Spherical Fuzzy Logarithmic Aggregation Operators Based on Entropy

SphericalFuzzyLogarithmicAggregationOperators BasedonEntropyandTheirApplicationinDecision SupportSystems

YunJin 1,ShahzaibAshraf 2 andSaleemAbdullah 2,*

1 SchoolofEconomicandManagement,NanjingUniversityofAeronauticsandAstronautics, Nanjing211106,China

2 DepartmentofMathematics,AbdulWaliKhanUniversity,Mardan23200,Pakistan

* Correspondence:saleemabdullah@awkum.edu.pk;Tel.:+92-333-9430925

Received:2June2019;Accepted:23June2019;Published:26June2019

Abstract: Keepinginviewtheimportanceofnewdefinedandwellgrowingsphericalfuzzysets, inthisstudy,weproposedanovelmethodtohandlethesphericalfuzzymulti-criteriagroup decision-making(MCGDM)problems.Firstly,wepresentedsomenovellogarithmicoperations ofsphericalfuzzysets(SFSs).Then,weproposedseriesofnovellogarithmicoperators,namely sphericalfuzzyweightedaverageoperatorsandsphericalfuzzyweightedgeometricoperators.We proposedthesphericalfuzzyentropytofindtheunknownweightsinformationofthecriteria.We studysomeofitsdesirablepropertiessuchasidempotency,boundaryandmonotonicityindetail. Finally,thedetailedstepsforthesphericalfuzzydecision-makingproblemsweredeveloped,anda practicalcasewasgiventocheckthecreatedapproachandtoillustrateitsvalidityandsuperiority. Besidesthis,asystematiccomparisonanalysiswithotherexistentmethodsisconductedtoreveal theadvantagesofourproposedmethod.Resultsindicatethattheproposedmethodissuitableand effectiveforthedecisionprocesstoevaluatetheirbestalternative.

Keywords: sphericalfuzzysets;logarithmicsphericaloperationallaws;logarithmicspherical aggregationoperators;entropy;multi-criteriagroupdecisionmaking(MCGDM)problems

1.Introduction

Thecomplicationofasystemisgrowingeverydayinreallifeandgettingthefinestoptionfrom thesetofpossibleonesisdifficultforthedecisionmakers.Toattainasingleobjectiveisdifficult tosummarizebutnotincredible.Manyorganizationsfounddifficultieswithsettingmotivations, goalsandopinions’complications.Thus,organizationaldecisionssimultaneouslyincludenumerous objectives,whethertheyregardindividualsorcommittees.Thisreflectionsuggests,accordingto criteriasolvedoptionally,restrictingeachdecisionmakertoattainanidealsolution-optimumunder eachcriterioninvolvedinpracticalproblems.Consequently,thedecisionmakerismoreconcentrated toestablishmoreapplicableandreliabletechniquestofindthebestoptions.

Tohandletheambiguityanduncertaintydataindecision-makingproblems,theclassicalorcrisp methodscannotbealwayseffective.Thus,dealingwithsuchuncertainsituations,Zadeh[1]in1965 presentedtheideaofthefuzzyset.Zadehassignsmembershipgradestoelementsofasetinthe interval[0,1]byofferingtheideaoffuzzysets(FSs).Zadeh’sworkinthisdirectionisremarkableas manyofthesettheoreticpropertiesofcrispcasesweredefinedforfuzzysets.Fuzzysettheorygotthe attentionofresearchersandfounditsapplicationsindecisionscience[2],artificialintelligence[3],and medicaldiagnosis[4],anditsenormousapplicationsarediscussedin[5].

Aftermanyapplicationsoffuzzysettheory,Atanassovobservedthattherearemanyshortcomings inthistheoryandintroducedthenotionofintuitionisticfuzzysets[6]togeneralizetheconceptof

Entropy 2019, 21,628;doi:10.3390/e21070628 www.mdpi.com/journal/entropy

Zadeh’sfuzzyset.Inintuitionisticfuzzysets,eachelementisexpressedbyanorderedpair,andeach pairischaracterizedbymembershipandnon-membershipgradesontheconditionthatthesumoftheir gradesarelessthanorequalto1.Duringthelastfewdecades,theintuitionisticfuzzysets(IFSs)are fruitfulandbroadlyutilizedbyresearcherstograsptheambiguityandimprecisiondata.Tocumulate alltheexecutiveofcriteriaforalternatives,aggregationoperatorsplayavitalrolethroughoutthe informationmergingprocedure.Xu[7]presentedaweightedaveragingoperatorwhileXuand Yager[8]developedageometricaggregationoperatorforaggregatingthedifferentintuitionisticfuzzy numbers.Verma[9]in2015proposedthegeneralizedBonferronimeanoperatorand,in[10],Vermaand Sharmaproposedthemeasureofinaccuracyusingintuitionisticfuzzyinformation.Deschrijver[11] developedtheIFSrepresentationoft-normsandt-conorms.Insomedecisiontheories,thedecision makersdealwiththesituationofparticularattributeswherevaluesoftheirsummationofmembership degreesexceeds1.Insuchconditions,IFShasnoabilitytoobtainanysatisfactoryresult.Toovercome thissituation,Yager[12]developedtheideaofaPythagoreanfuzzyset(PyFS)asageneralization ofIFS,whichsatisfiesthefactthatthevalueofsquaresummationofitsmembershipdegreesis lessthenorequalto1.Clearly,PyFSismoreflexiblethanIFStodealwiththeimprecisionand ambiguityinthepracticalmulti-criteriadecision-making(MCDM)problems.ZhangandXu[13] establishedanextensionofTOPSIStoMCDMwithPyFSinformation.Theerrorfortheproofof distancemeasureinZhangandXu[13]hasbeenpointedoutbyYangetal.[14].ForMCDMproblems inaPythagoreanfuzzyenvironment,YagerandAbbasov[15]developedaseriesofaggregation operators.PengandYang[16]explainedtheirrelationshipamongtheseaggregationoperatorsand establishedthesuperiorityandinferiorityranking(SIR)forthemulti-criteriagroupdecision-making (MCGDM)method.UsingEinsteinoperation,Garg[17]generalizedPythagoreanfuzzyinformation aggregation.Gouetal.[18]studiedmanyPythagoreanfuzzyfunctionsandinvestigatedtheir fundamentalpropertiessuchascontinuity,derivatively,anddifferentiabilityindetail.Zhang[19] putforwardarankedqualitativeflexible(QUALIFLEX)multi-criteriaapproachwiththecloseness index-basedrankingmethodsformulti-criteriaPythagoreanfuzzydecisionanalysis.Zengetal.[20] exploredahybridmethodforPythagoreanfuzzyMCDM.Zeng[21],appliedthePythagoreanfuzzy probabilisticOWA(PFPOWA)operatorforMAGDMproblems.Formorestudy,wereferto[22–24].

IFStheoryandPyFStheoryhavebeensuccessfullyappliedindifferentareas,butthereare situationsthatcannotberepresentedbyitinreallife,suchasvoting,wemayfacehumanopinions involvingmoreanswersofthetype:yes,abstain,noandrefusal(forexample,inademocraticelection station,thecouncilissues’ 500 votingpapersforacandidate.Thevotingresultsaredividedintofour groupsaccompaniedwiththenumberofpapersthatarevotefor (251),abstain (99),voteagainst (120) andrefusalofvoting (30).Here,groupabstainmeansthatthevotingpaperisawhitepaper rejectingbothagreeanddisagreeforthecandidatebutstilltakesthevote,grouprefusalofvotingis eitherinvalidvotingpapersordidnottakethevote.Thecandidateissuccessfulbecausethenumber ofsupportpapersisover50%(i.e.,250).

However,atleastfivepeoplesaidlateronintheirblogsthattheysupportthecandidateinthe lastmomentbecausetheyfindthatthesupportnumberseemslargerthantheagainstnumber.Such kindofexamples(inwhichthenumberofabstainsisakeyfactorandthegrouprefusalofvoting indeedexists)happenedinrealityandIFSandPyFScouldnothandleit).Thus,Cuong[25]proposed anewnotionnamedpicturefuzzysets,whichisanextensionoffuzzysetsandintuitionisticfuzzy sets.Picturefuzzysetsgivethreemembershipdegreesofanelementnamedthepositivemembership degree,theneutralmembershipdegree,andthenegativemembershipdegree,respectively.Thepicture fuzzysetsolvedthevotingproblemsuccessfully,andisappliedtoclustering[26],fuzzyinference[27], anddecision-making[28–35].

