Solving intuitionistic fuzzy multiobjective linear programming problem under neutrosophics

http://www.aimspress.com/journal/Math

Researcharticle

AIMSMathematics,6(5):4556–4580. DOI:10.3934/math.2021269 Received:24September2020 Accepted:03February2021 Published:22February2021

Solvingintuitionisticfuzzymultiobjectivelinearprogrammingproblem underneutrosophicenvironment

AbdullahAliH.Ahmadini1 andFirozAhmad2,3,*

1 DepartmentofMathematics,FacultyofSciences,JazanUniversity,Jazan,SAUDIARABIA

2 DepartmentofStatisticsandOperationsResearch,AligarhMuslimUniversity,Aligarh,INDIA

3 SQC&ORUnit,IndianStatisticalInstitute,203B.T.Road,Kolkata,700108,INDIA

* Correspondence: Email:firoz.ahmad02@gmail.com.

Abstract: Theexistenceofneutral /indeterminacydegreesreflectsthemorepracticalaspects ofdecision-makingscenarios.Thus,thispaperhasstudiedtheintuitionisticfuzzymultiobjective linearprogrammingproblems(IFMOLPPs)underneutrosophicuncertainty.Tohighlightthedegrees ofneutralityinIFMOLPPs,wehaveinvestigatedtheneutrosophicoptimizationtechniqueswith intuitionisticfuzzyparameters.Themarginalevaluationofeachobjectiveisdeterminedbythree differentmembershipfunctions,suchastruth,indeterminacy,andfalsitymembershipdegreesunder theneutrosophicenvironment.Themarginalevaluationofeachobjectivefunctioniselicitedby varioussortsofmembershipfunctionssuchaslinear,exponential,andhyperbolictypesofmembership functions,whichsignifiesanopportunityfordecision-makerstoselectthedesiredmembership functions.Thedevelopedneutrosophicoptimizationtechniqueisimplementedonexistingnumerical problemsthatrevealthevalidityandapplicabilityoftheproposedmethods.Acomparativestudyis alsopresentedwithotherapproaches.Atlast,conclusionsandfutureresearchdirectionsareaddressed basedontheproposedwork.

Keywords: intuitionisticfuzzyparameters;indeterminacymembershipfunction;neutrosophic optimizationtechnique;multiobjectivelinearprogrammingproblem

MathematicsSubjectClassification: 03B52,03F55,62J05,62J99

1.Introduction

Optimizationtechniqueshavemuchimportanceandpopularityinreallifewhilesolving optimizationproblems.Avarietyofmathematicalmodelsexistswhiledealingwithreal-lifescenarios ofdifferentproblems.Hence,themathematicalmodelscontainsomespecificobjectivefunction(s) alongwithasetofwell-definedconstraint(s)subjecttowhichtheproblemsaretobeoptimized(either

maximizedorminimized).Thus,amathematicalprogrammingproblemhavingmultipleconflicting objectivesunderasetofmixedconstraintsistermedasamultiobjectiveprogrammingproblem (MOPP).Inreallife,thedifferentfieldsofoptimizationsuchastransportationproblem,inventory control,assignmentproblems,portfoliooptimization,supplierselection,manufacturingsystem, supplychainmanagement,engineeringproblems,etc.containmorethanoneconflictingobjectives (eitherminimizationormaximization)andtaketheformofMOPP.Whiledealingwithmultiple objectives,itisnotalwayspossibletoobtainasinglesolutionthatoptimizeseachobjectiveefficiently. However,acompromisesolutionispossiblethatsatisfieseachobjectivesimultaneously.Therefore theconceptofcompromisesolutionisanimportantaspectandleadsinsearchoftheglobaloptimality criterion.Inthepastfewdecades,atremendousamountofresearchhasbeenpresentedinthecontext ofmultiobjectiveoptimizationtechniques.

Initially,thefuzzyset(FS)wasinvestigatedby[35],whichcontainsthedegreeofbelongingness (membershipfunctions)fortheelementintothefeasiblesolutionset.BasedonFS,[39]presentedthe fuzzyprogrammingapproach(FPA)forMOPPinwhichmembershipfunctionsrepresentthe marginalevaluationofeachobjective.InFPA,thedecision-makerssatisfactionlevelisachievedby maximizingthemembershipfunctionofeachobjective.Lateron,theextensionsofFPAsuchfuzzy intervalprogramming,fuzzystochasticprogramming,fuzzygoalprogramming,etc.havebeen presentedandsuccessfullyappliedtosolvetheMOPPaccordingtothenatureoftheproblemand appropriaterequirementofthetechniques.

Insomecases,onlythemembershipfunctionmaynotbethebestrepresentativeofthedegreeof belongingnessofanelementintothefeasiblesolutionset.However,thedegreeofnon-belongingness wouldbebettertorepresentthehesitationaspectsofanelementintothefeasiblesolutionset.Asa result,intuitionisticfuzzyset(IFS)waspresentedby[11],whichisthegeneralizationandextensionof theFS.TheIFSdealswithboththedegreeofbelongingnessandnon-belongingnessoftheelementinto thefeasiblesolutionsimultaneously.BasedontheIFS,P.P.Angelov[10]suggestedtheintuitionistic fuzzyprogrammingapproach(IFPA)tosolvetheMOPPinwhichthemarginalevaluationofeach objectivefunctionisdeterminedbythemembershipaswellasnon-membershipfunctions.InIFPA, thesatisfactionlevelofthedecision-makerisachievedbymaximizingthemembershipandminimizing thenon-membershipfunctionsofeachobjectivesimultaneously.Afterward,theextendedversionof IFPA,suchasintuitionisticfuzzystochasticprogramming,intuitionisticfuzzygoalprogramming,etc. havebeendevelopedandappliedtovariousdecision-makingproblems.

FurtherextensionofFSandIFShasbeenpresentedbecauseindeterminacydegreealsoexistin decision-makingprocesses.Tocopewiththedegreeofindeterminacy/neutrality,F.Smarandache[32] proposedtheneutrosophicset(NS).TheNSconsidersthreedifferentmembershipfunctionsnamely; thetruth(degreeofbelongingness),indeterminacy(degreeofbelongingnessuptosomeextent)and afalsity(degreeofnon-belongingness)membershipfunctionsfortheelementintoafeasiblesolution set.Recently,manyauthorshaveusedtheconceptofNSsuchas[2, 4, 5, 26]whilemakingtheoptimal decisions.Anindeterminacydegreearetheareasofignoranceofpropositions’pointsbetweenthetruth andafalsitydegree.Itmeansthattheindeterminacydegreesareindependentandcanbeoptimized simultaneously.Forexample,ifwecollectsomeinformationfromscholarsregardingajournaland thepossibilitythatthejournalisgoodis0.7,thejournalisnotgoodis0.6,anddonotknowaboutthe journalis0.3.Then,thiskindoflinguisticvaguenessorimprecisenessarebeyondthescopeofFSand IFSandconsequentlybeyondFPAandIFPAtodecisionmakingprocessrespectively.Therefore,itis worthtoincorporatetheindeterminacydegreewhiledealingwithMOPP.

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Acontinuouseffortisbeingmadebymanyresearchersorpractitionerstoobtainthebestpossible solutionofmultiobjectivelinearprogrammingproblems(MOLPPs).Alargepartoftheliteratureisfull offuzzy-basedoptimizationtechniquesforMOLPPs.Inthepastfewyears,thegeneralizedconcept ofafuzzysetisutilizedtosolvetheMOLPPs.Anintuitionisticfuzzy-basedoptimizationmethod isalsoimplementedbymanyresearchersandgainedawiderangeofapplicabilityandacceptability whileoptimizingtheMOLPPs.However,theexistingapproacheshavesomelimitationsandcanbe overcomebyapplyingtheproposedneutrosophicoptimizationtechnique.Thefollowingaresome essentialpointsthatensurearemarkablecontributiontothedomainofmultiobjectiveoptimization techniques.

