SymmetryMeasuresofSimplifiedNeutrosophicSets forMultipleAttributeDecision-MakingProblems
AngyanTu 1,2 ID ,JunYe 2 ID andBingWang 1,*
1 SchoolofMechatronicEngineeringandAutomation,ShanghaiUniversity,149YanchangRoad, Shanghai200072,China;lucytu@shu.edu.cn
2 DepartmentofElectricalandInformationEngineering,ShaoxingUniversity,508HuanchengWestRoad, Shaoxing312000,Zhejiang,China;yejun@usx.edu.cn
* Correspondence:susanbwang@shu.edu.cn;Tel.:+86-21-5633-1568
Received:27March2018;Accepted:4May2018;Published:5May2018
Abstract: Asimplifiedneutrosophicset(containingintervalandsingle-valuedneutrosophicsets)can beusedfortheexpressionandapplicationinindeterminatedecision-makingproblemsbecausethree elementsinthesimplifiedneutrosophicset(includingintervalandsinglevaluedneutrosophicsets) arecharacterizedbyitstruth,falsity,andindeterminacydegrees.Underasimplifiedneutrosophic environment,therefore,thispaperfirstlydefinessimplifiedneutrosophicasymmetrymeasures. Thenweproposeanormalizedsymmetrymeasureandaweightedsymmetrymeasureofsimplified neutrosophicsetsanddevelopasimplifiedneutrosophicmultipleattributedecision-makingmethod basedontheweightedsymmetrymeasure.Allalternativescanberankedthroughtheweighted symmetrymeasurebetweentheidealsolution/alternativeandeachalternative,andthenthe bestonecanbedetermined.Finally,anillustrativeexampleontheselectionofmanufacturing schemes(alternatives)intheflexiblemanufacturingsystemdemonstratestheapplicabilityofthe proposedmethodinasimplified(intervalandsinglevalued)neutrosophicsetting,andthenthe decision-makingmethodbasedontheproposedsymmetrymeasureisinaccordwiththeranking orderandbestchoiceofexistingprojectionandbidirectionalprojection-baseddecision-making methodsandstrengthenstheresolution/discriminationinthedecision-makingprocesscorresponding tothecomparativeexample.
Keywords: asymmetrymeasure;symmetrymeasure;simplifiedneutrosophicset;decisionmaking
1.Introduction
Torepresentinconsistentandindeterminateinformationintherealworld,Smarandache[1] introducedtheneutrosophicset(NS)conceptastheextensionofthefuzzysetand(interval-valued) intuitionisticfuzzysets.Becausethreefunctionvaluesoftruth-membership,falsity-membership, andindeterminacy-membershipinNSaredefinedintherealstandardinterval[0,1]or nonstandardinterval] 0,1+[,thenonstandardintervalshowsitsdifficultapplicationinthereal world.AsthesubclassofNS,Ye[2]presentedasimplifiedneutrosophicset(SNS),whereits indeterminacy-membership,truth-membership,andfalsity-membershipfunctionsareinthereal standardinterval[0,1]toconvenientlyapplyinengineeringfields.SNSincludesaninterval neutrosophicset(INS)[3]andasingle-valuedneutrosophicset(SVNS)[4].Afterthat,Ye[5] introducedthreesimplifiedneutrosophicsimilaritymeasuresinvectorspaceandtheirmulticriteria decision-makingmethods.Then,outrankingapproaches[6,7]wereusedforsimplifiedneutrosophic andintervalneutrosophicdecision-makingproblems.Someresearchersproposedcorrelation coefficients,crossentropymeasures,similaritymeasuresforINSs/SVNSs/SNSs,andtheirmultiple attributedecision-making(MADM)methods[8–12].Someresearcherspresentedvariousaggregation
Symmetry 2018, 10,144;doi:10.3390/sym10050144 www.mdpi.com/journal/symmetry
symmetry SS Article
operatorsofSVNSs/INSs/SNSsfordecision-makingfields[13–19].Furthermore,projectionand bidirectionalprojectionmeasuresofINSsandSVNSs[20,21]wereintroducedfortheirdecision-making. TOPSISmethod[22]waspresentedfordecision-makingwithSVNSinformation,alsoSVNSgraphs[23] wereusedfordecision-makingproblems.Then,somedecisionmakingmethodswerepresented basedontheneutrosophicMULTIMOORA,WASPAS-SVNS,andextendedTOPSISandVIKOR methods[24–26]underSNSenvironments.
