SOLVABILITY OF SYSTEM OF INTUITIONISTIC FUZZY LINEAR EQUATIONS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

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 A ⊗ x = b is said to be solvable if, for a definite solution x( A; b) , A ⊗ x( A; b) = b holds, otherwise unsolvable. In general A ⊗ x ( A; b ) ≤ b 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 A ⊗ x = b be consistent when the coefficient intuitionistic fuzzy matrix (IFM) A is regular.

DOI : 10.5121/ijfls.2014.4303

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

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 A ⊗ x ( A; b ) ≤ b holds always. So to distinguish the fact, we say a system of IFLEs is solvable if and only if A ⊗ x( A; b) = b and we define that particular solution x( A; b) 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, A ⊗ x = b 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 n > 0 , F (n ) denotes the set of all n-dimensional column vector and F (m, n) denotes the set of all IFM of order ( m × n ) over F . The respective IFM is defined as follows.

Definition 2.1 (Intuitionistic fuzzy matrices) An intuitionistic fuzzy matrix (IFM) A of order m× n is defined as A = [ xij , 〈 aijµ , aijν 〉 ]m×n where aijµ , aijν are called membership and non-membership values of xij in A , which maintains the condition 0 ≤ aijµ + aijν ≤ 1 . For simplicity, we write A = [ xij , aij ]m×n or simply

[ aij ]m×n where aij = 〈 aijµ , aijν 〉 .

In arithmetic operations, only the values of aijµ and aijν are needed so from here we only consider the values of aij = 〈 aijµ , aijν 〉 . All elements of an IFM are the members of

F = {〈 a, b〉 : 0 ≤ a + b ≤ 1} .

Comparison between intuitionistic fuzzy matrices have an important role in our work, which is defined below.

Definition 2.2 (Dominance of IFM) Let A, B ∈ Fm×n such that A = (〈 aijµ , aijν 〉 ) and B = (〈bijµ , bijν 〉 ) , then we write A ≤ B if,

aijµ ≤ bijµ and aijν ≥ bijν for all i, j , and we say that A is dominated by B or B dominates A . A and B are said to be comparable, if either A ≤ B or B ≤ A .

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

Let Vn denotes the set of all n -tuples (〈 x1µ , x1ν 〉 , 〈 x2 µ , x2ν 〉, K , 〈 xnµ , xnν 〉 ) over F . An element of Vn is called an intuitionistic fuzzy vector (IFV) of dimension n , where xiµ and xiν are the membership and non-membership values of the component xi . The system Vn together with the operations, componentwise addition and multiplication forms intuitionistic fuzzy vectors space (IVFS).

Definition 2.3 (Row space and column space) Let A = (〈 aijµ , aijν 〉 ) ∈ Fm×n be an IFM. Then the element 〈 aijµ , aijν 〉 is the ij th entry of A .

Let Aiå ( Aåj ) denote the i th row ( i th column) of A .

The row space R ( A) of A is the subspace of Vn generated by the rows { Aiå } of A . The column space C ( A) of A is the subspace of Vm generated by the columns { Aåj } of A .

Definition 2.4 (Linear combination of IFVs) Let S = {a1 , a2 , K , a p } be a set of intuitionistic fuzzy vectors of dimension n . The linear p

combination of elements of the set S is a finite sum

∑c a

i i

where ai ∈ S and ci ∈ [0,1] . The

i =1

set of all linear combinations of the elements of S is called the span of S , denoted by 〈 S 〉 .

An example of V3 and its spanning set is given below.

Example 2.5 Let S = {a1 , a2 , a3 } be a subset of V3 , where

a1 = (〈0.8,0.2〉, 〈0.6,0.3〉, 〈0.4,0.3〉 ), a2 = (〈0.5,0.3〉, 〈0.5,0.1〉, 〈0.4,0.2〉 ) and a3 = (〈 0.7,0.3〉 , 〈 0.7,0.2〉 , 〈 0.9,0.1〉 ) .Then, 〈 S 〉 = {c1 (〈 0.8,0.2〉 , 〈 0.6,0.3〉 , 〈 0.4,0.3〉 ) + c 2 (〈 0.5,0.3〉 , 〈 0.5,0.1〉 , 〈 0.4,0.2〉 ) + c3 (〈 0.7,0.3〉 , 〈0.7,0.2〉 , 〈 0.9,0.1〉 )} 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 s ∈ S is a linear combination of S \ {s} .

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.

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The examples of independent and dependent set of vectors are given below.

Example 2.7 Let S = {a1 , a2 , a3 } be a subset of V3 , where

a1 = (〈0.8,0.2〉, 〈0.6,0.3〉, 〈0.4,0.3〉 ), a2 = (〈0.5,0.3〉, 〈0.5,0.1〉, 〈0.4,0.2〉 ) and a3 = (〈 0.7,0.3〉 , 〈 0.7,0.2〉 , 〈 0.9,0.1〉 ) .