• Anindeterminacydegreeistheregionofignoranceofpropositions’valuesbetweenthetruthand falsitydegrees.Thisaspectcanonlybemanagedwiththeneutrosophicoptimizationmethod.

• TheexistingmethodsofsolvingMOLPPs[19, 28, 36]consideredonlythemembershipfunction whereas[22, 30, 31]includedthemembershipaswellasnon-membershipdegreesofeach objectivefunction.Theydonotcovertheindeterminacydegreewhilemakingdecisions.We havesuccessfullycopewiththeconceptofneutralityandhencesuggestedindeterminacydegree alongwithmembershipandnon-membershipdegreessimultaneously.

• ThemethodsofhandlingMOLPPs[13, 17]onlytakecareofthedegreeofbelongingnessamong theparametersanddonotconsiderthedegreeofnon-belongingness.Therefore,thisstudy capturedtheconceptsofbelongingnessandnon-belongingnessdegreesamongtheparameters anddealtwiththehesitationaspects.

• Thepresentedapproachby[18]isapplicableonlyforsingle-objectiveproblems.Hencethe addressedmethodcanbeeasilyappliedtoMOLPPswithdifferenttypesofmembershipfunctions.

• Thestudypresentedby[30, 36]doesnotpermittheversatilityofvaguenessdegree(shape parameters)inindeterminacydegrees,butwhileapplyingexponential-typemembershipfunction underneutrosophicenvironment,itcanbeavailed.

Therestofthepaperhasbeensummarizedasfollows:InSection 2,relatedresearchworkhave beendiscussedwhereasSection 3 representsthepreliminaryconceptsregardingintuitionisticfuzzy andneutrosophicsets.TheSection 4 representsmodelingofIFMOLPPs.Theproposedneutrosophic optimizationtechniqueispresentedinSection 5,whereasinSection 6,numericalillustration,and comparativestudywithotherexistingapproachesisdepictedtoverifytheperformanceofthe propoundedsolutionmethod.Finally,concludingremarksandfutureresearchdirectionispresented inSection 7

2.Literaturereview

Literaturesuggestslotsofresearchhasbeencarriedouttosolvethemultiobjectiveprogramming problemsusingfuzzy,intuitionisticfuzzy,andneutrosophicprogrammingapproaches.[20]presented asatisfyingsolutionapproachformultipleobjectives,whichcomprisethemorecriticalobjective attaintingthehighersatisfyingdegree.Amodifiedmethodbasedonthefuzzyrankingfunctionis suggestedby[33]andthecrispmodelsoobtained.Anuncertaingoalprogrammingapproachwas proposedby[21]forthesolutionofuncertainMOPP.P.Singh,etal.[28]discussedanefficient methodthatcarriesthequalityofthreedifferentapproaches,namely,fuzzyprogramming,goal programming,andinteractiveprogramming.Furthermore,thedevelopedalgorithmisappliedtothe

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4559 multiobjectivetransportationproblem.TheMOPPunderuncertaintywasaddressedby[38]and namedasanuncertainmultiobjectiveprogrammingproblem.Theknowledgeofefficiencyconcepts withdifferentforms,suchasexpected-value,properexpectedvalue,andtheirrelationshipswitheach other,havebeenestablishedundertheuncertainenvironment.Amultiobjectivefractional programmingproblemwaspresentedby[14],andanovellinearizationstrategictechniquehasbeen developedtosolvetheproblem.J.Dong,etal.[16]alsosuggestedanewapproachbasedonranking functionandTOPSIStosolvetheMOPPunderuncertainty.Anewapproachisproposedby[17]to solvethefullyfuzzyMOPPunderfuzzyparametersthatreducesthecomputationalcomplexityofthe problem.Asolutionapproachisalsodiscussedby[13]tosolvethefullyfuzzymultiobjectivelinear programmingproblemunderfuzzytrapezoidalparameters.S.L.Tilahu[34]developedafeasibility reductionapproachforhierarchicaldecision-makinginMOPP,andthepreferencefreemethod(ideal pointmethod)isappliedtosolveit.

Theintuitionisticfuzzygoalprogrammingapproachisdiscussedby[25]thathastakenthe advantagesofintuitionisticfuzzyset,goalprogramming,andinteractiveprocedures,tosolvethe MOPP.A.K.Nishad,etal.[23]addressedanintuitionisticfuzzygoalprogrammingtosolvethe MOPPunderimpreciseobjectivesandconstraintsand;implementedtosolvetheagricultural productionmanagementsystem.D.Rani,etal.[24]alsosuggestedanalgorithmtosolveMOPP underanintuitionisticfuzzyenvironment.Theproposedapproachhasbeencomparedwithother existingapproaches.S.K.Singh,etal.[30]alsoaddressedintuitionisticfuzzyMOPPundertriangular intuitionisticfuzzyparametersandmixedconstraintsand;solvedtheMOPPusingdifferent membershipfunctions.V.Singh,etal.[31]presentedthemodelingandoptimizationframeworkfor MOPPunderanintuitionisticfuzzyenvironmentwithanoptimisticandpessimisticpointofview. S.K.Singh,etal.[29]solvedthemulti-objectivemixedintegerprogrammingproblemunder intuitionisticfuzzyenvironmentandimplementedonthesupplychainplanningproblem.Recently,A. A.H.Ahmadini,etal.[9]investigateedanovelpreferenceschemeformultiobjectivegoal programmingproblemsunderdifferentsortsofmembershipfunctions.

Basedonneutrosophicsettheory,F.Ahmad,etal.[6]addressedamodifiedneutrosophic optimizationalgorithmforsupplychainmanagementinanuncertainsituation.S.Zeng,etal.[37] presentedmulti-criteriasupplierselectionframeworkbassedontheneutrosophicfuzzydataset. Recently,manyauthorshavecontributedtotheapplicationofneutrosophicoptimizaationtheoryin variousreal-lifeproblems.F.Ahmad[8]discussedthesupplierselectionproblemunderType-2fuzzy parametersandsolvedusingtheinteractiveneutrosophicprogrammingapproach.A.Y.Adhami,et al.[1]alsoproposedanovelPythagorean-hesitantfuzzyoptimizationtechniqueandapplieditto solvethetransportationproblem.F.Ahmad,etal.[7]performedastudyonEnergy-Food-Water securitynexusmanagementandappliedtheneutrosophicprogrammingapproachtosolvethe propoundedmodel.Hence,theproposedneutrosophicapproachemphasizesindeterminacyandtruth andfalsitydegreeswhilemakingthedecisions.Italsogeneralizesthefuzzyandintuitionistic fuzzy-basedapproachesunderdifferenttypesofmembershipfunctionssuchaslinear,exponential, andhyperbolicmembershipfunctions.

3.Preliminaryconcepts

Inthissection,wehavediscussedsomebasicconceptsrelatedtointuitionisticfuzzyand neutrosophicsets.

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Definition1. [11](IntuitionisticFuzzySet)Considerauniversaldiscourse W;thenanintuitionistic fuzzyset(IFS) Y in W isdepictedbytheorderedtripletsasfollows: ˜ Y = {w,µY (w),νY (w) | w ∈ W} where µY (w): W → [0, 1]denotesthemembershipfunctionand νY (w): W → [0, 1]denotesthe non-membershipfunctionoftheelement w intotheset Y,respectively,withtheconditions 0 ≤ µY (w) + νY (w) ≤ 1.

µY (w)

and νY (w) =



Remark1. If a2 = a2 = a3 = a3 thenTrIFN Y = a1 ≤ a1 ≤ a2 ≤ a2 ≤ a3 ≤ a3 ≤ a4 ≤ a4 isreducedinto triangularintuitionisticfuzzynumber(TIFN).(seeFigure 1)

Figure1. Trapeziodalintuitionisticfuzzynumber.

