Multicriteria Decision Making Using Double Refined Indeterminacy Neutrosophic Cross Entropy

Applied Mechanics and Materials

Submitted: 2016-07-21

ISSN: 1662-7482, Vol. 859, pp 129-143 Revised: 2016-08-22 doi:10.4028/www.scientific.net/AMM.859.129

Accepted: 2016-08-23

© 2017 Trans Tech Publications, Switzerland Online: 2016-12-01

MulticriteriaDecisionMakingUsingDoubleRefinedIndeterminacy NeutrosophicCrossEntropyandIndeterminacyBasedCrossEntropy

IlanthenralKandasamy1,a andFlorentinSmarandache2,b∗

1 SchoolofComputerScienceandEngineeringVITUniversity,Vellore,India632014 2DepartmentofMathematics,UniversityofNewMexico,USA ailanthenral.k@vit.ac.in, bfsmarandache@gmail.com *correspondingauthor

Keywords: Neutrosophicset,DoubleRefinedIndeterminacyNeutrosophicSet(DRINS),SingleValuedNeutrosophicSet(SVNS),Crossentropy,Multicriteriadecisionmaking.

Abstract.DoubleRefinedIndeterminacyNeutrosophicSet(DRINS)isaninclusivecaseoftherefined neutrosophicset,definedbySmarandache[1],whichprovidestheadditionalpossibilitytorepresent withsensitivityandaccuracytheuncertain,imprecise,incomplete,andinconsistentinformationwhich areavailableinrealworld.Moreprecisionisprovidedinhandlingindeterminacy;byclassifyingindeterminacy(I)intotwo,basedonmembership;asindeterminacyleaningtowardstruthmembership (IT )andindeterminacyleaningtowardsfalsemembership(IF ).ThiskindofclassificationofindeterminacyisnotfeasiblewiththeexistingSingleValuedNeutrosophicSet(SVNS),butitisaparticular caseoftherefinedneutrosophicset(whereeach T , I, F canberefinedinto T1, T2,…; I1, I2,…; F1, F2,…).DRINSisbetterequippedatdealingindeterminateandinconsistentinformation,withmore accuracythanSVNS,whichfuzzysetsandIntuitionisticFuzzySets(IFS)areincapableof.Basedon thecrossentropyofneutrosophicsets,thecrossentropyofDRINSs,knownasdoublerefinedIndeterminacyneutrosophiccrossentropy,isproposedinthispaper.Thisproposedcrossentropyisused foramulticriteriadecision-makingproblem,wherethecriteriavaluesforalternativesareconsidered underaDRINSenvironment.Similarly,anindeterminacybasedcrossentropyusingDRINSisalso proposed.ThedoublerefinedIndeterminacyneutrosophicweightedcrossentropyandindeterminacy basedcrossentropybetweentheidealalternativeandanalternativeisobtainedandutilizedtorankthe alternativescorrespondingtothecrossentropyvalues.Themostdesirableone(s)indecisionmaking processisselected.Anillustrativeexampleisprovidedtodemonstratetheapplicationoftheproposed method.Abriefcomparisonoftheproposedmethodwiththeexistingmethodsiscarriedout.

Introduction

FuzzysettheoryintroducedbyZadeh(1965)[2]providesaconstructiveanalytictoolforsoftdivision ofsets.Zadeh’sfuzzysettheory[2]wasextendedtointuitionisticfuzzyset(A-IFS),inwhicheach elementisassignedamembershipdegreeandanon-membershipdegreebyAtanassov(1986)[3]. A-IFSismoresuitableindealingwithdatathathasfuzzinessanduncertaintythanfuzzyset.A-IFS wasfurthergeneralizedintothenotionofintervalvaluedintuitionisticfuzzyset(IVIFS)byAtanassov andGargov(1989)[4].

Entropyisanessentialconceptformeasuringuncertaininformation.Zadehintroducedtheconcept offuzzyentropy[5].Thebeginningforthecrossentropyapproachwasfoundedininformationtheory byShannon[6].Ameasureofthecrossentropydistancebetweentwoprobabilitydistributionswas putforwardbyKullback-Leibler[7],lateramodifiedcrossentropymeasurewasproposedbyLin[8]. Afuzzycrossentropymeasureandasymmetricdiscriminationinformationmeasurebetweenfuzzy setswasproposedbyShangandJiang[9].Sinceintuitionisticfuzzysetisageneralizationofafuzzy set,anextensionoftheDe-Luca-Termininonprobabilisticentropy[10]knownasintuitionisticfuzzy cross-entropywasproposedbyVlachosandSergiadis[11]anditwasappliedtopatternrecognition, imagesegmentationandalsotomedicaldiagnosis.Vaguecross-entropybetweenVagueSets(VSs)by

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equivalencewiththecrossentropyofprobabilitydistributionswasdefinedbyZhangandJiang[12] anditsapplicationtothepatternrecognitionandmedicaldiagnosiswascarriedout.

ThefaultdiagnosisproblemofturbineusingthecrossentropyofVagueSetswasinvestigatedbyYe [13].Intuitionisticfuzzycrossentropywasappliedtomulticriteriafuzzydecision-makingproblems byYe[14].Aninterval-valuedintuitionisticfuzzycross-entropybasedonthegeneralizationofthe vaguecross-entropywasproposedandappliedtomulticriteriadecision-makingproblemsbyYe[15].

