Computing the Greatest X-eigenvector of Fuzzy Neutrosophic Soft Matrix

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InternationalJournalofMathematicsAnditsApplications

ComputingtheGreatestX-eigenvectorofFuzzy NeutrosophicSoftMatrix

M.Kavitha1 ∗ ,P.Murugadas2 andS.Sriram3

1DepartmentofMathematics,AnnamalaiUniversity,Annamalainagar,Tamilnadu,India.

2DepartmentofMathematics,GovernmentArtsCollege(Autonomous),Karur,Tamilnadu,India.

ResearchArticle

3Mathematicswing,DirectorateofDistanceEducation,AnnamalaiUniversity,Annamalainagar,Tamilnadu,India.

Abstract: AFuzzyNeutrosophicSoftVector(FNSV)xissaidtobeaFuzzyNeutrosophicSoftEigenvector(FNSEv)ofasquare max-minFuzzyNeutrosophicSoftMatrix(FNSM)Aif A ⊗ x = x.AFNSEvxofAiscalledthegreatestX-FNSEvof Aif x ∈ X = {x : x ≤ x ≤ x} and y ≤ x foreachFNSEv y ∈ X.Amax-minFNSMAiscalledstronglyX-robustifthe orbit x,A ⊗ x,A2 ⊗ x,... reachesthegreatestX-FNSEvwithanystartingFNSVofX.Wesuggestan O(n3)algorithm forcomputingthegreatestX-FNSEvofAandstudythestrongX-robustness.Thenecessaryandsufficientconditionfor strongX-robustnessareintroducedandanefficientalgarithmforverifyingtheseconditionsisdescribed.

MSC: Primary03E72,Secondary15B15.

Keywords: FuzzyNeutrosophicSoftSet(FNSS),FuzzyNeutrosophicSoftMatrices(FNSMs),FuzzyNeutrosophicSoftEigenvectors(FNSEv),Intervalvector,max-minFuzzyNeutrosophicSoftMatrix(FNSM). c JSPublication.

1.Introduction

Uncertaintyformshaveaveryimportantpartinourdailylife.Duringthetimewehandlereallifeproblemsinvolving uncertaintylikeMedicalfields,Engineering,IndustryandEconomicsandsoon.Theconventionaltechniquesmaynotbe enoughandeasy,soZadeh[34]gavetheintroductionoffuzzysettheoryandthiscameouttobeagiftforthestudyofsome uncertaintytypeswheneveroldtechniquesdidnotwork.Fuzzytheoryandthegeneralizationsregardingitcontributedto someremarkableresultsinreallifethatinvolveuncertaintiesofcertaintype.Ranjit[26]hasanalyzedthatwhetherthefuzzy theoryisanappropriatetoolforlargesizeprobleminimprecisionanduncertaintyininformationrepresentationandprocessingornot.Forthemotiveofhandlingdifferenttypeofuncertainties,severalgeneralizationsandmodificationregarding fuzzysettheorylikevaguesets,roughsets,softsets,theoryofIntuitionisticFuzzySet(IFS)andothergeneralizationIFS ismostuseful.Atanassov[2, 18]developedtheconceptofIFS.TheideasofIFSs,weredevelopedlaterin[19, 20].In1995, Smarandache[30]foundedatheorycalledneutrosophictheoryandneutrosophicsethascapabilitytodealwithuncertainty, imprecise,incompleteandinconsistentinformationwhichexistinrealworld.Thetheoryisapowerfultoolwhichgeneralizes theconceptoftheclassicalset,fuzzyset,interval-valuedfuzzyset,intuitionisticfuzzyset,interval-valuedintuitionistic fuzzyset,andsoon.In1999,aRussianresearcherMolodtsov[12]initiatedtheconceptofsoftsettheoryasageneral mathematicaltoolforhandlinguncertaintyandvagueness.AfterMolodtsov’sworkseveralresearcherswerestudiedonsoft

