a a= = + + = = 1, 2, , i m = ,1, 2, , j n =(10) where ( )2 2 2,3ij qj ij qj ij qjij qjT T I I F Fd a a + + =denotesthesinglevaluedneutrosophicEuclideandistance betweentwosingle-valuedneutrosophicvalues(SVNVs) ijaand qjadefined as in Eq. (4). Let ( )1 1 12 2 21 1,3m m mj ij ij qji i qm mij qj ij qj ij qji qD D d a aT T I I F F= = == == = = + + 1, 2, , j n =(11) then jD representsthedeviationvalueofallalternatives to other alternatives for the criterion jc C e . Further, let ( )2 2 21 1 1 13n n m mij qj ij qj ij qjj j jj j i qT T I I F FDw w D w= = = = + + = = (12) then( ) Dw representsthedeviationvalueofallalterna-tives to other alternatives for all the criteria. Basedontheaboveanalysis,wecanconstructanon-linearprogrammingmodeltoselecttheweightvectorw by maximizing( ) Dw , as follows:( )2 2 21 1 121max3s.t. 0, 1, 2, , , 1n m mij qj ij qj ij qjjj i qnj jjT T I I F FDw ww j n w= = == + + => = =(M-1) To solve this model, we construct the Lagrange function as follows: ( )2 2 21 1 121,312n m mij qj ij qj ij qjjj i qnjjT T I I F FLw ww= = == + + = +| | |\ . (13) whereis the Lagrange multiplier. DifferentiatingEq.(13)withrespectto jw( 1, 2, , j n = )and ,andsettingthesepartialderiva-tivesequaltozero,thenthefollowingsetofequationsis obtained: 2 2 21 103m mij qj ij qj ij qjji qjT T I I F FLww= = + + c= + =c (14) 2111 02njjLw=| |c= = |c\ .(15) It follows from Eq. (14) that 2 2 21 13m mij qj ij qj ij qji qjT T I I F Fw= = + + = (16) Putting Eq. (16) into Eq. (15), we get 39Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information 22 2 21 1 13n m mij qj ij qj ij qjj i qT T I I F F= = =| | + + |= | |\ . (17) Then, by combining Eqs. (16) and (17), we have 2 2 21 122 2 21 1 133m mij qj ij qj ij qji qjn m mij qj ij qj ij qjj i qT T I I F FwT T I I F F= == = = + + =| | + + | | |\ . (18) By normalizing jw( 1, 2, , j n = ), we make their sum into a unit, and get 2 2 21 12 2 211 1 133m mij qj ij qj ij qjj i qj nn m mij qj ij qj ij qj jjj i qT T I I F FwwT T I I F F w= = -== = = + + = = + + (19) whichcanbeconsideredastheoptimalweightvectorof criteria. However, it is noted that there are practical situations in which the information about the weight vector is not com-pletelyunknownbutpartiallyknown.Forsuchcases,we establish the following constrained optimization model: ( )2 2 21 1 11max3s.t. , 0, 1, 2, , , 1n m mij qj ij qj ij qjjj i qnj jjT T I I F FDw ww w j n w= = == + + =eA > = =(M-2) It is noted that the model (M-2) is a linear programming model that can be solved using the MATLAB mathematics software package. Suppose that the optimal solution to the model(M-2)is( )1 2, , ,Tnw ww w = ,whichcanbecon-sidered as the weight vector of criteria. 3.3.ExtendedTOPISmethodfortheMCDMwith single valued neutrosophic information TOPSISmethod,initiallyintroducedbyHwangand Yoon[7],isawidelyusedmethodfordealingwith MCDMproblems,whichfocusesonchoosingthealterna-tive with the shortest distance from the positive ideal solu-tion (PIS) and the farthest distance from the negative ideal solution (NIS).After obtainingthecriterion weightvalues onbasisofthemaximizingdeviationmethod,inthefol-lowing, we will extend the TOPSIS method to take single-valuedneutrosophicinformationintoaccountandutilize the distance measures of SVNVs to obtain the final ranking of the alternatives. In general, the criteria can be classified into two types: benefitcriteriaandcostcriteria.Thebenefitcriterion means that a higher value is better while for the cost crite-rion is valid the opposite. Let 1Cbe a collection of benefit criteriaand 2Cbeacollectionofcostcriteria,where 1 2C C C =and 1 2C C = C. Under single-valued neu-trosophicenvironments,thesingle-valuedneutrosophic PIS (SVNPIS), denoted byA+, can be identified by usingamaximumoperatorforthebenefitcriteriaandamini-mumoperatorforthecostcriteriatodeterminethebest value of each criterion among all alternatives as follows: { }1 2, , ,nA a a a+ + + +=(20) where { } { } { }{ } { } { }12max , min , min , if ,min , max , max , if .ij ij iji i ijij ij iji i iT I F j CaT I F j C+e= e (21) The single-valued neutrosophic NIS (SVNNIS), denoted byA, can be identified by using a minimum operator forthebenefitcriteriaandamaximumoperatorforthecost criteriatodeterminetheworstvalueofeachcriterion among all alternatives as follows: { }1 2, , ,nA a a a =(22) where { } { } { }{ } { } { }12min , max , max , if ,max , min , min , if .ij ij iji i ijij ij iji i iT I F j CaT I F j Ce= e (23) Theseparationmeasures, id+ and id,ofeachalterna-tivefromtheSVNPISA+ andtheSVNNISA,respec-tively, are derived from 40Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information ( ){ } { } { }{ } { } { }1212 2 22 2 2,max min min3min max max3ni j ij jjij ij ij ij ij iji i ijj Cij ij ij ij ij iji i ijj Cd w d a aT T I I F FwT T I I F Fw+ +=ee=| | + + |\ .