Computing the Greatest X-eigenvector of Fuzzy Neutrosophic Soft Matrix

2017·0.44 MB·English

by  M. Kavitha, P. Murugadas, S. Sriram;

IVnotleurmnaeti5o,nIaslsuJeou4r–nFal(2o0f1M7)a,th89em3–a9t0ic7s. AnditsApplications nationalJournal ofMathematicsAndits
IASvSaNil:ab23le47O-1n5li5n7e: http://ijmaa.in/ retnI•7551-7432:NSSI• snoitacilppA
International Journal of Mathematics And its Applications
Computing the Greatest X-eigenvector of Fuzzy
Neutrosophic Soft Matrix
Research Article
M. Kavitha1∗, P. Murugadas2 and S. Sriram3
1 DepartmentofMathematics,AnnamalaiUniversity,Annamalainagar,Tamilnadu,India.
2 DepartmentofMathematics,GovernmentArtsCollege(Autonomous),Karur,Tamilnadu,India.
3 Mathematicswing,DirectorateofDistanceEducation,AnnamalaiUniversity,Annamalainagar,Tamilnadu,India.
Abstract: A Fuzzy Neutrosophic Soft Vector(FNSV) x is said to be a Fuzzy Neutrosophic Soft Eigenvector(FNSEv) of a square
max-min Fuzzy Neutrosophic Soft Matrix (FNSM) A if A⊗x=x. A FNSEv x of A is called the greatest X-FNSEv of
Aifx∈X ={x:x≤x≤x}andy≤xforeachFNSEvy∈X. Amax-minFNSMAiscalledstronglyX-robustifthe
orbit x,A⊗x,A2⊗x,... reaches the greatest X-FNSEv with any starting FNSV of X. We suggest an O(n3) algorithm
forcomputingthegreatestX-FNSEvofAandstudythestrongX-robustness. Thenecessaryandsufficientconditionfor
strongX-robustnessareintroducedandanefficientalgarithmforverifyingtheseconditionsisdescribed.
MSC: Primary03E72,Secondary15B15.
Keywords: Fuzzy Neutrosophic Soft Set (FNSS), Fuzzy Neutrosophic Soft Matrices(FNSMs), Fuzzy Neutrosophic Soft Eigenvec-
tors(FNSEv),Intervalvector,max-minFuzzyNeutrosophicSoftMatrix(FNSM).
(cid:13)c JSPublication.
1. Introduction
Uncertainty forms have a very important part in our daily life. During the time we handle real life problems involving
uncertainty like Medical fields, Engineering, Industry and Economics and so on. The conventional techniques may not be
enoughandeasy,soZadeh[34]gavetheintroductionoffuzzysettheoryandthiscameouttobeagiftforthestudyofsome
uncertainty types whenever old techniques did not work. Fuzzy theory and the generalizations regarding it contributed to
someremarkableresultsinreallifethatinvolveuncertaintiesofcertaintype. Ranjit[26]hasanalyzedthatwhetherthefuzzy
theory is an appropriate tool for large size problem in imprecision and uncertainty in information representation and pro-
cessingornot. Forthemotiveofhandlingdifferenttypeofuncertainties,severalgeneralizationsandmodificationregarding
fuzzy set theory like vague sets, rough sets, soft sets, theory of Intuitionistic Fuzzy Set (IFS) and other generalization IFS
ismost useful. Atanassov[2, 18] developedthe concept ofIFS. Theideas ofIFSs, were developedlater in[19,20]. In 1995,
Smarandache[30]foundedatheorycalledneutrosophictheoryandneutrosophicsethascapabilitytodealwithuncertainty,
imprecise,incompleteandinconsistentinformationwhichexistinrealworld. Thetheoryisapowerfultoolwhichgeneralizes
the concept of the classical set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, interval -valued intuitionistic
fuzzy set, and so on. In 1999, a Russian researcher Molodtsov [12] initiated the concept of soft set theory as a general
mathematicaltoolforhandlinguncertaintyandvagueness. AfterMolodtsov’sworkseveralresearcherswerestudiedonsoft
∗ E-mail: [email protected]
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ComputingtheGreatestX-eigenvectorofFuzzyNeutrosophicSoftMatrix
set theory with applications. Maji et.al, [13] initiated the concept of fuzzy soft set with some properties regarding fuzzy
soft union, intersection, complement of fuzzy soft set. Moreover Maji et al. [14] extended soft sets to intuitionistic fuzzy
softsetsandneutrosophicsoftsets. Matricesinmax-minalgebra(theadditionandthemultiplicationareformallyreplaced
byoperationsofmaximumandminimum)canbeusedinarangeofpracticalproblemsrelatedtoscheduling,optimization,
modeling of fuzzy discrete dynamic system, graph theory, knowledge engineering, cluster analysis, fuzzy system and also
related to describing diagnosis of technical devices [33] or Medical diagnosis [28]. The research of max-min algebra can be
motivatedbyadaptingmax-plusmulti-processorinteractionsystems[3]. Inthesesystemswehavenprocessorswhichwork
in stages, and the algebraic model of their interactive work, entry x (k) of a vector x(k), represents the state of processor i
i
after some stage k, and the entry a of a matrix A encodes the influence of the work of processor j in the previous stage
ij
on the work of processor i in the current stage. For simplicity, the system is assumed to be homogeneous, so that A does
not change from stage to stage. Summing up all the influence effects multiplied by the results of previous stage, we have
(cid:76)
x (k+1) = a ⊗x (k). The summation is often interpreted as waiting till all works of the system are finished and all
i ij j
j
the necessary influence constraints are satisfied.
Thus the orbit x,A⊗x,...Ak⊗x where Ak =A⊗...⊗A, (k-times) represents the evolution of such a system. Regarding
the orbits, one wishes to know the set of starting vectors from which a steady state of multi-processor interaction system
(an eigenvector of A; A⊗x=x) can be achieved. The set of starting vectors from which a system reaches an eigenvector
(thegreatesteigenvector)ofAafterafinitenumberofstage,ingeneral,containsthesetofalleigenvector,butitcanbean
intervalvectorX =[x,x]:={x;x≤x≤x}andalsoasbigasthewholespace. KimandRouch[21]introducedtheconcept
of Fuzzy Matrix(FM). FM plays a vital role in various areas in Science and Engenering and solves the problems involving
varioustypesofuncertainties[15]. FMsdealonlywithmembershipvaluewhereasIntuitionisticFuzzyMatrices(IFMs)deals
withbothmembershipandnon-membershipvalues. Khanet.al,[16]introducedtheconceptofIFMsandseveralinteresting
propertiesonIFMshavebeenobtainedin[17]. YangandJi[32],introducedamatrixrepresentationoffuzzysetandapplied
itindecisionmakingproblems. Boraet.al,[4]introducedtheintuitionisticfuzzysoftmatricesandappliedintheapplication
of a Medical diagnosis. Sumathi and Arokiarani [1] introduced new operation on fuzzy neutrosophic soft matrices. Dhar
et.al, [6] have also defined neutrosophic fuzzy matrices and studied square neutrosophic fuzzy matrices. Uma et.al, [31]
introduced two types of fuzzy neutrosophic soft matrices.
