Thecomplexneutrosophicsoftexpertset anditsapplicationindecisionmaking
AshrafAl-QuranandNasruddinHassan∗ SchoolofMathematicalSciences,FacultyofScienceandTechnology,UniversitiKebangsaanMalaysia, UKMBangiSelangorDE,Malaysia
Abstract.Thispaperpresentsanovelcomplexneutrosophicsoftexpertset(CNSES)concept.TherangeofvaluesofCNSES isextendedtotheunitcircleinthecomplexplanebyaddinganadditionaltermcalledthephasetermwhichdescribes CNSES’selementsintermsofthetimeaspect.CNSESisahybridstructureofsoftsetsandsingle-valuedneutrosophicsets (SVNSs)definedinacomplexsettingwheretheexperts’opinionsareincluded,thusmakingithighlysuitableforusein decision-makingproblemsthatinvolveuncertainandindeterminatedatawherethetimefactorplaysakeyroleindetermining thefinaldecision.Basedonthisnewconceptwedefinesomeconceptsrelatedtothisnotionaswellassomebasicoperations namelythecomplement,union,intersection,ANDandOR.Thebasicpropertiesandrelevantlawspertainingtothisconcept suchastheDeMorgan’slawsarealsoverified.Lastly,weproposeanalgorithmtosolvecomplexneutrosophicsoftexpert decision-makingproblembyconvertingitfromthecomplexstatetotherealstateandsubsequentlyprovidedthedetailed decisionsteps.Thisstudyissupportedbythecomparisonwithotherexistingmethods.
Keywords:Complexneutrosophicset,decisionmaking,neutrosophicset,single-valuedneutrosophicset,softexpertset
1.Introduction
Smarandache[1]firstlyproposedthetheoryof neutrosophicsetasageneralizationoffuzzyset[2] andintuitionisticfuzzyset[3].Neutrosophicsetcan dealwithuncertain,indeterminateandincongruous informationwheretheindeterminacyisquantifiedexplicitlyandtruthmembership,indeterminacy membershipandfalsitymembershiparecompletely independent.Theneutrosophicsetwasintroducedfor thefirsttimebySmarandacheinhis1998book[4] whichisalsomentionedbyHoweinthefreeonline dictionaryofcomputing.Inordertoapplyneutrosophicsetinreal-lifeproblems,itsoperatorsneed tobespecified,therefore,thesingle-valuedneutrosophicsetanditsbasicoperationsweredefinedby
∗ Correspondingauthor.NasruddinHassan,SchoolofMathematicalSciences,FacultyofScienceandTechnology,Universiti KebangsaanMalaysia,43600UKMBangiSelangorDE,Malaysia. Tel.:+60389213710;E-mail:nas@ukm.edu.my.
Wangetal.[5]asaspecialcaseofneutrosophic set,sincesinglevalueisaninstanceofsetvalue. Subsequently,theworksonSVNSsandtheirhybrid structuresintheoriesandapplicationshavebeen progressingrapidly[6–9].Multi-criteriadecisionmaking(MCDM)isanimportantbranchofdecision theory,whichhasbeenextensivelystudiedinmany research[10–13].Duetothecomplexityofreal decision-makingproblems,thedecisioninformation isoftenincomplete,indeterminateandinconsistent information,thentheaforementioneduncertaintysets canofferusefultoolstohandlesuchdecision-making problems.Therefore,theintegrationoftheseuncertaintysetsinMCDMtechniqueshasincreasingly attractedtheattentionofmanyresearchers.Thislead toaproductiveoutputinrelevantresearchliterature [14–26].Softsettheory,ontheotherhand,wasinitiatedbyMolodtsov[27]asageneralmathematical toolusedtohandleuncertainties,imprecisionand vagueness.Sinceitsinception,alotofextensionsof softsetmodelhavebeendevelopedsuchasfuzzysoft
sets[28],vaguesoftsets[29],interval-valuedvague softsets[30–32],softexpertsets[33],softmultisettheory[34]andneutrosophicsoftset[35–39].At present,softsethasalluredwideattentionandmade manyachievements[40–42].Thedevelopmentof theuncertaintysetsthathavebeenmentionedabove arenotlimitedtotherealfieldbutextendedtothe complexfield.Theintroductionoffuzzysetswas followedbytheirextensiontothecomplexfuzzyset [43].Incomplexfuzzyset,thedegreeofmembership function μ istradedbyacomplex-valuedfunction oftheform rs (x)eiωs (x) i=√ 1 ,where rs (x)and ωs (x)arebothreal-valuedfunctionsand rs (x)eiωs (x) hastherangeincomplexunitcircle.Thereisalsoan addedadditionaltermcalledthephasetermtosolve theenigmaintranslatingsomecomplex-valuedfunctionsonphysicaltermstohumanlanguageandvice versa.AlkouriandSalleh[44]introducedtheconceptofcomplexintuitionisticfuzzysettorepresent theinformationwhichishappeningrepeatedlyovera periodoftime,whileSelvachandranetal.[45]introducedtheconceptofcomplexvaguesoftsetswhich combinethekeyfeaturesofsoftandcomplexfuzzy sets.Tohandleimprecise,indeterminate,inconsistent,andincompleteinformationthathasperiodic nature,AliandSmarandache[46]introducedcomplexneutrosophicset.Incomplexneutrosophicset, eachmembershipfunctionassociateswithaphase term.Thisfeaturegiveswave-likepropertiesthat couldbeusedtodescribeconstructiveanddestructive interferencedependingonthephasevalueofanelement,aswellasitsabilitytodealwithindeterminacy.
Overtheyears,manytechniquesandmethodshave beenproposedastoolstobeusedtofindthesolutions ofproblemsthatarenonlinearorvagueinnature,with everymethodintroducedsuperiortoitspredecessors. Followinginthisdirection,ourproposedmodelisan extensionofsoftexpertset,fuzzysoftexpertset[47], intutionisticfuzzysoftexpertset(IFSES)[48],vague softexpertset[49]andsingle-valuedneutrosophic softexpertset(SVNSES)[50].Thusitwillincorporatetheadvantagesofallofthesemodels.Tofacilitate ourdiscussion,wefirstreviewsomebackgroundon SVNSandcomplexneutrosophicsetinSection2.In Section3,wegivethemotivationforthispaper.In Section4,weintroducetheconceptofCNSESand giveitstheoreticoperations.InSection5,wediscuss anapplicationofthisconceptineconomy.InSection6,thecomparisonanalysisisconductedtoverify thevalidityoftheproposedapproach.Finally,conclusionsarepointedoutinSection7.Consequently,our proposedconceptwillenrichcurrentstudiesinneu-
trosophicsoftsets[51–55]andcomplexfuzzysets [43,56].
2.Preliminaries
Inthissection,werecapitulatetheconceptsofneutrosophicandcomplexneutrosophicsetsandpresent anoverviewoftheoperationsstructuresofthecomplexneutrosophicmodelthatarerelevanttothework inthispaper.Thecomplexneutrosophicsoftset (CNSS)isalsointroduced.
Definition2.1. (see[1])Let U beauniverseof discourse.Aneutrosophicset N in U isdefined as: A ={<u; TN (u); IN (u); FN (u) >; u ∈ U } where TN (u),IN (u)and FN (u)arethetruthmembership function,theindeterminacymembershipfunction andthefalsitymembershipfunction,respectively, suchthat T ; I ; F : X →] 0;1+ [and 0 ≤ TN (u) + IN (u) + FN (u) ≤ 3+
Inordertoapplyneutrosophicsetonthescientificfields,itsparametersshouldhavetobe specified.HenceWangetal.[5]providedthefollowingdefinition.
Definition2.2. (see[5])Let U beauniverseofdiscourse.Asingle-valuedneutrosophic set(SVNS) S in U definedas: S = U T (U ), I (U ),F (U ) /u,u ∈ U, when U iscontinuousand S = n i=1 T (Ui ),I (Ui ),F (Ui ) /ui ,ui ∈ U, when U isdiscrete,where TS , IS and FS arethetruth membershipfunction,theindeterminacymembershipfunctionandthefalsitymembershipfunction, respectivelyand TS ; IS ; FS : U → [0, 1]
Definition2.3. (see[50])Let U ={u1 ,u2 ,...,un } be auniversalsetofelements, E ={e1 ,e2 ,...,em } bea universalsetofparameters, X ={x1 ,x2 ,...,xi } be asetofexperts(agents)and O ={1 = agree, 0 = disagree} beasetofopinions.Let Z ={E × X × O} and A ⊆ Z .Thenthepair(U,Z )iscalledasoftuniverse.Let F : Z → SVN U , where SVN U denotesthe collectionofallsingle-valuedneutrosophicsubsets of U .Suppose F : Z → SVN U beafunctiondefined as:
F (z) = F (z)(ui ), ∀ui ∈ U.
