![International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014
DOI : 10.5121/ijfls.2014.4303 13
SOLVABILITY OF SYSTEM OF
INTUITIONISTIC FUZZY LINEAR EQUATIONS
Rajkumar Pradhan and Madhumangal Pal
Department of Applied Mathematics with Oceanology and Computer
Programming,Vidyasagar University, Midnapore – 721 102, India.
Abstract
In this paper, it is shown that for a system of intuitionistic fuzzy linear equations of the form bxA =⊗ is
said to be solvable if, for a definite solution );( bAx , bbAxA =);(⊗ holds, otherwise unsolvable. In
general bbAxA ≤⊗ );( holds always, so taking a tolerable solution of an unsolvable system, keeping
right hand side of the system constant, modification of the left hand side intuitionistic fuzzy matrix A has
been made, such that, the system will be solvable with the help of Chebychev Approximation. The maximum
solution of the system is also defined here.
Keywords
Intuitionistic fuzzy matrix, system of intuitionistic fuzzy linear equation, Principal solution, tolerable
solution, chebychev distance.
1. Introduction
Several problems in various areas such as economics, engineering and physics lead to the solution
of a system of linear equations. Linear systems of equations with uncertainty on the parameters,
plays a major role in several applications in the areas mentioned above. In many applications, the
parameters of the system (or at least some of them) should be represented by intuitionistic fuzzy
rather than crisp or fuzzy numbers. Hence it is important to develop mathematical procedures that
would appropriately treat intuitionistic fuzzy linear systems and solve them.
The solvability of fuzzy relational equations based upon max-min composition was first proposed
and investigated by Sanchez [11], and was further Studied by Czogala et al. [3, 4]. Higashi and Klir
[5] derived several alternative general schemes for solving the equations. Latter many other authors
contributes to this topic, by generalizing and extending the original results in various directions,
e.g. [6, 7]. Cechlarova [2] studied the unique solvability of linear system of equations over the
max-min fuzzy algebra on the unit real interval. First time Pradhan and Pal [9] established the
intuitionistic fuzzy relational equation of the form bxA =⊗ be consistent when the coefficient
intuitionistic fuzzy matrix (IFM) A is regular.](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-1-320.jpg)
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In our present paper we discuss about the solvability of the system of intuitionistic fuzzy linear
equations (IFLEs). Here we derived the conditions for which the system of IFLEs be solvable. We
also derived the maximum of the solutions for a system of IFLEs. For a consistent system for any
solution x , the inequality bbAxA ≤⊗ );( holds always. So to distinguish the fact, we say a
system of IFLEs is solvable if and only if bbAxA =);(⊗ and we define that particular solution
);( bAx as principal solution. In the last of this paper, we derived an algorithm by which we
modify the coefficient IFM A of an unsolvable system, bxA =⊗ to get a principal solution.
This paper is organized as follows. In Section 2, definitions of some basic terms are given. The
conditions for which a system of IFLEs be consistent is described in Section 3. Section 4 is about
the algorithm by which we can modify the coefficient IFM A so that the system be solvable. In
Section 5, we drawn the conclusion.
2. Preliminaries
In this section, some elementary aspects that are necessary for this paper are introduced.
By max-min intuitionistic fuzzy algebra F , we mean any linearly ordered set ),( ≤F with two
binary operations addition and multiplication denoted by ⊕ and ⊗ respectively. For any natural
number 0>n , )(nF denotes the set of all n-dimensional column vector and ),( nmF denotes
the set of all IFM of order )( nm× over F . The respective IFM is defined as follows.
Definition 2.1 (Intuitionistic fuzzy matrices)
An intuitionistic fuzzy matrix (IFM) A of order nm× is defined as nmijijij aaxA ×〉〈 ],,[= νµ
where µija , νija are called membership and non-membership values of ijx in A , which
maintains the condition 10 ≤+≤ νµ ijij aa . For simplicity, we write nmijij axA ×],[= or simply
nmija ×][ where 〉〈 νµ ijijij aaa ,= .
In arithmetic operations, only the values of µija and νija are needed so from here we only
consider the values of 〉〈 νµ ijijij aaa ,= . All elements of an IFM are the members of
1}0:,{= ≤+≤〉〈 babaF .
Comparison between intuitionistic fuzzy matrices have an important role in our work, which is
defined below.
Definition 2.2 (Dominance of IFM)
Let nmFBA ×∈, such that ),(= 〉〈 νµ ijij aaA and ),(= 〉〈 νµ ijij bbB , then we write BA ≤ if,
µµ ijij ba ≤ and νν ijij ba ≥ for all ji, , and we say that A is dominated by B or B dominates
A . A and B are said to be comparable, if either BA ≤ or AB ≤ .](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-2-320.jpg)
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Let nV denotes the set of all n -tuples ),,,,,,( 2211 〉〈〉〈〉〈 νµνµνµ nn xxxxxx K over F . An
element of nV is called an intuitionistic fuzzy vector (IFV) of dimension n, where µix and νix
are the membership and non-membership values of the component ix . The system nV together
with the operations, componentwise addition and multiplication forms intuitionistic fuzzy vectors
space (IVFS).
Definition 2.3 (Row space and column space)
Let nmijij FaaA ×∈〉〈 ),(= νµ be an IFM. Then the element 〉〈 νµ ijij aa , is the ij th entry of A .
Let )( ji AA åå denote the i th row (i th column) of A .
The row space )(AR of A is the subspace of nV generated by the rows }{ åiA of A . The
column space )(AC of A is the subspace of mV generated by the columns }{ jAå of A .
Definition 2.4 (Linear combination of IFVs)
Let },,,{= 21 paaaS K be a set of intuitionistic fuzzy vectors of dimension n . The linear
combination of elements of the set S is a finite sum ii
p
i
ac∑1=
where Sai ∈ and [0,1]∈ic . The
set of all linear combinations of the elements of S is called the span of S , denoted by 〉〈S .
An example of 3V and its spanning set is given below.
