Bipolarneutrosophicsoftsets andapplicationsindecisionmaking
MumtazAlia ,LeHoangSonb,∗ ,IrfanDelic andNguyenDangTiend
a UniversityofSouthernQueensland,Australia
b VNUUniversityofScience,VietnamNationalUniversity,Vietnam
c MuallimRıfatFacultyofEducation,7AralıkUniversity,Kilis,Turkey
d People’sPoliceUniversityofTechnologyandLogistics,BacNinh,Vietnam
Abstract.Neutrosophicset,proposedbySmarandacheconsidersatruthmembershipfunction,anindeterminacymembership functionandafalsitymembershipfunction.Softset,proposedbyMolodtsovisamathematicalframeworkwhichhasthe abilityofindependencyofparameterizationsinadequacy,syndromeoffuzzyset,roughset,probability.Thoseconceptshave beenutilizedsuccessfullytomodeluncertaintyinseveralareasofapplicationsuchascontrol,reasoning,gametheory,pattern recognition,andcomputervision.Nonetheless,therearemanyproblemsinreal-worldapplicationscontainingindeterminate andinconsistentinformationthatcannotbeeffectivelyhandledbytheneutrosophicsetandsoftset.Inthispaper,wepropose thenotationofbipolarneutrosophicsoftsetsthatcombinessoftsetsandbipolarneutrosophicsets.Somealgebraicoperations ofthebipolarneutrosophicsetsuchasthecomplement,union,intersectionareexamined.Wethenproposeanaggregation bipolarneutrosophicsoftoperatorofabipolarneutrosophicsoftsetanddevelopadecisionmakingalgorithmbasedonbipolar neutrosophicsoftsets.Numericalexamplesaregiventoshowthefeasibilityandeffectivenessofthedevelopedapproach.
Keywords:Algebraicoperations,bipolarneutrosophicsoftsets,decisionmaking,neutrosophicsets,softsets
1.Introduction
Tohandleuncertainty,Zadeh[34]proposedfuzzy setwhichischaracterizedbyamembershipdegree withrangeintheunitinterval [0, 1].Fromseveral decades,thisnovelconceptisutilizedsuccessfullyto modeluncertaintyinseveralareasofapplicationsuch ascontrol,reasoning,gametheory,patternrecognition,andcomputervision.Fuzzysets,especially, becomeanimportantareafortheresearchinmedicaldiagnosis,engineering,socialsciencesetc.Since infuzzyset,thedegreeofassociationofanelement issinglevalueintheunitinterval [0, 1],itmaynot beadequatethatthenon-associationofanelement
∗ Correspondingauthor.LeHoangSon,334NguyenTrai, ThanhXuan,Hanoi,Vietnam.Tel.:+84904171284;E-mail: sonlh@vnu.edu.vn.
isequalto1minustheassociationdegreeduetothe existenceofhesitationdegree.ThusAtanassov[4] coinedintuitionisticfuzzysetin1986toovercome thisissuebyincorporatingthehesitationdegreesocalledhesitationmarginwhichisdefineby1minus thesumofassociationdegreeandnon-association degree.Consequentlytheintuitionisticfuzzysetcapturedanassociationdegreeaswellasnon-association degreewhichbecamethegeneralizationoffuzzyset. Tojudgethehumandecisionmakingabilitybased onpositiveandnegativeeffects,BoscandPivert [5]saidthatbipolarityprovidesthepropensityof thehumanmindtoreasonandmakedecisionsthat dependsonpositiveandnegativeeffects.Theyargued thatbothpositiveinformationdepictswhatispossible,satisfactory,permitted,desired,orconsidered asbeingacceptablewhilethenegativestatements expresswhatisimpossible,restricted,rejected,or
forbiddenandnegativityofchoicescorrespondto constraints,sincetheyparticularizethatwhatkind ofvaluesorobjectshavetoberejected(i.e.,those thatdonotsatisfytheconstraintsortotallyopposite), whereaspositivepreferencescorrespondtowishes, astheyspecifywhichobjectsaremoredesirable thanothers(i.e.,satisfyuserwishes)withoutrejectingthosethatdonotmeetthewishes.Toutilize thisidea,Lee[24,25]definedbipolarfuzzysets whichgeneralizestheconceptfuzzysets.Kangand Kang[23]appliedthebipolarfuzzysettheoryto sub-semigroupswithoperatorsinsemigroups.
Smarandache[32]in1998,introducedneutrosophicsetandneutrosophiclogicbyconsideringa truthmembershipfunction,anindeterminacymembershipfunctionandafalsitymembershipfunction. Neutrosophicsethastheabilitytogeneralizeclassical sets,fuzzysets,intuitionisticfuzzysets.Smarandache[32]andWangetal.[33]furtherdeveloped singlevaluedneutrosophicsetsinordertousethem inaneasywayinscientificandengineeringfields. Then,Delietal.[16]developedbipolarneutrosophic setsandstudytheirapplicationindecicionmaking.Alietal.[2]proposedneutrosophiccubicset withapplicationinpatternrecognition.Broumietal. [36,37]introducedBipolarSingleValuedNeutrosophicGraphtheoryanditsShortestPathproblem. Recently,AliandSmarandache[1]definecomplex neutrosophicsettorepresenttheuncertain.Some moreliteratureonneutrosophicsetandapplications canbefoundin[7,8,17–20,38–58].
