ThegeneralizedDicemeasuresformultiple attributedecisionmakingundersimplified neutrosophicenvironments
JunYe∗
DepartmentofElectricalandInformationEngineering,ShaoxingUniversity,Shaoxing, ZhejiangProvince,P.R.China
Abstract.Asimplifiedneutrosophicset(SNS)isasubclassofneutrosophicsetandcontainsasingle-valuedneutrosophic set(SVNS)andanintervalneutrosophicset(INS).Itwasproposedasageneralizationofanintuitionisticfuzzyset(IFS) andaninterval-valuedintuitionisticfuzzyset(IVIFS)inordertodealwithindeterminateandinconsistentinformation.The paperproposesanotherformoftheDicemeasuresofSNSsandthegeneralizedDicemeasuresofSNSsandindicatesthat theDicemeasuresandasymmetricmeasures(projectionmeasures)arethespecialcasesofthegeneralizedDicemeasuresin someparametervalues.Then,wedevelopthegeneralizedDicemeasures-basedmultipleattributedecision-makingmethods withsimplifiedneutrosophicinformation.BytheweightedgeneralizedDicemeasuresbetweeneachalternativeandtheideal solution(idealalternative)correspondingtosomeparametervaluerequiredbydecisionmakers’preference,allthealternatives canberankedandthebestonecanbeobtainedaswell.Finally,arealexampleontheselectionofmanufacturingschemes demonstratestheapplicationsoftheproposeddecision-makingmethodsundersimplifiedneutrosophicenvironment.The effectivenessandflexibilityoftheproposeddecision-makingmethodsareshownbychoosingdifferentparametervalues.
Keywords:GeneralizedDicemeasure,Dicemeasure,decisionmaking,simplifiedneutrosophicset,asymmetricmeasure, projectionmeasure
1.Introduction
Multipleattributedecisionmakingisamainbranch ofdecisiontheory,whereneutrosophictheoryintroducedbySmarandache[1]hasbeensuccessfully appliedinrecentyears.Asageneralizationofan intuitionisticfuzzyset(IFS)[16]andanintervalvaluedintuitionisticfuzzy(IVIFS)[17],asimplified neutrosophicset(SNS)introducedbyYe[10]is asubclassofaneutrosophicsets[1],includinga single-valuedneutrosophicset(SVNSs)[4]andan intervalneutrosophicset(INSs)[3].Hence,SNSs
∗
Correspondingauthor.JunYe,DepartmentofElectricaland InformationEngineering,ShaoxingUniversity,508Huancheng WestRoad,Shaoxing,ZhejiangProvince312000,P.R.China.Tel.: +8657588327323;E-mail:yehjun@aliyun.com.
areverysuitableforhandlingdecisionmaking problemswithindeterminateandinconsistentinformation,whichIFSsandIVIFSscannotdescribeand dealwith.Recently,manyresearchershaveapplied SNSsandthesubclassesofSNSs(SVNSsandINSs) tothedecision-makingproblems.Variousmethods havebeendevelopedtosolvethemultipleattribute decision-makingproblemswithsimplifiedneutrosophicinformation.Forexample,Ye[9]proposed thecorrelationcoefficientofSVNSsandappliedit tomultipleattributedecisionmaking.ChiandLiu [18]andBiswasetal.[21]extendedTOPSISmethod tosingle-valuedandintervalneutrosophicmultiple attributedecision-makingproblems.Ye[11–13]presentedsomesimilaritymeasuresofSVNSs,INSs andSNSsandappliedthemtodecisionmaking.
Ye[14]putforwardacross-entropymeasureof SVNSsformultipleattributedecisionmakingproblems.Ye[10],Zhangetal.[5],Liuetal.[19],Liuand Wang[20],andPengetal.[8]developedsomesimplified,intervalandsingle-valuedneutrosophicnumber aggregationoperatorsandappliedthemtomultiple attributedecision-makingproblems.Pengetal.[7] andZhangetal.[6]proposedoutrankingapproaches formulticriteriadecision-makingproblemswithsimplifiedandintervalneutrosophicinformation.Sahin andKucuk[22]presentedasubsethoodmeasurefor SVNSsandappliedittomultipleattributedecision making.S¸ahinandLiu[23]introducedamaximizingdeviationmethodforneutrosophicmultiple attributedecisionmakingwithincompleteweight information.Ye[15]presentedamultipleattribute decision-makingmethodbasedonthepossibility degreerankingmethodandorderedweightedaggregationoperatorsofintervalneutrosophicnumbers.
SincetheDicemeasureisoneofvectorsimilarity measures,itisausefulmathematicaltoolforhandlingdecision-makingproblems.However,theDice measureofSNSs[13]usedfordecisionmakinglacks flexibilityindecision-makingprocess.Therefore,it isnecessarytoimprovetheDicemeasureofSNSs tohandlemultipleattributedecision-makingproblemstosatisfytherequirementsofdecisionmakers’ preferenceandflexibledecisionmaking.Inorder todoso,themainpurposesofthispaperare:(1) toproposeanotherformoftheDicemeasuresof SNSs,(2)topresentthegeneralizedDicemeasures ofSNSs,and(3)todevelopthegeneralizedDice measures-basedmultipleattributedecision-making methodswithsimplifiedneutrosophicinformation. Inthedecisionmakingprocess,themainadvantageoftheproposedmethodsismoregeneraland moreflexiblethanexistingdecision-makingmethods withsimplifiedneutrosophicinformationtosatisfy thedecisionmakers’preferenceand/orpractical requirements.