Example 2.5 Let },,{= 321 aaaS be a subset of 3V , where
)0.4,0.2,0.5,0.1,0.5,0.3(=),0.4,0.3,0.6,0.3,0.8,0.2(= 21 〉〈〉〈〉〈〉〈〉〈〉〈 aa and
)0.9,0.1,0.7,0.2,0.7,0.3(=3 〉〈〉〈〉〈a .Then,
)}0.9,0.1,0.7,0.2,0.7,0.3()0.4,0.2,0.5,0.1
,0.5,0.3()0.4,0.3,0.6,0.3,0.8,0.2({=
3
21
〉〈〉〈〉〈+〉〈〉〈
〉〈+〉〈〉〈〉〈〉〈
c
ccS
Definition 2.6 (Dependence of IFVs)
A set S of intuitionistic fuzzy vectors is independent if and only if each element of S can not be
expressed as a linear combination of other elements of S , that is, no element Ss∈ is a linear
combination of }{ sS .
A vector α may be expressed by some other vectors. If it is possible then the vector α is called
dependent otherwise it is called independent. These terminologies are similar to classical vectors.](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-3-320.jpg)
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The examples of independent and dependent set of vectors are given below.
Example 2.7 Let },,{= 321 aaaS be a subset of 3V , where
)0.4,0.2,0.5,0.1,0.5,0.3(=),0.4,0.3,0.6,0.3,0.8,0.2(= 21 〉〈〉〈〉〈〉〈〉〈〉〈 aa and
)0.9,0.1,0.7,0.2,0.7,0.3(=3 〉〈〉〈〉〈a .
Here the set S is an independent set. If not then 321 = aaa βα + for F∈βα, .
So,
)0.9,0.1,0.7,0.2,0.7,0.3()0.4,0.2,0.5,0.1,0.5,0.3(=1 〉〈〉〈〉〈+〉〈〉〈〉〈 βαa
.))}(0.1,1max),(0.2,1max{min)},(0.9,min),(0.4,min{max
,)}(0.2,1max),(0.1,1max{min)},(0.7,min),(0.5,min{max
,)}(0.3,1max),(0.3,1max{min)},(0.7,min),(0.5,min{max(=
〉−−〈
〉−−〈
〉−−〈
βαβα
βαβα
βαβα
It is not possible to find any F∈βα, such that the corresponding coefficients on both
sides will be equal. That is, 321 aaa βα +≠ . Similarly, 312 aaa βα +≠ and 123 aaa βα +≠
. So the set S is independent.
Let },{= 21 aaS be a subset of 3V , where )0.6,0.3,0.5,0.3,0.7,0.3(=1 〉〈〉〈〉〈a and
)0.6,0.2,0.5,0.1,0.8,0.2(=2 〉〈〉〈〉〈a . Here 21 = caa for 0.7=c . So S is a dependent
set.
Definition 2.8 (Basis)
Let W be an intuitionistic fuzzy subspace of nV and S be a subset of W such that the
elements of S are independent. If every element of W can be expressed uniquely as a linear
combination of the elements of S , then S is called a basis of intuitionistic fuzzy subspace W .
Definition 2.9 (Standard basis)
A basis B of an intuitionistic fuzzy vector space W is a standard basis if and only if whenever
jij
n
j
i bab ∑1=
= for Bbb ji ∈, and [0,1]∈ija then iiii bba = .
Example 2.10 Let },,{= 321 aaaS be a subset of 3V given by
)0.6,0.3,0.7,0.2,0.5,0.4(=1 〉〈〉〈〉〈a ,
)0.8,0.2,0.6,0.2,0.5,0.3(=2 〉〈〉〈〉〈a and )0.8,0.1,0.4,0.3,0.4,0.4(=3 〉〈〉〈〉〈a . Here
S is independent set, since 32211 acaca +≠ , 34132 acaca +≠ and 26153 acaca +≠ . So](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-4-320.jpg)
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all have the same left side with some different right side. Hence the system is not solvable.
Example 3.3 Let us consider the system of IFLEs bxA =⊗ where, =A
〉〈〉〈
〉〈〉〈
〉〈〉〈
0.4,0.40.8,0.1
0.6,0.20.6,0.3
0.5,0.40.7,0.2
and T
b ]0.5,0.5,1,0,0.4,0.5[= 〉〈〉〈〉〈 .
Here for 2=k , 〉〈〉〈〉〈〉〈〉〈 1,0<0.6,0.2=}0.4,0.4,0.6,0.2,0.5,0.4{max . Hence by Lemma 3.1,
the system of IFLEs bxA =⊗ is not solvable.
The solvability of a system of IFLEs of the form (1) depends upon the characteristics of the
coefficient IFM A . The following theorem deduce the fact.
Theorem 3.4 The system of IFLEs bxA =⊗ has a solution, that is, be solvability if the non-zero
rows of the coefficient IFM A forms a standard basis for the row space of itself.
Proof: As the non-zero rows of the IFM A forms a standard basis for the row space of A , then
the IFM A be regular (see [8, 10]). That is there exists a g-inverse −
A of A such that
AAAA =⊗⊗ −
. Now, bxA =⊗ gives bxAAA =⊗⊗⊗ −
.
That implies, bbAA =⊗⊗ −
. Which shows, )( bA ⊗−
is a solution of the given system. Hence
the system of IFLE is solvability.
Example 3.5 Let us consider the system of IFLEs bxA =⊗ with
=A
〉〈〉〈〉〈
〉〈〉〈〉〈
0.8,0.20.6,0.30.5,0.5
0.5,0.50.6,0.40.7,0.3
,
T
xxxxxxX ],,,,,[= 332211 〉〈〉〈〉〈 νµνµνµ and
T
b ]0.5,0.4,0.6,0.3[= 〉〈〉〈 .
Here the non-zero rows of the IFM A are linearly independent and form a standard basis also. So
A is regular and one of its g-inverse is =−
A
〉〈〉〈
〉〈〉〈
〉〈〉〈
0.8,0.20.5,0.5
0.5,0.50.5,0.5
0.5,0.50.8,0.2
. Then
T
bAx ]0.5,0.4,0.5,0.5,0.6,0.3[== 〉〈〉〈〉〈−
is one of the solution of the above system of IFLEs.
We know that g-inverse of an IFM A is not unique. So the solution of a system of IFLEs may
have many solutions. Among these solutions the maximum is defined by as follows.
Definition 3.6 Any element x of ),( bAS is called a maximum solution of the system bxA =⊗](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-6-320.jpg)
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if for all ),( bASx∈ , xx ≥ implies xx = .
The following theorem demonstrate how to find the maximum solution of the system of IFLEs.