Asmentionedabove,themeasuremethodofSNSsisanimportanttoolindecision-making.To developnewmeasuresinsimplifiedneutrosophicdecision-makingproblems,thisstudyproposes asymmetrymeasuresofSNSsandtheirnormalizedsymmetrymeasureofSNSsforthefirsttime, andthendevelopsaMADMmethodbyusingtheweightedsymmetrymeasureofSNSs.Therefore, thispaperispresentedasthefollowingframe.SomedefinitionsofasymmetrymeasuresofSNSsare presentedinSection 2.ThenormalizedsymmetrymeasureandweightedsymmetrymeasureofSNSs areproposedinSection 3.AMADMmethodusingtheweightedsymmetrymeasureisdeveloped inSection 4.InSection 5,thepracticalexampleisprovidedinsimplifiedneutrosophicenvironments toshowtheapplication,alongwiththesensitiveanalysisregardingtotheattributeweightvalues, andthenthefeasibilityandeffectivenessareindicatedbythecomparativeexample.Atlast,Section 6 indicatesconclusionsandfuturework.
2.AsymmetryMeasuresofSimplifiedNeutrosophicSets
Inthissection,asymmetrymeasuresofSNSsarepresented,includingasymmetrymeasuresof SVNSsandINSs.
Ye[2]presentedanSNSasasubclassofaNS[1]andgavethefollowingdefinition.
Definition1 [2]. ASNSisdefinedas A = { x, uA (x), vA (x), hA (x) |x ∈ U} intheuniverseofdiscourse U,suchthatuA(x):U → [0,1],vA(x):U → [0,1],and hA (x) :U → [0,1],whicharedescribedbythetruth, indeterminacyandfalsity-membershipdegrees,satisfying0 ≤ supuA(x)+supvA(x)+suphA(x) ≤ 3forINS or0 ≤ uA(x)+vA(x)+hA(x) ≤ 3forSVNSandx ∈ U.
Forconvenience,anelementintheSNSAisdenotedby a=(ua,va,ha),whichiscalledthe simplifiedneutrosophicnumber(SNN),includingasinglevaluedneutrosophicnumber(SVNN)and anintervalneutrosophicnumber(INN).
First,asymmetrymeasuresofSVNSsaredefinedinthefollowing.
Definition2. LetB={b1,b2, ,bn}andA={a1,a2, ,an}betwoSVNSs,wherebj =(ubj,vbj,hbj)and aj =(uaj,vaj,haj)arethej-thSVNNs(j=1,2, ,n)ofBandArespectively.Then
PB (A)= A · B B 2 =
PA (B)= A · B A 2 =
arecalledasymmetrymeasuresofBandA.
n ∑ j = 1 (uaj ubj + vaj vbj + haj hbj ) n ∑ j = 1 (u2 bj + v2 bj + h2 bj )
n ∑ j = 1 (uaj ubj + vaj vbj + haj hbj ) n ∑ j = 1 (u2 aj + v2 aj + h2 aj )
(1)
(2)
Ifoneconsiderstheweightofeachelement bj or aj (j =1,2, , n),theweightedasymmetry measureofSNSscanbeintroducedbelow.
