Vector Similarity Measures for Simplified Neutrosophic Hesitant Fuzzy Set and Their Applications

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VECTOR SIMILARITY MEASURES FOR SIMPLIFIED NEUTROSOPHIC HESITANT FUZZY

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JournalofInequalitiesandSpecialFunctions

ISSN:2217-4303,URL:http://ilirias.com/jiasf Volume7Issue4(2016),Pages176-194.

VECTORSIMILARITYMEASURESFORSIMPLIFIED NEUTROSOPHICHESITANTFUZZYSETANDTHEIR APPLICATIONS

TAHIRMAHMOOD,JUNYEANDQAISARKHAN

Abstract. Inthisarticlewepresentthreesimilaritymeasuresbetweensimplifiedneutrosophichesitantfuzzysets,whichcontaintheconceptofsingle valuedneutrosophichesitantfuzzysetsandintervalvaluedneutrosophichesitantfuzzysets,basedontheextensionofJaccardsimilaritymeasure,Dice similaritymeasureandCosinesimilarityinthevectorspace.Thenbasedon thesethreedefinedsimilaritymeasureswepresentamultipleattributedecision makingmethodtosolvethesimplifiedneutrosophichesitantfuzzymultiplecriteriadecisionmakingproblem,inwhichtheevaluatedvaluesofthealternative withrespecttothecriteriaisrepresentedbysimplifiedneutrosophichesitant fuzzyelements.Furtherweappliedtheproposedsimilaritymeasurestopatternrecognition.Attheendanumericalexamplesarediscussedtoshowthe effectivenessoftheproposedsimilaritymeasures.

1. Introduction

Theevidenceprovidedbypeopletoexamineissuesisfrequentlyfuzzyandcrisp, andthefuzzyset(FS)hasbeenverifiedtobeveryimportantinmulti-criteria decisionmaking. FS theorywasfirstpresentedbyL.A.Zadehin1965[49]seeing degreesofmembershiptoprogressivelyassesthemembershipofelementsinaset. Aftertheintroductionof FS,atanassov[1]presentedtheconceptofintuitionistic fuzzyset(IFS)byconsideringmembershipandnon-membershipdegreesofan elementtotheset.Atanassovetal.extended IFS tointervalvaluedintuitionistic fuzzyset(IVIFS)[2].Aftertheintroductionof IFS alotofsimilaritymeasures areproposedbymanyresearchersastheextensionofthesimilaritymeasuresfor FS,asasimilaritymeasureisanimportanttoolfordeterminingsimilaritybetween objects.Liandcheng[20]presentedsimilaritymeasurefor IFS whichisthefirstto appliedthemtopatternrecognition.S.KDeetal.gavesomeapplicationof IFS tomedicaldaignosis[19].Ye[36]presentedcosinesimilaritymeasurefor IFS and appliedittomedicaldiagnosisandpatternrecognition.Ye[37,38]alsopresented

2010 MathematicsSubjectClassification. 90B50,91B06,91B10,62C99.

Keywordsandphrases. Vectorsimilaritymeasure;Jaccardsimilaritymeasure;Dicesimilarity measure;Cosinesimilaritymeasure;hesitantfuzzysets;intervalvaluedhesitantfuzzysets;Single valuedneutrosophichesitantfuzzysets;intervalneutrosophichesitantfuzzysets;multi-criteria decisionmaking;patternrecognition.

c 2016UniversitetiiPrishtin¨es,Prishtin¨e,Kosov¨e. SubmittedMay1,2016.PublishedNovember17,2016.

cosinesimilaritymeasureandvectorsimilaritymeasuresfor IVIFS andtrapezoidal IFN andappliedthemtomultiplecriteriadecisionmakingproblem. Neutrosophicset( NS)wasfirstpresentedbyF.Smarandache[25,26]which simplifiestheconceptofcrispset, FS, IFS, IVIFS,andsoon.In NS itsindeterminacyiscomputedunambiguouslyanditstruth-membership,indeterminacymembershipandfalsity-membershiparesignifiedfreely. NS simplifiesalltheabove definedsetsfromphilosophicalpointofview.Severalsimilaritymeasuresfor NS hasbeendefinedbyBroumi.S.etal.[3].Broumi.S.etal.[6,7,8,9,10,11] singlevaluedneutrosophicgraph,BipolarSingleValuedNeutrosophicGraphs, IsolatedSingleValuedNeutrosophicGraphs,BipolarSingleValuedNeutrosophic GraphTheory,SingleValuedNeutrosophicGraphs:Degree,OrderandSize,and AnIntroductiontoBipolarSingleValuedNeutrosophicGraphTheory.Majumdar etal.[21]presentedsimilarityandentropyfor NS. NS isdifficulttoapplyinreallife andengineeringproblem.Theconceptofsinglevaluedneutrosophicsets SVNSs wasintroducedforthefirsttimebyF.Smarandachein1998[25].Wangetal. [30,31]introducedtheconceptofintervalneutrosophicsets( INSs)andprovided basicoperationandpropertiesof SVNSs and INSs.The SVNSs and INSs isasubclassof NS.SahinR.etal.[24]definedsubsethoodfor SVNSs.Zhang etal[50]alsodefinedsomeoperationfor INS.Ye[39,40,41,42]definedcorrelationcoefficient,improvedcorrelation,entropyandsimilaritymeasurefor SVNS and INS andgavetheirapplicationinmultipleattributedecisionmakingproblems(MCDM ).Broumi.S.etal.[4,5]presentedcorrelationandcosinesimilarity measurefor INS.Ye[43]presentedtheconceptofsimplifiedneutrosophicsets( SNSs)whichcontainedtheconceptof SVNS and INS anddefinedsomebasic operationfor SNSs andappliedthemtomulti-criteriadecisionmakingproblem. Ye[44,45]alsodefinedvectorsimilaritymeasuresfor SNSs andimprovedcosine similaritymeasuresfor SNSs.Pengetal[18]findsomedrawbacksoftheoperation for SNS definedbyYeandpresentednewoperationsfor SNS.