Theneutrosophicsetisanotherimportantgeneralizationsoftheclassicset,fuzzyset,intuitionistic fuzzysetandpicturefuzzysettodealwithuncertaintiesindecision-makingproblems.Manyauthors contributedinthedecision-makingtheoryusingneutrosophicinformation.Ashraf[36]proposed thelogarithmichybridaggregationoperatorsforsinglevalueneutrosophicsets.Draganetal.[37]

proposedthenovelapproachfortheselectionofpowergenerationtechnologyusingthecombinative distance-basedassessment(CODAS)method.Manydecision-makingapproacheslike[38,39]make importantcontributionsusingneutrosophicinformation.

Thepicturefuzzysetbecomesmorefamousbyintroducingvariouskindsofaggregationoperators. However,ithasashortcominginthatitisonlyvalidfortheenvironmentwhosesumofdegreesis lessthanorequaltoone.However,inday-to-daylife,therearemanysituationswherethiscondition isruledout.Forinstance,ifapersongivingtheirpreferenceintheformofpositive,neutraland negativemembershipdegreestowardsaparticularobjectis0.7,0.3and0.5,thenclearlythissituation isnothandlingwithpicturefuzzyset.Inordertoresolveit,Ashraf[40]proposedthenotionof thesphericalfuzzyset.Forinstance,correspondingtotheabove-consideredexample,weseethat (0.7)2 +(0.3)2 + (0.5)2 = 0.94 andhenceasphericalfuzzyset(SFS)isanextensionoftheexisting extensionsoffuzzysettheories.Inthesphericalfuzzyset,eachelementcanbewrittenintheformof tripletcomponent,andeachpairiscategorizedbyapositivemembershipdegree,aneutralmembership degreeandanegativemembershipdegreesuchthatthesumoftheirsquareislessthanorequalto one.Ashraf[41]proposedsomeseriesofsphericalaggregationoperatorsusingt-normandt-conorm andgaveitsapplicationstoshowtheeffectivenessofproposedoperators.Ashrafin[42]proposed theGRAmethodbasedonasphericallinguisticfuzzyChoquetintegralenvironmentandgaveits application.Formorestudy,wereferto[43–45]

Thelogarithmicoperationsbeinggoodalternatives,comparedwiththealgebraicoperations,have thepotentialtooffersimilarsmoothestimationsasthealgebraicoperations.However,thereislittle investigationonlogarithmicoperationsontheIFSsandPyFSs.Motivatedbytheseideas,wedevelopa sphericalfuzzyMCDMmethodbasedonthelogarithmicaggregationoperators,withthelogarithmic operationsofthesphericalfuzzysets(SFSs)handlingsphericalfuzzyMCDMwithinSFSs.

Thus,thegoalofthisarticleistoproposethedecision-makingmethodforMCDMproblemsin whichthereexisttheinterrelationshipsamongthecriteria.Thecontributionsofthisstudyare:

• Wedevelopsomenovellogarithmicoperationsforsphericalfuzzysets,whichcanovercomethe weaknessesofalgebraicoperationsandcapturetherelationshipbetweenvariousSFSs.

• Weextendlogarithmicoperatorstologarithmicsphericalfuzzyoperators,namelylogarithmic sphericalfuzzyweightedaveraging(L-SFWA),logarithmicsphericalfuzzyorderedweighted averaging(L-SFOWA),logarithmicsphericalfuzzyhybridweightedaveraging(L-SFHWA), logarithmicsphericalfuzzyweightedgeometric(L-SFWG),logarithmicsphericalfuzzyordered weightedgeometric(L-SFOWG)andlogarithmicsphericalfuzzyhybridweightedgeometric (L-SFHWG)toSFSs,whichcanovercomethealgebraicoperators’drawbacks.

• Wedevelopthesphericalfuzzyentropyforsphericalfuzzyinformation,whichcanhelptofind theunknownweightsinformationofthecriteria.

• Wedevelopanalgorithmtodealwithmulti-attributedecision-makingproblemsusingspherical fuzzyinformation.

• Toshowtheeffectivenessandreliabilityoftheproposedsphericalfuzzylogarithmicaggregation operators,theapplicationoftheproposedoperatorinemergingtechnologyenterprises isdeveloped.

• Resultsindicatethattheproposedtechniqueismoreeffectiveandgivesmoreaccurateoutputas comparedtoexistingstudies.

Inordertoattaintheresearchgoalthathasbeenstatedabove,theorganizationofthisarticleisoffered as:Section 2 concentratesonsomebasicnotionsandoperationsofexistingextensionsoffuzzyset theoriesandalsosomediscussiontoproposethesphericalfuzzyentropy.Section 3 presentssome novellogarithmicoperationallawsofSFSs.Section 4 definesthelogarithmicaggregationoperatorsfor SFNsanddiscussesitsproperties.Section 5 presentsanapproachforhandlingthesphericalfuzzy MCDMproblembasedontheproposedlogarithmicoperators.Section 5.1 usesanapplicationcaseto verifythenovelmethodandSection 5.2 presentsthecomparisonstudyaboutalgebraicandlogarithmic aggregationoperators.Section 6 concludesthestudy.

2.Preliminaries

Theconceptsandbasicoperationsofexistingextensionsoffuzzysetsarerecalledinthissection, andtheyarethefoundationofthisstudy.

Definition1 ([27]). Amapping T : Θ × Θ → Θ issaidtobetriangular-normifeachelement T satisfiesthat: (1) Tiscommutative,monotonicandassociative, (2) T (v∗,1) = v∗ , eachv∗ ∈ T, where Θ =[0,1] istheuniteinterval.

Definition2 ([27]). Amapping S : Θ × Θ → Θ issaidtobetriangular-conormifeachelement S satisfies that (1) Siscommutative,monotonicandassociative, (2) S (v∗,0) = v∗ , eachv∗ ∈ S, where Θ =[0,1] istheuniteinterval.

Asdifferentnormsarethecurialelementsforproposingaggregationoperatorsinfuzzysettheory, hereweenlistsomebasicnormsoperationsforfuzzysetsinFigure 1

t-normt-conormProposedBy min(x, y) max(x, y) Zadeh x.yx + y x.y GoguenandBandler (x1 ∧ y1, x2 ∧ y2, x3 ∨ y3)(x1 ∨ y1, x2 ∧ y2, x3 ∧ y3) Cuong (x1 ∧ y1, x2.y2, x3 ∨ y3)(x1 ∨ y1, x2.y2, x3 ∧ y3) Cuong (x1.y1, x2.y2, x3 + y3 x3.y3)(x1 + y1 x1.y1, x2.y2, x3, y3) Cuong Figure1. BasicNormsOperations.

Now,weenlistdifferenttypesofnormswithitsgeneratorstooinFigures 2 and 3

Namet-normAdditiveGenerators

Algebric TA (x, y) = xyt(r1)= log r1 Einstein TA (x, y) = xy 1+(1 x)(1 y) t(r1)= log 2 r1 r1 Hamacher TA (x, y) = xy γ+(1 γ)(x+y xy) , γ > 0 t(r1)= log γ+(1 γ)r1 r1 , γ > 0

Frank TA (x, y) = logγ 1 + (γx 1)(γy 1) γ 1 , γ > 0 γ = 1, t(r1)= log r1 γ = 1, t(r1)= log γ 1 γr1 1

Figure2. T-normwithitsGenerators.

Namet-conormAdditiveGenerators

Algebric SA (x, y) = x + y xys(r1)= log (1 r1) Einstein SA (x, y) = x+y 1+xy s(r1)= log 1+r1 1 r1 Hamacher SA (x, y) = x+y xy (1 γ)xy 1 (1 γ)xy , γ > 0 s(r1)= log γ+(1 γ)(1 r1 ) 1 r1 , γ > 0 Frank SA (x, y) = 1 logγ 1 + (γ1 x 1)(γ1 y 1) γ 1 , γ > 0 γ = 1, s(r1)= log (1 r1) γ = 1, s(r1)= log γ 1 γ1 r1 1

Figure3. T-conormwithitsGenerators.

