Neutrosophic Sets and Systems, Vol. 7, 2015KalyanMondal,andSurapatiPramanik,RoughNeutrosophicMulti-AttributeDecision-MakingBasedonGrey Relational AnalysisRough Neutrosophic Multi-Attribute Decision-MakingBased on Grey Relational Analysis Kalyan Mondal1, and Surapati Pramanik2*Birnagar High School (HS), Birnagar, Ranaghat, District: Nadia, Pin Code: 741127, West Bengal, India. E mail:kalyanmathematic@gmail.com *Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, PO-Narayanpur, and District: North 24 Parganas, Pin Code: 743126, West Bengal, India.*Correosponding Address: Email: sura_pati@yahoo.co.inAbstract.Thispaperpresentsroughnetrosophicmulti-attributedecisionmakingbasedongreyrelational analysis.Whiletheconceptofneutrosophicsetsisapowerfullogictodealwithindeterminateand inconsistentdata,thetheoryof rough neutrosophic sets isalsoapowerfulmathematicaltooltodealwith incompleteness.Theratingofallalternativesis expressedwiththeupperandlowerapproximation operatorandthepairofneutrosophicsetswhichare characterizedbytruth-membershipdegree, indeterminacy-membershipdegree,andfalsity-membershipdegree.Weightofeachattributeispartially knowntodecisionmaker.Weextendtheneutrosophic greyrelationalanalysismethodtoroughneutrosophic greyrelationalanalysismethodandapplyittomulti-attributedecisionmakingproblem.Informationentropy methodisusedtoobtainthepartiallyknownattribute weights.Accumulatedgeometricoperatorisdefinedto transform rough neutrosophic number (neutrosophic pair) to single valued neutrosophic number. Neutrosophic grey relationalcoefficientisdeterminedbyusingHamming distancebetweeneachalternativetoidealrough neutrosophicestimatesreliabilitysolutionandtheideal roughneutrosophicestimatesun-reliabilitysolution. Thenroughneutrosophicrelationaldegreeisdefinedto determine the ranking order of all alternatives. Finally, a numericalexampleisprovidedtoillustratethe applicability and efficiency of the proposed approach. Keywords:Neutrosophicset,RoughNeutrosophicset,Single-valuedneutrosophicset,Greyrelationalanalysis,Information Entropy, Multi-attribute decision making. Introduction The notion of rough set theory was originally proposed by Pawlak [1, 2].The concept of rough set theory [1, 2, 3, 4] isanextensionofthecrispsettheoryforthestudyof intelligentsystemscharacterizedbyinexact,uncertainor insufficient information. It is a useful tool for dealing with uncertaintyorimprecisioninformation.Ithasbeen successfully applied in the different fields such as artificial intelligence[5],patternrecognition[6,7],medical diagnosis[8,9,10,11],datamining[12,13,14],image processing[15],conflictanalysis[16],decisionsupport systems[17,18],intelligentcontrol[19],etc.Inrecent years,theroughsettheoryhascaughtagreatdealof attentionandinterestamongtheresearchers.Various notionsthatcombinetheconceptofroughsets[1],fuzzy sets[20],vagueset[21],greyset[22,23]intuitionistic fuzzy sets [24], neutrosophic sets [25]are developed such asroughfuzzysets[26],fuzzyroughsets[27,28,29],generalized fuzzy rough sets [30, 31], vague rough set [32], roughgreyset[33,34,35,36]roughintuitionisticfuzzy sets[37],intuitionisticfuzzyroughsets[38],rough neutrosophic sets [ 39, 40].However neutrosophic set [41, 42] isthegeneralizationoffuzzyset,intuitionisticfuzzyset,greyset,andvagueset.Amongthehybridconcepts, the concept of rough neutrosophic sets [39, 40] is recently proposedandveryinteresting.Literaturereviewreveals thatonlytwostudiesonroughneutrosophicsets[39,40] are done. Neutrosophicsetsandroughsetsaretwodifferent