Here the set S is an independent set. If not then a1 = αa2 + β a3 for α , β ∈ F .

So, a1 = α (〈0.5,0.3〉, 〈0.5,0.1〉, 〈0.4,0.2〉 ) + β (〈0.7,0.3〉, 〈0.7,0.2〉, 〈0.9,0.1〉 ) = (〈 max{min (0.5, α ), min (0.7, β )}, min{max (0.3,1 − α ), max (0.3,1 − β )}〉 , 〈 max{min (0.5, α ), min (0.7, β )}, min{max (0.1,1 − α ), max (0.2,1 − β )}〉 , 〈 max{min (0.4, α ), min (0.9, β )}, min{max (0.2,1 − α ), max (0.1,1 − β )}〉 ).

It is not possible to find any α , β ∈ F such that the corresponding coefficients on both sides will be equal. That is, a1 ≠ αa2 + β a3 . Similarly, a2 ≠ αa1 + β a3 and a3 ≠ αa2 + β a1 . So the set S is independent.

Let S = {a1 , a2 } be a subset of V3 , where a1 = (〈0.7,0.3〉, 〈0.5,0.3〉, 〈0.6,0.3〉 ) and

a2 = (〈0.8,0.2〉, 〈0.5,0.1〉, 〈0.6,0.2〉 ) . Here a1 = ca2 for c = 0.7 . So S is a dependent set.

Definition 2.8 (Basis) Let W be an intuitionistic fuzzy subspace of Vn 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 n

bi = ∑aij b j for bi , b j ∈ B and aij ∈ [0,1] then aii bi = bi . j =1

Example 2.10 Let S = {a1 , a2 , a3 } be a subset of V3 given by

a1 = (〈0.5,0.4〉, 〈0.7,0.2〉, 〈0.6,0.3〉 ) , a2 = (〈0.5,0.3〉, 〈0.6,0.2〉, 〈0.8,0.2〉 ) and a3 = (〈 0.4,0.4〉 , 〈 0.4,0.3〉 , 〈 0.8,0.1〉 ) . Here S is independent set, since a1 ≠ c1a2 + c2 a3 , a 2 ≠ c3 a1 + c 4 a3 and a3 ≠ c5 a1 + c6 a2 . So 16

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

{a1 , a2 , a3} is a basis for 〈 S 〉 . Now this is a standard basis also. For a1 = c11a1 + c12 a2 + c13a3 holds if c11 = 0.8 , c12 = 0.5 and c13 = 0.6 . Also a1 = c11a1 for c11 = 0.8 . Similarly for a2 and a3 .

3.Solvability In this section, we consider the system of IFLEs of the form, A ⊗ x = b that

.....(1)

〈 max min ( aijµ , x jkµ ), min max ( aijν , x jkν )〉 = 〈bikµ , bikν 〉

is,

j

j

....(2) where the IFM A ∈ F ( m × n) and the intuitionistic fuzzy vector b ∈ F (m) are given and the intuitionistic fuzzy vector x ∈ F (n) 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 S ( A, b) = { x ∈ F ( n) | A ⊗ x = b} .

Now our aim is to find whether the system (1) is solvable, that is, whether the solution set S ( A, b) is non-empty.

Lemma 3.1 Let us consider the system of IFLE A ⊗ x = b . If max (〈 a jkµ , a jkν 〉 ) < (〈bkµ , bkν 〉 ) j

for some k , then S ( A, b) = φ , that is the system is not solvable.

Proof: If max (〈 a jkµ , a jkν 〉 ) < (〈bkµ , bkν 〉 ) for some k , then

min (〈 a jkµ , a jkν 〉 ) ≤ 〈 a jkµ , a jkν 〉 ≤ max (〈 a jkµ , a jkν 〉 ) < (〈bkµ , bkν 〉 ) . j

j

Hence, 〈 max min ( a jkµ , x jµ ), min max ( a jkν , x jν )〉 < (〈bkµ , bkν 〉 ) for some k , and by j

j

equation (2) no values 〈 x jµ , x jν 〉 exists that satisfy the equation (1). Therefore S ( A, b) = φ .

Remark 3.2 Let us consider the condition of the Lemma 3.1 be max (〈 a jkµ , a jkν 〉 ) > (〈bkµ , bkν 〉 ) j

for some k . Then according to the proof of the Lemma 3.1,

min (〈 a jkµ , a jkν 〉 , 〈 x jµ , x jν 〉 ) ≥ 〈 a jkµ , a jkν 〉 ≥ max (〈 a jkµ , a jkν 〉 ) > (〈bkµ , bkν 〉 ) j

implies

j

the only possibility is, 〈 a jkµ , a jkν 〉 are same for all j . Then two cases may arise, Case-1: If 〈bkµ , bkν 〉 are equal for all k . Then the system reduce to one equation. Hence the system is solvable.