Therestofthepaperisorganizedasfollows. Section2reviewstheDicemeasuresofSNSs.Section3proposesanotherformoftheDicemeasures ofSNSs.InSection4,weproposethegeneralizedDicemeasuresofSNSsandindicatetheDice measuresandasymmetricmeasures(projectionmeasures)asthespecialcasesofthegeneralizedDice measuresinsomeparametervalues.InSection5,the generalizedDicemeasures-basedmultipleattribute decision-makingmethodsaredevelopedundersimplifiedneutrosophicenvironment.InSection6,areal exampleontheselectionofmanufacturingschemes
isgiventoshowtheapplicationoftheproposedmethods,andthentheeffectivenessandflexibilityofthe proposedmethodsareindicatedbychoosingdifferent parametervalues.Finally,Section7containsconclusionsandfuturework.
2.TheDicemeasuresofSNSs
Asasubsetofaneutrosophicset[1],Ye[10]introducedaSNSandgaveitsdefinition.
Definition1. [10]ASNS S intheuniverseof discourse X isdefinedas S ={ x,tS (x),uS (x), v S (x) |x ∈ X},where tS (x): X → [0, 1],us (x): X → [0, 1],and v s (x): X → [0, 1]areatruth-membership functionandanindeterminacy-membershipfunction,afalsity-membershipfunction,respectively, oftheelement x totheset S withthecondition 0 ≤ ts (x) + us (x) + v s (x) ≤ 3for x ∈ X
Infact,SNSscontaintheconceptsofSVNSs andINSs,whicharethesubclassesofSNSs.For convenience,acomponentelement <x,ts (x), us (x), v s (x) > inaSNS S isdenotedby sx =< tx ,ux , v x > forshort,whichiscalledthesimplifiedneutrosophicnumber(SNN),where tx ,ux , v x ∈ [0, 1]and0 ≤ tx , +ux + v x ≤ 3forasinglevaluedneutrosophicnumber(SVNN),andthen tx = [t L x ,t U x ] ⊆ [0, 1], ux = [uL x ,uU x ] ⊆ [0, 1], v x = [v L x , v U x ] ⊆ [0, 1]and0 ≤ t U x + uU x + v U x ≤ 3foran intervalneutrosophicnumber(INN).
Ye[13]presentedtheDicemeasuresofSNSs, whichwasdefinedbelow.
Definition2. [13]Let S1 ={s11 ,s12 ,...,s1n } and S2 ={s21 ,s22 ,...,s2n } betwoSNSs.If s1j =< t1j ,u1j , v 1j ,> and s2j =<t2j ,u2j , v 2j ,> (j = 1, 2,...,n)arethe j -thSVNNsin S1 and S2 respectively,thentheDicemeasurebetween S1 and S2 is definedas:
DSVNN 1 (S1 ,S2 ) = 1 n
n j =1
n j =1
2s1j s2j s1j 2 + s2j 2 = 1 n
2(t1j t2j + u1j u2j + v 1j v 2j ) (t 2 1j + u2 1j + v 2 1j ) + (t 2 2j + u2 2j + v 2 2j )
(1)
If s1j =<t1j ,u1j , v 1j > and s2j =<t2j ,u2j , v 2j > (j = 1, 2,...,n)arethe j -thINNsin S1 and S2 respectively,thentheDicemeasurebetween S1 and S2 isdefinedas:
DINN 1 (S1 ,S2 ) = 1 n
n j =1
3.AnotherformoftheDicemeasuresofSNSs
2s1j s2j s1j 2 + s2j 2 . = 1 n
n j =1
2 t L 1j t L 2j + t U 1j t U 2j + u L 1j u L 2j +u U 1j u U 2j + v L 1j v L 2j + v U 1j v U 2j ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(t L 1j )2 + (u L 1j )2 + (v L 1j )2 +(t U 1j )2 + (u U 1j )2 + (v U 1j )2 +(t L 2j )2 + (u L 2j )2 + (v L 2j )2 +(t U 2j )2 + (u U 2j )2 + (v U 2j )2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(2)