Definition3 ([15]). Foraset ,byaPythagoreanfuzzysetin ,wemeanastructure

ε = { Pσ (ˇ rγ ) , Nσ (ˇ rγ ) |ˇ rγ ∈ } , inwhich Pσ : → Θ and Nσ : → Θ indicatethatthepositiveandnegativegradesin , Θ = [0,1] arethe unitintervals.Inaddition,thefollowingconditionsatisfiedby ρσ and Nσ is 0 ≤ P2 σ (ˇ rγ ) + N2 σ (ˇ rγ ) ≤ 1;forall ˇ r ∈ Then, ε issaidtobeaPythagoreanfuzzysetin

Definition4 ([25]). Foraset ,byapicturefuzzysetin ,wemeanastructure

ε = { Pσ (ˇ rγ ) , Iσ (ˇ rγ ) , Nσ (ˇ rγ ) |ˇ rγ ∈ } ,

inwhich Pσ : → Θ, Iσ : → Θ and Nσ : → Θindicatethatthepositive,neutralandnegativegrades in , Θ = [0,1] aretheunitintervals.Inaddition,thefollowingconditionsatisfiedby Pσ , Iσ and Nσ is 0 ≤ Pσ (ˇ rγ ) + Iσ (ˇ rγ ) + Nσ (ˇ rγ ) ≤ 1,forall ˇ rγ ∈ . Then, ε issaidtobeapicturefuzzysetin .

Definition5 ([40]). Foraset ,byasphericalfuzzysetin ,wemeanastructure

ε = { Pσ (ˇ rγ ) , Iσ (ˇ rγ ) , Nσ (ˇ rγ ) |ˇ rγ ∈ } ,

inwhich Pσ : → Θ, Iσ : → Θ and Nσ : → Θ indicatethatthepositive,neutralandnegativegrades in , Θ = [0,1] aretheunitintervals.Inaddition,thefollowingconditionsatisfiesby Pσ , Iσ and Nσ is 0 ≤ P2 σ (ˇ rγ ) + I2 σ (ˇ rγ ) + N2 σ (ˇ rγ ) ≤ 1,forall ˇ rγ ∈ . Then, ε issaidtobeasphericalfuzzysetin . χσ (ˇ rγ ) = 1 P2 σ (ˇ rγ ) + I2 σ (ˇ rγ ) + N2 σ (ˇ rγ ) issaidtobearefusaldegreeof ˇ rγ in ,forSFS { Pσ (ˇ rγ ) , Iσ (ˇ rγ ) , Nσ (ˇ rγ ) |ˇ rγ ∈ },whichistriplecomponents Pσ (ˇ rγ ) , Iσ (ˇ rγ ) , andNσ (ˇ rγ ) issaidto SFNdenotedby e = Pe, Ie, Ne ,where Pe, Ie and Ne ∈ [0,1],withtheconditionthat: 0 ≤ P2 e + I2 e + N2 e ≤ 1.

AshrafandAbdullah[40]proposedthebasicoperationsofsphericalfuzzysetasfollows: Definition6. ForanytwoSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) and εq = Pξq (ˇ rγ ) , Iξq (ˇ rγ ) , Nξq (ˇ rγ ) in .Theunion,intersectionandcomplimentareproposedas: (1) ε ρ ⊆ εq iff ∀ˇ rγ ∈ , Pξρ (ˇ rγ ) ≤ Pξq (ˇ rγ ) , Iξρ (ˇ rγ ) ≤ Iξq (ˇ rγ ) andNξρ (ˇ rγ ) ≥ Nξq (ˇ rγ ); (2) ε ρ = εq iff ε ρ ⊆ εq and εq ⊆ ε ρ ; (3) ε ρ ∪ εq = max Pξρ , Pξq ,min Iξρ , Iξq ,min Nξρ , Nξq ; (4) ε ρ ∩ εq = min Pξρ , Pξq ,min Iξρ , Iξq ,max Nξρ , Nξq ; (5) ε ρ = Nξρ , Iξρ , Pξρ



 

(3) β1(ε ρ ⊕ εq)= β1ε ρ ⊕ β1εq, β1 > 0; (4) (ε ρ ⊗ εq)β1 = ε ρ ⊗ εq, β1 > 0; (5) β1ε ρ ⊕ β2ε ρ =(β1 + β2)ε ρ , β1 > 0, β2 > 0; (6) ε ρ ⊗ ε ρ = ε ρ , β1 > 0, β2 > 0; (7) (ε ρ ⊕ εq) ⊕ εl = ε ρ ⊕ (εq ⊕ εl ); (8) (ε ρ ⊗ εq) ⊗ εl = ε ρ ⊗ (εq ⊗ εl )

Definition9. ForanySFN, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) in .Then,scoreandaccuracyvaluesare definedas

(1) S(ε ρ )= 1 3 (2 + Pξρ Iξρ Nξρ ) ∈ [0,1] (2) A(ε ρ )= Pξρ Nξρ ∈ [0,1]

ThescoreandaccuracyvaluesdefinedabovesuggestwhichSFNisgreaterthanotherSFNs.The comparisontechniqueisdefinedinthenextdefinition.

Definition10. ForanySFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2) in .Then,thecomparison techniqueisproposedas

(1)If S(ε1) < S(ε2), then ε1 < ε2, (2)If S(ε1) > S(ε2), then ε1 > ε2, (3)If S(ε1)= S(ε2), then (a) A(ε1) < A(ε2), then ε1 < ε2, (b) A(ε1) > A(ε2), then ε1 > ε2, (c) A(ε1)= A(ε2), then ε1 ≈ ε2.

AshrafandAbdullah[40]proposedaggregationoperatorsforSFNsbasedondifferentnorms:

Definition11. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .The structureofthesphericalweightedaveraging(SFWA)operatoris

SFWA (ε1, ε2,..., εn) = n ∑ ρ=1 βρ ε ρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1.

Definition12. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .The structureofthesphericalorderweightedaveraging(SFOWA)operatoris

SFOWA (ε1, ε2,..., εn) = n ∑ ρ=1 βρ ε η(ρ), where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0, ∑n ρ=1 βρ = 1 andthe ρth biggestweightedvalueis ε η(ρ) consequentlybytotalorder ε η(1) ≥ ε η(2) ≥ ... ≥ ε η(n).

Definition13. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .The structureofthesphericalhybridweightedaveraging(SFHWA)operatoris

SFHWA (ε1, ε2,..., εn) = n ∑ ρ=1 βρ ε∗ η(ρ),

where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0, ∑n ρ=1 βρ = 1 andthe ρth biggestweightedvalue is ε∗ η(ρ) ε∗ η(ρ) = nβρ ε η(ρ), ρ ∈ N consequentlybytotalorder ε∗ η(1) ≥ ε∗ η(2) ≥ ≥ ε∗ η(n) Inaddition,the associatedweightsare ω =(ω1, ω2,..., ωn) with ωρ ≥ 0, Σn ρ=1ωρ = 1.

Definition14. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .The structureofsphericalweightedgeometric(SFWG)operatoris

SFWG (ε1, ε2,..., εn) = n ∏ ρ=1 ε ρ βρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1.

Definition15. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .The structureofthesphericalorderweightedgeometric(SFOWG)operatoris

SFOWG (ε1, ε2,..., εn) = n ∏ ρ=1 ε η(ρ) βρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0, ∑n ρ=1 βρ = 1 andthe ρth biggestweightedvalueis ε η(ρ) consequentlybytotalorder ε η(1) ≥ ε η(2) ≥ ≥ ε η(n) Definition16. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .The structureofsphericalhybridweightedgeometric(SFHWG)operatoris

SFHWG (ε1, ε2,..., εn) = n ∏ ρ=1 ε∗ η(ρ) βρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0, ∑n ρ=1 βρ = 1 andthe ρth biggestweightedvalue is ε∗ η(ρ) ε∗ η(ρ) = nβρ ε η(ρ), ρ ∈ N consequentlybytotalorder ε∗ η(1) ≥ ε∗ η(2) ≥ ≥ ε∗ η(n) Inaddition, associatedweightsare ω =(ω1, ω2,..., ωn) with ωρ ≥ 0, Σn ρ=1ωρ = 1.