Torepresentuncertain,imprecise,incomplete,andinconsistentinformationthatarepresentinreal world,theconceptofaneutrosophicsetfromphilosophicalpointofviewwasproposedbySmarandache[16].Theneutrosophicsetisaprevailingframeworkthatgeneralizestheconceptoftheclassic set,fuzzyset,intuitionisticfuzzyset,intervalvaluedfuzzyset,intervalvaluedintuitionisticfuzzyset, paraconsistentset,paradoxistset,andtautologicalset.Truthmembership,indeterminacymembership,andfalsitymembershiparerepresentedindependentlyintheneutrosophicset.However,the neutrosophicsetgeneralizestheabovementionedsetsfromthephilosophicalpointofview,and itsfunctions TA(x),IA(x),and FA(x) arerealstandardornonstandardsubsetsof ] 0, 1+[,thatis, TA(x): X →] 0, 1+[,IA(x): X →] 0, 1+[, and FA(x): X →] 0, 1+[, respectivelywiththe condition 0 ≤ supTA(x)+ supIA(x)+ supFA(x) ≤ 3+

Itisdifficulttoapplyneutrosophicsetinthisforminrealscientificandengineeringareas.Toovercomethisdifficulty,Wangetal.[17]introducedaSingleValuedNeutrosophicSet(SVNS),which isaninstanceofaneutrosophicset.SVNScandealwithindeterminateandinconsistentinformation, whichfuzzysetsandintuitionisticfuzzysetsareincapableof.Ye[18,19,20]presentedthecorrelation coefficientofSVNSsanditscross-entropymeasureandappliedthemtosingle-valuedneutrosophic decision-makingproblems.Recently,Ye[21]hadproposedaSingleValuedNeutrosophiccrossentropytododecisionmakinginmulticriteriadecisionmakingproblemswiththedatarepresentedby SVNSs.

Owingtothefuzziness,uncertaintyandindeterminatenatureofmanypracticalproblemsinthereal world,neutrosophyhasfoundapplicationinmanyfieldsincludingSocialNetworkAnalysis(Salama etal[22]),ImageProcessing(ChengandGuo[23],SengurandGuo[24],Zhangetal[25]),Social problems(VasanthaandSmarandache[26],[27])etc.

Toprovidemoreaccuracyandprecisiontoindeterminacy,theindeterminacyvaluepresentinthe neutrosophicsethasbeenclassifiedintotwo;basedonmembership;asindeterminacyleaningtowards truthmembershipandasindeterminacyleaningtowardsfalsemembership.Whentheindeterminacy I canbeidentifiedasindeterminacywhichismoreoftruthvaluethanfalsevalue,butitcannotbe classifiedastruthitisconsideredtobeindeterminacyleaningtowardstruth (IT ).Whentheindeterminacycanbeidentifiedtobeindeterminacywhichismoreofthefalsevaluethanthetruthvalue, butitcannotbeclassifiedasfalseitisconsideredtobeindeterminacyleaningtowardsfalse (IF ). Indeterminacyleaningtowardstruthandindeterminacyleaningtowardsfalsitymakestheindeterminacyinvolvedinthescenariotobemoreaccurateandprecise.Thismodifiedrefinedneutrosophic setwasdefinedasDoubleRefinedIndeterminacyNeutrosophicSet(DRINS)aliasdoublerefined IndeterminacyNeutrosophicSet(DVNS)byKandasamy[28].

ToprovideaillustrationofrealworldproblemwhereDRINScanbeusedtorepresenttheproblem; thefollowingscenariosaregiven:Considerthescenariowheretheexpert’sopinionisrequestedabout aparticularstatement,he/shemaystatethatthepossibilityinwhichthestatementistrueis0.6and thestatementisfalseis0.5,thedegreeinwhichhe/sheisnotsurebutthinksitistrueis0.2andthe degreeinwhichhe/sheisnotsurebutthinksitisfalseis0.1.UsingadoublerefinedIndeterminacy neutrosophicnotationordoublerefinedIndeterminacyneutrosophicrepresentationitcanbeexpressed as x(0.6,0.2,0.1,0.5).

Assumeanotherexample,supposethereare10votersduringavotingprocess.Twopeoplevote yes,twopeoplevoteno,threepeopleareforyesbutstillundecidedandtwopeoplearefavouring towardsanobutstillundecided.UsingadoublerefinedIndeterminacyneutrosophicnotation,itcan beexpressedas x(0.2,0.3,0.3,0.2).However,theseexpressionsarebeyondthescopeofrepresentation

130 Advanced Research in Area of Materials, Aerospace, Robotics and Modern Manufacturing Systems

usingtheexistingSVNS.Therefore,thenotionofaDoubleRefinedIndeterminacyneutrosophicset ismoregeneralanditovercomestheaforementionedissues.

Thispaperisorganisedintosevensections:Sectiononeisintroductoryinnature.Thebasicconceptsrelatedtothepaperisgiveninsectiontwo.Sectionthreeofthepaperintroducesanddefines thecrossentropyofDoubleRefinedIndeterminacyNeutrosophicSet(DRINS).Sectionfourdeals withthesolvingmulticriteriadecisionmakingproblemsusingthecrossentropyofDRINSundera DRINSbasedenvironment.Illustrativeexamplesareprovidedtodemonstratetheproposedapproach insectionfive.Sectionsixprovidesabriefcomparisonoftheproposedapproachwiththeexisting approach.Conclusionsandfuturedirectionofworkisgiveninthelastsection.

Preliminaries/BasicConcepts

NeutrosophyandSingleValuedNeutrosophicSet(SVNS). Neutrosophyisabranchofphilosophy, introducedbySmarandache[16],whichstudiestheorigin,nature,andscopeofneutralities,aswellas theirinteractionswithdifferentideationalspectra.Itconsidersaproposition,concept,theory,event, orentity,“A”inrelationtoitsopposite,“Anti-A”andthatwhichisnot A,“Non-A”,andthatwhich isneither“A”nor“Anti-A”,denotedby“Neut-A”.Neutrosophyisthebasisofneutrosophiclogic, neutrosophicprobability,neutrosophicset,andneutrosophicstatistics.

Theconceptofaneutrosophicsetfromphilosophicalpointofview,introducedbySmarandache [16],isasfollows.