Molodtsov[29]proposedsoftsettohandle uncertaintyinaparameterizedway.Softsetisa mathematicalframeworkwhichhastheabilityof independencyofparameterizationsinadequacy,syndromeoffuzzyset,roughset,probabilityetc..Soft setappliedsuccessfullyinseveralfieldstotackle theissuesandproblemssuchassmoothnessof functions,gametheory,operationreaserch,Riemann integration,Perronintegration,andprobability.Also, KaraaslanandKaratas[22]Aslametal.[3]studied bipolarsoftsetsandbipolarfuzzysoftsets,respectively.Ahugeamountofresearchworkonsoftset theorycanbeseenin[9–12,14,21,26,30].Also, someauthorsstudiedconceptofneutrosophicsoft setin[6,13,15,27,28].
Thispaperisdedicatedtoproposebipolarneutrosophicsetwhichisahybridstructureofsoftset andbipolarneutrosophicset.Firstly,weintroducethe bipolarneutrosophicsoftsetanddiscusssomebasic propertieswithillustrativeexamplesadoptingfrom KangandKang[23].Then,westudysomealgebraic
operationsofthebipolarneutrosophicsetsuchasthe complement,union,intersectionetc.Wethenpropose anaggregationbipolarneutrosophicsoftoperatorof abipolarneutrosophicsoftsetanddevelopadecision makingalgorithmbasedonbipolarneutrosophicsoft sets.Numericalexamplesaregiventoshowthefeasibilityandeffectivenessofthedevelopedapproach.
Theorganizationofthispaperisasfollows.InSection1,wepresentedtherelevantliteraturereview. Section2isdedicatedtothefundamentalconcepts. InSection3,bipolarneutrosophicsethasbeenpresented.Wealsostudiedcorepropertiesinthesame section.Section4isaboutaggregationbipolarneutrosophicsoftoperatorofabipolarneutrosophicsoftset. Inthissectiontheproposedalgorithmbasedonaggregationbipolarneutrosophicsoftoperatorofabipolar neutrosophicsoftsetispresentedwithanumerical example.ConclusionisgiveninSection5.
2.Preliminary
Inthissection,wegivethebasicdefinitionsand resultsofneutrosophicsettheory[32],softsettheory[29],neutrosophicsoftsettheory[13],bipolar fuzzyset[24],bipolarfuzzysoftset[3]andbipolar neutrosophicset[16]thatareusefulforsubsequent discussions.
Definition1. [32]Let U beauniverse.Aneutrosophicsets(NS) K in U ischaracterizedbya truth-membershipfunction TK ,anindeterminacymembershipfunction IK andafalsity-membership function FK . TK (x); IK (x)and FK (x)arerealstandardornon-standardelementsof]0 ,1+ [. Itcanbe writtenas:
K ={<x, (TK (x),IK (x),FK (x)) >: x ∈ U, TK (x),IK (x),FK (x) ∈] 0, 1[+ }
Thereisnorestrictiononthesumof TK (x), IK (x) and FK (x),so0 ≤ TK (x) + IK (x) + FK (x) ≤ 3+ Definition2. [33]Let E beauniverse.Asinglevalued neutrosophicsets(SVNS)A,whichcanbeusedin realscientificandengineeringapplications,in E is characterizedbyatruth-membershipfunction TA ,a indeterminacy-membershipfunction IA andafalsitymembershipfunction FA . TA (x), IA (x)and FA (x)are realstandardelementsof[0, 1].Itcanbewrittenas
A ={<x, (TA (x),IA (x),FA (x)) >: x ∈ E, TA (x),IA (x),FA (x) ∈ [0, 1]}.
Definition3. [29]Let U beauniverse, E beaset ofparametersthatdescribetheelementsof U ,and A ⊆ E .Then,asoftset FA over U isasetdefinedby asetvaluedfunction fA representingamapping
fA : E → P (U )s.t fA (x) =∅ if x ∈ E A (1)
where fA iscalledapproximatefunctionofthesoft set FA .Inotherwords,thesoftsetisaparameterized familyofsubsetsoftheset U ,andthereforeitcanbe writtenasetoforderedpairs
FA ={(x,fA (x)): x ∈ E,fA (x) =∅ if x ∈ E A}
Definition4. [13]Let U beauniverse, N (U )bethe setofallneutrosophicsetson U , E beasetofparametersthataredescribingtheelementsof U .Then,a neutrosophicsoftset N over U isasetdefinedbya setvaluedfunction fN representingamapping
fN : E → N (U )
where fN iscalledanapproximatefunctionofthe neutrosophicsoftset N .For x ∈ E ,theset fN (x)is called x-approximationoftheneutrosophicsoftset N whichmaybearbitrary,someofthemmaybe emptyandsomemayhaveanonemptyintersection. Inotherwords,theneutrosophicsoftsetisaparameterizedfamilyofsomeelementsoftheset N (U ),and thereforeitcanbewrittenasetoforderedpairs, N ={(x, {<u,TfN (x) (u),IfN (x) (u), FfN (x) (u) >: x ∈ U } : x ∈ E } where TfN (x) (u),IfN (x) (u),FfN (x) (u) ∈ [0, 1]
Definition5. [13]Let N1 and N2 betwoneutrosophic softsetsoverneutrosophicsoftuniverses(U,A)and (U,B ),respectively.