concepts. Literature review reflects that both are capable of handinguncertaintyandincompleteinformation.New hybridintelligentstructurecalledroughneutrosophic setsseemstobeveryinterestingandapplicablein realisticproblems.Itseemsthatthecomputational techniques based on any one of these structures alone will notalwaysprovidethebestresultsbutafusionoftwoor more of them can often offer better results [40].Decisionmakingprocessevolvesthroughcrisp environmenttothefuzzyanduncertainandhybridenvironment.Itsdynamics,adaptability,andflexibility continue to exist and reflect a high degree of survival value. Approximate reasoning, fuzziness, greyness, neutrosophics anddynamicreadjustmentcharacterizethisprocess.The decision making paradigm evolved in modern society must bestrategic,powerfulandpragmaticratherthanretarded.Realisticmodelcannotbeconstructedwithoutgenuine understandingofthemostadvanceddecisionmaking modelevolvedsofari.e.thehumandecisionmaking 8Neutrosophic Sets and Systems, Vol. 7, 2015KalyanMondal,andSurapatiPramanik,RoughNeutrosophicMulti-AttributeDecision-MakingBasedonGrey Relational Analysisprocess.In order to perform this, very new hybrid concept suchasroughneutrosophicsetmustbeintroducedin decision making model. Decisionmakingthatincludesmorethanonemeasureof performanceintheevaluationprocessistermedasmulti-attributedecisionmaking(MADM).Differentmethodsof MADM are available in the literature.Several methods of MADMhavebeenstudiedforcrisp,fuzzy,intuitionistic fuzzy,greyandneutrosophicenvironment.Amongthese, the most popular MADM methods are Technique for Order PreferencebySimilaritytoIdealSolution(TOPSIS) proposedbyHwang&Yoon[43],PreferenceRanking OrganizationMethodforEnrichmentEvaluations(PRO-METHEE)proposedbyBransetal.[44], VIekriterijumskoKOmpromisnoRangiranje(VIKOR) developedbyOpricovic&Tzeng[45],ELiminationEt ChoixTraduisantlaREalit(ELECTRE)studiedbyRoy [46],ELECTREIIproposedbyRoyandBertier[47], ELECTREEIIIproposedby(Roy[48],ELECTREIV proposedbyRoyandHugonnard[49),Analytical HierarchyProcess(AHP)developedbySatty[50],fuzzy AHPdevelopedbyBuckley[51],AnalyticNetwork Process (ANP)studied by Mikhailov [52], Fuzzy TOPSIS proposedbyChen[53],singlevaluedneutrosophicmulti criteriadecisionmakingstudiedbyYe[54,55,56], neutrosophicMADMstudiedbyBiswasetal.[57], Entropy based grey relational analysis method for MADM studiedbyBiswasetal.[58].Asmallnumberof applicationsofneutrosophicMADMareavailableinthe literature.MondalandPramanik[59]usedneutrosophic multicriteriadecisionmakingforteacherselectionin highereducation.MondalandPramanik[60]also developedmodelofschoolchoiceusingneutrosophic MADMbasedongreyrelationalanalysis.However, MADMinroughneutrosophicenvironmentisyetto appearintheliterature.Inthispaper,anattempthasbeen madetodeveloproughneutrosophicMADMbasedon grey relational analysis.Restofthepaperisorganizedinthefollowingway. Section2presentspreliminariesofneutrosophicsetsand roughneutrosophicsets.Section3isdevotedtopresent rough neutrosophic multi-attribute decision-makingbased on grey relational analysis. Section 4 presents a numerical exampleoftheproposedmethod.Finally,section5 presentsconcludingremarksanddirectionoffuture research. 