= +| | + + |\ .(24) ( ){ } { } { }{ } { } { }1212 2 22 2 2,min max max3max min min3ni j ij jjij ij ij ij ij iji i ijj Cij ij ij ij ij iji i ijj Cd w d a aT T I I F FwT T I I F Fw =ee=| | + + |\ .= +| | + + |\ .(25) Therelativeclosenesscoefficientofanalternative iAwithrespecttothesingle-valuedneutrosophicPISA+ isdefined as the following formula: iii idCd d+ =+ (26) where0 1iC s s ,1, 2, , i m = . Obviously, an alternative iAis closer to the single-valued neutrosophic PISA+ andfarther from the single-valued neutrosophic NISA as iCapproaches 1. The larger the value of iC , the more differ-entbetween iAandthesingle-valuedneutrosophicNIS A,whilethemoresimilarbetweeniAandthesingle-valuedneutrosophicPISA+.Therefore,thealternative(s)with the maximum relative closeness coefficient should be chosen as the optimal one(s). Based on the above analysis, we will develop a practi-calapproachfordealingwithMCDMproblems,inwhich theinformationaboutcriterionweightsisincompletely knownorcompletelyunknown,andthecriterionvalues take the form of single-valued neutrosophic information. The flowchart of the proposed approach for MCDM is provided in Fig. 1. The proposed approach is composed of the following steps: Step1.ForaMCDMproblem,thedecisionmaker(DM) constructsthesingle-valuedneutrosophicdecisionmatrix ( ) ( ), ,ij ij ij ijmnmnA a T I F= = ,where, ,ij ij ij ija T I F =isasingle-valuedneutrosophicvalue(SVNV),givenby the DM, for the alternative iAwith respect to the attribute jc . Step2.Iftheinformationaboutthecriterionweightsis completelyunknown,thenweuseEq.(19)toobtainthe criterionweights;iftheinformationaboutthecriterion weights is partly known, then we solve the model (M-2) to obtain the criterion weights. Step 3. Utilize Eqs. (20), (21), (22), and (23) to determine thesingle-valuedneutrosophicpositiveidealsolution (SVNPIS)A+ and the single-valued neutrosophic negativeideal solution (SVNNIS)A.Step4.UtilizeEqs.(24)and(25)tocalculatethesepara-tion measures id+ andid of each alternativeiAfrom the single-valuedneutrosophicpositiveidealsolution (SVNPIS)A+ and the single-valued neutrosophic negativeideal solution (SVNNIS)A, respectively.Step5.UtilizeEq.(26)tocalculatetherelativecloseness coefficient iC ofeachalternative iAtothesingle-valued neutrosophic positive ideal solution (SVNPIS)A+.Step6.Rankthealternatives iA( 1, 2, , i m = )accord-ing to the relative closeness coefficients iC( 1, 2, , i m = ) tothesingle-valuedneutrosophicpositiveidealsolution (SVNPIS)A+ and then select the most desirable one(s).4 A novel method for interval neutrosophic multi-criteriadecisionmakingwithincompleteweight information Inthissection,wewillextendtheresultsobtainedin Section 3 to interval neutrosophic environments. 4.1. Problem description Similar to Subsection 3.1, a MCDM problem under in-tervalneutrosophicenvironmentscanbesummarizedas follows:let{ }1 2, , ,mA AA A =beasetofmalterna-tivesand{ }1 2, , ,nC c c c =beacollectionofn criteria, whoseweightvectoris( )1 2, , ,Tnw ww w = ,with | | 0,1jw e ,1, 2, , j n = ,and 11njjw==.Inthiscase,thecharacteristicofthealternative iA( 1, 2, , i m = ) with respect to all the criteria is represented by the follow-ing INS: ( ) ( ) ( ){ }( ) ( ) ( ) ( )( ) ( ), , ,, inf ,sup , inf ,sup ,inf ,supi i ii j j j j j A A Aj j j j j A A A Ajj j A AA c T c I c F c c Cc T c T c I c I cc CF c F c= e (( = e ` ( )where ( ) ( ) ( ) | | inf ,sup 0,1ij j jA A AT c T c T c (= _ , ( ) ( ) ( ) | | inf ,sup 0,1ij j jA A AI c I c I c (= _ ,41Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information ( ) ( ) ( ) | | inf ,sup 0,1ij j jA A AF c F c F c (= _ ,and ( ) ( ) ( )sup sup sup 3j j jA A AT c I c F c + + s ( 1, 