Definition3. [22]ATrIFN Y = {(a1, a2, a3, a4; µY );(a1, a2, a3, a4; νY )} issaidtobeanon-negative TrIFNifandonlyiff a1 ≥ 0.

Definition4. [22]ArithmeticoperationsonTrIFNs. Assumethat A = {(a1, a2, a3, a4);(a1, a2, a3, a4)} and B = {(b1, b2, b3, b4);(b1, b2, b3, b4)} betwoTrIFNs. Then

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Addition : A + pB = a1 + pb1, a2 + pb2, a3 + pb3, a4 + pb4; a1 + pb1, a2 + pb2, a3 + pb3, a4 + pb4 , p ≥ 0 a1 + pb4, a2 + pb3, a3 + pb2, a4 + pb1; a1 + pb4, a2 + pb3, a3 + pb2, a4 + pb1 , p < 0

Multiplication : ˜ A × ˜ B = a1b1, a2b2, a3b3, a4b4; a1b1, a2b2, a3b3, a4b4 , when a1, b1 ≥ 0

Definition5. [22](Accuracyfunction)Theaccuracyfunction φ forTrIFN A = {(a1, a2, a3, a4);(a1, a2, a3, a4)} canberepresentedasfollows: φ( ˜ A) = (a1 + a2 + a3 + a4 + a1 + a2 + a3 + a4) 8

Theorem1. Supposethat ˜ A and ˜ B betwoTrIFNs.Thentheaccuracyfunction φ : IF( R ) → R isa linearfunctioni.e., φ(A + B) = φ(A) + pφ(B)forall p ∈ R . Proof: Letusconsiderthat ˜ A = {(a1, a2, a3, a4);(a1, a2, a3, a4)} and ˜ B = {(b1, b2, b3, b4);(b1, b2, b3, b4)} betwoTrIFNs.Then,basedonthenatureof p,twodifferentcaseswillarise: CaseI: when p ≥ 0,then A + pB = a1 + pb1, a2 + pb2, a3 + pb3, a4 + pb4; a1 + pb1, a2 + pb2, a3 + pb3, a4 + pb4 Therefore, φ( ˜ A + ˜ B) = a1 + pb1 + a2 + pb2 + a3 + pb3 + a4 + pb4 + a1 + pb1 + a2 + pb2 + a3 + pb3 + a4 + pb4 8 = a1 + a2 + a3 + a4 + a1 + a2 + a3 + a4 8 + pb1 + pb2 + pb3 + pb4 + pb1 + pb2 + pb3 + pb4 8

= φ( ˜ A) + pφ( ˜ B)

CaseII: when p < 0,then A + pB = a1 + pb4, a2 + pb3, a3 + pb2, a4 + pb1; a1 + pb4, a2 + pb3, a3 + pb2, a4 + pb1 Suchthat, φ(A + B) = a1 + pb4 + a2 + pb3 + a3 + pb2 + a4 + pb1 + a1 + pb4 + a2 + pb3 + a3 + pb2 + a4 + pb1 8 = a1 + a2 + a3 + a4 + a1 + a2 + a3 + a4 8 + pb1 + pb2 + pb3 + pb4 + pb1 + pb2 + pb3 + pb4 8 = φ( ˜ A) + pφ( ˜ B). Ineachcase,wehaveprovedthat φ(A + B) = φ(A) + pφ(B).Thusaccuracyfunction φ islinear. Definition6. [2](NeutrosophicSet(NS))Supposethat X beauniversaldiscoursesuchthat x ∈ X.A neutrosophicset A in X canbestatedbythreemembershipfunctionsnamely,truth µA(x),indeterminacy λA(x)andafalsity νA(x),whichcanberepresentedbythefollowingexpressions: A = {< x,µA(x),λA(x),νA(x) > |x ∈ X}

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4562 where µA(x),λA(x)and νA(x)arerealstandardornon-standardsubsetsbelongto]0 , 1+[,alsogivenas, µA(x): X → ]0 , 1+[, λA(x): X → ]0 , 1+[,and νA(x): X → ]0 , 1+[.Thereisnorestrictiononthe sumof µA(x),λA(x)and νA(x),sowehave, 0 ≤ sup µA(x) + λA(x) + sup νA(x) ≤ 3+

Definition7. [2]Supposethat X beauniverseofdiscourse.Asinglevaluedneutrosophicset A canbe representedasfollows: A = {< x,µA(x),λA(x),νA(x) > |x ∈ X} where µA(x),λA(x)and νA(x) ∈ [0, 1]and0 ≤ µA(x) + λA(x) + νA(x) ≤ 3foreach x ∈ X

Definition8. [5]Considerthattherebetwodifferentsinglevaluedneutrosophicsets W and Y.The unionof W and Y alsoresultsinasinglevaluedneutrosophicset Z,i.e., Z = (W ∪ Y),whosetruth

µZ (x),indeterminacy λZ (x)andfalsity νZ (x)membershipfunctionscanberepresentedasfollows:

µZ (x) = max(µW (x),µY (x))

λZ (x) = max(λW (x),λY (x))

νZ (x) = min(νW (x),νY (x))foreach x ∈ X.

Definition9. [5]Considerthattherebetwodifferentsinglevaluedneutrosophicsets W and Y.The intersectionof W and Y alsoresultsinasinglevaluedneutrosophicset Z,i.e., Z = (W ∩ Y),whose truth µZ (x),indeterminacy λZ (x)andfalsity νZ (x)membershipfunctionscanberepresentedasfollows:

µZ (x) = min(µW (x),µY (x))

λZ (x) = min(λW (x),λY (x))

νZ (x) = max(νW (x),νY (x))foreach x ∈ X

4.Intuitionisticfuzzymultiobjectivelinearprogrammingproblem

Mostoften,real-lifeproblemsexhibitsoptimizationofmorethanoneobjectivesatatime.The mostpromisingsolutionsetthatsatisfieseachobjectiveefficientlyistermedasthebestcompromise solution.Hencetheconventionalformofmultiobjectivelinearprogrammingproblem(MOLPP)with k objectivesisgivenasfollows(4.1):

Minimize O(x) = [O1(x), O2(x), , Ok(x)] s.t. J j=1 aij x j ≥ bi, i = 1, 2, ··· , I1, J j=1 aij x j ≤ bi, i = I1 + 1, I1 + 2, ··· , I2, J j=1 aij x j = bi, i = I2 + 1, I2 + 2, ··· , I. x j ≥ 0, j = 1, 2, ··· , J.

(4.1)

where Ok(x) = K k=1 ckj x j, ∀ k = 1, 2, ··· , K isthe k-thobjectivefunctionandislinearinnature, bi, ∀ i = 1, 2, , I and; x j, ∀ j = 1, 2, , J aretherighthandsidesandasetofdecision variables,respectively.

Definition10. Assumethat G bethesetoffeasiblesolutionfor(4.1).Thenapoint x∗ issaidtobean efficientorParetooptimalsolutionof(4.1)ifandonlyiftheredoesnotexistany x ∈ G suchthat, AIMSMathematics

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Ok(x∗) ≥ Ok(x), ∀ k and Ok(x∗) > Ok(x)forallatleastone k.

Definition11. Apoint x∗ ∈ G issaidtobeweakParetooptimalsolutionfor(4.1)ifandonlyifthere doesnotexistany x ∈ G suchthat, Ok(x∗) ≥ Ok(x), ∀ k = 1, 2, ··· , K.

InMOLPP(4.1),ifalltheparametersareintuitionisticfuzzynumberthenitistermedas intuitionisticfuzzymultiobjectivelinearprogrammingproblem(IFMOLPP).Itisassumedthatallthe parameterspresentinproblem(4.2)istrapeziodalintuitionisticfuzzynumber.Thusthemathematical formulationofIFMOLPPcanbestatedasfollows(4.2):

Minimize OIF (x) = OIF 1 (x), OIF 2 (x), ··· , OIF k (x) s.t.