3.Entropy

Basically,wefamiliarizetheconceptofentropy,whenprobabilitymeasuresthediscrimination ofcriteriabeingimposedonmulti-attributedecision-makingproblems.Non-probabilisticentropy firstlyapproximatedbyDeLucaandTermini[46]alsopresentedsomenecessitiestofindintuitive comprehensionofthedegreeoffuzziness.Manyresearchersaregettinginterestinthisfieldandhave donealotofworksuchasScmidtandKacprzyk[47]proposingsomeaxiomsfordistancebetween intuitionisticfuzzysetsandnon-probabilisticentropymeasureforthem.Inthissection,werecallthe conceptofShannonentropy,fuzzyentropy,entropyforPythagoreanfyzzynumbersandproposethe entropyforsphericalfuzzynumbers.

Definition17 ([48]). Let δ ηρ ρ ∈ {1,2,..., n} bethesetof n-completeprobabilitydistributions.Shannon entropyfor δ ηρ ρ ∈ {1,2,..., n} probabilitydistributionisdefinedas Es (δ) = n ∑ ρ=1 δ ηρ log δ ηρ .

Definition18 ([49]). LetFbeanyfuzzysetin ,fuzzyentropyforthesetF,wemeanastructure

F (δ) = 1 n

n ∑ ρ=1 P ηρ log P ηρ + 1 P ηρ log 1 P ηρ .

Definition19 ([49]). Let F beanyfuzzysetin ,and,forPythagoreanfuzzyentropyfortheset F,wemeana structure

Pyq = 1 + 1 n n ∑ ρ=1 (Pi log (Pi ) + Ni log (Ni )) n ∑ q=1 1 + 1 n n ∑ ρ=1 Pi log (Pi ) + Ni log (Ni ) .

Definition20. LetFbeanyfuzzysetin ,sphericalfuzzyentropyforthesetF,andwemeanastructure γq = 1 + 1 n n ∑ ρ=1 (Pi log (Pi ) + Ii log (Ii ) + Ni log (Ni )) n ∑ q=1 1 + 1 n n ∑ ρ=1 Pi log (Pi ) + Ii log (Ii ) + Ni log (Ni ) .

4.SphericalFuzzyLogarithmicOperationalLaws

Motivatedbythenovelconceptofsphericalfuzzyset,weintroducedsomenovellogarithmic operationallawsforSFNs.Asrealnumbersystem ogσ 0 ismeaninglessand ogσ 1 isnotdefined therefore,inourstudy,wetakenonemptysphericalfuzzysetsand σ = 1, where σ isanyrealnumber.

Definition21. ForanySFN, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) in .Thelogarithmicsphericalfuzzy numberisdefinedas

ogσ ε ρ = 1 ogσ Pξρ (ˇ rγ ) 2 , ogσ 1 I2 ξρ (ˇ rγ ) , ogσ 1 N2 ξρ (ˇ rγ ) |ˇ rγ ∈ inwhich Pσ : → Θ, Iσ : → Θ and Nσ : → Θ indicatethatthepositive,neutralandnegativegrades in , Θ = [0,1] aretheunitintervals.Inaddition,thefollowingconditionsatisfiesby Pσ , Iσ and Nσ is 0 ≤ P2 σ (ˇ rγ ) + I2 σ (ˇ rγ ) + N2 σ (ˇ rγ ) ≤ 1 forall ˇ rγ ∈ Therefore,themembershipgradeis 1 ogσ Pξρ (ˇ rγ ) 2 : → Θ, suchthat 0 ≤ 1 ogσ Pξρ (ˇ rγ ) 2 ≤ 1, ∀ˇ rγ ∈ , theneutralgradeis ogσ 1 I2 ξρ (ˇ rγ ) : → Θ, suchthat 0 ≤ ogσ 1 I2 ξρ (ˇ rγ ) ≤ 1, ∀ˇ rγ ∈ , andthenegativegradeis

ogσ 1 N2 ξρ (ˇ rγ ) : → Θ, suchthat 0 ≤ ogσ 1 N2 ξρ (ˇ rγ ) ≤ 1, ∀ˇ rγ ∈ . Therefore, ogσ ε ρ = 1 ogσ Pξρ (ˇ rγ ) 2 , ogσ 1 I2 ξρ (ˇ rγ ) , ogσ 1 N2 ξρ (ˇ rγ ) |ˇ rγ ∈ 0 < σ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ ≤ 1, σ = 1

isSPN. Definition22. ForanySFN,

thenthefunction ogσ ε ρ isknowntobealogarithmicoperatorforasphericalfuzzyset,anditsvalueiscalled logarithmicSFN(L-SFN).Here,wetake ogσ 0 = 0, σ > 0, σ = 1.

Theorem1. ForanySFN, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) in ,then ogσ ε ρ isalsoaspherical fuzzynumber.

Proof. SinceanySFN ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) in ,whichmeansthat Pσ : → Θ, Iσ : → Θ and Nσ : → Θ indicatethatthepositive,neutralandnegativegradesin , Θ = [0,1] arethe unitintervals.Inaddition,thefollowingconditionsatisfiedby Pσ , Iσ and Nσ is 0 ≤ P2 σ (ˇ rγ ) + I2 σ (ˇ rγ ) + N2 σ (ˇ rγ ) ≤ 1.Thefollowingtwocaseshappen.

Case-1When 0 < σ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1 andsince ogσ ε ρ isa decreasingfunctionw.r.t σ. Thus, 0 ≤ ogσ Pξρ , ogσ 1 I2 ξρ , ogσ 1 N2 ξρ ≤ 1 andhence

0 ≤ 1 ogσ Pξρ (ˇ rγ ) 2 ≤ 1,0 ≤ ogσ 1 I2 ξρ (ˇ rγ ) ≤ 1, 0 ≤ ogσ 1 N2 ξρ (ˇ rγ ) ≤ 1 and 0 ≤ 1 ogσ Pξρ (ˇ rγ ) 2 + ogσ 1 I2 ξρ (ˇ rγ ) + ogσ 1 N2 ξρ (ˇ rγ ) ≤ 1. Therefore, ogσ ε ρ is SFN.

Case-2When σ > 1, 0 < 1 σ < 1 and 1 σ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ ; similartotheabove, wecanfindthat ogσ ε ρ isSFN.Thus,theprocedureiseliminatedhere.

Example1. Supposethat,foranySFN, ε ρ = 0.8,0.5,0.3 in with σ = 0.4, then ogσ ε ρ = 1 ( og0.4 (0.8))2 , og0.4 1 (0.5)2 , og0.4 1 (0.3)2 = (0.969,0.156,0.051) .

Inaddition,if σ = 8,thenitfollows: og 1 σ ε ρ = 1 og 1 8 (0.8) 2 , og 1 8 1 (0.5)2 , og 1 8 1 (0.3)2 = (0.994,0.069,0.022)

Now,wegivesomediscussiononthebasicpropertiesoftheL-SFN.

(ˇ rγ )

      (ρ = 1,2) in ,with 0 < σ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Then, (1) ogσ ε1 ⊕ ogσ ε2 = ogσ ε2 ⊕ ogσ ε1, (2) ogσ ε1 ⊗ ogσ ε2 = ogσ ε2 ⊗ ogσ ε1 Proof. ThisisstraightforwardfromDefinition 23,sotheprocedureiseliminatedhere.

Theorem4. ForanytwoL-SFNs, ogσ ε ρ =      

1 ogσ Pξρ (ˇ rγ ) 2 , ogσ 1 I2 ξρ (ˇ rγ ) , ogσ 1 N2 ξρ (ˇ rγ )

      (ρ = 1,2,3) in ,with 0 < σ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Then, (1) ( ogσ ε1 ⊕ ogσ ε2) ⊕ ogσ ε3 = ogσ ε1 ⊕ ( ogσ ε2 ⊕ ogσ ε3) , (2)( ogσ ε1 ⊗ ogσ ε2) ⊗ ogσ ε3 = ogσ ε1 ⊗ ( ogσ ε2 ⊗ ogσ ε3) Proof. ThisisstraightforwardfromDefinition 23,sotheprocedureiseliminatedhere.