settheorywithapplications.Majiet.al,[13]initiatedtheconceptoffuzzysoftsetwithsomepropertiesregardingfuzzy softunion,intersection,complementoffuzzysoftset.MoreoverMajietal.[14]extendedsoftsetstointuitionisticfuzzy softsetsandneutrosophicsoftsets.Matricesinmax-minalgebra(theadditionandthemultiplicationareformallyreplaced byoperationsofmaximumandminimum)canbeusedinarangeofpracticalproblemsrelatedtoscheduling,optimization, modelingoffuzzydiscretedynamicsystem,graphtheory,knowledgeengineering,clusteranalysis,fuzzysystemandalso relatedtodescribingdiagnosisoftechnicaldevices[33]orMedicaldiagnosis[28].Theresearchofmax-minalgebracanbe motivatedbyadaptingmax-plusmulti-processorinteractionsystems[3].Inthesesystemswehavenprocessorswhichwork instages,andthealgebraicmodeloftheirinteractivework,entry xi(k)ofavectorx(k),representsthestateofprocessori aftersomestagek,andtheentry aij ofamatrixAencodestheinfluenceoftheworkofprocessorjinthepreviousstage ontheworkofprocessoriinthecurrentstage.Forsimplicity,thesystemisassumedtobehomogeneous,sothatAdoes notchangefromstagetostage.Summingupalltheinfluenceeffectsmultipliedbytheresultsofpreviousstage,wehave xi(k +1)= j aij ⊗ xj (k) Thesummationisofteninterpretedaswaitingtillallworksofthesystemarefinishedandall thenecessaryinfluenceconstraintsaresatisfied. Thustheorbit x,A ⊗ x,...Ak ⊗ x where Ak = A ⊗ ⊗ A,(k-times)representstheevolutionofsuchasystem.Regarding theorbits,onewishestoknowthesetofstartingvectorsfromwhichasteadystateofmulti-processorinteractionsystem (aneigenvectorof A; A ⊗ x = x)canbeachieved.Thesetofstartingvectorsfromwhichasystemreachesaneigenvector (thegreatesteigenvector)ofAafterafinitenumberofstage,ingeneral,containsthesetofalleigenvector,butitcanbean intervalvector X =[x, x]:= {x; x ≤ x ≤ x} andalsoasbigasthewholespace.KimandRouch[21]introducedtheconcept ofFuzzyMatrix(FM).FMplaysavitalroleinvariousareasinScienceandEngeneringandsolvestheproblemsinvolving varioustypesofuncertainties[15].FMsdealonlywithmembershipvaluewhereasIntuitionisticFuzzyMatrices(IFMs)deals withbothmembershipandnon-membershipvalues.Khanet.al,[16]introducedtheconceptofIFMsandseveralinteresting propertiesonIFMshavebeenobtainedin[17].YangandJi[32],introducedamatrixrepresentationoffuzzysetandapplied itindecisionmakingproblems.Boraet.al,[4]introducedtheintuitionisticfuzzysoftmatricesandappliedintheapplication ofaMedicaldiagnosis.SumathiandArokiarani[1]introducednewoperationonfuzzyneutrosophicsoftmatrices.Dhar et.al,[6]havealsodefinedneutrosophicfuzzymatricesandstudiedsquareneutrosophicfuzzymatrices.Umaet.al,[31] introducedtwotypesoffuzzyneutrosophicsoftmatrices.

Inthepresentpaper,weconsiderageneralizedversionoftheproblemtocomputethegreatestFNSEvofAbelongingtoan intervalFNSVX(calledthegreatestX-FNSEvofA),whichisthemainresultofthepaper.Weshowthatunderacertain naturalconditionthegreatestX-FNSEvofAcanbecomputedbyan0(n 3)algorithm.Thenextsectionwillbeoccupied bysomedefinitionandnotationofthemax-minFuzzyNeutosophicSoftAlgebra(FNSA),leadingtothediscussionofthe greatestX-FNSEvoftheFNSMAandstrongX-robustnessofA.Section-6isdevotedtothemainresultcharacterizing strongX-robustFNSMwithorbitofA.Letusconcludewithabriefoverviewoftheworkonmax-minFNSAtowhichthis paperisrelated.Theproblemofcomputingthegreatesteigenvectorofagivenmax-minFNSM.Theconceptsofrobustness (anFNSEvofAisreachedwithanystartingvector)andstrongrobustness(thegreatestFNSEvofAisreachedwithany startingvector)inmax-minalgebra.Someequivalentconditionsandefficientalgorithmsforsomepolynomialprocedures checkingtheweakrobustness(aneigenvectorisreachedonlyifastaringvectorisaneigenvectorofA)inmax-minalgebra.

2.Preliminaries

Inthissectionsomebasicdefinitionoffuzzyneutrosophicsoftset,fuzzyneutrosophicsoftmatrix,andfuzzyneutrosophic softmatrixType-I.

Definition2.1 ([30]) Aneutrosophicset A ontheuniverseofdiscourse X isdefinedas A = { x,TA(x),IA(x),FA(x) ,x ∈ X},where T,I,F : X → ] 0, 1+[ and 0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3+ (1)

Fromphilosophicalpointofviewtheneutrosophicsettakesthevaluefromrealstandardornon-standardsubsetsof ] 0, 1+[ ButinreallifeapplicationespeciallyinScientificandEngineeringproblemsitisdifficulttouseneutrosophicsetwithvalue fromrealstandardornon-standardsubsetof ] 0, 1+[.Henceweconsidertheneutrosophicsetwhichtakesthevaluefrom thesubsetof [0, 1] Thereforewecanrewriteequation (1) as 0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3 Inshortanelement a inthe neutrosophicset A, canbewrittenas a = a T ,a I ,a F , where a T denotesdegreeoftruth, a I denotesdegreeofindeterminacy, a F denotesdegreeoffalsitysuchthat 0 ≤ a T + a I + a F ≤ 3

Example2.2. Assumethattheuniverseofdiscourse X = {x1,x2,x3} where x1,x2 and x3 characterizethequality, reliability,andthepriceoftheobjects.Itmaybefurtherassumedthatthevaluesof {x1,x2,x3} arein[0,1]andtheyare obtainedfromsomeinvestigationsofsomeexperts.Theexpertsmayimposetheiropinioninthreecomponentsviz;thedegree ofgoodness,thedegreeofindeterminacyandthedegreeofpoornesstoexplainthecharacteristicsoftheobjects.SupposeA isaNeutrosophicSet(NS)of X, suchthat A = { x1, 0 4, 0 5, 0 3 , x2, 0 7, 0 2, 0 4 , x3, 0 8, 0 3, 0 4 } wherefor x1 the degreeofgoodnessofqualityis0.4,degreeofindeterminacyofqualityis0.5anddegreeoffalsityofqualityis0.3etc,.