Then F (z)iscalledasingle-valuedneutrosophic softexpertvalueoverthesoftuniverse(U,Z ).
AliandSmarandache[46]conceptualizedcomplex neutrosophicsetandgavethebasicoperationsinthe followingtwodefinitions.
Definition2.4. (see[46])Letauniverseofdiscourse U ,acomplexneutrosophicset S in U is characterizedbyatruthmembershipfunction TS (u), anindeterminacymembershipfunction IS (u),anda falsitymembershipfunction FS (u)thatassignsan element u ∈ U acomplex-valuedgradeof TS (u), IS (u),and FS (u)in S .Bydefinition,thevalues TS (u), IS (u), FS (u)andtheirsummayall bewithintheunitcircleinthecomplexplane andareoftheform, TS (u) = pS (u).ejμS (u) , IS (u) = qS (u).ejνS (u) and FS (u) = rS (u).ejωS (u) ,eachof pS (u),qS (u),rS (u)and μS (u),νS (u),ωS (u)are, respectively,realvaluedand pS (u),qS (u),rS (u) ∈ [0, 1]suchthat0 ≤ PS (u) + qS (u) + rS (u) ≤ 3+ Definition2.5. (see[46])Let A and B betwocomplex neutrosophicsetsontheuniverse U ,where A ischaracterizedbyatruthmembershipfunction TA (u) = pA (u).ejμA (u) ,anindeterminacymembershipfunction IA (u) = qA (u).ejνA (u) andafalsitymembership function FA (u) = rA (u).ejωA (u) and B ischaracterizedbyatruthmembershipfunction TB (u) = pB (u).ejμB (u) ,anindeterminacymembershipfunction IB (u) = qB (u).ejνB (u) andafalsitymembership function FB (u) = rB (u).ejωB (u)
Wedefinethethecomplement,subset,unionand intersectionoperationsasfollows.
(1)Thecomplementof A,denotedas˜c (A)isspecifiedbyfunctions:
Tc(A) (u) = pc(A) (u).ejμc(A) (u) = rA (u).ej (2π μA (u)) , I ˜ c(A) (u) = q ˜ c(A) (u).ejνc(A) (u) = (1 qA (u)).ej (2π νA (u)) , and F ˜ c(A) (u) = r ˜ c(A) (u).ejωc(A) (u) = pA (u).ej (2π ωA (u)) .
(2) A issaidtobecomplexneutrosophicsubsetof B (A ⊆ B )ifandonlyifthefollowingconditionsaresatisfied:
(a) TA (u) ≤ TB (u)suchthat pA (u) ≤ pB (u)and μA (u) ≤ μB (u).
(b) IA (u) ≥ IB (u)suchthat qA (u) ≥ qB (u) and νA (u) ≥ νB (u).
(c) FA (u) ≥ FB (u)suchthat rA (u) ≥ rB (u) and ωA (u) ≥ ωB (u).
(3)Theunion(intersection)of A and B ,denotedas A ∪ (∩)B andthetruthmembershipfunction
TA∪(∩)B (u),theindeterminacymembership function IA∪(∩)B (u),andthefalsitymembershipfunction FA∪(∩)B (u)aredefinedas:
TA∪(∩)B (u) = [(pA (u) ∨ (∧)pB (u))]
.ej (μA (u)∨(∧)μB (u)) ,
IA∪(∩)B (u) = [(qA (u) ∧ (∨)qB (u))]
.ej (νA (u)∧(∨)νB (u)) , and FA∪(∩)B (u) = [(rA (u) ∧ (∨)rB (u))]
.ej (ωA (u)∧(∨)ωB (u)) ,
where ∨ =maxand ∧ =min.
WewillnowintroducetheconceptofCNSS. Definition2.6. Let U beauniverse, E beasetof parametersand A ⊆ E .Let CNS (U )beasetofall complexneutrosophicsubsetsof U .Apair(H,A)is calledacomplexneutrosophicsoftset(CNSS)over U where H isamappinggivenby
H : A → CNS (U ).
Inotherwords,theCNSS(H,A)isaparameterized familyofallcomplexneutrosophicsetsof U
3.Motivationforcomplexneutrosophic softexpertset
Neutrosophicsetdealswithinformationordata whichcontainuncertainty,indeterminacyandfalsity. Fuzzysetandintuitionisticfuzzysetdonothandleindeterminacy,wherebytheinformationmight betrueandfalseorneithertruenorfalseatthe sametime.Thus,neutrosophicsetcansolvesome problemswhereindeterminacyisdeeplyembeddedinhumanthinkingduetotheimperfectionof knowledgethathumanreceivesorobservesfrom theexternalworld.Inreality,manyphenomenaand eventshappenedperiodicallyandalloftheabove modelscannotaddressthesesituations.Therefore, manyuncertaintyapproachesaredevelopedsuch ascomplexfuzzysetwhichischaracterizedbya complex-valuedmembershipfunctionthathandles informationwithuncertaintyandperiodicitysimultaneously.Consequently,complexintuitionisticfuzzy setwasthereafterdevelopedbyaddingacomplexvaluednonmembershipfunctionthathandlesthe falsityandperiodicitysimultaneously.Nonetheless,
thesemodelscannotdealwithindeterminateinformationwhichappearinaperiodicmannerinreal life.Toovercomethisdifficulty,complexneutrosophicsetisintroducedbyaddingacomplex-valued indeterminacymembershipfunctionwhichtackles theindeterminacyandperiodicitysimultaneously. Thecomplexneutrosophicsetissuperiortothese modelswiththreecomplex-valuedmembershipfunctionswhichholduncertainty,indeterminacyand falsitywithperiodicity.Further,thecomplexneutrosophicsetisessentiallyneutrosophicsetdefined inacomplexsetting.Thus,ithastheaddedadvantagesoftheneutrosophicsetbyvirtueofthe complexityfeaturewhichhastheabilitytocaptureinformationthatareperiodicinnature,whereas neutrosophicsetdoesnothavethisfeature.Thediscussionaboveshowstheascendancyofcomplex neutrosophicset.
However,complexneutrosophicsetlacksthe adequateparameterizationtooltofacilitatetherepresentationofparametersandititdoesnothavea mechanismtoincorporatetheopinionofallexperts inonemodel.Thisdecreasesthevalidityofthismodel asmostsituationsinthereal-wordareopentointerpretationsbydifferentpeople.Thus,theCNSESis proposedtoprovideamoreadequateparameterizationtoolthatcanrepresenttheproblemparameters inamorecomprehensiveandcompletemanner.It hasalsotheaddedadvantageofallowingtheusersto knowtheopinionofalltheexpertsinasinglemodel withouttheneedforanyadditionalcumbersome operations.TheproposedCNSESmodelhowever, providesamoreaccuraterepresentationoftwodimensionalinformationi.e.informationpresented bytheamplitudetermsandinformationpresentedby thephaseterms.Thephasetermrepresentsthetime factorthatmayinterfere,constructivelyordestructively,withtheassociatedamplitudeterminthe decisionprocess.Thismakesitmorevalidandreal inmodelingreallifeproblemswheretimefactor andthejudgmentsofhumanbeingsplayamajor role.
Anoveladjustableapproachtodecision-making problemsbasedonCNSESisalsointroduced. ThisapproachconvertstheCNSEStoaSVNSES usingapracticalandusefulalgorithmwhichhighlightstheroleofthetimefactorindetermining thefinaldecision.Thenewlyproposedapproach efficientlycapturestheincomplete,indeterminate, andinconsistentinformationandextendsexisting decision-makingmethodstoprovideamorecomprehensiveoutlookfordecision-makers.
4.Complexneutrosophicsoftexpertset
Inthissection,weintroducethedefinitionofcomplexneutrosophicsoftexpertset(CNSES)whichis acombinationofsoftexpertsetandsingle-valued neutrosophicsetdefinedinacomplexsetting.We definesomeoperationsonthisconcept,namelysubset,equality,complement,union,intersection,AND andOR.WealsoshowthatDeMorgan’slawand otherpertaininglawsalsoholdinCNSES.
WebeginbyproposingthedefinitionofCNSES, andgiveanillustrativeexampleofit.