Hesitantfuzzyset( HFS)isanothergeneralizationof FS proposedbyTorra andNarukawa[28,29].Theadvantegeof HFS isthatitauthorizationsthemembershipdegreeofanelementtoagivensetwithalimiteddifferentvalues,which canascendinagroupdecisionmakingproblem.Recentlya HFS hasacknowledgedmoreandmoreconsiderationsinceitspresence.XiaandXu[32,33,34,35] presentedhesitantfuzzyinformation,similaritymeasuresfor HFS andcorrelation for HFS.Chenetal[13]presentedsomecorrelationcoefficientfor HFS andappliedthemtoclusteringanalysis.Ye[46]presentedvectorsimilaritymeasuresfor HFS.Chenetal.[12]furtherextendedHFStointervalvaluedhesitantfuzzyset. ( IVHFS)Farhadinia.B[15,16]definedinformationmeasuresfor HFS,and IVHFS. anddistancemeasuresforhighorder HFS.

Dualhesitantfuzzysets(DHFS)waspresentedbyZhu[51,52]whichisthe generalizationof HFS, FS, IFS andfuzzymultisetsarespecialcasesof DHFS RecentlySu.Z[27]presentedsimilaritymeasurefor DHFSs andapplieditto

patternrecognition.Correlationof DHFS waspresentedbyYe[47]andapplied themto MCDM under DHF information.Asmentionedabovethathesitancy isthemostcommonproblemindecisionmakingforwhich HFS isasuitable meansbyallowingseveralpossiblevaluesforanelementtoaset.Howeverin HFS theyconsideronlyonetruth-membershipfunctionanditcannotexpressthis problemwithafewdifferentvaluesassignedbytruth-membershiphesitantdegree, indeterminacy-membershipdegree,andfalsity-membershipdegreesduetodoubtsof decisionmakers.andalsoin DHFS theyconsidertwofunctionsthatismembership andnon-membershipfunctionsandcannotconsiderindeterminacy-membership function.TooverawedthisproblemandthedecisionmakercangetmoreinformationYe[48]presentedtheconceptofsinglevaluedneutrosophichesitantfuzzysets (SVNHF )anddefinedsomebasicoperations,aggregationoperatorsandapplied itto MCDM under SVNHF .environment.LuiP.[22]presentedtheconceptofintervalneutrosophichesitantfuzzyset INHF anddefinedsomebasicoperations, aggregationoperatorsandappliedthemtomulticriteriadecisionmakingproblem.

Sincevectorsimilaritymeasuresplayedanimportantroleindecisionmaking,so wepresentvectorsimilaritymeasuresfor SNHF andthengiveitsapplicationsin MCDM andpatternrecognition.

Therestofthearticleisorganizedasfollows.Insection2wedefinedsomebasic definitionsrelatedtoourwork.Insection3wedefinedvectorsimilaritymeasures forsimplifiedneutrosophichesitantfuzzysets.Insection4weapplytheproposed similaritymeasurestomultiattributedecisionmakingproblemundersimplified neutrosophicenvironment.insection5weapplytheproposedsimilaritymeasures topatternrecognition.Attheendwegivesomenumericalexamplestoshowthe effectivenessoftheproposedsimilaritymeasures,conclusionandreferencesare given.

2. Preliminaries

Inthissectionwedefinesomebasicdefinitionsabouthesitantfuzzysets,interval valuedhesitantfuzzyset,singlevaluedneutrosophichesitantfuzzyset,interval neutrosophichesitantfuzzysetsandthevectorsimilaritymeasures,suchasJaccard similaritymeasure,Dicesimilaritymeasure,Cosinesimilaritymeasure.

Definition2.1. [28,29]Let U beafixedset,ahesitantfuzzyset ˆ M on U isdefined intermsofafunction f ˆ M (a) thatwhenappliedto U returnsafinitesubsetsof [0, 1]. Hesitantfuzzysetismathematicallyrepresentedas, ˆ M = { a,fM (a)|a ∈ U },

Where fM (a)isasetofsomedifferentvaluesin[0, 1], denotingthepossible membershipdegreesoftheelement a ∈ U to ˆ M.

Definition2.2. [12]Let U beafixedset,anintervalvaluedhesitantfuzzyset ˆ D on U isdefinedintermsofafunction f ˆ M (a) thatwhenappliedto U returnsafinite setofsubintervalsof [0, 1]. Anintervalvaluedhesitantfuzzysetismathematically representedas,

ˆ D = { a,fD(a)|a ∈ U },

Where fD(a)isasetofsomedifferentvaluesin[0, 1], denotingthepossible membershipdegreesoftheelement a ∈ U to ˆ D.

Definition2.3. [48]Let U beanonemptyfixedset.Asinglevaluedneutrosophic hesitantfuzzyset ˆ N isthestructureoftheform: ˆ N = { a, ˘ t(a), ˘ ı(a), ˘ f (a) } Inwhich ˘ t(a), ˘ ı(a)and ˘ f (a)arethreesetsofsomevaluesintherealunitinterval [0, 1], representingthepossibletruth-membershiphesitantdegrees,indeterminacymembershiphesitantdegreesandfalsity-membershiphesitantdegreeoftheelement a ∈ U totheset ˆ N, respectively,withtheconditionthat0 ≤ ϕ,χ,ψ ≤ 1and 0 ≤ ϕ+ +χ+ +ψ+ ≤ 3, where ϕ ∈ ˘ t(a),χ ∈ ˘ ı(a),ψ ∈ ˘ f (a),ϕ+ = ∪ϕ∈ ˘ t max{ϕ},χ+ = ∪ χ∈i max{χ},ψ+ = ∪ψ∈f max{ψ}.