Definition2.3 ([12]) Let U betheinitialuniversesetand E beasetofparameter.Consideranon-emptyset A,A ⊂ E Let P (U ) denotesthesetofallfuzzyneutrosophicsetsof U .Thecollection (F,A) istermedtothefuzzyneutrosophicsoft set(FNSS)over U ,where F isamappinggivenby F : A → P (U ) Hereafterwesimplyconsider A asFNSSover U instead of (F,A)

Definition2.4 ([1]). Let U = {c1,c2,...cm} betheuniversalsetand E bethesetofparametersgivenby E = {e1,e2,...em} Let A ⊂ E.Apair (F,A) bea FNSS over U .Thenthesubsetof U ×E isdefinedby RA = {(u,e); e ∈ A,u ∈ FA(e)} which iscalledarelationformof (FA,E) Themembershipfunction,indeterminacymembershipfunctionandnonmembership functionarewrittenby TRA : U × E → [0, 1],IRA : U × E → [0, 1] and FRA : U × E → [0, 1] where TRA (u,e) ∈ [0, 1],IRA (u,e) ∈ [0, 1] and FRA (u,e) ∈ [0, 1] arethemembershipvalue,indeterminacyvalueandnonmembershipvalue respectivelyof u ∈ U foreach e ∈ E.If [(

Definition2.6. Let A ∈Nm×n,B ∈Nn×p, thecompositionof A and B isdefinedas A ◦ B = n k=1 (a T ik ∧ bT kj ), n k=1 (a I ik ∧ bI kj ), n k=1 (a F ik ∨ bF kj )

equivalentlywecanwritethesameas = n k=1 (a T ik ∧ bT kj ), n k=1 (a I ik ∧ bI kj ), n k=1 (a F ik ∨ bF kj )

Theproduct A ◦ B isdefinedifandonlyifthenumberofcolumnsof A issameasthenumberofrowsof B A and B are saidtobeconformableformultiplication.Weshalluse AB insteadof A ◦ B. Where (a T ik ∧ bT kj ) meansmax-minoperation and n k=1 (a F ik ∨ bF kj ) meansmin-maxoperation.

3.MainResult

Let(N , ≤)beaboundedlinearlyorderedsetwiththeleastelementin N denotedby O =(0, 0, 1)andthegreatestoneby I =(1, 1, 0).Thesetofnaturals(naturalswithzero)isdenotedby N(N0).Forgivennaturals n,m ∈ N,weusethenotations N and M forthesetofallsmallerorequalnaturalnumbers,i.e., N = {1, 2,...,n} and M = {1, 2,...,m}, respectively.The setof n × m matricesover N isdenotedby N(n,m),speciallythesetof n × 1vectorsover N isdenotedby N(n).The max-minalgebraisatriple(N , ⊕, ⊗),where

a ⊕ b =max{a,b} a ⊗ b =min{a,b}.

Theoperations ⊕, ⊗ areextendedtotheFNSM-FNSValgebraover N bythedirectanalogytotheconventionallinear algebra.IfeachentryofaFNSM A ∈N(n,n) (aFNSV x ∈N(n))isequalto O weshalldenoteitas A = O(here O represent azeromatrix)(x = O)here O representzerovector.Thegreatestcommondivisorandtheleastcommonmultipleofaset S ⊆N isdenotedbygcd S,and lcmS respectively.For

∈N(n,n), C ∈N(n,n) wewrite

then p iscalledacycle.Acycleiselementaryifallnodesexcepttheterminalnodearedistinct.Adigraphiscalledstrongly connectedifanytwodistrinctnodesof G arecontainedinacommoncycle. Byastronglyconnectedcomponent K of G =(N,E)wemeanasubdigraph K generatedbyanon-emptysubset K ⊆ N suchthatanytwodistinctnodes i,j ∈ K arecontainedinacommoncycleandKisthemaximalsubsetwiththisproperty. Astronglyconnectedcomponent K ofadigraphiscallednon-trivial,ifthereisacycleofpositivelengthin K.Forany non-trivialstronglyconnectedcomponent K theperiodof K isdefinedasper K = gcd{l(c); c isacyclein K,l(c) > 0} If K istrivial,thenper K =1.Thereisawell-knownconnectionbetweentheentriesinpowersofFNSMsandpathsin associateddigraphs:the(i,j)thentry a T ij ,a I ij ,a F ij k in Ak isequaltothemaximumofweightsofpathsfrom P k ij ,where P k ij isthesetofallpathsoflength k beginningatnode i andendingatnode j.If Pij denotesthesetofallpathsfrom i to j,then a T ij ,a I ij ,a F ij ∗ =max{ a T ij ,a I ij ,a F ij k ; k =1, 2,...} isthemaximumweightofapathfrom Pij and a T jj ,a I jj ,a F jj ∗ isthemaximumweightofacyclecontainingnode j.ForagivenFNSM A ∈N (n,n),thenumber λ ∈N andthen-tuple x ∈N (n)aretheso-calledFNSEvalueof A andFNSEvof A,respectively,if A ⊗ x = λ ⊗ x.