1. N1 issaidtobeneutrosophicsoftsubset of N2 if A ⊆ B and TfN1 (x) (u) ≤ TfN2 (x) (u), IfN1 (x) (u) ≤ IfN2 (x) (u), FfN1 (x) (u) ≥ FfN2 (x) (u), ∀x ∈ A, u ∈ U .
2. N1 and N2 aresaidtobeequalif N1 neutrosophicsoftsubsetof N2 and N2 neutrosophic softsubsetof N2 Definition6. [13]Let N1 and N2 betwoneutrosophic softsets.Then,
1.Thecomplementofaneutrosophicsoftset N1 denotedby N c 1 andisdefinedby
N1 c ={(x, {<u,FfN1 (x) (u), 1 IfN1 (x) (u), TfN1 (x) (u) >: x ∈ U } : x ∈ E }
2.Theunionof N1 and N2 isdenotedby N3 = N1 ˜ ∪N2 andisdefinedby
N3 ={(x, {<u,TfN3 (x) (u),IfN3 (x) (u), FfN3 (x) (u) >: x ∈ U } : x ∈ E } where
TfN3 (x) (u) = max(TfN1 (x) (u),TfN2 (x) (u)), IfN3 (x) (u) = min(IfN1 (x) (u),IfN2 (x) (u)), FfN3 (x) (u) = min(FfN1 (x) (u),FfN2 (x) (u))
3.Theintersectionof N1 and N2 isdenotedby N4 = N1 ∩N2 andisdefinedby
N4 ={(x, {<u,TfN4 (x) (u),IfN4 (x) (u), FfN4 (x) (u) >: x ∈ U } : x ∈ E }
where
TfN4 (x) (u) = min(TfN1 (x) (u),TfN2 (x) (u)), IfN4 (x) (u) = max(IfN1 (x) (u),IfN2 (x) (u)), FfN4 (x) (u) = max(FfN1 (x) (u),FfN2 (x) (u)).
Definition7. [24]Let U beauniverse.Abipolar fuzzyset in U isdefinedas;
={(u,T + (u),T (u)): u ∈ U }
where T + → [0, 1]and T → [ 1, 0].Thepositivemembershipdegree T + (u),denotesthetruth membershipcorrespondingtoabipolarfuzzyset andthenegativemembershipdegree T (u)denotes thetruthmembershipofanelement u ∈ U tosome implicitcounter-propertycorrespondingtoabipolarfuzzyset
Definition8. [3]Let U beauniverseand E beaset ofparametersthataredescribingtheelementsof U . Abipolarfuzzysoftset in U isdefinedas; ={(e, {(u,T + (u),T (u)): u ∈ U }): e ∈ E }
where T + → [0, 1]and T → [ 1, 0].Thepositive membershipdegree T + (u),denotesthetruthmembershipcorrespondingtoabipolarfuzzysoftset andthenegativemembershipdegree T (u)denotes thetruthmembershipofanelement u ∈ U tosome implicitcounter-propertycorrespondingtoabipolar
fuzzysoftset .
Definition9. [16]Let U beauniverse.Abipolar neutrosophicset A in U isdefinedas;
A ={(u,T + (u),I + (u),F + (u), T (u),I (u),F (u)): u ∈ U }
where T + ,I + ,F + → [0, 1]and T ,I ,F → [ 1, 0].Thepositivemembershipdegree T + (u), I + (u),F + (u),denotesthetruthmembership,indeterminatemembershipandfalsemembershipof anelementcorrespondingtoabipolarneutrosophicset A andthenegativemembershipdegree T (u),I (u),F (u)denotesthetruthmembership, indeterminatemembershipandfalsemembershipof anelement u ∈ U tosomeimplicitcounter-property correspondingtoabipolarneutrosophicset A
3.Bipolarneutrosophicsoftsets
Inthissection,weproposetheconceptofneutrosophicsoftsetsandtheiroperations.
Definition10. Let U beauniverseand E beasetof parametersthataredescribingtheelementsof U .A bipolarneutrosophicsoftset B in U isdefinedas; B ={(e, {(u,T + (u),I + (u),F + (u),T (u), I (u),F (u)): u ∈ U }): e ∈ E }
where T + ,I + ,F + → [0, 1]and T ,I ,F → [ 1, 0].Thepositivemembershipdegree T + (u), I + (u),F + (u),denotesthetruthmembership,indeterminatemembershipandfalsemembershipofan elementcorrespondingtoabipolarneutrosophic softset B andthenegativemembershipdegree T (u),I (u),F (u)denotesthetruthmembership, indeterminatemembershipandfalsemembershipof anelement u ∈ U tosomeimplicitcounter-property correspondingtoabipolarneutrosophicsoftset B.