Theorem 3.7 If for a system of IFLEs bxA =⊗ has a solution denoted by ),( bAx and is
defined by =,= 〉〈 νµ xxx
∀≤〉〈
kjkk
kjk
baifbmin
jbaif
>}{
1,0
is the maximum solution.
Proof: As the system of IFLEs bxA =⊗ has a solution, so it is consistent, then x is a solution
of the system. If x is not a solution, then bxA ≠⊗ and therefore
),(),(min),,(max 00
〉〈≠〉〈 νµννµµ kkjjk
j
jjk
j
bbxamaxxamin for at least one 0k .
By definition of x , since 〉〈≤〉〈 νµνµ kkjj bbxx ,, for each k , so 〉〈≤〉〈 νµνµ
00
,, kkjj bbxx . By
our assumption, 〉〈〉〈 νµνµ 00
,<),(max kkjjk
j
bbaa for some 0k and by Lemma 3.1 it follows that
φ=),( bAS , which is a contradiction. Hence x is a solution of the system bxA =⊗ .
Now let us prove that x is a maximum solution. If possible let us assume that 〉〈 νµ yyy ,= be a
solution of the system such that xy > , that is 〉〈〉〈 νµνµ 0000
,>, jjjj xxyy for at least one 0j .
Therefore by definition of x , we have ),(>,
00
〉〈〉〈 νµνµ kkjj bbminyy when kkj ba >
0
for some
k . Again, since φ≠),( bAS , by Lemma 3.1, 〉〈〉〈 νµνµ 0000
,>,(max kkjkjk
j
bbaa for each 0k .
Hence, 〉〈≠〉〈 ),(min),,(max,
0000 ννµµνµ jjk
j
jjk
j
kk yamaxyaminbb , which contradicts our
assumption ),( bASy ∈ .
Therefore, x is the maximum solution of the system of IFLEs bxA =⊗ .
Example 3.8 Given =A
〉〈〉〈〉〈
〉〈〉〈〉〈
0.8,0.20.6,0.30.5,0.5
0.5,0.50.6,0.40.7,0.3
and T
b ]0.6,0.3,0.5,0.3[= 〉〈〉〈 .
Find out the maximum solution of the system bxA =⊗ .
Ans. From the definition of maximum solution, 〉〈0.5,0.3=1x , 〉〈0.6,0.3=2x and
〉〈0.5,0.3=3x . So .]0.5,0.3,0.6,0.3,0.5,0.3[= T
x 〉〈〉〈〉〈 Thus, φ≠),( bAS and bxA =⊗](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-7-320.jpg)
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holds. Hence xx T
=]0.5,0.3,0.6,0.3,0.5,0.3[= 〉〈〉〈〉〈 is the maximum solution.
Definition 3.9 (Moore-Penrose Inverse)
For an IFM nmFA ×∈ , an IFM nmFG ×∈ is said to be a Moore-Penrose inverse of A , if
AAGA = , GGAG = , AGAG T
=)( and GAGA T
=)( .
The Moore-Penrose inverse of A is denoted by +
A .
Theorem 3.10 Let us consider a system of IFLEs bxA =⊗ . The system must have a solution,
that is, must be consistent if the coefficient IFM A is a symmetric and idempotent of order n.
Proof: Since A is symmetric and idempotent square IFM, it is already prove in [10], that A
itself its Moore-Penrose inverse. That is, +
AA = . So in that case the solution will be
AbbAx == +
.
Example 3.11 Consider the system of IFLEs bxA =⊗ where, =A
〉〈〉〈
〉〈〉〈
0.7,0.10.6,0.2
0.6,0.20.8,0.2
and T
b ]0.6,0.2,0.8,0.2[= 〉〈〉〈 . Here, AAT
= and AA =2
, that is, the IFM A is symmetric
and idempotent. So the Moore-Penrose inverse +A of A is itself A . Then the solution will be
T
AbbAx ]0.6,0.2,0.8,0.2[=== 〉〈〉〈+
.
At a glance a system of IFLEs is solvable or not are depicted in following figure.
4 Chebychev Approximation
In this section, we describe an algorithm by which we approach the right hand side of the system of
IFLEs bxA =⊗ successively changing the original IFM )( nmFA ×∈ to an IFM
)( nmFD ×∈ such that bxD =⊗ is solvable.
Let us consider the solution or tolerable solution );( bAx′ of the system of IFLEs
bxA =⊗ as =);( bAx′
∀≤〉〈
iiji
iij
baifbmin
ibaif
>}{
1,0
. .....(3)
Now if we define that the system (1) is solvable if and only if (3) is its solution, that is,
bbAxA =);(′⊗ holds, but in general bbAxA ≤′⊗ );( holds always. So our aim is, by
changing the IFM A and retain the right hand side of the system same to make the system
solvable.
Before going to that, first we have to define some importent terms.
Definition 4.1 The Chebychev distance of two IFM )(, nmFBA ×∈ is denoted by ),( BAρ and](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-8-320.jpg)
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Algorithm MATRIX
begin 0=k ; 〉〈∆ 0,0=k ; AA k =)(∆ ;
compute );( bAx′ ;
If bbAxA ≠′⊗ );( then
〉≠−+∆
≠−+∆〈
〉∆∆〈∆ +++
})(|;)({|min
},)(|;)({|min=
,=
,
,
111
ννννν
µµµµµ
νµ
δδ
δδ
iijkiijk
ji
k
iijkiijk
ji
k
kkk
bAbA
bAbA
repeat
1= +kk ;
);(=)( bAA kk →∆ δ
until bbAxA kk =));(()( δδ ′⊗ ;
output: kkA ∆);(δ
end MATRIX.
The above algorithm can be illustrate by the following example.
Let us consider the system of IFLEs bxA =⊗ where,
=A
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
0.7,0.20.7,0.10.3,0.40.7,0.10.5,0.3
0.2,0.60.4,0.40.5,0.50.8,0.10.3,0.5
0.6,0.20.1,0.80.9,0.10.2,0.60.6,0.3
0.2,0.50.4,0.50.7,0.20.6,0.40.3,0.5
and T
b ]0.5,0.4,0.3,0.5,0.9,0.1,0.4,0.4[= 〉〈〉〈〉〈〉〈 .
The corresponding tolerable solution will be
T
bAx ]0.5,0.4,0.3,0.5,0.3,0.5,0.3,0.5,0.5,0.4[=);( 〉〈〉〈〉〈〉〈〉〈′ but
bbAxA ≤′⊗ );( so the system is unsolvable.