2, , i m = , 1, 2, , j n = ). Here, ( ) ( ) ( )inf ,supij j jA A AT c T c T c (= indicatesthe interval degree to which the alternative iAsatisfies the cri-terion jc , ( ) ( ) ( )inf ,supij j jA A AI c I c I c (= indicates theindeterminacyintervaldegreetowhichthealternative iAsatisfiesordoesnotsatisfythecriterion jc ,and ( ) ( ) ( )inf ,supij j jA A AF c F c F c (= indicatestheinter-val degree to which the alternative iAdoes not satisfy the criterion jc .Forthesakeofsimplicity,acriterionvalue ( ) ( ) ( ), , ,i i ij j j jA A Ac T c I c F cin iAis denoted by an in-tervalneutrosophicvalue(INV) , , , , , , ,L U L U L Uij ij ij ij ij ij ij ij ij ija T I F T T I I F F((( = = ( 1, 2, , i m = , 1, 2, , j n = ),whichisusuallyderived fromtheevaluationofanalternative iAwithrespecttoa criterion jcbymeansofascorelawanddataprocessing inpractice[19,22].All ija( 1, 2, , i m = , 1, 2, , j n = ) constituteanintervalneutrosophicdecisionmatrix ( ) ( ) ( ), , , , , , ,L U L U L Uij ij ij ij ij ij ij ij ij ijmn mn mnA a T I F T T I I F F ((( = = = (see Table 2): Table 2: Interval neutrosophic decision matrixA . 4.2.Obtainingtheoptimalweightsofcriteriaun-derintervalneutrosophicenvironmentsbythe maximizing deviation method In what follows, similar to Subsection 3.2, based on the maximizingdeviationmethod,weconstructanoptimiza-tion model to determine the optimal relative weights of cri-teria under interval neutrosophic environments. For the at-tribute jc C e ,thedeviationofthealternative iAtoall the other alternatives can be defined as below: ( )2 2 22 2 21 1,6L L U U L Lij qj ij qj ij qjU U L L U Um mij qj ij qj ij qjij ij qjq qT T T T I II I F F F FD d a a= = + + + + + = = 1, 2, , i m = ,1, 2, , j n =(27) where ( )2 2 22 2 2,6L L U U L Lij qj ij qj ij qjU U L L U Uij qj ij qj ij qjij qjT T T T I II I F F F Fd a a + + + + + =denotestheintervalneutrosophicEuclideandistancebe-tween two interval neutrosophic values (INVs) ijaand qjadefined as in Eq. (8). Let ( )1 1 12 2 22 2 21 1,6m m mj ij ij qji i qL L U U L Lij qj ij qj ij qjU U L L U Um mij qj ij qj ij qji qD D d a aT T T T I II I F F F F= = == == = = + + + + + 1, 2, , j n =(28) then jD representsthedeviationvalueofallalternatives to other alternatives for the criterion jc C e . Further, let ( )12 2 22 2 21 1 16nj jjL L U U L Lij qj ij qj ij qjU U L L U Un m mij qj ij qj ij qjjj i qDw w DT T T T I II I F F F Fw== = == =| | + + + | | + + | | | | |\ . (29) then( ) Dw representsthedeviationvalueofallalterna-tives to other alternatives for all the criteria. Fromtheaboveanalysis,wecanconstructanon-linear programmingmodeltoselecttheweightvectorwby maximizing( ) Dw , as follows:21cjcnc1A11 11 11, , T I F1 1 1, ,j j jT I F1 1 1, ,n n nT I FiA1 1 1, ,i i iT I F , ,ij ij ijT I F , ,in in inT I FmA1 1 1, ,m m mT I F , ,mj mj mjT I F , ,mn mn mnT I F42Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information ( )2 2 22 2 21 1 121max6s.t. 0, 1, 2, , , 1L L U U L Lij qj ij qj ij qjU U L L U Un m mij qj ij qj ij qjjj i qnj jjT T T T I II I F F F FDw ww j n w= = ==| | + + + | | + + |= | | | |\ .> = = (M-3) To solve this model, we construct the Lagrange function: ( )2 2 22 2 21 1 121,612L L U U L Lij qj ij qj ij qjU U L L U Un m mij qj ij qj ij qjjj i qnjjT T T T I II I F F F FLw ww= = ==| | + + + | | + + |= + | | | |\ .| | |\ . (30) whereis the Lagrange multiplier. DifferentiatingEq.(30)withrespectto jw( 1, 2, , j n = )and ,andsettingthesepartialderiva-tivesequaltozero,thenthefollowingsetofequationsis obtained: 2 2 22 2 21 160L L U U L Lij qj ij qj ij qjU U L L U Um mij qj ij qj ij qji qjjT T T T I II I F F F FLww = = + + + + + c= +c= (31) 2111 02njjLw=| |c= = |c\ . (32) It follows from Eq. (32) that 2 2 22 2 21 16L L U U L Lij qj ij qj ij qjU U L L U Um mij qj ij qj ij qji qjT T T T I II I F F F Fw= = + + + + + = (33) Putting Eq. (33) into Eq. (32), we get 22 2 22 2 21 1 16L L U U L Lij qj ij qj ij qjU U L L U Un m mij qj ij qj ij qjj i qT T T T I II I F F F F= = =| | + + | |+ + + |= | | | |\ . (34) Then, by combining Eqs. (33) and (34), we have 2 2 22 2 21 122 2 22 2 21 1 166L L U U L Lij qj ij qj ij qjU U L L U Um mij qj ij qj ij qji qjL L U U L Lij qj ij qj ij qjU U L L U Un m mij qj ij qj ij qjj i qT T T T I II I F F F FwT T T T I II I