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Definition3. LetB={b1,b2, ... ,bn}andA={a1,a2, ... ,an}betwoSVNSs,wherebj =(ubj,vbj,hbj)and aj =(uaj,vaj,haj)arethej-thSVNNs(j=1,2, ... ,n)ofBandArespectively,andlettheweightofanelementbj oraj bewj,wj ∈ [0,1],and ∑n j = 1 wj = 1 .Then
PwB (A)=
n ∑ j = 1 w2 j (uaj ubj + vaj vbj + haj hbj ) n ∑ j = 1 w2 j (u2 bj + v2 bj + h2 bj )
(3)
n ∑ j = 1 w2 j (uaj ubj + vaj vbj + haj hbj ) n ∑ j = 1 w2 j (u2 aj + v2 aj + h2 aj )
(4) arecalledtheweightedasymmetrymeasuresofBandA.
PwA (B)=
Bythesimilarway,thetwoasymmetrymeasuresofSVNSscanbefurtherextendedtothe asymmetrymeasuresofINSs,whicharegivenbythefollowingdefinition.
Definition4. LetB={b1,b2, ,bn}andA={a1,a2, ,an}betwoINSs,where bj =([uL bj, uU bj ], [vL bj, vU bj ], [hL bj, hU bj ]) and aj =([uL aj, uU aj ], [vL aj, vU aj ], [hL aj, hU aj ]) arethej-thINNs(j=1,2, ,n)ofBandArespectively.Then,twoasymmetrymeasuresofBandAaredefinedas
PB (A)= A B B 2 =
PA (B)= A B A 2 =
n ∑ j = 1 (uL aj uL bj + uU aj uU bj + vL aj vL bj + vU aj vU bj + hL aj hL bj + hU aj hU bj ) n ∑ j = 1 [(uL bj )2 +(uU bj )2 +(vL bj )2 +(vU bj )2 +(hL bj )2 +(hU bj )2] (5)
n ∑ j = 1 (uL aj uL bj + uU aj uU bj + vL aj vL bj + vU aj vU bj + hL aj hL bj + hU aj hU bj ) n ∑ j = 1 [(uL aj )2 +(uU aj )2 +(vL aj )2 +(vU aj )2 +(hL aj )2 +(hU aj )2]
(6)
Similarly,ifoneconsiderstheweightofeachelement bj or aj (j =1,2, ... , n),theweighted asymmetrymeasuresofINSscanbeintroducedbelow.
Definition5. LetB={b1,b2, ,bn}andA={a1,a2, ,an}betwoINSs,where bj =([uL bj, uU bj ], [vL bj, vU bj ], [hL bj, hU bj ]) and aj =([uL aj, uU aj ], [vL aj, vU aj ], [hL aj, hU aj ]) arethej-thINNs(j=1,2, ,n)ofBandArespectively,andlettheweightofanelementaj orbj bewj,wj ∈ [0,1],and ∑n j = 1 wj = 1 Thus,twoweightedasymmetrymeasuresofAonBaredefinedas
PwB (A)= (A B)w B 2 w
=
PwA (B)= (A · B)w A 2 w
n ∑ j = 1 w2 j (uL aj uL bj + uU aj uU bj + vL aj vL bj + vU aj vU bj + hL aj hL bj + hU aj hU bj ) n ∑ j = 1 w2 j [(uL bj )2 +(uU bj )2 +(vL bj )2 +(vU bj )2 +(hL bj )2 +(hU bj )2] (7)
=
n ∑ j = 1 w2 j (uL aj uL bj + uU aj uU bj + vL aj vL bj + vU aj vU bj + hL aj hL bj + hU aj hU bj ) n ∑ j = 1 w2 j [(uL aj )2 +(uU aj )2 +(vL aj )2 +(vU aj )2 +(hL aj )2 +(hU aj )2] (8)
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3.NormalizedSymmetryMeasuresofSimplifiedNeutrosophicSets
= B 2 A 2 B 2 A 2 + B 2 − A 2 B A
iscalledthenormalizedsymmetrymeasurebetweenBandA,where B = ∑n j = 1 (u2 bj + v2 bj + h2 bj ) and A = ∑n j = 1 (u2 aj + v2 aj + h2 aj) forSVNSsor B = ∑n j = 1 [(uL bj)2 +(uU bj)2 +(vL bj)2 +(vU bj)2 +(hL bj)2 +(hU bj)2] and A = ∑n j = 1 [(uL aj )2 +(uU aj )2 +(vL aj )2 +(vU aj )2 +(hL aj )2 +(hU aj )2] forINSsarethemodulesofB andArespectively,and B A = ∑n j = 1 (uL aj uL bj + uU aj uU bj + vL aj vL bj + vU aj vU bj + hL aj hL bj + hU aj hU bj ) istheinner productbetweenBandA.