Now by the above algorithm, in the first iteration,
〉〈∆ 0.1,0.1=1 , =)( 1∆A
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
0.6,0.30.6,0.20.4,0.40.6,0.20.5,0.4
0.3,0.50.3,0.50.4,0.50.7,0.20.3,0.5
0.7,0.10.2,0.70.9,0.10.3,0.50.7,0.2
0.3,0.40.4,0.40.6,0.30.5,0.40.4,0.4
and T
bAx ]0.5,0.4,0.5,0.4,0.3,0.5,0.3,0.5,1,0[=));(( 1 〉〈〉〈〉〈〉〈〉〈∆′ .](https://image.slidesharecdn.com/solvabilityofsystemofintuitionisticfuzzylinearequations-140813062723-phpapp01/85/Solvability-of-system-of-intuitionistic-fuzzy-linear-equations-11-320.jpg)
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Here also, bbAxA ≤∆′⊗ ));(( 1 .
In the second iteration,
〉〈∆ 0.2,0.2=2 , =)( 2∆A
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
〉〈〉〈〉〈〉〈〉〈
0.5,0.40.5,0.40.5,0.40.5,0.40.5,0.4
0.3,0.50.3,0.50.3,0.50.5,0.40.3,0.5
0.9,0.10.4,0.50.9,0.10.5,0.30.9,0.1
0.4,0.40.4,0.40.4,0.40.4,0.40.4,0.4
and T
bAx ]1,0,1,0,1,0,0.3,0.5,1,0[=));(( 2 〉〈〉〈〉〈〉〈〉〈∆′ .
In this case, bbAxA =));(( 2∆′⊗ . So )(= 2∆AD is the Chebychev best approximation of the
coefficient IFM A of the given system and ));(( 2 bAx ∆′ is the principal solution.
5 Conclusions
In this article, we try to find the conditions for which a system of IFLEs be solvable. We also shown
that, for a particular type of coefficient IFM the system of IFLEs must have a solution. Finally, we
try to modify the coefficient IFM of a system of IFLEs, keeping right hand side intutitionistic fuzzy
vector same, to make it solvable.
References
[1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986).
[2] K. Cechlarova, Unique solvability of max-min fuzzy equations and strong regularity of matrices over
fuzzy algebra, Fuzzy Sets and Systems, 75, 165-177 (1995).
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Solvability of system of intuitionistic fuzzy linear equations
- 1. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 DOI : 10.5121/ijfls.2014.4303 13 SOLVABILITY OF SYSTEM OF INTUITIONISTIC FUZZY LINEAR EQUATIONS Rajkumar Pradhan and Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming,Vidyasagar University, Midnapore – 721 102, India. Abstract In this paper, it is shown that for a system of intuitionistic fuzzy linear equations of the form bxA =⊗ is said to be solvable if, for a definite solution );( bAx , bbAxA =);(⊗ holds, otherwise unsolvable. In general bbAxA ≤⊗ );( holds always, so taking a tolerable solution of an unsolvable system, keeping right hand side of the system constant, modification of the left hand side intuitionistic fuzzy matrix A has been made, such that, the system will be solvable with the help of Chebychev Approximation. The maximum solution of the system is also defined here. Keywords Intuitionistic fuzzy matrix, system of intuitionistic fuzzy linear equation, Principal solution, tolerable solution, chebychev distance. 1. Introduction Several problems in various areas such as economics, engineering and physics lead to the solution of a system of linear equations. Linear systems of equations with uncertainty on the parameters, plays a major role in several applications in the areas mentioned above. In many applications, the parameters of the system (or at least some of them) should be represented by intuitionistic fuzzy rather than crisp or fuzzy numbers. Hence it is important to develop mathematical procedures that would appropriately treat intuitionistic fuzzy linear systems and solve them. The solvability of fuzzy relational equations based upon max-min composition was first proposed and investigated by Sanchez [11], and was further Studied by Czogala et al. [3, 4]. Higashi and Klir [5] derived several alternative general schemes for solving the equations. Latter many other authors contributes to this topic, by generalizing and extending the original results in various directions, e.g. [6, 7]. Cechlarova [2] studied the unique solvability of linear system of equations over the max-min fuzzy algebra on the unit real interval. First time Pradhan and Pal [9] established the intuitionistic fuzzy relational equation of the form bxA =⊗ be consistent when the coefficient intuitionistic fuzzy matrix (IFM) A is regular.
- 2. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 14 In our present paper we discuss about the solvability of the system of intuitionistic fuzzy linear equations (IFLEs). Here we derived the conditions for which the system of IFLEs be solvable. We also derived the maximum of the solutions for a system of IFLEs. For a consistent system for any solution x , the inequality bbAxA ≤⊗ );( holds always. So to distinguish the fact, we say a system of IFLEs is solvable if and only if bbAxA =);(⊗ and we define that particular solution );( bAx as principal solution. In the last of this paper, we derived an algorithm by which we modify the coefficient IFM A of an unsolvable system, bxA =⊗ to get a principal solution. This paper is organized as follows. In Section 2, definitions of some basic terms are given. The conditions for which a system of IFLEs be consistent is described in Section 3. Section 4 is about the algorithm by which we can modify the coefficient IFM A so that the system be solvable. In Section 5, we drawn the conclusion. 2. Preliminaries In this section, some elementary aspects that are necessary for this paper are introduced. By max-min intuitionistic fuzzy algebra F , we mean any linearly ordered set ),( ≤F with two binary operations addition and multiplication denoted by ⊕ and ⊗ respectively. For any natural number 0>n , )(nF denotes the set of all n-dimensional column vector and ),( nmF denotes the set of all IFM of order )( nm× over F . The respective IFM is defined as follows. Definition 2.1 (Intuitionistic fuzzy matrices) An intuitionistic fuzzy matrix (IFM) A of order nm× is defined as nmijijij aaxA ×〉〈 ],,[= νµ where µija , νija are called membership and non-membership values of ijx in A , which maintains the condition 10 ≤+≤ νµ ijij aa . For simplicity, we write nmijij axA ×],[= or simply nmija ×][ where 〉〈 νµ ijijij aaa ,= . In arithmetic operations, only the values of µija and νija are needed so from here we only consider the values of 〉〈 νµ ijijij aaa ,= . All elements of an IFM are the members of 1}0:,{= ≤+≤〉〈 babaF . Comparison between intuitionistic fuzzy matrices have an important role in our work, which is defined below. Definition 2.2 (Dominance of IFM) Let nmFBA ×∈, such that ),(= 〉〈 νµ ijij aaA and ),(= 〉〈 νµ ijij bbB , then we write BA ≤ if, µµ ijij ba ≤ and νν ijij ba ≥ for all ji, , and we say that A is dominated by B or B dominates A . A and B are said to be comparable, if either BA ≤ or AB ≤ .