Definition1. [16]Let X beaspaceofpoints(objects),withagenericelementin X denotedby x. Aneutrosophicset A in X ischaracterizedbyatruthmembershipfunction TA(x),anindeterminacy membershipfunction IA(x),andafalsitymembershipfunction FA(x).Thefunctions TA(x), IA(x), and FA(x) arerealstandardornonstandardsubsetsof ] 0, 1+[,thatis, TA(x): X →] 0, 1+[,IA(x): X →] 0, 1+[, and FA(x): X →] 0, 1+[, withthecondition 0 ≤ supTA(x)+supIA(x)+supFA(x) ≤ 3+

Thisdefinitionofneutrosophicsetisdifficulttoapplyinrealworldapplicationofscientificand engineeringfields.Therefore,theconceptofSingleValuedNeutrosophicSet(SVNS),whichisan instanceofaneutrosophicsetwasintroducedbyWangetal.[17].

Definition2. [17]Let X beaspaceofpoints(objects)withgenericelementsin X denotedby x.An SingleValuedNeutrosophicSet(SVNS) A in X ischaracterizedbytruthmembershipfunction TA(x), indeterminacymembershipfunction IA(x),andfalsitymembershipfunction FA(x).Foreachpoint x in X,thereare TA(x),IA(x),FA(x) ∈ [0, 1],and 0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3.Therefore,an SVNSAcanberepresentedby A = {⟨x,TA(x),IA(x),FA(x)⟩| x ∈ X} Thefollowingexpressions aredefinedin[17]forSVNSs A,B: • A ∈ B ifandonlyif TA(x) ≤ TB(x),IA(x) ≥ IB(x),FA(x) ≥ FB(x) forany x in X • A = B ifandonlyif A ⊆

Therefinedneutrosophiclogicdefinedby[1]isasfollows: Definition3. Tcanbesplitintomanytypesoftruths: T1,T2,...,Tp, and I intomanytypesofindeterminacies: I1, I2, ...,Ir, and F intomanytypesoffalsities: F1,F2,...,Fs, whereall p,r,s ≥ 1 areintegers,and p + r + s = n.Inthesameway,butallsubcomponents Tj ,Ik,Fl arenotsymbols, butsubsetsof[0,1],forall j ∈{1, 2,...,p} all k ∈{1, 2,...,r} andall l ∈{1, 2,...,s}.Ifall sourcesofinformationthatseparatelyprovideneutrosophicvaluesforaspecificsubcomponentare independentsources,theninthegeneralcaseweconsiderthateachofthesubcomponents Tj ,Ik,Fl is independentwithrespecttotheothersanditisinthenon-standardset ] 0, 1+[.

CrossEntropyofSVNSsandMulticriteriaDecisionMaking.

Theconceptsofcross-entropyandsymmetricdiscriminationinformationmeasuresbetweentwo fuzzysetsproposedbyShangandJiang[9]andbetweentwoSVNSswasproposedbyYe[20].

Definition4. Assumethat A =(A(x1),A(x2),...,A(xn)) and B =(B(x1),B(x2),...,B(xn)) are twofuzzysetsintheuniverseofdiscourse X = x1, x2,...,xn.Thefuzzycrossentropyof A from B isdefinedasfollows: H(A,B)= n ∑ i=1 {A(xi)log2 A(xi) 1 2 (A(xi)+ B(xi)) +(1 A(xi))log2 (1 A(xi)) 1 1 2 (A(xi)+ B(xi))} (1)

whichindicatesthedegreeofdiscriminationof A from B.

ShangandJiang[9]proposedasymmetricdiscriminationinformationmeasure

I(A,B)= H(A,B)+ H(B,A) since H(A,B) isnotsymmetricwithrespecttoitsarguments.Moreover,thereare I(A,B) ≥ 0and I(A,B)=0 ifandonlyif A = B.Thecrossentropyandsymmetric discriminationinformationmeasuresbetweentwofuzzysetswasextendedtoSVNSsbyYe[20].

Let A and B betwoSVNSsinauniverseofdiscourse X = {xl,x2,...,xn},whicharedenoted by A = {⟨xi,TA(xi),IA(xi),FA(xi)⟩| xi ∈ X} and B = {⟨xi,TB(xi),IB(xi),FB(xi)⟩| xi ∈ X}, where TA(xi),IA(xi),FA(xi),TB(xi),IB(xi),FB(xi) ∈ [0, 1] forevery xi ∈ X.

Theinformationcarriedbythetruth,indeterminacyandfalsitymembershipsinSVNSs, A and B isconsideredasfuzzyspaceswiththreeelements.BasedonEquation 1,theamountofinformation fordiscriminationof TA(xi) from TB(xi) (i =1,2, ...,n)isgivenas

ET (A,B; xi)= TA(xi)log2 TA(xi) TB(xi) +(1 TA(xi))log2 1 TA(xi) 1 1 2 (TA(xi)+ TB(xi))

Theexpectedinformationbasedonthesinglemembershipfordiscriminationof A against B is

ET (A,B)= n ∑ i=1 {TA(xi)log2 TA(xi) TB(xi) +(1 TA(xi))log2 1 TA(xi) 1 1 2 (TA(xi)+ TB(xi))}

Similarly,theindeterminacyandthefalsitymembershipfunction,havethefollowingamountsofinformation:

EI (A,B)= n ∑ i=1 {IA(xi)log2 IA(xi) IB(xi) +(1 IA(xi))log2 1 IA(xi) 1 1 2 (IA(xi)+ IB(xi))}

EF (A,B)= n ∑ i=1 {FA(xi)log2 FA(xi) FB(xi) +(1 FA(xi))log2

1 FA(xi) 1 1 2 (FA(xi)+ FB(xi))}

Thesinglevaluedneutrosophiccrossentropymeasurebetween A and B isobtainedasthesumof thethreemeasures:

E(A,B)= ET (A,B)+ EI (A,B)+ EF (A,B)

E(A,B) alsoindicatesdiscriminationdegreeof A from B. AccordingtoShannon’sinequality[6],itisseenthat E(A,B) ≥ 0,and E(A,B)=0 ifandonly if TA(xi)= TB(xi), IA(xi)= IB(xi), and FA(xi)= FB(xi) forany xi ∈ X.Moreover,itisseen that E(Ac,Bc)= E(A,B),where Ac and Bc arethecomplementofSVNSsof A and B,respectively.Then, E(A,B) isnotsymmetric,i.e., E(B,A) = E(A,B),soitismodifiedtoasymmetric discriminationinformationmeasureforSVNSsas

D(A,B)= E(A,B)+ E(B,A).