Therefore,thecloserthevalueof M(B,A)isto1,thecloser B isto A,andthenthereare M(B,A) =M(A,B) = 1if B=A, anditsatisfies0 ≤ M(B,A) ≤ 1forany B andany A,whichisanormalized symmetrymeasure.
ThebellowingweightedsymmetrymeasurebetweenSNSscanbeintroducedifoneconsidersthe weightofeachelement bj or aj (j =1,2,..., n).
Definition7. LetB={b1,b2, ... ,bn}andA={a1,a2, ... ,an}betwoSNSs,wherebj =(ubj,vbj,hbj)and aj =(uaj,vaj,haj)arethej-thSNNs(j=1,2, ... ,n)ofBandArespectively,andlettheweightofanelementbj oraj bewj,wj ∈ [0,1],and ∑n j = 1 wj = 1.Thus Mw (B, A)= 1 1 + (B A)w A 2 w
(B A)w B 2 w
= B 2 w A 2 w B 2 w A 2 w + B 2 w − A 2 w (B A)w (10) isknownastheweightedsymmetrymeasureofBandA,where B w = ∑n j = 1 w2 j (u2 bj + v2 bj + h2 bj ) and A w = ∑n j = 1 w2 j (u2 aj + v2 aj + h2 aj)forSVNSsor B w = ∑n j = 1 w2 j [(uL bj)2 +(uU bj)2 +(vL bj)2 +(vU bj)2 +(hL bj)2 +(hU bj)2] and A w = ∑n j = 1 w2 j [(uL aj )2 +(uU aj )2 +(vL aj )2 +(vU aj )2 +(hL aj )2 +(hU aj )2] forINSsarethe weightedmodulesofBandA,andthen (B A)w = ∑n j = 1 w2 j (uaj ubj + vaj vbj + haj hbj ) or (B A)w = ∑n j = 1 w2 j (uL aj uL bj + uU aj uU bj + vL aj vL bj + vU aj vU bj + hL aj hL bj + hU aj hU bj ) forINNsisknownasthe weightedinnerproductofBandA.
4.Decision-MakingMethodUsingtheWeightedSymmetryMeasure Inthissection,theproposedweightedsymmetrymeasureisutilizedforsimplifiedneutrosophic MADMproblems. Setasetofalternativesas S ={S1, S2, , Sm}andasetofattributesas A ={A1, A2, , An}ina MADMproblem.Assumethattheweightoftheattribute Aj is wj, wj ∈ [0,1],and ∑n j = 1 wj = 1.
InSNSsetting,thesatisfactionevaluationsofanalternative Si (i =1,2, , m)foranattribute Aj (j =1,2, , n)isexpressedbyanSNS Si = {si1, si2,..., sin },where sij =(uij, vij, hij)satisfies uij, vij, hij
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AnormalizedsymmetrymeasureofSNSsisproposedinthissection. Definition6. LetB={b1,b2, ,bn}andA={a1,a2, ,an}betwoSNSs,wherebj =(ubj,vbj,hbj)and aj =(uaj,vaj,haj)arethej-thSNNs(j=1,2, ,n)ofBandArespectively.Thus M(B, A)= 1 1 + B A A 2 B A B 2 (9)
Decision-MakingMethodUsingtheWeightedSymmetryMeasure
∈ [0,1]and0 ≤ uij + vij + hij ≤ 3forSVNNor uij, vij, hij ⊆ [0,1]and 0 ≤ uU ij + vU ij + hU ij ≤ 3 forINN. Thus,thedecisionmatrixofSNSscanbeestablishedas D =(sij)m×n.