- 3. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 15 Let nV denotes the set of all n -tuples ),,,,,,( 2211 〉〈〉〈〉〈 νµνµνµ nn xxxxxx K over F . An element of nV is called an intuitionistic fuzzy vector (IFV) of dimension n, where µix and νix are the membership and non-membership values of the component ix . The system nV together with the operations, componentwise addition and multiplication forms intuitionistic fuzzy vectors space (IVFS). Definition 2.3 (Row space and column space) Let nmijij FaaA ×∈〉〈 ),(= νµ be an IFM. Then the element 〉〈 νµ ijij aa , is the ij th entry of A . Let )( ji AA åå denote the i th row (i th column) of A . The row space )(AR of A is the subspace of nV generated by the rows }{ åiA of A . The column space )(AC of A is the subspace of mV generated by the columns }{ jAå of A . Definition 2.4 (Linear combination of IFVs) Let },,,{= 21 paaaS K be a set of intuitionistic fuzzy vectors of dimension n . The linear combination of elements of the set S is a finite sum ii p i ac∑1= where Sai ∈ and [0,1]∈ic . The set of all linear combinations of the elements of S is called the span of S , denoted by 〉〈S . An example of 3V and its spanning set is given below. Example 2.5 Let },,{= 321 aaaS be a subset of 3V , where )0.4,0.2,0.5,0.1,0.5,0.3(=),0.4,0.3,0.6,0.3,0.8,0.2(= 21 〉〈〉〈〉〈〉〈〉〈〉〈 aa and )0.9,0.1,0.7,0.2,0.7,0.3(=3 〉〈〉〈〉〈a .Then, )}0.9,0.1,0.7,0.2,0.7,0.3()0.4,0.2,0.5,0.1 ,0.5,0.3()0.4,0.3,0.6,0.3,0.8,0.2({= 3 21 〉〈〉〈〉〈+〉〈〉〈 〉〈+〉〈〉〈〉〈〉〈 c ccS Definition 2.6 (Dependence of IFVs) A set S of intuitionistic fuzzy vectors is independent if and only if each element of S can not be expressed as a linear combination of other elements of S , that is, no element Ss∈ is a linear combination of }{ sS . A vector α may be expressed by some other vectors. If it is possible then the vector α is called dependent otherwise it is called independent. These terminologies are similar to classical vectors.
- 4. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 16 The examples of independent and dependent set of vectors are given below. Example 2.7 Let },,{= 321 aaaS be a subset of 3V , where )0.4,0.2,0.5,0.1,0.5,0.3(=),0.4,0.3,0.6,0.3,0.8,0.2(= 21 〉〈〉〈〉〈〉〈〉〈〉〈 aa and )0.9,0.1,0.7,0.2,0.7,0.3(=3 〉〈〉〈〉〈a . Here the set S is an independent set. If not then 321 = aaa βα + for F∈βα, . So, )0.9,0.1,0.7,0.2,0.7,0.3()0.4,0.2,0.5,0.1,0.5,0.3(=1 〉〈〉〈〉〈+〉〈〉〈〉〈 βαa .))}(0.1,1max),(0.2,1max{min)},(0.9,min),(0.4,min{max ,)}(0.2,1max),(0.1,1max{min)},(0.7,min),(0.5,min{max ,)}(0.3,1max),(0.3,1max{min)},(0.7,min),(0.5,min{max(= 〉−−〈 〉−−〈 〉−−〈 βαβα βαβα βαβα It is not possible to find any F∈βα, such that the corresponding coefficients on both sides will be equal. That is, 321 aaa βα +≠ . Similarly, 312 aaa βα +≠ and 123 aaa βα +≠ . So the set S is independent. Let },{= 21 aaS be a subset of 3V , where )0.6,0.3,0.5,0.3,0.7,0.3(=1 〉〈〉〈〉〈a and )0.6,0.2,0.5,0.1,0.8,0.2(=2 〉〈〉〈〉〈a . Here 21 = caa for 0.7=c . So S is a dependent set. Definition 2.8 (Basis) Let W be an intuitionistic fuzzy subspace of nV and S be a subset of W such that the elements of S are independent. If every element of W can be expressed uniquely as a linear combination of the elements of S , then S is called a basis of intuitionistic fuzzy subspace W . Definition 2.9 (Standard basis) A basis B of an intuitionistic fuzzy vector space W is a standard basis if and only if whenever jij n j i bab ∑1= = for Bbb ji ∈, and [0,1]∈ija then iiii bba = . Example 2.10 Let },,{= 321 aaaS be a subset of 3V given by )0.6,0.3,0.7,0.2,0.5,0.4(=1 〉〈〉〈〉〈a , )0.8,0.2,0.6,0.2,0.5,0.3(=2 〉〈〉〈〉〈a and )0.8,0.1,0.4,0.3,0.4,0.4(=3 〉〈〉〈〉〈a . Here S is independent set, since 32211 acaca +≠ , 34132 acaca +≠ and 26153 acaca +≠ . So