ForanytwoL-SFNs, og

in .Then,scoreandaccuracy valuesaredefinedas (1) S( ogσ ε ρ )= 1 ogσ Pξρ (ˇ rγ ) 2 ogσ 1 I2 ξρ (ˇ rγ ) 2 ogσ 1 N2 ξρ (ˇ rγ ) 2 , (2) A( ogσ ε ρ )= 1 ogσ Pξρ (ˇ rγ ) 2 + ogσ 1 N2 ξρ (ˇ rγ ) 2





1 < ogσ ε2, (2)If

ogσ ε1) > S

(





5.1.LogarithmicAveragingOperators

Definition26. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Thestructureoflogarithmicsphericalweighted averaging(L-SFWA)operatoris L SFWA (ε1, ε2,..., εn) = n ∑ ρ=1 βρ ogσρ ε ρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1. Theorem6. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξ

1,whichis

, whichcompletestheproof. Remark1. If σ1 = σ2 = σ3 = = σn = σ, thatis 0 < σ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1, thentheL SFWAoperatorisreducedasfollows: L SFWA (ε1, ε2,..., εn) =

       

2βρ , n ∏ ρ=1 ogσ 1 I2 ξρ

1 n ∏ ρ=1 ogσ Pξρ

βρ , n ∏ ρ=1 ogσ 1 N2 ξρ

βρ Properties: The L SFWA operatorsatisfiessomepropertiesthatarelistedbelow: (1)Idempotency:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .Then,ifthecollectionofSFNs ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) isidentical, L SFWA (ε1, ε2,..., εn) = ε

(2)Boundedness:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in . ε ρ = minρ ρξρ ,maxρ Iξρ ,maxρ Nξρ and ε+ ρ = maxρ Pξρ ,minρ Iξρ ,minρ Nξρ (ρ = 1,2,..., n) in , therefore ε ρ ⊆ L SFWA (ε1, ε2,..., εn) ⊆ ε+ ρ .

(3)Monotonically:ForanycollectionofSFNs, ε ρ = Pξρ (rγ ) , Iξρ (rγ ) , Nξρ (rγ ) (ρ = 1,2,..., n) in .If ε ρ ⊆ ε∗ ρ for (ρ = 1,2,..., n) ,then L SFWA (ε1, ε2,..., εn) ⊆ L SFWA (ε∗ 1, ε∗ 2,..., ε∗ n)

Definition27. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Thestructureofthelogarithmicsphericalordered weightedaveraging(L-SFOWA)operatoris L SFOWA (ε1, ε2,..., εn) = n ∑ ρ=1 βρ ogσρ ε η(ρ), where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1 andthe ρth biggestweightedvalue is ε η(ρ) consequentlybytotalorder ε η(1) ≥ ε η(2) ≥ ... ≥ ε η(n).

Theorem7. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ

biggestweightedvalue is ε η(ρ) consequentlybytotalorder ε η(1) ≥ ε η(2) ≥ ... ≥ ε η(n). Proof. TheproofissimilartoTheorem 6.Thus,theprocedureiseliminatedhere. Remark2. If σ1 = σ2 = σ3 = = σn = σ, thatis, 0 < σ ≤ min ρξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1, thentheL SFOWAoperatorisreducedasfollows:

(2)Boundedness:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ε ρ = minρ Pξρ ,maxρ Iξρ ,maxρ Nξρ and ε+ ρ = maxρ Pξρ ,minρ Iξρ ,minρ Nξρ (ρ = 1,2,..., n) in ; therefore, ε ρ ⊆ L SFOWA (ε1, ε2,..., εn) ⊆ ε+ ρ

(3)Monotonically:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in . If ε ρ ⊆ ε∗ ρ for (ρ = 1,2,..., n) ,then L SFOWA (ε1, ε2,..., εn) ⊆ L SFOWA (ε∗ 1, ε∗ 2,..., ε∗ n) Definition28. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Thestructureofalogarithmicsphericalhybrid weightedaveraging(L-SFHWA)operatoris L SFHWA (ε1, ε2,..., εn) = n ∑ ρ=1 βρ ogσρ ε∗ η(ρ), where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1 andthe ρth biggestweighted valueis ε∗ η(ρ) ε∗ η(ρ) = nβρ ε η(ρ), ρ ∈ N consequentlybytotalorder ε∗ η(1) ≥ ε∗ η(2) ≥ ≥ ε∗ η(n) In addition,associatedweightsare ω =(ω1, ω2,..., ωn) with ωρ ≥ 0, Σn ρ=1ωρ = 1. Theorem8. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Then,byusinglogarithmicoperationsandDefinition 28, L SFHWAisdefinedas



     

(2)Boundedness:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in . ε ρ = minρ Pξρ ,maxρ Iξρ ,maxρ Nξρ and ε+ ρ = maxρ Pξρ ,minρ Iξρ ,minρ Nξρ (ρ = 1,2,..., n) in ; therefore, ε ρ ⊆ L SFHWA (ε1, ε2,..., εn) ⊆ ε+ ρ .

(3)Monotonically:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .If ε ρ ⊆ ε∗ ρ for (ρ = 1,2,..., n) ,then L SFHWA (ε1, ε2,..., εn) ⊆ L SFHWA (ε∗ 1, ε∗ 2,..., ε∗ n)

5.2.LogarithmicGeometricOperators

Definition29. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Thestructureoflogarithmicsphericalweighted geometric(L-SFWG)operatoris L SFWG (ε1, ε2,..., εn) = n ∏ ρ=1 ogσρ ε ρ βρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1.

Remark4. If σ1 = σ2 = σ3 = = σn = σ, thatis, 0 < σ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1, thentheL SFWGoperatorreducesto L SFWG (ε1, ε2,..., εn) =

Properties: The L SFWG operatorsatisfiessomepropertiesthatarelistedbelow: (1)Idempotency:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .Then,ifacollectionofSFNs ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) areidentical,thatis, L SFWG (ε1, ε2,..., εn) = ε

(2)Boundedness:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ε ρ = minρ ρξρ ,maxρ Iξρ ,maxρ Nξρ and ε+ ρ = maxρ Pξρ ,minρ Iξρ ,minρ Nξρ (ρ = 1,2,..., n) in , therefore ε ρ ⊆ L SFWG (ε1, ε2,..., εn) ⊆ ε+ ρ

(3)Monotonically:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in .If ε ρ ⊆ ε∗ ρ for (ρ = 1,2,..., n) ,then L SFWG (ε1, ε2,..., εn) ⊆ L SFWG (ε∗ 1, ε∗ 2,..., ε∗ n)

Definition30. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Thestructureoflogarithmicsphericalordered weightedgeometric(L-SFOWG)operatoris

L SFOWG (ε1, ε2,..., εn) = n ∏ ρ=1 ogσρ ε η(ρ) βρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1 andthe ρth biggestweightedvalue is ε η(ρ) consequentlybytotalorder ε η(1) ≥ ε η(2) ≥ ... ≥ ε η(n).