F F F F= == = = + + + + + =| | + + + | | + + | | | | |\ . (35) By normalizing jw( 1, 2, , j n = ), we make their sum into a unit, and get 2 2 22 2 21 12 2 212 2 21 1 166L L U U L Lij qj ij qj ij qjU U L L U Um mij qj ij qj ij qjj i qj nL L U U L Lij qj ij qj ij qj jjU U L L U Un m mij qj ij qj ij qjj i qT T T T I II I F F F FwwT T T T I I wI I F F F F= = -== = = + + + + + = = + + + + + (36) whichcanbeconsideredastheoptimalweightvectorof criteria. However, it is noted that there are practical situations in which the information about the weight vector is not com-pletelyunknownbutpartiallyknown.Forsuchcases,we establish the following constrained optimization model: ( )2 2 22 2 21 1 11max6s.t. , 0, 1, 2, , , 1L L U U L Lij qj ij qj ij qjU U L L U Un m mij qj ij qj ij qjjj i qnj jjT T T T I II I F F F FDw ww w j n w= = ==| | + + + | | + + |= | | | |\ .eA > = = (M-4) 43Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information It is noted that the model (M-4) is a linear programming model that can be solved using the MATLAB mathematics software package. Suppose that the optimal solution to the model(M-4)is( )1 2, , ,Tnw ww w = ,whichcanbecon-sidered as the weight vector of criteria. 4.3.ExtendedTOPISmethodfortheMCDMwith interval neutrosophic information Afterobtainingtheweightsofcriteriaonbasisofthe maximizingdeviationmethod,similartoSubsection3.3, we next extend the TOPSIS method to interval neutrosoph-ic environments and develop an extended TOPSIS method to obtain the final ranking of the alternatives. Underintervalneutrosophic environments, theinterval neutrosophic PIS (INPIS), denoted byA+, and the intervalneutrosophic NIS (INNIS), denoted byA, can be defined as follows: { }1 2, , ,nA a a a+ + + += (37) { }1 2, , ,nA a a a =(38) where { } { } { }{ } { } { }1max , min , minmax , max , min , min ,, if ,min , minmin , max , maxmin , min , max , max ,, ifmax , maxij ij iji i iL U L Uij ij ij iji i i iL Uij iji ijij ij iji i iL U L Uij ij ij iji i i iL Uij iji iT I FT T I Ij CF FaT I FT T I IF F+= (( e ( = (( = ( 2. j Ce(39) { } { } { }{ } { } { }1min , max , maxmin , min , max , max ,, if ,max , maxmax , min , minmax , max , min , min ,, ifmin , minij ij iji i iL U L Uij ij ij iji i i iL Uij iji ijij ij iji i iL U L Uij ij ij iji i i iL Uij iji iT I FT T I Ij CF FaT I FT T I IF F= (( e ( == (( ( 2. j Ce(40) Theseparationmeasures, id+ andid,ofeachalterna-tive iAfromtheINPISA+ andtheINNISA,respec-tively, are derived from ( ){ } { } { }{ } { } { }{ } { } { }{ }112 2 22 2 22 2 22,max max minmin min min6min min maxmaxni j ij jjL L U U L Lij ij ij ij ij iji i iU U L L U Uij ij ij ij ij iji i ijj CL L U U L Lij ij ij ij ij iji i iU U Lij ij ijijd w d a aT T T T I II I F F F FwT T T T I II I Fw+ +=e=| | + + + | | | + + |\ .= + + + + + { } { }22 2max max6L U Uij ij iji ij CF F Fe| | | | |+ |\ . (41) ( ){ } { } { }{ } { } { }{ } { } { }{ }112 2 22 2 22 2 22,min min maxmax max max6max max minminni j ij jjL L U U L Lij ij ij ij ij iji i iU U L L U Uij ij ij ij ij iji i ijj CL L U U L Lij ij ij ij ij iji i iU U Lij ij ijijd w d a aT T T T I II I F F F FwT T T T I II I Fw =e=| | + + + | | | + + |\ .= + + + + + { } { }22 2min min6L U Uij ij iji ij CF F Fe| | | | |+ |\ . (42) Therelativeclosenesscoefficientofanalternative iAwith respect to the interval neutrosophic PISA+ is definedas the following formula: iii idCd d+ =+ (43) where0 1iC s s ,1, 2, , i m = . Obviously, an alternative iAisclosertotheintervalneutrosophicPISA+ andfar-therfromtheintervalneutrosophicNISA asiC ap-proaches 1. The larger the value of iC , the more different between iAandtheintervalneutrosophicNISA,whilethe more similar between iAand the interval neutrosophic PISA+.Therefore,thealternative(s)withthemaximumrelativeclosenesscoefficientshouldbechosenastheop-timal one(s). Basedontheaboveanalysis,analogoustoSubsection 3.3,wewilldevelopapracticalapproachfordealingwith MCDM problems, in which the information about criterion weightsisincompletelyknownorcompletelyunknown, andthecriterionvaluestaketheformofintervalneutro-sophic information. The flowchart of the proposed approach for MCDM is 44Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information provided in Fig. 1. The proposed approach is composed of the following steps: Step1.ForaMCDMproblem,thedecisionmakercon-structstheintervalneutrosophicdecisionmatrix ( ) ( ), ,ij ij ij ijmnmnA a T I F= = ,where, ,ij ij ij ija T I F =is an interval neutrosophic value (INV), given by the DM, for the alternative iAwith respect to the criterion jc . Step2.Iftheinformationaboutthecriterionweightsis completelyunknown,thenweuseEq.