Example1. Let U ={u1 ,u2 ,u3 }, E ={e1 ,e2 } Then,bipolarneutrosophicsoftset B1 and B2 over U isgivenas,respectively;
B1 ={(e1 , {(u1 , 0 5, 0 8, 0 1, 0 5, 0 7, 0 2),
(u2 , 0.6, 0.8, 0.7, 0.5, 0.7, 0.2),
(u3 , 0.6, 0.8, 0.1, 0.5, 0.8, 0.8)}),
(e2 , {(u1 , 0.8, 0.8, 0.7, 0.5, 0.7, 0.2), (u2 , 0.4, 0.8, 0.7, 0.5, 0.7, 0.2),
(u3 , 0.7, 0.8, 0.1, 0.4, 0.7, 0.4)}) and
B2 ={(e1 , {(u1 , 0 4, 0 8, 0 5, 0 6, 0 7, 0 2), (u2 , 0.3, 0.6, 0.7, 0.3, 0.7, 0.2), (u3 , 0.6, 0.2, 0.6, 0.5, 0.5, 0.3)}),
(e2 , {(u1 , 0.1, 0.8, 0.7, 0.2, 0.7, 0.2), (u2 , 0 1, 0 8, 0 7, 0 5, 0 5, 0 5), (u3 , 0 7, 0 6, 0 1, 0 4, 0 7, 0 3)})
Definition11. Anemptybipolarneutrosophicsoft set B∅ in U isdefinedas; B∅ ={(e, {(u, 0, 0, 1, 1, 0, 0)): u ∈ U }): e ∈ E }
Definition12. Anabsolutebipolarneutrosophicsoft set BU in U isdefinedas; BU ={(e, {(u, 1, 1, 0, 0, 1, 1)): u ∈ U }): e ∈ E }
Itisnotedthattheemptyandabsoluteneutrosophic softsetsformtheunittotheproposedsystem.
Example2. Let U ={u1 ,u2 ,u3 }, E ={e1 ,e2 ,e3 } Then,
1.Emptybipolarneutrosophicsoftset B∅ in U is givenas;
B∅ ={(e1 , {(u1 , 0, 0, 1, 1, 0, 0), (u2 , 0, 0, 1, 1, 0, 0), (u3 , 0, 0, 1, 1, 0, 0)}), (e2 , {(u1 , 0, 0, 1, 1, 0, 0), (u2 , 0, 0, 1, 1, 0, 0), (u3 , 0, 0, 1, 1, 0, 0)}), (e3 , {(u1 , 0, 0, 1, 1, 0, 0), (u2 , 0, 0, 1, 1, 0, 0), (u3 , 0, 0, 1, 1, 0, 0)})}
2.Absolutebipolarneutrosophicsoftset BU in U isgivenas;
BU ={(e1 , {(u1 , 1, 1, 0, 0, 1, 1), (u2 , 1, 1, 0, 0, 1, 1), (u3 , 1, 1, 0, 0, 1, 1)}), (e2 , {(u1 , 1, 1, 0, 0, 1, 1), (u2 , 1, 1, 0, 0, 1, 1),
(u3 , 1, 1, 0, 0, 1, 1)}),
(e3 , {(u1 , 1, 1, 0, 0, 1, 1),
(u2 , 1, 1, 0, 0, 1, 1),
(u3 , 1, 1, 0, 0, 1, 1)})}
Definition13. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2betwobipolarneutrosophicsoftsetsover U Then, B1 isbipolarneutrosophicsoftsubsetof B2 , isdenotedby B1 B2 ,if T + 1 (u) ≤ T + 2 (u), I + 1 (u) ≥ I + 2 (u), F + 1 (u) ≥ F + 2 (u), T1 (u) ≥ T2 (u), I1 (u) ≤ I2 (u)and F1 (u) ≤ F2 (u)forall(e,u) ∈ E × U .
Example3. Let U ={u1 ,u2 }, E ={e1 ,e2 }.If B1 ={(e1 , {(u1 , 0.7, 0.8, 0.2, 0.5, 0.9, 0.3), (u2 , 0.6, 0.8, 0.7, 0.5, 0.7, 0.2)}), (e2 , {(u1 , 0.8, 0.8, 0.7, 0.5, 0.7, 0.2), (u2 , 0 4, 0 8, 0 7, 0 5, 0 7, 0 2)})} and B2 ={(e1 , {(u1 , 0 8, 0 1, 0 2, 0 6, 0 8, 0 3), (u2 , 0 9, 0 2, 0 3, 0 9, 0 7, 0 2)}), (e2 , {(u1 , 0 9, 0 8, 0 7, 0 5, 0 7, 0 2), (u2 , 0.5, 0.8, 0.7, 0.8, 0.7, 0.1)})} then,wehave B1 B2 .