Let U beauniverse, E asetofparameters, X asetofexperts(agents),and O ={1 = agree, 0 = disagree} asetofopinions.Let Z = E × X × O and A ⊆ Z
Definition4.1. Apair(H,A)iscalledacomplexneutrosophicsoftexpertset(CNSES)over U ,where H isamappinggivenby H : A → CN U , where CN U denotesthepowercomplexneutrosophic setof U
Itistobenotedthat ∀α ∈ A,H (α)representsthe degreeandthephaseofbelongingness,indeterminacyandnon-belongingnessoftheelementsof U in H (α).
TheCNSES(H,A)canbewrittenas: (H,A) = α,TH (a) (u),IH (α) (u),FH (α) (u) : α ∈ A,u ∈ U , where ∀u ∈ U , ∀α ∈ A, TH (α) (u) = pH (α) (u) .ejμH (α) (u) ,IH (α) (u) = qH (α) (u).ejνH (α) (u) and FH (α) (u) = rH (α) (u).ejωH (α) (u) with TH (α) (u),IH (α) (u) and FH (α) (u)representingthecomplex-valuedtruth membershipfunction,complex-valuedindeterminacymembershipfunctionandcomplex-valued falsitymembershipfunction,respectively ∀u ∈ U Thevalues TH (α) (u),IH (α) (u),FH (α) (u)arewithin theunitcircleinthecomplexplaneandboththe amplitudeterms pH (α) (u),qH (α) (u),rH (α) (u)andthe phaseterms μH (α) (u),νH (α) (u),ωH (α) (u)arereal valuedsuchthat pH (α) (u),qH (α) (u),rH (α) (u) ∈ [0, 1] and0 ≤ pH (α) (u) + qH (α) (u) + rH (α) (u) ≤ 3.
Example4.2. Supposethatapharmaceuticalcompanydevelopstwotypesofitsmedicineandwishes totaketheopinionofsomeexpertsconcerningthese
(H,A)
= (e1 ,p, 1),
0.7ej 2 (0 3) , 0.2ej 2 (0 3) , 0.1ej 2 (0 2) u1 , 0.9ej 2 (0 6) , 0.6ej 2 (0 4) , 0.3ej 2 (0) u2 ,
(e1 ,q, 1), 0 5ej 2 (0 3) , 0 2ej 2 (0 9) , 0 9ej 2 (0 8) u1 , 0 9ej 2 (0 9) , 0 4ej 2 (0 5) , 0 5ej 2 (0 6) u2 ,
(e2 ,p, 1), 0.3ej 2 (0 9) , 0.2ej 2 (0 6) , 0.9ej 2 (0 1) u1 , 0.9ej 2 (0 5) , 0.5ej 2 (0 7) , 0.4ej 2 (0 1) u2 ,
(e2 ,q, 1), 0 1ej 2 (0.8) , 0 2ej 2 (0.4) , 0 9ej 2 (0.8) u1 , 0 9ej 2 (0.5) , 0 1ej 2 (0.1) , 0 2ej 2 (0) u2 ,
(e3 ,p, 1), 0 1ej 2 (0 4) , 0 2ej 2 (0 6) , 0 9ej 2 (0 2) u1 , 0 9ej 2 (0 3) , 0 1ej 2 (0 5) , 0 3ej 2 (0 8) u2 , (e3 ,q, 1), 0.1ej 2 (0 3) , 0.2ej 2 (0 7) , 0.9ej 2 (0 9) u1 , 0.9ej 2 (0 4) , 0.1ej 2 (0 6) , 0.5ej 2 (0) u2 , (e1 ,p, 0), 0 1ej 2 (0 7) , 0 8ej 2 (0 7) , 0 7ej 2 (0 8) u1 , 0 3ej 2 (0 4) , 0 4ej 2 (0 6) , 0 9ej 2 (1) u2 , (e1 ,q, 0), 0.9ej 2 (0 7) , 0.8ej 2 (0 1) , 0.5ej 2 (0 2) u1 , 0.5ej 2 (0 1) , 0.6ej 2 (0 5) , 0.9ej 2 (0 4) u2 , (e2 ,p, 0), 0.9ej 2 (0.1) , 0.8ej 2 (0.4) , 0.3ej 2 (0.9) u1 , 0.4ej 2 (0.5) , 0.1ej 2 (0.3) , 0.4ej 2 (0.9) u2 , (e2 ,q, 0), 0 9ej 2 (0 2) , 0 8ej 2 (0 6) , 0 1ej 2 (0 2) u1 , 0 2ej 2 (0 5) , 0 9ej 2 (0 9) , 0 2ej 2 (1) u2 , (e3 ,p, 0), 0.9ej 2 (0 6) , 0.8ej 2 (0 4) , 0.1ej 2 (0 8) u1 , 0.3ej 2 (0 7) , 0.9ej 2 (0 5) , 0.9ej 2 (0 2) u2 , (e3 ,q, 0), 0 9ej 2 (0 7) , 0 8ej 2 (0 3) , 0 1ej 2 (0 1) u1 , 0 5ej 2 (0 6) , 0 9ej 2 (0 4) , 0 9ej 2 (1) u2 .
medicationsbytakingintoaccountthedegreeof effectivenessandthetimetakentoovercomethe diseasewhicharerepresentedbyamplitudeterms andphaseterms,respectively.Let U ={u1 ,u2 } be asetofmedication, E ={e1 ,e2 ,e3 } asetofparametersthatdescribesthedegreeofinfluencewhere ei (i = 1, 2, 3)denotesthedecisions“highinfluence”, “averageinfluence”and“lowinfluence”respectively andlet X ={p,q} beasetofexperts.
Supposethatthecompanyhasdistributedaquestionnairetothetwoexpertstomakedecisionson thesetwonewmedication,thentheCNSES(H,A) isdefinedasbelow:
IntheCNSES(H,A),boththeamplitudeterms andphasetermsliebetween0and1suchthatan amplitudetermwithvaluecloseto0(1)impliesthat amedicinehasaverylittle(strong)influenceona diseaseandaphasetermwithvaluecloseto0(1) impliesthatthismedicinetakesaveryshort(long) timetoovercomethedisease.
Inthefollowing,weintroducetheconceptof thesubsetoftwoCNSESsandtheequalityoftwo CNSESs.
Definition4.3. FortwoCNSESs(H,A)and(G,B ) over U ,(H,A)iscalledacomplexneutrosophicsoft expertsubsetof(G,B )if
1. A ⊆ B, 2. ∀ ∈ A,H ( )iscomplexneutrosophicsubset of G( ).
Definition4.4. TwoCNSESs(H,A)and(G,B )over U ,aresaidtobeequalif(H,A)isacomplexneutrosophicsoftexpertsubsetof(G,B )and(G,B )isa complexneutrosophicsoftexpertsubsetof(H,A).
Inthefollowing,weproposethedefinitionofthe complementofaCNSESalongwithanillustrative exampleandgiveapropositionofthecomplementof aCNSES.
Let U beauniverseofdiscourseand (H,A) bea CNSESon U, whichisasdefinedbelow:
(H,A) = α,TH (α) (u),IH (α) (u),FH (α) (u) : α ∈ A,u ∈ U .
Definition4.5. Thecomplementof (H,A) isdenoted by (H,A)c = (H c ,A) , andisdefinedas: (H,A)c = α,TH c (α) (u),IH c (α) (u),FH C (α) (u) : α ∈ A,u ∈ U , where TH c (α) (u) = pH c (α) (u).ejμH c (α) (u) = rH (α) (u) .ej (2π μH (α) (u)) ,IH c (α) (u) = qH c (α) (u).ejνH c (α) (u) = (1 qH (α) (u)).ej (2π νH (α) (u)) and FH c (α) (u) = rH c (α) (u).ejωH c (α) (u) = pH (α) (u).ej (2π ωH (α) (u)) .
Example4.6. Considertheapproximationgivenin Example4.2,where H (e1 ,p, 1)
= 0.7ej 2 (0 3) , 0.2ej 2 (0 3) , 0.1ej 2 (0 2) u1 , 0 9ej 2 (0 6) , 0 6ej 2 (0 4) , 0 3ej 2 (0) u2 .