Inthepresentpaper,weconsiderageneralizedversionoftheproblemtocomputethegreatestFNSEvofAbelongingtoan
interval FNSV X (called the greatest X-FNSEv of A), which is the main result of the paper. We show that under a certain
natural condition the greatest X-FNSEv of A can be computed by an 0(n3) algorithm. The next section will be occupied
by some definition and notation of the max-min Fuzzy Neutosophic Soft Algebra (FNSA), leading to the discussion of the
greatest X-FNSEv of the FNSM A and strong X-robustness of A. Section-6 is devoted to the main result characterizing
strongX-robustFNSMwithorbitofA.Letusconcludewithabriefoverviewoftheworkonmax-minFNSAtowhichthis
paperisrelated. Theproblemofcomputingthegreatesteigenvectorofagivenmax-minFNSM.Theconceptsofrobustness
(an FNSEv of A is reached with any starting vector) and strong robustness (the greatest FNSEv of A is reached with any
starting vector) in max-min algebra. Some equivalent conditions and efficient algorithms for some polynomial procedures
checkingtheweakrobustness(aneigenvectorisreachedonlyifastaringvectorisaneigenvectorofA)inmax-minalgebra.
2. Preliminaries
In this section some basic definition of fuzzy neutrosophic soft set, fuzzy neutrosophic soft matrix, and fuzzy neutrosophic
soft matrix Type-I.
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M.Kavitha,P.MurugadasandS.Sriram
Definition 2.1 ([30]). AneutrosophicsetAontheuniverseofdiscourseX isdefinedasA={(cid:104)x,T (x),I (x),F (x)(cid:105),x∈
A A A
X}, where T,I,F :X → ]−0,1+[ and
−0≤T (x)+I (x)+F (x)≤3+ (1)
A A A
From philosophical point of view the neutrosophic set takes the value from real standard or non-standard subsets of ]−0,1+[.
But in real life application especially in Scientific and Engineering problems it is difficult to use neutrosophic set with value
from real standard or non-standard subset of ]−0,1+[. Hence we consider the neutrosophic set which takes the value from
the subset of [0,1]. Therefore we can rewrite equation (1) as 0≤T (x)+I (x)+F (x)≤3. In short an element a in the
A A A (cid:101)
neutrosophicsetA,canbewrittenasa=(cid:104)aT,aI,aF(cid:105),whereaT denotesdegreeoftruth, aI denotesdegreeofindeterminacy,
(cid:101)
aF denotes degree of falsity such that 0≤aT +aI +aF ≤3.
Example 2.2. Assume that the universe of discourse X = {x , x , x } where x , x and x characterize the quality,
1 2 3 1 2 3
reliability, and the price of the objects. It may be further assumed that the values of {x , x , x } are in [0,1] and they are
1 2 3
obtainedfromsomeinvestigationsofsomeexperts. Theexpertsmayimposetheiropinioninthreecomponentsviz;thedegree
of goodness, the degree of indeterminacy and the degree of poorness to explain the characteristics of the objects. Suppose A
is a Neutrosophic Set (NS) of X, such that A={(cid:104)x , 0.4, 0.5, 0.3(cid:105),(cid:104)x , 0.7, 0.2, 0.4(cid:105),(cid:104)x ,0.8, 0.3, 0.4(cid:105)} where for x the
1 2 3 1
degree of goodness of quality is 0.4, degree of indeterminacy of quality is 0.5 and degree of falsity of quality is 0.3 etc,.
Definition 2.3 ([12]). Let U be the initial universe set and E be a set of parameter. Consider a non-empty set A,A⊂E.
Let P(U) denotes the set of all fuzzy neutrosophic sets of U. The collection (F,A) is termed to the fuzzy neutrosophic soft
set(FNSS)overU,whereF isamappinggivenbyF :A→ P(U).HereafterwesimplyconsiderAasFNSSoverU instead
of (F,A).
Definition2.4([1]). LetU ={c , c , ...c }betheuniversalsetandEbethesetofparametersgivenbyE ={e , e , ...e }.
1 2 m 1 2 m
LetA⊂E. Apair(F,A)beaFNSS overU. Thenthesubsetof U×E isdefinedbyR ={(u,e); e∈A, u∈F (e)}which
A A
is called a relation form of (F , E). The membership function, indeterminacy membership function and non membership
A
function are written by T : U ×E → [0,1], I : U ×E → [0,1] and F : U ×E → [0,1] where T (u,e) ∈
RA RA RA RA
[0,1], I (u,e) ∈ [0,1] and F (u,e) ∈ [0,1] are the membership value, indeterminacy value and non membership value
RA RA
respectively of u∈U for each e∈E. If [(T , I , F )]=[T (u ,e ), I (u ,e ) ,F (u ,e )] we define a matrix
ij ij ij ij i j ij i j ij i j
 
(cid:104)T , I , F (cid:105) ··· (cid:104)T , I , F (cid:105)
11 11 11 1n 1n 1n
 
 
 (cid:104)T21, I21, F21(cid:105) ··· (cid:104)T2n, I2n, F2n(cid:105) 
[(cid:104)T ,I , F (cid:105)] = .
ij ij ij m×n . . . 
.. .. .. 