IntheMADMproblem,thesimilaritymeasurebetweentheidealsolution/alternativeand analternativecanbeusedfordeterminingthebestoneamongallalternatives.Byconsidering s∗ j =(u∗ j , v∗ j , h∗ j )=(max i (uij ), min i (vij ), min i (hij )) forSVNNsor s∗ j =(u∗ j , v∗ j , h∗ j )= ([max i (uL ij ), max i (uU ij )], [min i (vL ij ), min i (vU ij )], [min i (hL ij ), min i (hU ij )) forINNs(j =1,2, , n;i =1,2, ... , m)astheidealsolution,ansimplifiedneutrosophicidealsolution/alternativecanbegiven as S∗ = s∗ 1 , s∗ 2 ,..., s∗ n . ThenbyapplyingEquation(10),theweightedsymmetrymeasurebetween S* and Si (i =1,2, ... , m)isyieldedby Mw (S∗ , Si )= 1 1 + (S∗ Si )w Si 2 w
(S∗ Si )w S∗ 2 w
= Si 2 w S∗ 2 w Si 2 w S∗ 2 w + Si 2 w − S∗ 2 w (S∗ · Si )w (11) where Si w = ∑n j = 1 w2 j (u2 ij + v2 ij + h2 ij), S∗ w = ∑n j = 1 w2 j (u∗2 j + v∗2 j + h∗2 j ),and (Si S∗)w = ∑n j = 1 w2 j (uiju∗ j + vijv∗ j + hijh∗ j ) forSVNNsor Si w = ∑n j = 1 w2 j ((uL ij)2 +(uU ij )2 +(vL ij)2 +(vU ij )2 +(hL ij)2 +(hU ij )2), S∗ w = ∑n j = 1 w2 j ((uL∗ j )2 +(uU∗ j )2 +(vL∗ j )2 +(vU∗ j )2 +(hL∗ j )2 +(hU∗ j )2),and (S∗ · Si )w = ∑n j = 1 w2 j (uL ij uL∗ j + uU ij uU∗ j + vL ij vL∗ j + vU ij vU∗ j + hL ij hL∗ j + hU ij hU∗ j ) forINNs. Thus,thegreaterthevalueof Mw(S* , Si)is,thecloser Si isto S*,andthenthebetterthealternative Si is.
5.Decision-MakingExamples
Apracticalexampleaboutselectingthemanufacturingschemes(alternatives)intheflexible manufacturingsystemisprovidedinSNS(SVNSandINS)environmentstoshowtheapplicationsof theweightedsymmetrymeasure-basedMADMmethodinrealisticscenarios,andthenacomparative examplewithexistingrelativemeasuresforSVNSisgiventoshowthefeasibilityandeffectivenessof theproposedmethod.
5.1.PracticalExample
AssumethatweconsideraMADMproblemintheflexiblemanufacturingsystemaboutthe selectionofmanufacturingschemes(alternatives).Setasetoffouralternativesfortheflexible manufacturingsystemas S= {S1, S2, S3, S4}.Theyneedtosatisfythethreeattributes:(i) A1 is theimprovementinquality;(ii) A2 isthemarketresponse;(iii) A3 isthemanufacturingcost.Inthe decision-makingproblem,thedecisionmaker/expertspecifiestheweightvectoroftheattributesas W =(0.36,0.3,0.34)correspondingtotheimportanceofthethreeattributes.
Thus,thedecisionmakergivesthesatisfactionevaluationofanalternative Si (i =1,2,3,4)foran attribute Aj (j =1,2,3)bytheevaluationinformationofSVNNs,andthensingle-valuedneutrosophic decisionmatrixcanbeconstructedas
(0.75,0.2,0.2)(0.7,0.24,0.26)(0.6,0.2,0.25) (0.8,0.1,0.1)(0.75,0.2,0.3)(0.7,0.3,0.1) (0.7,0.2,0.15)(0.8,0.2,0.1)(0.75,0.25,0.2) (0.8,0.1,0.2)(0.7,0.15,0.2)(0.7,0.2,0.3)
Thus,thedevelopedapproachisusedfortheMADMproblem.
