ANovelMCDMMethodBasedonPlithogenic HypersoftSetsunderFuzzyNeutrosophicEnvironment
MuhammadRayeesAhmad 1 ,MuhammadSaeed 1 ,UsmanAfzal 1 andMiin-ShenYang 2,∗
1 DepartmentofMathematics,UniversityofManagementandTechnology,Lahore54770,Pakistan; F2018265003@umt.edu.pk(M.R.A.);muhammad.saeed@umt.edu.pk(M.S.);usman.afzal@umt.edu.pk(U.A.)
2 DepartmentofAppliedMathematics,ChungYuanChristianUniversity,Chung-Li32023,Taiwan * Correspondence:msyang@math.cycu.edu.tw
Received:18September2020;Accepted:6November2020;Published:10November2020
Abstract: Inthispaper,weadvancethestudyofplithogenichypersoftset(PHSS).Wepresentfour classificationsofPHSSthatarebasedonthenumberofattributeschosenforapplicationandthenature ofalternativesorthatofattributevaluedegreeofappurtenance.ThesefourPHSSclassifications covermostofthefuzzyandneutrosophiccasesthatcanhaveneutrosophicapplicationsinsymmetry. Wealsomakeexplanationswithanillustrativeexamplefordemonstratingthesefourclassifications. Wethenproposeanovelmulti-criteriadecisionmaking(MCDM)methodthatisbasedonPHSS,asan extensionofthetechniquefororderpreferencebysimilaritytoanidealsolution(TOPSIS).Anumber ofrealMCDMproblemsarecomplicatedwithuncertaintythatrequireeachselectioncriteriaor attributetobefurthersubdividedintoattributevaluesandallalternativestobeevaluatedseparately againsteachattributevalue.TheproposedPHSS-basedTOPSIScanbeusedinordertosolvethesereal MCDMproblemsthatarepreciselymodeledbytheconceptofPHSS,inwhicheachattributevalue hasaneutrosophicdegreeofappurtenancecorrespondingtoeachalternativeunderconsideration, inthelightofsomegivencriteria.Forarealapplication,aparkingspotchoiceproblemissolved bytheproposedPHSS-basedTOPSISunderfuzzyneutrosophicenvironmentanditisvalidatedby consideringtwodifferentsetsofalternativesalongwithacomparisonwithfuzzyTOPSISineachcase. TheresultsarehighlyencouragingandaMATLABcodeofthealgorithmofPHSS-basedTOPSIS isalsocompliedinordertoextendthescopeoftheworktoanalyzetimeseriesandindeveloping algorithmsforgraphtheory,machinelearning,patternrecognition,andartificialintelligence.
Keywords: Softset;hypersoftset;plithogenichypersoftset(PHSS);multi-criteriadecision making(MCDM);PHSS-basedTOPSIS
1.Introduction
Astrongmathematicaltoolisalwaysneededinordertocombatrealworldproblemsinvolving uncertaintyinthedata.Thisnecessityhasurgedscholarstointroducedifferentmathematicaltools tofacilitatetheworldforsolvingsuchproblems.In1965,theconceptoffuzzysetwasintroduced byZadeh[1],inwhicheachelementisassignedamembershipdegreeintheformofasinglecrisp valueintheinterval [0,1].Ithasbeenstudiedextensivelybytheresearchersandanumberofreallife problemshavebeensolvedbyfuzzysets[2 5].However,insomepracticalsituations,itisseenthat thismembershipdegreeishardtobedefinedbyasinglenumber.Theuncertaintyinthemembership degreebecamethecausetointroducetheconceptofinterval-valuedfuzzysetinwhichthedegree ofmembershipisanintervalvaluein [0,1].Lateron,theconceptofintuitionisticfuzzyset(IFS)was proposedbyAtanassov[6]in1986,whichincorporatesthenon-membershipdegree.IFShadmany applications[7 10].
Symmetry 2020, 12,1855;doi:10.3390/sym12111855 www.mdpi.com/journal/symmetry
However,IFSisunabletodealwithindeterminateinformation,whichisverycommoninbelief systems.ThisinadequacywasaddressedbySmarandache[11]in2000,whointroducedtheconceptof neutrosophicsetinwhichmembership(T),indeterminacy(I)andnon-membership(F)degreeswere independentlyquantifiedi.e., T,I,F ∈ [0,1] andthesum T+I+F neednottobecontainedin [0,1]. Allofthesemathematicaltoolshavebeenthoroughlyexploredandsuccessfullyappliedtodeal withuncertainties[12 15],yetthesetoolsusuallyfailtohandleuncertaintyinavarietyofpractical situation,becausethesetoolsrequireallnotionstobeexactanddonotpossessaparametrizationtool. Consequently,softsetwasintroducedbyMolodstsov[16]in1999,whichcanberegardedasageneral mathematicaltooltodealwithuncertainty.Molodstsov[16]definedsoftsetasaparameterizedfamily ofsubsetsofauniverseofdiscourse.In2003,Majietal.[17]introducedaggregationoperationsonsoft sets.Softsetsandtheirhybridshavebeensuccessfullyappliedinvariousareas[18 21].
InavarietyofreallifeMCDMproblems,theattributesneedtobefurthersub-dividedinto attributevaluesforabetterdecision.ThisneedwasfulfilledbySmarandache[22],whointroducedthe conceptofhypersoftsetasageneralizationoftheconceptofsoftsetin2018.Besides,Smarandache[22] introducedtheconceptofplithogenichypersoftsetwithcrisp,fuzzy,intuitionisticfuzzy,neutrosophic, andplithogenicsets.In2020,Saeedetal.[23]presentedastudyonthefundamentalsofhypersoftset theory.Smarandache[24,25]developedtheaggregationoperationsonplithogenicsetandprovedthat theplithogenicsetisthemostgeneralizedstructurethatcanbeefficientlyappliedtoavarietyofreal lifeproblems[26 29].
APHSS-basedTOPSISisproposedinthearticletodealwithMCDMproblem,inwhichattribute mayhaveattributevaluesandeachattributevaluehasaneutrosophicdegreeofappurtenanceof eachalternative.Theproposedmethodisauthenticatedbytakingtwodifferentsetsofalternatives. AcomparisonwithfuzzyTOPSISismadeineachcase.Itshowsthattheresultsarehighly inspiring.AMATLABcodeofthealgorithmofPHSS-basedTOPSISisalsocompliedinorder toencompassthescopeoftheworktoanalyzetimeseriesandindevelopingalgorithmsforgraph theory,artificialintelligence,machinelearning,patternrecognition,andneutrosophicapplicationsin symmetry.ItappearsquitepertinenttopointoutthatthearticlegivesdetailedinsightonPHSSwith relateddefinitionsanditsimplementationinMCDMprocess.Thescopeoftheworkcanbeextended inothermathematicsdirectionsaswellbyintroducingimportanttheoremsandpropositions[24].
Theremainderofthisarticleisorganized,asfollows.InSection 2,webrieflyreviewsome basicnotions,leadingtothedefinitionsofsoftsets,hypersoftsets,plithogenicsets,andplithogenic hypersoftsets(PHSSs),alongwithanillustrativeexample.Section 3 consistsofthefourproposed classificationsofPHSSsbasedondifferentcriteria.Moreexplanationswithanillustrativeexample forthefourclassificationsarealsomade.InSection 4,thealgorithmoftheproposedPHSS-based TOPSISisgiven,alongwithitsapplicationtoareallifeparkingspotchoiceproblemunderfuzzy neutrosophicenvironmentanditscomparisonwithfuzzyTOPSIS.Section 5 providestheconclusion andfuturedirections.
2.Preliminaries
Thissectioncomprisesofsomenecessarybasicconceptsthatarerelatedtoplithogenichypersoft set(PHSS),whichisalsodefinedinthissectionalongwithanillustrativeexampleforaclear understanding.Throughoutthestudy,let U beanon-emptyuniversalset, P(U ) bethepowersetof U , X ⊆U beafinitesetofalternatives,and A beafinitesetof n distinctparametersorattributes, asgivenby
A = {a1, a2, , an }, n ≥ 1.
