Time-neutrosophicSoftExpertSets andItsDecisionMakingProblem
VakkasUlu¸cay∗,MehmetS . ahinandNecatiOlgun DepartmentofMathematics GaziantepUniversity,Gaziantep27310-Turkey ∗
Correspondingauthor:vulucay27@gmail.com
Articlehistory
Received:18April2017
Receivedinrevisedform:9January2018
Accepted:15February2018
Publishedonline:1December2018
Abstract Inthispaperwepresentanewconceptcalledtime-neutrosophicsoftexpertset (T-NSESs).Wealsodefineitsbasicoperations,namelycomplement,union,intersection, AND,ORandstudysomeoftheirproperties.Wegiveexamplesfortheseconcepts. Finallywepresentanapplicationofthisconceptinadecision-makingproblem.
Keywords Softexpertset;neutrosophicsoftset;neutrosophicsoftexpertset;timeneutrosophicsoftexpertsets.
MathematicsSubjectClassification 03B52
1Introduction
Insomereallifeproblemsinexpertsystem,beliefsystem,informationfusion,wemustconsider thetruth-membershipaswellasthefalsity-membershipfor properdescriptionofanobjectin uncertain,ambiguousenvironment.IntuitionisticfuzzysetsintroducedbyAtanassov[1].After Atanassov’swork,Smarandache[2,3]introducedtheconceptofneutrosophicsetwhichisa mathematicaltoolforhandlingproblemsinvolvingimprecise,indeterminacyandinconsistent data.In1999Molodtsov[4]initiatedanovelconceptofsoft settheoryasanewmathematical toolfordealingwithuncertainties.AfterMolodtsov’swork,somedifferentoperationand applicationofsoftsetswerestudiedbyChen etal. [5]andMaji etal. [6].Later,Maji[7] proposedneutrosophicsoftsetswithoperations.Alkhazaleh etal. [8]generalizedtheconcept offuzzysoftexpertsetswhichincludethepossibilityofeachelementintheuniverseisattached withtheparameterizationoffuzzysetswhiledefiningafuzzysoftexpertset.Alkhazaleh et al. [9]introducedgeneralizedtheconceptofparameterizedinterval-valuedfuzzysoftsets,where themappinginwhichtheapproximatefunctionaredefinedfromfuzzyparametersset,and theygaveanapplicationofthisconceptindecisionmaking. IntheotherstudyAlkhazaleh andSalleh[10]introducedtheconceptofsoftexpertsetswhereausercanknowtheopinion ofallexpertsets.AlkhazalehandSalleh[11]generalizedtheconceptofasoftexpertsetto fuzzysoftexpertset,whichisamoreeffectiveanduseful.Theyalsodefineitsbasicoperations, 34:2(2018)245–260 | www.matematika.utm.my | eISSN0127-9602 |
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namelycomplement,union,intersection,ANDandORandgive anapplicationofthisconcept indecisionmakingproblem.Theyalsostudiedamappingonfuzzysoftexpertclassesand itsproperties.Sahin etal. [12]introducedtheconceptofneutrosophicsoftexpertsets.They alsodefineitsbasicoperationsandgiveanapplicationofthisconceptindecision-making problem.Ourobjectiveistointroducetheconceptofgeneralizedneutrosophicsoftexpertset. InSection1,fromintuitionisticfuzzysetstoneutrosophicsoftexpertsetsarementions.In section2,preliminariesaregiven.Insection3,theconceptofgeneralizedneutrosophicsoft expertsetanditsbasicoperations,namelycomplement,union,intersectionANDandOR.In section4,wegiveanapplicationofthisconceptinadecision-makingproblem.InSection5 giveconclusions.
2Preliminary
Inthissectionwerecallsomerelateddefinitions.
Definition1 [3] LetUbeaspaceofpoints(objects),withagenericelementin Udenotedby u.Aneutrosophicsets(N-sets)AinUischaracterizedbyatruth-membershipfunction TA,an indeterminacy-membershipfunction IA andafalsity-membershipfunction FA. TA(u),IA(u) and FA(u) arerealstandardornonstandardsubsetsof [0, 1].Itcanbewrittenas A = { u, (TA(u),IA(u),FA(u)) : u ∈ U,TA(u),IA(u),FA(u) ∈ [0, 1]} Thereisnorestrictiononthesumof TA(u),IA(u) and FA(u) so, 0 ≤ TA(u)+ IA(u)+ FA(u) ≤ 3
Definition2 [7] LetUbeaninitialuniversesetandEbeasetofparameters.Consider A ⊆ E.Let P (U ) denotesthesetofallneutrosophicsetsofU.Thecollection (F,A) istermed tobethesoftneutrosophicsetover U ,where F isamappinggivenby F : A → P (U ).
Definition3 [6] Aneutrosophicset A iscontainedinanotherneutrosophicset B i.e. A ⊆ B ifforall TA(x) ≤ TB (x),IA(x) ≤ IB (x),FA(x) ≥ FB (x).Let U beauniverse, E isasetof parameters,and X isasoftexperts(agents).Let O beasetofopinion, Z = E × X × O and A ⊆ Z.
Definition4 [12] Apair (F,A) iscalledasoftexpertsetover U, where F isamappinggiven by F : A → P (U ) where P (U ) denotesthepowersetof U.
Definition5 [12] Apair (F,A) iscalledaneutrosophicsoftexpertsetover U ,where F is mappinggivenby
F : A → P (U )
Where P (U ) denotesthepowerneutrosophicsetof U.
Definition6 [12] Let (F,A) and (G,B) betwoneutrosophicsoftexpertsetsoverthecommonuniverse U. (F,A) issaidtobeneutrosophicsoftexpertsubsetof (G,B),if A⊆B and TF (e)(x)≤TG(e)(x),IF (e)(x)≤IG(e)(x), FF (e)(x) ˜ ≥FG(e)(x), forall e ∈ A,x ∈ U .Wedenoteitby (F,A) ˜ ⊆(G,B) (F,A) issaidtobeneutrosophicsoftexpertsupersetof (G,B) if (G,B) isaneutrosophicsoft expertsubsetof (F,A).Wedenoteby (F,A)⊇(G,B).
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Definition7 [12] Equalityoftwoneutrosophicsoftexpertsets.Two(NSES), (F,A) and (G,B) overthecommonuniverse U aresaidtobeequalif (F,A) isneutrosophicsoftexpert subsetof (G,B) and (G,B) isneutrosophicsoftexpertsubsetof (F,A).Wedenoteitby (F,A)=(G,B).
Definition8 [12] Complementofaneutrosophicsoftexpertset.Thecomplementofaneutrosophicsoftexpertset (F,A) denotedby (F,A)c andisdefinedas (F,A)c =(F c , ¬A) where F c = ¬A → P (U ) ismappinggivenby F c (x)=neutrosophicsoftexpertcomplementwith TF c (x) = FF (x),IF c (x) = IF (x),FF c(x) = TF (x).
Definition9 [12] Anagree-neutrosophicsoftexpertset (F,A)1 over U isaneutrosophicsoft expertsubsetof (F,A) definedasfollow: (F,A)1 = {F1(m): m ∈ E × X ×{1}}.
Definition10 [12] Andisagree-neutrosophicsoftexpertset (F,A)0 over U isaneutrosophic softexpertsubsetof (F,A) definedasfollow: (F,A)0 = {F0(m): m ∈ E × X ×{0}}.