Thelargerthedifferencebetween A and B is,thelarger D(A,B) is.ThecrossentropyofSVNSwas usedtohandlethemulticriteriadecisionmakingproblemundersinglevaluedneutrosophicenvironmentbymeansofthecrossentropymeasureofSVNSs. Theweightedcrossentropybetweenanalternative Ai andtheidealalternative A∗ iscalculatedas D(A∗,Ai)= n ∑ i=1 wj {log2 1 1 2 (1+ Tij ) + log2 1 1+ 1 2 (Iij ) + log2 1 1+ 1 2 (Fij )} + n ∑ i=1 wj {Tij log2 Tij 1 2 (1+ Tij ) +(1 Tij )log2 1 Tij 1 1 2 (1+ Tij )} + n ∑ i=1 wj {Iij +(1 Iij )log2 1 Iij 1 1 2 (Iij )} + n ∑ i=1 wj {Fij +(1 Fij )log2 1 Fij 1 1 2 (Fij )} .

Basedonthecrossentropyvaluetherankingiscarriedout.Thebestalternativeisselectedbased intherankingofthecrossentropyvalues.

DoubleRefinedIndeterminacyNeutrosophicSets(DRINSs)andTheirProperties.

Indeterminacydealswithuncertaintythatisfacedineverysphereoflifebyeveryone.Itmakes research/sciencemorerealisticandsensitivebyintroducingtheindeterminateaspectoflifeasaconcept.Therearetimesinrealworldwheretheindeterminacy I canbeidentifiedtobeindeterminacy whichhasmoreoftruthvaluethanfalsevalue,butitcannotbeclassifiedastruth.Similarlyinsome casestheindeterminacycanbeidentifiedtobeindeterminacywhichhasmoreoffalsevaluethantruth value,butitcannotbeclassifiedasfalse.Toprovidemoresensitivitytoindeterminacy,thiskindof indeterminacyisclassifiedintotwo.Whentheindeterminacy I canbeidentifiedasindeterminacy whichismoreoftruthvaluethanfalsevalue,butitcannotbeclassifiedastruth,itisconsideredtobe indeterminacyleaningtowardstruth (IT ).Whereasincasetheindeterminacycanbeidentifiedtobe indeterminacywhichismoreoffalsevaluethantruthvalue,butitcannotbeclassifiedasfalse,itis consideredtobeindeterminacyleaningtowardsfalse (IF ).

Indeterminacyleaningtowardstruthandindeterminacyleaningtowardsfalsitymakethehandling oftheindeterminacyinvolvedinthescenariotobemoremeaningful,logical,accurateandprecise.It providesabetteranddetailedviewoftheexistingindeterminacy. ThedefinitionofDoubleRefinedIndeterminacyNeutrosophicSet(DRINS)[28]isasfollows:

134 Advanced Research in Area of Materials, Aerospace, Robotics and Modern Manufacturing Systems

when X iscontinuous.Itisrepresentedas

A = n ∑ i=1 {⟨T (xi),IT (xi),IF (xi),F (xi)⟩| xi,xi ∈ X} when X isdiscrete.

ToillustratetheapplicationofDRINSintherealworldcondsiderparametersthatarecommonly usedtodefinequalityofserviceofsemanticwebserviceslikecapability,trustworthinessandprice forillustrativepurpose.Theevaluationofqualityofserviceofsemanticwebservices[29]isusedto illustratesettheoreticoperationonDoubleRefinedIndeterminacyNeutrosophicSets(DRINSs).

Definition6. ThecomplementofaDRINS A denotedby c(A) isdefinedas Tc(A)(x)= FA(x), ITc(A)(x)=1 ITA(x), IFc(A)(x)=1 IFA(x) and Fc(A)(x)= TA(x) forall x in X.

Definition7. ADRINS A iscontainedintheotherDRINS B,thatis A ⊆ B,ifandonlyif TA(x) ≤ TB(x), ITA(x) ≤ ITB (x), IFA(x) ≤ IFB(x) and FA(x) ≥ FB(x) ∀x in X

Notethatbythedefinitionofcontainmentrelation, X isapartiallyorderedsetandnotatotally orderedset.

Definition8. TwoDRINSs A and B areequal,denotedas A = B,ifandonlyif A ⊆ B and B ⊆ A.

TheunionoftwoDRINSs A and B isaDRINS C,denotedas C = A ∪ B,,theintersectionoftwo DRINSs A and B isaDRINS C,denotedas C = A ∩ B,andthedifferenceoftwoDRINSs A and B is D,writtenas D = A \ B,wasdefinedin[28].Threeoperatorscalledastruthfavourite(△),falsity favourite(▽)andindeterminacyneutral(∇)aredefinedoverDRINSs.Twooperatorstruthfavourite (△)andfalsityfavourite(▽)aredefinedtoremovetheindeterminacyintheDRINSsandtransform itintointuitionisticfuzzysetsorparaconsistentsets.SimilarlytheDRINScanbetransformedintoa SVNSbyapplyingindeterminacyneutral(∇)operatorthatcombinestheindeterminacyvaluesofthe DRINS.ThesethreeoperatorsareuniqueonDRINSs.

Definition9. ThetruthfavouriteofaDRINS A,writtenas B = △A,whosetruthmembershipand falsitymembershipfunctionsarerelatedtothoseofAby TB(x)= min(TA(x)+ITA(x), 1), ITB (x)= 0, IFB(x)=0 and FB(x)= FA(x) forall x in X

Definition10. ThefalsityfavouriteofaDRINS A,writtenas B = ▽A,whosetruthmembershipand falsitymembershipfunctionsarerelatedtothoseof A by TB(x)= TA(x), ITB(x)=0, IFB(x)=0 and FB(x)= min(FA(x)+ IFA(x), 1) forall x in X.