J j=1 ˜ aIF ij x j ≥ ˜ bIF i , i = 1, 2, ··· , I1, J j=1 aIF ij x j ≤ bIF i , i = I1 + 1, I1 + 2, , I2, J j=1 aIF ij x j = ˜ bIF i , i = I2 + 1, I2 + 2, ··· , I. x j ≥ 0, j = 1, 2, , J

(4.2)

where OIF k (x) = K k=1 ˜ ckj IF x j, ∀ k = 1, 2, ··· , K isthe k-thobjectivefunctionwithtrapeziodal intuitionisticfuzzyparameters.

Withtheaidofaccuracyfunction(Theorem1)whichislinear,theIFMOLPP(4.2)canbeconverted intothefollowingdeterministicMOLPP(4.3):

Minimize O (x) = O1(x), O2(x), , Ok(x) s.t. J j=1 aij x j ≥ bi , i = 1, 2, , I1, J j=1 aij x j ≤ bi , i = I1 + 1, I1 + 2, ··· , I2, J j=1 aij x j = bi , i = I2 + 1, I2 + 2, ··· , I. x j ≥ 0, j = 1, 2, , J

(4.3) where Ok(x) = φ ˜ OIF k (x) = K k=1 φ ˜ ckj IF x j, ∀ k = 1, 2, , K; bi = φ bIF i and aij = φ ˜ aIF ij ,for all i = 1, 2, , I, j = 1, 2, , J arethecrispversionofalltheobjectivefunctionsandparameters. Theorem2. AnefficientsolutionfordeterministicMOLPP(4.3)isalsoanefficientsolutionfor IFMOLPP(4.2). Proof: Assumethat X = (x1, x2, ··· , xn) beanefficientsolutionofproblem(4.3).Then X isfeasible forproblem(4.3),itmeansthatthefollowingconditionwillhold: J j=1 aij x j ≥ bi , i = 1, 2, , I1, J j=1 aij x j ≤ bi , i = I1 + 1, I1 + 2, ··· , I2, J j=1 aij x j = bi , i = I2 + 1, I2 + 2, ··· , I. x j ≥ 0, j = 1, 2, , J

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Sinceithasbeenprovedthat φ islinear(Theorem1),thenwehave

J j=1 φ aIF ij x j ≥ φ ˜ bIF i , i = 1, 2, ··· , I1, J j=1 φ ˜ aIF ij x j ≤ φ ˜ bIF i , i = I1 + 1, I1 + 2, , I2, J j=1 φ aIF ij x j = φ bIF i , i = I2 + 1, I2 + 2, , I x j ≥ 0, j = 1, 2, , J

Consequently,wehave

J j=1 ˜ aIF ij x j ≥ bIF i , i = 1, 2, , I1, J j=1 aIF ij x j ≤ bIF i , i = I1 + 1, I1 + 2, ··· , I2, J j=1 ˜ aIF ij x j = ˜ bIF i , i = I2 + 1, I2 + 2, , I. x j ≥ 0, j = 1, 2, ··· , J

Hence, X isafeasiblesolutionforproblem(4.2). Moreover,since X isanefficientsolutionforproblem(4.3),theredoesnotexistany X∗ = x∗ 1, x∗ 2, ··· , x∗ n suchthat Ok(X∗) ≤ Ok(X) ∀ k = 1, 2, ··· , K and Ok(X∗) < Ok(X)foratleastone k.Thuswehaveno X∗ suchthatMin K k=1 φ ˜ ckj IF x j ≤ Min K k=1 φ ˜ ckj IF x∗ j ∀ k = 1, 2, , K andMin K k=1 φ ckj IF x j < Min K k=1 φ ckj IF x∗ j foratleastone k. Since φ isalinearfunction(Theorem1),wehaveno X∗ suchthat Min K k=1 φ ˜ ckj IF x j ≤ Min K k=1 φ ˜ ckj IF x∗ j ∀ k = 1, 2, , K and Min K k=1 φ ckj IF x j < Min K k=1 φ ckj IF x∗ j foratleastone k Thus X isefficientsolutionforproblem(4.2).

Considerasingle-objectiveintuitionisticfuzzylinearprogrammingproblem(SOIFLPP)asgivenin problem(4.4):

Minimize ˜ OIF (x) s.t.

J j=1 aIF ij x j ≥ ˜ bIF i , i = 1, 2, ··· , I1, J j=1 ˜ aIF ij x j ≤ bIF i , i = I1 + 1, I1 + 2, , I2, J j=1 aIF ij x j = bIF i , i = I2 + 1, I2 + 2, ··· , I. x j ≥ 0, j = 1, 2, ··· , J.

(4.4)

Usingtheaccuracyfunctionwhichislinearinnature,SOIFLPP(4.4)canbeconvertedintothe followingdeterministicsingle-objectivelinearprogrammingproblem(SOLPP)(4.5):

Minimize O (x) s.t.

J j=1 aij x j ≥ bi , i = 1, 2, , I1, J j=1 aij x j ≤ bi , i = I1 + 1, I1 + 2, ··· , I2, J j=1 aij x j = bi , i = I2 + 1, I2 + 2, ··· , I. x j ≥ 0, j = 1, 2, , J

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Theorem3. AnoptimalsolutionforthedeterministicSOLPP(4.5)isalsoanoptimalsolutionfor SOIFLPP(4.4). Proof. SameasProof2. HencetheoptimalsolutionforanIFLPPcanbeeasillytransformedintodeterministicLPPwiththe aidofaccuracyfunction.Now,wewillproceedtowardsthesolutionproceduresoftheMOLPP(4.3) underneutrosophicenvironment.

5.Proposedneutrosophicfuzzyoptimizationtechnique

Manymultiobjectiveoptimizationtechniquesarepopularamongresearchers.Basedonthefuzzy set,differentfuzzyoptimizationmethodcameintoexistence.Inthefuzzyprogrammingapproach,the marginalevaluationofeachobjectivefunctionisdepictedbyonlythemembershipfunctionsandcan beachievedbymaximizingit.Theextensionofthefuzzyoptimizationmethodispresentedby introducingintuitionisticfuzzyoptimizationtechniques.Itiscomparativelymoreadvancedthanthe fuzzytechniquebecausethemarginalevaluationofeachobjectivefunctionisdepictedbythe membershipandnon-membershipfunctions,whichcanbeachievedbymaximizingthemembership andminimizingthenon-membershipfunctions,respectively.Thereal-lifecomplexitymostoften createstheindeterminacysituationwhilemakingoptimaldecisions.Apartfromtheacceptanceand rejectiondegreesinthedecision-makingprocess,theindeterminacydegreealsohasmuch importance.Thustocovertheindeterminacydegreeoftheelementintothefeasiblesolutionset,F. Smarandache[32]investigatedaneutrosophicset.Thename“neutrosophic”istheadvance combinationoftwoexplicitterms,namely;“neutre”extractedfromFrenchmeans,neutral,and “sophia”adoptedfromGreekmeans,skill/wisdom,thatunanimouslyprovidethedefinition “knowledgeofneutralitydegree”(see[2, 32]).TheNSconsidersthreesortsofmembershipfunctions, suchastruth(degreeofbelongingness),indeterminacy(degreeofbelongingnessuptosomeextent), andafalsity(degreeofnon-belongingness)degreesintothefeasiblesolutionset.Theideaof independentindetermincydegreedifferstheNSwithalltheuncertaindecisionsetssuchasFSand IFS.Theupdatedliteratureworksolelyhighlightsthatmanypractitionersorresearchershavetaken thedeepinterestintheneutrosophicresearchfield(see,[3 5, 26]).TheNSresearchdomainwould getexposureinthefutureandassistindealingwithindeterminacyinthedecision-makingprocess. ThisstudyalsofetchesthenovelideasofneutrosophicoptimizationtechniquesbasedontheNS.The marginalevaluationofeachobjectivefunctionisquantifiedbythetruth,indeterminacy,andfalsity membershipfunctionsundertheneutrosophicdecisionset.ThustheNSplaysavitalrolewhile optimizingthemultiobjectiveoptimizationproblemsbyincorporating,executing,andimplementing theindeterminacydegrees.