ThissectionproposesanotherformoftheDice measuresofSNSs,whichisdefinedasfollows. Definition3. Let S1 ={s11 ,s12 ,...,s1n } and S2 ={s21 ,s22 ,...,s2n } betwoSNSs.If s1j =< t1j ,u1j , v 1j > and s2j =<t2j ,u2j , v 2j > (j = 1, 2,...,n)arethe j -thSVNNsin S1 and S2 respectively,thentheDicemeasurebetween S1 and S2 is definedas:
Then,thetwoDicemeasures DSVNN 1 (S1 ,S2 )and DINN 1 (S1 ,S2 )satisfythefollowingproperties[13]: (P1) DSVNN 1 (S1 ,S2 ) = DSVNN 1 (S2 ,S1 )and DINN 1 (S1 ,S2 ) = DINN 1 (S2 ,S1 ); (P2)0 ≤ DSVNN 1 (S1 ,S2 ) ≤ 1and0 ≤ DINN 1 (S1 ,S2 ) ≤ 1; (P3) DSVNN 1 (S1 ,S2 ) = 1and DINN 1 (S1 ,S2 ) = 1,if S1 = S2 Especiallywhen tij = t L ij = t U ij , uij = uL ij = uU ij , and v ij = v L ij = v U ij for i = 1, 2and j = 1, 2,...,n arehold,Equation(2)isdegeneratedtoEquation(1). Inrealapplications,oneusuallytakesthe importantdifferencesofeachelement sij (i = 1, 2; j = 1, 2,...,n)intoaccount.Let W = (w1 ,w2 ,...,wn )T betheweightvectorfor sij (i = 1, 2; j = 1, 2,...,n), wj ≥ 0and n j =1 wj = 1. Then,basedonEquations(1)and(2),Ye[13]furtherintroducedtheweightedDicemeasuresofSNSs, respectively,asfollows:
DSVNN 2 (S1 ,S2 ) = 2(S1 · S2 ) |S1 |2 + |S2 |2 = 2 n j =1 (t1j t2j + u1j u2j + v 1j v 2j ) n j =1 (t 2 1j + u2 1j + v 2 1j ) + n j =1 (t 2 2j + u2 2j + v 2 2j ) (5)
If s1j =<t1j ,u1j , v 1j > and s2j =<t2j ,u2j , v 2j > (j = 1, 2,...,n)arethe j -thINNsin S1 and S2 respectively,thentheDicemeasurebetween S1 and S2 isdefinedas:
DINN 2 (S1 ,S2 ) = 2(S1 · S2 ) |S1 |2 + |S2 |2 = 2 n j =1 ⎛ ⎝ t L 1j t L 2j + t U 1j t U 2j + u L 1j u L 2j +u U 1j u U 2j + v L 1j v L 2j + v U 1j v U 2j
⎞ ⎠
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
n j =1 ⎡ ⎣ (t L 1j )2 + (t U 1j )2 + (u L 1j )2 +(u U 1j )2 + (v L 1j )2 + (v U 1j )2 ⎤ ⎦+ n j =1 ⎡ ⎣ (t L 2j )2 + (t U 2j )2 + (u L 2j )2 +(u U 2j )2 + (v L 2j )2 + (v U 2j )2 ⎤ ⎦
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
. (6)
DWSVNN 1 (S1 ,S2 ) = n j =1 w j 2s1j · s2j s1j 2 + s2j 2 , = n j =1 w j 2(t1j t2j + u1j u2j + v 1j v 2j ) (t 2 1j + u2 1j + v 2 1j ) + (t 2 2j + u2 2j + v 2 2j ) (3) DWINN 1 (S1 ,S2 ) = n j =1 wj 2s1j s2j s1j 2 + s2j 2 = n j =1 wj
2 t L 1j t L 2j + t U 1j t U 2j + u L 1j u L 2j +u U 1j u U 2j + v L 1j v L 2j + v U 1j v U 2j ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
(t L 1j )2 + (u L 1j )2 + (v L 1j )2 +(t U 1j )2 + (u U 1j )2 + (v U 1j )2 +(t L 2j )2 + (u L 2j )2 + (v L 2j )2 +(t U 2j )2 + (u U 2j )2 + (v U 2j )2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(4)
Obviously,thetwoDicemeasures DSVNN 2 (S1 ,S2 )and DINN 2 (S1 ,S2 )alsosatisfythefollowingproperties: (P1) DSVNN 2 (S1 ,S2 ) = DSVNN 2 (S2 ,S1 )and DINN 2 (S1 ,S2 ) = DINN 2 (S2 ,S1 ); (P2)0 ≤ DSVNN 2 (S1 ,S2 ) ≤ 1and0 ≤ DINN 2 (S1 ,S2 ) ≤ 1; (P3) DSVNN 2 (S1 ,S2 ) = 1and DINN 2 (S1 ,S2 ) = 1,if S1 = S2 .
Proof:
(P1)Itisobviousthatthepropertyistrue. (P2)Itisobviousthatthepropertyistrueaccordingtotheinequality a2 + b2 ≥ 2ab forEquations(5) and(6).