Case-2: If 〈bkµ , bkν 〉 are different for some k . Then the equation of the system will be such that, 17

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

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 A ⊗ x = b where, A =

〈0.7,0.2〉 〈0.5,0.4〉  〈0.6,0.3〉 〈0.6,0.2〉  and b = [〈 0.4,0.5〉 , 〈1,0〉 , 〈 0.5,0.5〉 ]T .    〈0.8,0.1〉 〈0.4,0.4〉  Here for k = 2 , max{〈 0.5,0.4〉 , 〈 0.6,0.2〉 , 〈 0.4,0.4〉} = 〈 0.6,0.2〉 < 〈1,0〉 . Hence by Lemma 3.1, the system of IFLEs A ⊗ x = b 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 A ⊗ x = b 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 A ⊗ A − ⊗ A = A . Now, A ⊗ x = b gives A ⊗ A − ⊗ A ⊗ x = b .

That implies, A ⊗ A − ⊗ b = b . Which shows, ( A− ⊗ b) is a solution of the given system. Hence the system of IFLE is solvability.

Example 3.5 Let us consider the system of IFLEs A ⊗ x = b with

〈 0.7,0.3〉 〈 0.6,0.4〉 〈 0.5,0.5〉  T , X = [〈 x1µ , x1ν 〉, 〈 x2 µ , x2ν 〉, 〈 x3µ , x3ν 〉 ] and A=   〈 0.5,0.5〉 〈 0.6,0.3〉 〈 0.8,0.2〉  b = [〈0.6,0.3〉 , 〈 0.5,0.4〉 ]T .

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

〈0.8,0.2〉 〈0.5,0.5〉  A = 〈0.5,0.5〉 〈0.5,0.5〉  . Then 〈0.5,0.5〉 〈0.8,0.2〉  −

x = A− b = [〈 0.6,0.3〉 , 〈 0.5,0.5〉 , 〈 0.5,0.4〉 ]T 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 S ( A, b) is called a maximum solution of the system A ⊗ x = b 18

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

if for all x ∈ S ( A, b) , x ≥ x implies x = x .

The following theorem demonstrate how to find the maximum solution of the system of IFLEs. Theorem 3.7 If for a system of IFLEs A ⊗ x = b has a solution denoted by x( A, b) and is

if a jk ≤ bk ∀ j  〈1,0〉   defined by x = 〈 x µ , xν 〉 = min{bk } if a jk > bk  is the maximum solution.     Proof: As the system of IFLEs A ⊗ x = b has a solution, so it is consistent, then x is a solution of the system. If x is not a solution, then A ⊗ x ≠ b and therefore 〈 max min ( a jkµ , x jµ ), min max ( a jkν , x jν )〉 ≠ (〈bk µ , bk ν 〉 ) for at least one k0 . j

0

j

0

By definition of x , since 〈 x jµ , x jν 〉 ≤ 〈bkµ , bkν 〉 for each k , so 〈 x jµ , x jν 〉 ≤ 〈bk µ , bk ν 〉 . By 0

0

our assumption, max (〈 a jkµ , a jν 〉 ) < 〈bk µ , bk ν 〉 for some k0 and by Lemma 3.1 it follows that 0 0 j

S ( A, b) = φ , which is a contradiction. Hence x is a solution of the system A ⊗ x = b .

Now let us prove that x is a maximum solution. If possible let us assume that y = 〈 y µ , yν 〉 be a solution of the system such that y > x , that is 〈 y j µ , y j ν 〉 > 〈 x j0 µ , x j0ν 〉 for at least one j0 .

0

0

Therefore by definition of x , we have 〈 y j µ , y j ν 〉 > min(〈bkµ , bkν 〉 ) when a j 0

0k

0

> bk for some

k . Again, since S ( A, b) ≠ φ , by Lemma 3.1, max (〈 a jk µ , a jk ν 〉 > 〈bk µ , bk ν 〉 for each k0 .

0 0 0 0 j

Hence,

〈bk µ , bk ν 〉 ≠ 〈 max min ( a jk µ , y jµ ), min max ( a jk ν , y jν )〉 , which contradicts our 0

0

j

0

j

0

assumption y ∈ S ( A, b) .

Therefore, x is the maximum solution of the system of IFLEs A ⊗ x = b .

〈 0.7,0.3〉 〈 0.6,0.4〉 〈 0.5,0.5〉  and b = [〈 0.5,0.3〉, 〈 0.6,0.3〉 ]T .  〈 0.5,0.5〉 〈 0.6,0.3〉 〈 0.8,0.2〉  Find out the maximum solution of the system A ⊗ x = b .