Byusingthecomplexneutrosophiccomplement, weobtainthecomplementoftheapproximation givenby
H (e1 ,p, 1)
=
0 1ej 2 (0.7) , 0 8ej 2 (0.7) , 0 7ej 2 (0.8) u1 , 0 3ej 2 (0 4) , 0 4ej 2 (0 6) , 0 9ej 2 (1) u2
Proposition4.7. If (H,A) isaCNSESover U ,then, ((H,A)c )c = (H,A)
Proof. FromDefinition4.5,wehave(H,A)c = (H c ,A)where (H,A)c
= α,TH c (α) (u),IH c (α) (u),FH C (α) (u) : α ∈ A,u ∈ U ,
= α,pH c (α) (u).ejμH c (α) (u) ,qH c (α) (u).ejνH c (α) (u) , rH c (α) (u).ejωH c (α) (u) : α ∈ A,u ∈ U ,
= α,rH (α) (u).ej (2π μH (α) (u)) , (1 qH (α) (u)) .ej (2π νH (α) (u)) ,pH (α) (u).ej (2π ωH (α) (u)) : α ∈ A,u ∈ U
Thus, ((H,A)c )c
= α,rH c (α) (u).ej (2π μH c (α) (u)) , (1 qH c (α) (u)).ej (2π νH c (α) (u)) , pH c (α) (u).ej (2π ωH c (α) (u)) : α ∈ A,u ∈ U ,
= α,pH (α) (u).e j 2π (2π μH (α) (u)) , 1 (1 qH (α) (u)) .e j 2π (2π νH (α) (u)) , rH (α) (u).e j 2π (2π ωH (α) (u)) : α ∈ A,u ∈ U ,
= α,pH (α) (u).ejμH (α) (u) ,qH (α) (u).ejνH (α) (u) ,
rH (α) (u).ejωH (α) (u) : α ∈ A,u ∈ U , = α,TH (α) (u),IH (α) (u),FH (α) (u) : α ∈ A,u ∈ U , = (H,A).
Thiscompletestheproof. Now,weputforwardthedefinitionofanagreeCNSESandthedefinitionofadisagree-CNSES.
Definition4.8. Anagree-CNSES(H,A)1 over U is acomplexneutrosophicsoftexpertsubsetof(H,A) wheretheopinionsofallexpertsareagreeandis definedasfollows: (H,A)1 = H (e): e ∈ Z × X ×{1}
Definition4.9. Adisagree-CNSES(H,A)0 over U is acomplexneutrosophicsoftexpertsubsetof(H,A) wheretheopinionsofallexpertsaredisagreeandis definedasfollows: (H,A)0 = H (e): e ∈ Z × X ×{0}
Inthefollowing,weintroducethedefinitionsof theunionandintersectionoftwoCNSESs.
Definition4.10. TheunionoftwoCNSESs (H,A) and (G,B ) overauniverse U isaCNSES (K,C ), where C = A ∪ B and ∀ ∈ C, ∀ u ∈ U,
Definition4.11. TheintersectionoftwoCNSESs (H,A) and (G,B ) overauniverse U isaCNSES (K,C ),where C = A ∪ B and ∀ ∈ C, ∀ u ∈ U,
TK ( ) (u) = ⎧
pH ( ) (u).ejμH ( ) (u) , if ∈ A B pG( ) (u).ejμG( ) (u) , if ∈ B A (pH ( ) (u) ∧ pG( ) (u)) .ej (μH ( ) (u)∧μG( ) (u)) , if ∈ A ∩ B,
rH ( ) (u).ejωH ( ) (u) , if ∈ A B rG( ) (u).ejωG( ) (u) , if ∈ B A (rH ( ) (u) ∧ rG( ) (u)) .ej (ωH ( ) (u)∧ωG( ) (u)) , if ∈ A ∩ B, where ∨= max,and ∧= min. Theunion (H,A) ˜ ∪ (G,B ) = (K,C ).
IK ( ) (u) = ⎧
qH ( ) (u).ejνH ( ) (u) , if ∈ A B qG( ) (u).ejνG( ) (u) , if ∈ B A (qH ( ) (u) ∨ qG( ) (u)) .ej (νH ( ) (u)∨νG( ) (u)) , if ∈ A ∩ B, FK ( ) (u) =
rH ( ) (u).ejωH ( ) (u) , if ∈ A B rG( ) (u).ejωG( ) (u) , if ∈ B A (rH ( ) (u) ∨ rG( ) (u)) .ej (ωH ( ) (u)∨ωG( ) (u)) , if ∈ A ∩ B, where ∨= max,and ∧= min. Theintersection (H,A) ˜ ∩ (G,B ) = (K,C ) WeshowthatDeMorgan’slawholdsforthe CNSESasfollows.
Proposition4.12. If (H,A) and (G,B ) aretwoCNSESsover U ,thenwehavethefollowingproperties: 1. ((H,A)∪(G,B ))c = (H,A)c ∩(G,B )c , 2. ((H,A)∩(G,B ))c = (H,A)c ∪(G,B )c .
Proof. (1)Assumethat(H,A)∪(G,B ) = (K,C ), where C = A ∪ B and ∀ ∈ C, TK ( ) (u) = ⎧
pH ( ) (u).ejμH ( ) (u) , if ∈ A B pG( ) (u).ejμG( ) (u) , if ∈ B A (pH ( ) (u) ∨ pG( ) (u) .ej (μH ( ) (u)∨μG( ) (u)) , if ∈ A ∩ B.
Since(H,A)∪(G,B ) = (K,C ),thenwehave ((H,A)∪(G,B ))c = (K,C )c = (K c ,C ). Hence ∀ ∈ C, TK c ( ) (u)
rH ( ) (u).ej (2π μH ( ) (u)) , if ∈ A B rG( ) (u).ej (2π μH ( ) (u)) , if ∈ B A (rH ( ) (u) ∧ rG( ) (u)) .ej ((2π μH ( ) (u))∧(2π μG( ) (u))) , if ∈ A ∩ B.
Since(H,A)c = (H c ,A)and(G,B )c = (Gc ,B ), thenwehave(H,A)c ∩(G,B )c = (H c ,A)∩(Gc ,B ). Supposethat(H c ,A)∩(Gc ,B ) = (I,D),where D = A ∪ B .Hence ∀ ∈ D,
pH c ( ) (u).ejμH c ( ) (u) , if ∈ A B pGc ( ) (u).ejμGc ( ) (u) , if ∈ B A (pH c ( ) (u) ∧ pGc ( ) (u)) .ej (μH c ( ) (u)∧μGc ( ) (u)) , if ∈ A ∩ B,
rH ( ) (u).ej (2π μH ( ) (u)) , if ∈ A B rG( ) (u).ej (2π μH ( ) (u)) , if ∈ B A (rH ( ) (u) ∧ rG( ) (u)) .ej ((2π μH ( ) (u))∧(2π μG( ) (u))) , if ∈ A ∩ B.
Therefore, K c and I arethesameoperatorsand D = C ,whichimplies, T(H ( )∪G( ))c (u) = TH c ( )∩Gc ( ) (u), ∀u ∈ U.
Similarly,onthesamelines,wecanshowitalso holdsfortheidentityandfalsityterms.Thusitfollowsthat((H,A)∪(G,B ))c = (H,A)c ∩(G,B )c and thiscompletestheproof.
(2)Theproofissimilartothatof(1). WewillnowgivethedefinitionsofANDandOR operationswithapropositiononthesetwooperations.
Definition4.13. Let(H,A)and(G,B )beanytwo CNSESsoverasoftuniverse(U,Z ).Thentheoperation(H,A)AND(G,B )denotedby(H,A)∧(G,B ) isdefinedby(H,A)∧(G,B ) = (K,A × B ),where (K,A × B ) = K (α,β ),suchthat K (α,β ) = H (α) ∩ G(β ),forall(α,β ) ∈ A × B ,and ∩ representsthe complexneutrosophicintersection.
Definition4.14. Let(H,A)and(G,B )beanytwo CNSESsoverasoftuniverse(U,Z ).Thentheoperation(H,A)OR(G,B )denotedby(H,A)∨(G,B ) isdefinedby(H,A)∨(G,B ) = (K,A × B ),where (K,A × B ) = K (α,β ),suchthat K (α,β ) = H (α) ∪ G(β ),forall(α,β ) ∈ A × B ,and ∪ representsthe complexneutrosophicunion.
Proposition4.15. If (H,A) and (G,B ) aretwoCNSESsover U ,thenwehavethefollowingproperties:
1. ((H,A)∨(G,B ))c = (H,A)c ∧(G,B )c , 2. ((H,A)∧(G,B ))c = (H,A)c ∨(G,B )c
Proof. Theproofof(1)and(2)issimilartotheproof ofPropositions4.12.
5.Decision-makingunderthecomplex neutrosophicsoftexpertenvironment
Inthissection,wepresentanapplicationofCNSES inadecision-makingproblembyconsideringthefollowingproblem.