Definition11. TheindeterminacyneutralofaDRINS A,writtenas B = ∇A,whosetruthmembership,indeterminatemembershipandfalsitymembershipfunctionsarerelatedtothoseof A by TB(x)= TA(x), ITB (x)= min(ITA(x)+ ITB (x), 1), IFB(x)=0 and FB(x)= FA(x) forall x in X

Allsettheoreticoperatorslikecommutativity,Associativity,Distributivity,Idempotency,AbsorptionandtheDeMorgan’sLawsweredefinedoverDRINSs[28].Thedefinitionofcomplement,union andintersectionofDRINSsandDRINSsitselfsatisfiesmostpropertiesoftheclassicalset,fuzzyset, intuitionisticfuzzysetandSNVS.Similartofuzzyset,intuitionisticfuzzysetandSNVS,itdoesnot satisfytheprincipleofmiddleexclude.

ConsidertwoDRINSs A and B inauniverseofdiscourse X = xl,x2,...,xn,whicharedenotedby A = {⟨xi,TA(xi),ITA(xi),IFA(xi),FA(xi)⟩| xi ∈ X} and B = {⟨xi,TB(xi),ITB (xi),IFB(xi),FB(xi)⟩| xi ∈ X}, where TA(xi),ITA(xi),IFA(xi),FA(xi),TB(xi),ITB(xi),IFB(xi),FB(xi) ∈ [0, 1] forevery xi ∈ X.

Theinformationcarriedbythetruthmembership,indeterminacyleaningtowardstruthmembership,indeterminacyleaningtowardsfalsitymembership,andthefalsitymembershipinDRINSs A and B areconsideredasfuzzyspaceswithfourelements.ThusbasedonEquation 1,theamountof informationfordiscriminationof TA(xi) from TB(xi) (i =1,2, ...,n)canbegivenby

ET (A,B; xi)= TA(xi)log2 TA(xi) TB(xi) +(1 TA(xi))log2

1 TA(xi) 1 1 2 (TA(xi)+ TB(xi))

Therefore,theexpectedinformationbasedonthesinglemembershipfordiscriminationofAagainst Bisexpressedby

ET (A,B)= n ∑ i=1 {TA(xi)log2 TA(xi) TB(xi)++(1 TA(xi))log2 1 TA(xi) 1 1 2 (TA(xi)+ TB(xi))}

Similarly,consideringtheindeterminacyleaningtowardstruthmembershipfunction,indeterminacyleaningtowardsfalsitymembershipfunctionandfalsitymembershipfunctionthefollowing amountsofinformationisgiven:

EIT (A,B)= n ∑ i=1 {ITA(xi)log2

ITA(xi) ITB (xi)+(1 ITA(xi))log2 1 ITA(xi) 1 1 2 (ITA(xi)+ ITB (xi))} ,

EIF (A,B)= n ∑ i=1 {IFA(xi)log2 IFA(xi) IFB(xi)+(1 IFA(xi))log2 1 IFA(xi) 1 1 2 (IFA(xi)+ IFB(xi))} ,

EF (A,B)= n ∑ i=1 {FA(xi)log2 FA(xi) FB(xi) +(1 FA(xi))log2 1 FA(xi) 1 1 2 (FA(xi)+ FB(xi))} .

TheDoubleRefinedIndeterminacyneutrosophiccrossentropymeasurebetween A and B isobtained asthesumofthefourmeasures: E(A,B)= ET (A,B)+ EIT (A,B)+ EIF (A,B)+ EF (A,B)

E(A,B) alsoindicatesdiscriminationdegreeof A from B.AccordingtoShannon’sinequality[5],it canbeeasilyprovedthat E(A,B) ≥ 0,and E(A,B)=0 ifandonlyif TA(xi)= TB(xi), ITA(xi)= ITB (xi),IFA(xi)= IFB(xi), and FA(xi)= FB(xi) forany xi ∈ X.Iteasilyseenthat E(Ac,Bc)= E(A,B),where Ac and Bc arethecomplementofDRINSs A and B,respectively.Since E(A,B) is notsymmetricitismodifiedtoasymmetricdiscriminationinformationmeasureforDRINSsas

D(A,B)= E(A,B)+ E(B,A) (2)

Thelarger D(A,B) is,thelargerthedifferencebetween A and B is.

Acrossentropymeasurebasedonlyontheindeterminacyinvolvedinthescenarioisintroduced inthispaper.Indeterminacybasedcrossentropyisdefinedasthesumofinformationofindeterminacyleaningtowardsfalsitymembershipandinformationofindeterminacyleaningtowardstruth

membership.Theindeterminacybasedcrossentropy IE(A,B) istheDoubleRefinedIndeterminacy neutrosophiccrossentropymeasurebasedonindeterminacybetween A and B;isobtainedasthesum ofthetwomeasures:

IE(A,B)= n ∑ i=1

{ITA(xi)log2 ITA(xi) ITB (xi)+(1 ITA(xi))log2 1 ITA(xi) 1 1 2 (ITA(xi)+ ITB (xi))} + n ∑ i=1 {IFA(xi)log2 IFA(xi) IFB(xi)+(1 IFA(xi))log2 1 IFA(xi) 1 1 2 (IFA(xi)+ IFB(xi))}

Itindicatesthediscriminationdegreeofindeterminacyof A from B.AccordingtoShannon’s inequality[5],itcanbeeasilyprovedthat IE(A,B) ≥ 0,and IE(A,B)=0 ifandonlyif TA(xi)= TB(xi), ITA(xi)= ITB(xi),IFA(xi)= IFB(xi), and FA(xi)= FB(xi) forany xi ∈ X.Itiseasilyseen that IE(Ac,Bc)= IE(A,B),where Ac and Bc arethecomplementofDRINSs A and B,respectively. Since IE(A,B) isnotsymmetric,itismodifiedtoasymmetricdiscriminationinformationmeasure forDRINSsas

ID(A,B)= IE(A,B)+ IE(B,A) (3) Thelargerthedifferenceinindeterminacybetween A and B is,thelarger ID(A,B) is.