Definition2.4. [22]Let U beanonemptyfixedset.Aintervalneutrosophichesitantfuzzyset ˆ H isstructureoftheform ˆ H = { a, ˘ t(a), ˘ ı(a), ˘ f (a) }

Inwhich ˘ t(a), ˘ ı(a)and ˘ f (a)arethreesetsofsomeintervalvaluesinthesetof realunitinterval[0, 1], representingthepossibletruth-membershiphesitantdegrees, indeterminacy-membershiphesitantdegreesandfalsity-membershiphesitantdegree oftheelement a ∈ U totheset ˆ H, respectively,withtheconditionthat ϕ = [ϕl,ϕu] ∈ ˘ t(a),χ =[χl,χu] ∈ ˘ ı(a),ψ =[ψl,ψu] ∈ ˘ f (a) ⊆ [0, 1]and0 ≤ sup ϕ+ + sup χ+ +sup ψ+ ≤ 3,ϕ+ = ∪ϕ∈ ˘ t max{ϕ},χ+ = ∪ χ∈i max{χ},ψ+ = ∪ψ∈f max{ψ}.

2.1. Vectorsimilaritymeasures. Let A =(c1,c2,...,cm) andB =(e1,e2,...,em) betwovectorsoflengthis m whereallthecoordinatesarepositive.Thejaccard similaritymeasureofthesetwovectors[17]isgivenby ˜ J(A,B)= A.B ||A||2 2 + ||B||2 2 A.B =

m k=1 ckek m k=1 c2 k + m k=1 e2 k m k=1 ckek (2.1)

Where A.B = m k=1 ckek istheinnerproductofthevectors A and B and ||A||2 = m k=1 c2 and ||B||2 = m k=1 e2 aretheEuclideannormsof A and B.

ThentheDicesimilaritymeasure[14]isdefinedasfollows: D(A,B)= 2A.B ||A||2 2 + ||B||2 2 = 2 m k=1 ckek m k=1 c2 k + m k=1 e2 k (2.2)

andtheCosinesimilaritymeasure[23]isdefinedasfollows:

C(A,B)= A.B ||A||2 ||B||2 =

m k=1 ckek m k=1 c2 k m k=1 e2 k

(2.3)

TheCosinesimilaritymeasureisnothingjustthecosineanglebetweentwo vectors.

theaboveformulasaresameinthesenseeachtakethevalueintheunitinterval [0, 1] Thejaccardanddiceandcosinesimilaritymeasuresarerespectivelyundefined if ck = ek =0,fork =1, 2...mandck =0 orek =0,k =1, 2...m. Nowweassume thatthecosinesimilaritymeasureequaltozerowhen ck =0 orek =0,k =1, 2...m. Thesevectorsimilaritymeasuresatisfythefollowingpropertiesforthevectors AandB.

(1) J(A,B)= J(B,A),D(A,B)= D(B,A),andC(A,B)= C(B,A) (2)0 ≤ J(A,B),D(A,B),C(A,B) ≤ 1

(3) J(A,B)= D(A,B)= C(A,B)=1 ifA = B

3. VectorSimilarityMeasureForSimplifiedNeutrosophichesitant fuzzySets

Ifthesimplifiedneutrosophichesitantfuzzysetcontainsfinitesetofsinglepoints, thatissinglevaluedneutrosophichesitantfuzzyset.ThentheJaccard,Diceand Cosinesimilaritymeasuresaredefinedasfollows: Definition3.1. Let A and B betwo SVNHFSs ontheuniversalset U = {u1,u2,...,um}, respectivelydenotedby A = { uj ,τA(uj ),ιA(uj ),κA(uj )|uj ∈ U } and B = { uj ,τB (uj ),ιB (uj ),κB (uj )|uj ∈ U } forall j =1, 2,...,m.Thenwedefinethejaccard,Diceandcosinesimilaritymeasuresfor SVNHFSs A and B asfollows: JSVNHFS (A, B)= 1 m

m j=1

lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(¯ uj ) + lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj ) lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 + lej i=1 τBσ(i)(˜ uj ) 2 + lej i=1 ιBσ(i)(˜ uj ) 2 + lej i=1 κBσ(i)(˜ uj ) 2     lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(˜ uj )+ lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj )

   

DSVNHFS (A, B)= 1 m

m j=1

CSVNHFS (A, B)= 1 m

m j=1

    lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 + lej i=1 τBσ(i)(˜ uj ) 2 + lej i=1 ιBσ(i)(˜ uj ) 2 + lej i=1 κBσ(i)(˜ uj ) 2 (2)

2     lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(˜ uj ) + lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj )

lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(˜ uj )+ lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj ) lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 + lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 (3)

Here lej =max(le(¯ τA(˜ uj ), ιA(˜ uj ), κA(˜ uj )),le(¯ τB (˜ uj ), ιB (˜ uj ), κB (˜ uj )))forall˜ uj ∈ U, where le(¯ τA(˜ uj ), ιA(˜ uj ), κA(˜ uj ))and le(¯ τB (˜ uj ), ιB (˜ uj ), κB (˜ uj )),respectivelyrepresentsthenumberofvaluesin¯ τA(˜ uj ), ιA(˜ uj ), κA(˜ uj )and¯ τB (˜ uj ), ιB (˜ uj ), κB (˜ uj ) Whenthenumberofvaluesin¯ τA(˜ uj ), ιA(˜ uj ), κA(˜ uj )and¯ τB (˜ uj ), ιB (˜ uj ), κB (˜ uj ) are notequalthatis le(¯ τA(˜ uj ), ιA(˜ uj ), κA(˜ uj )) = le(¯ τB (˜ uj ), ιB (˜ uj ), κB (˜ uj )). Thenthe pessimisticaddtheminimumvaluewhiletheoptimisticaddthemaximumvalue. Thisdependonthedecisionmakersthatwassuccessfullyappliedforhesitantfuzzy setsby[34].