 
 
(cid:104)T , I , F (cid:105) ··· (cid:104)T , I , F (cid:105)
m1 m1 m1 mn mn mn
Which is called an m×n FNSM of the FNSS (F ,E) over U
A
2.1. FNSMs of Type-I
Definition2.5([31]). LetA=((cid:104)aT, aI , aF(cid:105)), B =(cid:104)(bT, bI , bF(cid:105))∈N . Thecomponentwiseadditionandcomponent
ij ij ij ij ij ij m×n
wise multiplication is defined as
A⊕B =(sup{aT, bT},sup{aI , bI },inf{aF, bF})
ij ij ij ij ij ij
A⊗B =(inf{aT, bT},inf{aI , bI },sup{aF, bF})
ij ij ij ij ij ij
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ComputingtheGreatestX-eigenvectorofFuzzyNeutrosophicSoftMatrix
Definition 2.6. Let A∈N , B ∈N , the composition of A and B is defined as
m×n n×p
(cid:32) n n n (cid:33)
A◦B = (cid:88)(aT ∧ bT ), (cid:88)(aI ∧ bI ), (cid:89)(aF ∨ bF )
ik kj ik kj ik kj
k=1 k=1 k=1
equivalently we can write the same as
(cid:32) n n n (cid:33)
= (cid:95)(aT ∧bT ), (cid:95)(aI ∧bI ), (cid:94)(aF ∨bF ) .
ik kj ik kj ik kj
k=1 k=1 k=1
The product A◦B is defined if and only if the number of columns of A is same as the number of rows of B. A and B are
said to be conformable for multiplication. We shall use AB instead of A◦B. Where (cid:80)(aT ∧ bT ) means max-min operation
ik kj
n
and (cid:81)(aF ∨ bF ) means min-max operation.
ik kj
k=1
3. Main Result
Let (N,≤) be a bounded linearly ordered set with the least element in N denoted by O=(0,0,1) and the greatest one by
I =(1,1,0). Thesetofnaturals(naturalswithzero)isdenotedbyN(N ). Forgivennaturalsn,m∈N,weusethenotations
0
N and M for the set of all smaller or equal natural numbers, i.e., N ={1,2,...,n} and M ={1,2,...,m}, respectively. The
set of n×m matrices over N is denoted by N , specially the set of n×1 vectors over N is denoted by N . The
(n,m) (n)
max-min algebra is a triple (N, ⊕, ⊗), where
a⊕b=max{a,b}
a⊗b=min{a,b}.
The operations ⊕,⊗ are extended to the FNSM-FNSV algebra over N by the direct analogy to the conventional linear
algebra. IfeachentryofaFNSMA∈N (aFNSVx∈N )isequaltoO weshalldenoteitasA=O(hereO represent
(n,n) (n)
a zero matrix)(x=O) here O represent zero vector. The greatest common divisor and the least common multiple of a set
S ⊆N is denoted by gcd S, and lcm S respectively. For A,C ∈N , C ∈N we write
(n,n) (n,n)
A≤C(A<C) if (cid:104)aT,aI ,aF(cid:105)≤(cid:104)cT,cI ,cF(cid:105)((cid:104)aT,aI ,aF(cid:105)<(cid:104)cT,cI ,cF(cid:105))
ij ij ij ij ij ij ij ij ij ij ij ij
holds true for all i,j ∈N. Similarly, for
(cid:104)xT,xI,xF(cid:105)=((cid:104)xT,xI,xF(cid:105),(cid:104)xT,xI,xF(cid:105),...,(cid:104)xT,xI,xF(cid:105))t ∈N(n)
1 1 1 2 2 2 n n n
(cid:104)yT,yI,yF(cid:105)=((cid:104)yT,yI,yF(cid:105),(cid:104)yT,yI,yF(cid:105),...,(cid:104)yT,yI,yF(cid:105))t ∈N(n)
1 1 1 1 1 1 2 2 2 n n n
we write
(cid:104)xT,xI,xF(cid:105)≤(cid:104)yT,yI,yF(cid:105)((cid:104)xT,xI,xF(cid:105)<(cid:104)yT,yI,yF(cid:105)) if (cid:104)xT,xI,xF(cid:105)≤(cid:104)yT,yI,yF(cid:105)((cid:104)xT,xI,xF(cid:105)<(cid:104)yT,yI,yF(cid:105))
i i i i i i i i i i i i
for each i ∈ N. The rth power of a FNSM A is denoted by Ar with elements (cid:104)aT,aI ,aF(cid:105)r. By digraph we understand a
ij ij ij
pairG=(V ,E ),whereV isanon-emptyfiniteset,calledthenodeset,andE ⊆V ×V ,calledthearcset. Adigraph
G G G G G G
G(cid:48) is a subdigraph of G, if VG(cid:48) ⊆ VG and EG(cid:48) ⊆ EG. A path in G is the sequence of nodes p = (vo,v1,...,vt) such that
(v ,v ) ∈ E for all k = 1,2,...,l (denoted as (v ,v)-path). The number l(p) ≥ O is called the length of p. If v = v,
k−1 k G 0 l 0 l
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M.Kavitha,P.MurugadasandS.Sriram
thenpiscalledacycle. Acycleiselementaryifallnodesexcepttheterminalnodearedistinct. Adigraphiscalledstrongly
connected if any two distrinct nodes of G are contained in a common cycle.
By a strongly connected component K of G = (N,E) we mean a subdigraph K generated by a non-empty subset K ⊆ N
suchthatanytwodistinctnodesi,j ∈K arecontainedinacommoncycleandKisthemaximalsubsetwiththisproperty.
A strongly connected component K of a digraph is called non-trivial, if there is a cycle of positive length in K. For any
non-trivial strongly connected component K the period of K is defined as per K = gcd{l(c); c is a cycle in K,l(c) > 0}.
If K is trivial, then per K = 1. There is a well-known connection between the entries in powers of FNSMs and paths in
associated digraphs: the (i,j) th entry (cid:104)aT,aI ,aF(cid:105)k in Ak is equal to the maximum of weights of paths from Pk, where
ij ij ij ij
Pk is the set of all paths of length k beginning at node i and ending at node j. If P denotes the set of all paths from i
ij ij
to j, then (cid:104)aT,aI ,aF(cid:105)∗ = max{(cid:104)aT,aI ,aF(cid:105)k;k = 1,2,...} is the maximum weight of a path from P and (cid:104)aT,aI ,aF(cid:105)∗
ij ij ij ij ij ij ij jj jj jj
is the maximum weight of a cycle containing node j. For a given FNSM A∈N(n,n), the number λ∈N and the n-tuple
x∈N(n) are the so-called FNSE value of A and FNSEv of A, respectively, if
A⊗x=λ⊗x.
The FNSE space V(A,λ) is defined as the set of all FNSEvs of A with associated FNSE value λ, i.e.,
V(A,λ)={(cid:104)xT,xI,xF(cid:105)∈N(n); A⊗x=λ⊗x}.
In case λ=I let us denote V(A,I) by abbreviation V(A). Define the greatest FNSEV (cid:104)xT,xI,xF(cid:105)∗(A) corresponding to a
FNSM A and the FNSE value I as
(cid:104)xT,xI,xF(cid:105)∗(A)= (cid:88) (cid:104)xT,xI,xF(cid:105),
(cid:104)xT,xI,xF(cid:105)∈V(A)
For every FNSM A∈N(n,n) denote
(cid:104)cT,cI,cF(cid:105)(A)= (cid:77)(cid:104)aT,aI ,aF(cid:105),
i i i ij ij ij
j∈N
(cid:104)cT,cI,cF(cid:105)(A)=(cid:79)(cid:104)cT,cI,cF(cid:105)(A),
i i i
i∈N
(cid:104)cT,cI,cF(cid:105)∗(A)=((cid:104)cT,cI,cF(cid:105)(A),...,(cid:104)cT,cI,cF(cid:105)(A))t ∈N(n).
