Received 26 March 2014Accepted 24 November 2014Multi-valued Neutrosophic Sets and Power Aggregation Operators with Their Applications in Multi-criteria Group Decision-making Problems Juan-juan Peng School of Economics and Management, Hubei University of Automotive Technology,Shiyan 442002, China School of Business, Central South University,Changsha, 410083, China E-mail: xiaqing1981@126.com Jian-qiang Wang* School of Business, Central South University,Changsha, 410083, China E-mail: jqwang@csu.edu.cn Xiao-hui Wu School of Economics and Management, Hubei University of Automotive Technology,Shiyan 442002, China School of Business, Central South University, Changsha 410083, China E-mail: 254700361@qq.com Jing Wang School of Business, Central South University,Changsha, 410083, China E-mail: 30422815@qq.com Xiao-hong Chen School of Business, Central South University,Changsha 410083, China E-mail: 375104630@qq.com Abstract In recent years, hesitant fuzzy sets (HFSs) and neutrosophic sets (NSs) have become a subject of great interest for researchersandhavebeenwidelyappliedtomulti-criteriagroupdecision-making(MCGDM)problems.Inthis paper, multi-valued neutrosophic sets (MVNSs) are introduced, which allow the truth-membership, indeterminacy-membership and falsity-membership degree have a set of crisp values between zero and one, respectively. Then the operationsofmulti-valuedneutrosophicnumbers(MVNNs)basedonEinsteinoperationsaredefined,anda comparisonmethodforMVNNsisdevelopeddependingontherelatedresearchofHFSsandAtanassovs intuitionisticfuzzysets(IFSs).Furthermore,themulti-valuedneutrosophicpowerweightedaverage(MVNPWA) operator and the multi-valued neutrosophic power weighted geometric (MVNPWG) operator are proposed and the desirablepropertiesoftwooperatorsarealsodiscussed.Finally,anapproachforsolvingMCGDMproblemsis exploredbyapplyingthepoweraggregationoperators,andanexampleisprovidedtoillustratetheapplicationof the proposed method, together with a comparison analysis. Keywords: Multi-criteria group decision-making, multi-valued neutrosophic sets, power aggregation operators. *Corresponding author. Tel.:+8673188830594.E-mail: jqwang@csu.edu.cn (Jian-qiang WANG). International Journal of Computational Intelligence Systems, Vol. 8, No. 2 (2015) 345-363Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors345Downloaded by [Central South University], [jian-qiang Wang] at 17:33 29 December 2014 Peng et al. 1.Introduction Inmanycases,itisdifficultfordecision-makersto preciselyexpressapreferencewhensolvingmulti-criteriadecision-making(MCDM)andmulti-criteria groupdecision-making(MCGDM)problemswith inaccurate,uncertainorincompleteinformation.Under thesecircumstances,Zadehsfuzzysets(FSs)1,where themembershipdegreeis represented by arealnumber between zero and one, are regarded as an important tool forsolvingMCDMandMCGDMproblems23,fuzzy logicandapproximatereasoning4,andpattern recognition5.However, FSs can not handle certain cases where it is hard to define the membership degree using one specific value.Inordertoovercomethelackofknowledgeof non-membershipdegrees,Atanassovintroduced intuitionistic fuzzy sets (IFSs) 6, an extension of Zadehs FSs. Furthermore, Gau and Buehrer defined vague sets7 andsubsequentlyBustincepointedoutthatthevague setsandIFSsaremathematicallyequivalentobjects8. IFSssimultaneouslytakeintoaccountthemembership degree,non-membershipdegreeanddegreeof hesitation.Therefore,theyaremoreflexibleand practicalwhenaddressingfuzzinessanduncertainty