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D =
First,thesingle-valuedneutrosophicidealsolution/alternativeof s∗ j =(u∗ j , v∗ j , h∗ j )= (max i (uij ),min i (vij ),min i (hij )) for j =1,2,3and i =1,2,3,4canbedeterminedby S∗ = {s∗ 1 , s∗ 2 , s∗ 3 } = {(0.8,0.1,0.1), (0.8,0.15,0.1), (0.75,0.2,0.1)}
Then,accordingtoEquation(11),theweightedsymmetrymeasurevaluesof S* and Si canbe obtainedasfollows: Mw(S* , S1)=0.8945, Mw(S* , S2)=0.9964, Mw(S* , S3)=0.9717,and Mw(S* , S4)=0.9730
Sincethevaluesoftheweightedsymmetrymeasureare Mw(S* , S2)> Mw(S* , S4)> Mw(S* , S3)> Mw(S* , S1),thefouralternativesarerankedas S2 > S4 > S3 > S1.Obviously, S2 isthebestoneamong thefouralternatives.
Ifthefitjudgmentsofthealternatives Si (i =1,2,3,4)fortheattributesareexpressedbyinterval neutrosophicinformation,theintervalneutrosophicdecisionmatrixcanbeconstructedas D
([0.7,0.8], [0.1,0.2], [0.15,0.3])([0.7,0.8], [0.2,0.3], [0.1,0.3])([0.6,0.7], [0,0.2], [0.1,0.4]) ([0.75,0.9], [0.1,0.2], [0.1,0.2])([0.7,0.8], [0.1,0.2], [0.1,0.3])([0.6,0.7], [0.2,0.3], [0.1,0.3]) ([0.6,0.8], [0.1,0.3], [0.1,0.2])([0.7,0.8], [0.1,0.3], [0.1,0.2])([0.7,0.8], [0.2,0.4], [0.1,0.3]) ([0.8,0.9], [0.1,0.2], [0.1,0.2])([0.7,0.8], [0.1,0.2], [0.1,0.3])([0.6,0.8], [0.2,0.3], [0.2,0.4])
Then,anintervalneutrosophicidealsolution/alternativeof s∗ j =(u∗ j , v∗ j , h∗ j )=([max i (uL ij ), max i (uU ij )], [min i (vL ij ), min i (vU ij )], [min i (hL ij ), min i (hU ij )) for i =1, 2,3,4and j =1,2,3canbedeterminedby S∗ = s∗ 1, s∗ 2,..., s∗ n = {([0.8,0.9], [0.1,0.2], [0.1,0.2]), ([0.7,0.8], [0.1,0.2], [0.1,0.2]), ([0.7,0.8], [0,0.2], [0.1,0.3])}
ByusingEquation(11),theweightedsymmetrymeasurevaluesof S* and Si (i =1,2,3,4)canbe obtainedas Mw(S* , S1)=0.9053, Mw(S* , S2)=0.9423, Mw(S* , S3)=0.9401,and Mw(S* , S4)=0.9762
Sincetheweightedsymmetrymeasurevaluesare Mw(S4, S*)> Mw(S2, S*)> Mw(S3, S*)> Mw(S1, S*), thefouralternativesarerankedas S4 > S2 > S3 > S1.Thus, S4 isthebestoneamongthefouralternatives. Inthispracticalexample,thereislittledifferenceofrankingordersunderSVNSand INSenvironments.
ToindicatethesensitivityoftheproposedMADMmethod,thisworkonlyconsidersthatthe attributeweightsmayaffecttherankingofalternativesasthesensitiveanalysisbecausetheattribute weightsaregivenbythedecisionmaker’ssubjectivejudgment/preferenceinthisdecision-making problem.Iftheweightsofthethreeattributesarenotconsideredinthisdecision-makingproblem,the threeweightvaluesinEquation(11)arereducedto wj =1/n =1/3for j =1,2,3.