Definition14. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2betwobipolarneutrosophicsoftsetsover U .Then, B1 isbipolarneutrosophicsoftequalto B2 , isdenotedby B1 = B2 ,if T + 1 (u) = T + 2 (u), I + 1 (u) = I + 2 (u), F + 1 (u) = F + 2 (u), T1 (u) = T2 (u), I1 (u) = I2 (u)and F1 (u) = F2 (u)forall(e,u) ∈ E × U
Definition15. Let B beabipolarneutrosophicsoft setsover U .Then,thecomplementofabipolarneutrosophicsoftset B,isdenotedby Bc ,isdefined as;
Bc ={(e, {(u,F + (u), 1 I + (u),T + (u),F (u), 1 I (u),T (u)): u ∈ U }): e ∈ E }
Example4. ConsidertheExample1.Then, Bc ={(e1 , {(u1 , 0.1, 0.2, 0.5, 0.2, 0.3, 0.5), (u2 , 0.7, 0.2, 0.6, 0.2, 0.3, 0.5), (u3 , 0.1, 0.2, 0.6, 0.8, 0.2, 0.5)}),
(e2 , (u1 , 0.7, 0.2, 0.8, 0.2, 0.3, 0.5), (u2 , 0 7, 0 2, 0 4, 0 2, 0 3, 0 5), (u3 , 0 1, 0 2, 0 7, 0 4, 0 3, 0 4))}
Definition16. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2betwobipolarneutrosophicsoftsetsover U . Then,theunionof B1 and B2 ,isdenotedby B1 B2 , isdefinedas; B1 B2 ={(e, {(u,maxi {T + i (u)},mini {I + i (u)}, mini {F + i (u)},mini {Ti (u)},maxi {Ii (u)}, maxi {Fi (u)}): u ∈ U }): e ∈ E,andi = 1, 2}
Example5. ConsidertheExample1.Then, B1 B2 ={(e1 , {(u1 , 0.5, 0.8, 0.1, 0.6, 0.7, 0.2), (u2 , 0 6, 0 6, 0 7, 0 5, 0 7, 0 2), (u3 , 0 6, 0 2, 0 1, 0 5, 0 5, 0 3)}), (e2 , (u1 , 0 8, 0 8, 0 7, 0 5, 0 7, 0 2), (u2 , 0 4, 0 8, 0 7, 0 5, 0 7, 0 2), (u3 , 0.7, 0.6, 0.1, 0.4, 0.7, 0.3))}
Definition17. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2,...,n be n bipolarneutrosophicsoftsetsover U .Then,theunionof n bipolarneutrosophicsoftset Bi ,isdenotedby n i=1 Bi ,isdefinedas; n i=1 Bi ={(e, {(u,maxi {T + i (u)},mini {I + i (u)}, mini {F + i (u)},mini {Ti (u)},maxi {Ii (u)}, maxi {Fi (u)}): u ∈ U }): e ∈ E, i = 1, 2,...,n}
Definition18. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2betwobipolarneutrosophicsoftsetsover U Then,theintersectionof B1 and B2 ,isdenotedby B1 B2 ,isdefinedas; B1 B2 ={(e, {(u,mini {T + i (u)},maxi {I + i (u)}, maxi {F + i (u)},maxi {Ti (u)},mini {Ii (u)},
mini {Fi (u)}): u ∈ U }): e ∈ E,i = 1, 2}
Example6. ConsidertheExample1.Then, B1 B2 ={(e1 , {(u1 , 0.4, 0.8, 0.5, 0.5, 0.7, 0.2), (u2 , 0 3, 0 8, 0 7, 0 3, 0 7, 0 2),
(u3 , 0 6, 0 8, 0 6, 0 5, 0 8, 0 8)}),
(e2 , (u1 , 0 1, 0 8, 0 7, 0 2, 0 7, 0 2), (u2 , 0 1, 0 8, 0 7, 0 5, 0 7, 0 5), (u3 , 0.7, 0.8, 0.1, 0.4, 0.7, 0.4))}
Definition19. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2,...,n be n bipolarneutrosophicsoftsetsover U .Then,theintersectionof n bipolarneutrosophic softset Bi ,isdenotedby n i=1 Bi ,isdefinedas; n i=1 Bi ={(e, {(u,mini {T + i (u)},maxi {I + i (u)}, maxi {F + i (u)},maxi {Ti (u)}, mini {Ii (u)},mini {Fi (u)}): u ∈ U }) : e ∈ E,andi = 1, 2,...,n}
Proposition1. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2, 3 bethreebipolarneutrosophicsoftsets overU.Then,
1. B1 B2 = B2 B1
2. B1 B2 = B2 B1
3. B1 (B2 B3 ) = (B1 B2 ) B3
4. B1 (B2 B3 ) = (B1 B2 ) N3
Proof. Theproofscanbeeasilyobtainedsincethe maxfunctionsandminfunctionsarecommutative andassociative.