Definition11 [12] Let (H,A) and (G,B) betwoNSESsoverthecommonuniverse U. Then theunionof (H,A) and (G,B) isdenotedby “(H,A)∪(G,B)” andisdefinedby (H,A)∪(G,B)= (K,C),where C = A ∪ B andthetruth-membership,indeterminacy-membershipandfalsitymembershipof (K,C) areasfollows:
TK(e)(m)= TH (e)(m),ife ∈ A B
= TG(e)(m),ife ∈ B A
=max(TH (e)(m),TG(e)(m)),ife ∈ A ∩ B
IK(e)(m)= IH (e)(m),ife ∈ A B = IG(e)(m),ife ∈ B A = IH (e)(m)+ IG(e)(m) 2 ,ife ∈ A ∩ B
FK(e)(m)= FH (e)(m),ife ∈ A B = FG(e)(m),ife ∈ B A =min(TH (e)(m),TG(e)(m)),ife ∈ A ∩ B.
Definition12 [12] Let (H,A) and (G,B) betwoNSESsoverthecommonuniverse U. Then theintersectionof (H,A) and (G,B) denotedby “(H,A)˜∩(G,B)” isdefinedby (H,A)˜∩(G,B)= (K,C),where C = A ∩ B andthetruth-membership,indeterminacy-membershipandfalsitymembershipof (K,C) areasfollows:
TK(e)(m)=min(TH (e)(m),TG(e)(m))
IK(e)(m)= IH (e)(m)+ IG(e)(m) 2 FK(e)(m)=max(TH (e)(m),TG(e)(m))
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Definition13 [13] Let U beaninitialuniversalsetandlet E beasetofparameters.Let I U denotesthepowersetofallfuzzysubsetsof U ,let A ⊆ E and T beasetoftimewhere T = {t1,t2 ,t3,...,tn}. Acollectionofpairs (F,E)t forall t ∈ T iscalledatime-fuzzysoftset T FSS over U where F isamappinggivenby Ft : A → I U
3Time-neutrosophicSoftExpertSets
Inthissection,weintroducethedefinitionoftime-neutrosophicsoftexpertsetsandstudy someitsproperties.Throughoutthepaper, U isaninitialuniverse, E isasetofparameters, T beasetoftimewhere T = {t1 ,t2,t3,...,tn} , X isasetofexperts(agents),and O = {agree = 1,disagree =0} isasetofopinions.Let Z = E × X × O and A ⊆ Z.
Definition14 Apair (F,A)t iscalledatime-neutrosophicsoftexpertsetover U, where F isamappinggivenbywhere N (U ) bethesetofallneutrosophicsoftexpertsubsetsof U .Let A ⊆ Z and T beasetoftimewhere T = {t1,t2 ,t3,...,tn}.Atime-neutrosophicsoftexpertset F µ t over U .Atime-neutrosophicsoftexpertset F µ t over U isdefinedbythesetoforderedpairs
F µ t = {F (e),µ(e): e ∈ A,F (e) ∈ N (U ),µ(e) ∈ [0, 1]}
where F isamappinggivenby F : A → N (U ) and µ isafuzzysetsuchthat µ : A → I =[0, 1] Here F µ t isamappingdefinedby F µ t : A → N (U ).Foranyparameter e ∈ A,F (e) isreferred astheneutrosophicvaluesetofparameter e, i.e, F µ t (e)= { u t/(TF (e)(u),IF (e)(u),FF (e)(u) },
Where T,I,F : U → [0, 1] arethemembershipfunctionoftruth,indeterminacyandfalsityof theelement u ∈ U respectively.Forany u ∈ U and e ∈ A 0 ≤ TA(u)+ IA(u)+ FA(u) ≤ 3
Infact F µ t isaparameterizedfamilyofneutrosophicsoftexpertsetson U ,whichhasthedegree ofpossibilityoftheapproximatevaluesetwhichisprepresentedby µ(e) foreachparameter e. Sowecanwriteitasfollows:
F µ t (e)= {( ut 1 F (e)(u1) , ut 2 F (e)(u2) , ut 3 F (e)(u3) ,..., ut n F (e)(un)),µ(e)} Example1
F µ 2 (e1 ,q, 1)= {( u t2 1 0.3, 0.2, 0.5 , u t2 2 0.5, 0.6, 0.2 , u t2 3 0.8, 0.1, 0.4), 0 3}
F µ 3 (e1,r, 1)= {( u t3 1 0 8, 0 4, 0 3 , u t3 2 0 7, 0 3, 0 5 , u t3 3 0 2, 0 6, 0 5 ), 0 7}
F µ 1 (e2,p, 1)= {( u t1 1 0 7, 0 3, 0 6 , u t1 2 0 5, 0 1, 0 4 , u t1 3 0 8, 0 6, 0 3 ), 0 1}
F µ 2 (e2 ,q, 1)= {( u t2 1 0 6, 0 7, 0 1 , u t2 2 0 8, 0 4, 0 7 , u t2 3 0 5, 0 1, 0 7), 0 8}
F µ 3 (e2,r, 1)= {( u t3 1 0.5, 0.1, 0.8 , u t3 2 0.9, 0.3, 0.6 , u t3 3 0.4, 0.1, 0.7 ), 0 6}
F µ 1 (e3,p, 1)= {( u t1 1 0 6, 0 3, 0 2 , u t1 2 0 5, 0 6, 0 7 , u t1 3 0 8, 0 1, 0 4 ), 0 7}
F µ 2 (e3 ,q, 1)= {( u t2 1 0 7, 0 3, 0 4 , u t2 2 0 6, 0 2, 0 5 , u t2 3 0 7, 0 4, 0 6), 0.6}
F µ 3 (e3,r, 1)= {( u t3 1 0.7, 0.4, 0.6 , u t3 2 0.5, 0.3, 0.6 , u t3 3 0.1, 0.4, 0.2 ), 0 4}
F µ 1 (e1,p, 0)= {( u t1 1 0 4, 0 1, 0 2 , u t1 2 0 7, 0 3, 0 5 , u t1 3 0 4, 0 1, 0 6 ), 0 3}
F µ 2 (e1 ,q, 0)= {( u t2 1 0 7, 0 3, 0 5 , u t2 2 0 6, 0 2, 0 4 , u t2 3 0 4, 0 5, 0 1), 0.2}
F µ 3 (e1,r, 0)= {( u t3 1 0 6, 0 4, 0 3 , u t3 2 0 7, 0 2, 0 6 , u t3 3 0 4, 0 1, 0 3 ), 0.1}
F µ 1 (e2,p, 0)= {( u t1 1 0 5, 0 1, 0 7 , u t1 2 0 4, 0 1, 0 5 , u t1 3 0 7, 0 1, 0 4 ), 0 3}
F µ 2 (e2 ,q, 0)= {( u t2 1 0 4, 0 3, 0 6 , u t2 2 0 7, 0 2, 0 5 , u t2 3 0 8, 0 1, 0 4), 0.7}
F µ 3 (e2,r, 0)= {( u t3 1 0 3, 0 2, 0 6 , u t3 2 0 4, 0 3, 0 5 , u t3 3 0 5, 0 1, 0 4 ), 0.2}
F µ 1 (e3,p, 0)= {( u t1 1 0.4, 0.3, 0.6 , u t1 2 0.5, 0.1, 0.6 , u t1 3 0.6, 0.2, 0.5 ), 0.8}
F µ 2 (e3 ,q, 0)= {( u t2 1 0 6, 0 2, 0 7 , u t2 2 0 8, 0 1, 0 4 , u t2 3 0 5, 0 3, 0 4), 0 5}
F µ 3 (e3,r, 0)= {( u t3 1 0 5, 0 4, 0 6 , u t3 2 0 6, 0 4, 0 3 , u t3 3 0 7, 0 2, 0 1 ), 0.1}
Definition15 Fortwotime-neutrosophicsoftexpertsets {T NSESs} (F µ t ,A) and (Gη t ,B)
over U , (F µ t ,A) iscalledatime-neutrosophicsoftexpertsubsetof (Gη t ,B) if, (i) B ⊆ A, (ii) forall t ∈ T , ε ∈ B,Gη t (ε) istime-neutrosophicsoftexpertsubset F µ t (ε)
Example2 RecallExample1suchthat;
A = {(e1,p, 1), (e2 ,p, 1), (e2 ,q, 0), (e3,r, 1)},B = {(e1,p, 1), (e2 ,p, 1), (e3,r, 1)}
SinceBisneutrosophicsoftexpertsubsetof A,clearly B ⊂ A. Let(Gη t ,B)and(F µ t ,A)be definedasfollows:
(F µ t ,A)= {[(e1,p, 1), ( u t1 1 0 4,0 3,0 2 , u t1 2 0 6,0 1,0 8 , u t1 3 0 5,0 7,0 2 ), 0.5],
[(e2,p, 1), ( u t1 1 0 7,0 3,0 6 , u t1 2 0 5,0 1,0 4, u t1 3 0 8,0 6,0 3 ), 0 1],
[(e2,q, 0), ( u t2 1 0 4,0 3,0 6 , u t2 2 0 7,0 2,0 5, u t2 3 0 8,0 1,0 4 ), 0.7],
[(e3,r, 1), ( u t3 1 0 7,0 4,0 6 , u t3 2 0 5,0 3,0 6 , u t3 3 0 1,0 4,0 2 ), 0 4]}
(Gη t ,B)= {[(e1,p, 1), ( u t1 1 0 4,0 3,0 2 , u t1 2 0 6,0 1,0 8 , u t1 3 0 5,0 7,0 2 ), 0.5],
[(e2,p, 1), ( u t1 1 0 7,0 3,0 6 , u t1 2 0 5,0 1,0 4, u t1 3 0 8,0 6,0 3 ), 0 1], [(e3,r, 1), ( u t3 1 0 7,0 4,0 6 , u t3 2 0 5,0 3,0 6 , u t3 3 0 1,0 4,0 2 ), 0.4]}.