2 Mathematical Preliminaries 2.1 Definitions on neutrosophic Set The concept of neutrosophy set is originated from the new branchof philosophy,namely,neutrosophy.Neutrosophy [25] gains very popularity because of its capability to deal with the origin, nature, and scope of neutralities, as well as their interactions with different conceptional spectra.Definition2.1.1:LetEbeaspaceofpoints(objects)with generic element in E denoted by y. Then a neutrosophic set N1inEischaracterizedbyatruthmembershipfunction TN1,anindeterminacymembershipfunctionIN1anda falsity membership function FN1. The functions TN1 and FN1 arerealstandardornon-standardsubsetsof] [+ 1 , 0 thatisTN1: ] [+ 1 0, E ; IN1: ] [+ 1 0, E ; FN1:] [+ 1 0, E .It should be noted that there is no restriction on the sum of( ), yT N1( ), yI N1( ) yFN1 i.e.( ) ( ) ( )3 01 1 1++ + yFyIyT N N N-Definition2.1.2:(complement)Thecomplementofa neutrosophic set A is denoted by N1c and is defined by( ) { } ( ) yTyT N Nc1 11 =+;( ) { } ( ) yIyI N Nc1 11 =+( ) { } ( ) yFyF N Nc1 11 =+Definition2.1.3:(Containment)Aneutrosophicset N1iscontainedintheotherneutrosophicsetN2, 2 N 1 N if and only if the following result holds. ( ) ( ), inf inf y T y T2 N 1 N( ) ( )y T y T2 N 1 Nsup sup( ) ( ), inf inf y I y I2 N 1 N ( ) ( )y I y I2 N 1 Nsup sup ( ) ( ), inf inf y F y F2 N 1 N( ) ( )y F y F2 N 1 Nsup supfor all y in E. Definition2.1.4:(Single-valuedneutrosophicset). LetEbeauniversalspaceofpoints(objects)witha generic element of E denoted by y. A single valued neutrosophic set [61] S is characterized by atruthmembershipfunction( ) yT N,afalsitymembershipfunction ( ) yFN andindeterminacymembershipfunction( ) yI Nwith( ) yT N,( ) yFN,( ) yI N [ ]1 , 0 for all y in E.WhenEiscontinuous,aSNVSScanbewrittenas follows: ( ) ( ) ( ) =yS S SE y , y yI, yF, yTSandwhenEisdiscrete,aSVNSScanbewrittenas follows: ( ) ( ) ( ) E y , y yI, yF, yTSS S S =It should be observed that for a SVNS S, ( ) ( ) ( )E y 3 y I y F y T 0S S S, sup + sup + sup Definition2.1.5:Thecomplementofasinglevalued neutrosophic set S is denoted bycSand is defined by ( ) ( ) yFyT ScS= ;( ) ( ) yIyI ScS =1 ;( ) ( ) yTyF ScS=Definition2.1.6: A SVNS SN1 is contained in the other SVNSSN2 ,denotedasSN1 SN2 iff, ( ) ( ) yTyTS N S N 2 1 ;( ) ( ) yIyIS N S N 2 1 ;( ) ( ) yFyFS N S N 2 1 ,E y . 9Neutrosophic Sets and Systems, Vol.7, 2015 KalyanMondal,andSurapatiPramanik,RoughNeutrosophicMulti-AttributeDecision-MakingBasedonGrey Relational AnalysisDefinition2.1.7:Twosinglevaluedneutrosophicsets SN1andSN2areequal,i.e.SN1=SN2,iff,S S 2 N 1 N and S S 2 N 1 N Definition2.1.8: (Union) The union of two SVNSs SN1 and SN2 is a SVNS SN3 , written as S S S 2 N 1 N 3 N = . Itstruthmembership,indeterminacy-membershipand falsity membership functions are related toSN1 and SN2 as follows: ( ) ( ) ( ) ( ) yT, yTmax yTS 2 N S 1 N S 3 N= ; ( ) ( ) ( ) ( ) yI, yImax yIS 2 N S 1 N S 3 N= ; ( ) ( ) ( ) ( ) yF, yFmin yFS 2 N S 1 N 3 N S=for all y in E.Definition2.1.9: (Intersection) The intersection of two SVNSs N1 and N2 is a SVNS N3, written as. 2 N 1 N 3 N=Itstruthmembership,indeterminacymembershipand falsitymembershipfunctionsarerelatedtoN1anN2as follows: ( ) ( ) ( ) ( ) ; y T , y T min = y T2 NS1 NS3 NS( ) ( ) ( ) ( ) ; y I , y I max = y I2 NS1 NS3 NS ( ) ( ) ( ) ( ), y F , y F max = y F2 NS1 NS3 NS.Ey. Distance between two neutrosophic sets.The general SVNS can be presented in the follow form ( ) ( ) ( ) ( ) ( ) { } E y : yF, yI, yTy SS S S = Finite SVNSs can be represented as follows: ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )) 1 ( ,, ,, , , ,1 1 1 1E yy F y I y T yy F y I y T ySm S m S m S mS S S =LDefinition 2.1.10:Let ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )) 2 (, ,, , , ,1 1 11 1 1 1 1 1 11=y F y I y T yy F y I y T ySn NS n NS n NS nNSNSNSNL ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )) 3 (, ,, , , ,2 2 21 2 1 2 1 212=y F y I y T xy F y I y T xSn NS n NS n NS nNSNSNSNL betwosingle-valuedneutrosophicsets,thenthe Hammingdistance[57]betweentwoSNVSN1andN2is defined as follows: ( )( ) ( )( ) ( )( ) ( )+ + ==n1 i2 NS1 NS2 NS1 NS2 NS1 NS2 N 1 N Sy FyFyIyIyTyTS S d, (4) andnormalizedHammingdistance[58]betweentwo SNVSsSN1 and SN2 is defined as follows: ( )( ) ( )( ) ( )( ) ( )+ + ==n1 i2 NS1 NS2 NS1 NS2 NS1 NS2 N 1 NNSy FyFyIyIyTyTn 31SSd, (5)

