Vector Similarity Measures of Q-Linguistic Neutrosophic Variable Sets

VectorSimilarityMeasuresofQ-Linguistic NeutrosophicVariableSetsandTheir Multi-AttributeDecisionMakingMethod

JunYe 1,* ,ZeboFang 2 andWenhuaCui 1

1 DepartmentofElectricalEngineeringandAutomation,ShaoxingUniversity,508HuanchengWestRoad, Shaoxing312000,China;wenhuacui@usx.edu.cn

2 DepartmentofPhysics,ShaoxingUniversity,508HuanchengWestRoad,Shaoxing312000,China; csfzb@usx.edu.cn

* Correspondence:yejun@usx.edu.cnoryehjun@aliyun.com;Tel.:+86-575-8832-7323

Received:30September2018;Accepted:16October2018;Published:22October2018

Abstract: Sincelanguageisusedforthinkingandexpressinghabitsofhumansinreallife, thelinguisticevaluationforanobjectivethingisexpressedeasilyinlinguisticterms/values.However, existinglinguisticconceptscannotdescribelinguisticargumentsregardinganevaluatedobjectin two-dimensionaluniversalsets(TDUSs).Todescribelinguisticneutrosophicargumentsindecision makingproblemsregardingTDUSs,thisstudyproposesaQ-linguisticneutrosophicvariableset (Q-LNVS)forthefirsttime,whichdepictsitstruth,indeterminacy,andfalsitylinguisticvalues independentlycorrespondingtoTDUSs,andvectorsimilaritymeasuresofQ-LNVSs.Thereafter, alinguisticneutrosophicmulti-attributedecision-making(MADM)approachbyusingthepresented similaritymeasures,includingthecosine,Dice,andJaccardmeasures,isdevelopedunderQ-linguistic neutrosophicsetting.Lastly,theapplicabilityandeffectivenessofthepresentedMADMapproachis presentedbyanillustrativeexampleunderQ-linguisticneutrosophicsetting.

Keywords: Q-linguisticneutrosophicvariableset;vectorsimilaritymeasure;cosinemeasure; Dicemeasure;Jaccardmeasure;decisionmaking

1.Introduction

Sincelanguageisusedforthinkingandexpressinghabitsofhumansinreallife,thelinguistic evaluationforanobjectivethingisexpressedeasilyinlinguisticterms/values[1].Hence,theywere appliedtolinguisticfuzzyreason[1]andlinguisticdecision-making(DM)problems[2–9].Becauseof linguisticuncertaintyandhesitancyinthelinguisticevaluationforanobjectivething,thereexistthe representationsofinterval/uncertainlinguisticnumbersorhesitantlinguisticnumbers.Hence,onthe onehand,interval/uncertainlinguisticnumberswereproposedandappliedto(group)DMproblems inuncertainlinguisticsetting[10–14].Ontheotherhand,hesitantlinguisticvariables(LVs)andhesitant uncertainLVswerepresentedandappliedin(group)DMproblemsinhesitant(uncertain)linguistic setting[15–19].Inaddition,alinguisticcubicvariablewasputforwardbasedoncombininganinterval LVwithauniqueLVandusedforDMproblemsinlinguisticcubicsetting[20,21].Further,alinguistic cubichesitantfuzzynumber/variablewaspresentedtodepictthehybridinformationofitsuncertain linguisticargumentanditshesitantlinguisticargumentandutilizedforDMproblemsinlinguistic cubichesitantfuzzysetting[22].Bycombininganeutrosophicnumberwithlanguage,aneutrosophic linguisticnumberandsomeweightedaggregationoperatorsofneutrosophiclinguisticnumbers[23] wereintroducedforneutrosophiclinguisticnumberDMproblems,andthenthesimilaritymeasure andexpectedvalueofhesitantneutrosophiclinguisticnumbers[24]werefurtherpresentedforDM problemswithhesitantneutrosophiclinguisticnumbers.