Proposition2. Let B1 ={(e, {(u,T + 1 (u),I + 1 (u), F + 1 (u),T1 (u),I1 (u),F1 (u)): u ∈ U }): e ∈ E } be abipolarneutrosophicsoftsetsoverU.Then,
1. Bc 1 c = B1
2. (BU )c = B∅
3. B1 BU
4. B∅ B1
5. B1 B1
Proposition3. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2, 3 bethreebipolarneutrosophicsoftsets overU.Then,
1. B1 B2 ∧ B2 B3 ⇒ B1 B3 2. B1 = B2 ∧ B2 = B2 ⇔ B1 = B3 3. B1 B2 ∧ B2 B1 ⇔ B1 = B2
Proposition4. Let B1 ={(e, {(u,T + 1 (u),I + 1 (u), F + 1 (u),T1 (u),I1 (u),F1 (u)): u ∈ U }): e ∈ E } be abipolarneutrosophicsoftsetsoverU.Then, 1. B1 B1 = B1 2. B1 B∅ = B1 3. B1 BU = BU
Proposition5. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } be abipolarneutrosophicsoftsetsoverU.Then,
1. B1 B1 = B1 2. B1 B∅ = B∅ 3. B1 BU = B1
Proposition6. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2 betwobipolarneutrosophicsoftsetsoverU. Then,DeMorgan’slawsarevalid
1. (B1 B2 )c = Bc 1 Bc 2 2. (B1 B2 )c = Bc 1 Bc 2
Proof. i. (B1 B1 )c ={(e, {(u,maxi {T + i (u)},mini {I + i (u)}, mini {F + i (u)},mini {Ti (u)}, maxi {Ii (u)},maxi {Fi (u)}): u ∈ U }) : e ∈ E,andi = 1, 2}c ={(e, {(u,mini {F + i (u)}, 1 mini {I + i (u)}, maxi {T + i (u)},maxi {Fi (u)}, 1 maxi {Ii (u)},mini {Ti (u)}): u ∈ U }) : e ∈ E,andi = 1, 2} ={(e, {(u,mini {F + i (u)},maxi {1 I + i (u)}, maxi {T + i (u)},maxi {Fi (u)}, mini {−1 Ii (u)},mini {Ti (u)}): u ∈ U }) : e ∈ E,andi = 1, 2}{(e, {(u,T + 1 (u), I + 1 (u),F + 1 (u),T1 (u),I1 (u), F1 (u)): u ∈ U }): e ∈ E }c {(e, {(u,T + 2 (u),I + 2 (u),F + 2 (u), T2 (u),I2 (u),F2 (u)): u ∈ U }): e ∈ E }c
= Bc 1 Bc 2 ii. (B1 B1 )c
={(e, {(u,mini {T + i (u)},maxi {I + i (u)}, maxi {F + i (u)},maxi {Ti (u)}, mini {Ii (u)},mini {Fi (u)}): u ∈ U })
: e ∈ E,andi = 1, 2}c
={(e, {(u,maxi {F + i (u)}, 1 maxi {I + i (u)},mini {T + i (u)}, mini {Fi (u)}, 1 mini {Ii (u)}, maxi {Ti (u)}): u ∈ U })
: e ∈ E,andi = 1, 2}
={(e, {(u,maxi {F + i (u)}, mini {1 I + i (u)},mini {T + i (u)}, mini {Fi (u)},maxi {−1 Ii (u)}, maxi {Ti (u)}): u ∈ U })
: e ∈ E,andi = 1, 2}
={(e, {(u,T + 1 (u),I + 1 (u), F + 1 (u),T1 (u),I1 (u),F1 (u)) : u ∈ U }): e ∈ E }c {(e, {(u,T + 2 (u),I + 2 (u), F + 2 (u),T2 (u),I2 (u),F2 (u)) : u ∈ U }): e ∈ E }c
= Bc 1 Bc 2
Proposition7. Let Bi ={(e, {(u,T + i (u),I + i (u), F + i (u),Ti (u),Ii (u),Fi (u)): u ∈ U }): e ∈ E } for i = 1, 2, 3 bethreebipolarneutrosophicsoftsets overU.Then,
1. B1 (B2 B3 ) = (B1 B2 ) (B1 B3 )
2. B1 (B2 B3 ) = (B1 B2 ) (B1 B3 )
4.Aggregationbipolarneutrosophic softoperator
Inthissection,weproposeanaggregationbipolar neutrosophicsoftoperatorofabipolarneutrosophic softsets.Also,wedevelopeanalgorithmbasedon bipolarneutrosophicsoftsetsandgivenumerical
examplestoshowthefeasibilityandeffectivenessof thedevelopedapproach.
Definition20. Let B ={(e, {(u,T + (u),I + (u), F + (u),T (u),I (u),F (u)): u ∈ U }): e ∈ E }= {{(u,T + e (u),I + e (u),F + e (u),Te (u),Ie (u),Fe (u)): u ∈ U } : e ∈ E } beabipolarneutrosophicsoftsets over U .Then,aggregationbipolarneutrosophicsoft operator,denotedby Bagg ,isdefinedas; Bagg ={μB (u)/u : u ∈ U }
μB (u) = 1 2|E × U | e∈E (|1 I + e (u)(T + e (u) F + e (u)) +Ie (u)(Te (u) Fe (u))|) where |E × U | isthecardinalityof E × U .