with the following properties ( ) ) 6 ( 3 , 0 . 12 1nSS dN N S ( ) ) 7 ( 1 , 0 . 22 1 SSd N NNS 2.2 Definitions on rough neutrosophic set There exist two basic components in rough set theory, namely,crispsetandequivalencerelation,whicharethe mathematicalbasisofRSs.Thebasicideaofroughsetis basedontheapproximationofsetsbyacoupleofsets knownasthelowerapproximationandtheupper approximationofaset.Here,thelowerandupper approximation operators are based on equivalence relation. Roughneutrosophicsets[39,40]isthegeneralizationof rough fuzzy set [26] and rough intuitionistic fuzzy set [ 37]. Definition2.2.1:LetZbeanon-nullsetandRbean equivalencerelationonZ.LetPbeneutrosophicsetinZ withthemembershipfunction ,PTindeterminacy functionPIandnon-membershipfunctionPF .Thelower and the upper approximations of P in the approximation (Z, R) denotedby ( ) P Nand ( ) P Narerespectivelydefinedasfollows: ( )[ ]) 8 ( ,,/ ) ( ), ( ), ( ,) ( ) ( ) (Z x x zx F x I x T xP NRP N P N P N > < = ) ( ) ( , ) ( ) ( ) ( ) (2 1 2 1 2 1P N P N P N P N P N P N U U U > < = ) ( ) ( , ) ( ) ( ) ( ) (2 1 2 1 2 1P N P N P N P N P N P N I I I > + + < = + ) ( ) ( , ) ( ) ( ) ( ) (2 1 2 1 2 1P N P N P N P N P N P N > < = ) ( . ) ( , ) ( . ) ( ) ( . ) (2 1 2 1 2 1P N P N P N P N P N P N IfN,M,Lareroughneutrosophicsetsin(Z,R),then the following proposition [40] are stated from definitions Proposition i: N N N = ) (~ ~ . 1M N N M N M M N U U U U = = , . 2) ( ) (, ) ( ) ( . 3N M L N M LN M L N M LI I I IU U U U==) ( ) ( ) (, ) ( ) ( ) ( . 4N L M L N M LN L M L N M LI U I U IU I U I U==Proposition ii: De Morgans Laws are satisfied for neutrosophic sets )) ( (~ ) ) ( ~ ( )) ( ) ( ( ~ . 12 1 2 1P N P N P N P N I U =)) ( (~ ) ) ( (~ )) ( ) ( ( ~ . 22 1 2 1P N P N P N P N U I =Proposition iii: IfP1andP2aretwoneutrosophicsetsinUsuchthat then P P ,2 1 ) ( ) (2 1P N P N ) ( ) ( ) ( . 12 2 2 1P N P N P P N I I ) ( ) ( ) ( . 22 2 2 1P N P N P P N U U Proposition iv: ) (~ ~ ) ( . 1 P N P N =) (~ ~ ) ( . 2 P N P N =) ( ) ( . 3 P N P N rough neutrosophic multi-attribute decision-makingbased on grey relational analysis 3. Roughneutrosophicmulti-attributedecision-making based on grey relational analysis We consider a multi-attribute decision making problem with m alternatives and n attributes. Let A1, A2, ..., Am and C1,C2,...,Cnrepresentthealternativesandattributes respectively.The rating reflects the performance of the alternative Ai against the attribute Cj. For MADM weight vector W = {w1, w2,...,wn } is assigned to the attributes. The weight wj ( j = 1,2,...,n)reflectstherelativeimportanceoftheattribute Cj(j=1,2,...,m)tothedecisionmakingprocess.The weightsoftheattributesareusuallydeterminedon subjectivebasis.Theyrepresenttheopinionofasingle decisionmakeroraccumulatetheopinionsofagroupof expertsusinggroupdecisiontechnique.Thevalues associatedwiththealternativesforMADMproblemsare presented in the Table 1.Table1:Roughneutrosophicdecision matrix) 11 (, ... , ,... ... ... ... .... ... ... ... ., ... , ,, ... , ,,2 2 1 12 2 22 22 21 21 21 1 12 12 11 11 12 1mn mn m m m m mn nn nnn m ij ijd d d d d d Ad