ByusingEquation(11)underaSVNSenvironment,theweightedsymmetrymeasurevalues between S* and Si (i =1,2,3,4)canbeobtainedas
Mw(S* , S1)=0.8933, Mw(S* , S2)=0.9987, Mw(S* , S3)=0.9811,and Mw(S* , S4)=0.9592
ByusingEquation(11)underanINSenvironment,theweightedsymmetrymeasurevalues between S* and Si (i =1,2,3,4)canbeobtainedas
Mw(S* , S1)=0.9207, Mw(S* , S2)=0.9482, Mw(S* , S3)=0.9550,and Mw(S* , S4)=0.9742
Thus,theirrankingorderis S4 > S3 > S2 > S1 accordingtotheabovemeasurevalues.
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=
Clearly,thereexistsalittledifferenceoftherankingorderswiththegivenattributeweightsand withouttheattributeweightsunderSVNSandINSenvironments,andthenthebestalternatives S2 and S4 inallrankingordersarestillidenticalinthesecases.
Bythesensitiveanalysisregardingtotheweightsofthethreeattributes,itisobviousthatthe attributeweightscanaffecttherankingordersoffouralternativestosomeextent,whichshowsome sensitivitytotheattributeweightsspecifiedbythedecisionmakerorexpert.
5.2.ComparativeExamplewithExistingRelativeMeasuresforSingle-ValuedNeutrosophicSets
Forconvenientcomparison,letusadoptaMADMproblemaboutselectingdesignschemes (alternatives)ofpunchingmachinefromliterature[21].Inthedesignschemesofpunchingmachine[21], asetoffourdesignschemes S= {S1, S2, S3, S4}needstosatisfyasetofthefiveattributes A= {A1, A2, A3, A4, A5},where A1, A2, A3, A4,and A5 arethemanufacturingcost,structurecomplexity,transmission effectiveness,reliability,andmaintainabilityrespectively.TheSVNSdecisionmatrixofevaluatingthe fouralternativesoverthefiveattributesisadoptedfromliterature[21],whichisgivenas D
(0.75,0.1,0.4)(0.80,0.1,0.3)(0.85,0.1,0.2)(0.85,0.1,0.3)(0.9,0.1,0.2) (0.70,0.1,0.5)(0.75,0.1,0.1)(0.75,0.2,0.1)(0.8,0.1,0.1)(0.8,0.2,0.3) (0.80,0.2,0.3)(0.78,0.1,0.2)(0.80,0.1,0.2)(0.8,0.2,0.2)(0.75,0.1,0.3) (0.9,0.1,0.2)(0.85,0.1,0.1)(0.9,0.1,0.2)(0.85,0.1,0.3)(0.85,0.2,0.3)
Then,theweightvectorofthefiveattributesisgivenas W =(0.25,0.2,0.25,0.15,0.15).Theideal solution/alternativein[21]is S∗ = {s∗ 1 , s∗ 2 , s∗ 3 , s∗ 4 , s∗ 5 } = {(0.9,0.1,0.2), (0.85,0.1,0.1), (0.9,0.1,0.1), (0.85,0.1,0.1), (0.9,0.1,0.2)}
Thus,wegettheweightedsymmetrymeasurevaluesbyEquation(11)as Mw(S* , S1)=0.9452, Mw(S* , S2)=0.8390, Mw(S* , S3)=0.8770,and Mw(S* , S4)=0.9803.
Allthemeasurevaluesofboththeproposedweightedsymmetrymeasure Mw(S* , Si)andthe variousmeasureslikethecosinemeasuresof Cosw(S* , Si)and Cw(S* , Si),theDicemeasureof Dw(S* , Si), theJaccardmeasureof Jw(S* , Si),theprojectionmeasureof Projws*(Si),andthebidirectionalprojection measureof BProjw(S* , Si)intheliterature[21]areshowninTable 1,wheretheaveragevalue(AV)and thestandarddeviation(SD)of Si for i =1,2,3,4arealsogiven.