4. Orbit Periodicity of Fuzzy Neutrosophic Soft Matrix
ThenotionsofanorbitofaFNSMAgeneratedbyFNSVxandknownpropertiesoftheorbitperiodicityareintroducedin
this section.
Definition 4.1. For any A ∈ N(n,n) and x ∈ N(n) the orbit of A generated by x is the FNSV sequence O(A,x) =
(x(r);r∈N ) whose initial FNSV is x(0)=x and successive members are defined by the formula
0
x(r+1)=A⊗x(r).
Definition 4.2. ThesequenceS =(S(r); r∈N)isultimatalyperiodicifthereisanaturalnumberpsuchthatthefollowing
holds for some natural number R:S(k+p)=S(k) for all k ≥R. The smallest natural number p with the above property
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ComputingtheGreatestX-eigenvectorofFuzzyNeutrosophicSoftMatrix
is called the period of S, denoted by per (S). Both operations ⊕ and (cid:12) in max-min algebra are idempotent, hence no new
numbers are created in the process of generating an orbit. Therefore any orbit in max-min algebra contains only a finite
number of different FNSVs. Thus an orbit is always ultimately periodic. The same holds true for the power sequence
(Ak;k ∈ N). Hence a power sequence and an orbit O(A,x) are always ultimately periodic sequences. Their periods will be
called the period of A and the orbit period of O(A,x), in notation per (A), per (A,x).
Gavalec has proved the following theorem for per(A), when A is square matrix. This theorem can be extended to square
FNSM by routine proceduers.
Theorem 4.3. Let A ∈ N and x ∈ N . Then per(A) = lcm per(A,x). A fuzzy neutrosophic soft matrix (fuzzy
(n,n) (n) x∈N
neutrosophic soft vector) is called binary if (cid:104)aT,aI ,aF(cid:105)∈{O,I}((cid:104)xT,xI,xF(cid:105)∈{O,I}) for each i,j ∈N.
ij ij ij j j j
Definition 4.4. Let A∈N be a binary FNSM and x∈N , be a binary FNSV. Then by G(A)=(V ,E ) we
(n,n) (n) G(A) G(A)
understand the digraph with V =N,E ={(i,j);(cid:104)aT,aI ,aF(cid:105)=I} and by G(A,x) we understand the corresponding
G(A) G(A) ij ij ij
node-weighteddigraphobtainedfromG(A)byappendingweight(cid:104)xT,xI,xF(cid:105)toeachnodei(denotedbyiO if(cid:104)xT,xI,xF(cid:105)=O
i i i i i i
and iI if (cid:104)xT,xI,xF(cid:105)=I). A path in G(A,x) is called an orbit path if the weight of its terminal node is I.
i i i
Definition 4.5. For A∈N ,(cid:104)xT,xI,xF(cid:105)∈N and (cid:104)hT,hI,hF(cid:105)∈N, the threshold FNSM A and threshold FNSV
(n,n) (n) (h)
(cid:104)xT,xI,xF(cid:105) corresponding to the threshold (cid:104)hT,hI,hF(cid:105) is a binary FNSM of the same type as A and a binary FNSV of
(h)
the same type as (cid:104)xT,xI,xF(cid:105), defined as follows:
 
(cid:104)aTij,aIij,aFij(cid:105)(h)=0I,, if (cid:104)aTij,aIijo,tahFiejr(cid:105)w≥is(cid:104)eh,T,hI,hF(cid:105), 
(2)

(cid:104)xTi ,xIi,xFi (cid:105)(h)=I0,, if (cid:104)xTi ,xIi,xoFith(cid:105)e≥rw(cid:104)ihsTe,hI,hF(cid:105), 
respectively. TheassociateddigraphsG(A )andG(A ,(cid:104)xT,xI,xF(cid:105)(h))willbecalledthethresholddigraphscorresponding
(h) (h)
to the threshold (cid:104)hT,hI,hF(cid:105).
Definition 4.6. Let k ∈ N be a node of G(A ,(cid:104)xT,xI,xF(cid:105) ). k is called removable if there is a node j ∈ N such that
(h) (h)
(j,k)-pathisanorbitpathinG(A ,(cid:104)xT,xI,xF(cid:105) )with((cid:104)xT,xI,xF(cid:105) )=O.If((cid:104)xT,xI,xF(cid:105) )=O thenk isremovable
(h) (h) j j j (h) k k k (h)
by the definition. Let G(A ,(cid:104)xT,xI,xF(cid:105) ) be the threshold digraph corresponding to the threshold (cid:104)hT,hI,hF(cid:105). Denote
(h) (h)
G˜(A ,(cid:104)xT,xI,xF(cid:105) )thedigraphwhicharosefromG(A ,(cid:104)xT,xI,xF(cid:105) )bydeletingallremovablenodes. Anodek∈N
(h) (h) (h) (h)
iscalledprecyclicinG˜(A ,(cid:104)xT,xI,xF(cid:105)(h))ifthereisapathpthatisfinishedbyacycle,i.e.,p=(k,v ,v ...,v ...,v ,v ).
(h) 1 2 t t+s t
If k is lying on a cycle then k is precyclic by the definition. A characterization of G˜(A ,(cid:104)xT,xI,xF(cid:105) ) and a description
(h) (h)
of precyclic node is necessary for computing the greatest fuzzy neutrosophic soft eigenvector belonging to an interval fuzzy
neutrosophic soft vector X.
5. Iterval Fuzzy Neutrosophic Soft Vectors
In this section we shall deal with properties of the greatest FNSEv belonging to an interval FNSV.
Definition 5.1. Let (cid:104)xT,xI,xF(cid:105),(cid:104)xT,xI,xF(cid:105) ∈ N(n), (cid:104)xT,xI,xF(cid:105) ≤ (cid:104)xT,xI,xF(cid:105). An interval FNSM X with bounds
(cid:104)xT,xI,xF(cid:105) and (cid:104)xT,xI,xF(cid:105) is defined as follows X = [(cid:104)xT,xI,xF(cid:105),(cid:104)xT,xI,xF(cid:105)] = {(cid:104)xT,xI,xF(cid:105) ∈ N(n); (cid:104)xT,xI,xF(cid:105) ≤
(cid:104)xT,xI,xF(cid:105) ≤ (cid:104)xT,xI,xF(cid:105)}. For a given A ∈ N and X ⊆ N define the greatest X-FNSEv (cid:104)xT,xI,xF(cid:105)∗(A,X)
(n,n) (n)
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M.Kavitha,P.MurugadasandS.Sriram
corresponding to a FNSM A and an intervel FNSV X as (cid:104)xT,xI,xF(cid:105)∗(A,X) = (cid:80) (cid:104)xT,xI,xF(cid:105). As it has been
x∈V(A)∩X
written, if X = N then (cid:104)xT,xI,xF(cid:105)∗(A,X) exists for every FNSM A, this is in a contrast with the case that X ⊂ N
(n) (n)
for which (cid:104)xT,xI,xF(cid:105)∗(A,X) exists if only if V(A)∩X (cid:54)=φ.