Therefore(Gη t ,B) ⊆ (F µ t ,A) Definition16 Two {T NSESs}, (F µ t ,A) and (Gη t ,B) over U, aresaidtobeequalif (F µ t ,A) isa {T NSESs} subsetof (Gη t ,B) and (Gη t ,B) isa {T NSESs} subsetof (F µ t ,A) Definition17 Agree-{T NSESs}, (F µ t ,A)1 over U isa {T NSESs} subsetof (F µ t ,A) definedas (F µ t ,A)1 = {F1(α): α ∈ E × X ×{1}} Example3 RecallExample1.Thentheagree-time-neutrosophicsoftexpertsets(F µ t ,Z)1 over U is (F µ t ,Z)1 = {[( u t1 1 0 4,0 3,0 2 , u t1 2 0 6,0 1,0 8 , u t1 3 0 5,0 7,0 2 ), 0.5], [( u t2 1 0 3,0 2,0 5 , u t2 2 0 5,0 6,0 2 , u t2 3 0 8,0 1,0 4), 0 3], [( u t3 1 0 8,0 4,0 3 , u t3 2 0 7,0 3,0 5 , u t3 3 0 2,0 6,0 5), 0.7], [( u t1 1 0 7,0 3,0 6 , u t1 2 0 5,0 1,0 4 , u t1 3 0 8,0 6,0 3), 0 1], [( u t2 1 0 6,0 7,0 1 , u t2 2 0 8,0 4,0 7 , u t2 3 0 5,0 1,0 7), 0.8], [( u t3 1 0 5,0 1,0 8 , u t3 2 0 9,0 3,0 6 , u t3 3 0 4,0 1,0 7), 0 6], [( u t1 1 0 6,0 3,0 2 , u t1 2 0 5,0 6,0 7 , u t1 3 0 8,0 1,0 4), 0.7], [( u t2 1 0 7,0 3,0 4 , u t2 2 0 6,0 2,0 5 , u t2 3 0 7,0 4,0 6), 0 6], [( u t3 1 0 7,0 4,0 6 , u t3 2 0 5,0 3,0 6 , u t3 3 0 1,0 4,0 2), 0.4]}.
Definition18 Adisagree-{T NSESs}, (F µ t ,A)0 over U isa {T NSESs} subsetof (F µ t ,A) isdefinedas (F µ t ,A)0 = {F0(α): α ∈ E × X ×{0}}.
Example4 ConsiderExample1.Thentheagree-time-neutrosophicsoft expertsets(F µ t ,Z)0 over U is (F µ t ,Z)0 = {[( u t1 1 0 4,0 1,0 2 , u t1 2 0 7,0 3,0 5 , u t1 3 0 4,0 1,0 6 ), 0 3],
[( u t2 1 0 7,0 3,0 5 , u t2 2 0 6,0 2,0 4 , u t2 3 0 4,0 5,0 1), 0 2], [( u t3 1 0 6,0 4,0 3 , u t3 2 0 7,0 2,0 6 , u t3 3 0 4,0 1,0 3), 0.1],
[( u t1 1 0.5,0.1,0.7 , u t1 2 0.4,0.1,0.5 , u t1 3 0.7,0.1,0.4), 0 3], [( u t2 1 0 4,0 3,0 6 , u t2 2 0 7,0 2,0 5 , u t2 3 0 8,0 1,0 4), 0.7], [( u t3 1 0 3,0 2,0 6 , u t3 2 0 4,0 3,0 5 , u t3 3 0 5,0 1,0 4), 0 2], [( u t1 1 0 4,0 3,0 6 , u t1 2 0 5,0 1,0 6 , u t1 3 0 6,0 2,0 5), 0 8], [( u t2 1 0 6,0 2,0 7 , u t2 2 0 8,0 1,0 4 , u t2 3 0 5,0 3,0 4), 0.5], [( u t3 1 0.5,0.4,0.6 , u t3 2 0.6,0.4,0.3 , u t3 3 0.7,0.2,0.1), 0 1]}
Definition19 Thecomplementofatime-neutrosophicsoftexpertset (F µ t ,A) isdenotedby (F µ t ,A)c forall t ∈ T andisdefinedby (F µ t ,A)c =(F µ(c) t , ¬A) where F µ(c) t : ¬A → I N (U ) is mappinggivenby F µ(c) t (α)= {TF (α)c = FF (α),IF (α)c = 1 IF (α),FF (α)c = TF (α) and µc(α)= 1 µ(α) foreachα ∈ E.}
Example5 RecallExample1.Complementofthetime-neutrosophicsoft expertset F µ t denotedby F µ(c) t isgivenasfollows:
(F µ(c) t ,Z)= {[(¬e1 ,p, 1), ( u t1 1 0.2,0.7,0.4 , u t1 2 0.8,0.9,0.6 , u t1 3 0.2,0.3,0.5 ), 0.5],
[(¬e2 ,p, 1), ( u t2 1 0 5,0 8,0 3 , u t2 2 0 2,0 4,0 5, u t2 3 0 4,0 9,0 8), 0 7],
[(¬e3 ,p, 1), ( u t3 1 0 2,0 6,0 8 , u t3 2 0 5,0 7,0 7, u t3 3 0 5,0 4,0 2), 0.3],
[(¬e1 ,q, 1), ( u t1 1 0 6,0 7,0 7, u t1 2 0 4,0 9,0 5, u t1 3 0 3,0 4,0 8), 0 9],
[(¬e2 ,q, 1), ( u t2 1 0 1,0 3,0 6, u t2 2 0 7,0 6,0 8, u t2 3 0 7,0 9,0 5), 0.2],
[(¬e3 ,q, 1), ( u t3 1 0 8,0 9,0 5, u t3 2 0 6,0 7,0 9, u t3 3 0 7,0 9,0 4), 0 4],
[(¬e1 ,r, 1), ( u t1 1 0 2,0 7,0 6 , u t1 2 0 7,0 4,0 5 , u t1 3 0 4,0 9,0 8 ), 0.3],