thanFSs.Moreover,insomeactualcases,the membershipdegree,non-membershipdegreeand hesitationdegreeofanelementinIFSsmaynotbea specificnumber;hence,theywereextendedtothe interval-valuedintuitionisticfuzzysets(IVIFSs)9.To date,IFSsandIVIFSshavebeenwidelyappliedin solving MCDM and MCGDM problems1021. In order to handlesituationswherepeoplearehesitantin expressingtheirpreferenceregardingobjectsina decision-makingprocess,hesitantfuzzysets(HFSs) wereintroducedbyTorraandNarukawa2223.Then someworkonHFSsandtheirextensionshavebeen undertaken,includingtheaggregationoperators,the correlationcoefficient,distance,correlationmeasures and outranking relations for HFSs2430.AlthoughthetheoryofFSshasbeendevelopedand generalized,itcannotdealwithalltypesof uncertainties in differentreal-worldproblems. Typesof uncertainties, such as the indeterminate information and inconsistentinformation,cannotbemanaged.For example,whenanexpertisaskedfortheiropinion aboutacertainstatement,heorshemaysaythe possibilitythatthestatementistrueis0.5,the possibilitythatthestatementisfalseis0.6andthe degreethatheorsheisnotsureis0.231.Thisissueis beyondthescopeoftheFSsandIFSs.Then Smarandacheproposedneutrosophiclogicand neutrosophic sets (NSs)3233 and subsequently Rivieccio pointedoutthatanNSisasetwhereeachelementof theuniversehasadegreeoftruth,indeterminacyand falsityrespectivelyanditliesin]0 ,1 [ +,thenon-standardunitinterval34.Clearly, thisistheextension of thestandardinterval[0,1] .Furthermore,the uncertaintypresentedhere,i.e.indeterminacyfactor,is dependentonoftruthandfalsityvalues,whereasthe incorporateduncertaintyisdependenton thedegreesof belongingnessanddegreeofnon-belongingnessof IFSs35.Additionally,theaforementionedexampleof NSscanbeexpressedasx(0.5,0.2,0.6).However, withoutspecificdescription,NSsaredifficulttoapply toreal-lifesituations.Therefore,single-valued neutrosophic sets (SVNSs) were proposed, which are an extensionofNSs31,35.Majumdaretalintroduceda measureofentropyofSVNSs35.Furthermore,the correlation coefficients of SVNSs as well as a decision-makingmethodusingSVNSswereintroduced36.In addition,Yealsointroducedtheconceptofsimplified neutrosophicsets(SNSs),whichcanbedescribedby threerealnumbersintherealunitinterval[0,1],and proposedanMCDMmethodusingtheaggregation operators of SNSs37. Wang et al and Lupiez proposed theconceptofintervalneutrosophicsets(INSs)and provided the set-theoretic operators of INSs38,39. Broumi andSmarandachediscussedthecorrelationcoefficient ofINSs40.Furthermore,Yeproposedthecross-entropy ofSVNSsandsimilarityofINSsrespectively4142. However,incertaincases,theoperationsofSNSs providedbyYemaybeunreasonable37.Forexample, the sum of any element and themaximumvalue should beequaltothemaximumvalue,butthisisnotalways the case during operations. The similaritymeasures and distancesofSVNSsthatarebasedonthoseoperations mayalsobeunrealistic.Pengetaldevelopednovel operations,outrankingrelationsandaggregation operatorsofSNSs4344,whichwerebasedonthe operationsinYe37andappliedthemtoMCGDM problems. Zhang et al introduced a MCDM method with INSs45.LiuandWanginvestigatedsingle-valued neutrosophic normalized weighted Bonferroni mean and appliedittoMCDMproblems46.Liuetaldeveloped some Hamacher aggregation operators with NSs47. Tian Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors346Downloaded by [Central South University], [jian-qiang Wang] at 17:33 29 December 2014 MVNSs and Power Aggregation Operators