Now, we define an auxiliary FNSEv (cid:104)xT,xI,xF(cid:105)⊕(A,X) of A belonging to X which allows us to use properties of digraphs
andtocharacterizethestructureofthegreatestX-FNSEv(cid:104)xT,xI,xF(cid:105)∗(A,X)correspondingtoaFNSMAandaninterval
FNSV X.
Definition 5.2. For a given A ∈ N(n,n) and X ⊆ N(n) define a FNSV (cid:104)xT,xI,xF(cid:105)⊕(A,X) = ((cid:104)xT,xI,xF(cid:105)⊕(A,X),...,
1 1 1
(cid:104)xT,xI,xF(cid:105)⊕(A,X))t as follows
n n n
(cid:104)xT,xI,xF(cid:105)⊕(A,X)=max{(cid:104)hT,hI,hF(cid:105)∈[(cid:104)xT,xI,xF(cid:105),(cid:104)xT,xI,xF(cid:105)]; k is precyclic in G˜(A ,(cid:104)xT,xI,xF(cid:105) }.
k k K k k k k k k (h) (h)
Noticethatifthereisk∈N suchthat{(cid:104)hT,hI,hF(cid:105)∈[(cid:104)xT,xI,xF(cid:105),(cid:104)xT,xI,xF(cid:105)];kisprecyclicinG˜(A ,(cid:104)xT,xI,xF(cid:105) )}=
k k k k k k (h) (h)
φ then (cid:104)xT,xI,xF(cid:105)⊕(A,X) does not exist.
k k k
Example 5.3. Let A and X =[(cid:104)xT,xI,xF(cid:105),(cid:104)xT,xI,xF(cid:105)] have the forms.
     
(cid:104)0.1 0.1 0.2(cid:105) (cid:104)0.3 0.2 0.4(cid:105) (cid:104)0.8 0.7 0.1(cid:105) (cid:104)0.6 0.5 0.1(cid:105) (cid:104)0.4 0.3 0.2(cid:105) (cid:104)0.8 0.7 0.1(cid:105)
     
     
(cid:104)0.0 0.0 1(cid:105) (cid:104)0.4 0.3 0.2(cid:105) (cid:104)0.6 0.5 0.1(cid:105) (cid:104)0.2 0.3 0.4(cid:105) (cid:104)0.4 0.3 0.2(cid:105) (cid:104)0.5 0.4 0.2(cid:105)
A= , x= , x= .
     
(cid:104)0.5 0.4 0.2(cid:105) (cid:104)0.4 0.3 0.2(cid:105) (cid:104)0.6 0.5 0.1(cid:105) (cid:104)0.1 0.1 0.2(cid:105) (cid:104)0.3 0.2 0.4(cid:105) (cid:104)0.6 0.5 0.1(cid:105)
     
     
(cid:104)0.9 0.8 0.1(cid:105) (cid:104)0.4 0.3 0.2(cid:105) (cid:104)0.3 0.2 0.4(cid:105) (cid:104)0.2 0.1 0.3(cid:105) (cid:104)0.4 0.3 0.2(cid:105) (cid:104)0.9 0.8 0.1(cid:105)
Figure 1:
Figure 2: G(A ,X )
G(A(0.5,0.4,0.2),X(0.5,0.4,0.2))=G(cid:101)(A(0.5,0.4,0.2),X(0.5,0.4,0.2)) (0.6,0.5,0.1) (0.6,0.5,0.1)
Figure 3: G(cid:101)(A(0.6,0.5,0.1),X(0.6,0.5,0.1)) Figure 4: G(A(0.7,0.6,0.1),X(0.7,0.6,0.1))
Figure 5: G(cid:101)(A(0.7,0.6,0.1),X(0.7,0.6,0.1))
By the definition of (cid:104)xT,xI,xF(cid:105)⊕(A,X) we get that node 2 and node 3 are precyclic in G˜(A ,x )
k (cid:104)0.5,0.4,0.2(cid:105) (cid:104)0.5,0.4,0.2(cid:105)
and nodes 1, 4 are precyclic in G˜(A ,x ). The graph G˜(A ,x ) (similarly as
(cid:104)0.6,0.5,0.1(cid:105) (cid:104)0.6,0.5,0.1(cid:105) (cid:104)0.7,0.6,0.1(cid:105) (cid:104)0.7,0.6,0.1(cid:105)
G˜(A ,x ) and G˜(A ,x ) is acyclic and hence we get that (cid:104)xT,xI,xF(cid:105)⊕(A,X) =
(cid:104)0.8,0.7,0.1(cid:105) (cid:104)0.8,0.7,0.1(cid:105) (cid:104)0.9,0.8,0.1(cid:105) (cid:104)0.9,0.8,0.1(cid:105)
((cid:104)0.6,0.5,0.1(cid:105),(cid:104)0.5,0.4,0.2(cid:105),(cid:104)0.5,0.4,0.2(cid:105),(cid:104)0.6,0.5,0.1(cid:105))t.Fromnowweshallsupposethat(cid:104)xT,xI,xF(cid:105)⊕(A,X)exists. Then
k
from the last definition the next lemma straightly follows.
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Lemma 5.4. Let A and X be given. Then (cid:104)xT,xI,xF(cid:105)≤(cid:104)xT,xI,xF(cid:105)⊕(A,X)≤(cid:104)xT,xI,xF(cid:105).
Theorem 5.5. Let A and X be given . Then (cid:104)xT,xI,xF(cid:105)⊕(A,X)∈V(A).