Symmetry 2018, 10,531;doi:10.3390/sym10100531 www.mdpi.com/journal/symmetry

Inreallifeenvironments,thetruth,indeterminacy,andfalsitylinguisticargumentsregardingan objectivethingarepresentedinahuman’sthinkingandexpressingprocessandlinguisticneutrosophic variables/numbers(LNVs)werepresentedtodepicttruth,falsity,andindeterminacylinguisticdegrees independently[25].Then,someaggregationoperatorsofLNVs[25,26],cosinemeasuresofLNVs[27], andcorrelationcoefficientsofLNVs[28]wereproposed,respectively,forDMproblemsinLNVsetting. RegardingthecombinationofaneutrosophiclinguisticnumberandanLNV,linguisticneutrosophic uncertainnumbersandtheirweightedaggregationoperatorswerepresentedforDMinuncertain linguisticsetting[29].BythehybridformofanintervalLV(anuncertainlinguisticargument)anda single-valuedLNV(anargumentofconfidentdegree),single-valuedlinguisticneutrosophicinterval LVs,andtheirweightedaggregationoperatorswereproposedforDMalongwithuncertain/interval linguisticargumentsandtheirlinguisticneutrosophicconfidentdegrees[30].Regardinghesitant LNVenvironment,similaritymeasuresbetweenhesitantLNVswerepresentedbytheleastcommon multiplecardinalityandappliedtohesitantlinguisticneutrosophicDMproblems[31].Bythehybrid formofLNV[25]andlinguisticcubicnumbers[20],linguisticneutrosophiccubicnumbersandtheir aggregationoperatorswereintroducedforlinguisticneutrosophiccubicDMproblems[32,33].

However,thevariouslinguisticconceptsarealldescribedinauniqueuniversalset,andthenin somedecisionsituationsthereexisttheassessmentproblemsofalternativesovertwo-dimensional universalsets(TDUSs).Forexample,supposeapersonwouldliketopurchaseahouseinagroup offourhouses(asetoffouralternatives H ={H1, H2, H3, H4}).Inhis/herattractiveevaluation ofhouses,theprice(x1),environment(x2),andtraffic(x3)ofthefourhousesareconsideredasa universalset X ={x1, x2, x3},andselectingtwocities c1 and c2 areconsideredasanotheruniversalset C ={c1, c2}.Obviously,theabovevariouslinguisticargumentscannotrepresentsuchanassessment problemforeachalternative Hj (j =1,2,3,4)overtheTDUSs X ={x1, x2, x3}and C ={c1, c2}in linguisticDMsetting.Then,aQ-neutrosophicsetandaQ-neutrosophicsoftsetwereputforward regardingTDUSsandappliedtoQ-neutrosophicsoftDMproblems[34].Althoughtheycanexpress andhandletheassessmentproblemswithTDUSsinneutrosophicDMenvironments,theycannot carryoutlinguisticneutrosophicDMproblemsoverTDUSs.Tosolvethisproblem,thisstudypresents aQ-linguisticneutrosophicvariableset(Q-LNVS)forthefirsttimetoexpressthelinguisticevaluation problemsofthetruth,falsity,andindeterminacyoverTDUSsfromthepredefinedlinguistictermset (LTS).ItthenputsforwardthevectorsimilaritymeasuresofQ-LNVSs,includingthecosine,Dice, andJaccardmeasuresofQ-LNVSs,andthenestablishesamulti-attributeDMapproachofQ-LNVSs bythevectorsimilaritymeasuresofQ-LNVstosolvelinguisticneutrosophicDMproblemsalong withTDUSs.ItisobviousthattheproposedDMapproachshowstheadvantageofcarryingoutthe linguisticneutrosophicDMproblemsregardingTDUSs,whichexistinglinguisticneutrosophicDM approaches[25–28]andQ-linguisticneutrosophicsoftDMapproaches[34]cannotsolve.

Theframeworkofthisstudyisorganizedbelow.ThesecondsectionproposesQ-LNVSsand vectorsimilaritymeasuresbetweenQ-LNVSs,includingthecosine,Dice,andJaccardmeasuresof Q-LNVSs.Thethirdsectiondevelopsamulti-attributeDMapproachofQ-LNVSsbyusingthevector similaritymeasuresinQ-linguisticneutrosophicsetting.Anillustrativeexampleanditssensitivity analysistoweightsarepresentedinthefourthsection.Thelastsectioncontainsconclusionsand futurestudy.

2.VectorSimilarityMeasuresofQ-LNVSs

First,wepresenttheconceptofQ-LNVStodepictalinguisticneutrosophicevaluationproblem bythetruth,falsity,andindeterminacylinguisticargumentsoverTDUSsinlinguisticsetting.