R.E.Bellman,etal.[12]firstpropoundedtheideaofafuzzydecisionset.Afterthat,itiswidely adoptedbymanyresearchers.Thefuzzydecisionconceptcomprisesfuzzydecision(D),fuzzygoal (G),andfuzzyconstraints(C),respectively.Herewerecallthemostextensivelyusedfuzzydecision setwiththeaidoffollowingmathematicalexpressions:

Consequently,wealsodepicttheneutrosophicdecisionset DN ,whichcontemplateoverneutrosophic objectivesandconstraintsasfollows:

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where

µD(x) = min

λD(x) = max

µO1 (x),µO2 (x),...,µOK (x)

µC1 (x),µC2 (x),...,µCI (x) ∀ x ∈ X

λO1 (x),λO2 (x),...,λOK (x) λC1 (x),λC2 (x),...,λCI (x) ∀ x ∈ X

νD(x) = max

νO1 (x),νO2 (x),...,νOK (x) νC1 (x),νC2 (x),...,νCI (x) ∀ x ∈ X

where µD(x),λD(x)and νD(x)arethetruth,indeterminacyandafalsitymembershipfunctionsof neutrosophicdecisionset DN respectively.

InordertodepictthedifferentmembershipfunctionsforMOLPP(4.3),theminimumandmaximum valuesofeachobjectivefunctionshavebeenrepresentedby Lk and Uk and;canbeobtainedasfollows:

Uk = max[Ok(x)]and Lk = min[Ok(x)] ∀ k = 1, 2, 3,..., K. (5.1)

Theboundsfor k-thobjectivefunctionundertheneutrosophicenvironmentcanbeobtainedas follows:

U µ k = Uk, Lµ k = Lk fortruthmembership (5.2) U λ k = Lµ k + sk, Lλ k = Lµ k forindeterminacymembership (5.3) U ν k = U µ k , Lν k = Lµ k + tk forfalsitymembership (5.4) where sk and tk ∈ (0, 1)arepredeterminedrealnumbersprescribedbydecision-makers.

5.1.Variousmembershipfunctions

InMOPPs,themarginalevaluationofeachobjectivefunctionisdepictedbyitsrespective membershipfunctions.Ingeneral,themostextensiveandwidelyusedmembershipfunctionisa linearoneduetoitssimplestructureandmoreaccessibleimplications.Thelinear-typemembership functioncontemplatesovertheconstantmarginalrateofsatisfactionordissatisfactiondegrees towardsanobjective.Mostcommonly,themarginalevaluationshavebeenevaluatedbyusinglinear membershipfunctiontoobtaintheobjective.However,itmaybepossibletorepresenteach objective’saspirationlevelwiththeaidofnon-linearmembershipfunction.Theflexiblebehaviorof non-linearmembershipalsofunctionswellenoughtodeterminethemarginalevaluationofobjectives satisfactiondegree.Italsodependsonsomeparameters’value,whichiswellenoughtoefficiently executetheDM(s)strategy.Anexponentialmembershipreflectsthesituationwhenthe decision-makerisworseoff concerninganobjectiveandchoosesahighermarginalrateof satisfaction.Thusbypreferringanexponentialmembershipfunction,thedecision-makercanalso reducedualitygapsbyselectingappropriateshapeparametersinvolvedinthedevelopmentof membershipfunction.Ahyperbolicmembershipfunctionshowstheflexiblecharacteristicbehavior fortheobjectivefunction.Thehyperbolicmembershipfunctionshowstheconvexitybehaviorovera portionoftheobjectiveandrevealsconcavitycharacteristicfeaturesfortheremainingportion.A worse-off situationofthedecisionconcerningagoalisdepictedbytheconvexportionofthe hyperbolicmembershipfunction,andthedecision-makerintendstogainahighermarginalrateof satisfactionlevel.Despitethis,abetter-off conditionofthedecisionconcerningagoalisdepictedby

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theconcaveportionofthehyperbolicmembershipfunction,andthedecision-makerintendstogaina lowermarginalrateofsatisfactionlevelconcerningthatgoal.Therefore,todealwiththeIFMOLPPs, onecanapplylinear,exponential,orhyperbolicmembershipfunctions,dependinguponthechoiceof decision-maker.

Thusthelinear,exponential,andhyperbolicmembershipfunctionsareconstructedunderthe neutrosophicenvironment.Eachofthemisdefinedforthetruth,indeterminacy,andafalsity membershipfunction,whichseemstobemorerealistic.

(O

(5.5)

(Ok(x

ν

(5.7)

Intheabovecase, L(.) k U (.) k forall k objectivefunction.Ifforanymembership L(.) k = U (.) k ,thenthe valueofthesemembershipwillbeequalto1(seeFigure 2).

Figure2. Diagrammaticrepresentationoftruth,indeterminacyandafalsitymembership degreeforobjectivefunction.

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Usingtheconceptof[12],wemaximizetheobjectivefunctiontoreachtheoptimalsolutionofeach objectives.Themathematicalexpressionforobjectivefunctionisdefinedasfollows:

Maxmink=1,2,3,...,K µ( ) k (Ok(x)) Minmaxk=1,2,3,...,K λ( ) k (Ok(x)) Minmaxk=1,2,3,...,K ν( ) k (Ok(x)) subjectto alltheconstraintsof(4.3)

(5.8)

Also,assumethat µL k (Ok(x)) ≥ α, λL k (Ok(x)) ≤ β and νL k (Ok(x)) ≤ γ,forall k. Usingauxiliaryparameters α,β and γ,theproblem(5.8)canbetransformedintothefollowingproblem (5.9): (LTMFA) Max α β γ subjectto Ok(x) + (U µ k Lµ k )α ≤ U µ k , Ok(x) (U λ k Lλ k )β ≤ Lλ k , Ok(x) (U ν k Lν k )γ ≤ Lν k , α ≥ β,α ≥ γ,α + β + γ ≤ 3 α,β,γ ∈ (0, 1) alltheconstraintsof(4.3)

(5.9)

Remark2. Theultimateaimofproblem(5.9)(LTMFA)manifeststhemaximizationofminimum possibilitylevel(truth)toacceptthebestpossiblesolutionand;minimizationofmaximumpossibility level(indeterminacy)and(afalsity)torejecttheworstpossiblesolutionbyconsideringallthe objectivefunctionsatatime.Itmeansthatwetrytodetermineasolutioninsuchawaythatit maximizestheminimumtruthdegree(acceptance)andminimizethemaximumindeterminacy (rejectionuptosomeextent)andafalsity(rejection)degreesbytakingallobjectivessimultaneously, toattaintheoptimalcompromisesolution.

Theorem4. Auniqueoptimalsolutionofproblem(5.9)(LTMFA)isalsoanefficientsolutionforthe problem(4.3). Proof: Supposethat x, α, β, γ beauniqueoptimalsolutionofproblem(5.9)(LTMFA).Then, α β γ > (α β γ) forany (x,α,β,γ) feasibletotheproblem(5.9)(LTMFA).Onthecontrary, assumethat x, α, β, γ isnotanefficientsolutionoftheproblem(4.3).Forthat,thereexists x

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Assumethat

Lk ,thisgives α β γ < (α

β

γ∗) whichmeansthatthe solutionisnotuniqueoptimal.Thus,wehavearrivedatacontradictionwiththefactthat x, α, β, γ is theuniqueoptimalsolutionof(LTMFA).Therefore,itisalsoanefficientsolutionoftheproblem (5.9).ThiscompletestheproofofTheorem4.