Theattributevaluesof a1, a2, ··· , an belongtothesets A1, A2, , An,respectively,where Ai ∩ Aj = φ, for i = j,and i, j ∈{1,2, , n}.Moreover,weconsiderafinitenumberofuni-dimensionalattributes andeachattributehasafinitediscretesetofattributevalues.However,itisworthmentioningthat
theattributesmayhaveaninfinitenumberofattributevalues.Insuchacase,everystructurewith non-Archimedeanmetricscanbedealtindepth[30,31].
2.1.SoftSets
Asoftsetover U isamapping F : B→ P(U ), B⊆A withthevalue FB (α) ∈ P(U ) at α ∈B and FB (α)= φ if α ∈B.Itisdenotedby (F , B) andwrittenasfollows[16]:
(F , B)= {(α, FB (α)) : α ∈B, FB (α) ∈ P(U )}
Moreover,asoftsetover U canberegardedasaparameterizedfamilyofthesubsetsof U .Foran attribute α ∈B, FB (α) isconsideredasthesetof α-approximateelementsofthesoftset (F , B)
2.2.HypersoftSets
Let C denotethecartesianproductofthesets A1, A2, , An,i.e., C = A1 × A2 × × An, n ≥ 1 Subsequently,ahypersoftset (H, C ) over U isamappingdefinedby H : C → P(U ) [22].Foran n-tuple (γ1, γ2,..., γn ) ∈ C,,where γi ∈ Ai, i = 1,2,3,..., n,ahypersoftsetiswrittenas (H, C )= {(γ, H(γ)) : γ =(γ1, γ2,..., γn ) ∈C, H(γ) ∈ P(U )}. Itmaybenotedthathypersoftsetisageneralizationofsoftset.
2.3.PlithogenicSets
Aset X iscalledaplithogenicsetifallofitsmembersarecharacterizedbytheattributesunder considerationandeachattributemayhaveanynumberofattributevalues[24].Eachattributevalue possessesacorrespondingappurtenancedegreeoftheelement x,totheset X,withrespecttosome givencriteria.Moreover,acontradictiondegreefunctionisdefinedbetweeneachattributevalueand thedominantattributevalueofanattributeinordertoobtainaccuracyforaggregationoperationson plithogenicsets.Thesedegreesofappurtenanceandcontradictionmaybefuzzy,intuitionisticfuzzyor neutrosophicdegrees.
Remark1. Plithogenicsetisregardedasageneralizationofcrisp,fuzzy,intuitionisticfuzzy.andneutrosophic sets,sincetheelementsoflatersetsarecharacterizedbyacombinedsingleattributevalue(degreeofappurtenance), whichhasonlyonevalueforcrispandfuzzysetsi.e.,membership,twovaluesincaseofintuitionisticfuzzyset i.e.,membershipandnon-membership,andthreevaluesforneutrosophicseti.e.,membership,indeterminacy, andnon-membership.Inthecaseofplithogenicset,eachelementisseparatelycharacterizedbyallattribute valuesunderconsiderationintermsofdegreeofappurtenance.
2.4.PlithogenicHypersoftSet(PHSS)
Let
areforfuzzy, intuitionisticfuzzy,andneutrosophicdegreeofappurtenance,respectively.
Furthermore,thedegreeofcontradiction(dissimilarity)betweenanytwoattributevaluesofthe sameattributeisafunctiongivenby
c : Ai × Ai → P([0,1]j ),1 ≤ i ≤ n, j = 1,2,3.
Foranytwoattributevalues γ1 and γ2 ofthesameattribute,itisdenotedby c(γ1, γ2) andsatisfiesthe followingaxioms: c(γ1, γ1)= 0, c(γ1, γ2)= c(γ2, γ1).
Subsequently, (X, A, C, d, c) iscalledaplithogenichypersoftset(PHSS)[22].Foran n-tuple (γ1, γ2, , γn ) ∈ C, γi ∈ Ai,1 ≤ i ≤ n,aplithogenichypersoftset F : C → P(U ) ismathematically writtenas F {γ1, γ2,..., γn } = {x(dx (γ1), dx (γ2),..., dx (γn )), x ∈ X} Remark2. Plithogenichypersoftsetisageneralizationofcrisphypersoftset,fuzzyhypersoftset,intuitionistic fuzzyhypersoftset,andneutrosophichypersoftset.
2.5.IllustrativeExample
Let U = {m1, m2, m3, ... , m10} beauniversecontainingmobilephones.Apersonwantstobuy amobilephoneforwhichthemobilephonesunderconsideration(alternatives)arecontainedin X ⊆U ,givenby X = {m2, m3, m5, m8}
Thecharacteristicsorattributesofthemobilephonesbelongtotheset A = {a1, a2, a3, a4},suchthat
a1 =Processorpower, a2 =RAM, a3 =Frontcameraresolution, a4 =Screensizeininches.
Theattributevaluesof a1, a2, a3, a4 arecontainedinthesets A1, A2, A3, A4 givenbelow.
A1 = {dual-core,quad-core,octa-core}, A2 = {2GB,4GB,8GB,16GB}, A3 = {2MP,5MP,8MP,16MP}, A4 = {4,4.5,5,5.5,6}
1.Softset
Consider B = {a2, a3}⊆A.Afterwards,asoftset (F , B),definedbythemapping F : B→ P(U ), isgivenby (F , B)= {(a2, {m2, m5}), (a3, {m2, m3, m8})}
Element-wise,itmaybewrittenas FB (a2)= {m2, m5}, FB (a3)= {m2, m3, m8}
2.Hypersoftset
Let C = A1 × A2 × A3 × A4.Then,ahypersoftsetover U isafunction f : C → P(U ).Foranelement (octa-core,8GB,16MP,5.5) ∈ C,itisgivenby f ({octa-core,8GB,16MP,5.5})= {m5, m8}
3.Plithogenichypersoftset
Forthesametuple (octa-core,8GB,16MP,5.5) ∈ C,aplithogenichypersoftset F : C → P(U ) isgivenby
F({octa-core,8GB,16MP,5.5})={m5 (dm5 (octa-core), dm5 (8GB), dm5 (16MP), dm5 (5.5)) , m8 (dm8 (octa-core), dm8 (8GB), dm8 (16MP), dm8 (5.5))}, where dm5 (γ) standsforthedegreeofappurtenanceoftheattributevalue γ ∈ (octa-core,8GB,16MP,5.5) totheelement m5 ∈ X.Asimilarmeaningappliesto dm8 (γ)
3.TheFourClassificationsofPHSS
Inthissection,weproposethefourdifferentclassificationsofPHSSthatarebasedonthenumber ofattributeschosenforapplicationandthecharacteristicsofalternativesunderconsiderationorthat oftheattributevaluedegreeofappurtenancefunction.ThesameexamplefromSection 2 isconsidered toeachclassificationforapracticalunderstanding.Figure 1 showsadiagramfortheseclassifications. Figure1. Flowchartoffourclassificationsofplithogenichypersoftsets(PHSS).
3.1.TheFirstClassification
Thisclassificationisbasedonthenumberofattributesthatarechosenbythedecisionmakers forapplication.
3.1.1.Uni-AttributePlithogenicHypersoftSet
Let α ∈A beanattributerequiredbytheexpertsforapplicationpurposeandtheattributevalues of α belongtothefinitediscreteset Y = {y1, y2, , ym }, m ≥ 1.Hence,thedegreeofappurtenance functionisgivenby d : X × Y → P([0,1]j ), ∀ x ∈ X,
suchthat d(x, y) ⊆ [0,1]j,where P([0,1]j ) denotesthepowersetof [0,1]j and j = 1,2,3 standsfor fuzzy,intuitionisticfuzzy,orneutrosophicdegreeofappurtenance,respectively.
Thecontradictiondegreefunctionbetweenanytwoattributevaluesof α,isgivenby
c : Y × Y → P([0,1]j ), ∀ y ∈ Y, j = 1,2,3.