(P3)If S1 = S2 ,thereare s1j = s2j (j = 1, 2,..., n)and |S1 |=|S2 |.Sothereare DSVNN 2 (S1 ,S2 ) = 1 and DINN 2 (S1 ,S2 ) = 1. Inpracticalapplications,theelementsfor sij (i = 1, 2; j = 1, 2,...,n)havedifferentweights. Let W = (w1 ,w2 ,...,wn )T betheweightvectorfor sij (i = 1, 2; j = 1, 2,...,n), wj ≥ 0and n j =1 wj = 1.Then,basedonEquations(5)and(6) wefurtherintroducetheweightedDicemeasuresof SNSs,respectively,asfollows:
DWSVNN 2 (S1 ,S2 ) = 2(S1 S2 )w |S1 |2 w + |S2 |2 w = 2 n j =1 w2 j (t1j t2j + u1j u2j + v 1j v 2j ) n j =1 w2 j (t 2 1j + u2 1j + v 2 1j ) + n j =1 w2 j (t 2 2j + u2 2j + v 2 2j ) , (7)
DWINN 2 (S1 ,S2 ) = 2(S1 S2 )w |S1 |2 w + |S2 |2 w = 2 n j =1 w2 j ⎛ ⎝ t L 1j t L 2j + t U 1j t U 2j + u L 1j u L 2j +u U 1j u U 2j + v L 1j v L 2j + v U 1j v U 2j
= 1 n
n j =1
(t1j t2j + u1j u2j + v 1j v 2j ) λ(t 2 1j + u2 1j + v 2 1j ) + (1 λ)(t 2 2j + u2 2j + v 2 2j ) , (9)
GSVNN 2 (S1 ,S2 ) = S1 S2 λ |S1 |2 + (1 λ) |S2 |2 = n j =1 (t1j t2j + u1j u2j + v 1j v 2j ) λ n j =1 (t 2 1j + u2 1j + v 2 1j ) + (1 λ) n j =1 (t 2 2j + u2 2j + v 2 2j ) , (10)
where λ isapositiveparameterfor0 ≤ λ ≤ 1. Then,thegeneralizedDicemeasuresimplysome specialcasesbychoosingsomevaluesoftheparameter λ.If λ = 0 5,thetwogeneralizedDicemeasures (9)and(10)aredegeneratedtotheDicemeasures(1) and(5);if λ = 0,1,thetwogeneralizedDicemeasuresaredegeneratedtothefollowingasymmetric measuresrespectively:
⎞ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
n j =1 w 2 j ⎡ ⎣ (t L 1j )2 + (t U 1j )2 + (u L 1j )2 +(u U 1j )2 + (v L 1j )2 + (v U 1j )2 ⎤ ⎦+ n j =1 w 2 j ⎡ ⎣ (t L 2j )2 + (t U 2j )2 + (u L 2j )2 +(u U 2j )2 + (v L 2j )2 + (v U 2j )2 ⎤ ⎦
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (8)
4.ThegeneralizedDicemeasuresofSNSs
Inthissection,weproposethegeneralizedDice measuresofSNSstoextendtheDicemeasuresof SNSs.
AsthegeneralizationoftheDicemeasuresof SNSs,thegeneralizedDicemeasuresbetweenSNSs aredefinedbelow.
Definition4. Let S1 ={s11 ,s12 ,...,s1n } and S2 ={s21 ,s22 ,...,s2n } betwoSNSs,where s1j = (t1j ,u1j , v 1j )and s2j = (t2j ,u2j , v 2j )(j = 1, 2,...,n)areconsideredasthe j -thSVNNsin theSNSs S1 and S2 .ThenthegeneralizedDicemeasuresbetween S1 and S2 aredefined,respectively,as follows:
GSVNN 1 (S1 ,S2 ) = 1 n
n j =1
s1j s2j λ s1j 2 + (1 λ) s2j 2
GSVNN 1 (S1 ,S2 ) = 1 n
n j =1
s1j · s2j s2j 2 = 1 n
n j =1
t1j t2j + u1j u2j + v 1j v 2j t 2 2j + u2 2j + v 2 2j for λ = 0, (11)
GSVNN 1 (S1 ,S2 ) = 1 n
n j =1
s1j · s2j s1j 2 = 1 n
n j =1
t1j t2j + u1j u2j + v 1j v 2j t 2 1j + u2 1j + v 2 1j for λ = 1, (12)
GSVNN 2 (S1 ,S2 ) = S1 S2 |S2 |2 = n j =1 (t1j t2j + u1j u2j + v 1j v 2j ) n j =1 (t 2 2j + u2 2j + v 2 2j ) for λ = 0, (13)
GSVNN 2 (S1 ,S2 ) = S1 · S2 |S1 |2 = n j =1 (t1j t2j + u1j u2j + v 1j v 2j ) n j =1 (t 2 1j + u2 1j + v 2 1j ) for λ = 1 (14)
Obviously,thefourasymmetricmeasuresarethe extensionoftherelativeprojectionmeasure(the improvedprojectionmeasure)ofintervalnumbers [2],hencethefourasymmetricmeasurescanbeconsideredastheprojectionmeasuresofSNSs.