E k (Ok(x)),indeterminacy λE k (Ok(x))andafalsity νE k (Ok(x))membership functionsunderneutrosophicenvironmentcanbestatedasfollows:

(5.11)

ν

(5.12) where d isthemeasuresofvaguenessdegree(shapeparameter)andassignedbythedecision-makers.

νE k (Ok(x)) =

(5.13)

Followedbyproblem(5.8),weassumethat µE k (Ok(x)) ≥ α, λE k (Ok(x)) ≤ β and νE k (Ok(x)) ≤ γ,for all k.Usingauxiliaryparameters α,β and γ,theproblem(5.8)canbeconvertedintothefollowing problem(5.13): (ETMFA) Max α β γ subjectto e d Ok (x) Lµ k Uµ k Lµ k e d 1 e d ≥ α, e d Uλ k Ok (x) Uλ k Lλ k e d 1 e d ≤ β, e d Uν k Ok (x) Uν k Lν k e d 1 e d ≤ γ, α ≥ β,α ≥ γ,α + β + γ ≤ 3 α,β,γ ∈ (0, 1) alltheconstraintsof(4.3)

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Remark3. If d → 0,thentheexponential-typemembershipfunctionswillbereducedintolinear-type membershipfunctions.

k ≤ O

 e

d Ok (¯ x) Lk Uk Lk e d 1 e d



 . If α∗ = min k 

) ≤ Ok(¯ x)forall k = 1, 2, ··· , K and Ok(x∗) < Ok(¯ x)foratleastone k. Consequently,wehave     

 ,then α∗ ≥ (>)¯α. Similarly,wehave Uk Ok (x∗) Uk Lk ≥ Uk Ok (¯ x) Uk Lk forall k = 1, 2, , K and Uk Ok (x∗) Uk Lk > Uk Ok (¯ x) Uk Lk foratleastone k Consequently,itgives e d Uk Ok (x ∗) Uk Lk e d 1 e d ≤ e d Uk Ok (¯ x) Uk Lk e d 1 e d forall k = 1, 2, ··· , K and e d Uk Ok (x ∗) Uk Lk e d 1 e d < e d Uk Ok (¯ x) Uk Lk e d 1 e d foratleastone k

k        e d

Hence,max k x) Uk

       ≤ (<        ,wehave β∗ ≤ (<) β. Again,byconsidering γ∗ = max k

Uk Ok (¯        ,weget γ∗ ≤ (<)¯γ. Thisgives α β γ < (α∗ β∗ γ∗),thatcontradictsthefactthat x, α, β, γ istheuniqueoptimal solutionoftheproblem(5.13)(ETMFA).Hence,theTheorem5isproved. 5.1.3.Hyperbolic-typemembershipfunctionsapproach(HTMFA) Thehyperbolic-typetruth µH k (Ok(x)),indeterminacy λH k (Ok(x))andafalsity νH k (Ok(x))membership functionsunderneutrosophicenvironmentcanbedepictedasfollows: µ H k (Ok(x)) =

Lk e d 1 e                1 ifOk(x) ≤ Lµ k 1 2 1 + tanh θk U µ k + Lµ k 2 Ok(x) ifLµ k ≤ Ok(x) ≤ U µ k 0 ifOk(x) ≥ U µ k

)max        e d Uk Ok (x ∗) Uk Lk e d 1 e d

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0 ifOk(x) ≤ Lν k 1 2 1 + tanh θk Ok(x) U ν k + Lν k 2 ifLν k ≤ Ok(x) ≤ U ν k 1 ifOk(x) ≥ U ν k

(5.16) where θk = 6 Uk Lk , ∀ k = 1, 2,..., K. Movingwiththesameprocedureuptoproblem(5.8),wesupposethat µH k (Ok(x)) ≥ α, λH k (Ok(x)) ≤ β and νH k (Ok(x)) ≤ γ,forall k.Usingauxiliaryparameters α,β and γ,theproblem(5.8)canbe transformedintothefollowingproblem(5.17): Max α β γ subjectto 1 2 1 + tanh θk U µ k + Lµ k 2 Ok(x) ≥ α, 1 2 1 + tanh θk U λ k + Lλ k 2 Ok(x) ≤ β, 1 2 1 + tanh θk Ok(x) U ν k + Lν k 2 ≤ γ, α ≥ β,α ≥ γ,α + β + γ ≤ 3 α,β,γ ∈ (0, 1),θk = 6 Uk Lk , ∀ k = 1, 2,..., K alltheconstraintsof(4.3)

Equivalently,wehaveproblem(5.18)asfollows:

(HTMFA) Max α β γ subjectto θkOk(x) + tanh 1 (2α 1) ≤ θk 2 U µ k + Lµ k , θkOk(x) tanh 1 (2β 1) ≤ θk 2 U µ k + Lµ k , θkOk(x) tanh 1 (2γ 1) ≤ θk 2 U µ k + Lµ k , α ≥ β,α ≥ γ,α + β + γ ≤ 3, α,β,γ ∈ (0, 1),θk = 6 Uk Lk , ∀ k = 1, 2,..., K alltheconstraintsof(4.3)

(5.17)

(5.18)

Theorem6. Auniqueoptimalsolutionofproblem(5.18)(HTMFA)isalsoanefficientsolutionforthe problem(4.3).

Proof. Letusconsiderthat x, α, β, γ beauniqueoptimalsolutionofproblem(5.18)(HTMFA),but notanefficientsolutionfortheproblem(4.3).Thisgives,thereexists x∗ (x∗ x) feasibletoproblem (4.3),suchthat Ok(x∗) ≤ Ok(¯ x)forall k = 1, 2, ··· , K and Ok(x∗) < Ok(¯ x)foratleastone k. AIMSMathematics

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5.2.Proposedsolutionalgorithm

Thestep-wisesolutionalgorithmissummarizedasfollows:

Step-1. FormulatetheIFMOLPPasgiveninproblem(4.2).

Step-2. Usingaccuracyfunction,obtainthecrispversionofIFMOLPPasgiveninproblem(4.3). Step-3. Solveeachobjectivefunctionindividuallyanddeterminetheupperandlowerboundbyusing Eq(5.1).

Step-4. Withtheaidof Uk and Lk,calculatetheupperandlowerboundfortruth,indeterminacyanda falsitymembershipunderneutrosophicenvirnmentasgiveninEqs(5.2)–(5.4).

Step-5. Elicitthedifferent-typesofmembershipfunctionsunderneutrosophicenvironmentbyusing Eqs(5.5 5.7),(5.10 5.12)and(5.14 5.16)accordingtodecision-makers’preferencerespectively.

Step-6. Developtheneutrosophicoptimizationmodelssuchas(LTMFA)or(ETMFA)or(HTMFA) underthegivensetofwell-definedconstraints.

Step-7. SolvetheneutrosophicMOPPs(5.9),(5.13)and(5.18)todeterminetheoptimalcompromise outcomesbyapplyingtheappropriatemethodsordifferentoptimizationsoftwarepackages.

6.Numericalillustrations

Followingnumericalexamplesaresolvedusingdifferentmethodsbymanyresearchers.Allthe crispMOLPPsarecodedinAMPLlanguageandsolvedwithhelpofsolverKintro10.3.0viaNEOS serverversion5.0on-linefacilityexecutedbyWisconsinInstitutesforDiscoveryattheUniversityof WisconsininMadisonforsolvingoptimizationproblems,see[15, 27].