Foranytwoattributevalues y1, y2 ∈ Y,itisdenotedby c(y1, y2) andthefollowingpropertieshold: c(y1, y1)= 0, c(y1, y2)= c(y2, y1).
Subsequently, (X, α, Y, d, c) istermedasauni-attributeplithogenichypersoftset.Foranattributevalue y ∈ Y,auni-attributeplithogenichypersoftset F : Y → P(U ) ismathematicallywrittenas
F(y)= {x(dx (y)) : x ∈ X}.
3.1.2.Multi-AttributePlithogenicHypersoftSet
Considerasubset B of A,consistingofallattributesthatwerechosenbytheexperts,givenby B = {b1, b2,..., bm }, m > 1.
Lettheattributevaluesof b1, b2,..., bm belongtothesets B1, B2,..., Bm,respectively,and Ym = B1 × B2 × ... × Bm.
Afterwards,theappurtenancedegreefunctionis
d : X × Ym → P([0,1]j ), ∀ x ∈ X,
suchthat d(x, y) ⊆ [0,1]j , j = 1,2,3.Inthiscase,thecontradictiondegreefunctionisgivenby c : Bi × Bi → P([0,1]j ),1 ≤ i ≤ m, j = 1,2,3.
Thedegreeofcontradictionbetweenanytwoattributevalues y1 and y2,isdenotedby c(y1, y2) andit satisfiesthefollowingaxioms: c(y1, y1)= 0, c(y1, y2)= c(y2, y1).
Subsequently, (X, B, Ym, d, c) iscalledamulti-attributeplithogenichypersoftset.Foran m-tuple (y1, y2, , ym ) ∈ Ym, yi ∈ Bi,1 ≤ i ≤ m,amulti-attributeplithogenichypersoftset F : Ym → P(U ) is mathematicallywrittenas F({y1, y2,..., ym })= {x(dx (y1), dx (y2),..., dx (ym )), x ∈ X}.
Example1. Considerthepreviousexampleinwhich U = {m1, m2, m3, ... , m10} and X ⊆U isgivenby X = {m2, m3, m5, m8}.Theattributesbelongtotheset A = {a1, a2, a3, a4},suchthat a1 =Processorpower, a2 =RAM, a3 =Frontcameraresolution, a4 =Screensizeininches.
Theattributevaluesofa1, a2, a3, a4 arecontainedinthesetsA1, A2, A3, A4 givenbelow:
A1 = {dual-core,quad-core,octa-core},
A2 = {2GB,4GB,8GB,16GB},
A3 = {2MP,5MP,8MP,16MP},
A4 = {4,4.5,5,5.5,6}.
1.Uni-attributeplithogenichypersoftset
Considerthemostdemandingfeatureofamobilephonegivenbytheattribute a3 thatstandsfor frontcameraresolution.Thesetofattributevaluesof a3 is A3 = {2MP,5MP,8MP,16MP}.Then, theuni-attributeplithogenichypersoftset F : A3 → P(U ) isgivenby
F(γ)= {x(dx (γ)), ∀ γ ∈ A3, x ∈ X}, where dx (γ) denotesthedegreeofappurtenanceof x ∈ X,totheset X,w.r.t.theattributevalue γ ∈ A3.Foranattributevalue16MP ∈ A3,wehave
F(16MP)= {m5(dm5 (16MP)), m8(dm8 (16MP))},
2.Multi-attributeplithogenichypersoftset
Let B = {a3, a4} bethesetofattributesrequiredbythecustomer.Therefore,weneed A3 and A4 givenby A3 = {2MP,5MP,8MP,16MP}, A4 = {4,4.5,5,5.5,6}
Supposethatthecustomerisinterestedtobuyamobilephonewithspecificrequirementsof16MPfront camerawith5.5inchscreensize.Inthiscase,wetake (16MP,5.5) ∈ A3 × A4 andamulti-attribute plithogenichypersoftset F : A3 × A4 → P(U ) isgivenby F({16MP,5.5})= {m5 (dm5 (16MP), dm5 (5.5)) , m8 (dm8 (16MP), dm8 (5.5))}, where dm5 (γ) standsforthedegreeofappurtenanceof m5 totheset X w.r.t.theattributevalue γ ∈ (16MP,5.5)
3.2.TheSecondClassification
Thisclassificationisbasedonthenatureoftheattributevaluedegreeofappurtenancethatmay becrisp,fuzzy,intuitionisticfuzzy,orneutrosophicdegreeofappurtenance.
3.2.1.PlithogenicCrispHypersoftSet
Aplithogenichypersoftset X iscrispiftheappurtenancedegree dx (γ) ofeachmember x ∈ X, w.r.t.eachattributevalue γ,iscrisp,i.e., dx (γ) iseither0or1.
3.2.2.PlithogenicFuzzyHypersoftSet
Iftheappurtenancedegree dx (γ) ofeachmember x ∈ X,w.r.t.eachattributevalue γ,isfuzzy, thenitiscalledtheplithogenicfuzzyhypersoftset.Mathematically, dx (γ) ∈ P([0,1])
3.2.3.PlithogenicIntuitionisticFuzzyHypersoftSet
Iftheattributevalueappurtenancedegree dx (γ) ofeach x ∈ X,w.r.t.eachattributevalue, isintuitionisticfuzzydegree,thenitiscalledtheplithogenicintuitionisticfuzzyhypersoftset. Mathematically,itiswrittenas dx (γ) ∈ P([0,1]2)
3.2.4.PlithogenicNeutrosophicHypersoftSet
Aplithogenichypersoftset X iscalledplithogenicneutrosophichypersoftsetif dx (γ) ∈ P([0,1]3)
Example2. For (octa-core,8GB,16MP,5.5) ∈ C,wehavethefollowingresults:
1.Plithogeniccrisphypersoftset
F({octa-core,8GB,16MP,5.5})= {m5(1,1,1,1), m8(1,1,1,1)}.
2.Plithogenicfuzzyhypersoftset
F({octa-core,8GB,16MP,5.5})= {m5(0.9,0.2,1,0.75), m8(0.5,0.5,0.25,0.9)}.
3.Plithogenicintuitionisticfuzzyhypersoftset
F({octa-core,8GB,16MP,5.5})={m5 ((0.9,0.1), (0.2,0.6), (1,0), (0.75,0.1)) , m8((0.5,0.25), (0.5,0.5), (0.25,0.1), (0.9,0))}
4.Plithogenicneutrosophichypersoftset
F({octa-core,8GB,16MP,5.5})={m5((0.9,0.7,0.1), (0.2,0.3,0.6), (1,0.25,0), (0.75,0.3,0.1)), m8((0.5,1,0.25), (0.5,0.9,0.5), (0.25,0.7,0.1), (0.9,0.8,0))}
3.3.TheThirdClassification
Thisclassificationisbasedonthepropertiesofattributevaluesanddegreeofappurtenancefunction.
3.3.1.PlithogenicRefinedHypersoftSet
Let (X, A, C, d, c) beaplithogenichypersoftsetand A denotethesetofattributevaluesof anattribute a.Ifanattributevalue γ ∈ A oftheattribute a issubdividedorsplitintoatleast twoormoreattributesub-values γ1, γ2, γ3, ... ∈ A,suchthattheattributesub-valuedegreeof appurtenancefunction d(x, γi ) ∈ P([0,1]j ),for i = 1,2,3, ... and j = 1,2,3 forfuzzy,intuitionistic fuzzy,neutrosophicdegreeofappurtenance,respectively,then X iscalledarefinedplithogenic hypersoftset.Itisrepresentedas (Xr, A, C, d, c).