Forpracticalapplications,theelementsof sij (i = 1, 2; j = 1, 2,...,n)implydifferentweights. Assumethat W = (w1 ,w2 ...,wn )T istheweight vectorfor sij (i = 1, 2; j = 1, 2,...,n), wj ≥ 0
and n j =1 wj = 1.Thus,basedonEquations(9)and (10)wefurtherintroducethefollowingweighted generalizedDicemeasuresofSNSs,respectively,as follows:
GWSVNN 1 (S1 ,S2 ) = n j =1 wj s1j · s2j λ s1j 2 + (1 λ) s2j 2 = n j =1 wj t1j t2j + u1j u2j + v 1j v 2j λ(t 2 1j + u2 1j + v 2 1j ) + (1 λ)(t 2 2j + u2 2j + v 2 2j ) , (15) GWSVNN 2 (S1 ,S2 ) = (S1 S2 )w λ |S1 |2 w + (1 λ) |S2 |2 w = n j =1 w2 j (t1j t2j + u1j u2j + v 1j v 2j ) ⎛ ⎜ ⎝ λ n j =1 w 2 j (t 2 1j + u 2 1j + v 2 1j )+ (1 λ) n j =1 w 2 j (t 2 2j + u 2 2j + v 2 2j ) ⎞ ⎟ ⎠ . (16)
Definition5. Let S1 ={s11 ,s12 ,...,s1n } and S2 ={s21 ,s22 ,...,s2n } betwoSNSs,where s1j = (t1j ,u1j , v 1j )and s2j = (t2j ,u2j , v 2j )(j = 1, 2,...,n)areconsideredasthe j -thINNsinthe SNSs S1 and S2 .ThenthegeneralizedDicemeasuresbetween S1 and S2 aredefined,respectively, asfollows:
where λ isapositiveparameterfor0 ≤ λ ≤ 1.Especially,when tij = t L ij = t U ij , uij = uL ij = uU ij ,and v ij = v L ij = v U ij for i = 1, 2and j = 1, 2,...,n arehold, Equations(17)and(18)aredegeneratedtoEquations (9)and(10).
Similarly,if λ = 0.5,thetwogeneralizedDice measures(17)and(18)aredegeneratedtotheDice measures(2)and(6);if λ = 0,1,thenthetwogeneralizedDicemeasuresaredegeneratedtothefollowing asymmetricmeasuresrespectively:
GINN 3 (S1 ,S2 ) = 1 n
n j =1
s1j s2j s2j 2 = 1 n
n j =1
t L 1j t L 2j + t U 1j t U 2j + uL 1j uL 2j + uU 1j uU 2j + v L 1j v L 2j + v U 1j v U 2j (t L 2j )2 + (uL 2j )2 + (v L 2j )2 + (t U 2j )2 + (uU 2j )2 + (v U 2j )2 for λ = 0, (19)
GINN 3 (S1 ,S2 ) = 1 n
n j =1
s1j s2j s1j 2 = 1 n
n j =1 t L 1j t L 2j + t U 1j t U 2j + uL 1j uL 2j + uU 1j uU 2j + v L 1j v L 2j + v U 1j v U 2j (t L 1j )2 + (uL 1j )2 + (v L 1j )2 + (t U 1j )2 + (uU 1j )2 + (v U 1j )2 for λ = 1, (20)
GINN 3 (S1 ,S2 ) = 1 n
n j =1
s1j · s2j λ s1j 2 + (1 λ) s2j 2 = 1 n n j =1
t L 1j t L 2j + t U 1j t U 2j + uL 1j uL 2j + uU 1j uU 2j + v L 1j v L 2j + v U 1j v U 2j ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ λ (t L 1j )2 + (u L 1j )2 + (v L 1j )2 +(t U 1j )2 + (u U 1j )2 + (v U 1j )2 +(1 λ) (t L 2j )2 + (u L 2j )2 + (v L 2j )2 +(t U 2j )2 + (u U 2j )2 + (v U 2j )2
GINN 4 (S1 ,S2 ) = S1 S2 |S2 |2 = n j =1 (t L 1j t L 2j + t U 1j t U 2j + uL 1j uL 2j + uU 1j uU 2j + v L 1j v L 2j + v U 1j v U 2j ) n j =1 [(t L 2j )2 + (t U 2j )2 + (uL 2j )2 + (uU 2j )2 + (v L 2j )2 + (v U 2j )2 ] for λ = 0, (21)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, (17) GINN 4 (S1 ,S2 ) = S1 S2 λ |S1 |2 + (1 λ) |S2 |2 = n j =1 (t L 1j t L 2j + t U 1j t U 2j + uL 1j uL 2j + uU 1j uU 2j + v L 1j v L 2j + v U 1j v U 2j ) ⎛
)2
GINN 4 (S1 ,S2 ) = S1 · S2 |S1 |2 = n j =1 (t L 1j t L 2j + t U 1j t U 2j + uL 1j uL 2j + uU 1j uU 2j + v L 1j v L 2j + v U 1j v U 2j ) n j =1 [(t L 1j )2 + (t U 1j )2 + (uL 1j )2 + (uU 1j )2 + (v L 1j )2 + (v U 1j )2 ] for λ = 1 (22)
Then,thefourasymmetricmeasuresarealsoconsideredastheextensionoftherelativeprojection measure(theimprovedprojectionmeasure)ofintervalnumbers[2],whicharealsocalledtheprojection measuresofSNSs.