Definition1. SetX={x1,x2, ... ,xn}andQ={q1,q2, ... ,qm}asTDUSsandletLTSbeS={sl|l ∈ [0,g]}, whereg+1isanoddcardinality.ThenaQ-LNVSLinXandQisdefinedbythefollowingform:

L = (xi, qj ), st (xi, qj ), su (xi, qj ), sv (xi, qj ) xi ∈ X, qj ∈ Q, st (xi, qj ), si (xi, qj ), s f (xi, qj ) ∈ S, j = 1,2,..., m; i = 1,2,..., n where st (xi, qj ), su (xi, qj ), sv (xi, qj ) arethetruth,indeterminacy,andfalsityLVs,respectively,inTDUSsfort, u,v ∈ [0,g]. Then,thebasiclinguisticelement (xi, qj ), st (xi, qj ), su (xi, qj ), sv (xi, qj ) in L issimplydenotedby lij = (xi, qj ), stij , suij , svij ,whichiscalledaQ-linguisticneutrosophicelement(Q-LNE).

Example1. Supposeapersonwouldliketobuyahousefromacity.Thereisasetoftwopotentialhouses H={L1,L2} intwocities.Then,settheirprice(x1),environment(x2),andtraffic(x3)asauniversalset X={x1,x2,x3} andsetthetwocitiesasanotheruniversalsetQ={q1,q2}.BasedonthepredefinedLTS S={s0 =extremelylow,s1 =verylow,s2 =low,s3 =slightlylow,s4 =medium,s5 =slightlyhigh,s6 =high, s7 =veryhigh,s8 =extremelyhigh},thetwoQ-LNEsetsobtainedfromSaregivenasfollows:

L1 ={<(x1,q1),s6,s1,s2>,<(x1,q2),s5,s2,s3>,<(x2,q1),s4,s3,s2>,<(x2,q2),s7,s1,s3>, <(x3,q1),s6,s2,s1>,<(x3,q2),s6,s1,s1>},

L2 ={<(x1,q1),s3,s4,s5>,<(x1,q2),s4,s2,s1>,<(x2,q1),s4,s2,s2>,<(x2,q2),s5,s1,s2>, <(x3,q1),s5,s2,s1>,<(x3,q2),s6,s3,s2>}.

Inthefollowing,wegivethevectorsimilaritymeasuresbetweenQ-LNVSs,includingthecosine, Dice,andJaccardmeasuresofQ-LNVSs.

Definition2. LetTDUSsbeX={x1,x2, ,xn}andQ={q1,q2, ,qp}andlet l1 ij = (xi, qj ), st1 ij , s u1 ij , s v1 ij and l2 ij = (xi, qj ), st2 ij , su2 ij , sv2 ij (j=1,2, ,p;i=1,2, ,n)betwogroupsofQ-LNEsintwoQ-LNVSs L1 andL2 regardingtheLTSS={sl|l ∈ [0,g]}.Then,thecosine,Dice,andJaccardmeasuresoftheQ-LNVSs L1 andL2 aredefined,respectively,asfollows:

C(L1, L2)= s g∑ p j 1 ∑n i 1 (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij ) ∑ p j=1 ∑n i=1 [(t1 ij )2 +(u1 ij )2 +(v1 ij )2 ]× ∑ p j=1 ∑n i=1 [(t2 ij )2 +(u2 ij )2 +(v2 ij )2 ]

D(L1, L2)= s 2g∑ p j=1 ∑n i=1 (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij ) ∑ p j 1 ∑n i 1 [(t1 ij )2 +(u1 ij )2 +(v1 ij )2 ]+∑ p j 1 ∑n i 1 [(t2 ij )2 +(u2 ij )2 +(v2 ij )2 ]

,(1)

,(2)

J(L1, L2)= s g∑ p j 1 ∑n i 1 (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij ) ∑ p j=1 ∑n i=1 [(t1 ij )2 +(u1 ij )2 +(v1 ij )2 ]+∑ p j=1 ∑n i=1 [(t2 ij )2 +(u2 ij )2 +(v2 ij )2 ] ∑ p j=1 ∑n i=1 (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij )

.(3)

Obviously,theabovecosine,Dice,andJaccardmeasuressatisfythefollowingproperties:

Whentheimportanceofelements xi (i =1,2, , n)and qj (j =1,2, , p)istakeninto account,theweightvectorscorrespondingto X ={x1, x2, , xn}and Q ={q1, q2, , qp}aregiven

as w ={w1, w2,..., wn} and w ={w1, w2, ... , wp},respectively.Thus,theweightedcosine,Dice,and Jaccardmeasuresof L1 and L2 canbepresented,respectively,asfollows:

Cw (L1, L2) = s g∑ p j=1 ωj ∑n i=1 wi (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij )

∑ p j 1 ωj ∑n i 1 wi [(t1 ij )2 +(u1 ij )2 +(v1 ij )2 ]× ∑ p j 1 ωj ∑n i 1 wi [(t2 ij )2 +(u2 ij )2 +(v2 ij )2 ]

Dw (L1, L2) = s 2g∑ p j 1 ωj ∑n i 1 wi (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij ) ∑ p j=1 ωj ∑n i=1 wi [(t1 ij )2 +(u1 ij )2 +(v1 ij )2 ]+∑ p j=1 ωj ∑n i=1 wi [(t2 ij )2 +(u2 ij )2 +(v2 ij )2 ]

,(4)

,(5)

Jw (L1, L2) = s g∑ p j 1 ωj ∑n i 1 wi (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij ) ∑ p j=1 ωj ∑n i=1 wi [(t1 ij )2 +(u1 ij )2 +(v1 ij )2 ]+∑ p j=1 ωj ∑n i=1 wi [(t2 ij )2 +(u2 ij )2 +(v2 ij )2 ] ∑ p j=1 ωj ∑n i=1 wi (t1 ij t2 ij +u1 ij v2 ij +v1 ij v2 ij )

.(6)

Example2. LetusconsidertwoQ-LNEsetsL1 ={<(x1,q1),s6,s1,s2>,<(x1,q2),s5,s2,s3>,<(x2,q1),s4,s3, s2>,<(x2,q2),s7,s1,s3>,<(x3,q1),s6,s2,s1>,<(x3,q2),s6,s1,s1>}andL2 ={<(x1,q1),s3,s4,s5>,<(x1,q2), s4,s2,s1>,<(x2,q1),s4,s2,s2>,<(x2,q2),s5,s1,s2>,<(x3,q1),s5,s2,s1>,<(x3,q2),s6,s3,s2>}intheLTS S={s0,s1,s2, ... ,s8}withg=8andtheTDUSsX={x1,x2,x3}andQ={q1,q2}.Supposetheweightvectors forX={x1,x2,x3}andQ={q1,q2}aregivenasw=(0.4,0.25,0.35)and ω =(0.4,0.6),respectively.Then, wecomputethemeasurevaluesofCw(L1,L2),Dw(L1,L2),Jw(L1,L2) ByusingEquations(4)–(6),theircalculationalprocessesareshownasfollows:

0.4[0.4(6 × 3 + 1 × 4 + 2 × 5)+ 0.25(4 × 4 + 3 × 2 + 2 × 2)+ 0.35(6 × 5 + 2 × 2 + 1 × 1)]+ 0.6[0.4(5 × 4 + 2 × 2 + 3 × 1)+ 0.25(7 × 5 + 1 × 1 + 3 × 2)+ 0.35(6 × 6 + 1 × 3 + 1 × 2)]          0.4[0.4(62 + 12 + 22)+ 0.25(42 + 32 + 22)+ 0.35(62 + 22 + 12)]+ 0.6[0.4(52 + 22 + 32)+ 0.25(72 + 12 + 32)+ 0.35(62 + 12 + 12)]     ×     0.4[0.42(32 + 42 + 52)+ 0.25(42 + 22 + 22)+ 0.35(52 + 22 + 12)]+ 0.6[0.42(42 + 22 + 12)+ 0.25(52 + 12 + 22)+ 0.35(62 + 32 + 22)]     = s7.0913, Dw(L1, L2) = s 2g∑2 j 1 ωj∑3 i 1 wi(t1 ijt2 ij+u1 ijv2 ij+v1 ijv2 ij) ∑2 j=1 ωj∑3 =1 wi[(t1 ij)2+(u1 ij)2+(v1 ij)2]+∑2 j=1 ωj∑3 i=1 wi[(t2 ij)2+(u2 ij)2+(v2 ij)2] = s 2×8     0.4[0.4(6 × 3 + 1 × 4 + 2 × 5)+ 0.25(4 × 4 + 3 × 2 + 2 × 2)+ 0.35(6 × 5 + 2 × 2 + 1 × 1)]+ 0.6[0.4(5 × 4 + 2 × 2 + 3 × 1)+ 0.25(7 × 5 + 1 × 1 + 3 × 2)+ 0.35(6 × 6 + 1 × 3 + 1 × 2)]          0.4[0.4(62 + 12 + 22)+ 0.25(42 + 32 + 22)+ 0.35(62 + 22 + 12)]+ 0.6[0.4(52 + 22 + 32)+ 0.25(72 + 12 + 32)+ 0.35(62 + 12 + 12)]      +      0.4[0.42(32 + 42 + 52)+ 0.25(42 + 22 + 22)+ 0.35(52 + 22 + 12)]+ 0.6[0.42(42 + 22 + 12)+ 0.25(52 + 12 + 22)+ 0.35(62 + 32 + 22)]      = s7.0777, Jw(L1, L2) = s g∑2 j=1 ωj ∑3 i=1 wi (t1 ijt2 ij +u1 ijv2 ij +v1 ijv2 ij ) ∑2 j 1 ωj ∑3 i 1 wi [(t1 ij )2+(u1 ij )2+(v1 ij )2]+∑2