d d d d d Ad d d d d d AC C Cd d DL= =Where ij ijd d , is rough neutrosophic number according to the i-th alternative and the j-th attribute. Greyrelationalanalysis[GRA][62]isamethodofmea-suringdegreeofapproximationamongsequencesaccord-ingtothegreyrelationalgrade.Greysystemtheorydeals with primarily on multi-input, incomplete, or uncertain in-formation.GRAissuitableforsolvingproblemswith complicated relationships between multiple factors and va-riables.Thetheoriesofgreyrelationalanalysishaveal-readycaughtmuchattentionandinterestamongthere-searchers[63,64].Ineducationalfield,Pramanikand 11Neutrosophic Sets and Systems, Vol.7, 2015 KalyanMondal,andSurapatiPramanik,RoughNeutrosophicMulti-AttributeDecision-MakingBasedonGrey Relational AnalysisMukhopadhyaya[65]studiedgreyrelationalanalysis basedintuitionisticfuzzymulticriteriagroupdecision-making approach for teacher selection in higher education. Rough neutrosophic multi-attribute decision-making based ongreyrelationalanalysisispresentedbythefollowing steps. Step1: Determination the most important criteria. Manyattributesmaybeinvolvedindecisionmaking problems. However, all attributes are not equally important. Soitisimportanttoselectthepropercriteriafordecision makingsituation.Themostimportantcriteriamaybe selected based on experts opinions. Step2: Data pre-processing Consideringamultipleattributedecisionmaking problem having m alternatives and n attributes, the general formofdecisionmatrixcanbepresentedasshownin Table-1.ItmaybementionedherethattheoriginalGRA methodcaneffectivelydealmainlywithquantitative attributes.Thereexistssomecomplexityinthecaseof qualitativeattributes.Inthecaseofaqualitativeattribute (i.e.quantitativevalueisnotavailable),anassessment value is taken as rough neutrosophic number. Step3:Constructionofthedecisionmatrixwith rough neutrosophic form For multi-attribute decision making problem, the rating of alternative Ai (i = 1, 2,m ) with respect to attribute Cj (j = 1, 2,n) is assumed to be rough neutrosophic sets. It can be represented with the following form. ( ) ( )( ) ( )( ) ( )=C CF I T N F I T NCF I T N F I T NCF I T N F I T NCAjin in in n in in in nni i i i i ii i i i i ii:,, ,,,,2 2 2 2 2 2 2 221 1 1 1 1 1 1 11L ( ) ( )n jfor C CF I T N F I T NCjij ij ijjij ij ijjj, , 2 , 1:,L = =(12)Here N and N are neutrosophic sets withij ij ij ij ij ijF I T and F I T , , , ,arethedegreesoftruthmembership,degreeof indeterminacyanddegreeoffalsitymembershipofthe alternative Ai satisfying the attribute Cj, respectively where , 1 , 0 ij ijT T , 1 , 0 ij ijI I 1 , 0 ij ijF F , 3 0 + + ij ij ijF I T 3 0 + + ij ij ijF I T Theroughneutrosophicdecisionmatrix(seeTable2) can be presented in the following form: Table 2. Rough neutrosophic decision matrix ) 13 (, ... , ,... ... ... ... .... ... ... ... ., ... , ,, ... , ,...) ( ), (2 2 1 12 2 22 22 21 21 21 1 12 12 11 11 12 1~mn mn n n n n mn nn nnn m ij ijNN N N N N N AN N N N N N AN N N N N N AC C CF N F N d = =Where ij ijN and N arelowerandupperapproximationsof the neutrosophic set P. Step4: Determination of the accumulated geometric operator. Letusconsideraroughneutrosophicsetas ( ) ( )ij ij ijijij ij ijijF I T N F I T N , , , , , Wetransformtheroughneutrosophicnumberto SVNSsbythefollowingoperator.TheAccumulated GeometricOperator(AGO)isdefinedinthefollowing form: =ij ij ijijF I T N , , ( ) ( ) ( ) , . , . , .5 . 0 5 . 0 5 . 