TheFNSEspace V (A,λ)isdefinedasthesetofallFNSEvsof A withassociatedFNSEvalue λ,i.e., V (A,λ)= { x T ,x I ,x F ∈N (n); A ⊗ x = λ ⊗ x}

Incase λ = I letusdenote V (A,I)byabbreviation V (A).DefinethegreatestFNSEV x T ,x I ,x F ∗(A)correspondingtoa FNSM A andtheFNSEvalue I as x T ,x I ,x F ∗(A)= xT ,xI ,xF ∈V (A) x T ,x I ,x F ,

ForeveryFNSM A ∈N (n,n)denote

c T i ,c I i ,c F i (A)= j∈N a T ij ,a I ij ,a F ij , c T ,c I ,c F (A)= i∈N c T i ,c I i ,c F i (A), c T ,c I ,c F ∗(A)=( c T ,c I ,c F (A),..., c T ,c I ,c F (A))t ∈N (n)

ThenotionsofanorbitofaFNSM

iscalledtheperiodof S,denotedbyper (S).Bothoperations ⊕ and inmax-minalgebraareidempotent,hencenonew numbersarecreatedintheprocessofgeneratinganorbit.Thereforeanyorbitinmax-minalgebracontainsonlyafinite numberofdifferentFNSVs.Thusanorbitisalwaysultimatelyperiodic.Thesameholdstrueforthepowersequence (Ak ; k ∈ N).Henceapowersequenceandanorbit O(A,x) arealwaysultimatelyperiodicsequences.Theirperiodswillbe calledtheperiodof A andtheorbitperiodof O(A,x),innotationper (A),per (A,x).

Gavalechasprovedthefollowingtheoremforper(A),whenAissquarematrix.Thistheoremcanbeextendedtosquare FNSMbyroutineproceduers.









InthissectionweshalldealwithpropertiesofthegreatestFNSEvbelongingtoanintervalFNSV. Definition5.1. Let x T ,x I ,x F , x T , x I , x F ∈N (n), x T ,x I ,x F ≤ x T , x I , x F .AnintervalFNSM X withbounds x T ,x I ,x F and x T , x I , x F isdefinedasfollows X =[ x T ,x I ,x F , x T , x I , x F ]= { x T ,x I ,x F ∈N (n); x T ,x I ,x F ≤ x T ,x I ,x F ≤ x T , x I , x F }.Foragiven A ∈N(n,n) and X ⊆N(n) definethegreatest X-FNSEv x T ,x I ,x F

898

∗(A,X)

correspondingtoaFNSM A andanintervelFNSV X as x T ,x I ,x F ∗(A,X)= x∈V (A)∩X x T ,x I ,x F .Asithasbeen written,if X = N(n) then x T ,x I ,x F ∗(A,X) existsforeveryFNSM A,thisisinacontrastwiththecasethat X ⊂N(n) forwhich x T ,x I ,x F ∗(A,X) existsifonlyif V (A) ∩ X = φ Now,wedefineanauxiliaryFNSEv x T ,x I ,x F ⊕(A,X)of A belongingto X whichallowsustousepropertiesofdigraphs andtocharacterizethestructureofthegreatest X-FNSEv x T ,x I ,x F ∗(A,X)correspondingtoaFNSM A andaninterval FNSV X Definition5.2. Foragiven A ∈N (n,n) and X ⊆N (n) defineaFNSV x T ,x I ,x F ⊕(A,X)=( x T 1 ,x I 1 ,x F 1 ⊕(A,X),..., x T n ,x I n,x F n ⊕(A,X))t asfollows x T k ,x I k,x F K ⊕(A,X)= max{ hT ,hI ,hF ∈ [ x T k ,x I k,x F k , x T k , x I k, x F k ]; k isprecyclicin G(A(h), x T , x I , x F (h)} Noticethatifthereis k ∈ N suchthat {

Figure1: G(A(0 5,0 4,0 2),X(0 5,0 4,0 2))= G(A(0 5,0 4,0 2),X(0 5,0 4,0 2)) Figure2: G(A(0 6,0 5,0 1),X(0 6,0 5,0 1)) Figure3: G(A(0 6,0 5,0 1),X(0 6,0 5,0 1)) Figure4: G(A(0 7,0 6,0 1),X(0 7,0 6,0 1)) Figure5: G(A(0 7,0 6,0 1),X(0 7,0 6,0 1))

Bythedefinitionof x T ,x I ,x F ⊕ k (A,X)wegetthatnode2andnode3areprecyclicin G(A 0 5,0 4,0 2 ,x 0 5,0 4,0 2 ) andnodes1,4areprecyclicin ˜ G(A 0 6,0 5,0 1 ,x 0 6,0 5,0 1 ) Thegraph ˜ G(A 0 7,0 6,0 1 ,x 0 7,0 6,0 1 )(similarlyas G(A 0 8,0 7,0 1 ,x 0 8,0 7,0 1 )and G(A 0 9,0 8,0 1 ,x 0 9,0 8,0 1 )isacyclicandhencewegetthat x T ,x I ,x F ⊕(A,X)= ( 0 6, 0 5, 0 1 , 0 5, 0 4, 0 2 , 0 5, 0 4, 0 2 , 0 6, 0 5, 0 1 )t Fromnowweshallsupposethat x T ,x I ,x F ⊕ k (A,X)exists.Then fromthelastdefinitionthenextlemmastraightlyfollows.