(36)toobtainthe criterionweights;iftheinformationaboutthecriterion weights is partly known, then we solve the model (M-4) to obtain the criterion weights. Step 3. Utilize Eqs. (37), (38), (39), and (40) to determine theintervalneutrosophicpositiveidealsolution(INPIS) A+ andtheintervalneutrosophicnegativeidealsolution(INNIS)A. Step4.Utilize Eqs. (41) and (42) to calculate the separa-tion measures id+ andid of each alternativeiAfrom the intervalneutrosophicpositiveidealsolution(INPIS)A+andtheintervalneutrosophicnegativeidealsolution(IN-NIS)A, respectively. Step5.Utilize Eq. (43) to calculate the relative closeness coefficient iC ofeachalternative iAtotheintervalneu-trosophic positive ideal solution (INPIS)A+.Step6.Rankthealternatives iA( 1, 2, , i m = )accord-ing to the relative closeness coefficients iC( 1, 2, , i m = ) to the interval neutrosophic positive ideal solution (INPIS) A+ and then select the most desirable one(s).5 Illustrative examples 5.1. A practical example under single-valued neu-trosophic environments Example5.1[13]. In order to demonstrate the application of the proposed approach, a multi-criteria decision making problemadaptedfromTanandChen[13]isconcerned withamanufacturingcompanywhichwantstoselectthe best global supplier according to the core competencies of suppliers. Now suppose that there are a set of foursuppli-ers{ }1 2 3 4 5, , , , A AA A A A =whosecorecompetenciesareevaluated by means of the following four criteria: (1) the level of technology innovation (1c ), (2) the control ability of flow (2c ), (3) the ability of management (3c ), (4) the level of service (4c ). Itisnotedthatallthecriteria jc( 1, 2, 3, 4 j = )arethe benefittypeattributes.Theselectionofthebestglobal suppliercanbemodeledasahierarchicalstructure,as showninFig.2.Accordingto[21],wecanobtainthe evaluationofanalternative iA( 1, 2,3, 4,5 i = )withre-specttoacriterion jc( 1, 2, 3, 4 j = )fromthequestion-naireofadomainexpert.Take 11aasanexample.When weasktheopinionofanexpertaboutanalternative 1Awithrespecttoacriterion 1c ,heorshemaysaythatthe possibilityinwhichthestatementisgoodis0.5andthe statementispooris 0.3and the degreein whichhe orshe is not sure is 0.1. In this case, the evaluation of the alterna-tive 1Awithrespecttothecriterion 1cisexpressedasa single-valuedneutrosophicvalue 110.5, 0.1, 0.3 a = . Through the similar method from the expert, we can obtain all the evaluations of all the alternatives iA( 1, 2,3, 4,5 i = ) with respect to all the criteria jc( 1, 2, 3, 4 j = ), which are listed in the following single valued neutrosophic decision matrix ( ) ( ), ,ij ij ij ijmnmnA a T I F= =(see Table 3). Table 3: Single valued neutrosophic decision matrixA.Selection of the best global supplier3A4A2A1A5A1c2c3c4cFig. 2: Hierarchical structure. Inwhatfollows,weutilizethedevelopedmethodto findthebestalternative(s).Wenowdiscusstwodifferent cases. Case1: Assume that the information about the criterion weightsiscompletelyunknown;inthiscase,weusethe following steps to get the most desirable alternative(s). Step 1. Considering that the information about the criterion weightsiscompletelyunknown, we utilizeEq.(19)to get the optimal weight vector of attributes: 3 1c2c3c4c1A 0.5,0.1,0.3 0.5,0.1,0.4 0.3,0.2,0.3 0.7,0.2,0.12A 0.6,0.1,0.2 0.5,0.2,0.2 0.5,0.4,0.1 0.4,0.2,0.33A0.9,0.0,0.1 0.3,0.2,0.3 0.2,0.2,0.5 0.4,0.3,0.24A 0.8,0.1,0.1 0.5,0.0,0.4 0.6,0.2,0.1 0.2,0.3,0.45A 0.7,0.2,0.1 0.4,0.3,0.2 0.6,0.1,0.3 0.5,0.4,0.145Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information ( ) 0.2184,0.2021,0.3105,0.2689Tw=Step2. Utilize Eqs. (20), (21), (22), and (23) to determine thesinglevaluedneutrosophicPISA+ andthesingleval-ued neutrosophic NISA, respectively:{ } 0.9, 0.0, 0.1 , 0.5, 0.0, 0.2 , 0.6, 0.1, 0.1 , 0.7, 0.2, 0.1 A+={ } 0.5, 0.2, 0.3 , 0.3, 0.3, 0.4 , 0.2, 0.4, 0.5 , 0.2, 0.4, 0.4 A=Step3:UtilizeEqs.