0ij ij ij ij ij ijijF F I I T T N (14)The decision matrix (see Table 3)is transformed in the form of SVNSs as follows: Table 3. Transformed decision matrix in the form SVNSmn mn mn m m m m m m mn n nn n nnn m ij ij ij SF I T F I T F I T AF I T F I T F I T AF I T F I T F I T AC C CF I T d, , ... , , , ,... ... ... ... .... ... ... ... ., , ... , , , ,, , ... , , , ,..., ,2 2 2 1 1 12 2 2 22 22 22 21 21 21 21 1 1 12 12 12 11 11 11 12 1= =(15) Step5: Determination of the weights of criteria.During decision-making process, decision makers may often encounter with partially known or unknown attribute weights.So,itiscrucialtodetermineattributeweightfor proper decision making. Many methods are available in the literatre to determine the unknown attribute weight such as maximizingdeviationmethodproposedbyWuandChen [66], entropy method proposed by Wei and Tang [67], and Xu and Hui [68]), optimization method proposed by Wang and Zhang [69], Majumder and Samanta [70]. Biswas et al. [57] used entropy method [70] for single valed neutrosophic MADM. 12Neutrosophic Sets and Systems, Vol. 7, 2015KalyanMondal,andSurapatiPramanik,RoughNeutrosophicMulti-AttributeDecision-MakingBasedonGrey Relational AnalysisInthispaperweuseanentropymethodfor determiningattributeweight.AccordingtoMajumderand Samanta [70], the entropy measure of a SVNS) ( ), ( ), (1 1 1 1 i NS i NS i NS NxFxIxTS = + ==mi icNS iNS iNS iNSN ixIxIxFxTnS En11 1 1 11) ( ) ( )) ( ) ( (11) ((16)which has the following properties: . 0 ) (0 ) ( . 111 1E x xIand set crisp a is S S EniNSN N i = =. 5 . 0 , 5 . 0 , 5 . 0) ( ), ( ), ( 1 ) ( . 21111111E xxFxIxTS EnNSNSNS N i = =) ( ) ( ) ( ) () ( ) ( ( ) ( ) ( () ( ) ( . 312121111121211112 1xIxIxIxIand xFxTxFxTS En S EncNS NScNS NSNSNSNSNSN i N i + + . ) ( ) ( . 411E x S En S EncNi N i = InordertoobtaintheentropyvalueEnjofthej-th attribute Cj (j = 1, 2,, n), equation (16) can be written as follows: + ==mi iCij i ij i ij i ij jx I xIx F x TnEn1) ( ) ( )) ( ) ( (11For i = 1, 2, , n; j = 1, 2, , m(17) It is observed that Ej [0,1] . Due to Hwang and Yoon [71],and Wang andZhang [69]the entropyweightofthe j-th attibute Cj is presented as:( ) ==nj jjjEnEnW111(18)WehaveweightvectorW=(w1,w2,,wn)Tof attributes Cj (j = 1, 2, , n) withwj 0 and 11= =ni jwStep6:Determinationoftheidealrough neutrosophicestimatesreliabilitysolution(IRNERS) andtheidealroughneutrosophicestimatesun-reliabilitysolution(IRNEURS)forroughneutrosophic decision matrix. Basedontheconceptoftheneutrosophiccube[72], maximumreliabilityoccurswhentheindeterminacy membershipgradeandthedegreeoffalsitymembership grade reach minimum simultaneously. Therefore, the ideal neutrosophic estimates reliability solution (INERS) [ ]r r r RnS S S S+ + + + = , , ,2 1L is definedas the solution in whichevery componentF I T r j j jjS+ + + + = , , is defined as follows: { },max T T ijij =+{ }I min Iijij=+and{ }F F ijij min=+in theneutrosophic decision matrix n m ij ij ij S F I T D= , ,(see the Table 1) for i = 1, 2, , n and j = 1, 2, , m. Basedontheconceptoftheneutrosophiccube[72], maximumun-reliabilityoccurswhentheindeterminacy membershipgradeandthedegreeoffalsitymembership gradereachmaximumsimultaneously.So,theideal neutrosophicestimatesunreliabilitysolution(INEURS) [ ]r r r RnS S S S = , , ,2 1L isthesolutioninwhichevery component F I T r j j jjS = , , is defined as follows: { },max T T ijij ={ } andI I ijij min={ }F F ijij min= in the neutrosophic decision matrix n m ij ij ij S F I T D= , ,(see the Table 1)for i = 1, 2, , n and j = 1, 2, , m. Fortheroughneutrosophicdecisionmakingmatrix D= n m ij ij ij F I T , , (see Table 1), Tij, Iij, Fij are the degrees of membership, degree of