Theorem10. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Then,byusinglogarithmicoperationsandDefinition 30, L SFOWGdefinedas

1, where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1 andthe ρth biggestweightedvalue is ε η(ρ) consequentlybytotalorder ε η(1) ≥ ε η(2) ≥ ≥ ε η(n) Proof. ThisproofissimilartoTheorem 9,sotheprocedureiseliminatedhere. Remark5. If σ1 = σ2 = σ3 = ... =

(3)Monotonically:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in If ε ρ ⊆ ε∗ ρ for (ρ = 1,2,..., n) ,then L SFOWG (ε1, ε2,..., εn) ⊆ L SFOWG (ε∗ 1, ε∗ 2,..., ε∗ n) Definition31. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 < σρ ≤ min Pξρ , 1 I2 ξρ , 1 N2 ξρ < 1, σ = 1. Thestructureoflogarithmicsphericalhybrid weightedgeometric(L-SFHWG)operatoris L SFHWG (ε1, ε2,..., εn) = n ∏ ρ=1 ogσρ ε∗ η(ρ) βρ , where βρ (ρ = 1,2,..., n) areweightvectorswith βρ ≥ 0 and ∑n ρ=1 βρ = 1 andthe ρth biggestweighted valueis ε∗ η(ρ) ε∗ η(ρ) = nβρ ε η(ρ), ρ ∈ N consequentlybytotalorder ε∗ η(1) ≥ ε∗ η(2) ≥ ≥ ε∗ η(n) Inaddition, associatedweightsare ω =(ω1, ω2,..., ωn) with ωρ ≥ 0, Σn ρ=1ωρ = 1. Theorem11. ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in ,with 0 <

31,

Properties: The L SFHWG operatorsatisfiessomepropertiesthatarelistedbelow: (1)Idempotency:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in . Then,ifcollectionofSFNs ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) areidentical,thatis, L SFHWG (ε1, ε2,..., εn) = ε (2)Boundedness:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in . ε ρ = minρ Pξρ ,maxρ Iξρ ,maxρ Nξρ and ε+ ρ = maxρ ρξρ ,minρ Iξρ ,minρ Nξρ (ρ = 1,2,..., n) in , therefore ε ρ ⊆ L SFHWG (ε1, ε2,..., εn) ⊆ ε+ ρ .

(3)Monotonically:ForanycollectionofSFNs, ε ρ = Pξρ (ˇ rγ ) , Iξρ (ˇ rγ ) , Nξρ (ˇ rγ ) (ρ = 1,2,..., n) in If ε ρ ⊆ ε∗ ρ for (ρ = 1,2,..., n) ,then L SFHWG (ε1, ε2,..., εn) ⊆ L SFHWG (ε∗ 1, ε∗ 2,..., ε∗ n)

6.ProposedTechniqueforSolvingDecision-MakingProblems

Inthissection,weproposeanewapproachtodecision-makingbasedonthesphericalfuzzyset. Thisapproachwillusedatainformationprovidedbythedecisionproblemonlyanddoesnotneed anyadditionalinformationprovidedbydecisionmakers,inordertoavoidtheeffectofsubjective informationinfluencingthedecisionresults.Inthefollowing,wewillintroduceasphericalfuzzyset decision-makingmatrixasindicatedbelow.

Step1: Let H =(h1, h2,..., hm) beadistinctsetof m probablealternativesand Y =(y1, y2,..., yn) bea finitesetof n criteria,where hi indicatesthe ith alternativesand yj indicatesthe jth criteria.Let D =(d1, d2,..., dt) beafinitesetof t experts,where dk indicatesthe kthexpert.Theexpert dk suppliesherappraisalofanalternative hi onanattribute yj asaSFNs (i = 1,2,..., m; j = 1,2,..., n) Theexperts’informationisrepresentedbythesphericalfuzzysetdecision-makingmatrix Ds = E(s)

ρ m×n .Assumethat βρ (ρ = 1,2,..., m) isaweightvectoroftheattribute yj suchthat 0 ≤

and ψ =(ψ1, ψ2,..., ψm) istheweightvectorofthedecisionmakers dk suchthat

1. Whenweconstructthesphericalfuzzydecision-makingmatrices, Ds = E(s) iρ m×n fordecisions. Basically,criteriahavetwotypes:oneisbenefitcriteriaandtheotheroneiscostcriteria.If thesphericalfuzzydecisionmatriceshavecosttypecriteriamatrices, Ds = Es iρ m×n can

=1

beconvertedintothenormalizedsphericalfuzzydecisionmatrices, ˇ rs = ˇ r(s) iρ m×n ,where ˇ rs ip = Es iρ ,forbenefitcriteria Ap Es iρ ,forcostcriteria Ap, j = 1,2,..., n, and Es iρ isthecomplementof Es iρ Ifallthe criteriahavethesametype,thenthereisnoneedfornormalization. Takingthedecisioninformationfromthegivenmatrix ˇ rk andusingthe SFWA/SFWG operator, theindividualtotalsphericalfuzzypreferencevalue ˇ rk i ofthealternative hi isderivedasfollows: ˇ rk i = L SFWA(εk i1, εk i2,..., εk in), (i = 1,2,..., m; k = 1,2,..., t), where β =(β1, β2,..., βn)T istheweightvectoroftheattribute.

Step2: Inthisstep,wefindthecollectivesphericalinformationusingasphericalfuzzyweighted averagingaggregationoperator.

Step3: Inthisstep,wefindtheweightsofeachofthecriteriabyusingthesphericalfuzzyentropy: γq = 1 + 1 n n ∑ ρ=1 (Pi log (Pi ) + Ii log (Ii ) + Ni log (Ni )) n ∑ q=1 1 + 1 n n ∑ ρ=1 Pi log (Pi ) + Ii log (Ii ) + Ni log (Ni )

Step4: Inthisstep,wecalculatetheaggregatedinformationusingallthelogarithmicaggregation operatorsofsphericalfuzzysets.

Step5: Wefindthescorevalue S( ogσ ε ρ ) andtheaccuracyvalue A( ogσ ε ρ ) ofthecumulativeoverall preferencevalue hi (i = 1,2,..., m)

Step6: Bythedefinition,rankthealternatives hi (i = 1,2,..., m) andchoosethebestalternativethat hasthemaximumscorevalue.

TheAlgorithmicStepsareshowninFigure 4

6.1.NumericalExample

Assumethatthereisacommitteethatselectsfiveapplicableemergingtechnologyenterprises Hg(g = 1,2,3,4,5),whicharegivenasfollows: (1)AugmentedReality (H1), (2)PersonalizedMedicine (H2) , (3)ArtificialIntelligence (H3), (4)GeneDrive (H4), (5)QuantumComputing (H5

Toassessthepossiblerisingtechnologyenterprisesaccordingtothefourattributes,whichare (1)Advancement (D1), (2)marketrisk (D2), (3)financialinvestments (D3) and (4)progressofscienceandtechnology (D4)

Toavoidtheconflictbetweenthem,thedecisionmakersgivestheweight β =(0.314,0.355,0.331)T

ConstructthesphericalfuzzysetdecisionmakingmatricesasshowninTables 1–3:

Figure4. AlgorithmicSteps

Table1. EmergingtechnologyEenterprises F1

D1 D2 D3 D4

H1 (0.84,0.34,0.40)(0.43,0.39,0.78)(0.67,0.50,0.30)(0.31,0.21,0.71) H2 (0.60,0.11,0.53)(0.23,0.35,0.59)(0.72,0.31,0.41)(0.11,0.25,0.82)

H3 (0.79,0.19,0.39)(0.11,0.21,0.91)(0.71,0.41,0.13)(0.34,0.25,0.51) H4 (0.63,0.51,0.13)(0.49,0.33,0.42)(0.61,0.43,0.45)(0.49,0.37,0.59) H5 (0.57,0.36,0.29)(0.50,0.15,0.60)(0.70,0.32,0.40)(0.33,0.44,0.65)

Table2. Emergingtechnologyenterprises F2

D1 D2 D3 D4

H1 (0.61,0.15,0.53)(0.16,0.35,0.62)(0.61,0.35,0.47)(0.55,0.17,0.74) H2 (0.66,0.11,0.51)(0.43,0.23,0.77)(0.93,0.08,0.09)(0.02,0.06,0.99)

H3 (0.88,0.09,0.07)(0.05,0.06,0.89)(0.56,0.17,0.44)(0.43,0.13,0.61)

H4 (0.59,0.32,0.34)(0.24,0.48,0.51)(0.68,0.53,0.39)(0.34,0.21,0.61)

H5 (0.71,0.31,0.24)(0.35,0.41,0.69)(0.73,0.44,0.21)(0.22,0.49,0.74)

Table3. Emergingtechnologyenterprises F3

D1 D2 D3 D4

H1 (0.85,0.25.0.15)(0.14,0.23,0.88)(0.78,0.38,0.18)(0.29,0.39,0.83)