[(¬e2 ,r, 1), ( u t2 1 0 4,0 7,0 7 , u t2 2 0 5,0 8,0 6 , u t2 3 0 6,0 6,0 7 ), 0 4],
[(¬e3 ,r, 1), ( u t3 1 0 6,0 6,0 7 , u t3 2 0 6,0 7,0 5 , u t3 3 0 2,0 6,0 1 ), 0 6],
[(¬e1 ,p, 0), ( u t1 1 0 2,0 9,0 4 , u t1 2 0 5,0 7,0 7, u t1 3 0 6,0 9,0 4), 0.7],
[(¬e2 ,p, 0), ( u t2 1 0 5,0 7,0 7 , u t2 2 0 4,0 8,0 6, u t2 3 0 1,0 5,0 4), 0 8],
[(¬e3 ,p, 0), ( u t3 1 0.3,0.6,0.6 , u t3 2 0.6,0.8,0.7, u t3 3 0.3,0.9,0.4), 0.9],
[(¬e1 ,q, 0), ( u t1 1 0 7,0 9,0 5, u t1 2 0 5,0 9,0 4, u t1 3 0 4,0 9,0 7), 0 7],
[(¬e2 ,q, 0), ( u t2 1 0 6,0 7,0 4, u t2 2 0 5,0 8,0 7, u t2 3 0 4,0 9,0 1), 0.3],
[(¬e3 ,q, 0), ( u t3 1 0 6,0 8,0 3, u t3 2 0 5,0 7,0 4, u t3 3 0 4,0 9,0 5), 0 8],
[(¬e1 ,r, 0), ( u t1 1 0 6,0 7,0 4 , u t1 2 0 6,0 9,0 5 , u t1 3 0 5,0 8,0 6 ), 0.2],
[(¬e2 ,r, 0), ( u t2 1 0 7,0 8,0 6 , u t2 2 0 4,0 9,0 8 , u t2 3 0 4,0 7,0 5 ), 0.5],
[(¬e3 ,r, 0), ( u t3 1 0 6,0 6,0 5 , u t3 2 0 3,0 6,0 6 , u t3 3 0 1,0 8,0 7 ), 0 9]}
Proposition1 If (F µ t ,A) isatime-neutrosophicsoftexpertsetover U ,then (i)((F µ t ,A)c)c =(F µ t ,A) (ii)((F µ t ,A)1)c =(F µ t ,A)0 (iii)((F µ t ,A)0)c =(F µ t ,A)1 Proof (i)FromDefinition19wehave(F µ t ,A)c =(F µ(c) t , ¬A)where, F µ(c) t (α)= TF (α)c = FF (α),I(F (α)c = 1 IF (α),FF (α)c = TF (α) and µc(α)= 1 µ(α) ∀α ∈ E, ∀t ∈ T. Now ((F µ t ,A)c)c =((F µ(c) t )c , ¬A)where F µ(c) t (α)=[TF (α)c = FF (α),IF (α)c = 1 IF (α),FF (α)c = TF (α),µc(α)= 1 µ(α)]c =[TF (α) = FF (α)c ,IF (α) = 1 IF (α)c ,FF (α) = TF (α)c ,µ(α)= 1 µc(α)] =[TF (α) = FF (α)c ,IF (α) = 1 (1 IF (α)),FF (α) = TF (α)c ,µ(α)= 1 (1 µ(α))] =[TF (α) = FF (α)c ,IF (α) = IF (α),FF (α) = TF (α)c ,µ(α)= µ(α)] =(F µ t ,A), ∀α ∈ E, ∀t ∈ T.
Theproofofthepropositions(ii)-(iii)areobvious. ✷ Definition20 Theunionoftwoentities {T NSESs} (F µ t ,A) and (Gη t ,B) over U, denoted by “(F µ t ,A)˜∪(Gη t ,B)” isthe {T-NSESs}(F µ t ,A)˜∪(Gη t ,B)=(H Ω t ,C),where C = A ∪ B and thetruth-membership,indeterminacy-membershipandfalsity-membershipof (H Ω t ,C) areas follows:
TH Ω t (e)(m)=
IH Ω t (e)(m)=
TF µ t (e)(m), if e ∈ A B ; TGη t (e)(m), if e ∈ B A; max(TF µ t (e)(m),TGη t (e)(m)), if e ∈ A ∩ B.
IF µ t (e)(m), if e ∈ A B ; IGη t (e)(m), if e ∈ B A; min(IF µ t (e)(m),IGη t (e)(m)), if e ∈ A ∩ B
FF µ t (e)(m), if e ∈ A B ; FGη t (e)(m), if e ∈ B A; min(FF µ t (e)(m),FGη t (e)(m)), if e ∈ A ∩ B andwhere Ω(m)=max(µ(e) (m),η(e)(m)) Example6 Supposethat(F µ t ,A)and(Gη t ,B)aretwo {T NSESs} over U ,suchthat
FH Ω t (e)(m)=
(F µ t ,A)= {[(e1,p, 1), ( u t1 1 0 4,0 3,0 2 , u t1 2 0 6,0 1,0 8 , u t1 3 0 5,0 7,0 2 ), 0 3], [(e2,p, 1), ( u t1 1 0 7,0 3,0 6 , u t1 2 0 5,0 1,0 4 , u t1 3 0 8,0 6,0 3 ), 0.2], [(e2,q, 0), ( u t2 1 0 4,0 3,0 6 , u t2 2 0 7,0 2,0 5 , u t2 3 0 8,0 1,0 4 ), 0 6], [(e3,r, 1), ( u t3 1 0.7,0.4,0.6, u t3 2 0.5,0.3,0.6 , u t3 3 0.1,0.4,0.2 ), 0 5]}
(Gη t ,B)= {[(e1,p, 1), ( u t1 1 0 6,0 5,0 1 , u t1 2 0 8,0 2,0 3 , u t1 3 0 9,0 2,0 3 ), 0.1],
[(e2,p, 1), ( u t1 1 0 6,0 7,0 1 , u t1 2 0 8,0 4,0 7, u t1 3 0 5,0 1,0 7 ), 0 4],
[(e3,r, 1), ( u t3 1 0 4,0 1,0 2 , u t3 2 0 5,0 4,0 2 , u t3 3 0 3,0 6,0 4 ), 0.8]}.