3.3.2.PlithogenicHypersoftOverset
Ifthedegreeofappurtenanceofanyelement x ∈ X w.r.t.anyattributevalue γ ∈ A of anattribute a isgreaterthan1,i.e., d(x, γ) > 1,then X iscalledaplithogenichypersoftoverset. Itisrepresentedas (Xo, A, C, d, c)
3.3.3.PlithogenicHypersoftUnderset
Ifthedegreeofappurtenanceofanyelement x ∈ X w.r.t.anyattributevalue γ ∈ A of anattribute a lessthan0,i.e., d(x, γ) < 0,then X iscalledaplithogenichypersoftunderset. Itisrepresentedas (Xu, A, C, d, c)
3.3.4.PlithogenicHypersoftOffset
Aplithogenichypersoftset (X, A, C, d, c) iscalledaplithogenichypersoftoffsetifitisboth anoversetandanunderset.Mathematically,if d(x1, γ1) > 1 and d(x2, γ2) < 0 forthesameor differentattributevalues γ1, γ2 ∈ A thatcorrespondtothesameordifferentmembers x1, x2 ∈ X, then (Xoff, A, C, d, c) isaplithogenichypersoftoffset.
3.3.5.PlithogenicHypersoftMultiset
Ifanelement x ∈ X repeatsitselfintotheset X withsameplithogeniccomponentsgivenby
x(c1, c2,..., cn ), x(c1, c2,..., cn ), orwithdifferentplithogeniccomponentsgivenby x(c1, c2,..., cn ), x(d1, d2,..., dn ), then (Xn, A, C, d, c) iscalledaplithogenichypersoftmultiset.
3.3.6.PlithogenicBipolarHypersoftSet
Iftheattributevalueappurtenancedegreefunctionisgivenby d : X × C → P([ 1,0]j ) × P([0,1]j ), ∀ x ∈ X, where j = 1,2,3, then, (Xb, A, C, d, c) iscalledplithogenicbipolarhypersoftset.Itmaybenotedthat, foranattributevalue γ, d(x, γ) allotsanegativedegreeofappurtenancein [ 1,0] andapositive degreeofappurtenancein [0,1] toeachelement x ∈ X withrespecttoeachattributevalue γ Remark3. Theconceptofplithogenicbipolarhypersoftsetcanbeextendedtoplithogenictripolarhypersoftset andsoonuptoplithogenicmultipolarhypersoftset.
3.3.7.PlithogenicComplexHypersoftSet
Ifforany x ∈ X,theattributevalueappurtenancedegreefunction,withrespecttoanyattribute value γ,isgivenby d : X × C → P([0,1]j ) × P([0,1]j ), j = 1,2,3, suchthat d(x, γ) isacomplexnumberoftheform c1 eic2 ,where c1 (amplitude)and c2 (phase)are subsetsof [0,1],then (Xcom, A, C, d, c) iscalledaplithogeniccomplexhypersoftset.
Example3. Considerthesameexampleofchoosingasuitablemobilephonefromtheset X = {m2, m3, m5, m8} Theattributesarea1, a2, a3, a4,whoseattributevaluesarecontainedinthesetsA1, A2, A3, A4
1.Plithogenicrefinedhypersoftset
Consideranattribute a4 = screensizeininches whoseattributevaluesbelongtotheset A4 = {4,4.5,5,5.5,6}.Arefinementof A4 isgivenby A4 = {4,4.5,4.7,5,5.5,5.8,6}, suchthatforall x ∈ X, d(x, γ) ∈ P([0,1]j ), ∀ γ ∈ A4.
Therefore,aplithogenicrefinedhypersoftset Fr : A4 → P(U ) isgivenby Fr ({4,4.5,4.7,5,5.5,5.8,6})={m5 dm5 (4), dm5 (4.5), dm5 (4.7), dm5 (5), dm5 (5.5), dm5 (5.8), dm5 (6) , m8 dm8 (4), dm8 (4.5), dm8 (4.7), dm8 (5), dm8 (5.5), dm8 (5.8), dm8 (6) }.
2.Plithogenichypersoftoverset
Leteachattributevaluehasasingle-valuedfuzzydegreeofappurtenancetoalltheelementsof X.
Subsequently,for (octa-core,8GB,16MP,5.5) ∈ C,aplithogenichypersoftoverset Fo : C → P(U ) is givenby
Fo ({octa-core,8GB,16MP,5.5})= {m5(0.9,0.2,1.3,0.75), m8(0.5,0.5,0.25,0.9)}
Itmaybenotedthat dm5 (16MP) > 1.
3.Plithogenichypersoftunderset
Aplithogenichypersoftundersetdefinedbythefunction Fu : C → P(U ) isgivenby
Fu ({octa-core,8GB,16MP,5.5})= {m5(0.9,0.2, 0.3,0.75), m8(0.5,0.5,0.25,0.9)}.
Itmaybenotedthat dm5 (16MP) < 0.
4.Plithogenichypersoftoffset
Aplithogenichypersoftoffsetisafunction Foff : C → P(U ),asgivenby
Foff({octa-core,8GB,16MP,5.5})= {m5(0.9,0.2, 0.3,0.75), m8(0.5,1.5,0.25,0.9)}.
Notethat dm5 (16MP) < 0and dm8 (8GB) > 1.
5.Plithogenichypersoftmultiset
Aplithogenichypersoftmultiset Fm : C → P(U ) isgivenby
Fm ({octa-core,8GB,16MP,5.5})= {m5(0.9,0.2,0.3,0.75), m5(0.7,0.1,0.9,1), m8(0.5,0.5,0.25,0.9)}
Itshouldbenotedthattheelement m5 repeatsitselfwithdifferentplithogeniccomponents.
6.Plithogenicbipolarhypersoftset
Aplithogenicbipolarhypersoftset F2 : C → P(U ) isgivenby
F2({octa-core,8GB,16MP,5.5})={m5({−0.1,0.9}, {−1,0.2}, {−0.9,0.3}, {−0.5,1}), m8({−0.5,0}, {−0.9,1}, {−0.2,0.2}, {−1,0.8})}
7.Plithogeniccomplexhypersoftset
Aplithogeniccomplexhypersoftset Fcom : C → P(U ) isgivenby
Fcom({octa-core,8GB,16MP,5.5})={m5(0.9e0.5i,0.2e0.9i,0.3e0.25i,0.75ei ), m8(0.5e0.5i , e0.3i,0.25e0.75i,0.9e0.1i )}
3.4.TheFourthClassification
Theattributevaluedegreeofappurtenancemaybeasinglecrispvaluein [0,1],afinitediscrete setoranintervalvaluein [0,1].Therefore,wehavethefollowingclassificationofPHSS.
3.4.1.Single-ValuedPlithogenicHypersoftSet
Aplithogenichypersoftsetiscalledasingle-valuedplithogenichypersoftsetiftheattributevalue appurtenancedegreeisasinglenumberin [0,1].
3.4.2.HesitantPlithogenicHypersoftSet
Iftheattributevaluedegreeofappurtenanceisafinitediscretesetoftheform {m1, m2, , mi }, 1 ≤ i < ∞,includedin [0,1],thensuchaplithogenichypersoftsetiscalledahesitantplithogenic hypersoftset.
3.4.3.Interval-ValuedPlithogenicHypersoftSet
Aplithogenichypersoftsetisknownasaninterval-valuedplithogenichypersoftsetiftheattribute valueappurtenancedegreefunctionisanintervalvaluein [0,1].Theintervalvaluemaybeanopen, closed,orsemiopeninterval.
Example4. For (octa-core,8GB,16MP,5.5) ∈ C,witheachattributevaluehavingfuzzydegreeof appurtenance,wehavethefollowingresults:
1.Single-valuedplithogenichypersoftset
F({octa-core,8GB,16MP,5.5})= {m5(0.9,0.2,1,0.75), m8(0.5,0.5,0.25,0.9)}
Eachattributevalueisassignedasinglevaluein [0,1] asadegreeofappurtenanceto m5 and m8
2.Hesitantplithogenichypersoftset
F({octa-core,8GB,16MP,5.5})={m5({0.9,0.75}, {0.2,0.7}, {1,0.9}, {0.75,0.5}), m8({0.5,0.1}, {0.5,0.9}, {0.25,0}, {0.9,1})}.