Example5.1. Supposeweareinterestedinunderstandingthemostimportanteconomicfactors (indicators)affectingMalaysianeconomyin2016. Supposewetakefourfactorswhicharerepresentedin theuniversalset U ={u1 ,u2 ,u3 ,u4 } where u1 =the plungeincommodityandoilprices, u2 =China’seconomicslowdown, u3 =goodsandservicestax(GST) implementedinthisyearand u4 =theexchangerate variability.Ourproblemistoarrangethesefourfactorsindescendingorderfrommostimportanttoleast important.Let E ={e1 ,e2 ,e3 } betheparametersset thatrepresentsthemajorsectorsoftheMalaysian economy,where e1 =industrysector, e2 =tourism sector, e3 =externaltradesector.Suppose X ={p,q} beasetofeconomicexpertswhoareassignedtoanalyzethesefourfactorsbydeterminingthedegreeand thetotaltimeoftheinfluenceofthesefactorsonthe mentionedsectorsoftheMalaysianeconomyasin thefollowingCNSES: (H,A
, 0.4ej 2 (3/12) , 0.2ej 2 (3/12) , 0.6ej 2 (11/12) u3 , 0.9ej 2 (8/12) , 0.5ej 2 (5/12) , 0.3ej 2 (0) u4 , (e1 ,q, 1), 0 9ej 2 (8/12) , 0 1ej 2 (2/12) , 0 3ej 2 (1/12) u1 , 0 7ej 2 (6/12) , 0 4ej 2 (5/12) , 0 9ej 2 (8/12) u2 , 0.3ej 2 (1/12) , 0.9ej 2 (6/12) , 0.9ej 2 (10/12) u3 , 0.8ej 2 (8/12) , 0.4ej 2 (6/12) , 0.3ej 2 (2/12) u4 , (e2 ,p, 1), 0 8ej 2 (8/12) , 0 2ej 2 (4/12) , 0 3ej 2 (1/12) u1 , 0 4ej 2 (6/12) , 0 5ej 2 (1/12) , 0 4ej 2 (1/12) u2 ,
0.3ej 2 (6/12) , 0.5ej 2 (7/12) , 0.9ej 2 (8/12) u3 , 0.6ej 2 (7/12) , 0.3ej 2 (5/12) , 0.2ej 2 (6/12) u4 ,
(e2 ,q, 1),
0 5ej 2 (11/12) , 0 2ej 2 (1/12) , 0 1ej 2 (1/12) u1 , 0 7ej 2 (6/12) , 0 1ej 2 (1/12) , 0 9ej 2 (4/12) u2 ,
0.2ej 2 (6/12) , 0.6ej 2 (4/12) , 0.9ej 2 (6/12) u3 , 0.5ej 2 (7/12) , 0.5ej 2 (4/12) , 0.2ej 2 (4/12) u4 ,
(e3 ,p, 1), 0 7ej 2 (8/12) , 0 1ej 2 (2/12) , 0 4ej 2 (2/12) u1 , 0 9ej 2 (3/12) , 0 1ej 2 (6/12) , 0 5ej 2 (9/12) u2 ,
0.4ej 2 (1/12) , 0.2ej 2 (7/12) , 0.9ej 2 (2/12) u3 , 0.8ej 2 (9/12) , 0.3ej 2 (1/12) , 0.2ej 2 (1/12) u4 , (e3 ,q, 1), 0 7ej 2 (6/12) , 0 5ej 2 (4/12) , 0 2ej 2 (2/12) u1 , 0 4ej 2 (7/12) , 0 1ej 2 (8/12) , 0 4ej 2 (6/12) u2 ,
0.3ej 2 (3/12) , 0.5ej 2 (8/12) , 0.9ej 2 (6/12) u3 , 0.4ej 2 (7/12) , 0.7ej 2 (5/12) , 0.7ej 2 (6/12) u4 , (e1 ,p, 0), 0 1ej 2 (1/12) , 0 8ej 2 (11/12) , 0 9ej 2 (12/12) u1 , 0 7ej 2 (7/12) , 0 4ej 2 (8/12) , 0 5ej 2 (8/12) u2 , 0.6ej 2 (9/12) , 0.8ej 2 (9/12) , 0.4ej 2 (1/12) u3 , 0.3ej 2 (4/12) , 0.5ej 2 (7/12) , 0.9ej 2 (12/12) u4 , (e1 ,q, 0), 0 3ej 2 (4/12) , 0 9ej 2 (10/12) , 0 1ej 2 (11/12) u1 , 0 9ej 2 (6/12) , 0 6ej 2 (7/12) , 0 7ej 2 (4/12) u2 , 0 9ej 2 (11/12) , 0 1ej 2 (6/12) , 0 7ej 2 (2/12) u3 , 0 3ej 2 (4/12) , 0 6ej 2 (6/12) , 0 8ej 2 (10/12) u4 , (e2 ,p, 0), 0 3ej 2 (4/12) , 0 8ej 2 (8/12) , 0 8ej 2 (11/12) u1 , 0 4ej 2 (6/12) , 0 5ej 2 (11/12) , 0 4ej 2 (11/12) u2 , 0.9ej 2 (6/12) , 0.5ej 2 (5/12) , 0.7ej 2 (4/12) u3 , 0.2ej 2 (5/12) , 0.7ej 2 (7/12) , 0.6ej 2 (6/12) u4 , (e2 ,q, 0), 0.1ej 2 (1/12) , 0.8ej 2 (11/12) , 0.5ej 2 (11/12) u1 , 0.9ej 2 (6/12) , 0.9ej 2 (11/12) , 0.7ej 2 (8/12) u2 , 0 9ej 2 (6/12) , 0 4ej 2 (8/12) , 0 2ej 2 (6/12) u3 , 0 2ej 2 (5/12) , 0 5ej 2 (8/12) , 0 5ej 2 (8/12) u4 , (e3 ,p, 0), 0.4ej 2 (4/12) , 0.9ej 2 (10/12) , 0.3ej 2 (10/12) u1 , 0.5ej 2 (9/12) , 0.9ej 2 (6/12) , 0.9ej 2 (3/12) u2 , 0.9ej 2 (11/12) , 0.8ej 2 (5/12) , 0.6ej 2 (10/12) u3 , 0.2ej 2 (3/12) , 0.7ej 2 (11/12) , 0.8ej 2 (11/12) u4 , (e3 ,q, 0), 0 2ej 2 (6/12) , 0 5ej 2 (8/12) , 0 3ej 2 (10/12) u1 , 0 4ej 2 (5/12) , 0 9ej 2 (4/12) , 0 4ej 2 (6/12) u2 , 0.9ej 2 (9/12) , 0.5ej 2 (4/12) , 0.7ej 2 (6/12) u3 , 0.7ej 2 (5/12) , 0.3ej 2 (7/12) , 0.4ej 2 (6/12) u4
Inthecontextofthisexample,theamplitudeterms measuretheinfluencedegreeofthementionedfactorsontheMalaysianeconomy,whilethephaseterm representsthephaseofthisinfluenceortheperiodof thisinfluence.
Followinginthisdirection,weprovideanexample ofscenariosthatcouldpossiblyoccurinthiscontext. Forexample,intheapproximation
H (e1 ,p, 1) = 0.9ej 2 (11/12) , 0.2ej 2 (1/12) , 0.1ej 2 (0) u1 , 0 5ej 2 (5/12) , 0 6ej 2 (4/12) , 0 7ej 2 (4/12) u2 ,... ,
thecomplexneutrosophicsoftexpertvalue(CNSEV) 0.9ej 2 (11/12) , 0.2ej 2 (1/12) , 0.1ej 2 (0) u1
indicatesthattheplungeincommodityandoil priceshasabiginfluenceontheMalaysianeconomy.Thecomplex-valuedtruthmembershipfunction 0 9ej 2 (11/12) indicatesthattheexpert p agreesthat thereisastronginfluenceoftheplungeincommodityandoilpricesontheindustrialsector,sincethe amplitudeterm0 9isveryclosetooneandthisinfluencespanof11monthsisconsideredaverylong timeofinfluence(phasetermwithvalueveryclose toone),thecomplex-valuedindeterminacymembershipfunction0.2ej 2 (1/12) canbeinterpretedasthe expert p isunabletodetermineifthereisinfluence ornotwithadegreeof0.2andthisinfluenceisnot evidentforamonth.Forthecomplex-valuedfalsity membershipfunction0 1ej 2 (0) ,expert p presumes withadegreeof0.1thatthereisnoinfluenceandthe timewithnoinfluenceis0.