MulticriteriaDecisionMakingMethodBasedontheCrossEntropyofDRINS

Inamulticriteriadecisionmakingproblemallthealternativesareevaluateddependingonanumberof criteriaorsomeattributes,andthebestalternativeisselectedfromallthepossiblealternatives.Mostly multicriteriadecisionmakingproblemhavetobeinclusiveofuncertain,imprecise,incomplete,and inconsistentinformationthatarepresentinrealworldtomakeitmorerealistic.DRINScanbeusedto representthisinformationwithaccuracyandprecision.Inthissection,bymeansofutilizingthecross entropymeasureofDRINSsandindeterminacybasedcrossentropyamethodforsolvingthemulticriteriadecisionmakingproblemwhenconsideredinaDoubleRefinedIndeterminacyneutrosophic environment,isproposed.

Let A = {A1,A2,...,Am} beasetoffeasiblealternativesand C = {C1,C2,...,Cn} bethe setofcriteriaunderconsideration.Theweightofthecriterion Cj (j =1, 2,...,n), providedbythe decisionmaker,is wj , wj ∈ [0, 1] and ∑n i=1wi =1 Thecharacteristicofthealternative Ai(i = 1, 2,...,m) isgivenbyDRINS Ai = {⟨Cj ,TAi (Cj ),ITAi (Cj ),IFAi (Cj ),FAi (Cj )⟩| Cj ∈ C} where TAi (Cj ),ITAi (Cj ),IFAi (Cj ),FAi (Cj ) ∈ [0, 1], j =1, 2,...,n and i =1, 2,...,m. Now, TAi (Cj ) specifiesthedegreetowhichthealternative Ai fulfilsthecriterion Cj , ITAi (Cj ) specifiestheindeterminacyleaningtowardstruthdegreetowhichthealternative Ai fulfilsordoesnot fulfilthecriterion Cj .Similarly IFAi (Cj ) specifiestheindeterminacyleaningtowardsfalsedegree(or falseleaningindeterminacy)towhichthealternative Ai fulfilsordoesnotfulfilthecriterion Cj ,and FAi (Cj ) specifiesthedegreetowhichthealternative Ai doesnotfulfilthecriterion Cj Acriterionvalueisgenerallyobtainedfromthecalculationofanalternative Ai withrespectto acriteria Cj bymeansofascorelawanddataprocessinginpractice[12,17].Itisrepresentedas ⟨Cj ,TAi (Cj ),ITAi (Cj ),IFAi (Cj ),FAi (Cj )⟩ in Ai,isdenotedbythesymbol aij = ⟨Tij ,ITij ,IFij ,Fij ⟩ (j =1, 2,...,n, and i =1, 2,...,m), Therefore,aDoubleRefinedIndeterminacyneutrosophicdecisionmatrix A =(aij )m×n isobtained.

a11 a12 ...a1n a21 a22 ...a2n . . . am1 am2 ...amn

Theconceptofidealpointisutilizedtoaidtheidentificationofthebestalternativeinthedecision set,inmulticriteriadecisionmakingenvironments.Itisknownthatanidealalternativecannotexist intherealworld;butitdoesservesasausefultheoreticalconstructagainstwhichalternativescanbe evaluated[12].

Thereforeanidealcriterionvalue a∗ j = ⟨T ∗ j ,I∗ Tj ,I∗ Fj ,F ∗ j ⟩ = ⟨1, 0, 0, 0⟩(j =1, 2,...,n) isdefined intheidealalternative A∗.ByapplyingEquation 2 theweightedcrossentropybetweenanalternative Ai andidealalternative A∗ isobtainedtobe D(A∗,Ai)= n ∑ j=1 wj {log2 1 1 2 (1+ Tij ) + log2 1 1 1 2 (ITij ) +log2 1 1 1 2 (IFij ) + log2 1 1 1 2 (Fij )} + n ∑ j=1 wj {Tij log2 Tij 1 2 (1+ Tij ) +(1 Tij )log2 1 Tij 1 1 2 (1+ Tij )} + n ∑ j=1 wj {ITij +(1 ITij )log2 1 ITij 1 1 2 (ITij )} + n ∑ j=1 wj {IFij +(1 IFij )log2 1 IFij 1 1 2 (IFij )} + n ∑ j=1 wj {Fij +(1 Fij )log2 1 Fij 1 1 2 (Fij )} . (4)

Thesmallerthevalueof Di(A∗,Ai) is,thebetterthealternative Ai is,itimpliesthatthealternative Ai isclosetotheidealalternative A∗.Therankingorderofallalternativesisdeterminedandthebest oneisidentified,throughthecalculationoftheweightedcrossentropy Di(A∗,Ai)(i =1, 2,...,m) betweeneachalternativeandtheidealalternative.

Forcalculatingtheindeterminatebasedcrossentropymeasure IT D betweenalternative A and theindeterminateidealalternative A∗ IT theindeterminateidealalternative A∗ IT isdefinedasanideal criterionvalue a ∗ j = ⟨T ∗ j ,I∗ Tj ,I∗ Fj ,F

Calculatingtheindeterminatebasedcrossentropymeasure IF D betweenalternative A andthe indeterminateidealalternative A∗ IF

IF D(A∗ IF ,Ai)= n ∑ i=1 wj {log2 1 1 2 (1+ IFij ) + log2 1 1 1 2 (ITij )} + n ∑ i=1 wj {IFij log2 IFij 1 2 (1+ IFij ) +(1 IFij )log2 1 IFij 1 1 2 (1+ IFij )} + n ∑ i=1 wj {ITij +(1 ITij )log2 1 ITij 1 1 2 (ITij )} (6)

Theaverageof IT D(A∗ IT ,Ai) and IF D(A∗ IF ,Ai) istakenas ID(A∗ I ,Ai) ID(A∗,Ai)= IT D(A∗ IT ,Ai)+ IF D(A∗ IF ,Ai) 2 (7)

Thelargerthevalueof ID(A∗,Ai) is,thebetterthealternative Ai is,itimpliesthatthealternative Ai is farthertotheidealalternative A∗.Therankingorderofallalternativesisdeterminedandthebestone isidentified,throughthecalculationoftheindeterminacybasedweightedcrossentropy ID(A∗,Ai) (i =1, 2,...,m) betweeneachalternativeandtheidealalternative.