3.Interval-valuedplithogenichypersoftset
F({octa-core,8GB,16MP,5.5})={m5([0.25,0.75], [0.2,0.6], [0.1,0.9], [0.75,1]), m8([0.5,0.6], [0.3,0.9], [0.25,0.8], [0.9,1])}
Eachattributevaluehasanintervalvaluedegreeofappurtenancein [0,1] toeachelement m5 and m8
4.TheProposedPHSS-BasedTOPSISwithApplicationtoaParkingProblem
Inthissection,weusetheconceptofPHSSinordertoconstructanovelMCDMmethod, calledPHSS-basedTOPSIS,inwhichweextendTOPSISbasedonPHSSunderfuzzyneutrosophic environment.Moreover,aparkingspotchoiceproblemisconstructedinordertoemploythenewly developedPHSS-basedTOPSIStoproveitsvalidityandefficiency.Twodifferentsetsofalternatives areconsideredfortheapplicationandacomparisonisperformedwithfuzzyTOPSISinbothcases.
4.1.ProposedPHSS-BasedTOPSISAlgorithm
Let U beanon-emptyuniversalset,andlet X ⊆U bethesetofalternativesunderconsideration, givenby X = {x1, x2, ... , xm }.Let C = A1 × A2 × ... × An,where n ≥ 1 and Ai isthesetofall attributevaluesoftheattribute ai, i = 1,2,3, ... , n.Eachattributevalue γ hasacorresponding appurtenancedegree d(x, γ) ofthemember x ∈ X,inaccordancewithsomegivenconditionorcriteria. Ouraimistochoosethebestalternativeoutofthealternativeset X.Theconstructionstepsforthe proposedPHSS-basedTOPSISareasfollows:
S1:Chooseanorderedtuple (γ1, γ2, , γn ) ∈ C andconstructamatrixoforder n × m,whoseentries aretheneutrosophicdegreeofappurtenanceofeachattributevalue γ,withrespecttoeachalternative x ∈ X underconsideration.
S2:Employthenewlydevelopedplithogenicaccuracyfunction Ap,toeachelementofthematrix obtainedinS1,inordertoconverteachelementintoasinglecrispvalue,asfollows: Ap (Tγ , Iγ , Fγ )= Tγ + Iγ + Fγ 3 + Tγd + Iγd + Fγd 3 × cF (γ, γd ), (1) where Tγ , Iγ , Fγ representthemembership,indeterminacy,andnon-membershipdegreesof appurtenanceoftheattributevalue γ totheset X,and Tγd , Iγd , Fγd standforthemembership, indeterminacy,andnon-membershipdegreesofcorrespondingdominantattributevalue,whereas cF (γ, γd ) denotesthefuzzydegreeofcontradictionbetweenanattributevalue γ anditscorresponding dominantattributevalue γd.Thisgivesustheplithogenicaccuracymatrix.
S3:Applythetransposeontheplithogenicaccuracymatrixtoobtaintheplithogenicdecisionmatrix Mp =[mij ]m×n ofalternativesversuscriteria.
S4:Aplithogenicnormalizeddecisionmatrix Np =[yij ]m×n isconstructed,whichrepresentsthe relativeperformanceofalternativesandwhoseelementsarecalculatedasfollows: yij = mij m ∑ i=1 m2 ij
, j = 1,2,3,..., n
S5:Constructaplithogenicweightednormalizeddecisionmatrix Vp =[vij ]m×n = NpWn, where Wn =[w1 w2 ... wn ] isarowmatrixofallocatedweights wk assignedtothecriteria ak, k = 1,2,3, ... , n and ∑ wk = 1, k = 1,2, ... , n.Moreover,alloftheselectioncriteriaareassigned differentweightsbythedecisionmaker,dependingontheirimportanceinthedecisionmakingprocess.
S6:Determinetheplithogenicpositiveidealsolution V + p andplithogenicnegativeidealsolution Vp bythefollowingformula:
V + p = m max i=1 (vij ) if aj ∈ benefitcriteria, m min i=1 (vij ) if aj ∈ costcriteria, j = 1,2,3,..., n , Vp = m min i=1 (vij ) if aj ∈ benefitcriteria, m max i=1 (vij ) if aj ∈ costcriteria, j = 1,2,3,..., n
S7:Calculateplithogenicpositivedistance S+ i andplithogenicnegativedistance Si ofeachalternative from V + p and Vp ,respectively,whileusingthefollowingformulas:
S+ i = n ∑ j=1 (vij v+ i )2 , i = 1,2,3,..., m, Si = n ∑ j=1 (vij vi )2 , i = 1,2,3,..., m
S8:Calculatetherelativeclosenesscoefficient Ci ofeachalternativebythefollowingexpression: Ci = Si S+ i + Si , i = 1,2,3,..., m.
S9:Thehighestvaluefrom {C1, C2, , Cm } belongstothemostsuitablealternative.Similarly, thelowestvaluegivesustheworstalternative.
4.2.ParkingSpotChoiceProblem
Basedontheproposedmethod,aparkingspotchoiceproblemisconstructed.Parkingavehicleat somesuitableparkingspotisaninterestingreallifeMCDMproblem.Anumberofquestionsarisesin mind,forinstance,howmuchwilltheparkingfeebe,howfarisit,willitbeanopenorcoveredarea, howmanytrafficsignalswillbeontheway,etc.Thus,itbecomesachallengingtaskinthepresenceof somanyconsiderablecriteria.Thistaskisformulatedintheformofamathematicalmodelinorder toapplytheproposedtechniquetochoosethemostsuitableparkingspot.Considerapersonata particularlocationontheroad,whowantstoparkhiscaratasuitableparkingplace.Keepinginmind theperson’svariouspreferences,afewnearbyavailableparkingspotsareconsidered,havingdifferent specificationsintermsofparkingfee,distancebetweentheperson’slocationandeachparkingspot, thenumberofsignalsbetweenthecarandtheparkingspot,andtrafficdensityonthewaybetween thecarandtheparkingspot.Figure 2 showsthelocationofcartobeparkedatasuitableparkingspot.
A4 = {low(d1),high(d2),veryhigh(d3)}.
Thedominantattributevaluesof a1, a2, a3, a4 arechosentobe f1, r1, s1 and d1,respectively,andthe single-valuedfuzzydegreeofcontradictionbetweenthedominantattributevalueandallother attributevaluesisgivenbelow.
cF ( f1, f2)= 1 3 , cF ( f1, f3)= 2 3 ,
cF (r1, r2)= 1 6 , cF (r1, r3)= 2 6 , cF (r1, r4)= 3 6 , cF (r1, r5)= 4 6 , cF (r1, r6)= 5 6 ,
cF (s1, s2)= 1 2 ,
cF (d1, d2)= 1 3 , cF (d1, d3)= 2 3
TwodifferentsetsofalternativesareconsideredfortheapplicationofPHSS-basedTOPSIS,alongwith acomparisonwithfuzzyTOPSISineachcase.
4.2.1.Case1
Inthiscase,theparkingspotsunderconsideration(alternatives)arecontainedintheset X ⊆U , givenby X = {P1, P2, P3, P4}
Theneutrosophicdegreeofappurtenanceofeachattributevaluecorrespondingtoeachalternative P1, P2, P3, P4 isgiveninTable 1 Let C = A1 × A2 × A3 × A4 andconsideranelement ( f2, r1, s2, d1) ∈ C forwhichthe correspondingmatrixthatwasobtainedfromTable 1 isgivenbelow:
(0.7,0.9,0.1)(0.6,0.5,0.2)(0.2,0.3,0.6)(0.7,0.9,0.3) (0.8,0.1,0.7)(0.9,0.4,0.5)(0.9,0.4,0.0)(0.8,0.4,0.2) (1.0,0.8,0.6)(0.7,0.5,0.5)(0.4,0.4,0.7)(0.6,0.5,0.7) (0.1,0.2,1.0)(0.3,1.0,0.6)(0.7,0.9,0.2)(0.9,0.7,0.5)
Table1. Degreeofappurtenanceofeachattributevaluew.r.t.toeachalternative.