NexttheCNSES(H,A)isusedtogetherwitha generalizedalgorithmtosolvethedecision-making problemstatedatthebeginningofthissection.This algorithmisemployedtorankthefactorsthataffect theMalaysianeconomycorrespondingtotheirinfluencestrength.Inthisdecisionprocessthetimeof influenceplaysakeyrolewherethefactorwhichhasa largedegreeofinfluenceandalongtimeofinfluence willbemoreimportantthanothers.Thealgorithm givenbelowconvertsthecomplexneutrosophicsoft expertvalues(CNSEVs)tonormalizedsingle-valued neutrosophicsoftexpertvalues(SVNSEVs)andproceedstothefinaldecisionusingthesingle-valued neutrosophicsoftexpertmethod(SVNSEM)[50].
Thealgorithmstepsaregivenasfollows.
Algorithm:
1.InputtheCNSES(H,A)
2.ConverttheCNSES(H,A)totheSVNSES(H,A) byobtainingtheweightedaggregationvaluesof T H (α) (u),IH (α) (u)and F H (α) (u), ∀α ∈ A and ∀u ∈ U asthefollowingformulas:
T H (α) (u) = w1 pH (α) (u) + w2 (1/2π )μH (α) (u), I H (α) (u) = w1 qH (α) (u) + w2 (1/2π )νH (α) (u), F H (α) (u) = w1 rH (α) (u) + w2 (1/2π )ωH (α) (u),
where pH (α) (u),qH (α) (u), rH (α) (u)and μH (α) (u),νH (α) (u),ωH (α) (u)aretheamplitudeand phasetermsintheCNSES(H,A),respectively. T H (α) (u),IH (α) (u)and F H (α) (u)arethetruth membershipfunction,indeterminacymembership functionandfalsitymembershipfunctioninthe SVNSES(H,A),respectivelyand w1 , w2 are theweightsfortheamplitudeterms(degreesof influence)andthephaseterms(timesofinfluence),respectively,where w1 and w2 ∈ [0, 1]and w1 + w2 = 1
3.Findthevaluesof Z H (α) (u) = T H (α) (u)+(1 I H (α) (u))+(1 F H (α) (u)) 3 , ∀u ∈ U and ∀α ∈ A
Notethat Z H (α) (u)isthenormalizedvaluesof S H (α) (u) = T H (α) (u) I H (α) (u) F H (α) (u), ∀u ∈ U and ∀α ∈ A.Wenormalizetheelementsof S ={S H (α) (u), ∀u ∈ U and ∀α ∈ A},sinceitrepresentsthedegreeoftheinfluence,where S takes itsminimumvalueat 2when(T H (α) (u),IH (α) (u), F H (α) (u)) = (0, 1, 1),whileitsmaximumtakesthe value1at(T H (α) (u),IH (α) (u),FH (α) (u)) = (1, 0, 0).
4.FindthehighestnumericalgradefortheagreeSVNSESanddisagree-SVNSES.
5.Computethescoreofeachelement ui ∈ U by takingthesumofthenumericalgradeofeachelementfortheagree-SVNSESanddisagree-SVNSES, denotedby Ki and Si ,respectively.
6.Findthevaluesofthescore ri = Ki Si foreach element ui ∈ U .
7.Determinethevalueofthehighestscore
m = maxui ∈U {ri }.Thenthedecisionistochoose element ui astheoptimalsolutiontotheproblem.If therearemorethanoneelementwiththehighest ri score,thenanyoneofthoseelementscanbechosen astheoptimalsolution.
Itistobenotedthatthismethodisusedtodeal withdecision-makingproblemswithknownweight information(completeweightinformation).Toexecutetheabovesteps,weassumethattheweightvector fortheamplitudetermsis w1 = 0 7andtheweight vectorforthephasetermsis w2 = 0 3 Now,toconverttheCNSES(H,A)totheSVNSES (H,A),obtaintheweightedaggregationvaluesof T H (α) (u),IH (α) (u)and F H (α) (u), ∀α ∈ A and ∀u ∈ U
Toillustratethisstep,wecalculate T H (α ) (u ),IH (α ) (u ) and F H (α ) (u ),when α = (e1 ,p, 1)and u = u1 as shownbelow:
T H (e1 ,p,1) (u1 ) = w1 pH (e1 ,p,1) (u1 ) + w2 (1/2π )μH (e1 ,p,1) (u1 )
= 0.7(0.9) + 0.3(1/2π )(2π )(11/12)
= 0 9
I H (e1 ,p,1) (u1 )
= w1 qH (e1 ,p,1) (u1 ) + w2 (1/2π )νH (e1 ,p,1) (u1 )
= 0 7(0 2) + 0 3(1/2π )(2π )(1/12)
= 0 165
F H (e1 ,p,1) (u1 ) = w1 rH (e1 ,p,1) (u1 ) + w2 (1/2π )ωH (e1 ,p,1) (u1 )
= 0 7(0 1) + 0 3(1/2π )(2π )(0) = 0 07
Then,for α = (e1 ,p, 1)and u = u1 ,theSVNSEV (T H (α ) (u ),IH (α ) (u ),FH (α ) (u )) = (0 9, 0 165, 0 07).
Inthesamemanner,wecalculatetheotherSVNSEVs, ∀α ∈ A and ∀u ∈ U asintheTable1, whichgivesthevaluesof Z H (α) (u), ∀α ∈ A and ∀u ∈ U
Itistobenotedthattheupperandlowerterms foreachelementinTable1representtheSVNSEVs,
Table1 Valuesof(H,A)and Z H (α) (u)
Uu1 u2 u3 u4 (e1 ,p, 1) 0 905, 0 165, 0 07 0 475, 0 52, 0 59 0 355, 0 215, 0 695 0 83, 0 475, 0 21 0.890.4550.4820.715 (e1 ,q, 1) 0 83, 0 12, 0 235 0 64, 0 405, 0 83 0 235, 0 78, 0 88 0 76, 0 43, 0 26 0.8250.4680.1920.69 (e2 ,p, 1) 0 76, 0 24, 0 235 0 43, 0 375, 0 305 0 36, 0 525, 0 83 0 595, 0 335, 0 29 0.7620.5830.3350.657 (e2 ,q, 1) 0.625, 0.165, 0.095 0.64, 0.095, 0.73 0.29, 0.52, 0.78 0.525, 0.45, 0.24 0.7880.6050.330.612 (e3 ,p, 1) 0 83, 0 12, 0 33 0 705, 0 22, 0 757 0 305, 0 315, 0 68 0 785, 0 235, 0 165 0.7930.5760.4370.795 (e3 ,q, 1) 0 64, 0 45, 0 19 0 455, 0 27, 0 43 0 285, 0 55, 0 78 0 455, 0 615, 0 64 0.6670.5850.3180.4 (e1 ,p, 0) 0 095, 0 835, 0 93 0 665, 0 48, 0 55 0 645, 0 785, 0 305 0 31, 0 525, 0 93 0.110.5450.5450.285 (e1 ,q, 0) 0 31, 0 88, 0 345 0 78, 0 715, 0 59 0 905, 0 22, 0 54 0 31, 0 57, 0 81 0.3620.4920.7150.31 (e2 ,p, 0) 0 31, 0 76, 0 835 0 43, 0 625, 0 555 0 78, 0 475, 0 59 0 265, 0 665, 0 57 0.2380.4170.5720.343 (e2 ,q, 0) 0.095, 0.835, 0.625 0.78, 0.905, 0.69 0.78, 0.48, 0.29 0.265, 0.55, 0.55 0.2120.3950.670.389 (e3 ,p, 0) 0.38, 0.88, 0.46 0.575, 0.78, 0.705 0.905, 0.685, 0.67 0.215, 0.765, 0.835 0.3470.3630.5170.205 (e3 ,q, 0) 0 29, 0 55, 0 46 0 405, 0 73, 0 43 0 855, 0 45, 0 64 0 615, 0 385, 0 43 0.4270.4150.5880.6
Table2
Numericalgradeforagree-SVNSES
Uui Highest numerical grade
(e1 ,p, 1) u1 0.89 (e1 ,q, 1) u1 0.825 (e2 ,p, 1) u1 0.762 (e2 ,q, 1) u1 0.788 (e3 ,p, 1) u4 0.795 (e3 ,q, 1) u1 0.667
Table3
Numericalgradefordisagree-SVNSES
Uui Highest numerical grade
(e1 ,p, 0) u2 ,u3 0 545 (e1 ,q, 0) u3 0.715 (e2 ,p, 0) u3 0.572 (e2 ,q, 0) u3 0.67 (e3 ,p, 0) u3 0.517 (e3 ,q, 0) u4 0.6
Table4
Thescore ri = Ki Si
Ki Si ri
Score(u1 ) = 3 932Score(u1 ) = 03.932
Score(u2 ) = 0Score(u2 ) = 0 545 0.545 Score(u3 ) = 0Score(u3 ) = 3 019 3.019 Score(u4 ) = 0 795Score(u4 ) = 0 60.195
∀α ∈ A and ∀u ∈ U andthevaluesof Z H (α) (u), ∀α ∈ A and ∀u ∈ U ,respectively.