Theabovedefinedthreevectorsimilaritymeasuressatisfytheconditiondefined inYe[46].

(A1) JSVNHFS (A, B)= JSVNHFS (B, A), DSVNHFS (A, B)= DSVNHFS (B, A), CSVNHFS (A, B)= CSVNHFS (B, A)

(A2)0 ≤ JSVNHFS (A, B), DSVNHFS (A, B), CSVNHFS (A, B) ≤ 1

(A3) JSVNHFS (A, B)= DSVNHFS (A, B)= CSVNHFS (A, B)=< 1, 0, 0 > iff A = B

Proof. (A1)obviouslyitistrue.

(A2)obviouslythepropertyistrueduetotheinequality˜ x2 +˜ y2 ≥ 2˜xy foreq1 andeq2andthecosinevalueforeq3.

(A3)When A = B then¯ τA(˜ uj )=¯ τB (˜ uj ), ιA(˜ uj )=¯ ιB (˜ uj )and¯ κA(˜ uj )=¯ κB (˜ uj ) foreach˜ uj ∈ U,j =1, 2,...,m. Sothereare JSVNHFS (A, B)=1, DSVNHFS (A, B)=1, and CSVNHFS (A, B)=1 Inreallifeproblem,theelements˜ uj (j =1, 2,...,m)inauniversalset U = {˜ u1, ˜ u2,..., ˜ uj } havedifferentweights.Let =( 1, 2,..., j )T betheweightvectorof˜ uj (j =1, 2,...,m)with j ≥ 0,j =1, 2,...,m,and m j=1 j =1. Thereforewe

extendtheabovethreesimilaritymeasurestoweightedvectorsimilaritymeasures for SVNHFSs.

TheweightedJaccardsimilaritymeasurefor SVNHFSs A and B asfollows: JSVNHFS (A, B)= m j=1 j

lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(˜ uj )+ lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj ) lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 + lej i=1 τBσ(i)(˜ uj ) 2 + lej i=1 ιBσ(i)(˜ uj ) 2 + lej i=1 κBσ(i)(˜ uj ) 2 lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(˜ uj )+ lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj ) (4)

TheDicesimilaritymeasurefor SVNHFSs A and B asfollows:

DSVNHFS (A, B)= m j=1 j

2     lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(˜ uj ) + lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj )

    lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 + lej i=1 τBσ(i)(˜ uj ) 2 + lej i=1 ιBσ(i)(˜ uj ) 2 + lej i=1 κBσ(i)(˜ uj ) 2 (5)

TheCosinesimilaritymeasurefor SVNHFSs A and B asfollows:

lej i=1 τAσ(i)(˜ uj )¯ τBσ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ιBσ(i)(˜ uj )+ lej i=1 κAσ(i)(˜ uj )¯ κBσ(i)(˜ uj )   lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2    lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2  (6) Ifthe SNHFS containasetofintervalvaluesinsteadofasetofsinglepoints. Then

CSVNHFS (A, B)= m j=1 j

m j=1

lej i=1 τ L Aσ(i)(˜ uj )¯ τ L Bσ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ U Bσ(i)(˜ uj ) + lej i=1 ιL Aσ(i)(˜ uj )¯ ιL Bσ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(˜ uj )¯ ιU Bσ(i)(˜ uj )+ lej i=1 κL Aσ(i)(˜ uj )¯ κL Bσ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κU Bσ(i)(˜ uj ) lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2 + lej i=1 τ L Bσ(i)(˜ uj ) 2 + lej i=1 τ U Bσ(i)(˜ uj ) 2 + lej i=1 ιL Bσ(i)(˜ uj ) 2 + lej i=1 ιU Bσ(i)(˜ uj ) 2 + lej i=1 κL Bσ(i)(˜ uj ) 2 + lej i=1 κU Bσ(i)(˜ uj ) 2         

DSVNHFS (A, B 2

          

le

j i

+

+

lej i=1 τ L Aσ(i)(˜ uj )¯ τ L Bσ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ U Bσ(i)(˜ uj ) + lej i=1 ιL Aσ(i)(˜ uj )¯ ιL Bσ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(˜ uj )¯ ιU Bσ(i)(˜ uj )+ lej i=1 κL Aσ(i)(˜ uj )¯ κL Bσ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κU Bσ(i)(˜ uj ) )= 1 4m

         (7) Thedicesimilaritymeasurefor INHFSs m j=1

           lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2 + lej i=1 τ L Bσ(i)(˜ uj ) 2 + lej i=1 τ U Bσ(i)(˜ uj ) 2 + lej i=1 ιL Bσ(i)(˜ uj ) 2 + lej i=1 ιU Bσ(i)(˜ uj ) 2 + lej i=1 κL Bσ(i)(uj ) 2 + lej i=1 κU Bσ(i)(uj ) 2 (8)