H2 (0.94,0.04,0.07)(0.39,0.19,0.61)(0.63,0.18,0.35)(0.48,0.49,0.56)

H3 (0.73,0.13,0.46)(0.19,0.39,0.88)(0.87,0.35,0.18)(0.41,0.13,0.81)

H4 (0.82,0.12,0.43)(0.55,0.21,0.63)(0.53,0.33,0.47)(0.46,0.23,0.51)

H5 (0.61,0.33,0.29)(0.28,0.41,0.63)(0.74,0.34,0.14)(0.37,0.32,0.65)

Since D1, D3 arebenefittypecriteriaand D2, D4 iscosttypecriteria,weneedtohavenormalized thedecisionmatrices.NormalizeddecisionmatricesareshowninTables 4–6:

Table4. Emergingtechnologyenterprises r1

D1 D2 D3 D4

H1 (0.84,0.34,0.40)(0.78,0.39,0.43)(0.67,0.50,0.30)(0.71,0.21,0.31)

H2 (0.60,0.11,0.53)(0.59,0.35,0.23)(0.72,0.31,0.41)(0.82,0.25,0.11)

H3 (0.79,0.19,0.39)(0.91,0.21,0.11)(0.71,0.41,0.13)(0.51,0.25,0.34)

H4 (0.63,0.51,0.13)(0.42,0.33,0.49)(0.61,0.43,0.45)(0.59,0.37,0.49)

H5 (0.57,0.36,0.29)(0.60,0.15,0.50)(0.70,0.32,0.40)(0.65,0.44,0.33)

Table5.

Emergingtechnologyenterprises

r2

D1 D2 D3 D4

H1 (0.61,0.15,0.53)(0.62,0.35,0.16)(0.61,0.35,0.47)(0.74,0.17,0.55)

H2 (0.66,0.11,0.51)(0.77,0.23,0.43)(0.93,0.08,0.09)(0.99,0.06,0.02)

H3 (0.88,0.09,0.07)(0.89,0.06,0.05)(0.56,0.17,0.44)(0.61,0.13,0.43)

H4 (0.59,0.32,0.34)(0.51,0.48,0.24)(0.68,0.53,0.39)(0.61,0.21,0.34)

H5 (0.71,0.31,0.24)(0.69,0.41,0.35)(0.73,0.44,0.21)(0.74,0.49,0.22)

Table6. Emergingtechnologyenterprises r3

D1 D2 D3 D4

H1 (0.85,0.25.0.15)(0.88,0.23,0.14)(0.78,0.38,0.18)(0.83,0.39,0.29)

H2 (0.94,0.04,0.07)(0.61,0.19,0.39)(0.63,0.18,0.35)(0.56,0.49,0.48)

H3 (0.73,0.13,0.46)(0.88,0.39,0.19)(0.87,0.35,0.18)(0.81,0.13,0.41) H4 (0.82,0.12,0.43)(0.63,0.21,0.55)(0.53,0.33,0.47)(0.51,0.23,0.46) H5 (0.61,0.33,0.29)(0.63,0.41,0.28)(0.74,0.34,0.14)(0.65,0.32,0.37)

Step2: Usethe SFWA operatortoaggregatealltheindividualnormalizedsphericalfuzzydecision matrices.TheaggregatedsphericalfuzzydecisionmatrixisshowninTable 7

Table7. Collectivesphericalfuzzydecisioninformationmatrix ˇ r.

D1 D2 D3 D4

H1 (0.788,0.229,0.319)(0.785,0.315,0.208)(0.696,0.402,0.297)(0.767,0.239,0.371) H2 (0.807,0.078,0.279)(0.674,0.246,0.342)(0.818,0.160,0.227)(0.919,0.188,0.097) H3 (0.814,0.128,0.223)(0.893,0.165,0.099)(0.748,0.284,0.223)(0.677,0.159,0.393) H4 (0.702,0.267,0.271)(0.533,0.324,0.395)(0.615,0.424,0.433)(0.573,0.258,0.421) H5 (0.639,0.331,0.271)(0.644,0.298,0.363)(0.724,0.365,0.224)(0.685,0.411,0.296) Step3:Theentropyofeachattributecanbecalculatedbythefollowingequation:

Step4:Now,weapplyalltheproposedlogarithmicaggregationoperatorstocollectivespherical fuzzyinformationtofindtheaggregatedinformationasfollows:

Case-1: Usinglogarithmicsphericalfuzzyweightedaveragingaggregationoperator,weobtained AggregatedinformationusingL-SFWAOperator

H1 (0.91958,0.06259,0.06493)

H2 (0.95991,0.01743,0.03415)

H3 (0.94502,0.02250,0.03299) H4 (0.70096,0.07377,0.10852) H5 (0.82221,0.09375,0.06097)

Case-2: Usingalogarithmicsphericalfuzzyorderedweightedaveragingaggregationoperator,weobtained

AggregatedinformationusingtheL-SFOWAOperator

H1 (0.91927,0.06301,0.06474) H2 (0.96015,0.01779,0.03378) H3 (0.94556,0.02260,0.03256) H4 (0.70099,0.07411,0.10850) H5 (0.82340,0.09389,0.06056)

Case-3: Usingalogarithmicsphericalfuzzyhybridweightedaveragingaggregationoperator,weobtained

AggregatedinformationusingtheL-SFHWAOperator

H1 (0.99067,0.002598,0.00418) H2 (0.99894,0.00017,0.000327) H3 (0.99191,0.000335,0.001807) H4 (0.81516,0.003897,0.010210) H5 (0.960757,0.007889,0.002174) .

Case-4: Usingalogarithmicsphericalfuzzyweightedgeometricaggregationoperator,weobtained

AggregatedinformationusingtheL-SFWGOperator

H1 (0.91282,0.07890,0.07579) H2 (0.92862,0.02762,0.05696) H3 (0.91705,0.03351,0.06621) H4 (0.62676,0.08927,0.12240) H5 (0.81260,0.10000,0.06923) .

Case-5: Usingalogarithmicsphericalfuzzyorderedweightedgeometricaggregationoperator,weobtained

AggregatedinformationusingtheL-SFOWGOperator

H1 (0.91238,0.07945,0.07551) H2 (0.92843,0.02780,0.05688) H3 (0.91729,0.03354,0.06613) H4 (0.62644,0.08963,0.12237) H5 (0.81366,0.10007,0.06891) .

Case-6: Usingalogarithmicsphericalfuzzyhybridweightedgeometricaggregationoperator,weobtained

AggregatedinformationusingtheL-SFHWGOperator

H1 (0.99151,0.00543,0.00525)

H2 (0.99837,0.00037,0.00121)

H3 (0.98317,0.00118,0.00641)

H4 (0.76785,0.00679,0.01147)

H5 (0.97567,0.00868,0.00233) .

Step5:Wefindthescorevalue S( ogσ ε ρ ) andtheaccuracyvalue A( ogσ ε ρ ) ofthecumulativeoverall preferencevalue hi (i = 1,2,3,4,5).

Case-1: ScoreofaggregatedinformationfortheL-SFWAOperator,andweobtained

ScoreofaggregatedinformationfortheL-SFWAOperator

S( og0.5 H1) 0.985356

S( og0.5 H2) 0.996515

S( og0.5 H3) 0.993345

S( og0.5 H4) 0.737170 S( og0.5 H5) 0.920198

A( og0.5 H1) 0.985391

A( og0.5 H2) 0.996516

A( og0.5 H3) 0.993347

A( og0.5 H4) 0.737347

A( og0.5 H5) 0.920293

Case-2: ScoreofaggregatedinformationfortheL-SFOWAOperator,andweobtained

ScoreofaggregatedinformationfortheL-SFOWAOperator

S( og0.5 H1) 0.985236

S( og0.5 H2) 0.996558

S( og0.5 H3) 0.993479

S( og0.5 H4) 0.737233

S( og0.5 H5) 0.921374

A( og0.5 H1) 0.985271

A( og0.5 H2) 0.99656

A( og0.5 H3) 0.99348

A( og0.5 H4) 0.73741

A( og0.5 H5) 0.92147.