Tables2and3givethehighestnumericalgrade fortheelementsintheagree-SVNSESanddisagreeSVNSES,respectively.
Let Ki and Si ,representthescoreofeachnumerical gradefortheagree-SVNSESanddisagree-SVNSES, respectively.ThesevaluesaregiveninTable4.
Thus,fromTable4 maxui ∈U {ri }= r1 ,followed by r4 and r2 ,where minui ∈U {ri }= r3 . Therefore, theplungeincommodityandoilpricesisthemost importantfactorthataffectstheMalaysianeconomy, followedbytheexchangeratevariabilityandChina’s economicslowdown,wherethegoodsandservices taxlagsbehindthesefactors.
6.Comparisonanddiscussion
Inthissection,wewillcompareourproposed complexneutrosophicsoftexpertmethod(CNSEM)
totheSVNSEM[50]whichisageneralization ofintuitionisticfuzzysoftexpertmethod(IFSEM) [48],fuzzysoftexpertmethod[47]andsoftexpert method[33].
ComparedwithSVNSEMwhichusesthe SVNSEStodepictthedecision-makinginformation, theproposedCNSEMintroducesanewdescriptor, thatis,CNSEStopresentactualdecision-making information.FromExample5.1,itcanbeseenthat theSVNSEScannotrepresentthedegreeoftheinfluenceandthetimeoftheinfluencesimultaneously, sinceitisunabletorepresentvariablesintwodimensions.However,thestructureofCNSESprovides theabilitytodescribethesetwovariablessimultaneouslybyvirtueofthephasetermsandamplitude terms.ThustheSVNSEMcannotdirectlysolvesuch adecision-makingproblemwithcomplexneutrosophicsoftexpertinformation.
Incontrast,theCNSEMcandirectlyaddressthe single-valuedneutrosophicsoftexpertproblem, sincetheSVNSESisaspecialcaseofCNSES andcanbeeasilyrepresentedintheformof CNSES.Inotherwords,theSVNSESisaCNSES withphasetermsequalzeros.Forexamplethe SVNSEV(0.3,0.5,0.7)canberepresentedas (0 3ej 2 (0) , 0 5ej 2 (0) , 0 7ej 2 (0) )bymeansof CNSES.
Furthermore,ourmethodisapplicableforintuitionisticfuzzysoftexpertproblem,sinceIFSES isaspecialcaseofSVNSESandconsequentlyof CNSES.Forexampletheintuitionisticfuzzysoft expertvalue(0.3,0.5)canbe(0.3,0.2,0.5)by meansofSVNSESandhencecanbe(0 3ej 2 (0) , 0 2ej 2 (0) , 0 5ej 2 (0) )bymeansofCNSES,sincethe sumofthedegreesofmembership,nonmembership andindeterminacyofanintuitionisticfuzzyvalue equalto1.Notethattheindeterminacydegreeinintuitionisticfuzzysetisprovidedbydefaultandcannot bedefinedaloneunliketheneutrosophicsetwhere theindeterminacyisdefinedindependentlyandquantifiedexplicitly.
Thus,theproposedmethodhascertainadvantages. Firstly,thismethodusestheCNSEStorepresentthe decisioninformationandasanextensionofSVNSES andIFSES,CNSESincludesevaluationinformation missinginthefirsttwomodels,suchasthetime framewhichispresentedbythephaseterms.Our methodhighlightstheimpactthatthetimeframehas onthefinaldecision.Secondly,apracticalformula isemployedtoconverttheCNSEVstotheSVNSEVs,whichmaintainstheentiretyoftheoriginal datawithoutreducingordistortingthem.Thirdly,
ourmethodgivesadecision-makingwithasimple computationalprocesswithouttheneedtocarryout directedoperationsoncomplexnumbers.Finally,the CNSESthatisusedinourmethodhastheability tohandletheimprecise,indeterminate,inconsistent, andincompleteinformationthatiscapturedbythe amplitudetermsandphasetermssimultaneously.As aresult,theproposedmethodiscapableofdealing withdeeperuncertaindata.
7.Conclusion
WeestablishedtheconceptofCNSESbyextendingthetheoriesofSVNSandsoftexpertsetto complexnumbers.ThebasicoperationsonCNSES, namelycomplement,subset,union,intersection, AND,andORoperations,weredefined.Subsequently,thebasicpropertiesoftheseoperationssuch asDeMorgan’slawsandotherrelevantlawspertainingtotheconceptofCNSESwereproven.Finally, ageneralizedalgorithmisintroducedandappliedto theCNSESmodeltosolveahypotheticaldecisionmakingproblemanditssuperiorityandfeasibilityare furtherverifiedbycomparisonwithotherexisting methods.Thisnewextensionwillprovideasignificantadditiontoexistingtheoriesforhandling indeterminacy,wheretimeplaysavitalruleinthe decisionprocess,andspursmoredevelopmentsof furtherresearchandpertinentapplications.Forfurtherresearch,weintendtotakeintoaccountunknown weightinformationtodevelopsomerealapplications ofCNSESinotherareas,wherethephasetermmay representothervariablessuchasdistance,speedand temperature.
Acknowledgments
TheauthorswouldliketoacknowledgethefinancialsupportreceivedfromCentreforResearch andInstrumentationManagement(CRIM)Universiti KebangsaanMalaysia.
References
[1]F.Smarandache,Neutrosophicset-Ageneralisationofthe intuitionisticfuzzysets, InternationalJournalofPureand AppliedMathematics 24(3)(2005),287–297.
[2]L.A.Zadeh,Fuzzyset, InformationandControl 8(3)(1965), 338–353.
[3]K.Atanassov,Intuitionisticfuzzysets, FuzzySetsandSystems 20(1)(1986),87–96.
[4]F.Smarandache,Neutrosophy.NeutrosophicProbability, Set,andLogic,AmericanResearchPress,Rehoboth,USA, 1998.
[5]H.Wang,F.Smarandache,Y.Q.ZhangandR.Sunderraman, Singlevaluedneutrosophicsets, MultispaceandMultistructure 4 (2010),410–413.
[6]H.L.Yang,Z.L.Guo,Y.SheandX.Liao,Onsinglevaluedneutrosophicrelations, JournalofIntelligentandFuzzy Systems 30(2)(2016),1045–1056.
[7]R.SahinandA.Kucuk,Subsethoodmeasureforsingle valuedneutrosophicsets, JournalofIntelligentandFuzzy Systems 29(2)(2015),525–530.
[8]J.Ye,AnextendedTOPSISmethodformultipleattribute groupdecisionmakingbasedonsinglevaluedneutrosophic linguisticnumbers, JournalofIntelligentandFuzzySystems 28(1)(2015),247–255.
[9]J.Q.Wang,Y.YangandL.Li,Multi-criteriadecisionmakingmethodbasedonsingle-valuedneutrosophiclinguistic Maclaurinsymmetricmeanoperators, NeuralComputing andApplications (2016).doi:10.1007/s00521-016-2747-0
[10]H.Zhou,J.Q.WangandH.Y.Zhang,Stochasticmulticriteriadecision-makingapproachbasedonSMAA-ELECTRE withextendedgraynumbers, InternationalTransactionsin OperationalResearch (2016).doi:10.1111/itor.12380
[11]P.LiuandF.Teng,AnextendedTODIMmethodformultipleattributegroupdecision-makingbasedon2-dimension uncertainlinguisticvariable, Complexity 21(5)(2016), 20–30.
[12]P.Liu,L.HeandX.Yu,Generalizedhybridaggregation operatorsbasedonthe2-dimensionuncertainlinguistic informationformultipleattributegroupdecisionmaking, GroupDecisionandNegotiation 25(1)(2016),103–126.
[13]Z.P.Tian,J.Wang,H.Y.ZhangandJ.Q.Wang,Multi-criteria decision-makingbasedongeneralizedprioritizedaggregationoperatorsundersimplifiedneutrosophicuncertain linguisticenvironment, InternationalJournalofMachine LearningandCybernetics (2016).doi:10.1007/s13042016-0552-9
[14]P.Liu,Multipleattributegroupdecisionmakingmethod basedoninterval-valuedintuitionisticfuzzypowerHeronianaggregationoperators, Computers&Industrial Engineering 108 (2017),199–212.