2

0.25(4 × 4 + 3 × 2 + 2 × 2)+ 0.35(6 × 5 + 2 × 2 + 1 × 1)]+ 0.6[0.4(5 × 4 + 2 × 2 + 3 × 1)+ 0.25(7 × 5 + 1 × 1 + 3 × 2)+ 0.35(6 × 6 + 1 × 3 + 1 × 2)]

0.4[0.4(62 + 12 + 22)+ 0.25(42 + 32 + 22)+ 0.35(62 + 22 + 12)]+ 0.6[0.4(52 + 22 + 32)+ 0.25(72 + 12 + 32)+ 0.35(62 + 12 + 12)] 0.4[0.42(32 + 42 + 52)+ 0.25(42 + 22 + 22)+ 0.35(52 + 22 + 12)]+ 0.6[0.42(42 + 22 + 12)+ 0.25(52 + 12 + 22)+ 0.35(62 + 32 + 22)] 0.4

Supposethereisamulti-attributeDMproblem,inwhich L ={L1, L2,..., Lm}isdenotedbyaset of m alternatives.Then,TDUSs(twokindsofattributesets)arespecifiedas X ={x1, x2,..., xn} and Q ={q1, q2,... ,qp},respectively,andthentheircorrespondingweighvectorsaregivenas w =(w1, w2,..., wn) and w =(w1, w2,..., wp).Whereas,adecisionmakerisrequiredtoassessthe alternatives Lk (k =1,2, , m)ontheattributes xi (i =1,2, , n)and qj (j =1,2, , p)by Q-LNEsregardingthegivenLTS S ={sl|l ∈ [0, g]}withtheoddcardinality g +1.Intheassessment process,thedecisionmakergivesthetruth,falsity,andindeterminacylinguisticvaluesfor xi and qj onanalternative Lk bycorrespondinglinguistictermsin S,whichareconstructedasaQ-LNE lk ij = (xi, qj ), s tk ij , s uk ij , s vk ij (j =1,2, , p;i =1,2, , n; k =1,2, , m).Hence,alltheQ-LNEs

providedbythedecisionmakercanbecomposedofadecisionmatrixofQ-LNEs L = lk ij m×nq Thus,theproposedDMmethodusingthevectorsimilaritymeasuresofQ-LNVSsisappliedto themulti-attributeDMproblemwithQ-LNVSinformation.Whereas,theDMstepsaredepictedin detailbelow:

Step1: Since l∗ ij = (xi, qj ), st∗ ij , s u∗ ij , s v∗ ij = (xi, qj ), max k s tk ij ,min k s uk ij ,min k s vk ij isan idealQ-LNEasthebestQ-LNE,wecanestablishthefollowingidealQ-LNVS: L∗ = (xi, qj), st∗ ij , s u∗ ij , s v∗ ij xi ∈ X, qj ∈ Q, k = 1,2,..., m, j = 1,2,..., p, i = 1,2,..., n (7)

Step2: ByapplyingEquations(4)–(6),thecosine/Dice/Jaccardmeasurebetween Lk (k =1,2, ... , m) and L* isgivenbyusingthefollowingformula:

Cw (Lk, L∗ ) = s g∑ p j=1 ωj ∑n i=1 wi (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij ) ∑ p j=1 ωj ∑n i=1 wi [(tkij )2 +(ukij )2 +(vkij )2 ]× ∑ p j=1 ωj ∑n i=1 wi [(t∗ ij )2 +(u∗ ij )2 +(v∗ ij )2 ]

,(8) or

Dw (Lk, L∗ ) = s 2×g∑ p j=1 ω2 j ∑n i=1 w2 i (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij ) ∑ p j 1 ω2 j ∑n i 1 w2 i [(tkij )2 +(ukij )2 +(vkij )2 ]+∑ p j 1 ω2 j ∑n i 1 w2 i [(t∗ ij )2 +(u∗ ij )2 +(v∗ ij )2 ]

(10)

,(9) or Jw(Lk, L∗ ) = s g∑p j=1 ωj ∑n i=1 wi (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij ) ∑p j=1 ωj ∑n i=1 wi [(tkij )2 +(ukij )2 +(vkij )2 ]+∑p j=1 ωj ∑n i=1 wi [(t∗ ij )2 +(u∗ ij )2 +(v∗ ij )2 ] ∑p j=1 ωj ∑n i=1 wi (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij )

Step3: Accordingtothelinguisticvaluesofthevectorsimilaritymeasures,thealternativesareranked andthebestalternative Lk∗ ischosenregardingthebiggestlinguisticvaluefor X and Q

Step4: Basedon xj ∈ X (j =1,2, ... , p)or qi ∈ Q (i =1,2, ... , n),weneedtocalculatethemeasure valuesbetween Lk∗ xi, qj and L∗ xi, qj :

C Lk∗ (xi, qj), L∗ (xi, qj) = s g∑n i 1 (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij ) ∑n i 1 [(tkij )2 +(ukij )2 +(vkij )2 ]× ∑n i 1 [(t∗ ij )2 +(u∗ ij )2 +(v∗ ij )2 ]

for i = 1,2,..., n; (12)

for j = 1,2,..., p, (11) or C Lk∗ (xi, qj), L∗ (xi, qj) = s g∑p j 1 (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij ) ∑p j 1 [(tkij )2 +(ukij )2 +(vkij )2 ]× ∑p j 1 [(t∗ ij )2 +(u∗ ij )2 +(v∗ ij )2 ]

D Lk∗ (xi, qj ), L∗ (xi, qj ) = s 2g∑n i 1 (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij ) ∑n i=1 [(tkij )2 +(ukij )2 +(vkij )2 ]+∑n i=1 [(t∗ ij )2 +(u∗ ij )2 +(v∗ ij )2 ]

for j = 1,2,..., p,(13)

or D Lk∗ (xi, qj ), L∗ (xi, qj ) = s 2g∑ p j=1 (tkij t∗ ij +ukij v∗ ij +vkij v∗ ij ) ∑ p j=1 [(tkij )2 +(ukij )2 +(vkij )2 ]+∑ p j=1 [(t∗ ij )2 +(u∗ ij )2 +(v∗ ij )2 ]

for i = 1,2,..., n;(14)

J Lk∗ (xi, qj), L∗(xi, qj) = s g∑n i=1 (tkijt∗ ij +ukijv∗ ij +vkijv∗ ij ) ∑n i 1 [(tkij )2+(ukij )2+(vkij )2]+∑n i 1 [(t∗ ij )2+(u∗ ij )2+(v∗ ij )2] ∑n i 1 (tkijt∗ ij +ukijv∗ ij +vkijv∗ ij )

for j = 1,2,..., p, (15) or J Lk∗ (xi, qj), L∗(xi, qj) = s g∑p j=1 (tkijt∗ ij +ukijv∗ ij +vkijv∗ ij ) ∑p j 1 [(tkij )2+(ukij )2+(vkij )2]+∑p j 1 [(t∗ ij )2+(u∗ ij )2+(v∗ ij )2] ∑p j 1 (tkijt∗ ij +ukijv∗ ij +vkijv∗ ij )

for i = 1,2,..., n (16) Step5: Accordingtothelinguisticvaluesof C Lk∗ (xi, qj ), L∗ (xi, qj ) or D Lk∗ (xi, qj ), L∗ (xi, qj ) or J Lk∗ (xi, qj ), L∗ (xi, qj ) for X or Q (dependingonsomeactualsituation),wecandeterminethe bestone xi* or qj* correspondingtothebiggestlinguisticvalue. Step6: End. 4.IllustrativeExampleandSensitivityAnalysistoWeights