Nowwegiveadecisionalgorithmforbipolarneutrosophicsoftsets.
Algorithm.
1.Constructthebipolarneutrosophicsoftseton U 2.Computetheaggregationbipolarneutrosophic softoperator.
3.Findanoptimumalternativeseton U .
Example7. (Itisadoptedfrom[14])Assumethatthat aworkplacewantstofillaposition.Thereare5candidateswhofillinaforminordertoapplyformally fortheposition.Thereisadecisionmaker(DM),that isfromthedepartmentofhumanresources.
Hewanttointerviewthecandidates,butitis verydifficulttomakeitallofthem.Therefore,by usingthebipolarneutrosophicsoftdecisionmakingmethod,thenumberofcandidatesarereduced toasuitableone.Assumethatthesetofcandidates U ={u1 ,u2 ,u3 ,u4 ,u5 } whichmaybecharacterized byasetofparameters E ={e1 ,e2 ,e3 } whichis “ e1 = experience”,“e2 = technicalinformation” and“e3 = age”.Now,wecanapplythemethodas follows:
1.DMconstructsabipolarneutrosophicsoft B overthealternativesset U as;
B ={(e1 , {(u1 , 0 8, 0 9, 0 4, 0 5, 0 7, 0 6), (u2 , 0 5, 0 4, 0 8, 0 5, 0 7, 0 5), (u3 , 0.5, 0.5, 0.8, 0.5, 0.8, 0.9), (u4 , 0.9, 0.8, 0.3, 0.5, 0.2, 0.7),
(u5 , 0.5, 0.5, 0.4, 0.9, 0.8, 0.8)}), (e2 , {(u1 , 0.8, 0.4, 0.7, 0.4, 0.2, 0.6),
(u2 , 0.5, 0.3, 0.7, 0.9, 0.7, 0.8),
(u3 , 0 5, 0 9, 0 8, 0 5, 0 7, 0 6)),
(u4 , 0 5, 0 7, 0 8, 0 9, 0 3, 0 7),
(u5 , 0 4, 0 1, 0 8, 0 5, 0 8, 0 9)}),
(e3 , {(u1 , 0.7, 0.8, 0.6, 0.5, 0.1, 0.8),
(u2 , 0.8, 0.9, 0.4, 0.5, 0.4, 0.8),
(u3 , 0.2, 0.9, 0.5, 0.1, 0.9, 0.4),
(u4 , 0 5, 0 4, 0 2, 0 5, 0 6, 0 9),
(u5 , 0 9, 0 8, 0 8, 0 5, 0 7, 0 1)})}
2.DMfindstheaggregationbipolarneutrosophic softoperator Bagg of B as;
Bagg ={0 0793/u1 , 0 0923/u2 , 0 1010/u3 , 0 0797/u4 , 0 0983/u5 }
3.Finally,DMchooses u3 forthepositionfrom Bagg sinceithasthemaximumdegree0.1010 amongtheothers.
Example8. (Itisadoptedfrom[31])Let U = {o1 ,o2 ,o3 ,o4 ,o5 ,o6 } bethesetofobjectshavingdifferentcolors,sizesandsurfacetexture features.Theparameterset, E ={e1 ,e2 ,e3 } in which“e1 = colorspace”,“e2 = size”and“e3 = surfacetexture”.Wecanapplythealgorithmasfollows:
1.DMconstructsabipolarneutrosophicsoft B overthealternativesset U as;
B ={(e1 , {(o1 , 0 3, 0 4, 0 6, 0 3, 0 5, 0 4),
(o2 , 0 3, 0 9, 0 3, 0 6, 0 7, 0 4),
(o3 , 0 4, 0 5, 0 8, 0 5, 0 6, 0 7),
(o4 , 0.8, 0.2, 0.4, 0.7, 0.3, 0.5),
(o5 , 0.7, 0.3, 0.6, 0.7, 0.6, 0.6),
(o6 , 0.9, 0.2, 0.4, 0.7, 0.6, 0.6)}),
(e2 , {(o1 , 0 4, 0 2, 0 8, 0 6, 0 4, 0 8),
(o2 , 0 8, 0 6, 0 3, 0 7, 0 5, 0 6),
(o3 , 0 6, 0 4, 0 4, 0 3, 0 7, 0 8),
(o4 , 0.9, 0.8, 0.2, 0.7, 0.5, 0.6),
(o5 , 0.2, 0.1, 0.9, 0.3, 0.6, 0.7),
(o6 , 0.3, 0.2, 0.8, 0.3, 0.5, 0.7)}),
(e3 , {(o1 , 0.3, 0.4, 0.1, 0.7, 0.3, 0.6),
(o2 , 0.8, 0.9, 0.4, 0.5, 0.4, 0.8), (o3 , 0 5, 0 6, 0 3, 0 3, 0 7, 0 6), (o4 , 0 7, 0 6, 0 6, 0 3, 0 4, 0 7), (o5 , 0 6, 0 8, 0 5, 0 3, 0 5, 0 3), (o6 , 0.8, 0.7, 0.7, 0.3, 0.5, 0.3)})}
2.DMfindstheaggregationbipolarneutrosophic softoperator Bagg of B as;
Bagg ={0.1007/o1 , 0.0803/o2 , 0.0773/o3 , 0.0750/o4 , 0.0927/o5 , 0.930/o6 }
3.Finally,DMchooses o1 forthepositionfrom Bagg sinceithasthemaximumdegree0.1007 amongtheothers.