Lemma5.4. Let A and X begiven.Then x T ,x I ,x F ≤ x T ,x I ,x F ⊕(A,X) ≤ x T , x I , x F

Theorem5.5. Let A and X begiven.Then x T ,x I ,x F ⊕(A,X) ∈ V (A) Proof. Consideranarbitrarybutfixed k ∈ N andsupposethat k isprecyclicin G(A(h), x T , x I , x F (h))for hT ,hI ,hF = x T k ,x I k,x F k ⊕(A,X).Thenthereisapath p =(k,v1,...,vt,...,vt+s,vt)suchthat(vt,...,vt+s,vt)isacycleandeachnode vj ∈ p isprecyclicin G(A(h), x T , x I , x F (h)).Moreover x T v1 ,x I v1 ,x F v1 ⊕(A,X) ≥ x T k ,x I k,x F k ⊕(A,X)= hT ,hI ,hF and a T kv1 ,a I kv1 ,a

Henceweget

) ≥ x T k ,x I k,x F k ⊕(A,X)

Toprovethereverseinequalitysupposeforacontrarythatthereissomeindex l suchthat a T kl,a I kl,a F kl ⊗ x T l ,x I l ,x F l ⊕(A,X) > x T k ,x I k,x F k ⊕(A,X).Node l isprecyclicin G(A(h), x T , x I , x F (h))for hT ,hI ,hF = x T l ,x I l ,x F l ⊕(A,X), i.e.,thereisapath p =(l,u1,...,ut,...,ut+s,ut).Thenwecanconstructanewpath p = (k,l,u1,...ut,...,ut+s,ut)whichguaranteesthat k isprecyclicin G(A(h ), x T , x I , x F (h ))for hT ,hI ,hF = a T kl,a I kl,a F kl ⊗ x T l ,x I l ,x F l ⊕(A,X).Thisisacontradictionwiththedefinitionof x T k ,x I k,x F k ⊕(A,X).

Theorem5.6. Let A and X begiven.Then (∀ x T ,x I ,x F ∈ V (A) ∩ X)[ x T ,x I ,x F ≤ x T ,x I ,x F ⊕(A,X)] Proof. Supposethat A , X and x T ,x I ,x F ∈ V (A) ∩ X aregiven.Then(A ⊗ x T ,x I ,x F )k = x T k ,x I k,x F k implies thatthereis j1 ∈ N suchthat a T kj1 ,a I kj1 ,a F kj1 ⊗ x T j1 ,x I j1 ,x F j1 = x T k ,x I k,x F k and a T kj1 ,a I kj1 ,a F kj1 ≥ x T k ,x I k,x F k ∧ x T j1 ,x I j1 ,x F j1 ≥ x T k ,x I k,x F k Forrowindex j1 thereis j2 suchthat a T j1j2 ,a I j1j2 ,a F j1j2 ⊗ x T j2 ,x I j2 ,x F j2 = x T j1 ,x I j1 ,x F j1 and a T j1j2 ,a I j1j2 ,a F j1j2 ≥ x T j1 ,x I j1 ,x F j1 ∧ x T j2 ,x I j2 ,x F j2 ≥ x T j1 ,x I j1 ,x F j1 .Byrepeatingtheaboveprocessatmost n timeswe obtainapath p =(k,j1,...,js,js+1,...,js),i.e., k isprecyclicin G(A(xk ), x T , x I , x F (xk )).Noticethatnonodeof p can beremovable(if jl isremovablethenthereisapath p =(i1,...is = jl)in G(A(xk ), x T , x I , x F (xk ))with iO v and iI v+1 and henceinequality a T isis+1 ,a I isis+1 ,a F isis+1 ⊗ x T is+1 ,x I is+1 ,x F is+1 > x T is ,x I is ,x F is contradictstheassumptionthat A ⊗ x = x). Theassertionfollowsfromthefactthat x T k ,x I k,x F k ⊕(A,X)isequaltothegreatest hT ,hI ,hF forwhich k isprpecyclic in G(A(h), x T , x I , x F (h)).Inthelastthreeassertionswehaveshownthat x T ,x I ,x F ⊕(A,X)isanFNSEvbelongingto X andfulfillingamaximalitycondition(∀ x T ,x I ,x F ∈ V (A) ∩ X)[ x T ,x I ,x F ≤ x T ,x I ,x F ⊕(A,X)].Thisisareason toformulatethenextcorollary. Corollary5.7. If V (A) ∩ X = φ then x T ,x I ,x F ∗(A,X)= x T ,x I ,x F ⊕(A,X).

6.TheGreatestx-fuzzyNeutrosophicSoftEigenvector

Theeffectivecomputationof x T ,x I ,x F ⊕(A,X)consideredinthissection. Definition6.1. LetX begiven. X isinvariantunder A if x T ,x I ,x F ∈ X implies A ⊗ x T ,x I ,x F ∈ X. Since A isorder preservingand X isinvariantunder A,thefollowingLemmaisadmitable.