κ

Thecosinesimilaritymeasurefor INHFSs. CSVNHFS (A, B)= 1 2m

m j=1

lej i=1 τ L Aσ(i)(˜ uj )¯ τ L Bσ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ U Bσ(i)(˜ uj )+ lej i=1 ιL Aσ(i)(˜ uj )¯ ιL Bσ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(˜ uj )¯ ιU Bσ(i)(˜ uj )+ lej i=1 κL Aσ(i)(˜ uj )¯ κL Bσ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κU Bσ(i)(˜ uj )      lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2 +



       lej i=1 τ L Bσ(i)(˜ uj ) 2 + lej i=1 τ U Bσ(i)(˜ uj ) 2 + lej i=1 ιL Bσ(i)(˜ uj ) 2 + lej i=1 ιU Bσ(i)(˜ uj ) 2 + lej i=1 κL Bσ(i)(˜ uj ) 2 + lej i=1 κU Bσ(i)(˜ uj ) 2

TheweightedJaccardsimilaritymeasuresfor INHFS

lej i=1 τ L Aσ(i)(˜ uj )¯ τ L Bσ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ U Bσ(i)(˜ uj )+ lej i=1 ιL Aσ(i)(˜ uj )¯ ιL Bσ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(˜ uj )¯ ιU Bσ(i)(˜ uj )+ lej i=1 κL Aσ(i)(˜ uj )¯ κL Bσ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κU Bσ(i)(˜ uj ) lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2 + lej i=1 τ L Bσ(i)(˜ uj ) 2 + lej i=1 τ U Bσ(i)(˜ uj ) 2 + lej i=1 ιL Bσ(i)(˜ uj ) 2 + lej i=1 ιU Bσ(i)(˜ uj ) 2 + lej i=1 κL Bσ(i)(˜ uj ) 2 + lej i=1 κU Bσ(i)(˜ uj ) 2





        (10) TheweightedDicesimilaritymeasure



WDSVNHFS (A, B)= m j=1 j

2

          

lej i=1 τ L Aσ(i)(˜ uj )¯ τ L Bσ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ U Bσ(i)(˜ uj ) + lej i=1 ιL Aσ(i)(˜ uj )¯ ιL Bσ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(¯ uj )¯ ιU Bσ(i)(˜ uj ) + lej i=1 κL Aσ(i)(˜ uj )¯ κL Bσ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κU Bσ(i)(˜ uj )





    lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2 + lej i=1 τ L Bσ(i)(˜ uj ) 2 + lej i=1 τ U Bσ(i)(˜ uj ) 2 + lej i=1 ιL Bσ(i)(˜ uj ) 2 + lej i=1 ιU Bσ(i)(˜ uj ) 2 + lej i=1 κL Bσ(i)(uj ) 2 + lej i=1 κU Bσ(i)(uj ) 2 (11)

TheweightedCosinemeasure

lej i=1 τ L Aσ(i)(˜ uj )¯ τ L Bσ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ U Bσ(i)(˜ uj )+ lej i=1 ιL Aσ(i)(˜ uj )¯ ιL Bσ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(˜ uj )¯ ιU Bσ(i)(˜ uj )+ lej i=1 κL Aσ(i)(˜ uj )¯ κL Bσ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κU Bσ(i)(˜ uj )      lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2 +

 



(12) 4.

    MultiattributedecisionmakingWithsinglevaluedneutrosophic hesitantinformation

Ye[46]toutilizethevectorsimilaritymeasuresfor SVNHFSs and INHFSs with SVNHF and INHF informationrespectively. Fora MCDM problemwith SVNHF and INHF informationrespectively,let A = {A1, A2,..., An} beasetofalternativesand ˘ C = { ˘ C1, ˘ C2,..., ˘ Cm} beaset ofattributes.Ifforthealternatives ˇ Aj (forj =1, 2,...,n)thedecisionmakers provideseveralvaluesunderthecriteria ˘ Ci (fori =1, 2,...,m), theneachvalue isconsideredasa SVNHF elementand INHF element(¯ τji, ιji, κji)( forj = 1, 2,...,n,i =1, 2,...,m).

    

Thereforewecanstimulatea SVNHF and INHF decisionmatrixrespectively D =(¯ τji, ιji, κji)n×m, where(¯ τji, ιji, κji)( forj =1, 2,...,n,i =1, 2,...,m) isin theformof SVNHF elementsor INHF elements.

Fortheselectionofbestalternativesinmultiple-criteriadecisionmakingenvironments,theconceptofidealpointhasbeenusedinthedecisionset.Although inrealworldtheidealpointdoesnotexists.Theidealpointdeliverasuitable theoreticalhypothesistoevaluatealternatives.Thereforewedefineeachvalue ineachideal SVNHF or INHF element(¯ τ ∗ j , ι∗ j , κ∗ j )fortheidealalternative

C} as¯ τ ∗ jσ(k) =1, ι∗ jσ(k) =0, κ∗ jσ(k) =0,or τ ∗ jσ(k) = [1, 1], ι∗ jσ(k) =[0, 0], κ∗ jσ(k) =[0, 0],fork =1, 2,...,lei, where lei isthenumberof valuesorintervalvaluesin(¯ τji, ιji, κji)( forj =1, 2,...,n,i =1, 2,...,m). Forthedifferentimportanceofeachcriteriatheweightingvectorofcriteriais givenasˇ =( 1, 2,..., m)T , where i ≥ 0,i =1, 2,...,m,and m i=1 i =1 Thenweexploitthethreeweightedvectorsimilaritymeasuresfor SVNHFSs or INHFSs for MCDM under SVNHF or INHF information,whichcanbe definedasfollows: Step1. Calculateoneofthethreevectorsimilaritymeasuresbetweenthe alternative Aj (forj =1, 2,...,n)andtheidealalternative ˘ A∗ byusingoneofthe threeformulas:



lej i=1 τAσ(i)(˜ uj )¯ τ ∗ ˘ A∗ σ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ι∗ ˘ A∗ σ(i)(˜ uj ) + lej i=1 κAσ(i)(˜ uj )¯ κ∗ A∗ σ(i)(˜ uj ) lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 + lej i=1 τ ∗ ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 ι∗ ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 κ∗ ˘ A∗ σ(i)(˜ uj ) 2    (13)