Example9. (Itisadoptedfrom[27])Weconsidertheproblemtoselectthemostsuitablehouse whichMr.Xisgoingtochooseonthebasisof hismnumberofparametersoutofnnumberof houses(wechoose n = 5and m = 5).Let U = {h1 ,h2 ,h3 ,h4 ,h5 } bethesetofhouseshaving differentfeatures E ={e1 ,e2 ,e3 ,e4 ,e5 } inwhich inwhich“e1 = beautiful”,“e2 = cheap”,“e3 = ingoodrepairing”,“e4 = moderate”and“e5 = wooden”.Wecanapplythealgorithmasfollow:
1.DMconstructsabipolarneutrosophicsoft B overthealternativesset U as;
B ={(e1 , {(h1 , 0.6, 0.3, 0.8, 0.5, 0.7, 0.6), (h2 , 0 7, 0 2, 0 6, 0 5, 0 7, 0 5), (h3 , 0 8, 0 3, 0 4, 0 5, 0 8, 0 9), (h4 , 0 7, 0 5, 0 6, 0 5, 0 2, 0 7), (h5 , 0 8, 0 6, 0 7, 0 9, 0 8, 0 8)}), (e2 , {(h1 , 0.5, 0.2, 0.6, 0.4, 0.2, 0.6), (h2 , 0.6, 0.3, 0.7, 0.9, 0.7, 0.8), (h3 , 0.8, 0.5, 0.1, 0.6, 0.8, 0.6)), (h4 , 0 6, 0 8, 0 7, 0 9, 0 3, 0 7), (h5 , 0 5, 0 6, 0 8, 0 5, 0 8, 0 9)}), (e3 , {(h1 , 0 7, 0 3, 0 4, 0 5, 0 1, 0 8), (h2 , 0.7, 0.5, 0.6, 0.5, 0.4, 0.8), (h3 , 0.3, 0.5, 0.6, 0.1, 0.9, 0.4), (h4 , 0.7, 0.6, 0.8, 0.5, 0.6, 0.9), (h5 , 0.8, 0.7, 0.6, 0.5, 0.7, 0.1)}),
(e4 , {(h1 , 0.8, 0.5, 0.6, 0.5, 0.1, 0.8),
(h2 , 0 6, 0 8, 0 3, 0 5, 0 4, 0 8),
(h3 , 0 7, 0 2, 0 1, 0 1, 0 9, 0 4),
(h4 , 0 8, 0 3, 0 6, 0 5, 0 6, 0 9),
(h5 , 0.7, 0.8, 0.3, 0.5, 0.7, 0.1)}),
(e5 , {(h1 , 0.6, 0.7, 0.2, 0.5, 0.1, 0.8),
(h2 , 0.8, 0.1, 0.8, 0.5, 0.4, 0.8),
(h3 , 0 7, 0 2, 0 6, 0 1, 0 9, 0 4),
(h4 , 0 8, 0 3, 0 8, 0 5, 0 6, 0 9),
(h5 , 0 7, 0 2, 0 6, 0 5, 0 7, 0 1)})}
2.DMfindstheaggregationbipolarneutrosophic softoperator Bagg of B as;
Bagg ={0 1470/h1 , 0 1477/h2 , 0 1137/h3 , 0.1443/h4 , 0.1747/h5 }
3.Finally,DMchooses h5 forthepositionfrom Bagg sinceithasthemaximumdegree0.1747 amongtheothers.
IthasbeenobservedinExamples7–9thattheproposedmethodrequireslessstepsofcomputationthan therelevantworksin[14,27,31]whilstprovides moreinformationonmembershipdegrees(positive andnegative)fordecision.
5.Conclusion
Inthispaper,weintroducedthebipolarneutrosophicsoftsetthatcombinessoftsetsandbipolar neutrosophicsets.Somenewoperationsonbipolar neutrosophicsoftsetsweredesigned.Wedeveloped adecisionmakingmethodbasedonbipolarneutrosophicsoftsets.Numericalexamplestakenfrom theexistingworks[14,27,31]wereperformedto showthefeasibilityandelectivenessofthedeveloped approach.Forfurtherstudy,wewillapplyourwork torealworldproblemswithrealisticdataandextend proposedalgorithmtootherdecisionmakingmodels withvaguenessanduncertainty.Anextensionfrom BipolartoTripolarNeutrosophicSoftSetsandeven MultipolarNeutrosophicSoftSetsasinspiredin[35] willbeournexttargets.
Acknowledgments
ThisresearchisfundedbyVietnamNationalFoundationforScienceandTechnologyDevelopment (NAFOSTED)undergrantnumber102.01-2017.02.
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