Example1. [30]Thisnumericalexampleassumesthatthetrapezoidalintuitionisticfuzzynumber representsalltheparameters.Thusthemathematicalformulationoftheproblemcanbestatedas

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follows:

Min OIF 1 (x) = ˜ 5IF x1 + ˜ 3IF x2 Min ˜ OIF 2 (x) = ˜ 2∗IF x1 + ˜ 7IF x2 s.t. ˜ 2@IF x1 + ˜ 4IF x2 ≥ 25IF 1∗IF x1 + 1@IF x2 ≥ 10IF ˜ 4IF x1 + 5IF x2 ≥ 50IF x1, x2 ≥ 0

(6.1)

TheintuitionisticfuzzyparametersrelatedtoExample1arefurnishedinTable 1.Usingthe accuracyfunction(Definition5),thecrispversionoftheProblem(6.1)canbedepictedasfollows:

Min O1(x) = 5 125x1 + 3 33x2 Min O2(x) = 2.37x1 + 6.2x2 s.t. 1 8x1 + 3 62x2 ≥ 23 75 0 9x1 + 1 06x2 ≥ 9 6 3.62x1 + 5.125x2 ≥ 51.8 x1, x2 ≥ 0.

(6.2) Table1. Example1:Intuitionisticfuzzyparameters. 25IF = (22, 25, 25, 25;18, 25, 25, 25) 10IF = (9, 10, 10, 10;8, 10, 10, 10) 50IF = (50, 50, 50, 55;50, 50, 50, 60) 5IF = (4, 5, 5, 6;4, 5, 5, 7) 3IF = (3, 3, 3, 4;3, 3, 3, 4 5) 2@IF = (1 5, 2, 2, 2;1, 2, 2, 2) 2∗IF = (2, 2, 2, 3;2, 2, 2, 4) 7IF = (7, 7, 7, 7 5;6, 7, 7, 8) 4IF = (3, 4, 4, 4;2, 4, 4, 4) ˜ 1@IF = (1, 1, 1, 1;0 5, 1, 1, 2) ˜ 1∗IF = (0 5, 1, 1, 1;0 2, 1, 1, 1 5) • UsingtheLTMFA(5.9),Problem(6.2)willbeequivalenttothefollowing(6.3): Max α β γ s.t. 5.125x1 + 3.33x2 + (U µ 1 Lµ 1 )α ≤ U µ 1 , 5.125x1 + 3.33x2 (U λ 1 Lλ 1 )β ≤ Lλ 1 , 5 125x1 + 3 33x2 (U ν 1 Lν 1)γ ≤ Lν 1, 2 37x1 + 6 2x2 + (U µ 2 Lµ 2 )α ≤ U µ 2 , 2.37x1 + 6.2x2 (U λ 2 Lλ 2 )β ≤ Lλ 2 , 2.37x1 + 6.2x2 (U ν 2 Lν 2)γ ≤ Lν 2, 1 8x1 + 3 62x2 ≥ 23 75 0 9x1 + 1 06x2 ≥ 9 6 3.62x1 + 5.125x2 ≥ 51.8 α ≥ β,α ≥ γ,α + β + γ ≤ 3 α,β,γ ∈ (0, 1) x1, x2 ≥ 0

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•

UsingtheETMFA(5.13),Problem(6.2)willbeequivalenttothefollowing(6.4):

Max α β γ s.t. e d 5 125x1+3 33x2 Lµ 1 Uµ 1 Lµ 1 e d 1 e d ≥ α, e d Uλ 1 5 125x1 3 33x2 Uλ 1 Lλ 1 e d 1 e d ≤ β, e d Uν 1 5 125x1 3 33x2 Uν 1 Lν 1 e d 1 e d ≤ γ, e d 2 37x1+6 2x2 Lµ 2 Uµ 2 Lµ 2 e d 1 e d ≥ α, e d Uλ 2 2 37x1 6 2x2 Uλ 2 Lλ 2 e d 1 e d ≤ β, e d Uν 2 2 37x1 6 2x2 Uν 2 Lν 2 e d 1 e d ≤ γ, 1.8x1 + 3.62x2 ≥ 23.75 0.9x1 + 1.06x2 ≥ 9.6 3 62x1 + 5 125x2 ≥ 51 8 α ≥ β,α ≥ γ,α + β + γ ≤ 3 α,β,γ ∈ (0, 1) x1, x2 ≥ 0.

UsingtheHTMFA(5.18),Problem(6.2)willbeequivalenttothefollowing(6.5):

Max α β γ s.t.

θ1(5 125x1 + 3 33x2) + tanh 1 (2α 1) ≤ θ1 2 U µ 1 + Lµ 1 , θ1(5.125x1 + 3.33x2) tanh 1 (2β 1) ≤ θ1 2 U µ 1 + Lµ 1 , θ1(5.125x1 + 3.33x2) tanh 1 (2γ 1) ≤ θ1 2 U µ 1 + Lµ 1 , θ2(2 37x1 + 6 2x2) + tanh 1 (2α 1) ≤ θ2 2 U µ 2 + Lµ 2 , θ2(2.37x1 + 6.2x2) tanh 1 (2β 1) ≤ θ2 2 U µ 2 + Lµ 2 , θ2(2.37x1 + 6.2x2) tanh 1 (2γ 1) ≤ θ2 2 U µ 2 + Lµ 2 , 1.8x1 + 3.62x2 ≥ 23.75 0 9x1 + 1 06x2 ≥ 9 6 3 62x1 + 5 125x2 ≥ 51 8

α ≥ β,α ≥ γ,α + β + γ ≤ 3 α,β,γ ∈ (0, 1) x1, x2 ≥ 0

(6.4)

(6.5)

Onsolvingtheneutrosophicoptimizationmodel(6.3),(6.4)and(6.5),thesolutionresultsare summarizedinTable 2.Itisclearthattheobjectivevaluesarequitebetterthan[30]methodforallthe

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4575

membershipfunctions.Also,theproposedneutrosophic’sperformancemeasuresofvarious membershipfunctionscanbedepictedasHyperbolic>Exponential>Linear.Since[30]didnotusethe Exponentialmembershipfunctions;therefore,itisunjustifiabletoperformcomparisonsamongthe membershipfunction.However,ourproposedmethod’smaximumattaintmentofacceptancedegree isbetterachievedandrevealsitssuperiorityover[30]method.

Table2. Example1:Optimalsolutionresults.

MethodsMembershipObjectiveDeviationsfrom αβγ functionsvaluesidealsolutions

Linear(38.97,37.12)(8.82,5.86)0.76NANA

[30]Parabolic(43.02,39.79)(12.87,8.53)0.88NANA Hyperbolic(39.82,38.50)(9.67,7.24)1.00NANA Linear(38.56,38.28)(8.41,7.02)0.770.710.87

ProposedneutrosophicapproachExponential(39.42,38.15)(9.27,6.89)0.910.670.81 Hyperbolic(39.07,38.03)(8.92,6.77)1.000.510.76

NA:NotApplicable.

Example2. [22]Considerthefollowingnumericalillustration.

Min OIF 1 (x) = 2 i=1 3 j=1 c(1)IF ij xij

Min ˜ OIF 2 (x) = 2 i=1 3 j=1 c(2)IF ij xij

s.t.

2 i=1 3 j=1 xij ≤ (50, 70, 90, 110;40, 60, 100, 120)

2 i=1 3 j=1 xij ≤ (35, 55, 75, 95;25, 45, 85, 105) 2 i=1 3 j=1 xij ≥ (25, 45, 65, 85;15, 35, 75, 95) 2 i=1 3 j=1 xij ≥ (15, 25, 35, 45;10, 20, 40, 50) 2 i=1 3 j=1 xij ≥ (45, 55, 65, 75;40, 50, 70, 80) xij ≥ 0 ∀ i = 1, 2& j = 1, 2, 3

TheintuitionisticfuzzyparametersrelatedtoExample2aresummarizedinTable 3.Onsolving eachobjectiveindividually(Example2),wegettheupperandlowerbounds U1 = 8631.25, U2 = 3163.75, L1 = 7981.25and L2 = 2923.75forthefirstandsecondobjective functionsrespectively.OnimplementingtheneutrosophicMOPPs(5.9),(5.13)and(5.18),the solutionresultsaresummarizedinTable 4.Forthisexample,theobjectivevaluesarequitebetter than[22]methodforallthemembershipfunctions.Theperformancemeasuresoftheproposed neutrosophicofvariousmembershipfunctionscanbedepictedasHyperbolic>Exponential>Linear. Thus,theproposedneutrosophicoptimizationtechniqueoutperforms[22]methodforallthe membershipfunctions.However,themaximumattaintmentofacceptancedegreeisbetterachievedby ourproposedtechniquethatshowsthemostpromisingmethodover[22]method.