Forpracticalapplications,theelementsof sij (i = 1, 2; j = 1, 2,...,n)implydifferentweights. Assumethat W = (w1 ,w2 ,...,wn )T istheweight vectorfor sij (i = 1, 2; j = 1, 2,...,n), wj ≥ 0 and n j =1 wj = 1.Similarly,basedonEquations (17)and(18)wealsofurtherintroducetheweighted
generalizedDicemeasuresofSNSs,respectively,as follows:
GWINN 3 (S1 ,S2 ) = n j =1 wj s1j s2j λ s1j 2 + (1 λ) s2j 2 = n j =1 wj
t L 1j t L 2j + t U 1j t U 2j + u L 1j u L 2j +u U 1j u U 2j + v L 1j v L 2j + v U 1j v U 2j ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ λ (t L 1j )2 + (u L 1j )2 + (v L 1j )2 +(t U 1j )2 + (u U 1j )2 + (v U 1j )2 +(1 λ) (t L 2j )2 + (u L 2j )2 + (v L 2j )2 +(t U 2j )2 + (u U 2j )2 + (v U 2j )2
beusedtohelpidentifythebestalternativeinthe decisionset[13].Hence,byanidealSVNN
s ∗ j =<t ∗ j ,u ∗ j , v ∗ j >=< max i (tij ), min i (uij ), min i (v ij ) > oranidealINN
, (23)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
s ∗ j =<t ∗ j ,u ∗ j , v ∗ j >=< [max i (t L ij ), max i (t U ij )], [min i (u L ij ), min i (u U ij )], [min i (v L ij ), min i (v U ij )] >
for j = 1, 2,...,n and i = 1, 2,...,m,wecan determineasimplifiedneutrosophicidealsolution(idealalternative) S ∗ ={s∗ 1 ,s∗ 2 ,...,s∗ n },where s∗ j =<t ∗ j ,u∗ j , v ∗ j > isthe j -thidealSNN.
GWINN 4 (S1 ,S2 ) = (S1 S2 )w λ |S1 |2 w + (1 λ) |S2 |2 w =
t L 1j t L 2j + t U 1j t U 2j + u L 1j u L 2j +u U 1j u U 2j + v L 1j v L 2j + v U 1j v U 2j ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ λ n j =1 w 2 j (t L 1j )2 + (t U 1j )2 + (u L 1j )2 +(u U 1j )2 + (v L 1j )2 + (v U 1j )2 + (1 λ) n j =1 w 2 j (t L 2j )2 + (t U 2j )2 + (u L 2j )2 +(u U 2j )2 + (v L 2j )2 + (v U 2j )2
n j =1 w2 j
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (24)
5.Decisionmaking-methodsbased onthegeneralizedDicemeasures
Inthissection,weproposemultipleattribute decision-makingmethodsbyusingthegeneralized DicemeasuresofSNSsundersimplifiedneutrosophicenvironment.
Formultipleattributedecision-makingproblems, let S ={S1 ,S2 ,...,Sm } beasetofalternativesand R ={R1 ,R2 ,...,Rn } beasetofattributes.Then, theweightoftheattribute Rj (j = 1, 2,...,n)is wj , wj ∈ [0, 1]and n j =1 wj = 1.Thus,thefitjudgment(satisfactionevaluation)ofanattribute Rj (j = 1, 2,...,n)foranalternative Si (i = 1, 2,...,m) isrepresentedbyaSNS Si ={si1 ,si2 ,...,sin }, where sij =<tij ,uij , v ij > isaSVNNfor0 ≤ tij + uij + v ij ≤ 3oranINNfor0 ≤ t U ij + uU ij + v U ij ≤ 3(j = 1, 2,...,n and i = 1, 2,...,m).Therefore, wecanestablishasimplifiedneutrosophicdecision matrix D = (sij )m×n .
Inthemultipleattributedecision-makingproblem, theconceptofanidealsolution(idealalternative)can
Inthedecision-makingprocess,decisionmakers takesomevalueoftheparameter λ ∈ [0, 1]accordingtotheirpreferenceand/orrealrequirements,the weightedgeneralizedDicemeasurebetween Si (i = 1, 2,...,m)and S ∗ isobtainedbyusingoneofEquations(15),(16),(23)and(24)torankthealternatives.
Thus,thegreaterthevalueoftheweightedgeneralizedDicemeasurebetween Si (i = 1, 2,...,m) and S ∗ is,thebetterthealternative Si is.
6.Decision-makingexample ofmanufacturingschemes
Arealexampleaboutthedecision-makingproblemofmanufacturingschemeswithsimplified neutrosophicinformationisgiventodemonstrate theapplicationsandeffectivenessoftheproposed decision-makingmethodsinrealisticscenarios.
Toselectthebestmanufacturingscheme(alternative)fortheflexiblemanufacturingsystemina manufacturingcompany,thetechniquedepartmentof thecompanyprovidesfourmanufacturingschemes (alternatives)withrespecttosomeproductasaset ofthealternatives S ={S1 ,S2 ,S3 ,S4 } fortheflexiblemanufacturingsystem.Adecisionmustbemade accordingtothefourattributes:(1) R1 istheimprovementofquality;(2) R2 isthemarketresponse;(3) R3 isthemanufacturingcost;(4) R4 isthemanufacturingcomplexity.Theweightvectorofthefour attributes W = (0.3, 0.25, 0.25, 0.2)T isgivenby decisionmakers.