[15]P.LiuandS.M.Chen,Groupdecisionmakingbased onHeronianaggregationoperatorsofintuitionisticfuzzy numbers, IEEETransactionsonCybernetics 47(9)(2017), 2514–2530.
[16]Y.Y.Li,H.Y.ZhangandJ.Q.Wang,Linguisticneutrosophic setsandtheirapplicationinmulticriteriadecision-making problems, InternationalJournalforUncertaintyQuantification 7(2)(2017),135–154.
[17]P.Liu,S.M.ChenandJ.Liu,Multipleattributegroup decisionmakingbasedonintuitionisticfuzzyinteraction partitionedBonferronimeanoperators, InformationSciences 411 (2017),98–121.
[18]R.X.Liang,J.Q.WangandH.Y.Zhang,Amulti-criteria decision-makingmethodbasedonsingle-valuedtrapezoidal neutrosophicpreferencerelationswithcompleteweight information, NeuralComputingandApplications (2017). doi:10.1007/s00521-017-2925-8
[19]P.LiuandH.Li,Interval-valuedintuitionisticfuzzypower Bonferroniaggregationoperatorsandtheirapplicationto groupdecisionmaking, CognitiveComputation 9(4)(2017), 494–512.
[20]J.J.Peng,J.Q.WangandX.HWu,Anextensionofthe ELECTREapproachwithmulti-valuedneutrosophicinfor-
A.Al-QuranandN.Hassan/Thecomplexneutrosophicsoftexpertsetanditsapplicationindecisionmaking
mation, NeuralComputingandApplications (2016).doi: 10.1007/s00521-016-2411-8
[21]P.LiuandP.Wang,Someimprovedlinguisticintuitionisticfuzzyaggregationoperatorsandtheirapplicationsto multipleattributedecisionmaking, InternationalJournalof InformationTechnology&DecisionMaking 16(3)(2017), 817–850.
[22]H.GPeng,H.Y.ZhangandJ.Q.Wang,Probabilitymultivaluedneutrosophicsetsanditsapplicationinmulti-criteria groupdecision-makingproblems, NeuralComputingand Applications (2016).doi:10.1007/s00521-016-2702-0
[23]R.X.Liang,J.Q.WangandL.Li,Multi-criteriagroup decisionmakingmethodbasedoninterdependentinputsof singlevaluedtrapezoidalneutrosophicinformation, Neural ComputingandApplications (2016).doi:10.1007/s00521016-2672-2
[24]Z.P.Tian,J.Wang,J.Q.WangandH.Y.Zhang,An improvedMULTIMOORAapproachformulti-criteriadecisionmakingbasedoninterdependentinputsofsimplified neutrosophiclinguisticinformation, NeuralComputingand Applications (2016).doi:10.1007/s00521-016-2378-5
[25]P.Liu,L.Zhang,X.LiuandP.Wang,Multi-valuedneutrosophicnumberBonferronimeanoperatorswiththeir applicationsinmultipleattributegroupdecisionmaking, InternationalJournalofInformationTechnology&DecisionMaking 15(5)(2016),1181–1210.
[26]Y.X.Ma,J.Q.Wang,J.WangandX.H.Wu,Anintervalneutrosophiclinguisticmulti-criteriagroupdecision-making methodanditsapplicationinselectingmedicaltreatment options, NeuralComputingandApplications (2017).doi: 10.1007/s00521-016-2203-1
[27]D.Molodtsov,Softsettheory–firstresults, Computersand MathematicswithApplications 37(2)(1999),19–31.
[28]P.K.Maji,R.BiswasandA.R.Roy,Fuzzysoftsettheory, TheJournalofFuzzyMathematics 3(9)(2001),589–602.
[29]W.Xu,J.Ma,S.WangandG.Hao,Vaguesoftsetsandtheir properties, ComputersandMathematicswithApplications 59(2)(2010),787–794.
[30]K.AlhazaymehandN.Hassan,Interval-valuedvaguesoft setsanditsapplication, AdvancesinFuzzySystems 2012, ArticleID208489.
[31]K.AlhazaymehandN.Hassan,Generalizedinterval-valued vaguesoftset, AppliedMathematicalSciences 7(140) (2013),6983–6988.
[32]K.AlhazaymehandN.Hassan,Possibilityinterval-valued vaguesoftset, AppliedMathematicalSciences 7(140) (2013),6989–6994.
[33]S.AlkhazalehandA.R.Salleh,Softexpertsets, Advances inDecisionSciences 2011,ArticleID757868.
[34]S.Alkhazaleh,A.R.SallehandN.Hassan,Softmultisetstheory, AppliedMathematicalSciences 5(72)(2011), 3561–3573.
[35]P.K.Maji,Neutrosophicsoftset, AnnalsofFuzzyMathematicsandInformatics 5(1)(2013),157–168.
[36]I.DeliandS.Broumi,Neutrosophicsoftrelationsandsome properties, AnnalsofFuzzyMathematicsandInformatics 9(1)(2015),169–182.
[37]I.DeliandS.Broumi,NeutrosophicsoftmatricesandNSM decisionmaking, JournalofIntelligentandFuzzySystems 28(5)(2015),2233–2241.
[38]I.Deli,Interval-valuedneutrosophicsoftsetsanditsdecisionmaking, InternationalJournalofMachineLearning andCybernetics 8(2)(2017),665–676.
[39]S.Alkhazaleh,Time-neutrosophicsoftsetanditsapplications, JournalofIntelligentandFuzzySystems 30(2)(2016), 1087–1098.
[40]K.AlhazaymehandN.Hassan,Vaguesoftmultisettheory, InternationalJournalofPureandAppliedMathematics 93(4)(2014),511–523.
[41]Y.Al-QudahandN.Hassan,Bipolarfuzzysoftexpert setanditsapplicationindecisionmaking, InternationalJournalofAppliedDecisionSciences 10(2)(2017), 175–191.
[42]R.Chatterjee,P.MajumdarandS.K.Samanta,Type-2soft sets, JournalofIntelligentandFuzzySystems 29(2)(2015), 885–898.
[43]D.Ramot,R.Milo,M.FriedmanandA.Kandel,Complexfuzzysets, IEEETransactionsonFuzzySystems 10(2) (2002),171–186.
[44]A.AlkouriandA.R.Salleh,Complexintuitionistic fuzzysets, AIPConferenceProceedings 1482 (2012), 464–470.
[45]G.Selvachandran,P.K.Maji,I.E.AbedandA.R.Salleh, Complexvaguesoftsetsanditsdistancemeasures, Journal ofIntelligentandFuzzySystems 31(1)(2016),55–68.
[46]M.AliandF.Smarandache,Complexneutrosophic set, NeuralComputingandApplications (2016).doi: 10.1007/s00521-015-2154-y
[47]S.AlkhazalehandA.R.Salleh,Fuzzysoftexpertsetandits application, AppliedMathematics 5(9)(2014),1349–1368.
[48]S.BroumiandF.Smarandache,Intuitionisticfuzzysoft expertsetsanditsapplicationindecisionmaking, Journal ofNewTheory 1 (2015),89–105.
[49]K.AlhazaymehandN.Hassan,Vaguesoftexpertsetand itsapplicationindecisionmaking, MalaysianJournalof MathematicalSciences 11(1)(2017),23–39.
[50]S.BroumiandF.Smarandache,Singlevaluedneutrosophic softexpertsetsandtheirapplicationindecisionmaking, JournalofNewTheory 3 (2015),67–88.
[51]A.Al-QuranandN.Hassan,Neutrosophicvaguesoftexpert settheory, JournalofIntelligentandFuzzySystems 30(6) (2016),3691–3702.
[52]S.Alkhazaleh,n-Valuedrefinedneutrosophicsoftsettheory, JournalofIntelligentandFuzzySystems 32(6)(2017), 4311–4318.
[53]A.Al-QuranandN.Hassan,Neutrosophicvaguesoftset anditsapplications, MalaysianJournalofMathematical Sciences 11(2)(2017),141–163.
[54]N.HassanandA.Al-Quran,Possibilityneutrosophicvague softexpertsetfordecisionunderuncertainty, AIPConferenceProceedings 1830 (2017),ArticleID070007.
[55]A.Al-QuranandN.Hassan,Fuzzyparameterisedsingle valuedneutrosophicsoftexpertsettheoryanditsapplicationindecisionmaking, InternationalJournalofApplied DecisionSciences 9(2)(2016),212–227.
[56]Y.Al-QudahandN.Hassan,Operationsoncomplexmultifuzzysets, JournalofIntelligentandFuzzySystems 33(3) (2017),1527–1540.