4.1.IllustrativeExample

Supposeapersonwouldliketobuyahouseinoneoftwocities.Therearefourpotentialhouses (alternatives)of Lk (k =1,2,3,4)intwocities.Then,settheirprice(x1),environment(x2),andtraffic (x3)asauniversalset X ={x1, x2, x3}andsetthetwocitiesasanotheruniversalset Q ={q1, q2}.Thus, theQ-LNEscanindicatetheinfluenceofboththethreeattributesofhousesandthetwocitieson his/herbuyingattractivedegreeofahouse.Herewith,thetwoweighvectorsof X and Q aregiven as w =(0.4,0.25,0.35) and w =(0.4,0.6),respectively.Whereas,thealternative Lk (k =1,2,3,4)are assessedovertheTDUSs X ={x1, x2, x3}and Q ={q1, q2}fromthegivenLTS S ={s0 =extremelylow, s1 =verylow, s2 =low, s3 =slightlylow, s4 =medium, s5 =slightlyhigh, s6 =high, s7 =veryhigh, s8 =extremelyhigh}with g =8.Intheassessmentprocess,thedecisionmaker/buyercangivethe truth,indeterminacy,andfalsityvaluesfor xi and qj onanalternative Lk bycorrespondinglinguistic termsin S,andthenestablishQ-LNEs lk ij = (xi, qj ), s tk ij , s uk ij , s vk ij (j =1,2; i =1,2,3; k =1,2,3,4),which areconstructedastheDMmatrixofQ-LNEs:

(x1, q1), s6, s2, s1 (x1, q2), s7, s2, s3 (x2, q1), s5, s2, s1 (x2, q2), s4, s1, s1 (x3, q1), s5, s2, s2 (x3, q2), s7, s2, s2 (x1, q1), s6, s1, s2 (x1, q2), s7, s1, s1 (x2, q1), s6, s1, s1 (x2, q2), s5, s1, s2 (x3, q1), s6, s1, s3 (x3, q2), s7, s2, s1 (x1, q1), s5, s2, s3 (x1, q2), s6, s2, s4 (x2, q1), s4, s1, s1 (x2, q2), s5, s2, s2 (x3, q1), s4, s2, s2 (x3, q2), s5, s2, s3 (x1, q1), s5, s1, s1 (x1, q2), s6, s3, s5 (x2, q1), s5, s3, s3 (x2, q2), s6, s2, s4 (x3, q1), s5, s1, s1 (x3, q2), s6, s2, s3



Thus,theproposedmulti-attributeDMapproachcanbeusedforthisQ-linguisticneutrosophic DMproblem.TheDMstepsaredepictedbelow: Firstly,weestablishtheidealalternativefromtheDMmatrix L bytheidealQ-LNEset: L∗ = (x1, q1), s6, s1, s1 , (x1, q2), s7, s1, s1 , (x2, q1), s6, s1, s1 , (x2, q2), s6, s1, s1 , (x3, q1), s6, s1, s1 , (x3, q2), s7, s2, s1

Then,byEquations(8)–(10),themeasureresultsandrankingofthefouralternativesaregivenin Table 1

Table1. Measureresultsandrankingofthefouralternatives.

MeasureMethod MeasureValuebetween Lk (k =1,2,3,4)and L* Ranking Cw (Lk, L∗ ) s7.7472, s7.9088, s7.3465, s7.2437 L2 > L1 > L3 > L4 Dw (Lk, L∗ ) s7.7470, s7.9087, s7.3207, s7.2387 L2 > L1 > L3 > L4 Jw (Lk, L∗ ) s7.5095, s7.8194, s6.7478, s6.6097 L2 > L1 > L3 > L4

BasedonTable 1,alltherankingordersareidenticalregardingthecosine,Dice,andJaccard measures.Then,thebestalternativeis L2 Next,themeasurevaluesofEquations(11),(13),and(15)regarding Q, andthebestcityregarding L2 aregiveninTable 2

Table2. Measureresultsregarding Q andthebestcity.