Proof. Consideranarbitrarybutfixedk∈N andsupposethatkisprecyclicinG˜(A ,(cid:104)xT,xI,xF(cid:105) )for(cid:104)hT,hI,hF(cid:105)=
(h) (h)
(cid:104)xT,xI,xF(cid:105)⊕(A,X). Then there is a path p = (k,v ,...,v ,...,v ,v ) such that (v ,...,v ,v ) is a cycle and each node
k k k 1 t t+s t t t+s t
v ∈p is precyclic in G˜(A ,(cid:104)xT,xI,xF(cid:105) ). Moreover
j (h) (h)
(cid:104)xT ,xI ,xF (cid:105)⊕(A,X)≥(cid:104)xT,xI,xF(cid:105)⊕(A,X)=(cid:104)hT,hI,hF(cid:105) and (cid:104)aT ,aI ,aF (cid:105)≥(cid:104)hT,hI,hF(cid:105).
v1 v1 v1 k k k kv1 kv1 kv1
Hence we get
(A⊗(cid:104)xT,xI,xF(cid:105)⊕(A,X)) = (cid:88)(cid:104)aT ,aI ,aF (cid:105)⊗(cid:104)xT,xI,xF(cid:105)⊕(A,X)≥(cid:104)aT ,aI ,aF (cid:105)⊗(cid:104)xT ,xI ,xF (cid:105)⊕(A,X)
k kj kj kj j j j kv1 kv1 kv1 v1 v1 v1
j∈N
≥(cid:104)xT,xI,xF(cid:105)⊕(A,X).
k k k
To prove the reverse inequality suppose for a contrary that there is some index l such that (cid:104)aT,aI ,aF(cid:105) ⊗
kl kl kl
(cid:104)xT,xI,xF(cid:105)⊕(A,X) > (cid:104)xT,xI,xF(cid:105)⊕(A,X). Node l is precyclic in G˜(A ,(cid:104)xT,xI,xF(cid:105) ) for (cid:104)hT,hI,hF(cid:105) =
l l l k k k (h) (h)
(cid:104)xT,xI,xF(cid:105)⊕(A,X), i.e., there is a path p(cid:48) = (l,u ,...,u ,...,u ,u ). Then we can construct a new path p˜(cid:48) =
l l l 1 t t+s t
(k,l,u1,...ut,...,ut+s,ut)whichguaranteesthatkisprecyclicinG˜(A(h(cid:48)),(cid:104)xT,xI,xF(cid:105)(h(cid:48)))for(cid:104)hT,hI,hF(cid:105)(cid:48) =(cid:104)aTkl,aIkl,aFkl(cid:105)⊗
(cid:104)xT,xI,xF(cid:105)⊕(A,X). This is a contradiction with the definition of (cid:104)xT,xI,xF(cid:105)⊕(A,X).
l l l k k k
Theorem 5.6. Let A and X be given. Then (∀(cid:104)xT,xI,xF(cid:105)∈V(A)∩X) [(cid:104)xT,xI,xF(cid:105)≤(cid:104)xT,xI,xF(cid:105)⊕(A,X)].
Proof. Suppose that A , X and (cid:104)xT,xI,xF(cid:105) ∈ V(A)∩X are given. Then (A⊗(cid:104)xT,xI,xF(cid:105)) = (cid:104)xT,xI,xF(cid:105) implies
k k k k
that there is j ∈ N such that (cid:104)aT ,aI ,aF (cid:105) ⊗ (cid:104)xT ,xI ,xF(cid:105) = (cid:104)xT,xI,xF(cid:105) and (cid:104)aT ,aI ,aF (cid:105) ≥ (cid:104)xT,xI,xF(cid:105) ∧
1 kj1 kj1 kj1 j1 j1 j1 k k k kj1 kj1 kj1 k k k
(cid:104)xT ,xI ,xF(cid:105)≥(cid:104)xT,xI,xF(cid:105). For row index j there is j such that (cid:104)aT ,aI ,aF (cid:105)⊗(cid:104)xT ,xI ,xF(cid:105)=(cid:104)xT ,xI ,xF(cid:105) and
j1 j1 j1 k k k 1 2 j1j2 j1j2 j1j2 j2 j2 j2 j1 j1 j1
(cid:104)aT ,aI ,aF (cid:105) ≥ (cid:104)xT ,xI ,xF(cid:105)∧(cid:104)xT ,xI ,xF(cid:105) ≥ (cid:104)xT ,xI ,xF(cid:105). By repeating the above process at most n times we
j1j2 j1j2 j1j2 j1 j1 j1 j2 j2 j2 j1 j1 j1
obtain a path p = (k,j ,...,j ,j ,...,j ), i.e., k is precyclic in G˜(A ,(cid:104)xT,xI,xF(cid:105) ). Notice that no node of p can
1 s s+1 s (xk) (xk)
be removable (if j is removable then there is a path p(cid:48) =(i ,...i =j) in G(A ,(cid:104)xT,xI,xF(cid:105) ) with iO and iI and
l 1 s l (xk) (xk) v v+1
henceinequality(cid:104)aT ,aI ,aF (cid:105)⊗(cid:104)xT ,xI ,xF (cid:105)>(cid:104)xT,xI ,xF(cid:105)contradictstheassumptionthatA⊗x=x).
isis+1 isis+1 isis+1 is+1 is+1 is+1 is is is
The assertion follows from the fact that (cid:104)xT,xI,xF(cid:105)⊕(A,X) is equal to the greatest (cid:104)hT,hI,hF(cid:105) for which k is prpecyclic
k k k
in G˜(A ,(cid:104)xT,xI,xF(cid:105) ). In the last three assertions we have shown that (cid:104)xT,xI,xF(cid:105)⊕(A,X) is an FNSEv belonging to
(h) (h)
X and fulfilling a maximality condition (∀(cid:104)xT,xI,xF(cid:105)∈V(A)∩X) [(cid:104)xT,xI,xF(cid:105)≤(cid:104)xT,xI,xF(cid:105)⊕(A,X)]. This is a reason
to formulate the next corollary.
Corollary 5.7. If V(A)∩X (cid:54)=φ then (cid:104)xT,xI,xF(cid:105)∗(A,X)=(cid:104)xT,xI,xF(cid:105)⊕(A,X).
6. The Greatest x-fuzzy Neutrosophic Soft Eigenvector
The effective computation of (cid:104)xT,xI,xF(cid:105)⊕(A,X) considered in this section.
Definition 6.1. LetX be given. X is invariant under A if (cid:104)xT,xI,xF(cid:105)∈X implies A⊗(cid:104)xT,xI,xF(cid:105)∈X. Since A is order
preserving and X is invariant under A, the following Lemma is admitable.