indeterminacy and degree of non membershipofthealternativeAiofAsatisfyingthe attribute Cj of C. Step7:Calculationoftheroughneutrosophicgrey relationalcoefficientofeachalternativefromIRNERS and IRNEURS. Roughgreyrelationalcoefficientofeachalternative from IRNERS is: + +=+ ++ ++ijj iijijj iijj iij Gmax maxmax max min min, where ( ) q q dijSjS ij,+ + =, i=1, 2, ,m and j=1, 2, .,n(19) Roughgreyrelationalcoefficientofeachalternative from IRNEURS is: + += ijj iijijj iijj iij Gmax maxmax max min min, where ( ) q q dijSijS ij =, , i=1, 2, ,m and j=1, 2, .,n(20) [ ] 1 , 0 isthedistinguishablecoefficientorthe identificationcoefficientusedtoadjusttherangeofthe comparisonenvironment,andtocontrollevelof differencesoftherelationcoefficients.When 1 = ,the comparisonenvironmentisunchanged;when 0 = ,the comparisonenvironmentdisappears.Smallervalueof distinguishingcoefficientrefleststhelargerangeofgrey relationalcoefficient.Generally, 5 . 0 = isfixedfor decision making . Step8:Calculationoftheroughneutrosophicgrey relational coefficient.13Neutrosophic Sets and Systems, Vol.7, 2015 KalyanMondal,andSurapatiPramanik,RoughNeutrosophicMulti-AttributeDecision-MakingBasedonGrey Relational AnalysisRough neutrosophic grey relational coefficient of each alternativefromIRNERSandIRNEURSaredefined respectively as follows: G w G ijnj j i+=+ =1for i=1, 2, ,m (21)G w G ijnj j i= =1for i=1, 2, ,m(22) Step9:Calculationoftheroughneutrosophic relative relational degree.Roughneutrosophicrelativerelationaldegreeofeach alternativefromIndeterminacyTrthfullnessFalsity Positive Ideal Soltion (ITFPIS) is defined as follows: G GGi iii+ ++=, for i=1, 2, ,m(23) Step 10: Ranking the alternatives. The ranking order of all alternatives can be determined accordingtothedecreasingorderoftheroughrelative relationaldegree.Thehighestvalueof i indicatesthe best alternative. 4 Numerical exampleInthissection,roughneutrosophicMADMis consideredtodemonstratetheapplicationandthe effectivenessoftheproposedapproach.Letusconsidera decision-makingproblemstatedasfollows.Supposethere isaconsciousguardian,whowantstoadmithis/herchild toasuitableschoolforpropereducation.Therearethree schools (possible alternatives) to admit his/her child: (1) A1 is a Christian Missionary School; (2) A2 is a Basic English Medium School; (3) A3 is a Bengali Medium Kindergarten. Theguardian musttakeadecision basedonthefollowing four criteria: (1) C1 is the distance and transport; (2) C2is thecost;(3)C3isstuffandcurriculum;and(4)C4isthe administration and other facilites. We obtain the following rough neutrosophic decision matrix (see the Table 4) based on the experts assessment: Table4.Decisionmatrixwithroughneutrosophic number ( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( ) 1 . , 1 . , 9 ., 2 . , 3 . , 8 .1 . , 2 . , 9 . ,2 . , 2 . , 7 .2 . , 2 . , 8 . ,3 . , 4 . , 7 .1 . , 2 . , 8 ., 3 . , 2 . , 7 .2 . , 2 . , 8 ., 4 . , 4 . , 7 .1 . , 2 . , 8 . ,3 . , 2 . , 7 .1 . , 1 . , 8 . ,3 . , 3 . , 6 .2 . , 1 . , 8 ., 3 . , 4 . , 7 .2 . , 2 . , 9 ., 4 . , 4 . , 7 .1 . , 2 . , 8 . ,3 . , 4 . , 6 .2 . , 2 . , 8 . ,3 . , 4 . , 6 .2 . , 2 . , 8 ., 4 . , 3 . , 6 .) ( ), (3214 3 2 14 3AAAC C C CP N P N dij ij S= = (23) Step2:Determinationofthedecisionmatrixinthe form SVNS Usingaccumulatedgeometricoperator(AGO)from equation (13) we have the decision matrix in SVNS form is presented as follows:) 24 (1414 ., 1732 ., 8485 .1414 ., 2000 ., 7937 .2449 ., 2828 ., 7483 .1732 ., 2000 ., 7483 .2000 ., 2449 ., 7483 .1414 ., 2000 ., 7483 .1732 ., 1732 ., 6928 .2449 ., 2000 ., 7483 .2828 ., 2828 ., 7937 .1732 ., 2828 ., 6928 .2449 ., 2828 ., 6928 .2828 ., 2449 ., 6928 .3214 3 2 1AAAC C C C Step3: Determination of the weights of attribute Entropy value Enj of the j-th (j = 1, 2, 3) attributes can bedeterminedfromthedecisionmatrix dS(15)and equation (17) as: En1= 0.4512, En2 = 0.5318, En3 = 0.5096, En4 = 0.4672. Then the corresponding entropy weights w1, w2, w3, w4of all attributes according to equation (17) are obtained by w1 = 0.2700, w2 =0.2279, w3 =0.2402, w4 =0.2619 such that 11= =nj jwStep4:Determinationoftheidealrough neutrosophic estimates reliability solution (IRNERS): { } { } { }{ } { } { }{ } { } { }{ } { } { } = =+ + + + +4 4 43 3 32 2 21 1 14 3 2 1min , min , max, min , min , max, min , min , max, min , min , max, , ,iiiiiiiiiiiiiiiiiiiiiiiiS S S S SF I TF I TF I TF I Tq q q q Q =1414 . 0 , 1732 . 0 , 8485 . 0 , 1414 . 0 , 2000 . 0 , 7937 . 0, 1732 . 0 , 1732 . 0 , 7483 . 0 , 1732 . 0 , 2000 . 0 , 7483 . 0Step5:Determinationoftheidealroughneutrosophic estimates un-reliability solution (IRNEURS): { } { } { }{ } { } { }{ } { } { }{ } { } { } = = 4 4 43 3 32 2 21 1 143 2 1max , max , min, max , max , min, max , max , min, max , max , min, , ,iiiiiiiiiiiiiiiiiiiiiiiiS S S S SF I TF I TF I TF I Tq q q q Q=2828 . 0 , 2828 . 0 , 7483 . 0 , 1732 . 0 , 2828 . 0 , 6928 . 0, 2449 . 0 , 2828 . 0 , 6928 . 0 , 2828 . 0 , 2449 . 0 , 6928 . 014Neutrosophic Sets and Systems, Vol. 7, 2015 KalyanMondal,andSurapatiPramanik,RoughNeutrosophicMulti-AttributeDecision-MakingBasedonGrey Relational AnalysisStep6:Calculationoftheroughneutrosophicgrey relationalcoefficientofeachalternativefromIRNERS and IRNEURS.ByusingEquation(19)theroughneutrosophicgrey relational coefficient of each alternative from IRNERS can be obtained [ ]=+0000 . 1 0000 . 1 6399 . 0 0000 . 16305 . 0 8368 . 0 8075 . 0 7645 . 05544 . 0 6341 . 0 6207 . 0 3333 . 04 3Gij(25)Similarly,fromEquation(20)theroughneutrosophic greyrelationalcoefficientofeachalternativefrom IRNEURS is [ ]=3333 . 0 4690 . 0 6812 . 0 4755 . 05266 . 0 5314 . 0 4752 . 0 5948 . 05403 . 0 0000 . 1 0000 . 1 0000 . 14 3Gij(26)Step7:Determinethedegreeofroughneutrosophic grey relational co-efficient of each alternative from INERS andIRNEURS.Therequiredroughneutrosophicgrey relationalco-efficientcorrespondingtoIRNERSis obtained by using equations (20) as: G+1= 0.5290, G+2= 0.7566, G+3= 0.9179 (27) andcorrespondingtoIRNEURSisobtainedwiththehelp of equation (21) as: G1= 0.8796, G2= 0.5345, G3= 0.4836 (28) Step8:Thusroughneutrosophicrelativedegreeof eachalternativefromIRNERScanbeobtainedwiththe help of equation (22) as: 1= 0.3756, 2=0.5860, 3 =0.6549(29) Step9:Therankingorderofallalternativescanbe determinedaccordingtothevalueofroughneutrosophic relationaldegreei.e. 3 > 2> 1.Itisseenthatthe highest value of rough neutrosophic relational degree is R3 thereforeA3(BengaliMediumKindergarten)isthebest alternative (school) to admit the child.Conclusion Inthispaper,weintroduceroughneutrosophicmulti-attributedecision-makingbasedonmodifiedGRA.The conceptofroughset,netrosophicsetandgreysystem theory are fused to conduct the study first time.We define theAccumulated Geometric Operator (AGO) to transform rough neutrosophic matrix to SVNS.Here all the attribute weightsinformationarepartiallyknown.Entropybased modified GRA analysis method is introduced to solve this MADMproblem.Roughneutrosophicgreyrelation coefficientisproposedforsolvingmultipleattribute decision-making problems. 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