(24)and(25)tocalculatethesepara-tion measures id+ andid of each alternativeiAfrom the singlevaluedneutrosophicPISA+ andthesinglevaluedneutrosophic NISA, respectively:10.1510 d+= , 10.1951 d= , 20.1778 d+= , 20.1931 d= , 30.1895 d+= , 30.1607 d= , 40.1510 d+= , 40.2123 d= , 50.1523 d+= , 50.2242 d= . Step4:UtilizeEq. (26)tocalculatetherelativecloseness coefficient iCofeachalternative iAtothesinglevalued neutrosophic PISA+:10.5638 C = ,20.5205 C = , 30.4589 C = , 40.5845 C = , 50.5954 C = . Step5:Rankthealternatives iA( 1, 2,3, 4,5 i = )accord-ing to the relative closeness coefficient iC( 1, 2,3, 4,5 i = ). Clearly, 5 4 1 2 3A A A A A ,andthusthebestalter-native is 5A . Case2:Theinformationaboutthecriterionweightsis partlyknownandtheknownweightinformationisgiven as follows: 1 2 34410.15 0.25, 0.25 0.3, 0.3 0.4,0.35 0.5, 0, 1, 2, 3, 4, 1j jjw w ww w j w=s s s s s s A = `s s > = = ) Step1:Utilizethemodel(M-2)toconstructthesingle-objective model as follows: ( )1 2 3 4max 2.9496 2.7295 4.1923 3.6315s.t.Dw w w w ww = + + +eA By solvingthis model, we get the optimal weight vec-tor of criteria( ) 0.15,0.25,0.3,0.35Tw= .Step2. Utilize Eqs. (20), (21), (22), and (23) to determine thesinglevaluedneutrosophicPISA+ andthesingleval-ued neutrosophic NISA, respectively:{ } 0.9, 0.0, 0.1 , 0.5, 0.0, 0.2 , 0.6, 0.1, 0.1 , 0.7, 0.2, 0.1 A+={ } 0.5, 0.2, 0.3 , 0.3, 0.3, 0.4 , 0.2, 0.4, 0.5 , 0.2, 0.4, 0.4 A=Step3:UtilizeEqs.(24)and(25)tocalculatethesepara-tion measures id+ and id of each alternativeiAfrom the singlevaluedneutrosophicPISA+ andthesinglevaluedneutrosophic NISA, respectively:10.1368 d+= , 10.2260 d= , 20.1852 d+= , 20.2055 d= , 30.2098 d+= , 30.1581 d= , 40.1780 d+= , 40.2086 d= , 50.1619 d+= , 50.2358 d= . Step4:UtilizeEq. (26)tocalculatetherelativecloseness coefficient iCofeachalternative iAtothesinglevalued neutrosophic PISA+:10.6230 C = ,20.5260 C = , 30.4297 C = , 40.5396 C = , 50.5928 C = . Step5:Rankthealternatives iA( 1, 2,3, 4,5 i = )accord-ing to the relative closeness coefficient iC( 1, 2,3, 4,5 i = ). Clearly, 1 5 4 2 3A A A A A ,andthusthebestalter-native is 1A . 5.2.Theanalysisprocessunderintervalneutro-sophic environments Example5.2.LetsrevisitExample5.1.Supposethatthe fivepossiblealternativesaretobeevaluatedunderthe abovefourcriteriabytheformofINVs,asshowninthe followingintervalneutrosophicdecisionmatrixA(see Table 4). Table 4: Interval neutrosophic decision matrixA . Inwhatfollows,weproceedtoutilizethedeveloped method to find the most optimal alternative(s), which con-sists of the following two cases: Case1: Assume that the information about the criterion weightsiscompletelyunknown;inthiscase,weusethe following steps to get the most desirable alternative(s). Step 1. Considering that the information about the criterion weightsiscompletelyunknown, we utilizeEq.(36)to get the optimal weight vector of attributes: 4 1c2c3c4c1A 2A < [0.5, 0.7], [0.2, 0.3], [0.7, 0.8]> 3A< [0.4, 0.5], [0.2, 0.3], [0.4, 0.6]> 4A< [0.2,0.3], [0.1, 0.2], [0.4, 0.5]> < [0.2,0.3], [0.3, 0.5], [0.6, 0.8]> 5A < [0.6, 0.7], [0.1, 0.2], [0.7, 0.9]> 46Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information ( ) 0.2490,0.2774,0.2380,0.2356Tw=Step2.Utilize Eqs. (37), (38), (39), and (40) to determine theintervalneutrosophicPISA+ andtheintervalneutro-sophic NISA, respectively:0.7, 0.9, 0.1, 0.2, 0.2, 0.4 , 0.6, 0.7, 0.1, 0.2, 0.2, 0.3 ,0.8, 0.9, 0.1, 0.2, 0.3, 0.4 , 0.8, 0.9, 0.1, 0.2, 0.4, 0.5A+ = ` ) { } { }{ } { }0.2, 0.3, 0.3, 0.4, 0.6, 0.7 , 0.2, 0.3, 0.3, 0.5, 0.7, 0.9 ,0.2, 0.3, 0.2, 0.3, 0.7, 0.8 , 0.2, 0.3, 0.3, 0.5, 0.6, 0.8A = ` )Step3:UtilizeEqs.