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M.Kavitha,P.MurugadasandS.Sriram
Lemma6.2. X isinvariantunderAifandonlyif(cid:104)xT,xI,xF(cid:105)≤A⊗(cid:104)xT,xI,xF(cid:105)∧A⊗(cid:104)xT,xI,xF(cid:105)≤(cid:104)xT,xI,xF(cid:105).Suppose
thatX isinvariantunderAandO(A,(cid:104)xT,xI,xF(cid:105))=((cid:104)xT,xI,xF(cid:105)(r);r∈N )andO(A,(cid:104)xT,xI,xF(cid:105))=((cid:104)xT,xI,xF(cid:105)(r);r∈
0
N ) are orbits of A generated by (cid:104)xT,xI,xF(cid:105) and (cid:104)xT,xI,xF(cid:105) respectively. Then for each k∈N we have the following.
0 0
(cid:104)xT,xI,xF(cid:105)(k+1)=Ak+1⊗(cid:104)xT,xI,xF(cid:105)=Ak⊗(A⊗(cid:104)xT,xI,xF(cid:105))≤Ak⊗(cid:104)xT,xI,xF(cid:105)=(cid:104)xT,xI,xF(cid:105)(k) (3)
(cid:104)xT,xI,xF(cid:105)(k+1)=Ak+1⊗(cid:104)xT,xI,xF(cid:105)=Ak⊗(A⊗(cid:104)xT,xI,xF(cid:105))≥Ak⊗(cid:104)xT,xI,xF(cid:105)=(cid:104)xT,xI,xF(cid:105)(k). (4)
Lemma 6.3. Let X be invariant under A. Then
(∀ k∈N )[(cid:104)xT,xI,xF(cid:105)(n)=(cid:104)xT,xI,xF(cid:105)(n+k)], (5)
0
(∀ k∈N )[(cid:104)xT,xI,xF(cid:105)(n)=(cid:104)xT,xI,xF(cid:105)(n+k)]. (6)
0
Proof. Accordingto(3)itissufficienttoprovethat(cid:104)xT,xI,xF(cid:105)(n)≤(cid:104)xT,xI,xF(cid:105)(n+1). Forthesakeofacontradiction
assume that (cid:104)xT,xI,xF(cid:105)(n)(cid:54)≤(cid:104)xT,xI,xF(cid:105)(n+1), i.e., there is i∈N such that (cid:104)xT,xI,xF(cid:105)(n)>(cid:104)xT,xI,xF(cid:105)(n+1), i.e.,
i i i i i i
(An⊗(cid:104)xT,xI,xF(cid:105)) >(An+1⊗(cid:104)xT,xI,xF(cid:105)) . Then there is s∈N such that for each k∈N the following inequality holds
i i
true (cid:104)aT,aI ,aF(cid:105)n⊗(cid:104)xT,xI,xF(cid:105)>(cid:104)aT,aI ,aF(cid:105)n+1⊗(cid:104)xT,xI,xF(cid:105), or equivalently, there is a path p∈Pn such that next
is is is s s s ik ik ik k k k is
formula
ω(p)⊗(cid:104)xT,xI,xF(cid:105)=(cid:104)aT,aI ,aF(cid:105)n⊗(cid:104)xT,xI,xF(cid:105)>(cid:104)aT,aI ,aF(cid:105)n+1⊗(cid:104)xT,xI,xF(cid:105)≥ω(p˜)⊗(cid:104)xT,xI,xF(cid:105)
s s s is is is s s s ik ik ik k k k k k k
holdsforeachk∈N andeachp˜∈Pn+1. Sincel(p)=nthenthereisatleastonerepeatednode,i.e.,p=p(cid:48)∪candl(c)≥1.
ik
Now, we shall consider a path p(cid:48)(cid:48) such that p(cid:48)(cid:48) =p(cid:48)∪c∪c. Let l(p(cid:48)(cid:48))=v. Then v≥n+1 and we get
(cid:104)xT,xI,xF(cid:105)(v)≥(cid:104)aT,aI ,aF(cid:105)v⊗(cid:104)xT,xI,xF(cid:105)≥ω(p(cid:48)(cid:48))⊗(cid:104)xT,xI,xF(cid:105)=ω(p)⊗(cid:104)xT,xI,xF(cid:105)
i i i is is is s s s s s s s s s
=(cid:104)aT,aI ,aF(cid:105)n⊗(cid:104)xT,xI,xF(cid:105)>(cid:104)aT,aI ,aF(cid:105)n+1⊗(cid:104)xT,xI,xF(cid:105)=(cid:104)xT,xI,xF(cid:105)(n+1).
is is is s s s ik ik ik k k k i i i
This is a contradiction with (3) and (5) follows. To prove (6) it is enough (by (5)) to show that (cid:104)xT,xI,xF(cid:105)(n) ≥
(cid:104)xT,xI,xF(cid:105)(n+1). Forthesakeofacontradictionassumethat(cid:104)xT,xI,xF(cid:105)(n)(cid:54)≥(cid:104)xT,xI,xF(cid:105)(n+1),i.e.,thereisi∈N such
that(cid:104)xT,xI,xF(cid:105)(n)<(cid:104)xT,xI,xF(cid:105)(n+1)orequivalently(An⊗(cid:104)xT,xI,xF(cid:105)) <(An+1⊗(cid:104)xT,xI,xF(cid:105)) . Thenthereisk∈N
i i i i i i i i
such that for each s∈N the following inequality holds true (cid:104)aT,aI ,aF(cid:105)n⊗(cid:104)xT,xI,xF(cid:105)<(cid:104)aT,aI ,aF(cid:105)n+1⊗(cid:104)xT,xI,xF(cid:105),
is is is s s s ik ik ik k k k
or again equivalently, there is a path p∈Pn+1 such that next formula
ik
ω(p(cid:48))⊗(cid:104)xT,xI,xF(cid:105)≤(cid:104)aT,aI ,aF(cid:105)n⊗(cid:104)xT,xI,xF(cid:105)<(cid:104)aT,aI ,aF(cid:105)n+1⊗(cid:104)xT,xI,xF(cid:105)=ω(p)⊗(cid:104)xT,xI,xF(cid:105)
s s s is is is s s s ik ik ik k k k k k k
holds for each s∈N and each p(cid:48) ∈Pn. Since l(p)=n+1 then there is at least one repeated node, i.e., p=p˜∪c, where
is
p˜∈Pl(p˜) with l(p˜)≤n and l(c)≥1. Now, we shall consider a path p˜. It is clear that ω(p)≤ω(p˜) and hence we get
ik
(cid:104)xT,xI,xF(cid:105)(l(p˜))≥(cid:104)aT,aI ,aF(cid:105)l(p˜)⊗(cid:104)xT,xI,xF(cid:105)≥ω(p˜)⊗(cid:104)xT,xI,xF(cid:105)≥ω(p)⊗(cid:104)xT,xI,xF(cid:105)
ik ik ik k k k k k k k k k
=(cid:104)aT,aI ,aF(cid:105)n+1⊗(cid:104)xT,xI,xF(cid:105)> (cid:88)(cid:104)aT,aI ,aF(cid:105)n⊗(cid:104)xT,xI,xF(cid:105)=(cid:104)xT,xI,xF(cid:105)(n).
ik ik ik k k k is is is s s s
s∈N
This is a contradiction with (4) and (6) follows.