IllustrativeExamples

Toillustratetheapplicationoftheproposedmethod,themulticriteriadecisionmakingproblemfrom TanandChen[30]andYe[19]isadapted.Itisrelatedwithamanufacturingcompanythatwantsto selectthebestglobalsupplieraccordingtothecorecompetenciesofsuppliers.Supposethatthere arefoursuppliers A = A1,A2,A3,A4 enlisted;whosecorecompetenciesareevaluatedbasedofthe followingfourcriteria (C1,C2,C3,C4):

Theweightvectorrelatedtothefourcriteriais w =(0.3,0.25,0.25,0.2).

Theproposedmulticriteriadecisionmakingapproachisappliedtoselectthebestsupplier.From thequestionnaireofadomainexpert,theevaluationofanalternative Ai (i =1,2,3,4)withrespect toacriterion Cj (j =1,2,3,4),isobtained.Forinstance,whentheopinionofanexpertaboutan alternative A1 withrespecttoacriterion C1 isasked,heorshemaysaythatthepossibilityinwhich thestatementistrueis0.5,thedegreeinwhichheorshefeelsittruebutisnotsureis0.07,thedegree inwhichheorshefeelsitisfalsebutisnotsureis0.03andthepossibilitythestatementisfalseis0.3. Itcanbeexpressedas a11 = ⟨0 5, 0 07, 0 03, 0 2⟩,usingtheneutrosophicexpressionofDRINS.The possiblealternativeswithrespecttothegivenfourcriteriaisevaluatedbythesimilarmethodfromthe expert,thefollowingDoubleRefinedIndeterminacyneutrosophicdecisionmatrixAisobtained.

, 0.16, 0.4⟩⟨0.9, 0.06, 0.04, 0.1⟩⟨0.5, 0.1, 0.1, 0.2⟩ ⟨0.4, 0.17, 0.03, 0.1⟩⟨0.5, 0.18, 0.02, 0.3⟩⟨0.5, 0.06, 0.04, 0.4⟩⟨0.6, 0.14, 0.06, 0.1⟩ ⟨0.6, 0.07, 0.03, 0.2⟩⟨0.2, 0.15, 0.05, 0.5⟩⟨0.4, 0.16, 0.04, 0.2⟩⟨0.7, 0.11, 0.09, 0.1⟩

Thecrossentropyvaluesbetweenanalternative Ai (i =1,2,3,4)andtheidealalternative A∗ is obtainedbyapplyingEquation 4 is D(A∗,A1) =1.5054, D(A∗,A2) =1.1056, D(A∗,A3) =1.0821 and D(A∗,A4) =1.1849.Therankingorderofthefoursuppliersaccordingtothecrossentropyvalues is D(A∗,A1) ≤ D(A∗,A3) ≤ D(A∗,A2) ≤ D(A∗,A4)

Thetruth-indeterminacybasedcrossentropyvaluesbetweenanalternative Ai (i =1,2,3,4)andthe idealalternative A∗ IT isobtainedbyapplyingEquation 5,are IT D(A∗,A1) =1.5054, IT D(A∗,A2) =1.6920, IT D(A∗,A3) =1.4392and IT D(A∗,A4) =1.5067.Thefalse-indeterminacybasedcross entropyvaluesbetweenanalternative Ai (i =1,2,3,4)andtheidealalternative A∗ IF isobtained byapplyingEquation 6,are IF D(A∗,A1) =1.9000, IF D(A∗,A2) =1.6348, IF D(A∗,A3) =1.9256 and IF D(A∗,A4) =1.8447.TheindeterminacybasedcrossentropyvaluesbasedonEquation 7are ID(A∗,A1) =1.7027, ID(A∗,A2) =1.6634, ID(A∗,A3) =1.6824and ID(A∗,A4) =1.6757. Therankingorderofthefoursuppliersaccordingtothecrossentropyvaluesis ID(A∗,A1) ≥ ID(A∗,A3) ≥ ID(A∗,A4) ≥ ID(A∗,A2)

TheDRINScrossentropyandindeterminacybasedcrossentropyresultsofthedifferentalternatives andtheidealalternativesaretabulatedinTable1.

Table1: DRINCrossEntropyandindeterminacybasedCrossEntropyResults CrossEntropyValueDRINCrossEntropyIndeterminatebasedcrossentropy Ai D(A∗,Ai) ID(A∗ I ,Ai) A1 1.0793 1.7027 A2 1.1056 1.6634 A3 1.0821 1.6824 A4 1.1845 1.6757 Result A1 A1

AnalternativeisconsideredtobebestifithastheleastDRINcrossentropyvalueandthemaximum indeterminatebasedDRINcrossentropy.Thereforeitisseenthat A1 isthebestsupplier.

ItisclearlyseenthattheproposedDoubleRefinedIndeterminacyneutrosophicmulticriteriadecisionmakingmethodismorepreferableandsuitableforrealscientificandengineeringapplications becauseitcanhandlenotonlyincompleteinformationbutalsotheindeterminateinformationand inconsistentinformationwhichexistcommonlyinrealsituationsmorelogicallywithmuchmoreaccuracyandprecisionthatSVNSareincapableofdealing.