etaldevelopedsimplifiedneutrosophiclinguistic normalizedweightedBonferronimeanoperatorand applied it to MCDM problems48. However,decision-makerscanalsobehesitantwhen expressing their evaluation values for each parameter in SNSs.Forexample,ifthepossibilityofastatement being true is 0.6 or 0.7, the possibility of it being false is 0.2 or 0.3 and the degree that he or she is not sure is 0.1 or 0.2, this will be beyond the capability of SNSs. If the operationsandcomparisonmethodofSNSswere extended to multiple values, the shortcomings discussed earlierwouldstillexist.Therefore,WangandLi developedthedefinitionofmulti-valuedneutrosophic sets(MVNSs)49,basedonwhich,theEinstein operationsandcomparisonmethod,andpower aggregationoperatorsformulti-valuedneutrosophic numbers(MVNNs)aredefinedinthispaper. Consequently,aMCGDMmethodisestablishedbased ontheproposedoperators.Anillustrativeexampleis alsogiventodemonstratetheapplicabilityofthe proposed method. The rest of paper is organized as follows. In Section 2 some basic concepts and operations of SNSs are briefly reviewed.ThenthedefinitionofMVNSsis introduced, andtheoperations,acomparisonmethodanddistance of MVNNs are defined inSection 3. Section 4 contains twoMVNNpoweraggregationoperatorsanda MCGDMapproachwithMVNNs.InSection5,an illustrativeexampleandacomparisonanalysisare presentedtoverifytheproposedapproach.Finally,the conclusions are drawn in Section 6. 2.Preliminaries In this section, the definitions and operations of NSs and SNSs are introduced, which will be utilized in the latter analysis. Definition 1. LetXbe a space of points (objects), with a generic element inXdenoted byx . An NSAinXis characterized by a truth-membership function( )AT x , aindeterminacy-membershipfunction( )AI xanda falsity-membership function( )AF xas follows32: { }, ( ), ( ), ( )A A AA x T x I x F x x X = ,(1) ( )AT x ,( )AI xand( )AF xarerealstandardor nonstandardsubsetsof]0 ,1 [ +,thatis, ( ) : ]0 ,1 [AT x X + ,( ) : ]0 ,1 [AI x X + ,and ( ) : ]0 ,1 [AF x X + . There is no restriction on the sum of( )AT x ,( )AI xand( )AF x ,therefore ( ) ( ) ( ) 0 sup sup sup 3A A AT x I x F x + + + . Considering the applicability of NSs, Ye reduced NSs ofnonstandardintervalsintoSNSsofstandard intervals37,whichcanpreservetheoperationsofNSs properly. Definition 2. LetXbe a space of points (objects), with a generic element inXdenoted byx . An NSAinXischaracterizedby( )AT x ,( )AI xand( )AF x ,which are singleton subintervals/subsets in the real standard [0, 1],thatis( ) : [0,1]AT x X ,( ) : [0,1]AI x X ,and ( ) : [0,1]AF x X .Then,asimplificationofAis denoted by37: ( ) ( ) ( ) { }, | , ,A A AA x T x I x F x x X = ,(2) whichiscalledanSNSandisasubclassofNSs.For convenience,theSNSsisdenotedbythesimplified symbol { }( ), ( ), ( )A A AA T x I x F x = . The set of all SNSs is represented as SNSS. The operations of SNSs are also defined by Ye37. Definition3.LetA , 1Aand 2AbethreeSNSs.For any x X , the following operations can be true37. (1)( ) ( ) ( ) ( )1 2 1 21 2,A A A AA A T x T x T x T x + = + ( ) ( ) ( ) ( )1 2 1 2,A A A AI x I x I x I x + ( ) ( ) ( ) ( )1 2 1 2A A A AF x F x F x F x + ; (2)( ) ( ) ( ) ( )1 2 1 21 2, ,A A A AA A T x T x I x I x = ( ) ( )1 2A AF x F x ; (3)( ) ( ) ( ) ( )1 1 ,1 1 ,A AA T x I x = ( ) ( )1 1 , 0AF x > ; (4)( ) ( ) ( ) , , , 0A A AA T x I x F x = > . There are some limitations related to Definition 3 and these are now outlined. (i) Insomesituations,operationssuchas 1 2A A +and 1 2A A mightbeimpractical.Thisisdemonstrated in Example 1. Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors347Downloaded by [Central South University], [jian-qiang Wang] at 17:33 29 December 2014 Peng et al. Example1.Let{ }1, 0.5, 0.5, 0.5 A x = and { }2,1, 0, 0 A x = betwoSNSs.Clearly, { }2,1, 0, 0 A x = canbethelargeroftheseSNSs. Theoretically, the sum of any number and the maximum number should be equal to the maximum one. However, accordingtoDefinition3,{ }1 2,1, 0.5, 0.5 A A x + = 2A ,thereforetheoperation+cannotbeaccepted. Similarcontradictionsexistinotheroperationsof Definition 3, and thus those defined above are incorrect. (ii) ThecorrelationcoefficientofSNSs36,whichis basedontheoperationsofDefinition3,cannotbe accepted in some specific cases. Example2.Let{ }1, 0.8, 0, 0 A x = and { }2, 0.7, 0, 0 A x = betwoSNSs,and{ } ,1, 0, 0 A x = bethelargestoneoftheSNSs.Accordingtothe correlationcoefficientofSNSs36, ( ) ( )1 2 2, , 1 W A A W A A = =canbeobtained,butthis doesnotindicatewhichoneisthebest.However,itis clear that 1Ais superior to 2A . (iii) Inaddition,thecross-entropymeasureforSNSs41, whichisbasedontheoperationsofDefinition3, cannot be accepted in special cases. Example3.Let{ }1, 0.1, 0, 0 A x = and { }2, 0.9, 0, 0 A x = betwoSNSs,and{ } ,1, 0, 0 A x = be the largestoneoftheSNSs.Accordingtothecross-entropymeasureforSNSs41,( ) ( )1 1 2 2, , 1 S A A S A A = =can be obtained, which indicates that 1Ais equal to 2A . Yetitisnotpossibletodiscernwhichoneisthebest. Since( ) ( )2 1A AT x T x > ,( ) ( )2 1A AI x I x >and ( ) ( )2 1A AF x F x >foranyxinX ,itisclearthat 2Ais superior to 1A . (iv) If( ) ( )1 20A AI x I x = =foranyxinX ,then 1Aand 2AarebothreducedtoIFSs.However,the operationspresentedinDefinition3arenotin accordance with the operations of two IFSs6, 8, 10-21. 3.Multi-valued Neutrosophic Sets Inthissection,MVNSsareintroduced,andthe correspondingoperationsandcomparisonmethodare developed in termsofthoseofIFSs6, 8, 10-21 and HFSs22, 23. 3.1.MVNSs and theirs Einstein operations Definition 4. LetXbe a space of points (objects), with agenericelementinXdenotedbyx .AnMVNSsAinXis characterized by48: ( ) ( ) ( ){ }, , ,A A AA x T x I x F x x X = ,(3) where( )AT x,( )AI x,and( )AF xarethreesetsof precisevaluesin[0,1],denotingthetruth-membership degree, indeterminacy-membership function and falsity-membershipdegreerespectively,satisfying 0 , , 1, 0 3 + + + + + ,where ( ) ( ) ( ) , ,A A AT x I x F x ,( ) supAT x +=, ( ) supAI x += and( ) supAF x +=.IfX has only one element, thenAis called a multi-valuedneutrosophicnumber(MVNN),denotedby ( ) ( ) ( ) , ,A A AA T x I x F x = .Forconvenience,an MVNNcanbedenotedby, ,A A AA T I F = .Thesetof all MVNNs is represented as MVNNS. Obviously,MVNSsaregenerallyconsideredasan extensionofNSs.Ifeachof( ) ( ) ,A AT x I x and( )AF x foranyxhasonlyonevalue,i.e., and ,and 0 + 3 + , then MVNSs are reduced to SNSs; if ( )AI x = foranyx ,thenMVNSsarereducedto DHFSs; if( ) ( )A AI x F x = = for anyx , then MVNSs arereducedtoHFSs.Inaword,MVNSsarethe extensions of SNSs, DHFSs and HFSs. Inthefollowing,theoperationsofMVNNscanbe defined based on the operations of IFSs and HFSs. Definition 5. LetA MVNNS , then the complement of aMVNNcanbedenotedby CA ,whichcanbe defined as follows: { } { } { } , 1 ,A A ACF I TA = .