Lemma6.2. X isinvariantunder A ifandonlyif x T ,x I ,x F ≤ A⊗ x T ,x I ,x F ∧A⊗ x T , x I , x F ≤ x T , x I , x F Suppose that X isinvariantunder A and O(A, x T , x I , x F )=( x T , x I , x F (r); r ∈ N0) and O(A, x T ,x I ,x F )=( x T ,x I ,x F (r); r ∈ N0) areorbitsof A generatedby x T , x I , x F and x T ,x I ,x F respectively.Thenforeach k ∈ N0 wehavethefollowing.

x T , x I , x F (k +1)= Ak+1 ⊗ x T , x I , x F = Ak ⊗ (A ⊗ x T , x I , x F ) ≤ Ak ⊗ x T , x I , x F = x T , x I , x F (k)(3)

x T ,x I ,x F (k +1)= Ak+1 ⊗ x T ,x I ,x F = Ak ⊗ (A ⊗ x T ,x I ,x F ) ≥ Ak ⊗ x T ,x I ,x F = x T ,x I ,x F (k) (4) Lemma6.3. Let X beinvariantunder A.Then (∀ k ∈ N0)[ x T , x I , x F (n)= x T , x I , x F (n + k)], (5) (∀ k ∈N0)[ x T ,x I ,x F (n)= x T ,x I ,x F (n + k)] (6) Proof. Accordingto(3)itissufficienttoprovethat x T , x I , x F (n) ≤ x T , x I , x F (n +1).Forthesakeofacontradiction assumethat x T , x I , x F (n) ≤ x T , x I , x F (n +1),i.e.,thereis i ∈ N suchthat x T i , x I i , x F i (n) > x T i , x I i , x F i (n +1),i.e., (An ⊗ x T , x I , x F )i > (An+1 ⊗ x T , x I , x F )i.Thenthereis s ∈ N suchthatforeach k ∈ N thefollowinginequalityholds true a T is,a I is,a F is n ⊗ x T s , x I s , x F s > a T ik,a I ik,a F ik n+1 ⊗ x T k , x I k, x F k ,orequivalently,thereisapath p ∈P n is suchthatnext formula ω(p) ⊗ x T s , x I s , x F s = a T is,a I is,a F is n ⊗ x T s , x I s , x F s > a T ik,a I ik,a F ik n+1 ⊗ x T k , x I k, x F k ≥ ω(˜ p) ⊗ x T k , x I k, x F k holdsforeach k ∈ N andeach˜ p ∈P n+1 ik .Since l(p)= n thenthereisatleastonerepeatednode,i.e., p = p ∪ c and l(c) ≥ 1. Now,weshallconsiderapath p suchthat p = p ∪ c ∪ c.Let l(p )= v.Then v ≥ n +1andweget x T i , x I i , x F i (v) ≥ a T is,a I is,a F is v ⊗ x T s , x I s , x F s ≥ ω(p ) ⊗ x T s , x I s , x F s = ω(p) ⊗ x T s ,x I s ,x F s = a T is,a I is,a F is n ⊗ x T s , x I s , x F s > a T ik,a I ik,a F ik n+1 ⊗ x T k , x I k, x F k = x T i , x I i , x F i (n +1)

Thisisacontradictionwith(3)and(5)follows.Toprove(6)itisenough(by(5))toshowthat x T ,x I ,x F (n) ≥ x T ,x I ,x F (n+1).Forthesakeofacontradictionassumethat x T ,x I ,x F (n) ≥ x T ,x I ,x F (n+1),i.e.,thereis i ∈ N such that x T i ,x I i ,x F i (n) < x T i ,x I i ,x F i (n +1)orequivalently(An ⊗ x T ,x I ,x F )i < (An+1 ⊗ x T ,x I ,x F )i.Thenthereis k ∈ N suchthatforeach s ∈ N thefollowinginequalityholdstrue a T is,a I is,a F is n ⊗ x T s ,x I s ,x F s < a T ik,a I ik,a F ik n+1 ⊗ x T k ,x I k,x F k , oragainequivalently,thereisapath p ∈P n+1 ik suchthatnextformula ω(p ) ⊗ x T s ,x I s ,x F s ≤ a T is,a I is,a F is n ⊗ x T s ,x I s ,x F s < a T ik,a I ik,a F ik n+1 ⊗ x T k ,x I k,x F k = ω(p) ⊗ x T k ,x I k,x F k holdsforeach s ∈ N andeach p ∈P n is.Since l(p)= n +1thenthereisatleastonerepeatednode,i.e., p =˜ p ∪ c,where p ∈P

Proof. Theassertionsfollowsfromthefactsthat

A ⊗ x T ,x I ,x F (n)= A ⊗ (An ⊗ x T ,x I ,x F )= An+1 ⊗ x T ,x I ,x F = x T ,x I ,x F (n +1)= x T ,x I ,x F (n),