VakkasUlu¸cay etal. /MATEMATIKA34:2(2018)245–260
Then(F µ t ,A)˜∪(Gη t ,B)=(H Ω t ,C)where (H Ω t ,C)= {[(e1,p, 1), ( u t1 1 0 6,0 3,0 1 , u t1 2 0 8,0 1,0 3 , u t1 3 0 9,0 2,0 2 ), 0 3],
[(e2,p, 1), ( u t1 1 0 6,0 3,0 1, u t1 2 0 8,0 2,0 5 , u t1 3 0 7,0 1,0 4 ), 0 4],
[(e2,q, 0), ( u t2 1 0 4,0 3,0 6 , u t2 2 0 7,0 2,0 5 , u t2 3 0 8,0 1,0 4 ), 0.6], [(e3,r, 1), ( u t3 1 0.8,0.1,0.2 , u t3 2 0.5,0.3,0.2 , u t3 3 0.3,0.4,0.2 ), 0 8]}
Proposition2 If (F µ t ,A), (Gη t ,B)and (H Ω t ,C) arethree {T NSESs} over U, then (i)(F µ t ,A)∪(Gη t ,B)∪(H Ω t ,C)=(F µ t ,A)∪((Gη t ,B)∪(H Ω t ,C)) (ii)(F µ t ,A)∪(F µ t ,A) ˜ ⊆(F µ t ,A).
Proof (i)Wewanttoprovethat (F µ t ,A)∪(Gη t ,B)∪(H Ω t ,C)=(F µ t ,A)∪((Gη t ,B)∪(H Ω t ,C))
byusingDefinition20.Weconsiderthecasewhen e ∈ A ∩ B asothercasestrivial.Then wehave (F µ t ,A)∪(Gη t ,B)= {(u t max(TF µ t (e)(m),TGη t (e)(m)), min(IF µ t (e)(m),IGη t (e)(m)), min(FF µ t (e)(m),FGη t (e)(m))), max(µ(e)(m),η(e)(m))u ∈ U }
Wealsoconsiderthecasewhen e ∈ H astheothercasesaretrivial.Thenwehave
(F µ t ,A)∪(Gη t ,B)∪(H Ω t ,C)
= {(u t/ max(TF µ t (e)(m),TGη t (e)(m)), min(IF µ t (e)(m),IGη t (e)(m)), min(FF µ t (e)(m),FGη t (e)(m))), max(µ(e)(m),η(e)(m)), (ut/TH Ω t (e)(m),IH Ω t (e)(m),FH Ω t (e)(m), max(µ(e) (m),η(e)(m), Ω(e)(m))u ∈ U }
= {ut/TF µ(e) t (m),IF µ(e) t (m),FF µ(e) t (m), (ut/ max(TH Ω t (e)(m),TGη t (e)(m)), min(IH Ω t (e)(m),IGη t (e)(m)), min(FH Ω t (e)(m),FGη t (e)(m))), max(µ(e)(m),η(e)(m)), max(µ(e)(m),η(e)(m), Ω(e)(m))u ∈ U } =(F µ t ,A)˜∪((Gη t ,B)˜∪(H Ω
FK δ t (e)(m)=
FF µ t (e)(m), if e ∈ A B ; FGη t (e)(m), if e ∈ B A; max(FF µ t (e)(m),FGη t (e)(m)), if e ∈ A ∩ B
andwhere δ(m)=min(µ(e)(m),η(e)(m)).
Example7 Supposethat(F µ t ,A)and(Gη t ,B)aretwo {T NSESs} over U ,suchthat
(F µ t ,A)= {[(e1,p, 1), ( u t1 1 0 4,0 3,0 2 , u t1 2 0 6,0 1,0 8 , u t1 3 0 5,0 7,0 2 ), 0.3], [(e2,q, 1), ( u t1 1 0 7,0 3,0 6 , u t1 2 0 5,0 1,0 4 , u t1 3 0 8,0 6,0 3 ), 0 2], [(e2,q, 0), ( u t2 1 0 4,0 3,0 6 , u t2 2 0 7,0 2,0 5 , u t2 3 0 8,0 1,0 4 ), 0.6].
(Gη t ,B)= {[(e1,p, 1), ( u t1 1 0 6,0 5,0 1 , u t1 2 0 8,0 2,0 3 , u t1 3 0 9,0 2,0 3 ), 0.1], [(e3,r, 1), ( u t3 1 0 4,0 1,0 2 , u t3 2 0 5,0 4,0 2 , u t3 3 0 3,0 6,0 4 ), 0.8]}.
Then(F µ t ,A)∩(Gη t ,B)=(K δ t ,C)where (K δ t ,C)= {[(e1,p, 1), ( u t1 1 0 4,0 3,0 2 , u t1 2 0 6,0 1,0 8 , u t1 3 0 5,0 2,0 3 ), 0.1]}.
Proposition3 If (F µ t ,A), (Gη t ,B)and (K δ t ,C) arethree {T NSESs} over U, then (i)(F µ t ,A)∩(Gη t ,B)∩(K δ t ,C)=(F µ t ,A)∩((Gη t ,B)∩(K δ t ,C)) (ii)(F µ t ,A)˜∩(F µ t ,A)⊆(F µ t ,A)
Proof
(i)Wewanttoprovethat(F µ t ,A)˜∩(Gη t ,B)˜∩(K δ t ,C)=(F µ t ,A)˜∩((Gη t ,B)˜∩(K δ t ,C))byusing Definition21.Weconsiderthecasewhen e ∈ A ∩ B asothercasestrivial.Thenwehave (F µ t ,A)∩(Gη t ,B)= {(ut/ min(TF µ t (e)(m),TGη t (e)(m)), max(IF µ t (e)(m),IGη t (e)(m)), max(FF µ t (e)(m),FGη t (e)(m))), min(µ(e)(m),η(e)(m))u ∈ U }. Wealsoconsiderherethecasewhen e ∈ K astheothercasesaretrivial.Thenwehave (F µ t ,A)˜∩(Gη t ,B)˜∩(K δ t ,C) = {(ut/ min(TF µ t (e)(m),TGη t (e)(m)), max(IF µ t (e)(m),IGη t (e)(m)),max(FF µ t (e)(m),FGη t (e)(m))), min(µ(e)(m),η(e)(m)), (u t/TK δ t (e)(m),IK δ t (e)(m),FK δ t (e)(m), min(µ(e)(m),η(e)(m),δ(e)(m))u ∈ U } = {u t/TF µ(e) t (m),IF µ(e) t (m),FF µ(e) t (m), (u t/max(TK δ t (e)(m),TGη t (e)(m)), max(IK δ t (e)(m),IGη t (e)(m)), max(FK δ t (e)(m),FGη t (e)(m))), min(µ(e)(m),η(e)(m)), min(µ(e)(m),η(e)(m),δ(e)(m))u ∈ U } =(F µ t ,A)∩((Gη t ,B)∩(K δ t ,C)). (ii)Theproofisstraightforward. ✷
Proposition4 If (F µ t ,A), (Gη t ,B)and (K δ t ,C) arethree {T NSESs} over U, then (i)(F µ t ,A)∪(Gη t ,B)∩(K δ t ,C)=((F µ t ,A)∩(K δ t ,C))∪((Gη t ,B)∩(K δ t ,C)). (ii)(F µ t ,A)∩(Gη t ,B)∪(K δ t ,C)=((F µ t ,A)∪(K δ t ,C))∩((Gη t ,B)∪(K δ t ,C))
VakkasUlu¸cay etal. /MATEMATIKA34:2(2018)245–260 255
Proof Theproofscanbeeasilyobtainedfromrelativedefinitions. ✷
Definition22 If (F µ t ,A) and (Gη t ,B) aretwo {T-NSESs} overU,then”(F µ t ,A) AND (Gη t ,B)” denotedby (F µ t ,A) ∧ (Gη t ,B) isdenotedby
(F µ t ,A) ∧ (Gη t ,B)=(H Ω t ,A × B) suchthat H Ω t (α,β)= F µ t (α)∩Gη t (β) andthetruth-membership,indeterminacy-membershipand falsity-membershipof (H Ω t ,A × B) areasfollows:
TH Ω t (α,β)(m)=min(TF µ t (α)(m),TGη t (β)(m)),