7.8409

Lastly,theresultsbasedonTable 2 indicatethatthebuyershouldbuythehouse L2 inthebestcity q2

4.2.SensitivityAnalysistoWeights

Toindicatetheinfluenceoftheweightsonrankingordersintheillustrativeexample,weconsider thatthetwoweighvectorsof X and Q aregivenas w =(1/3,1/3,1/3)and w =(1/2,1/2),respectively, toanalyzethesensitivityoftheweightswithrespecttotherankingordersofthefouralternatives. Inthiscase,byEquations(8)–(10)themeasureresultsandrankingofthefouralternativesareindicated inTable 3.

Table3. Measureresultsandrankingofthefouralternativeswith w =(1/3,1/3,1/3)and w =(1/2,1/2).

MeasureMethod MeasureValuebetween Lk (k =1,2,3,4)and L* Ranking Cw (Lk, L∗ ) s7.7470, s7.8918, s7.3878, s7.2922 L2 > L1 > L3 > L4 Dw (Lk, L∗ ) s7.7430, s7.8917, s7.3448, s7.2892 L2 > L1 > L3 > L4 Jw (Lk, L∗ ) s7.5019, s7.7863, s6.7888, s6.6944 L2 > L1 > L3 > L4

Inthiscase,thereexiststhesamerankingorderinTables 1 and 3 regardingthecosine,Dice, andJaccardmeasures.Then,thebestalternativeissill L2,whichmeansthebuyershouldalsobuy house L2 inthebestcity q2 basedonTable 2.Itisobviousthatalltherankingordersimplythedecision robustnessbasedonthecosine,Dice,andJaccardmeasuresregardingthechangeofweightsinthis illustrativeexample,whichalsoshownosensitivityofalltherankingorderswithrespecttothechange oftheweights.IntheactualDMapplications,however,oneofthreevectormeasurescanbeselected bydecisionmakers’preferenceoractualrequirements.

However,existingvariouslinguisticneutrosophicDMapproaches[25–28]cannothandlethe DMproblemsinQ-LNVSsetting;whileourproposedDMmethodcancarryoutboththeexisting DMproblemswithLNVinformation[25–28]andtheDMproblemswithQ-LNVSinformation,which showsitsadvantageinQ-LNVSsettingbecausetheLNVsetisaspecialcaseofQ-LNVSundera universalset.Furthermore,theexistingQ-neutrosophicsoftDMapproach[34]cannotdealwiththe DMproblemswithQ-LNVSinformationbecausetheQ-neutrosophicsoftset[34]cannotexpress Q-linguisticneutrosophicinformation.Hence,ourproposedQ-linguisticneutrosophicDMmethod providesanewwayforlinguisticneutrosophicDMwithTDUSs.

5.Conclusions

ThisstudypresentedtheconceptofQ-LNVSforthefirsttimetodescribethetruth,falsity, andindeterminacylinguisticargumentsinTDUSs,andthenthecosine,Dice,andJaccardmeasuresof Q-LNVSsinvectorspace.Next,aQ-linguisticneutrosophicmulti-attributeDMapproachinQ-LNVS settingwasestablishedbyusingthecosine,Dice,andJaccardmeasuresofQ-LNVSstosolvelinguistic neutrosophicDMproblemsregardingTDUSs.Lastly,theapplicationofthedevelopedDMapproach wasgivenbyanillustrativeexampleinQ-LNVSsetting.Thedecisionresultsshowthattheestablished multi-attributeDMapproachofQ-LNVSscansolvelinguisticneutrosophicDMproblemsregarding TDUSs(two-dimensionalattributesets)inQ-LNVSsetting,whichindicatesitsmainadvantageand contribution.Basedonthefirststudy,thethreevectormeasuresofQ-LNVSswillbefurtherusedfor medicaldiagnosis,datamining,andclusteringanalysisforfutureresearchinQ-LNVSsetting.

AuthorContributions: J.Y.proposedtheQ-LNVSconceptandthevectorsimilaritymeasures;Z.F.andW.C. gavetheDMapproachandthecalculationandanalysis;alltheauthorswrotethemanuscriptandrevisedthe finalversion.

Funding: ThispaperwassupportedbytheNationalNaturalScienceFoundationofChina(Nos.61703280, 51872186).

ConflictsofInterest: Theauthorsdeclarenoconflictsofinterest.

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