(41)and(42)tocalculatethesepara-tion measures id+ andid of each alternativeiAfrom the interval neutrosophic PISA+ and the interval neutrosophicNISA, respectively:10.2044 d+= , 10.2541 d= , 20.2307 d+= , 20.2405 d= , 30.2582 d+= , 30.1900 d= , 40.2394 d+= , 40.2343 d= , 50.2853 d+= , 50.2268 d= . Step4:UtilizeEq. (43)tocalculatetherelativecloseness coefficient iCofeachalternative iAtotheintervalneu-trosophic PISA+:10.5543 C = ,20.5104 C = , 30.4239 C = , 40.4946 C = , 50.4429 C = . Step5:Rankthealternatives iA( 1, 2,3, 4,5 i = )accord-ing to the relative closeness coefficient iC( 1, 2,3, 4,5 i = ). Clearly, 1 2 4 5 3A A A A A ,andthusthebestalter-native is 1A . Case2:Theinformationabouttheattributeweightsis partlyknownandtheknownweightinformationisgiven as follows: 1 2 34410.25 0.3, 0.25 0.35, 0.35 0.4,0.4 0.45, 0, 1, 2,3, 4, 1j jjw w ww w j w=s s s s s s A = `s s > = = )Step1:Utilizethemodel(M-4)toconstructthesingle-objective model as follows: ( )1 2 3 4max 4.2748 4.7627 4.0859 4.0438s.t.Dw w w w ww= + + + eA By solving this model, we get the optimal weight vector of criteria( ) 0.25,0.25,0.35,0.4Tw= . Step2.Utilize Eqs. (37), (38), (39), and (40) to determine theintervalneutrosophicPISA+ andtheintervalneutro-sophic NISA, respectively:0.7, 0.9, 0.1, 0.2, 0.2, 0.4 , 0.6, 0.7, 0.1, 0.2, 0.2, 0.3 ,0.8, 0.9, 0.1, 0.2, 0.3, 0.4 , 0.8, 0.9, 0.1, 0.2, 0.4, 0.5A+ = ` ) { } { }{ } { }0.2, 0.3, 0.3, 0.4, 0.6, 0.7 , 0.2, 0.3, 0.3, 0.5, 0.7, 0.9 ,0.2, 0.3, 0.2, 0.3, 0.7, 0.8 , 0.2, 0.3, 0.3, 0.5, 0.6, 0.8A = ` )Step3:UtilizeEqs.(41)and(42)tocalculatethesepara-tion measures id+ andid of each alternativeiAfrom the interval neutrosophic PISA+ and the interval neutrosophic NISA, respectively:10.2566 d+= , 10.3152 d= , 20.2670 d+= , 20.3228 d= , 30.2992 d+= , 30.2545 d= , 40.3056 d+= , 40.2720 d= , 50.3503 d+= , 50.2716 d= . Step4:UtilizeEq. (43)tocalculatetherelativecloseness coefficient iCofeachalternative iAtotheintervalneu-trosophic PISA+:10.5513 C = ,20.5474 C = , 30.4596 C = , 40.4709 C = , 50.4368 C = . Step5:Rankthealternatives iA( 1, 2,3, 4,5 i = )accord-ing to the relative closeness coefficient iC( 1, 2,3, 4,5 i = ). Clearly, 1 2 4 3 5A A A A A ,andthusthebestalter-native is 1A . 5.3. Comparison analysis with the existing single-valuedneutrosophicorintervalneutrosophic multi-criteria decision making methods Recently,somemethods[20,21,22,23]havebeende-velopedforsolvingtheMCDMproblemswithsingle-valuedneutrosophicorintervalneutrosophicinformation. In this section, we will perform a comparisonanalysis be-tweenournewmethodsandtheseexistingmethods,and thenhighlighttheadvantagesofthenewmethodsover these existing methods. Itisnotedthattheseexistingmethodshavesomein-herent drawbacks, which are shown as follows: (1)Theexistingmethods[20,21,22,23]needthedecision maker to provide the weights of criteria in advance, which is subjective and sometime cannot yield the persuasive re-sults. In contrast, our methods utilize the maximizing devi-ationmethodtodeterminetheweightvaluesofcriteria, whichismoreobjectiveandreasonablethantheotherex-isting methods [20,21,22,23]. (2)InRef.[23],Yeproposedasimplifiedneutrosophic weighted arithmetic average operator and a simplified neu-trosophicweightedgeometricaverageoperator,andthen utilized two aggregation operators to develop a method for multi-criteriadecisionmakingproblemsundersimplified neutrosophic environments. However, it is noted that these 47Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information operatorsandmethodneedtoperformanaggregationon theinputsimplifiedneutrosophicarguments,whichmay increasethecomputationalcomplexityandthereforelead to the loss of information. In contrast, our methodsdo not need to perform such an aggregation but directly deal with theinputsimplifiedneutrosophicarguments,therebycan retaintheoriginaldecisioninformationasmuchaspossi-ble. (3)InRef.[22],YedefinedtheHammingandEuclidean distancesbetweenintervalneutrosophicsets(INSs)and proposed the similarity measures between INSs on the ba-sis of the relationship between similarity measures and dis-tances. Moreover, Ye [22] utilized the similaritymeasures betweeneachalternativeandtheidealalternativetorank thealternativesandtodeterminethebestone.Inorderto clearlydemonstratethecomparisonresults,weusethe methodproposedin[22]torevisitExample5.2,whichis shown as follows: First, we identify anideal alternative by using a maxi-mum operator for the benefit criteria and a minimum oper-ator for the cost criteria to determine the best value of each criterion among all alternatives as: 0.7, 0.9, 0.1, 0.2, 0.2, 0.4 , 0.6, 0.7, 0.1, 0.2, 0.2, 0.3 ,0.8, 0.9, 0.1, 0.2, 0.3, 0.4 , 0.8, 0.9, 0.1, 0.2, 0.4, 0.5A+ = ` ) In order to be consistent with Example 5.2, the same dis-tance measure and the same weights for criteria are adopt-edhere.Then,weapplyEq. (8)tocalculatethesimilarity