IH Ω t (α,β)(m)=max(IF µ t (α)(m),IGη t (β)(m)), FH Ω t (α,β)(m)=max(FF µ t (α)(m),FGη t (β)(m)), and Ω(m)=min(µ(e) (m),η(e)(m)), ∀α ∈ A, ∀β ∈ B,forallt ∈ T . Example8 Supposethat(F µ t ,A)and(Gη t ,B)aretwo {T NSESs} over U ,suchthat (F µ t ,A)= {[(e1,p, 1), ( u t1 1 0 2,0 3,0 5 , u t1 2 0 4,0 1,0 2 , u t1 3 0 6,0 3,0 7 ), 0.4], [(e3,r, 0), ( u t1 1 0 5,0 2,0 1, u t1 2 0 6,0 3,0 7 , u t1 3 0 2,0 1,0 8 ), 0 3]} (Gη t ,B)= {[(e1,p, 1), ( u t1 1 0 3,0 2,0 6 , u t1 2 0 6,0 3,0 2 , u t1 3 0 8,0 1,0 2 ), 0 5], [(e2,q, 0), ( u t1 1 0 1,0 3,0 5 , u t1 2 0 7,0 1,0 6, u t1 3 0 4,0 3,0 6 ), 0 6]}
Then(F µ t ,A) ∧ (Gη t ,B)=(H Ω t ,A × B)where (H Ω t ,A × B)= {[(e1,p, 1), (e1 ,p, 1), ( u t1 1 0 2,0 2,0 6 , u t1 2 0 4,0 1,0 2 , u t1 3 0 6,0 1,0 7 ), 0.4], [(e1,p, 1), (e2 ,q, 0), ( u t1 1 0 1,0 3,0 5 , u t1 2 0 4,0 1,0 6 , u t1 3 0 4,0 3,0 7 ), 0 4], [(e3,r, 0), (e1,p, 1), ( u t2 1 0 3,0 2,0 6 , u t2 2 0 6,0 3,0 7 , u t2 3 0 2,0 1,0 8), 0.3], [(e3,r, 0), (e2,q, 0), ( u t3 1 0 1,0 2,0 5 , u t3 2 0 6,0 1,0 7 , u t3 3 0 2,0 1,0 8), 0 3]}
Definition23 If (F µ t ,A) and (Gη t ,B) aretwo {T-NSESs} overU,then”(F µ t ,A) OR (Gη t ,B)” denotedby (F µ t ,A) ∨ (Gη t ,B) isdenotedby (F µ t ,A) ∨ (Gη t ,B)=(K δ t ,A × B)
suchthat K δ t (α,β)= F µ t (α) ∪ Gη t (β) andthetruth-membership,indeterminacy-membershipand falsity-membershipof (K δ t ,A × B) areasfollows:
TK δ t (α,β)(m)=max(TF µ t (α)(m),TGη t (β)(m)), IK δ t (α,β)(m)=min(IF µ t (α)(m),IGη t (β)(m)), FK δ t (α,β)(m)=min(FF µ t (α)(m),FGη t (β)(m)), and δ(m)=max(µ(e)(m),η(e)(m)), ∀α ∈ A, ∀β ∈ B,forallt ∈ T .
Example9 Supposethat(F µ t ,A)and(Gη t ,B)aretwo {T NSESs} over U ,suchthat
(F µ t ,A)= {[(e1,p, 1), ( u t1 1 0 2,0 3,0 5 , u t1 2 0 4,0 1,0 2 , u t1 3 0 6,0 3,0 7 ), 0.4],
[(e3,r, 0), ( u t1 1 0 5,0 2,0 1 , u t1 2 0 6,0 3,0 7 , u t1 3 0 2,0 1,0 8 ), 0 3]}
(Gη t ,B)= {[(e1,p, 1), ( u t1 1 0 3,0 2,0 6 , u t1 2 0 6,0 3,0 2 , u t1 3 0 8,0 1,0 2 ), 0.5],
[(e2,q, 0), ( u t1 1 0 1,0 3,0 5 , u t1 2 0 7,0 1,0 6, u t1 3 0 4,0 3,0 6 ), 0.6]}.
Then(F µ t ,A) ∨ (Gη t ,B)=(K δ t ,A × B)where
(K δ t ,A × B)= {[(e1,p, 1), (e1 ,p, 1), ( u t1 1 0 3,0 2,0 5 , u t1 2 0 6,0 1,0 2 , u t1 3 0 8,0 1,0 2 ), 0 5],
[(e1,p, 1), (e2 ,q, 0), ( u t1 1 0 2,0 3,0 5 , u t1 2 0 7,0 1,0 2 , u t1 3 0 6,0 3,0 6), 0.6], [(e3,r, 0), (e1,p, 1), ( u t2 1 0 5,0 2,0 1 , u t2 2 0 7,0 3,0 6 , u t2 3 0 8,0 1,0 2 ), 0 5], [(e3,r, 0), (e2,q, 0), ( u t3 1 0 5,0 2,0 1 , u t3 2 0 7,0 1,0 6 , u t3 3 0 4,0 1,0 6 ), 0.6]}.
Proposition5 If (F µ t ,A) and (Gη t ,B) aretime-neutrosophicsoftexpertsetsover U .Then (i)((F µ t ,A) ∧ (Gη t ,B))c =(F µ t ,A)c ∨ (Gη t ,B)c (ii)((F µ t ,A) ∨ (Gη t ,B))c =(F µ t ,A)c ∧ (Gη t ,B)c
Proof Theproofscanbeeasilyobtainedfromrelativedefinitions. ✷
4AnApplicationofTime-NeutrosophicSoftExpertSet
Inthissection,wepresentanapplicationoftime-neutrosophicsoftexpertsettheoryina decision-makingproblemwhichdemonstratesthatthismethodcanbesuccessfullyappliedto problemsofmanyfieldsthatcontainuncertainty.Wesuggest thefollowingalgorithmtosolve time-neutrosophicsoftexpertbaseddecisionmakingproblems.
Supposeyouwanttogetworkplaceworker.Fivealternatives areasfollows: U = {u1,u2 ,u3,u4,u5 },supposetherearefourparameters E = {e1,e2 ,e3,e4} wheretheparameters ei(i =1, 2, 3, 4)standfor“education,”“age,”“capability”and“experience”respectively and T = {t1,t2,t3} besetoftime.LetX={p,q,r} beasetofexperts.Fromthosefindings wecanfindthemostsuitablechoiceforthedecision.Afteraseriousdiscussion,theexperts constructthefollowingtime-nuetrosophicsoftexpertset (F µ t ,Z)giveninthenextpage.
InTables1and2wepresenttheagree-timeneutrosophicsoft expertsetanddisagreeneutrosophicsoftexpertset.Nowtodeterminethebestchoices,wefirstmarkthehighest numericaldegreeunderlineineachrowinagree-time-neutrosophicsoftexpertsetanddisagreetime-neutrosophicsoftexpertsetexcludingthelastcolumnwhichisthedegreeofsuchbelongingnessofanexpertagainstofparameters.Thenwecalculatethescoreofeachofsuch expertinagree-time-neutrosophicsoftexpertsetanddisagree-time-neutrosophicsoftexpert setbytakingthesumoftheproductsofthesenumericaldegreeswiththecorrespondingvalues of λ.Thenwecalculatethefinalscorebysubtractingthescoreof expertintheagree-timeneutrosophicsoftexpertsetfromthescoresofexpertindisagree-time-neutrosophicsoftexpert set.Theexpertwiththehighestscoreisthedesiredexpert.