Sr.Variables P1 P2 P3 P4
1 f1 (0.5,0.1,0.3)(0.5,0.0,0.7)(0.1,0.4,0.5)(0.2,0.1,0.6)
2 f2 (0.7,0.9,0.1)(0.6,0.5,0.2)(0.2,0.3,0.6)(0.7,0.9,0.3)
3 f3 (0.5,0.5,0.1)(0.0,0.1,0.5)(0.1,0.1,0.9)(0.5,0.7,0.2)
4 r1 (0.8,0.1,0.7)(0.9,0.4,0.5)(0.9,0.4,0.0)(0.8,0.4,0.2)
5 r2 (0.9,0.3,0.2)(0.6,0.1,0.0)(0.5,0.2,0.4)(0.9,0.1,0.4)
6 r3 (0.9,0.1,0.3)(0.8,0.3,0.1)(0.6,0.0,0.6)(0.2,0.2,0.5)
7 r4 (0.8,0.3,0.2)(1.0,0.1,0.5)(0.8,0.5,0.1)(0.7,0.3,0.6)
8 r5 (1.0,0.3,0.2)(1.0,0.3,0.2)(0.8,0.2,0.8)(0.6,0.5,0.6)
9 r6 (0.8,0.1,0.0)(0.6,0.8,0.5)(0.9,0.7,0.1)(0.4,0.8,0.7)
10 s1 (0.0,0.5,0.5)(0.4,0.1,0.6)(0.2,0.2,0.7)(0.8,0.3,0.4)
11 s2 (1.0,0.8,0.6)(0.7,0.5,0.5)(0.4,0.4,0.7)(0.6,0.5,0.7)
12 d1 (0.1,0.2,1.0)(0.3,1.0,0.6)(0.7,0.9,0.2)(0.9,0.7,0.5)
13 d2 (0.1,0.4,0.8)(0.2,0.2,0.8)(0.2,0.6,0.3)(0.2,0.8,0.5)
14 d3 (0.5,0.6,0.9)(0.9,0.6,0.3)(0.9,0.7,0.5)(0.6,0.7,0.6)
A.ApplicationofPHSS-basedTOPSISforCase1
Applytheplithogenicaccuracyfunction(1)tothematrix(2)inordertoobtaintheplithogenic accuracymatrixgivenby:
Theplithogenicdecisionmatrix Mp isconstructedbytakingthetransposeoftheplithogenic accuracymatrix.Itisasquarematrixoforder4,givenby Mp =
Acorrespondingtable,asshowninTable 2,ofalternativesversuscriteriamayalsobedrawnto seethesituationinaclearway.
Table2. Alternativesversuscriteriatable. Al/Crf2 r1 s2 d1 P1 0.66670.53330.96670.4333 P2 0.56670.60000.75000.6333 P3 0.47780.43330.68330.6000 P4 0.73330.46670.85000.7000
Aplithogenicnormalizeddecisionmatrix
p isobtainedas:
Aweightednormalizedmatrix W4 isconstructedas: W4 =[ 0.4,0.22,0.15,0.23 ],(3) whereastheplithogenicweightednormalizeddecisionmatrix Vp =[vij ]4×4 isgiven,asfollows: Vp
Theplithogenicpositiveidealsolution V + p andplithogenicnegativeidealsolution Vp are determined,asfollows: V + p = {0.1544,0.0930,0.0625,0.0831}, Vp = {0.2370,0.1288,0.0885,0.1342}.
Theplithogenicdistanceofeachalternativefromthe V + p and Vp ,respectively,isdeterminedas: S+ = 0.0697 0.0601 0.0320 0.0986
, S = 0.0573 0.0588 0.0956 0.0305
Therelativeclosenesscoefficient Ci, i = 1,2,3,4,ofeachalternativeiscomputedas: C1 = 0.4511, C2 = 0.4944, C3 = 0.7494, C4 = 0.2366.
Thehighestvaluecorrespondstothemostsuitablealternative.Since C3 = 0.7494 isthemaximum valueanditcorrespondsto P3,therefore,themostsuitableparkingspotis P3.TheTable 3 isconstructed torankallalternativesunderconsideration.
Table3. PHSS-basedTOPSISrankingtable. S+ i Si Ci Ranking P1 0.06970.05730.45113 P2 0.06010.05880.49442 P3 0.03200.09560.74941 P4 0.09860.03050.23664
AbargraphpresentedinFigure 3 isgiven,inwhichallalternatives P1, P2, P3, P4 arerankedby PHSS-basedTOPSIS.
Figure3. RankingofParkingSpotsbyPHSS-basedTOPSISforCase1. Itisevidentthattheparkingspot P3 isthemostsuitableplacetoparkthecarwhile P4 isnota goodchoiceforparkingbasedontheselectioncriteria.
B.ApplicationofFuzzyTOPSISforCase1
InordertoseetheimplementationoffuzzyTOPSIS[32 34]forthecurrentscenariooftheparking problem,weapplytheaverageoperator[27,35]toeachelementofthematrix 2 andtakethetranspose oftheresultingmatrixinordertoobtainthedecisionmatrixgivenby: M =
0.56670.53330.80000.4333 0.43330.60000.56670.6333 0.36670.43330.50000.6000 0.63330.46670.60000.7000
ApplyingthefuzzyTOPSIStothedecisionmatrix M,alongwiththesameweights giveninmatrix(3),weobtainthevaluesofpositivedistance S+,negativedistance S ,relative closeness Ci andrankingofeachalternative,asgiveninTable 4
Table4. FuzzyTOPSISrankingtable. S+ i Si Ci Ranking
P1 0.08880.05920.40003 P2 0.05910.08410.58722 P3 0.03200.11760.78631 P4 0.11700.03730.24174
AbargraphinFigure 4 isgiveninwhichallalternatives P1, P2, P3, P4 arerankedbyFuzzyTOPSIS. AcomparisonisshowninTable 5,inwhichitcanbeseenthattheresultobtainedbytheproposed PHSS-basedTOPSISisalignedwiththatoffuzzyTOPSIS.
Figure4. RankingofParkingSpotsbyFuzzyTOPSISforCase1.
Comparisonanalysisforcase1.
ItisobservedinTable 5 thattheresultsobtainedbybothmethodscoincideintermsoftheranking ofeachalternative,butdifferinthevaluesoftherelativeclosenessofeachalternative.Itisdueto thenatureoftheMCDMprobleminhandinwhicheachalternativeneedstobeevaluatedagainst
eachattributevaluepossessinganeutrosophicdegreeofappurtenancew.r.t.eachalternativeanda contradictiondegreeisdefinedbetweeneachattributevalueanditscorrespondingdominantattribute valuetobetakenintoconsiderationinthedecisionprocess.Insuchacase,theproposedPHSS-based TOPSISproducesamorereliablerelativeclosenessofeachalternative,asitcanbeenseeninthe parkingspotchoiceproblemthatwaschosenforthestudy.Therefore,itisworthnotingthatthe proposedPHSS-basedTOPSIScanberegardedasageneralizationoffuzzyTOPSIS[32],because thefuzzyTOPSIScannotbedirectlyappliedtoMCDMproblemsinwhichtheattributevalueshave aneutrosophicdegreeofappurtenancewithrespecttoeachalternative.Inthecaseoftheparking problem,fuzzyTOPSISisappliedafterapplyingsimpleaverageoperatortotheneutrosophicelements ofthematrix(2).However,itdoesnottakesintoaccountthedegreeofcontradictionbetweenthe attributevalues,whichisthelimitationoffuzzyTOPSIS.Thisconcernispreciselyaddressedbythe proposedPHSS-basedTOPSIS.
4.2.2.Case2
Inthiscase,thesetofparkingspotsunderconsiderationisgivenby
X = {P1, P5, P6, P7}
Theneutrosophicdegreeofappurtenanceofeachattributevaluethatcorrespondstoeachalternative of {P1, P5, P6, P7} isgiveninTable 6.
Table6. Degreeofappurtenanceofeachattributevaluew.r.teachalternative.