A ⊗ x T , x I , x F (n)= A ⊗ (An ⊗ x T , x I , x F )= An+1 ⊗ x T , x I , x F = x T , x I , x F (n +1)= x T , x I , x F (n) Theorem6.5. Let X beinvariantunder A.Then x T ,x I ,x F ⊕(A,X)= x T , x I , x F (n) Proof. Supposethat X isinvariantunder A and x T ,x I ,x F ⊕(A,X)=( x T 1 ,x I 1 ,x F 1 ⊕(A,X),... x T n ,x I n,x F n ⊕(A,X))T Thenbythedefinitionof x T k ,x I k,x F k ⊕(A,X)thereisapath p = p ∪ c (p isfinishedby c)in G(A(h), x T , x I x F (h))for hT ,hI ,hF = x T k ,x I k,x F k ⊕(A,X)and ω(p)= hT ,hI ,hF .Considernowapath p = p ∪ c ∪ ∪ c with l(p ) ≥ n and ω(p)= ω(p ).Thenweobtain x T k , x I k, x F k (n)= x T k , x I k, x F k (l(p )) ≥ ω(p )= x T k ,x I k,x F k ⊕(A,X) Inverseinequalityfollowsfromthefactthatapath p ∈P n kj beginningin k containsacycleand x T k , x I k, x F k (n)= j a T kj ,a I kj ,a F kj n ⊗ x T j , x I j , x F j = a T kl,a I kl,a F kl n ⊗ x T l , x I l , x F l = ω(p) ⊗ x T j , x I j , x F j ,i.e., k isprecyclicin G(A(h), x T , x I , x F (h))for hT ,hI ,hF = x T k , x I k, x F k (n).Hence x T ,x I ,x F ⊕(A,X) ≥ x T , x I , x F (n).

6.1.ComputingtheGreatest X-fuzzyNeutrosophicSoftEigenvector-GeneralCase

Inthissectionwecompute x T ,x I ,x F ⊕(A,X)when X isnotinvariantunder A Let A ∈N (n,n)and X ⊆N (n)begiven.Supposethat A ⊗ x T , x I , x F ≤ x T , x I , x F and V (A) ∩ X = φ,i.e., x T ,x I ,x F ⊕(A,X)exists.WelookforthegreatestFNSV x T , x I , x F ∈ X withthepropertythat x T ,x I ,x F ≤ A ⊗ x T , x I , x F ≤ x T , x I , x F .Forthispurpose x T , x I , x F willbeconstructedbythefollowingalgorithm.

Invariantupperbound

X, A

,

,

,M

,

,

φ and

j ,

I j ,

0 40 30 5 ,M

j

{j

N ; m = x T j , x I j , x F j } = {5, 6} instep2.Since a T 63,a I 63,a F 63 ⊗ x T 3 , x I 3 , x F 3 > x T 6 , x I 6 , x F 6 , put x T 3 , x I 3 , x F 3 := m = 0 40 30 5 (≥ x T 3 ,x I 3 ,x F 3 ), M = {j ∈ N ; m = x T j , x I j , x F j } = {3, 5, 6} and x T , x I , x F =( 0.6, 0.5, 0.2 , 0.7, 0.6, 0.1 , 0.4, 0.3, 0.5 , 0.5, 0.4, 0.3 , 0.4, 0.3, 0.5 0.4, 0.3, 0.5 , 0.7, 0.6, 0.1 )t . Instep4theset M = {j ∈ N ; m ≥ x T j , x I j , x F j } = {3, 5, 6},instep5weobtain N \ M = φ andthealgorithmgoesonstep 2. Thesecondrunofthealgorithm: Instep2weget m :=min j∈N/M x T j , x I j , x F j = 0 5, 0 4, 0 3 ,M = {j ∈ N ; m = x T j , x I j , x F j } = { 0 4, 0 3, 0 5 } Sincetheconditionofstep3isnotfulfilled ((A ⊗ x T , x I , x F )4 ≤ x T 4 , x I 4 , x F 4 ),wecontinuebystep4andstep5,i.e., M = {j ∈ N ; m ≥ x T j , x I j , x F j } = {3, 4, 5, 6}, N \ M = φ andthealgorithmgoesonstep2.

Thefourthrunofthealgorithm: Instep2weget m :=min j∈N \M x T j , x I j , x F j = 0 70 60 1 ,M = {j ∈ N ; m = x T j , x I j , x F j } = {2} Sincetheconditionofstep3isnotfulfilledand M = N thealgorithmterminatesinstep5withthevariablegr:=“yes”andoutput x T , x I , x F =( 0 6, 0 5, 0 2 , 0 7, 0 6, 0 1 , 0 4, 0 3, 0 5 , 0 5, 0 4, 0 3 , 0 4, 0 3, 0 5 , 0 4, 0 3, 0 5 , 0 6, 0 5, 0 2 )t Theorem6.7. Let A ∈N (n,n) and X ⊆N (n) begiven.Thenthealgorithminvariantupperboundiscorrect,itsoutput isthegreatestFNSV x T , x I , x F suchthat x T ,x I ,x F ≤ A ⊗ x T , x I , x F ≤ x T , x I , x F anditscomputationalcomplexity is O(n 2)

Proof. Thealgorithmfinisheswiththepositiveanswerinstep5,whereithascomputedFNSV x T , x I , x F suchthat x T ,x I ,x F ≤ A⊗ x T , x I , x F ≤ x T , x I , x F If x T , x I , x F doesnotexists(thiscorrespondstoelsebranchofthealgorithm instep3)thenitisimpossibletodecrease x T j , x I j , x F j onthelevel x T k , x I k, x F k if a T kj ,a I kj ,a F kj ⊗ x T j , x I j , x F j > x T k , x I k, x F k and x T k , x I k, x F k < x T j ,x I j ,x F j .Further,thesmallestpossibledecreaseof x T j , x I j , x F j instep3guaranteesthemaximality of x T , x I , x F ∈ X withproperty x T ,x I ,x F ≤ A ⊗ x T , x I , x F ≤ x T , x I , x F Fortheestimationofthecomputationalcomplexityobservethatthealgorithmchecksforeachcoordinate x T k , x I k, x F k at most n products a T kj ,a I kj ,a F kj ⊗ x T j , x I j , x F j andcomparesitwith x T k , x I k, x F k instep3.Thenumberofoperationsinstep 3is O(n 2)andinnootherstepitexceedsthebound O(n),hencetheoverallcomplexityis O(n 2).Now,wecansummarize theaboveresultsandsuggestanalgorithmforcomputingthegreatest X-FNSEvof A