measurebetweenanalternative iA( 1, 2,3, 4,5 i = )and the ideal alternativeA+ as follows:( ) ( )1 1, 1 , 1 0.2044 0.7956 s A A d A A+ += = =( ) ( )2 2, 1 , 1 0.2307 0.7693 s A A d A A+ += = =( ) ( )3 3, 1 , 1 0.2582 0.7418 s A A d A A+ += = =( ) ( )4 4, 1 , 1 0.2394 0.7606 s A A d A A+ += = =( ) ( )5 5, 1 , 1 0.2853 0.7147 s A A d A A+ += = =Finally,throughthesimilaritymeasure ( ),is A A+( 1, 2,3, 4,5 i = )betweeneachalternativeandtheidealal-ternative,therankingorderofallalternativescanbede-terminedas: 1 2 4 3 5A A A A A .Thus,theoptimal alternative is 1A . It is easy to see that the optimal alternative obtained by theYemethod[22]isthesameasourmethod,which shows the effectiveness, preciseness, and reasonableness of ourmethod.However,itisnoticedthattherankingorder ofthealternativesobtainedbyourmethodis 1 2 4 5 3A A A A A ,whichisdifferentfromthe ranking order obtained by the Ye method [22]. Concretely, therankingorderbetween 3Aand 5Aobtainedbytwo methodsarejustconverse,i.e., 5 3A Aforourmethod while 3 5A Afor the Ye method [22]. The main reason is thattheYemethoddeterminesasolutionwhichisthe closest to the positive ideal solution (PIS), while our meth-oddeterminesasolutionwiththeshortestdistancefrom thepositiveidealsolution(PIS)andthefarthestfromthe negative ideal solution (NIS). Therefore, the Ye method is suitableforthosesituationsinwhichthedecisionmaker wants to have maximum profit and the risk of the decisions islessimportantfor him,whileourmethodissuitablefor cautious(riskavoider)decisionmaker,becausethedeci-sionmakermightliketohaveadecisionwhichnotonly makes as much profit as possible, but also avoids as much risk as possible. Conclusions Consideringthatsomemulti-criteriadecisionmaking problems contain uncertain, imprecise, incomplete, and in-consistent information, and the information about criterion weightsisusuallyincomplete,thispaperhasdevelopeda novelmethodforsingle-valuedneutrosophicorinterval neutrosophicmulti-criteriadecisionmakingwithincom-plete weight information. First, motivated by the idea that a largerweightshouldbeassignedtothecriterionwitha largerdeviationvalueamongalternatives,amaximizing deviationmethodhasbeenpresentedtodeterminetheop-timal criterion weights under single-valued neutrosophic or intervalneutrosophicenvironments,whichcaneliminate theinfluenceofsubjectivityofcriterionweightsprovided bythedecisionmakerinadvance.Then,asingle-valued neutrosophic or interval neutrosophic TOPSIS is proposed to calculate the relative closeness coefficient of each alter-native to the single-valued neutrosophic or interval neutro-sophicpositiveidealsolution,basedonwhichtheconsid-eredalternativesarerankedandthenthemostdesirable oneisselected.Theprominentadvantagesofthedevel-opedmethodsarethattheycannotonlyrelievetheinflu-ence ofsubjectivityofthedecisionmaker butalsoremain theoriginaldecisioninformationsufficiently.Finally,the effectivenessandpracticalityofthedevelopedmethods havebeenillustratedwithabestglobalsupplierselection example,andtheadvantagesofthedevelopedmethods havebeendemonstratedwithacomparisonwiththeother existing methods. . 48Neutrosophic Sets and Systems, Vol. 4, 2014 Zhiming Zhang and Chong Wu, A novel method for single-valued neutrosophic multi-criteria decision making with incom-plete weight information Acknowledgment Theauthorswouldliketothanktheanonymousreviewersfortheirvaluablesuggestionsastohowto improvethispaper.ThisworkwassupportedbytheNational Natural Science Foundation of China (Grant Nos. 61073121,71271070and61375075)andtheNaturalScienceFoundationofHebeiProvinceofChina(Grant Nos. 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The conceptofa linguisticvariable andits appli-cationtoapproximatereasoning,Part1,InformationSci-ences, 8 (1975), 199249. Stage 3: Single valued neutrosophic or interval neutrosophic TOPSISDetermine the final rank and select the optimal alternative(s)Determine the criterion weightsStage 2: The maximizing deviation methodStage 1: Problem descriptionConstructsingle valued neutrosophic or interval neutrosophic decision matrixForm single valued neutrosophic or interval neutrosophic MCDMThe developed methodFig. 1: The flowchart of the developed methodsReceived: June 21, 2014. Accepted: July 1, 2014.49