(4) Itisnotedthatdifferentaggregationoperatorsareall basedondifferentt-conormsandt-normsandareused todealwithdifferentrelationshipsoftheaggregated arguments,whichsatisfytherequirementsofthe conjunctionanddisjunctionoperators,respectively. EinsteinoperationsincludetheEinsteinsum ( ) ( ) 1 a b a b a b = + + andEinsteinproduct ( ) ( ) ( ) ( )1 1 1 a b a b a b = + ( ) , [0,1] a b50,which are examples of t-norms and t-conorms, respectively. In the following, the operations of MVNNs can be defined based on Einstein operations. Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors348Downloaded by [Central South University], [jian-qiang Wang] at 17:33 29 December 2014 MVNSs and Power Aggregation Operators Definition6.Let, ,A A AA T I F = ,, ,B B BB T I F = be two MVNNs and0 > . The operations of MVNNs can be defined as follows: (1) ( ) ( )( ) ( )1 1,1 1A AA ATA AA + = `+ + ) ( )( ) ( )2,2A AAIA A ` + ) ( )( ) ( )22A AAFA A ` + ) ; (2) ( )( ) ( )2,2A AATA AA = ` + ) ( ) ( )( ) ( )1 1,1 1A AA AIA A + `+ + ) ( ) ( )( ) ( )1 11 1A AA AFA A + `+ + ) ; (3),,1A A B BA BT TA BA B + = `+ ) ( ) ( ),,1 1 1A A B BA BI IA B `+ ) ( ) ( ),1 1 1A A B BA BF FA B `+ ) ; (4)( ) ( ),,1 1 1A A B BA BT TA BA B = `+ ) ,,1A A B BA BI IA B + `+ ) ,1A A B BA BF FA B + `+ ) . Ifthereisonlyonespecificnumberin,A AT I and AF,thentheoperationsinDefinition6arereducedto the operations of SNNs as follows: (5) ( ) ( )( ) ( )1 1,1 1A AA AA + =+ + ( )( ) ( )2,2AA A +

( )( ) ( )22AA A +; (6) ( )( ) ( )2,2AA AA = + ( ) ( )( ) ( )1 1,1 1A AA A + + + ( ) ( )( ) ( )1 11 1A AA A + + + ; (7) ,1A BA BA B + =+ ( ) ( ),1 1 1A BA B + ( ) ( ) 1 1 1A BA B + ; (8)( ) ( ),1 1 1A BA BA B =+ ,1A BA B ++ 1A BA B ++ . Note that the operations of MVNNs coincide with the operations of IFSs6, 8, 10-21. Example4.Let { } { } { } 0.6 , 0.1, 0.2 , 0.2 A =and { } { } { } 0.5 , 0.3 , 0.2, 0.3 B =betwoMVNNs,and 2 = , then the following results can be achieved. (1){ } { } { } 2 0.8824 , 0.1105, 0.2439 , 0.2439 A = ; (2){ } { } { }21 , 0.1980, 0.3846 , 0.3846 A = ; (3){ } { } { } 0.8462 , 0.0184, 0.0385 , 0.0244, 0.0385 A B = ; (4){ } { } { } 0.2500 , 0.3884, 0.4717 , 0.3884, 0.4717 A B = . Theorem1.Let, ,A A AA T I F = ,, ,B B BB T I F = ,and , ,C C CC T I F = bethreeMVNNs,thenthefollowing equations can be true. (1) 1 2 2 1A A A A = ; (2) 1 2 2 1A A A A = ; (3)( ) , 0 A B A B = > ; (4)( ) , 0 A B A B = > ; (5)( )1 2 1 2 1 2, 0, 0 A A A = + > > ; (6) 1 2 1 21 2, 0, 0 A A A + = > > ; (7)( ) ( ) A B C A B C = ; (8)( ) ( ). A B C A B C = Proof. (1), (2), (7) and (8) can be easily obtained. (3) Since0 > , Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors349Downloaded by [Central South University], [jian-qiang Wang] at 17:33 29 December 2014 Peng et al. ( ),1 11 1,1 11 1A A B BA B A BA B A BT TA B A BA B A BA B | | | | + + + ||+ + \ . \ .= `| | | | + + + + || + + \ . \ . ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ),21 1 1,21 1 1 1 1 121 1 121 1 1 1 1 1A A B BA BA BI IA B A BA B A BA BA BA B A BA B A B | | | |+ \ . `| | | | + || ||+ + \ . \ . ) | | | |+ \ . `| | | | + || ||+ + \ . \ . ) ,.A A B BF F ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ),,,1 1 1 1,1 1 1 12,2 222 2A A B BA A B BA A B BA B A BT TA B A BA BI IA B A BA BF FA B A B + + = `+ + + ) ` + ) ` + ) and ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ),1 1 1 11 1 1 1,1 1 1 111 1 1 12 22 22 21 1 12 2A A B BA A B BA A B BT TA A B BA A B BA BA A B BA BA A B BA B + + + + + + + = `+ + + + + + + ) + +| | | + | + +\ . ,,A A B BI I `| | | | \ . ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ),2 22 2.2 21 1 12 2A A B BA BA A B BF FA BA A B B + + `| | | | || + || + + \ . \ . ) Thus,( ) A B A B = can be obtained. Similarly, (4), (5) and (6) can be true.3.2. Comparison method Basedonthescorefunctionandaccuracyfunctionof IFSs10-21,thescorefunctionandaccuracyfunctionofa MVNN can be provided below. Definition7.Let, ,A A AA T I F = beanMVNN,and thenscorefunction( ) s Aandaccuracyfunction( ) a Aof an MVNN can be defined as follows: (1)( ) ( ), ,13A A A A A AA A AA A A T I FT I Fs Al l l = ; (2)( ) ( ), ,13A A A A A AA A AA A A T I FT I Fa Al l l = + + . Here,A A A AT I and A AF ;,A AT Il l and AFldenotethenumberofelementin,A AT I and AF, respectively.Thescorefunctionisanimportantindexinranking MVNNs.ForanMVNNA,thebiggerthetruth-membership AT is,thegreatertheMVNNwillbe;the smallertheindeterminacy-membership AI is,the greatertheMVNNwillbe;similarly,thesmallerthe false-membership AF is, the greater the MVNN will be. Forthescorefunction,ifthegreatertheresultof A A A is,themoreaffirmativethestatementwill be. For the accuracy function, the bigger the sum of the truth,indeterminacyandfalsity,themoreaffirmative the statement will be.OnthebasisofDefinition7,themethodfor comparing MVNNs can be defined as follows. Definition8.LetAandBbetwoMVNNs.The comparision method can be defined as follows: (1)If( ) ( ) s A s B >or( ) ( ) s A s B =and ( ) ( ) a A a B > ,thenAissuperiortoB ,denotedby A B ; (2)If( ) ( ) s A s B =and( ) ( ) a A a B = ,thenAis indifferent toB , denoted by~ A B . (3)If( ) ( ) s A s B =and( ) ( ) a A a B ,so A B . (2)If{ } { } { } 0.6, 0.5 , 0.4 , 0.2 A =and{ } 0.5 , B ={ } { } 0.1, 0.2 , 0.4aretwoMVNNs,then( ) ( ) s A s B =0.017 = ,( ) 0.383 a A = and( ) 0.35 a B = , ( ) ( ) a A a B > , soA B . (3)If { } { } { } 0.6, 0.7 , 0.3 , 0.2 A =and { } 0.6, 0.7 B = ,{ } { } 0.2 , 0.3aretwoMVNNs,then ( ) ( ) s A s B = 0.05 = and( ) ( ) 0.3833 a A a B = = .So ~ A B . (4)If{ } { } { } 0.5 , 0.1, 0.2 , 0.1 A =and{ } 0.6 , B ={ } { } 0.2 , 0.1aretwoMVNNs,then( ) 0.0833 s A =and ( ) 0.1 s B = .( ) ( ) s A s B < , soA B B . (5)If{ } { } { } 0.5 , 0.1, 0.2 , 0.1 A =and{ } 0.7 , B ={ } { } 0.2, 0.3 , 0.2aretwoMVNNs,then( ) ( ) s A s B =0.0833 = ,( ) 0.25 a A =and( ) 0.3833 a B = . ( ) ( ) a A a B < , soA B B . Definition9.Let, ,A A AA T I F = and, ,B B BB T I F = be two MVNNs, then the HammingHausdorff distance between A and B can be defined as follows: ( )()1, max min max min6max min max minmax min max min .B B A A A A B BB B A A A A B BB B A A A A B BA B B AT T T TA B B AI I I IA B B AF F F Fd A B = + + + + + (5) Example6.Let{ } { } { } 0.4, 0.5 , 0.2 , 0.3 A =and { } { } { } 0.8 , 0.8 , 0.5 B =betwoMVNNs,then according to Eq. (5),( ) , 0.25 d A B =can be determined. 4.Power Operators and MCGDM Approach Inthissection,thepoweraggregationoperatorsof MVNNsarepresentedandanapproachforMCGDM problemsthatutilizestheseaggregationoperatorsis proposed. 4.1. Power aggregation operator Thepoweraverage(PA)operatorwasdevelopedby Yagerintheformofnonlinearweightedaverage aggregation operator51. Definition10.ThePAoperatoristhemappingPA: nR R , which is defined as follows51: ( )( ) ( )( ) ( )11 211, , ,1ni i in ni iSPAS ==+=+2 .(6) Here( ) ( )1,,ni i j i j iS Supp = = ,and ( ),i jSupp isthesupportfor ifrom j .Thenthefollowing properties are true. (1) ( ), [0,1]i jSupp ; (2) ( ) ( ), ,i j j iSupp Supp = ; (3) ( ) ( ), ,i j p q i j p qSupp Supp iff < . Apparently,theclosertwovaluesget,themorethey support each other. 