   

D( ˇ Aj , ˘ A∗)= m j=1 i

2     lej i=1 τAσ(i)(˜ uj )¯ τ ∗ A∗ σ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ι∗ A∗ σ(i)(˜ uj ) + lej i=1 κAσ(i)(˜ uj )¯ κ∗ A∗ σ(i)(˜ uj )

    lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 + lej i=1 τ ∗ ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 ι∗ ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 κ∗ ˘ A∗ σ(i)(˜ uj ) 2 (14)

C( ˇ Aj , ˘ A∗)= m j=1 i

lej i=1 τAσ(i)(˜ uj )¯ τ ∗ A∗ σ(i)(˜ uj )+ lej i=1 ιAσ(i)(˜ uj )¯ ι∗ A∗ σ(i)(˜ uj )+ lej i=1 κAσ(i)(˜ uj )¯ κ∗ A∗ σ(i)(˜ uj ) lej i=1 τAσ(i)(˜ uj ) 2 + lej i=1 ιAσ(i)(˜ uj ) 2 + lej i=1 κAσ(i)(˜ uj ) 2 lej i=1 τ ∗ ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 ι∗ ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 κ∗ ˘ A∗ σ(i)(˜ uj ) 2 (15)

WeusethefollowingformulasundertheSVNHF information. Under INHF information,weusethefollowingformulas:

DINHFS (A, B)=

j=1 i 2

lej i=1 τ L Aσ(i)(˜ uj )¯ τ ∗L A∗ σ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ ∗U A∗ σ(i)(˜ uj )+ lej i=1 ιL Aσ(i)(˜ uj )¯ ι∗L A∗ σ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(˜ uj )¯ ι∗U A∗ σ(i)(˜ uj ) + lej i=1 κL Aσ(i)(˜ uj )¯ κ∗L A∗ σ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κ∗U A∗ σ(i)(˜ uj )

 lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2 + lej i=1 τ ∗L ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 τ ∗U ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 ι∗L ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 ι∗U ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 κ∗L ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 κ∗U Bσ(i)(˜ uj ) 2

         (16) (17)

CINHFS (A, B)= m j=1 i

lej i=1 τ L Aσ(i)(˜ uj )¯ τ ∗L A∗ σ(i)(˜ uj )+ lej i=1 τ U Aσ(i)(˜ uj )¯ τ ∗U A∗ σ(i)(˜ uj )+ 2 lej i=1 ιL Aσ(i)(˜ uj )¯ ι∗L A∗ σ(i)(˜ uj )+ lej i=1 ιU Aσ(i)(˜ uj )¯ ι∗U A∗ σ(i)(˜ uj ) + lej i=1 κL Aσ(i)(˜ uj )¯ κ∗L A∗ σ(i)(˜ uj )+ lej i=1 κU Aσ(i)(˜ uj )¯ κ∗U A∗ σ(i)(˜ uj ) 



lej i=1 τ L Aσ(i)(˜ uj ) 2 + lej i=1 τ U Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 ιL Aσ(i)(˜ uj ) 2 + lej i=1 κL Aσ(i)(˜ uj ) 2 + lej i=1 κU Aσ(i)(˜ uj ) 2

              

 

lej i=1 τ ∗L ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 τ ∗U ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 ι∗L ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 ι∗U ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 κ∗L ˘ A∗ σ(i)(˜ uj ) 2 + lej i=1 κ∗U Bσ(i)(˜ uj ) 2

          (18) 4.1. Practicalexample. ThisexampleisadoptedfromYe[46]isusedasthe demonstrationoftheeffectivenessoftheproposeddecisionmakingmethodinreal lifeproblem.

ForSVNHinformation

Thereisaninvestmentcompany,whichwantstoinvestsomemoneyinthebest option.Thereisapanelwithfourpossiblealternativestoinvestthemoney: (1) A1 isacarcompany(2) A2 isafoodcompany,(3) A4 isacomputers company,(4) A4 isanarmscompany.Thefollowingthreecriterionsonwhichthe investmentcompanymusttakedecisionare(1) C1 istherisk,(2) C2 isthegrowth, (3) C3 istheenvironmentalimpact.Theweightvectorofthecriteriaisgivenas (0.35, 0.25, 0.4)T thefourpossiblealternative

Thedecisionmatrixforattributesandalternativethedataisrepresentedby SVNHFN

C1 C2 C3 A1 {0 3, 0 4}, {0 2, 0 3}, {0.5, 0.6} {0 4, 0 5}, {0 1, 0 3}, {0.4, 0.6} {0 1, 0 2}, {0 3, 0 4}, {0.6, 0.7}

2 {0.9, 1}, {0.05, 0.07}, {0.01, 0.02} {0.8, 0.9}, {0.1, 0.2}, {0.1, 0.2} {0.7, 0.9}, {0.1, 0.2}, {0.01, 0.02}