Comparison

ThispaperproposesatechniquethatextendsexistingSVNSandfuzzydecisionmakingmethodsand providesanimprovementindealingindeterminateandinconsistentinformationwithaccuracywhich isnewfordecisionmakingproblems.Forcomparativepurpose,theresultsofcrossentropyofSVNS [20]andtheproposedmethodaregiveninTable 2

FromTable 2,itisseenthattheresultsarequitedifferent.Theimportantreasoncanbeobtained bythefollowingcomparativeanalysisofthemethodsandtheircapacitytodealindeterminate,inconsistentandincompleteinformation.

DoubleRefinedIndeterminacyneutrosophicinformationisageneralizationofneutrosophicinformation.Itisobservedthatneutrosophicinformation/singlevaluedneutrosophicinformationis

140 Advanced Research in Area of Materials, Aerospace, Robotics and Modern Manufacturing Systems

Table2: CrossEntropyresultsofdifferentcrossentropybetweenidealalternativeandalternative CrossEntropySVNcrossEntropyDRINCrossEntropyIndeterminatebasedcrossentropy Value Di(A∗,Ai) D(A∗,Ai) ID(A∗ I ,Ai)

A1 1.1101 1.0793 1.7027 A2 1.1801 1.1056 1.6634 A3 0.9962 1.0821 1.6824 A4 1.2406 1.1850 1.6757 Result A3 A1 A1 generalizationofintuitionisticfuzzyinformation,andintuitionisticfuzzyinformationisitselfageneralizationoffuzzyinformation.

DRINSisaninstanceofaneutrosophicset,whichapproachestheproblemmorelogicallywith accuracyandprecisiontorepresenttheexistinguncertainty,imprecise,incomplete,andinconsistent information.Ithastheadditionalfeatureofbeingabletodescribewithmoresensitivitytheindeterminateandinconsistentinformation.While,theSVNScanhandleindeterminateinformationand inconsistentinformation,itiscannotdescribewithaccuracyabouttheexistingindeterminacy.

Itisknownthattheconnectorinfuzzysetisdefinedwithrespectto T (membershiponly)sothe informationofindeterminacyandnonmembershipislost.Theconnectorsinintuitionisticfuzzyset aredefinedwithrespecttotruthmembershipandfalsemembershiponly;heretheindeterminacyis takenaswhatisleftafterthetruthandfalsemembership.

Theintuitionisticfuzzysetcannotdealwiththeindeterminateandinconsistentinformationbut ithasprovisionstodescribeanddealwithincompleteinformation.InSVNS,truth,indeterminacy andfalsitymembershiparerepresentedindependently,andtheycanalsobedefinedwithrespectto anyofthem(norestriction)andtheapproachismorelogical.ThismakesSVNSequippedtodeal informationbetterthanIFS,whereasinDRINS,morescopeisgiventodescribeanddealwiththe existingindeterminateandinconsistentinformationbecausetheindeterminacyconceptisclassified astwodistinctvalues.ThisprovidesmoreaccuracyandprecisiontoindeterminacyinDRINS,than SVNS.

ItisclearlynotedthatinthecaseoftheSVNcrossentropybasedmulticriteriadecisionmaking methodthatwasproposedin[20],thattheindeterminacyconcept/valueisnotclassifiedintotwo,but itisrepresentedasasinglevaluedneutrosophicdataleadingtoalossofaccuracyoftheindeterminacy. SVNSareincapableofgivingthisamountoflogicalapproachwithaccuracyorprecisionaboutthe indeterminacyconcept.Similarlywhentheintuitionisticfuzzycrossentropywasconsidered,itwas notpossibletodealwiththeindeterminacymembershipfunctionindependentlyasitisdealtinSVN orDRINcrossentropybasedmulticriteriadecisionmakingmethod,leadingtoalossofinformation abouttheexistingindeterminacy.Inthefuzzycrossentropy,onlythemembershipdegreeisconsidered, detailsofnonmembershipandindeterminacyarecompletelylost.ItisclearlyobservedthattheDRINs representationandtheDRIN-crossentropybasedmulticriteriadecision-makingmethodarebetter logicallyequippedtodealwithindeterminate,inconsistentandincompleteinformation.

Conclusions

Inthispaperaspecialcaseofrefinedneutrosophicset,calledasDoubleRefinedIndeterminacyNeutrosophicSet(DRINS),withtwodistinctindeterminatevalueswasutilizedinmulticriteriadecision makingproblem.Betterlogicalapproachandprecisionisprovidedtoindeterminacysincetheindeterminateconcept/valueisclassifiedintotwobasedonmembership:oneasindeterminacyleaning towardstruthmembershipandanotherasindeterminacyleaningtowardsfalsemembership.Thiskind

ofclassificationofindeterminacyisnotfeasiblewithSVNS.DRINSisbetterequippedatdealing indeterminateandinconsistentinformation,withmoreaccuracythanSingleValuedNeutrosophicSet (SVNS),whichfuzzysetsandIntuitionisticFuzzysetsareincapableof.

InthispaperthecrossentropyofDRINSwasdefinedanditwasappliedsolvethemulticriteria decisionmakingproblem,thisapproachiscalledasDRINcrossentropybasedmulticriteriadecisionmakingmethod.ThroughtheillustrativecomputationalsampleoftheDRINcrossentropybasedmulticriteriadecision-makingmethodandothermethods,theresultshaveshownthattheDRIN-cross entropybasedmulticriteriadecision-makingmethodismoregeneralandmorereasonablethanthe others.Furthermore,insituationsthatarerepresentedbyindeterminateinformationandinconsistent information,theDRINcrossentropybasedmulticriteriadecision-makingmethodexhibitsitsgreat superiorityinclusteringthoseDoubleRefinedIndeterminacyneutrosophicdatabecausetheDRINSs areapowerfultooltodealwithuncertain,imprecise,incomplete,andinconsistentinformationwith accuracy.Inthefuture,DRINSsetsandtheDRINcrossentropybasedmulticriteriadecision-making methodcanbeappliedtomanyareassuchasonlinesocialnetworks,informationretrieval,investment decisionmaking,anddataminingwherefuzzytheoryhasbeenused.

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