Table3. Example2:Intuitionisticfuzzyparametersfor˜

, 50, 70, 90)˜

(75, 85, 105, 115;70, 80, 110, 120)

13

(40,

,

,

,

,

,

(50, 60, 80, 100;45, 55, 95, 105)˜ c(1)IF 23 = (25, 35, 55, 65;20, 30, 40, 70) c(2)IF 11 = (6, 10, 12, 16;4, 8, 14, 18)˜ c(2)IF 12 = (25, 35, 40, 50;20, 30, 45, 55)˜ c(2)IF 13 = (14, 22, 26, 34;10, 18, 30, 38) c(2)IF 21 = (20, 30, 35, 45;15, 25, 40, 50)˜ c(2)IF 22 = (8, 14, 17, 23;5, 11, 20, 26)˜ c(2)IF 23 = (16, 24, 28, 36;12, 20, 32, 40)

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Table4. Example2:Optimalsolutionresults.

MethodsMembershipObjectiveDeviationsfrom αβγ functionsvaluesidealsolutions

Linear(9575,3270)(1593.75,346.25)0.50.5NA [22]Exponential(9563,3252)(1581.75,328.25)0.37750.0NA Hyperbolic(9575,3270)(1593.75,346.25)0.50.5NA Linear(9471,3189)(1489.75,265.25)0.740.610.67

ProposedneutrosophicapproachExponential(9302,3126)(1320.75,202.25)0.8350.560.53

Hyperbolic(9249,3107)(1267.75,183.25)0.880.210.36

NA:NotApplicable.

Example3. [22]Letusconsiderthefollowingnumericalexample.

Min ˜ OIF 1 (x) = 2 i=1 3 j=1 c(1)IF ij xij

Min ˜ OIF 2 (x) = 2 i=1 3 j=1 ˜ c(2)IF ij xij

s.t.

2 i=1 3 j=1 xij ≤ (20, 24, 28;18, 24, 32)

2 i=1 3 j=1 xij ≤ (15, 18, 24;12, 18, 30)

2 i=1 3 j=1 xij ≥ (16, 18, 22;14, 18, 24)

2 i=1 3 j=1 xij ≥ (8, 12, 16;6, 12, 20)

2 i=1 3 j=1 xij ≥ (11, 12, 14;10, 12, 18) xij ≥ 0 ∀ i = 1, 2& j = 1, 2, 3.

TheintuitionisticfuzzyparametersrelatedtoExample3aresummarizedinTable 5.Onsolving eachobjectiveindividually(Example3),wegettheupperandlowerbounds U1 = 428.25, U2 = 531.75, L1 = 378.00and L2 = 484.12forthefirstandsecondobjectivefunctions respectively.ByapplyingtheneutrosophicMOPPs(5.9),(5.13)and(5.18),thesolutionresultsare summarizedinTable 6.Onsolvingthefinalneutrosophicoptimizationmodel,theobtainedobjective valuesaremorepromisingthan[22]methodforallthemembershipfunctions.Theworkingefficiency oftheproposedneutrosophicmethodundervariousmembershipfunctionscanbedepictedas Hyperbolic >Exponential>Linear.Thus,theproposedneutrosophictechniqueisthemostprominent methodcomparedto[22]methodunderallthemembershipfunctions.Moreover,themaximum attaintmentofacceptancedegreeisbetterachievedbyourproposedtechniquethatcanbeapreferable methodover[22]method.

Table5. Example3:Intuitionisticfuzzyparametersfor˜

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Table6. Example3:Optimalsolutionresults.

MethodsMembershipObjectiveDeviationsfrom αβγ functionsvaluesidealsolutions

Linear(454,546)(76,61.88)0.720.278NA [22]Exponential(437,531)(59,46.88)0.61380.0NA Hyperbolic(454,546)(76,61.88)0.93350.5NA Linear(427,526)(49,41.88)0.7430.250.47

ProposedneutrosophicapproachExponential(421,519)(43,34.88)0.7870.160.53 Hyperbolic(418,513)(40,28.88)0.9670.340.41

NA:NotApplicable.

7.Conclusions

Inthispaper,aneffectivemodelingandoptimizationframeworkfortheIFMOLPPispresented underneutrosophicfuzzyuncertainty.Theproposedoptimizationtechniquehasbeendesignedunder theneutrosophicenvironment,consistingofindependentindeterminacydegreeindecision-making processes.Thesatisfactorydegreeofdecision-makerisalwaysmaximizedindifferentformstoachieve thebestpossiblecompromisesolutionofeachobjectivefunction.Wehavealsodevelopedavarietyof membershipfunctionstodeterminethebestpossibleoptimalsolution.Theuseoflinear,exponential, andhyperbolicmembershipfunctionsmoreflexibleandrealisticindecision-makingprocessesdue totheindeterminacydegreewhileobtainingtheoptimalcompromisesolution.Theproposedmodels areimplementedonthreedifferentexistingnumericalexamples,andacomparativestudyisdone. Thesolutionresultsobtainedbyourproposedneutrosophicoptimizationtechniqueoutperformsas comparedtoothersforallthediscussednumericalexamples.Anampleopportunitytoselectthe differentsolutionresultsisalsodepictedbyincorporatingvarioussortsofmembershipfunctionsunder theneutrosophicenvironment.Inthecaseofexponential-typemembershipfunction,asetofdifferent optimalsolutionresultscanbegeneratedbytuningtheshapeparameters.

Severaladvantagescanbeavailedbyapplyingtheproposedapproachefficientlyinbothnumerical problemsaswellasreal-lifeapplications.

• Theproposedneutrosophicoptimizationtechniquesdealwiththeindeterminacydegree,whichis theareaofignoranceofpropositions’valuesbetweentheacceptanceandrejectiondegrees.

• Theproposedmultiobjectivemodelingapproachdealswiththeintuitionisticfuzzyparameters, whichinvolvesmembershipaswellasnon-membershipfunctionsandaremorerealisticas comparedtofuzzyparameters.

• Differentsortsofmembershipfunctionsundertheneutrosophicenvironmentprovidean opportunitytoselectthedesiredmembershipfunctionsaccordingtodecision-makers’choices.

• Theshapeparameter(d)inexponential-typemembershipfunctionsdepictsthedegreeof uncertaintyatadifferentlevelandprovidesmoreflexibilityindecision-makingprocesses.

• Anamplescopeforgeneratingavarietyofcompromisesolutionsetsbytuningtheparametersin indeterminacydegreesisalsoabenchmarkingadvantageoftheproposedapproach.

• Itcanbeeasilyimplementedondifferentreal-lifeproblemssuchastransportation,supplier selection,supplychain,manufacturing,inventorycontrol,assignmentproblems,etc.

Thepropoundedstudyhassomelimitationsthatcanbeaddressedinfutureresearch.Various metaheuristicapproachesmaybeappliedtosolvetheproposedneutrosophicmodelasafuture

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4578 researchscope.Thediscussedmodelscanbeappliedtovariousreal-lifeapplicationssuchas transportationproblems,supplierselectionproblems,inventorycontrol,portfoliooptimization,etc.

Acknowledgments

AllauthorsareverythankfultotheEditor-in-Chief,anonymousGuestEditor,andpotential reviewersforprovidingin-depthcommentsandsuggestionsthatimprovedthereadabilityandclarity ofthemanuscript.ThefirstauthorisverythankfultoJazanUniversityforprovidingtheonlineaccess ofvariouspaidjournals.

Conflictofinterest

Allauthorsdeclarenoconflictofinterest.

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