AlgorithmGreatest X-FNSEv

Input. X,A. output.“yes”invariablegrif x T ,x I ,x F ⊕(A,X) ∈ X exists;“no”ingrotherwise. begin 1.If A ⊗ x T , x I , x F ≤ x T , x I , x F thencompute x T , x I , x F bythealgorithmInvariantupperbound;put x T , x I , x F := x T , x I , x F ; 2.Compute x T , x I , x F (n)= An ⊗ x T , x I , x F ; 3.If x T ,x I ,x F ≤ An ⊗ x T , x I , x F then x T ,x I ,x F ⊕(A,X):= x T , x I , x F (n)else x T ,x I ,x F ⊕(A,X)doesnotexist; end Theorem6.8.

InthissectionwewillanalyzeconditionsforFNSMunderwhichmulti-processorinteractionsystemsreachthegreatest steadystatewithanystartingFNSVbelongingtoanintervalFNSV X.ThesetofstartingFNSVfromwhichamultiprocessorinteractionsystemreachesanFNSEv(thegreatestFNSEv)of A afterafinitenumberofstages,iscalledattraction set(stronglyattractionset)of A.Ingeneral,attractionset(stronglyattractionset)containsthesetofallFNSEv,theset ofallFNSEvsbelongingto X butitcanbealsoasbigasthewholespace.Letusdenotethesets attr(A)and attr∗(A)as follows. attr(A)= { x T ,x I ,x F ∈N (n); O(A, x T ,x I ,x F ) ∩ V (A) = φ},

attr∗(A)= { x T ,x I ,x F ∈N (n); x T ,x I ,x F ∗(A) ∈ O(A, x T ,x I ,x F )}

Theset attr(A)(attr∗(A))allowsustocharacterizeFNSMsforwhichanFNSEv(thegreatestFNSEv)isreachedwithany startingFNSV.Itiseasytoseethat x T ,x I ,x F ∗(A) ≥ c T ,c I ,c F ∗(A)holdstrueand x T ,x I ,x F ∗(A)cannotbereached withaFNSV x T ,x I ,x F ∈N (n), x T ,x I ,x F < c T ,c I ,c F ∗(A).Letusdenotetheset { x T ,x I ,x F ∈N (n); x T ,x I ,x F < c T ,c I ,c F ∗(A)} by M (A).

Definition6.9. A ∈N (n,n) iscalledstronglyrobustif attr∗(A)= N (n) \ M (A)

Theorem6.10. Let A ∈N (n,n) beaFNSM.Then A isstronglyrobustifandonlyif x T ,x I ,x F ∗(A)= c T ,c I ,c F ∗(A) and G(A(c(A))) isastronglyconnecteddigraphwithperiodequalto1.

Definition6.11. Let A, X begiven.Thenthestronglyattractionset attr∗(A,X) isdefinedasfollows. attr∗(A,X)= { x T ,x I ,x F ∈N (n); x T ,x I ,x F ∗(A,X) ∈ O(A, x T ,x I ,x F )}

Definition6.12. Let A,X begiven. A iscalledstrongly X-robustif X ⊆ attr∗(A,X). Lemma6.13. If A isstrongly X-robustthen (∀ x T ,x I ,x F ∈ X)[per(A, x T ,x I ,x F )= 1, 1, 0 ]

Proof. Supposethat A isstrongly X-robustand x T ,x I ,x F ∈ X isanarbitraryFNSV.Thenthereis k ∈ N suchthat x T ,x I ,x F (k)= x T ,x I ,x F ∗(A,X)andweobtainthefollowing. x T ,x I ,x F (k)= x T ,x I ,x F ∗(A,X)= A ⊗ x T ,x I ,x F ∗(A,X)= A ⊗ x T ,x I ,x F (k)= x T ,x I ,x F (k +1) andtheassertionfollows. Theorem6.14. Let A ∈N (n,n) and X begiven.Then A isstrongly X-robustifandonlyif An ⊗ x T ,x I ,x F = x T ,x I ,x F ∗(A,X)= An ⊗ x T , x I , x F Proof. Supposethat A ∈N (n,n)and X aregivenand An ⊗ x T ,x I ,x F = x T ,x I ,x F ∗(A,X)= An ⊗ x T , x I , x F Then foranarbitraryFNSV x T ,x I ,x F ∈ X weget(bymonotonicityof ⊗)thefollowing x T ,x I ,x F ∗(A,X)= An ⊗ x T ,x I ,x

7.Conclusion

Inthiswork,theauthorsobtainacomputingthegreatestX-FNSEvofaFNSMinmax-minalgebraandstudytheir orbitperiodicity,intervalvectorofFNSM.ThenX-FNSEvprocedureforcomputingthegreatest,generalcaseofgreatest X-FNSEv,AlgorithmInvariantupperbound,AlgorithmofgreatestX-FNSEs,andApplicationofthegreatestX-FNSE.

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