4.2. Power weighted average operator Definition11.Let, ,j j jj A A AA T I F = ( ) 1, 2, , j n = 2beacollectionofMVNNs,and( )1 2, , ,nw w w w = 2be theweightvectorof jA ( ) 1, 2, , j n = 2 ,with 0jw ( ) 1, 2, , j n = 2and 11njjw==.Themulti-valued neutrosophicpowerweightedaverage(MVNPWA) operatorofdimensionnisthemapping nMVN MV PWA: NN MVNN , and ( )( ) ( )( ) ( )11 211PWA , , ,1MVNnj j jjw nnj j jw S A AA A Aw S A== +=+2 .(7) Here ( ) ( )1,,nj j j ii j iS A w Supp A A= = and ( ),j iSupp A Aisthesupportfor jAfrom iA ,whichsatisfiesthe following conditions: (1) ( ), [0,1]i jSupp A A ; (2) ( ) ( ), ,i j j iSupp A A Supp A A = ; (3) ( )( , ) ,i j p qSupp A A Supp A A iff ( ),i jd A A > >canbeobtained.Sofor MVNPWAoperator,thefinalrankingis 1 2 4 3 .Clearly,thebestalternativeis 1while the worst alternative is 3 .IftheMVNPWGoperatorisutilizedinStep4and Step7,thenthescorefunctionvalue ( )is canbe obtained: ( ) ( ) ( ) ( )1 2 3 40.0301; 0.0259; 0.0860; 0.0572 s s s s = = = = Since( ) ( ) ( ) ( )2 1 4 3s s s s > > >andthescore values are different. Therefore, for MVNPWG operator, thefinalrankingis 2 1 4 3 ,andthebest alternative is 2while the worst alternative is 3 .Fromtheresultsgivenabove,thebestoneis 1or 2 ,andtheworstoneis 3 .Inmostcases,in order to calculatetheactualaggregationvaluesofthe alternatives, different aggregation operators can be used. Moreover,wecanfindthattwoaggregationoperators mentioned in the manuscript, the MVNPWA operator or theMVNPWGoperator,areallusedtodealwith differentrelationshipsoftheaggregatedarguments, whichcanprovidemorechoicesfordecision-makers. Theycanchoosedifferentaggregationoperator according to their preference.5.2. Comparison analysis Inordertoverifythefeasibilityoftheproposed decision-making approach basedontheMVNNs power aggregationoperators,acomparisonanalysisbasedon the same illustrative example is conducted here.Thecomparisonanalysisincludestwocases.Oneis the other methods that were outlined in Ye36, 37, 41, which arecomparedtotheproposedmethodusingsingle-valuedneutrosophicinformation.Intheother,the methodthatwasintroducedinWangandLi48are comparedwiththeproposedapproachusingmulti-valued neutrosophic information. Theproposedapproachiscomparedwithsome methods using single-valued neutrosophic information.Theproposedapproachiscomparedwithsome methodsusingsingle-valuedneutrosophicinfor-mation. WithregardtothethreemethodsinYe3637,41,all multi-valuedneutrosophicevaluationvaluesare translatedintosingle-valuedneutrosophicvaluesby usingthemeanvaluesoftruth-membership, indeterminacy-membershipandfalsity-membership respectively. Then two aggregation operators were used to aggregate the single-valued neutrosophic information first; and the correlation coefficient and weighted cross-entropybetweeneachalternativeandtheideal alternativewerecalculatedandusedtodeterminethe final ranking order of all the alternatives. If the methods inYe3637,41andtheproposedmethodareutilizedto solve the same MCDM problem, then the results can be obtained and are shown in Table 1. Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors360Downloaded by [Central South University], [jian-qiang Wang] at 17:33 29 December 2014 MVNSs and Power Aggregation Operators Table 1. The compared results utilizing the different methods with SNSs MethodsThe final ranking The best alternative(s) The worst alternative(s) Ye [36] 1 4 2 3 13Ye [37] 4 1 2 3 or 1 4 2 3 4or 13Ye [41] 4 2 1 3 43The proposed method 1 2 4 3 or 2 1 4 3 1or 23IftheaggregationoperatorsproposedbyYe37are used,fortheweightedaverageoperator,thefinal rankingis 4 1 2 3 .Clearly,thebest alternativeis 4whiletheworstalternativeis 3 .For