- 5. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 17 },,{ 321 aaa is a basis for 〉〈S . Now this is a standard basis also. For 3132121111 = acacaca ++ holds if 0.8=11c , 0.5=12c and 0.6=13c . Also 1111 = aca for 0.8=11c . Similarly for 2a and 3a . 3.Solvability In this section, we consider the system of IFLEs of the form, bxA =⊗ .....(1) that is, 〉〈〉〈 νµννµµ ikikjkij j jkij j bbxamaxxamin ,=),(min),,(max ....(2) where the IFM )( nmFA ×∈ and the intuitionistic fuzzy vector )(mFb∈ are given and the intuitionistic fuzzy vector )(nFx∈ is unknown. The solution set of the system defined in (1) for a given IFM A and an intuitionistic fuzzy vector b will be denoted by }=|)({=),( bxAnFxbAS ⊗∈ . Now our aim is to find whether the system (1) is solvable, that is, whether the solution set ),( bAS is non-empty. Lemma 3.1 Let us consider the system of IFLE bxA =⊗ . If ),(<),(max 〉〈〉〈 νµνµ kkjkjk j bbaa for some k , then φ=),( bAS , that is the system is not solvable. Proof: If ),(<),( 〉〈〉〈 νµνµ kkjkjk bbaamax for some k , then ),(<),(max,),(min 〉〈〉〈≤〉〈≤〉〈 νµνµνµνµ kkjkjk j jkjkjkjk j bbaaaaaa . Hence, ),(<),(min),,(max 〉〈〉〈 νµννµµ kkjjk j jjk j bbxamaxxamin for some k , and by equation (2) no values 〉〈 νµ jj xx , exists that satisfy the equation (1). Therefore φ=),( bAS . Remark 3.2 Let us consider the condition of the Lemma 3.1 be ),(>),(max 〉〈〉〈 νµνµ kkjkjk j bbaa for some k . Then according to the proof of the Lemma 3.1, ),(>),(max,),,,(min 〉〈〉〈≥〉〈≥〉〈〉〈 νµνµνµνµνµ kkjkjk j jkjkjjjkjk j bbaaaaxxaa implies the only possibility is, 〉〈 νµ jkjk aa , are same for all j . Then two cases may arise, Case-1: If 〉〈 νµ kk bb , are equal for all k . Then the system reduce to one equation. Hence the system is solvable. Case-2: If 〉〈 νµ kk bb , are different for some k . Then the equation of the system will be such that,
- 6. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 18 all have the same left side with some different right side. Hence the system is not solvable. Example 3.3 Let us consider the system of IFLEs bxA =⊗ where, =A 〉〈〉〈 〉〈〉〈 〉〈〉〈 0.4,0.40.8,0.1 0.6,0.20.6,0.3 0.5,0.40.7,0.2 and T b ]0.5,0.5,1,0,0.4,0.5[= 〉〈〉〈〉〈 . Here for 2=k , 〉〈〉〈〉〈〉〈〉〈 1,0<0.6,0.2=}0.4,0.4,0.6,0.2,0.5,0.4{max . Hence by Lemma 3.1, the system of IFLEs bxA =⊗ is not solvable. The solvability of a system of IFLEs of the form (1) depends upon the characteristics of the coefficient IFM A . The following theorem deduce the fact. Theorem 3.4 The system of IFLEs bxA =⊗ has a solution, that is, be solvability if the non-zero rows of the coefficient IFM A forms a standard basis for the row space of itself. Proof: As the non-zero rows of the IFM A forms a standard basis for the row space of A , then the IFM A be regular (see [8, 10]). That is there exists a g-inverse − A of A such that AAAA =⊗⊗ − . Now, bxA =⊗ gives bxAAA =⊗⊗⊗ − . That implies, bbAA =⊗⊗ − . Which shows, )( bA ⊗− is a solution of the given system. Hence the system of IFLE is solvability. Example 3.5 Let us consider the system of IFLEs bxA =⊗ with =A 〉〈〉〈〉〈 〉〈〉〈〉〈 0.8,0.20.6,0.30.5,0.5 0.5,0.50.6,0.40.7,0.3 , T xxxxxxX ],,,,,[= 332211 〉〈〉〈〉〈 νµνµνµ and T b ]0.5,0.4,0.6,0.3[= 〉〈〉〈 . Here the non-zero rows of the IFM A are linearly independent and form a standard basis also. So A is regular and one of its g-inverse is =− A 〉〈〉〈 〉〈〉〈 〉〈〉〈 0.8,0.20.5,0.5 0.5,0.50.5,0.5 0.5,0.50.8,0.2 . Then T bAx ]0.5,0.4,0.5,0.5,0.6,0.3[== 〉〈〉〈〉〈− is one of the solution of the above system of IFLEs. We know that g-inverse of an IFM A is not unique. So the solution of a system of IFLEs may have many solutions. Among these solutions the maximum is defined by as follows. Definition 3.6 Any element x of ),( bAS is called a maximum solution of the system bxA =⊗
- 7. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 19 if for all ),( bASx∈ , xx ≥ implies xx = . The following theorem demonstrate how to find the maximum solution of the system of IFLEs. Theorem 3.7 If for a system of IFLEs bxA =⊗ has a solution denoted by ),( bAx and is defined by =,= 〉〈 νµ xxx ∀≤〉〈 kjkk kjk baifbmin jbaif >}{ 1,0 is the maximum solution. Proof: As the system of IFLEs bxA =⊗ has a solution, so it is consistent, then x is a solution of the system. If x is not a solution, then bxA ≠⊗ and therefore ),(),(min),,(max 00 〉〈≠〉〈 νµννµµ kkjjk j jjk j bbxamaxxamin for at least one 0k . By definition of x , since 〉〈≤〉〈 νµνµ kkjj bbxx ,, for each k , so 〉〈≤〉〈 νµνµ 00 ,, kkjj bbxx . By our assumption, 〉〈〉〈 νµνµ 00 ,<),(max kkjjk j bbaa for some 0k and by Lemma 3.1 it follows that φ=),( bAS , which is a contradiction. Hence x is a solution of the system bxA =⊗ . Now let us prove that x is a maximum solution. If possible let us assume that 〉〈 νµ yyy ,= be a solution of the system such that xy > , that is 〉〈〉〈 νµνµ 0000 ,>, jjjj xxyy for at least one 0j . Therefore by definition of