A3 {0 5, 0 6}, {0 2, 0 3}, {0 1, 0 2}

A4 {0 9, 1}, {0 1, 0 2}, {0 15, 0 25}

{0 5, 0 7}, {0 1, 0 2}, {0 2, 0 3} {0 5, 0 7}, {0 1, 0 2}, {0 2, 0 3}

{0 8, 0 9}, {0 1, 0 2}, {0 1, 0 2} {0 4, 0 5}, {0 2, 0 4}, {0 3, 0 5}

Table1: Thevaluesobtainedthroughthedefinedweightedvectorsimilarity measuresfor SVNHFS WJSVNHF (A∗ , ˆ Ak) WDSVNHF (A∗ , ˆ Ak) WCSVNHF (A∗ , ˆ Ak) A1 0.22540.33840.4099 A2 0.94200.97620.8752 A3 0.681250.811200.8812 A4 0 730400 82050 8395 Rankingorder ˜ A2 ˜ A4 ˜ A3 ˜ A1 ˜ A2 ˜ A4 ˜ A3 ˜ A1, ˜ A3 ˜ A2 ˜ A4 ˜ A1

Thevaluesobtainedintable1showsthattheJaccardandDicesimilaritymeasureshavethesameresultthatis A2 isthebestalternativewhiletheCosinesimilaritymeasureshowsthat A3 isthebestalternative.Fromtable1weconcludethat theJaccardandDicesimilaritymeasurearebettertouseinmulti-criteriadecision making.

ForINSinformation Nowweconsidertheaboveexamplefor INHFSs, Thedecisionmatrixforattributesandalternativethedataisrepresentedby INHFN

C1 ˜ C2 ˜ C3 A1  

{[0 3, 0 4], [0 4, 0 5]}, {[0 2, 0 3], [0 3, 0 4]}, {[0 5, 0 6], [0 6, 0 7]}       {[0 4, 0 5], [0 5, 0 6]}, {[0 1, 0 2], [0 3, 0 4]}, {[0 4, 0 5], [0 5, 0 6]}       {[0 1, 0 2], [0 2, 0 3]}, {[0 3, 0 4], [0 4, 0 5]}, {[0 6, 0 7], [0 7, 0 8]}    A2    {[0 8, 0 9], [0 9, 1]}, {[0 01, 0 02], [0 02, 0 03]}, {[0.01, 0.02], [0.02, 0.03]}       {[0 8, 0 9], [0 9, 1]}, {[0 1, 0 2], [0 2, 0 3]}, {[0.1, 0.2], [0.1, 0.2]}       {[0 7, 0 8], [0 9, 1]}, {[0 05, 0 15], [0 15, 0 25]}, {[0.01, 0.02], [0.05, 0.09]}    A3    {[0.5, 0.6], [0.6, 0.7]}, {[0.2, 0.3], [0.3, 0.4]}, {[0.1, 0.2], [0.2, 0.3]}      

{[0.4, 0.5], [0.5, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.3], [0.3, 0.4]}      

{[0 7, 0 8], [0 9, 1]}, {[0 1, 0 2], [0 2, 0 3]}, {[0 1, 0 2], [0 2, 0 3]}       {[0 4, 0 5], [0 5, 0 6]}, {[0 2, 0 3], [0 3, 0 5]}, {[0 3, 0 4], [0 4, 0 5]}   

{[0.4, 0.5], [0.5, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.3], [0.3, 0.4]}    A4    {[0 8, 0 9], [0 9, 1]}, {[0 1, 0 2], [0 2, 0 3]}, {[0 15, 0 25], [0 25, 0 35]}      

Wearealsogivenanunknownpattern ˘ Q.Ouraimistoclassify ˘ Q inoneofthe givenpattern.Theprincipleofrecognizingpatternisthatofthemaximumdegree ofsimilaritybetween SVNHFSs isdescribedby N =arg Max1≤k≤3 CSVNHFS (Ak, ˜ Q)

A1 = { a1, {0.6, 0.7}, {0.2, 0.3}, {0.3, 0.4} , a2, {0.4, 0.5}, {0.3, 0.4}, {0.4, 0.5} , a3, {0 3, 0 4}, {0 2, 0 4}, {0 5, 0 7} }

A2 = { a1, {0 4, 0 5}, {0 3, 0 5}, {0 5, 0 6} , a2, {0 6, 0 7}, {0 2, 0 3}, {0 3, 0 4} , a3, {0 5, 0 6}, {0 2, 0 3}, {0 2, 0 4} } A3 = { a1, {0.2, 0.3}, {0.3, 0.4}, {0.6, 0.7} , a2, {0.4, 0.5}, {0.3, 0.4}, {0.4, 0.5} , a3, {0.6, 0.7}, {0.2, 0.3}, {0.3, 0.4} }

˘ Q = { a1, {0 1, 0 2}, {0 3, 0 4}, {0 7, 0 8} , a2, {0 2, 0 3}, {0 3, 0 4}, {0 6, 0 7} , a3, {0 4, 0 5}, {0 3, 0 5}, {0 5, 0 6} }

(A1, ˘ Q)=0.3501, (A2, ˘ Q)=0.4317, (A3, ˘ Q)= 0.4723

Onecanobservethatthepatternshouldbeclassifiedin A3 accordingtothe patternrecognitionprinciple.ThisresultisthesameobtainedbyYe[36].

For INHFS

Example5.2. For INHFS weusethesameexampledefinedabovejustthepatternsarerepresentedby INHFSs.