theweightedgeometricoperator,thefinalrankingis 1 4 2 3 ,andthebestalternativeis 1while theworst alternative is 3 .However,if themethodsof Ye36,41areused,thenthefinalrankingis 1 4 2 3 or 4 2 1 3 andthebest alternative is 1or 4a . It can be seen that the results of theproposed approachare differentfrom those thatuse the earlier methods of Ye3637, 41.Therearethreereasonswhydifferencesexistinthe finalrankingsofallthecomparedmethodsandthe proposedapproach.Firstly,theaggregationoperators thatareinvolvedinthemethodofYe37arerelatedto someimpracticaloperationsaswasdiscussedin Examples1-3.Secondly,ifthecorrelationcoefficient andcross-entropyproposed36,41,proposedonthebasis oftheoperations37,areextendtoMVNNs,the shortcomingsdiscussedinSection2wouldstillexist. Finally,theaggregationvalues,correlationcoefficients andcross-entropymeasuresofSNSswereobtained firstly in Ye3637, 41 and the differences were amplified in the final results due to the use of criteria weights.The proposed approach is compared with the method using multi-valued neutrosophic information. If the method in Wang and Li48 is utilized to solve the sameMCDMproblem,thentheMVNPWAand MVNPWGoperatorswereusedtoaggregatetheeva-luation informationofeachexpertrespectively;andthe finalrankingcanbedeterminedbyusingtheTODIM methodinRef.48.IftheMVNPWAoperatorisused first,thenthefinalrankingis 1 2 4 3 ,and thebestalternativeis 1whiletheworstalternativeis always 3 ;iftheMVNPWGoperatorisused,thenthe finalrankingis 2 1 4 3 ,andthebest alternative is 2 . Apparently, the result of the proposed approachisthesameasthatusingWangandLis method48,andthebestalternativeisalways 1or 2while the worst alternative is always 3 . Fromtheanalysispresentedabove,itcanbe concludedthatthemainadvantagesoftheapproach developedinthispaperovertheothermethodsarenot onlyduetoitsabilitytoeffectivelyovercomethe shortcomingsofthecomparedmethods,butalsodue to itsabilitytorelievetheinfluenceofunfairassessments providedbydifferentdecision-makersonthefinal aggregatedresults.Thismeansthatitcanavoidlosing anddistortingthepreferenceinformationprovided whichmakesthefinalresultsmorepreciseandreliable correspond with real life decision-making problems. 6.Conclusions MVNSscanbeappliedinsolvingproblemswith uncertain,imprecise,incompleteandinconsistent informationthatexistinscientificandengineering situations.BasedontherelatedresearchofIFSsand HFSs,theoperationsofMVNNsweredefinedinthis paperandthecomparisonmethodwasalsodeveloped. Furthermore,twoaggregationoperators,namelythe MVNPWAoperatorandMVNPWGoperator,were provided.Thus,aMCGDMapproachwasestablished thatwasbasedonproposedoperators.Anillustrative exampledemonstratedtheapplicationoftheproposed decision-makingapproach.Moreover,thecomparison analysisshowedthatthefinalresultproducedbythe Co-published by Atlantis Press and Taylor & FrancisCopyright: the authors361Downloaded by [Central South University], [jian-qiang Wang] at 17:33 29 December 2014 Peng et al. proposedmethodismorepreciseandreliablethanthe resultsproducedbytheexistingmethods.The contribution ofthisstudyisthatthe proposedapproach for MCDM problems with MVNNs could overcome the shortcomingsoftheexistingmethodsaswasdiscussed earlierandrelievestheinfluenceofunfairassessments providedbydifferentdecision-makersonthefinal aggregatedresults.Infutureresearch,theauthorswill continuetostudytherelatedmeasuresofMVNNsand applied them to solve more decision-making problems. 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