(F µ t ,Z)= {[(e1,p, 1), ( u t1 1 0 2,0 3,0 4 , u t1 2 0 8,0 2,0 6, u t1 3 0 6,0 3,0 5 , u t1 4 0 4,0 2,0 3 , u t1 5 0 6,0 3,0 1), 0.8],
[(e1,q, 1), ( u t2 1 0 3,0 1,0 4 , u t2 2 0 2,0 1,0 5, u t2 3 0 4,0 3,0 2 , u t2 4 0 4,0 2,0 3 , u t2 5 0 7,0 2,0 5), 0 5],
[(e1,r, 1), ( u t3 1 0 3,0 5,0 1 , u t3 2 0 6,0 2,0 5 , u t3 3 0 1,0 4,0 2, u t3 4 0 5,0 2,0 3 , u t3 5 0 4,0 3,0 2 ), 0 3],
[(e2,p, 1), ( u t1 1 0 6,0 2,0 3 , u t1 2 0 4,0 2,0 5 , u t1 3 0 3,0 4,0 1 , u t1 4 0 7,0 3,0 6 , u t1 5 0 5,0 2,0 4 ), 0.6],
[(e2,q, 1), ( u t2 1 0.1,0.3,0.6 , u t2 2 0.7,0.3,0.1, u t2 3 0.6,0.2,0.5 , u t2 4 0.3,0.1,0.6 , u t2 5 0.4,0.3,0.2), 0 5],
[(e2,r, 1), ( u t3 1 0 6,0 3,0 5 , u t3 2 0 7,0 2,0 6 , u t3 3 0 5,0 3,0 4, u t3 4 0 2,0 1,0 3 , u t3 5 0 6,0 2,0 5 ), 0.4],
[(e3,p, 1), ( u t1 1 0 2,0 4,0 6 , u t1 2 0 7,0 4,0 2 , u t1 3 0 4,0 1,0 2 , u t1 4 0 8,0 4,0 3 , u t1 5 0 7,0 3,0 4 ), 0 1],
[(e3,q, 1), ( u t2 1 0 4,0 2,0 6 , u t2 2 0 5,0 3,0 6, u t2 3 0 6,0 2,0 7 , u t2 4 0 8,0 2,0 4 , u t2 5 0 6,0 2,0 3), 0.7],
[(e3,r, 1), ( u t3 1 0 3,0 6,0 5 , u t3 2 0 6,0 2,0 5 , u t3 3 0 2,0 1,0 4, u t3 4 0 5,0 3,0 2 , u t3 5 0 4,0 1,0 5 ), 0 4],
[(e4,p, 1), ( u t1 1 0 2,0 3,0 6 , u t1 2 0 7,0 1,0 5 , u t1 3 0 4,0 2,0 8 , u t1 4 0 9,0 2,0 4 , u t1 5 0 3,0 4,0 6 ), 0.5],
[(e4,q, 1), ( u t2 1 0 5,0 2,0 3 , u t2 2 0 2,0 3,0 4, u t2 3 0 4,0 1,0 5 , u t2 4 0 6,0 3,0 2 , u t2 5 0 7,0 3,0 4), 0.5],
[(e4,r, 1), ( u t3 1 0 5,0 2,0 1 , u t3 2 0 6,0 3,0 5 , u t3 3 0 2,0 5,0 3, u t3 4 0 5,0 1,0 4 , u t3 5 0 3,0 2,0 5 ), 0 4],
[(e1,p, 0), ( u t1 1 0 2,0 3,0 4 , u t1 2 0 5,0 3,0 1 , u t1 3 0 6,0 3,0 4 , u t1 4 0 6,0 2,0 4 , u t1 5 0 7,0 5,0 6 ), 0.8],
[(e1,q, 0), ( u t2 1 0 5,0 1,0 7 , u t2 2 0 4,0 2,0 3, u t2 3 0 8,0 5,0 4 , u t2 4 0 7,0 3,0 6 , u t2 5 0 5,0 3,0 4), 0 6],
[(e1,r, 0), ( u t3 1 0 3,0 1,0 6 , u t3 2 0 6,0 3,0 7 , u t3 3 0 3,0 2,0 4, u t3 4 0 8,0 1,0 4 , u t3 5 0 6,0 4,0 5 ), 0.7],
[(e2,p, 0), ( u t1 1 0 7,0 3,0 5 , u t1 2 0 6,0 2,0 4 , u t1 3 0 4,0 3,0 5 , u t1 4 0 3,0 2,0 5 , u t1 5 0 4,0 3,0 5 ), 0 9],
[(e2,q, 0), ( u t2 1 0 6,0 2,0 4 , u t2 2 0 5,0 3,0 7, u t2 3 0 8,0 1,0 3 , u t2 4 0 2,0 3,0 6 , u t2 5 0 6,0 2,0 4), 0.3],
[(e2,r, 0), ( u t3 1 0 6,0 3,0 4 , u t3 2 0 5,0 2,0 4 , u t3 3 0 7,0 4,0 5, u t3 4 0 5,0 2,0 4 , u t3 5 0 4,0 3,0 5 ), 0 1],
[(e3,p, 0), ( u t1 1 0.6,0.2,0.4 , u t1 2 0.6,0.1,0.5 , u t1 3 0.5,0.4,0.6 , u t1 4 0.8,0.3,0.6 , u t1 5 0.7,0.2,0.4 ), 0 4],
[(e3,q, 0), ( u t2 1 0 7,0 1,0 6 , u t2 2 0 4,0 5,0 8, u t2 3 0 4,0 3,0 5 , u t2 4 0 6,0 2,0 5 , u t2 5 0 4,0 3,0 1), 0.2],
[(e3,r, 0), ( u t3 1 0 2,0 3,0 6 , u t3 2 0 7,0 4,0 5 , u t3 3 0 4,0 2,0 8, u t3 4 0 9,0 1,0 4 , u t3 5 0 6,0 3,0 2 ), 0 3],
[(e4,p, 0), ( u t1 1 0 4,0 2,0 6 , u t1 2 0 5,0 2,0 6 , u t1 3 0 9,0 5,0 1 , u t1 4 0 3,0 2,0 6 , u t1 5 0 4,0 3,0 5 ), 0 6],
[(e4,q, 0), ( u t2 1 0 3,0 2,0 1 , u t2 2 0 6,0 1,0 5, u t2 3 0 6,0 2,0 5 , u t2 4 0 8,0 3,0 2 , u t2 5 0 2,0 3,0 4), 0.5],
[(e4,r, 0), ( u t3 1 0.6,0.2,0.5 , u t3 2 0.7,0.1,0.6 , u t3 3 0.5,0.3,0.1, u t3 4 0.3,0.2,0.6 , u t3 5 0.4,0.3,0.1 ), 0 1]}
Thefollowingalgorithmmaybefollowedbyyouwanttogetworkplaceworker. Nowcalculatethescoreof ui byusingthedatainTable3:
Score(u1 )=0
Score(u2 )=(0.8 ∗ 0.7)+(0.7 ∗ 0.4)+(0.6 ∗ 0.2)+(0.7 ∗ 0.5)+(0.6 ∗ 0.5)+(0.6 ∗ 0.3)=1.79
Score(u3 )=0
Score(u4 )=(0 7 ∗ 0 8)+(0 8 ∗ 0 3)+(0 9 ∗ 0 6)+(0 8 ∗ 0 4)=1 66
Score(u5 )=(0.7 ∗ 0.6)+(0.7 ∗ 0.6)=0.84
Table1:Agree-time-neutrosophicSoftExpertSet