x , we have ),(>, 00 〉〈〉〈 νµνµ kkjj bbminyy when kkj ba > 0 for some k . Again, since φ≠),( bAS , by Lemma 3.1, 〉〈〉〈 νµνµ 0000 ,>,(max kkjkjk j bbaa for each 0k . Hence, 〉〈≠〉〈 ),(min),,(max, 0000 ννµµνµ jjk j jjk j kk yamaxyaminbb , which contradicts our assumption ),( bASy ∈ . Therefore, x is the maximum solution of the system of IFLEs bxA =⊗ . Example 3.8 Given =A 〉〈〉〈〉〈 〉〈〉〈〉〈 0.8,0.20.6,0.30.5,0.5 0.5,0.50.6,0.40.7,0.3 and T b ]0.6,0.3,0.5,0.3[= 〉〈〉〈 . Find out the maximum solution of the system bxA =⊗ . Ans. From the definition of maximum solution, 〉〈0.5,0.3=1x , 〉〈0.6,0.3=2x and 〉〈0.5,0.3=3x . So .]0.5,0.3,0.6,0.3,0.5,0.3[= T x 〉〈〉〈〉〈 Thus, φ≠),( bAS and bxA =⊗
- 8. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 20 holds. Hence xx T =]0.5,0.3,0.6,0.3,0.5,0.3[= 〉〈〉〈〉〈 is the maximum solution. Definition 3.9 (Moore-Penrose Inverse) For an IFM nmFA ×∈ , an IFM nmFG ×∈ is said to be a Moore-Penrose inverse of A , if AAGA = , GGAG = , AGAG T =)( and GAGA T =)( . The Moore-Penrose inverse of A is denoted by + A . Theorem 3.10 Let us consider a system of IFLEs bxA =⊗ . The system must have a solution, that is, must be consistent if the coefficient IFM A is a symmetric and idempotent of order n. Proof: Since A is symmetric and idempotent square IFM, it is already prove in [10], that A itself its Moore-Penrose inverse. That is, + AA = . So in that case the solution will be AbbAx == + . Example 3.11 Consider the system of IFLEs bxA =⊗ where, =A 〉〈〉〈 〉〈〉〈 0.7,0.10.6,0.2 0.6,0.20.8,0.2 and T b ]0.6,0.2,0.8,0.2[= 〉〈〉〈 . Here, AAT = and AA =2 , that is, the IFM A is symmetric and idempotent. So the Moore-Penrose inverse +A of A is itself A . Then the solution will be T AbbAx ]0.6,0.2,0.8,0.2[=== 〉〈〉〈+ . At a glance a system of IFLEs is solvable or not are depicted in following figure. 4 Chebychev Approximation In this section, we describe an algorithm by which we approach the right hand side of the system of IFLEs bxA =⊗ successively changing the original IFM )( nmFA ×∈ to an IFM )( nmFD ×∈ such that bxD =⊗ is solvable. Let us consider the solution or tolerable solution );( bAx′ of the system of IFLEs bxA =⊗ as =);( bAx′ ∀≤〉〈 iiji iij baifbmin ibaif >}{ 1,0 . .....(3) Now if we define that the system (1) is solvable if and only if (3) is its solution, that is, bbAxA =);(′⊗ holds, but in general bbAxA ≤′⊗ );( holds always. So our aim is, by changing the IFM A and retain the right hand side of the system same to make the system solvable. Before going to that, first we have to define some importent terms. Definition 4.1 The Chebychev distance of two IFM )(, nmFBA ×∈ is denoted by ),( BAρ and
- 9. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 21 is defined by 〉−−〈 ||min|,|max=),( ,, ννµµρ ijij ji ijij ji babaBA . The Chebychev distance of an IFM )( nmFA ×∈ and the set )( nmFS ×∈ is defined by ),(inf=),( BASA SB ρρ ∈ . Definition 4.2 We say that an IFM )( nmFB ×∈ is closer to an intuitionistic fuzzy vector )(mFv∈ than an IFM )( nmFA ×∈ if 〉〈≥〉〈≥〉〈 νµνµνµ iiijijijij vvbbaa ,,, or 〉〈≤〉〈≤〉〈 νµνµνµ iiijijijij vvbbaa ,,, for all indices Mi∈ and Nj ∈ and we denote by vBA ←→ . Lemma 4.3 Let us consider two IFM )(, nmFCA ×∈ and the intuitionistic fuzzy vector )(mFb∈ such that bCA ←→ . Then );();( bAxbCx ′≥′ . Proof: From the definition of the solution of the system of IFLEs of the form bxA =⊗ we have, =);( bCx′ ∀≤〉〈 iiji iij bcifbmin ibcif >}{ 1,0 and =);( bAx′ ∀≤〉〈 iiji iij baifbmin ibaif >}{ 1,0 . Now, as bCA ←→ , we have },>,;{},>,;{ 〉〈〉〈⊆〉〈〉〈 νµνµνµνµ iiijijiiijij bbaaibbcci for each Nj ∈ . So, );();( bAxbCx ′≥′ . Lemma 4.4 Let A and C be two IFM of order )( nm× and )(mFb∈ be an intuitionistic fuzzy vector with bCA ←→ . If bxA =⊗ is solvable then bxC =⊗ is also solvable. Proof: From our assumption, solvability of bxA =⊗ means that bbAxA =);(′⊗ . The i -th equation of which gives, ijij n j bbAxa =);( 1= ′⊗∑ . .....(4) Let us suppose that in (4) the equality has been achieved in term k . Thus, iik bbAxa =);(′⊗ , which is only possible if iik ba ≥ as well as ik bbAx ≥′ );( . Since, bCA ←→ , we get iikik bca ≥≥ and Lemma 4.3 gives, ikk bbAxbCx ≥≥ ′′ );();( . This implies, ikik bbCxC ≥⊗ ′ );( . Again for any IFM C , bbCxC ≤′⊗ );( . Hence the only possibility is, bbCxC =);(′⊗ , that is, bbC =⊗ is also solvable.