Case-3: ScoreofaggregatedinformationfortheL-SFHWAOperator,andweobtained

ScoreofaggregatedinformationfortheL-SFHWAOperator

S( og0.5 H1) 0.999817

S( og0.5 H2) 0.999998

S( og0.5 H3) 0.999863

S( og0.5 H4) 0.913067

S( og0.5 H5) 0.996664

A( og0.5 H1) 0.999817

A( og0.5 H2) 0.999998

A( og0.5 H3) 0.999863

A( og0.5 H4) 0.913067

A( og0.5 H5) 0.996664.

Case-4: ScoreofaggregatedinformationfortheL-SFWGOperator,andweobtained

ScoreofaggregatedinformationfortheL-SFWGOperator

S( og0.5 H1) 0.982645

S( og0.5 H2) 0.988581

S( og0.5 H3) 0.984383

S( og0.5 H4) 0.545559

S( og0.5 H5) 0.910307

A( og0.5 H1) 0.98272

A( og0.5 H2) 0.988593

A( og0.5 H3) 0.984404

A( og0.5 H4) 0.545863

A( og0.5 H5) 0.910436.

Case-5: ScoreofaggregatedinformationfortheL-SFOWGOperator,andweobtained

ScoreofaggregatedinformationforL-SFOWGOperator

S( og0.5 H1) 0.982461

S( og0.5 H2) 0.988519

S( og0.5 H3) 0.984477

S( og0.5 H4) 0.544563

S( og0.5 H5) 0.911435

A( og0.5 H1) 0.982537

A( og0.5 H2) 0.98853

A( og0.5 H3) 0.984498

A( og0.5 H4) 0.544868

A( og0.5 H5) 0.911564

Case-6: ScoreofaggregatedinformationfortheL-SFHWGOperator,andweobtained

ScoreofaggregatedinformationfortheL-SFHWGOperator

S( og0.5 H1) 0.999914

S( og0.5 H2) 0.999997

S( og0.5 H3) 0.999657

S( og0.5 H4) 0.91689

S( og0.5 H5) 0.999278

A( og0.5 H1) 0.999914

A( og0.5 H2) 0.999997

A( og0.5 H3) 0.999657

A( og0.5 H4) 0.91689

A( og0.5 H5) 0.999278

Step6:Wefindthebest(suitable)alternativethathasthemaximumscorevaluefromsetofalternatives hi (i = 1,2,3,4,5).OverallpreferencevalueandrankingofthealternativesaresummarizedinTable 8. Table8. Overallpreferencevalueandrankingofthealternativesfor σ = 0.5 > 0. S( ogσ H1 ) S( ogσ H2 ) S( ogσ H3 ) S( ogσ H4 ) S( ogσ H5 ) Ranking

L-SFWA 0.9853560.9965150.9933450.7371700.920198 H2 > H3 > H1 > H5 > H4 L-SFOWA 0.9852360.9965580.9934790.7372330.921374 H2 > H3 > H1 > H5 > H4 L-SFHWA 0.9998170.9999980.9998630.9130670.996664 H2 > H3 > H1 > H5 > H4 L-SFWG 0.9826450.9885810.9843830.5455590.910307 H2 > H3 > H1 > H5 > H4 L-SFOWG 0.9824610.9885190.9844770.5445630.911435 H2 > H1 > H3 > H5 > H4 L-SFHWG 0.9999140.9999970.9996570.916890.999278 H2 > H1 > H3 > H5 > H4

Inthissection,wegivethecomparisonanalysisonhowourproposedlogarithmicaggregation operatorsareeffectiveandreliabletoaggregatethesphericalfuzzyinformation.Ashraf[40,41]

proposedthesphericalaggregationoperatorstoaggregatethesphericalfuzzyinformatio;inthis partofourstudy,wegiveacomparisonbetweenproposedandnovellogarithmicsphericalfuzzy aggregationoperators.Wetakethesphericalfuzzyinformationfrom[40]asfollows:

Collectivesphericalfuzzyinformationmatrix (Ashraf,(2018))

H1 (0.658,0.427,0.294)(0.574,0.361,0.339)(0.492,0.548,0.436) H2 (0.733,0.489,0.290)(0.452,0.677,0.249)(0.658,0.307,0.499) H3 (0.388,0.663,0.441)(0.684,0.276,0.273)(0.443,0.266,0.670) H4 (0.765,0.332,0.443)(0.571,0.564,0.367)(0.314,0.349,0.632)

Now,weutilizedalogarithmicsphericalfuzzyweightedaveragingoperatortochoosethebest alternativeasfollows:

AggregatedinformationusingtheL-SFWAOperator

H1 (0.802348,0.128406,0.073504) H2 (0.887498,0.116768,0.072182) H3 (0.585977,0.092906,0.144143) H4 (0.793431,0.084318,0.154080) O

TherankingofthealternativesusingAshraf[40]informationisshownintheFigure 6

Figure6. ComparisonRanking.

Thebastalternativeis H2 Theobtainedresultutilizingalogarithmicsphericalfuzzyweighted averagingoperatoristhesameasresultsshownbyAshraf[40].Hence,thisstudyproposedthenovel logarithmicaggregationoperatorstoaggregatethesphericalfuzzyinformation.Thisstudygivesa morereliabletechniquetoaggregateandtodealwithuncertaintiesindecision-makingproblemsusing sphericalfuzzysets.Utilizingproposedsphericalfuzzylogarithmicaggregationoperators,wefindthe bestalternativefromasetofalternativesgivenbythedecisionmaker.Hence,theproposedMCGDM techniquebasedonsphericalfuzzylogarithmicaggregationoperatorsgivesanothertechniquetofind thebestalternativeasanapplicationindecisionsupportsystems.

7.Conclusions

Inthispaper,wehaverevealedanovellogarithmicoperationofSFSswiththerealbasenumber σ.Additionally,wehaveanalyzedtheirpropertiesandrelationships.Inviewoftheselogarithmic operations,webuiltuptheweightedaveragingandgeometricaggregationoperatorsnamedL-SFWA, L-SFOWA,L-SFHWA,L-SFWG,L-SFOWGandL-SFHWG.AsphericalfuzzyMCDMproblemwith interactivecriteria,anapproachbasedontheproposedoperators,wasproposed.Finally,thismethod wasappliedtoMCDMproblems.Fromthedecisionresultsdisplayedinnumericalexamples,wecan findthatourcreatedapproachcanovercomethedrawbacksofexistingalgebraicaggregationoperators. Inthesucceedingwork,weshallcombinetheproposedoperatorwithsomenovelfuzzysets,suchas type-2fuzzysets,neutrosophicsets,andsoon.Inaddition,wemayinvestigateourcreatedapproach inthefieldofdifferentareas,suchaspersonnelevaluation,medicalartificialintelligence,energy managementandsupplierselectionevaluation.Inaddition,wecandevelopmoredecision-making approacheslikeGRA,TODAM,TOPSIS,VIKORandsoontodealwithuncertaintiesintheformof sphericalfuzzyinformation.

AuthorContributions: Conceptualization,S.A.(ShahzaibAshraf)andS.A.(SaleemAbdullah);methodology,S.A. (ShahzaibAshraf);software,S.A.(ShahzaibAshraf);validation,S.A.(ShahzaibAshraf),S.A.(SaleemAbdullah) andY.J.;formalanalysis,S.A.(SaleemAbdullah)andY.J.;investigation,S.A.(ShahzaibAshraf);writing—original draftpreparation,S.A.;writing—reviewandediting,S.A.(ShahzaibAshraf);visualization,Y.J.;supervision,S.A. (SaleemAbdullah);fundingacquisition,Y.J.

Funding: ThispaperissupportedbyMajorHumanitiesandSocialSciencesResearchProjectsinZhejiang Universities(No.2018QN058),theChinaPostdoctoralScienceFoundation(No.2018QN058),theZhejiang ProvinceNaturalScienceFoundation(No.LY18G010007)andtheNationalNaturalScienceFoundationofChina (No.71761027).

Acknowledgments: Theauthorswouldliketothanktheeditorinchief,associateeditorandtheanonymous refereesfordetailedandvaluablecommentsthathelpedtoimprovethismanuscript.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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