Sr.Variables P1 P5 P6 P7
1 f1 (0.5,0.1,0.3)(0.6,0.6,0.8)(0.7,0.2,0.4)(0.9,0.5,0.2)
2 f2 (0.7,0.9,0.1)(0.8,0.8,0.5)(0.4,0.4,0.7)(0.7,0.2,0.1)
3 f3 (0.5,0.5,0.1)(0.4,0.2,0.5)(1.0,0.5,0.9)(1.0,0.7,0.6)
4 r1 (0.8,0.1,0.7)(0.9,0.5,0.2)(0.5,0,0.9)(0.8,0.6,0.1)
5 r2 (0.9,0.3,0.2)(0.5,0.4,0.2)(0.7,0.5,0.4)(0.9,0.6,0.8)
6 r3 (0.9,0.1,0.3)(0.5,0.7,0.3)(0.9,1.0,0.6)(0.2,0,1.0)
7 r4 (0.8,0.3,0.2)(1.0,0.2,1.0)(1.0,0.5,0.7)(0.8,0.8,0.9)
8 r5 (0.2,0.3,0.9)(1.0,0.1,0.8)(0.4,0.6,0.8)(0.8,0.6,0.6)
9 r6 (0.5,0.7,0.5)(0.8,0.2,0.0)(0.6,0.3,0.7)(0.0,0.9,0.9)
10 s1 (0,0.5,0.5)(0.8,0.4,0.6)(0.9,0.2,0.2)(0.8,0.4,0.7)
11 s2 (1.0,0.8,0.6)(0.7,1.0,0.2)(0.2,0.4,0.7)(0.9,0,1.0)
12 d1 (0.1,0.2,1.0)(1.0,0.4,0.3)(0.7,0.5,0.6)(0.8,0.5,0.7)
13 d2 (0.1,0.4,0.8)(0.7,1.0,0.8)(0.6,0.6,1.0)(1.0,0.8,0.8)
14 d3 (0.5,0.6,0.9)(1.0,0.6,0.5)(1.0,1.0,0.5)(1.0,0.5,0.8)
Let C = A1 × A2 × A3 × A4 andconsideranelement ( f2, r1, s2, d1) ∈ C forwhichthe correspondingmatrixobtainedfromTable 6,isgivenbelow:
(0.7,0.9,0.1)(0.8,0.8,0.5)(0.4,0.4,0.7)(0.7,0.2,0.1) (0.8,0.1,0.7)(0.9,0.5,0.2)(0.5,0.0,0.9)(0.8,0.6,0.1) (1.0,0.8,0.6)(0.7,1.0,0.2)(0.2,1.0,0.7)(0.9,0.0,1.0) (0.1,0.2,1.0)(1.0,0.4,0.3)(0.7,0.5,0.6)(0.8,0.5,0.7)
TheproposedPHSS-basedTOPSISandfuzzyTOPSISareemployed,asfollows: A.ApplicationofPHSS-BasedTOPSISforCase2
Theplithogenicaccuracymatrixinthiscaseisgivenby
0.66670.92220.64440.5111
Plithogenicdecisionmatrix Mp isgivenby Mp =
0.66670.53330.96670.4333
Aplithogenicnormalizeddecisionmatrix Np isthenconstructedas: Np =
0.47480.52050.54640.3612
Theplithogenicweightednormalizeddecisionmatrix Vp isgiven,asfollows: Vp =
0.18990.11450.08200.0831 0.26270.12880.07910.1214 0.18360.09300.05510.1150 0.14560.10020.08050.1342
Theplithogenicpositiveidealsolution V + p andplithogenicnegativeidealsolution Vp are determined,suchthat V + p = {0.1456,0.0930,0.0551,0.0831}, Vp = {0.2627,0.1288,0.0820,0.1342}.
Theplithogenicpositivedistance S+,plithogenicnegativedistance S ,relativecloseness Ci,and rankingofeachalternativeisshowninTable 7
Table7. PHSS-basedTOPSISrankingtable. S+ i Si Ci Ranking P1 0.05610.09010.61633 P5 0.13060.01310.09124 P6 0.04960.09290.65182 P7 0.05760.12060.67691
AgraphicalrepresentationoftherankingofallalternativesobtainedbyPHSS-basedTOPSIS, isshowninFigure 5
Figure5. RankingofParkingSpotsbyPHSS-basedTOPSISforCase2.
Itcanbeseenthattheparkingspot P7 isthemostsuitablealternativeinthelightofchosencriteria.
B.ApplicationofFuzzyTOPSISforCase2
Inthiscase,thedecisionmatrix M fortheimplementationoffuzzyTOPSISisgivenby
0.56670.53330.80000.4333 0.70000.60000.63330.6333 0.50000.43330.63330.6000 0.33330.46670.63330.7000
ByimplementingthefuzzyTOPSIStothematrix M,withthesameweightsgivenin(3),thevalues ofpositivedistance S+,negativedistance S ,relativecloseness Ci,andrankingofeachalternativeare showninTable 8.
Table8. FuzzyTOPSISrankingtable. S+ i Si Ci Ranking P1 0.09080.07240.44393 P5 0.14530.02240.13374 P6 0.06940.08630.55432 P7 0.05160.13970.73011
TherankingofallalternativescanalsobeenvisualizedasabargraphinFigure 6,inwhichall alternatives P1, P5, P6, P7 arerankedbyFuzzyTOPSIS.
ThemostsuitableparkingspotobtainedbyfuzzyTOPSISisalso P7
AcomparisonofrankingsobtainedbyPHSS-basedTOPSISandfuzzyTOPSISisshownin Table 9 forcase2.
Itmaybenotedthatsimilarresultsareobtainedincase2,withthehelpofproposedPHSS-based TOPSISandfuzzyTOPSISwithexactlysamerankingofeachalternative,butwithaconsiderably differentvaluesoftherelativeclosenessofeachalternativeasshowninTable 9.Therefore,itis accomplishedthattheresultsthatwereobtainedbythePHSS-basedTOPSISarevalidandmore reliableandPHSS-basedTOPSIScanberegardedasthegeneralizationoffuzzyTOPSISonthebasisof thestudyconductedinthearticle.
Figure6. RankingofParkingSpotsbyFuzzyTOPSISforCase2.
Table9. Comparisonanalysisforcase2.
Sr.ParkingsPHSS-BasedTOPSISRankingFuzzyTOPSISRanking 1 P1 3rd3rd 2 P5 4th4th 3 P6 2nd2nd 4 P7 1st1st
5.Conclusions
IthasalwaysbeenachallengingtasktodealwithreallifeMCDMproblems,duetothe involvementofmanycomplexitiesanduncertainties.Inparticular,somereallifeMCDMproblems aredesignedinawaythatthegivenattributesneedtobefurtherdecomposedintotwoormore attributevaluessuchthateachalternativeisthenrequiredtobeevaluatedagainsteachattribute valueinordertoperformadetailedanalysistoreachafairconclusion.Todealwithsuchsituations, anovelPHSS-basedTOPSISisproposedinthepresentstudy,anditisappliedtoaMCDMparking problemwithdifferentchoicesofthesetofalternativesandacomparisonwithfuzzyTOPSISisdone toprovethevalidityandefficiencyoftheproposedmethod.Alloftheresultsarequitepromisingand graphicallydepictedforaclearunderstanding.Moreover,thealgorithmoftheproposedmethodis producedinMATLABinordertobroadenthescopeofthestudytootherresearchareas,including graphtheory,machinelearning,patternrecognition,etc.
AuthorContributions: Conceptualization,M.S.andU.A.;methodology,M.R.A.andM.S.;validation,M.R.A.and U.A.;formalanalysis,M.R.A.andU.A.;investigation,M.S.andM.-S.Y.;writing—originaldraftpreparation,M.R.A. andM.S.;writing—reviewandediting,U.A.andM.-S.Y.;supervision,M.S.andM.-S.Y.;fundingacquisition, M.-S.Y..Allauthorshavereadandagreedtothepublishedversionofthemanuscript.