- 10. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 22 Lemma 4.5 Let us consider the system of IFLEs bxA =⊗ and );( bAx′ be its tolerable solution. If there exists an IFM D such that, bxD =⊗ is solvable with δρ =),( DA , then there also exists an IFM C such that, bCA ←→ and δρ ≤),( CA with bxC =⊗ is solvable. Proof: We can choose the IFM C in three different way. Case-1: If 〉〈≤〉〈≤〉〈 νµνµνµ iiiiii ddaabb ,,, or 〉〈≥〉〈≥〉〈 νµνµνµ iiiiii ddaabb ,,, , we set )}(2,{)},(2,{= )}(,{)},(,{=,= )}(2,{)},(2,{= )}(,{)},(,{=,= νννµµµ ννννµµµµνµ νννµµµ ννννµµµµνµ ijijiijiji ijijijiijijijiijijij ijijiijiji ijijijiijijijiijijij dabmaxdabmin adabmaxdaabminccc or dabmindabmax daabminadabmaxccc −−〈 〉−−−+〈〉〈 −−〈 〉−+−−〈〉〈 respectively. Case-2: If 〉〈≤〉〈≤〉〈 νµνµνµ iiiiii bbddaa ,,, or 〉〈≥〉〈≥〉〈 νµνµνµ iiiiii bbddaa ,,, , then take ijij dc = . Case-3: If 〉〈≤〉〈≤〉〈 νµνµνµ iiiiii ddbbaa ,,, or 〉〈≥〉〈≥〉〈 νµνµνµ iiiiii ddbbaa ,,, , then take ijij bc = . Now from the construction of C by the above three cases, it is obvious that δρ ≤);( CA and bCA ←→ . More over, bCD ←→ , hence by Lemma 4.4, bxC =⊗ is solvable. Definition 4.6 For a given IFM )( nmFA ×∈ and the intuitionistic fuzzy vector )(nFb∈ we denote the IFM )( nmFD ×∈ by ),( bA →∆ such that for each Mi∈ and Nj ∈ , =,= 〉〈 νµ ijijij ddd ≥∆+∆−〈 ∆−∆+〈 iijiijiij iijiijiij baifbaminbamax baifbamaxbamin },{},,{ <},{},,{ νννµµµ νννµµµ . It is obvious that, bbAA ←→∆→ ),( for any non-negative 〉∆∆〈∆ νµ ,= . More over as ∆ increases, we finally arrive at a matrix D such that iij bd = for all NjMi ∈∈ , , which satisfy the condition , bbDxD =);(′⊗ . So computation of the IFM D is an iterative process, which can be describe by the following algorithm.
- 11. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 23 Algorithm MATRIX begin 0=k ; 〉〈∆ 0,0=k ; AA k =)(∆ ; compute );( bAx′ ; If bbAxA ≠′⊗ );( then 〉≠−+∆ ≠−+∆〈 〉∆∆〈∆ +++ })(|;)({|min },)(|;)({|min= ,= , , 111 ννννν µµµµµ νµ δδ δδ iijkiijk ji k iijkiijk ji k kkk bAbA bAbA repeat 1= +kk ; );(=)( bAA kk →∆ δ until bbAxA kk =));(()( δδ ′⊗ ; output: kkA ∆);(δ end MATRIX. The above algorithm can be illustrate by the following example. Let us consider the system of IFLEs bxA =⊗ where, =A 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 0.7,0.20.7,0.10.3,0.40.7,0.10.5,0.3 0.2,0.60.4,0.40.5,0.50.8,0.10.3,0.5 0.6,0.20.1,0.80.9,0.10.2,0.60.6,0.3 0.2,0.50.4,0.50.7,0.20.6,0.40.3,0.5 and T b ]0.5,0.4,0.3,0.5,0.9,0.1,0.4,0.4[= 〉〈〉〈〉〈〉〈 . The corresponding tolerable solution will be T bAx ]0.5,0.4,0.3,0.5,0.3,0.5,0.3,0.5,0.5,0.4[=);( 〉〈〉〈〉〈〉〈〉〈′ but bbAxA ≤′⊗ );( so the system is unsolvable. Now by the above algorithm, in the first iteration, 〉〈∆ 0.1,0.1=1 , =)( 1∆A 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 0.6,0.30.6,0.20.4,0.40.6,0.20.5,0.4 0.3,0.50.3,0.50.4,0.50.7,0.20.3,0.5 0.7,0.10.2,0.70.9,0.10.3,0.50.7,0.2 0.3,0.40.4,0.40.6,0.30.5,0.40.4,0.4 and T bAx ]0.5,0.4,0.5,0.4,0.3,0.5,0.3,0.5,1,0[=));(( 1 〉〈〉〈〉〈〉〈〉〈∆′ .
- 12. International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014 24 Here also, bbAxA ≤∆′⊗ ));(( 1 . In the second iteration, 〉〈∆ 0.2,0.2=2 , =)( 2∆A 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 〉〈〉〈〉〈〉〈〉〈 0.5,0.40.5,0.40.5,0.40.5,0.40.5,0.4 0.3,0.50.3,0.50.3,0.50.5,0.40.3,0.5 0.9,0.10.4,0.50.9,0.10.5,0.30.9,0.1 0.4,0.40.4,0.40.4,0.40.4,0.40.4,0.4 and T bAx ]1,0,1,0,1,0,0.3,0.5,1,0[=));(( 2 〉〈〉〈〉〈〉〈〉〈∆′ . In this case, bbAxA =));(( 2∆′⊗ . So )(= 2∆AD is the Chebychev best approximation of the coefficient IFM A of the given system and ));(( 2 bAx ∆′ is the principal solution. 5 Conclusions In this article, we try to find the conditions for which a system of IFLEs be solvable. We also shown that, for a particular type of coefficient IFM the system of IFLEs must have a solution. Finally, we try to modify the coefficient IFM of a system of IFLEs, keeping right hand side intutitionistic fuzzy vector same, to make it solvable. References [1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986). [2] K. Cechlarova, Unique solvability of max-min fuzzy equations and strong regularity of matrices over fuzzy algebra, Fuzzy Sets and Systems, 75, 165-177 (1995). [3] E. Czogala and W. Pedrycz, On identification in fuzzy systems and its applications in control problems, Fuzzy Sets and Systems, 6, 257-273 (1982) . [4] E. Czogala and W. Pedrycz, Control problems in fuzzy systems, Fuzzy Sets and Systems, 7, 257-273 (1982). [5] M. Higashi and G,J. Klir, Resolution of finite finite fuzzy relation equatios, Fuzzy Sets and Systems, 13, 65-82 (1984). [6] H. Imai, K. Kikuchi and M. Miyakoshi, Unattainable solution of a fuzzy relation equation, Fuzzy Sets and Systems, 99, 193-196 (1998). [7] Li Jian-Xin, The smallest solution of max-min fuzzy equations, Fuzzy sets and Systems, 41, 317-327 (1990). [8] S.K. Khan and A. Pal, The generalized inverse of intuitionistic fuzzy matrix, Journal of Physical Sciences, 11, 62-67 (2007). [9] R. Pradhan and M. Pal, Intuitionistic fuzzy linear transformations, Annals of Pure and Applied Mathematics, 1(1), 57-68 (2012). [10] R. Pradhan and M. Pal, The generalized inverse of intuitionistic fuzzy matrices, Communicated. [11] E. Sanchez, Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48 (1976). [12] L.A. Zadeh, Fuzzy sets, Information and Control, 8, 338-353 (1965).