Inthedecision-makingproblem,thedecisionmakersarerequiredtomakethefitjudgment(satisfaction evaluation)ofanattribute Rj (j = 1, 2, 3, 4)foran alternative Si (i = 1, 2, 3, 4)andtogivesimplified neutrosophicevaluationinformation,whichisshown inthefollowingdecisionmatrixwithSVNNs:
(0 75, 0 2, 0 3)(0 7, 0 2, 0 3)(0 65, 0 2, 0 25) (0 8, 0 1, 0 2)(0 75, 0 2, 0 1)(0 75, 0 2, 0 1) (0 7, 0 2, 0 2)(0 78, 0 2, 0 1)(0 85, 0 15, 0 1) (0 8, 0 2, 0 1)(0 85, 0 2, 0 2)(0 7, 0 2, 0 2)
(0 75, 0 2, 0 1) (0 85, 0 1, 0 2) (0 76, 0 2, 0 2) (0 86, 0 1, 0 2)
⎤ ⎥ ⎥ ⎦
Then,thedevelopeddecision-makingmethodscan beusedforthedecisionmakingproblem. Accordingto s ∗ j =<t ∗ j ,u ∗ j , v ∗ j >= < max i (tij ), min i (uij ), min i (v ij ) > for j = 1, 2, 3, 4and i = 1, 2, 3, 4,wecanobtain anidealsolution(idealalternative)asfollows: S ∗ ={s ∗ 1 ,s ∗ 2 ,s ∗ 3 ,s ∗ 4 } = < 0 8, 0 1, 0 1 >,< 0 85, 0 2, 0 1 >, < 0 85, 0 15, 0 1 >,< 0 86, 0 1, 0 1 >
ByusingEquation(15)or(16)anddifferentvalues oftheparameter λ,theweightedgeneralizedDice measurevaluesbetween Si (i = 1, 2, 3, 4)and S ∗ canbeobtained,whichareshowninTables1and2 respectively.
FromTables1and2,wecanseethatdifferentrankingordersareindicatedbytakingdifferentvaluesof theparameter λ anddifferentgeneralizedDicemeasures.Thenwecanobtainthatthebestalternativeis S2 or S3 or S4 .
Furthermore,forthespecialcasesofthetwogeneralizedDicemeasuresweobtainthefollowingresults:
(1)When λ = 0,thetwoweightedgeneralized Dicemeasuresarereducedtotheweighted projectionmeasuresof Si on S ∗ .Thus,the alternative S2 isthebestchoiceamongallthe alternatives.
< [0 7, 0 8], [0 1, 0 2], [0 2, 0 3] >
< [0 7, 0 9], [0 1, 0 2], [0 2, 0 3] > < [0 7, 0 8], [0 1, 0 3], [0 2, 0 3] >
< [0 8, 0 9], [0 2, 0 3], [0 1, 0 2] >
< [0.6, 0.7], [0.1, 0.2], [0.2, 0.4] >
< [0 7, 0 8], [0 2, 0 3], [0 1, 0 2] >
< [0 8, 0 9], [0 1, 0 2], [0 1, 0 1] >
< [0 7, 0 8], [0 1, 0 2], [0 1, 0 3] >
Table1
ThemeasurevaluesofEquation(15)andrankingorders
λGWSVNN 1 GWSVNN 1 GWSVNN 1 GWSVNN 1 Rankingorder (S1 ,S ∗ )(S2 ,S ∗ )(S3 ,S ∗ )(S4 ,S ∗ )
00.88950.95170.93610.9287 S2 S3 S4 S1 0.20.91570.96670.95580.9472 S2 S3 S4 S1 0.50.96120.99240.98760.9816 S2 S3 S4 S1 0.70.99661.01191.01041.0100 S2 S3 S4 S1 11.05941.04551.04751.0641 S4 S1 S3 S2
Table2
ThemeasurevaluesofEquation(16)andrankingorders
λGWSVNN 2 GWSVNN 2 GWSVNN 2 GWSVNN 2 Rankingorder (S1 ,S ∗ )(S2 ,S ∗ )(S3 ,S ∗ )(S4 ,S ∗ ) 00.89080.95030.93870.9372 S2 S3 S4 S1 0.20.91750.96670.95870.9557 S2 S3 S4 S1 0.50.96050.99240.99030.9849 S2 S3 S4 S1 0.70.99151.01021.01261.0054 S3 S2 S4 S1 11.04191.03831.04791.0378 S3 S1 S2 S4
thealternative S2 isthebestchoiceamongall thealternatives.
(3)When λ = 1,thetwoweightedgeneralized Dicemeasuresarereducedtotheweighted projectionmeasuresof S ∗ on Si .Thus,the alternative S3 or S4 isthebestchoiceamong allthealternatives.
Obviously,accordingtodifferentvaluesofthe parameter λ anddifferentmeasures,rankingorders maybedifferent.Thustheproposeddecision-making methodscanbeassignedsomevalueof λ andsome measuretosatisfythedecisionmakers’preference and/orrealrequirements.