˜ A1 = { a1, {[0 4, 0 5], [0 6, 0 7]}, {[0 1, 0 2], [0 2, 0 3]}, {[0 2, 0 3], [0 3, 0 4]} , a2, {[0.3, 0.4], [0.4, 0.5]}, {[0.3, 0.4], [0.3, 0.4]}, {[0.3, 0.4], [0.4, 0.5]} , a3, {[0.2, 0.3], [0.3, 0.4]}, {[0.1, 0.3], [0.2, 0.4]}, {[0.4, 0.6], [0.5, 0.7]} }

A2 = { a1, {[0 3, 0 4], [0 4, 0 5]}, {[0 2, 0 3], [0 4, 0 5]}, {[0 3, 0 5], [0 4, 0 6]} , a2, {[0 4, 0 6], [0 5, 0 7]}, {[0 2, 0 3], [0 3, 0 4]}, {[0 3, 0 4], [0 4, 0 5]} , a3, {[0 4, 0 5], [0 6, 0 7]}, {[0 1, 0 2], [0 2, 0 3]}, {[0 1, 0 3], [0 2, 0 4]} }

A3 = { a1, {[0 1, 0 2], [0 2, 0 3]}, {[0 2, 0 3], [0 3, 0 4]}, {[0 5, 0 6], [0 6, 0 7]} , a2, {[0.3, 0.4], [0.4, 0.5]}, {[0.2, 0.3], [0.3, 0.4]}, {[0.3, 0.4], [0.4, 0.5]} , a3, {[0.5, 0.6], [0.6, 0.7]}, {[0.1, 0.2], [0.2, 0.3]}, {[0.2, 0.3], [0.3, 0.4]} }

˘

Q = { a1, {[0 1, 0 3], [0 2, 0 4]}, {[0 2, 0 3], [0 3, 0 4]}, {[0 6, 0 7], [0 7, 0 8]} , a2, {[0 1, 0 2], [0 2, 0 3]}, {[0 2, 0 4], [0 4, 0 5]}, {[0 5, 0 7], [0 6, 0 8]} , a3, {[0 2, 0 4], [0 4, 0 6]}, {[0 3, 0 4], [0 4, 0 5]}, {[0 4, 0 5], [0 5, 0 6]} }

(A1, ˘ Q)=0.1954, (A2, ˘ Q)=0.1960, (A3, ˘ Q)= 0.2150

Onecanobservethatthepatternshouldbeclassifiedin A3 accordingtothe patternrecognitionprinciple.ThisresultisthesameobtainedbyYe[36].

6. Discussion

Assimplifiedneutrosophichesitantfuzzysetisanimportantextensionofthe Fuzzyset(FS),Intuitionisticfuzzyset(IFS),Singlevaluedneutrosophicset(SVNS), intervalneutrosophicset(INS),Hesitantfuzzyset(HFS)andDualhesitantfuzzy set(DHFS),singlevaluedneutrosophichesitantfuzzyset(SVNHFS),interval neutrosophichesitantfuzzyset(INHFS).Asmentionedabovethathesitancy isthemostcommonproblemindecisionmakingforwhich HFS isasuitable meansbyallowingsevrespossiblevaluesforanelementtoaset.Howeverin HFS theyconsideronlyonetruth-membershipfunctionanditcannotexpressthis problemwithafewdifferentvaluesassignedbytruth-membershiphesitantdegree, indeterminacy-membershipdegree,andfalsity-membershipdegreesduetodoubtsof decisionmakers.andalsoin DHFS theyconsidertwofunctionsthatismembership andnon-membershipfunctionsandcannotconsiderindeterminacy-membership function.Thereforesimplifiedneutrosophicsetsisthemoresuitablefordecision makingandcanhandleincomplete,inconsistenceandindeterminateinformation whichoccurindecisionmaking.Theotheradvantegeofthesimplifiedneutrosophic setisthatitcontaintheconceptofsinglevaluedneutrosophichesitantfuzzysets andaswellasintervalneutrosophichesitantfuzzysets.Ascomparingtoother setsthesimplifiedneutrosophichesitantfuzzysetcontainmoreinformationand thedecisionmakerscangetmoreinformationtotaketheirdecision.

7. Conclusion

Assimplifiedneutrosophichesitantfuzzyset(SNHFS)isanewextensionwhich consistsoftheconceptofFuzzyset(FS),Intuitionisticfuzzyset(IFS),Singlevaluedneutrosophicset(SVNS),intervalneutrosophicset(INS),Hesitantfuzzy set(HFS)andDualhesitantfuzzyset(DHFS),singlevaluedneutrosophicset (SVNHFS),intervalneutrosophichesitantfuzzyset(INHFS).Inthisarticle wedefinedvectorsimilaritymeasuresforsimplifiedneutrosophichesitantfuzzyset andappliedthemtomulti-criteriadecisionmakingandpatternrecognition.We alsoobservethattheJaccardandDicesimilaritymeasuresarebettertoapply inmulti-criteriadecisionmakingproblemthentheCosinesimilaritymeasure.In futureweshallapplytheabovedistancemeasurestomedicaldiagnosisandalso definedforNeutrosophicOverset,NeutrosophicUnderset,andNeutrosophicOffset.

CompliancewithEthicalStructure

Wedeclarethatwehavenocommercialorassociativeinterestthatrepresentsa conflictofinterestinconnectionwithworksubmitted.

Acknowledgments. Theauthorswouldliketothanktheanonymousrefereefor his/hercommentsthathelpedusimprovethisarticle.

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TahirMahmood

DepartmentofMathematicsandStatistics,InternationalIslamicUniversity,44000Islamabad,Pakistan

E-mailaddress: tahirbakht@yahoo.com

JunYe

DepartmentofElectricalandInformationEngineering,ShoaxingUniversity,China

E-mailaddress: yehjun@aliyun.com

QaisarKhan

DepartmentofMathematicsandStatistics,InternationalIslamicUniversity,44000Islamabad,Pakistan

E-mailaddress: qaisarkhan421@gmail.com