U u1 u2 u3 u4 u5 λ (e1,p) 0.2,0.3,0.4 0.8,0.2,0.6 0.6,0.3,0.5 0.4,0.2,0.3 0.6,0.3,0.1 0.8 (e2,p) 0.3,0.1,0.4 0.2,0.1,0.5 0.4,0.3,0.2 0.4,0.3,0.2 0.7,0.2,0.5 0.5 (e3,p) 0.3,0.5,0.1 0.6,0.2,0.5 0.1,0.4,0.2 0.5,0.2,0.3 0.4,0.3,0.2 0.3 (e4,p) 0.6,0.2,0.3 0.4,0.2,0.5 0.3,0.4,0.1 0.7,0.3,0.6 0.5,0.2,0.4 0.6 (e1,q) 0.1,0.3,0.6 0.7,0.3,0.1 0.6,0.2,0.5 0.3,0.1,0.6 0.4,0.3,0.2 0.5 (e2,q) 0.6,0.3,0.5 0.7,0.2,0.6 0.5,0.3,0.4 0.2,0.1,0.3 0.6,0.2,0.5 0.4 (e3,q) 0.2,0.4,0.6 0.7,0.4,0.2 0.4,0.1,0.2 0.8,0.4,0.3 0.7,0.3,0.4 0.1 (e4,q) 0.4,0.2,0.6 0.5,0.3,0.6 0.6,0.2,0.7 0.8,0.2,0.4 0.6,0.2,0.3 0.7 (e1,r) 0.3,0.6,0.5 0.6,0.2,0.5 0.2,0.1,0.4 0.5,0.3,0.2 0.4,0.1,0.5 0.4 (e2,r) 0.2,0.3,0.6 0.2,0.3,0.4 0.4,0.2,0.8 0.2,0.5,0.3 0.3,0.4,0.6 0.5 (e3,r) 0.5,0.2,0.3 0.6,0.3,0.5 0.4,0.1,0.5 0.6,0.3,0.2 0.7,0.3,0.4 0.5 (e4,r) 0.5,0.2,0.1 0.5,0.3,0.1 0.2,0.5,0.3 0.5,0.1,0.4 0.3,0.2,0.5 0.1
Table2:Disagree-time-neutrosophicSoftExpertSet
U u1 u2 u3 u4 u5 λ (e1,p) 0.2,0.3,0.4 0.8,0.2,0.6 0.6,0.3,0.5 0.4,0.2,0.3 0.6,0.3,0.1 0.8 (e2,p) 0.3,0.1,0.4 0.2,0.1,0.5 0.4,0.3,0.2 0.4,0.3,0.2 0.7,0.2,0.5 0.5 (e3,p) 0.3,0.5,0.1 0.6,0.2,0.5 0.1,0.4,0.2 0.5,0.2,0.3 0.4,0.3,0.2 0.3 (e4,p) 0.6,0.2,0.3 0.4,0.2,0.5 0.3,0.4,0.1 0.7,0.3,0.6 0.5,0.2,0.4 0.6 (e1,q) 0.1,0.3,0.6 0.7,0.3,0.1 0.6,0.2,0.5 0.3,0.1,0.6 0.4,0.3,0.2 0.5 (e2,q) 0.6,0.3,0.5 0.7,0.2,0.6 0.5,0.3,0.4 0.2,0.1,0.3 0.6,0.2,0.5 0.4 (e3,q) 0.2,0.4,0.6 0.7,0.4,0.2 0.4,0.1,0.2 0.8,0.4,0.3 0.7,0.3,0.4 0.1 (e4,q) 0.4,0.2,0.6 0.5,0.3,0.6 0.6,0.2,0.7 0.8,0.2,0.4 0.6,0.2,0.3 0.7 (e1,r) 0.3,0.6,0.5 0.6,0.2,0.5 0.2,0.1,0.4 0.5,0.3,0.2 0.4,0.1,0.5 0.4 (e2,r) 0.2,0.3,0.6 0.2,0.3,0.4 0.4,0.2,0.8 0.2,0.5,0.3 0.3,0.4,0.6 0.5 (e3,r) 0.5,0.2,0.3 0.6,0.3,0.5 0.4,0.1,0.5 0.6,0.3,0.2 0.7,0.3,0.4 0.5 (e4,r) 0.5,0.2,0.1 0.5,0.3,0.1 0.2,0.5,0.3 0.5,0.1,0.4 0.3,0.2,0.5 0.1
Nowcalculatethescoreof ui byusingthedatainTable4:
Score(u1 )=(0.7 ∗ 0.8)+(0.7 ∗ 0.3)=0.77
Score(u2 )=(0 7 ∗ 0 1)=0 07 Score(u3 )=(0.9 ∗ 0.6)+(0.8 ∗ 0.7)+(0.8 ∗ 0.4)+(0.7 ∗ 0.2)=1.56 Score(u4 )=(0.8 ∗ 0.5)+(0.8 ∗ 0.5)+(0.8 ∗ 0.6)+(0.9 ∗ 0.3)=1.55
Score(u5 )=(0 7 ∗ 0 9)=0 63
Thefinalscoreof ui isasfollows:
Score(u1 )=0 0 77= 0 77, Score(u2 )=1.79 0.07=1.72, Score(u3 )=0 1 56= 1 56, Score(u4 )=1 66 1 55=0 11, Score(u5 )=0 84 0 63=0 21
Table3:DegreeTableofAgree-time-neutrosophicSoftExpertSet
R ui Highestnumericaldegree λ
(e1,p) u2 0.8 0.8 (e2,p) u4 0.7 0.5 (e3,p) u4 0.8 0.3 (e4,p) u4 0.9 0.6 (e1,q) u5 0.7 0.5 (e2,q) u2 0.7 0.4 (e3,q) u4 0.8 0.1 (e4,q) u5 0.7 0.7 (e1,r) u2 0.6 0.4 (e2,r) u2 0.7 0.5 (e3,r) u2 0.6 0.5 (e4,r) u2 0.6 0.1
Table4:DegreeTableofDisagree-time-neutrosophicSoftExpertSet
R ui Highestnumericaldegree λ
(e1,p) u5 0.7 0.8 (e2,p) u1 0.7 0.5 (e3,p) u4 0.8 0.3 (e4,p) u3 0.9 0.6 (e1,q) u3 0.8 0.5 (e2,q) u3 0.8 0.4 (e3,q) u1 0.7 0.1 (e4,q) u4 0.8 0.7 (e1,r) u4 0.8 0.4 (e2,r) u3 0.7 0.5 (e3,r) u4 0.9 0.5 (e4,r) u2 0.7 0.1
Clearly,themaximumscoreisthescore1.72,shownintheaboveforthe u2 .Hencethebest decisionforexpertsaretoselect u2,followedby u5 .
5Conclusion
Theaimofourwork,”time-neutrosophicsoftexpertset”aid inallareas,willgivethebest decisionwithouttheneedforexpertsinthefield.Inthisstudy,weaddtheindeterminacy parametersanddifferentlypreviousstudieswehaveshownin thisstudy,couldincreasethe numberofspecialists.Wehaveintroducedtheconceptoftime-neutrosophicsoftexpertset whichismoreeffectiveandusefulandstudiedsomeofitsproperties.Alsothebasicoperations onneutrosophicsoftexpertsetnamelycomplement,union,intersection,ANDandORhave beendefined.Finally,wepresentanapplicationofT-NSESsinadecisionmakingproblem.
VakkasUlu¸cay etal. /MATEMATIKA34:2(2018)245–260 260
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