Ifthefitjudgment(satisfactionevaluation)ofan attribute Rj (j = 1, 2, 3, 4)foranalternative Si (i = 1, 2, 3, 4)isgiveninthedecisionmaking problembythefollowingdecisionmatrixwithINNs:
< [0 7, 0 8], [0 1, 0 2], [0 2, 0 3] >
< [0 7, 0 8], [0 1, 0 3], [0 1, 0 2] >
< [0 8, 0 9], [0 1, 0 2], [0 1, 0 2] >
< [0 8, 0 9], [0 2, 0 3], [0 2, 0 3] >
< [0.7, 0.8], [0.1, 0.2], [0.1, 0.2] >
< [0 8, 0 9], [0 0, 0 1], [0 2, 0 3] >
< [0 7, 0 8], [0 1, 0 2], [0 1, 0 2] >
< [0 7, 0 9], [0 0, 0 1], [0 1, 0 3] >
(2)When λ = 0.5,thetwoweightedgeneralized Dicemeasuresarereducedtotheweighted Dicesimilaritymeasuresof Si and S ∗ .Thus,
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪
< [0 8, 0 9], [0 1, 0 2], [0 1, 0 2] >, < [0 7, 0 8], [0 1, 0 2], [0 1, 0 2] >, < [0 8, 0 9], [0 1, 0 2], [0 1, 0 1] >, < [0.8, 0.9], [0.0, 0.1], [0.1, 0.2] >
⎫ ⎪ ⎪ ⎪
⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ .
FromTables3and4,differentrankingordersare shownbytakingdifferentvaluesof λ anddifferent measures.Thenwecanobtainthatthebestalternative is S1 or S2 or S4 Furthermore,forthespecialcasesofthetwogeneralizedDicemeasuresweobtainthefollowingresults:
(1)When λ = 0,thetwoweightedgeneralized Dicemeasuresarereducedtotheweighted projectionmeasuresof Si on S ∗ .Thus,the alternative S4 isthebestchoiceamongallthe alternatives.
(2)When λ = 0 5,thetwoweightedgeneralized Dicemeasuresarereducedtotheweighted Dicesimilaritymeasuresof Si and S ∗ .Thus, thealternative S2 isthebestchoiceamongall thealternatives.
(3)When λ = 1,thetwoweightedgeneralized Dicemeasuresarereducedtotheweighted projectionmeasuresof S ∗ on Si .Thus,the alternative S1 isthebestchoiceamongallthe alternatives.
Therefore,accordingtodifferentvaluesofthe parameter λ anddifferentmeasures,rankingorders maybealsodifferent.Thustheproposeddecisionmakingmethodscanbeassignedsomevalueof λ andsomemeasuretosatisfythedecisionmakers’ preferenceand/orrealrequirements.
Obviously,thedecision-makingmethodsbased theDicemeasuresandtheprojectionmeasuresare thespecialcasesoftheproposeddecision-making
Table3
ThemeasurevaluesofEquation(23)andrankingorders
λGWINN 3 GWINN 3 GWINN 3 GWINN 3 Rankingorder (S1 ,S ∗ )(S2 ,S ∗ )(S3 ,S ∗ )(S4 ,S ∗ )
00.90850.97700.98611.0159 S4 S3 S2 S1 0.20.93250.98190.98601.0015 S4 S3 S2 S1 0.50.97370.99030.99010.9863 S2 S3 S4 S1 0.71.00530.99660.99550.9797 S1 S2 S3 S4 11.06011.00721.00750.9746 S1 S3 S2 S4
Table4
ThemeasurevaluesofEquation(24)andrankingorders
λGWINN 4 GWINN 4 GWINN 4 GWINN 4 Rankingorder (S1 ,S ∗ )(S2 ,S ∗ )(S3 ,S ∗ )(S4 ,S ∗ )
00.90350.97260.97901.0109 S4 S3 S2 S1 0.20.93060.97950.98331.0011 S4 S3 S2 S1 0.50.97430.99010.99000.9867 S2 S3 S4 S1 0.71.00590.99720.99450.9774 S1 S2 S3 S4 11.05721.00821.00130.9637 S1 S2 S3 S4 methodsbasedongeneralizedDicemeasures. Therefore,inthedecision-makingprocess,the decision-makingmethodsdevelopedinthispaper aremoregeneralandmoreflexiblethanexisting decision-makingmethodsundersimplifiedneutrosophicenvironment.
7.Conclusion
ThispaperproposedanotherformoftheDice measuresbetweenSNSsandthegeneralizedDice measuresofSNSsandindicatedtheDicemeasures ofSNSsandtheprojectionmeasures(asymmetric measures)ofSNSsarethespecialcasesofthegeneralizedDicemeasuresofSNSscorrespondingto someparametervalues.Then,wedevelopedmultipleattributedecision-makingmethodsbasedonthe generalizedDicemeasuresofSNSsundersimplified neutrosophicenvironment.Accordingtodifferent parametervaluesandsomemeasurepreferredby decisionmakers,bytheweightedgeneralizedDice measurebetweeneachalternativeandtheideal solution(idealalternative),allalternativescanbe rankedandthebestalternativecanbeselectedas well.Finally,arealexampleabouttheselection ofmanufacturingschemes(alternatives)demonstratedtheapplicationsofthedevelopedmethods undersimplifiedneutrosophicenvironment,andthen theeffectivenessandflexibilityofthedeveloped decision-makingmethodswereshowncorresponding todifferentparametervalues.Inthedecision-making processundersimplifiedneutrosophicenvironment,
themainadvantageismoregeneralandmoreflexible thanexistingdecision-makingmethodstosatisfythe decisionmakers’preferenceand/orpracticalrequirements.
Inthefuturework,weshallextendthegeneralized DicemeasuresofSNSstootherareassuchaspattern recognition,faultdiagnosis,andimageprocessing.
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