Corollary 6.4. Let X be invariant under A. Then {(cid:104)xT,xI,xF(cid:105)(n),(cid:104)xT,xI,xF(cid:105)(n)}⊆V(A)∩X.
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Proof. The assertions follows from the facts that
A⊗(cid:104)xT,xI,xF(cid:105)(n)=A⊗(An⊗(cid:104)xT,xI,xF(cid:105))=An+1⊗(cid:104)xT,xI,xF(cid:105)=(cid:104)xT,xI,xF(cid:105)(n+1)=(cid:104)xT,xI,xF(cid:105)(n),
A⊗(cid:104)xT,xI,xF(cid:105)(n)=A⊗(An⊗(cid:104)xT,xI,xF(cid:105))=An+1⊗(cid:104)xT,xI,xF(cid:105)=(cid:104)xT,xI,xF(cid:105)(n+1)=(cid:104)xT,xI,xF(cid:105)(n).
Theorem 6.5. Let X be invariant under A. Then (cid:104)xT,xI,xF(cid:105)⊕(A,X)=(cid:104)xT,xI,xF(cid:105)(n).
Proof. Suppose that X is invariant under A and (cid:104)xT,xI,xF(cid:105)⊕(A,X) = ((cid:104)xT,xI,xF(cid:105)⊕(A,X),... (cid:104)xT,xI,xF(cid:105)⊕(A,X))T.
1 1 1 n n n
Then by the definition of (cid:104)xT,xI,xF(cid:105)⊕(A,X) there is a path p = p(cid:48)∪c (p is finished by c) in G˜(A ,(cid:104)xT,xIxF(cid:105) ) for
k k k (h) (h)
(cid:104)hT,hI,hF(cid:105) =(cid:104)xT,xI,xF(cid:105)⊕(A,X) and ω(p) = (cid:104)hT,hI,hF(cid:105). Consider now a path p(cid:48)(cid:48) = p(cid:48)∪c∪...∪c with l(p(cid:48)(cid:48)) ≥ n and
k k k
ω(p)=ω(p(cid:48)(cid:48)). Then we obtain
(cid:104)xT,xI,xF(cid:105)(n)=(cid:104)xT,xI,xF(cid:105)(l(p(cid:48)(cid:48)))≥ω(p(cid:48)(cid:48))=(cid:104)xT,xI,xF(cid:105)⊕(A,X).
k k k k k k k k k
Inverse inequality follows from the fact that a path p ∈ Pn beginning in k contains a cycle and (cid:104)xT,xI,xF(cid:105)(n) =
kj k k k
(cid:80)(cid:104)aT ,aI ,aF (cid:105)n ⊗ (cid:104)xT,xI,xF(cid:105) = (cid:104)aT,aI ,aF(cid:105)n ⊗ (cid:104)xT,xI,xF(cid:105) = ω(p) ⊗ (cid:104)xT,xI,xF(cid:105), i.e., k is precyclic in
kj kj kj j j j kl kl kl l l l j j j
j
G˜(A ,(cid:104)xT,xI,xF(cid:105) ) for (cid:104)hT,hI,hF(cid:105)=(cid:104)xT,xI,xF(cid:105)(n). Hence (cid:104)xT,xI,xF(cid:105)⊕(A,X)≥(cid:104)xT,xI,xF(cid:105)(n).
(h) (h) k k k
6.1. Computing the Greatest X-fuzzy Neutrosophic Soft Eigenvector-General Case
In this section we compute (cid:104)xT,xI,xF(cid:105)⊕(A,X) when X is not invariant under A.
Let A ∈ N(n,n) and X ⊆ N(n) be given. Suppose that A ⊗ (cid:104)xT,xI,xF(cid:105) (cid:54)≤ (cid:104)xT,xI,xF(cid:105) and V(A) ∩ X (cid:54)= φ, i.e.,
(cid:104)xT,xI,xF(cid:105)⊕(A,X) exists. We look for the greatest FNSV (cid:104)x˜T,x˜I,x˜F(cid:105) ∈ X with the property that (cid:104)xT,xI,xF(cid:105) ≤ A⊗
(cid:104)x˜T,x˜I,x˜F(cid:105)≤(cid:104)x˜T,x˜I,x˜F(cid:105). For this purpose (cid:104)x˜T,x˜I,x˜F(cid:105) will be constructed by the following algorithm.
Algorithm Invariant upper bound
Input. X, A.
Output. “yes” in variable gr if (cid:104)x˜T,x˜I,x˜F(cid:105)∈X with the property that
(cid:104)xT,xI,xF(cid:105)≤A⊗(cid:104)x˜T,x˜I,x˜F(cid:105)≤(cid:104)x˜T,x˜I,x˜F(cid:105) exists;
“no” in gr otherwise.
begin
1. (cid:104)x˜T,x˜I,x˜F(cid:105):=(cid:104)xT,xI,xF(cid:105); M :=φ;
2. m:= min (cid:104)x˜T,x˜I,x˜F(cid:105); M ={j ∈N; m=(cid:104)x˜T,x˜I,x˜F(cid:105)};
j j j j j j
j∈N|M
3. While (∃k∈M)(∃j ∈N)[(cid:104)aT ,aI ,aF (cid:105)⊗(cid:104)x˜T,x˜I,x˜F(cid:105)>(cid:104)x˜T,x˜I,x˜F(cid:105)] do
kj kj kj j j j k k k
If (cid:104)x˜T,x˜I,x˜F(cid:105)≥(cid:104)xT,xI,xF(cid:105) then (cid:104)x˜T,x˜I,x˜F(cid:105):=m∧M :=M ∪{j} else gr:=“no”;
k k k j j j j j j
(comment: iftheconditionA⊗(cid:104)x˜T,x˜I,x˜F(cid:105)≤(cid:104)x˜T,x˜I,x˜F(cid:105)isnotfulfilledinrowk, correspondingvariables(cid:104)x˜T,x˜I,x˜F(cid:105)are
modified)
4. M ={j ∈N;m≥(cid:104)x˜T,x˜I,x˜F(cid:105)};
j j j
(comment: M consists of row indices for which A⊗(cid:104)x˜T,x˜I,x˜F(cid:105)≤(cid:104)x˜T,x˜I,x˜F(cid:105) holds true)
5. If N \M =φ then gr:=“ yes” else go to 2;
end.
902