Table1. Variousmeasureresultsandrankingorders.AV:averagevalue;SD:standarddeviation.
Measure S1 S2 S3 S4 AVSDRankingOrder Cosw(S* , Si) 0.97850.96850.98700.99420.98210.0096 S4 > S3 > S1 > S2 Cw(S* , Si) 0.97980.97500.98750.99290.98380.0069 S4 > S3 > S1 > S2 Dw(S* , Si) 0.97870.96960.98450.99270.98140.0084 S4 > S3 > S1 > S2 Jw(S* , Si) 0.95860.94270.96940.98570.96410.0157 S4 > S3 > S1 > S2 Projws*(Si)0.39330.36320.38060.41580.38820.0192 S4 > S1 > S3 > S2 BProjw(S* , Si) 0.98830.96360.97280.99580.98010.0126 S4 > S1 > S3 > S2 Mw(S* , Si) 0.94520.83900.87700.98030.91040.0555 S4 > S1 > S3 > S2
FromTable 1,therankingorderbasedontheproposedsymmetrymeasureisthesameasthe onesoftheprojectionandbidirectionalprojectionmeasures,butindicatesalittledifferenceofother rankingorders.However,thebestoneinallrankingordersisthesame.Itisobviousthatthe proposedsymmetrymeasureisfeasibleandeffective.Accordingtotheirstandarddeviations,theSD valueoftheproposedsymmetrymeasureis0.0555,whichisthebiggestoneamongtheSDvaluesof variousmeasures.IntheMADMprocess,however,thecosine,Dice,andJaccardmeasuresindicate
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smallerSDvalues,whichshowlowerresolution/discrimination;whiletheMADMmethodsusingthe projectionandbidirectionalprojectionmeasuresalsoimplylowresolution/discriminationduetotheir smallSDvalues.Obviously,thenewMADMmethodcanstrengthentheresolution/discrimination intheMADMprocessofthefouralternativessoastoprovideeffectivedecisioninformationfor decisionmakers.
6.Conclusions
ThispaperfirstlydefinedasymmetrymeasureofSNSs(SVNSsandINSs),andthendeveloped thenormalizedsymmetrymeasureandweightedsymmetrymeasureofSNSs(SVNSsandINSs)and theirMADMmethodwithSNSinformation(intervalandsinglevaluedneutrosophicinformation). Thentherankingofallalternativesandthebestonecanbegiventhroughtheweightedsymmetry measurebetweentheidealsolution/alternativeandeachalternative.Finally,apracticalexample demonstratedtheapplicationsofthedevelopedmethodforselectingthemanufacturingschemes (alternatives)intheflexiblemanufacturingsystemundersingle-valuedandintervalneutrosophic environments,alongwiththesensitiveanalysisregardingtotheattributeweights,andthenthe feasibilityandeffectivenessoftheproposedmethodwereindicatedbythecomparativeexamplein single-valuedneutrosophicsetting.
SincetheMADMmethodproposedinthisstudycontainsthebiggeststandarddeviations amongtheseexistingrelatedMADMmethods,thehigherresolution/discriminationgiveninthe decision-makingprocessisitsmainadvantage.However,thisstudyonlyproposesthesimplified neutrosophicsymmetrymeasureanditsMADMmethodwiththegiven(subjective)attribute weightsforthefirsttime,butitcannothandlesimplifiedneutrosophicgroupdecision-making problems.Therefore,inthefuturethisstudywillbeextendedtosimplifiedneutrosophicorsimplified neutrosophiccubicgroupdecision-makingproblemswithgiven/unknownweights.
AuthorContributions: AngyanTuproposedtheasymmetryandsymmetrymeasuresofSNSsandtheirMADM method;BingWangandJunYepresentedthedecision-makingexampleandcomparativeanalysis;wewrotethis papertogether.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
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