Funding: TheAPCwasfundedinpartbytheMinistryofScienceandtechnology(MOST)ofTaiwanunderGrant MOST-109-2118-M-033-001-.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
References
1. Zadeh,L.A.Fuzzysets. Inform.Control 1965, 8,338–353.[CrossRef]
2. Yang,M.S.;Hung,W.L.;Chang-Chien,S.J.OnasimilaritymeasurebetweenLR-typefuzzynumbersandits applicationtodatabaseacquisition. Int.J.Intell.Syst. 2005, 20,1001–1016.[CrossRef]
3. Meng,F.;Tang,J.;Fujita,H.Consistency-basedalgorithmsfordecision-makingwithintervalfuzzypreference relations. IEEETrans.FuzzySyst. 2019, 27,2052–2066.[CrossRef]
Symmetry 2020, 12,1855
4. Ruiz-Garca,G.;Hagras,H.;Pomares,H.;Ruiz,I.R.Towardafuzzylogicsystembasedongeneralformsof intervaltype-2fuzzysets. IEEETrans.FuzzySyst. 2019, 27,2381–2395.[CrossRef]
5. Ullah,K.;Hassan,N.;Mahmood,T.;Jan,N.;Hassan,M.Evaluationofinvestmentpolicybasedon multi-attributedecision-makingusingintervalvaluedT-sphericalfuzzyaggregationoperators. Symmetry 2019, 11,357.[CrossRef]
6. Atanassov,K.T.Intuitionisticfuzzysets. FuzzySetsSyst. 1986, 20,87–96.[CrossRef]
7. Hwang,C.M.;Yang,M.S.;Hung,W.L.Newsimilaritymeasuresofintuitionisticfuzzysetsbasedonthe Jaccardindexwithitsapplicationtoclustering. Int.J.Intell.Syst. 2018, 33,1672–1688.[CrossRef]
8. Garg,H.;Kumar,K.Linguisticinterval-valuedatanassovintuitionisticfuzzysetsandtheirapplicationsto groupdecisionmakingproblems. IEEETrans.FuzzySyst. 2019, 27,2302–2311.[CrossRef]
9. Roy,J.;Das,S.;Kar,S.;Pamucar,D.AnextensionoftheCODASapproachusinginterval-valuedintuitionistic fuzzysetforsustainablematerialselectioninconstructionprojectswithincompleteweightinformation. Symmetry 2019, 11,393.[CrossRef]
10. Yang,M.S.;Hussian,Z.;Ali,M.Beliefandplausibilitymeasuresonintuitionisticfuzzysetswithconstruction ofbelief-plausibilityTOPSIS. Complexity 2020,1–12.[CrossRef]
11. Smarandache,F. AUnifyingFieldinLogics:NeutrosophicLogic.Neutrosophy,NeutrosophicSet,Probability, andStatistics,2nded.;AmericanResearchPress:Rehoboth,DE,USA,2000.
12. Majumdar,P.;Samanta,S.K.Onsimilarityandentropyofneutrosophicsets. J.Intell.FuzzySyst. 2014, 26,1245–1252.[CrossRef]
13. Li,X.;Zhang,X.;Park,C.GeneralizedintervalneutrosophicChoquetaggregationoperatorsand theirapplications. Symmetry 2018, 10,85.[CrossRef]
14. Abdel-Basset,M.;Mohamed,M.Anovelandpowerfulframeworkbasedonneutrosophicsetstoaid patientswithcancer. FutureGener.Comput.Syst. 2019, 98,144–153.[CrossRef]
15. Vasantha,W.B.;Kandasamy,I.;Smarandache,F.Neutrosophiccomponentssemigroupsandmultiset neutrosophiccomponentssemigroups. Symmetry 2020, 12,818.
16. Molodtsov,D.Softsettheory-firstresults. Comput.Math.Appl. 1999, 37,19–31.[CrossRef]
17. Maji,P.K.;Biswas,R.;Roy,A.R.Softsettheory. Comput.Math.Appl. 2003, 45,555–562.[CrossRef]
18. Ali,M.I.;Feng,F.;Liu,X.;Min,W.K.;Shabir,M.Onsomenewoperationsinsoftsettheory. Comput.Math.Appl. 2009, 57,1547–1553.[CrossRef]
19. Inthumathi,V.;Chitra,V.;Jayasree,S.Theroleofoperatorsonsoftsetsindecisionmakingproblems. Int.J.Comput.Appl.Math. 2017, 12,899–910.
20. Feng,G.;Guo,X.Anovelapproachtofuzzysoftset-basedgroupdecision-making. Complexity 2018, 2018,2501489.[CrossRef]
21. Biswas,B.;Bhattacharyya,S.;Chakrabarti,A.;Dey,K.N.;Platos,J.;Snasel,V.Colonoscopycontrast-enhanced byintuitionisticfuzzysoftsetsforpolypcancerlocalization. Appl.SoftComput. 2020, 95,106492.[CrossRef]
22.
Smarandache,F.Extensionofsoftsettohypersoftset,andthentoplithogenichypersoftset. Neutrosophic SetsSyst. 2018, 22,168–170.
23. Saeed,M.;Ahsan,M.;Siddique,M.K.;Ahmad,M.R.Astudyofthefundamentalsofhypersoftsettheory. Int.Sci.Eng.Res. 2020, 11,320–329.
24. Smarandache,F. Plithogeny,PlithogenicSet,Logic,Probobility,andStatistics;Pons:Brussels,Belgium,2017.
25.
Smarandache,F.Plithogenicset,anextensionofcrisp,fuzzy,intuitionisticfuzzy,andneutrosophic sets-revisited. NeutrosophicSetsSyst. 2018, 21,153–166.
26. Collan,M.;Luukka,P.EvaluatingR&Dprojectsasinvestmentsbyusinganoverallrankingfromfournew fuzzysimilaritymeasure-basedTOPSISvariants. IEEETrans.FuzzySyst. 2014, 22,505–515.
27. Saqlain,M.;Saeed,M.;Ahmad,M.R.;Smarandache,F.GeneralizationofTOPSISforneutrosophichypersoft setusingaccuracyfunctionanditsapplication. NeutrosophicSetsSyst. 2019, 27,131–137.
28. Khalil,A.M.;Cao,D.;Azzam,A.;Smarandache,F.;Alharbi,W.R.Combinationofthesingle-valued neutrosophicfuzzysetandthesoftsetwithapplicationsindecision-making. Symmetry 2020, 12,1361. [CrossRef]
29. Abdel-Basset,M.;Mohamed,R.AnovelplithogenicTOPSIS-CRITICmodelforsustainablesupplychain riskmanagement. J.Clean.Prod. 2020, 247,119586.[CrossRef]
30. Schumann,A.p-Adicvaluedlogicalcalculiinsimulationoftheslimemouldbehaviour. J.Appl.Non-Class.Logics 2015, 25,125–139.[CrossRef]
31. Schumann,A.p-Adicmultiple-validityandp-adicvaluedlogicalcalculi. J.Mult.-ValuedLog.SoftComput. 2007, 13,29–60.
32. Yang,T.;Hung,C.C.Multiple-attributedecisionmakingmethodsforplantlayoutdesignproblem. Robot.Comput.-Integr. 2007, 23,126–137.[CrossRef]
33. Kabir,G.;Hasin,M.Comparativeanalysisoftopsisandfuzzytopsisfortheevaluationoftravelwebsite servicequality. Int.Qual.Res. 2012, 6,169–185.
34. Zhang,L.;Zhan,J.;Yao,Y.IntuitionisticfuzzyTOPSISmethodbasedonCVPIFRSmodels:Anapplicationto biomedicalproblems. Inf.Sci. 2020, 517,315–339.[CrossRef]
35. Yager,R.R.Thepoweraverageoperator. IEEETrans.Syst.ManCybern.Syst.Hum. 2001, 31,724–731. [CrossRef]
Publisher’sNote: MDPIstaysneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutional affiliations.
c 2020bytheauthors.LicenseeMDPI,Basel,Switzerland.Thisarticleisanopenaccess articledistributedunderthetermsandconditionsoftheCreativeCommonsAttribution (CCBY)license(http://creativecommons.org/licenses/by/4.0/).