Example 3.8 Given A = 

Ans. From the definition of maximum solution,

x1 = 〈0.5,0.3〉 , x2 = 〈0.6,0.3〉 and

x3 = 〈0.5,0.3〉 . So x = [〈 0.5,0.3〉 , 〈 0.6,0.3〉 , 〈 0.5,0.3〉 ]T . Thus, S ( A, b) ≠ φ and A ⊗ x = b 19

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

holds. Hence x = [〈 0.5,0.3〉 , 〈0.6,0.3〉 , 〈 0.5,0.3〉 ]T = x is the maximum solution.

Definition 3.9 (Moore-Penrose Inverse) For an IFM A ∈ Fm×n , an IFM G ∈ Fm×n is said to be a Moore-Penrose inverse of A , if

AGA = A , GAG = G , ( AG )T = AG and (GA)T = GA .

The Moore-Penrose inverse of A is denoted by A + .

Theorem 3.10 Let us consider a system of IFLEs A ⊗ x = b . 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, A = A+ . So in that case the solution will be x = A + b = Ab .

〈 0.8,0.2〉 〈 0.6,0.2〉   〈 0.6,0.2〉 〈 0.7,0.1〉  and b = [〈 0.8,0.2〉 , 〈 0.6,0.2〉 ]T . Here, AT = A and A 2 = A , 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 x = A+ b = Ab = [〈 0.8,0.2〉 , 〈0.6,0.2〉 ]T .

Example 3.11 Consider the system of IFLEs A ⊗ x = b where, A = 

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 A ⊗ x = b successively changing the original IFM A ∈ F ( m × n) to an IFM D ∈ F ( m × n) such that D ⊗ x = b is solvable.

Let us consider the solution or tolerable solution x′( A; b) of the system of IFLEs

if aij ≤ bi ∀ i  〈1,0〉   A ⊗ x = b as x′( A; b) = min{bi } if aij > bi .    

.....(3)

Now if we define that the system (1) is solvable if and only if (3) is its solution, that is, A ⊗ x′( A; b) = b holds, but in general A ⊗ x′( A; b) ≤ b 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 A, B ∈ F ( m × n) is denoted by ρ ( A, B ) and 20

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

is defined by ρ ( A, B ) = 〈 max | aijµ − bijµ |, min | aijν − bijν |〉 . i, j

i, j

The Chebychev distance of an IFM A ∈ F ( m × n) and the set S ∈ F ( m × n) is defined by ρ ( A, S ) = inf ρ ( A, B ) .

B∈S

Definition 4.2 We say that an IFM B ∈ F ( m × n) is closer to an intuitionistic fuzzy vector v ∈ F (m) than an IFM A ∈ F ( m × n) if

〈 aijµ , aijν 〉 ≥ 〈bijµ , bijν 〉 ≥ 〈 viµ , viν 〉 or 〈 aijµ , aijν 〉 ≤ 〈bijµ , bijν 〉 ≤ 〈 viµ , viν 〉 for all indices

i ∈ M and j ∈ N and we denote by A → B ← v .

Lemma 4.3 Let us consider two IFM A, C ∈ F ( m × n) and the intuitionistic fuzzy vector b ∈ F (m) such that A → C ← b . Then x′(C ; b) ≥ x′( A; b) .

Proof: From the definition of the solution of the system of IFLEs of the form A ⊗ x = b we have,

if cij ≤ bi ∀ i  〈1,0〉   x′(C ; b) = min{bi } if cij > bi      if aij ≤ bi ∀ i  〈1,0〉   and x′( A; b) = min{bi } if aij > bi .     Now, as A → C ← b , we have {i; 〈 cijµ , cijν 〉 > 〈biµ , biν 〉} ⊆ {i; 〈 aijµ , aijν 〉 > 〈biµ , biν 〉} for each j ∈ N . So, x′(C ; b) ≥ x′( A; b) .

Lemma 4.4 Let A and C be two IFM of order ( m × n) and b ∈ F (m ) be an intuitionistic fuzzy vector with A → C ← b . If A ⊗ x = b is solvable then C ⊗ x = b is also solvable.

Proof: From our assumption, solvability of A ⊗ x = b means that A ⊗ x′( A; b) = b . The i -th n

equation of which gives,

∑a

ij

⊗ x′j ( A; b) = bi .

.....(4)

j =1

Let us suppose that in (4) the equality has been achieved in term k . Thus, aik ⊗ x′( A; b) = bi , which is only possible if aik ≥ bi as well as xk ′ ( A; b) ≥ bi .

Since, A → C ← b , we get aik ≥ cik ≥ bi and Lemma 4.3 gives, xk ′ (C ; b) ≥ xk ′ ( A; b) ≥ bi .

This implies, Cik ⊗ xk ′ (C ; b) ≥ bi . Again for any IFM C , C ⊗ x′(C ; b) ≤ b .

Hence the only possibility is, C ⊗ x′(C ; b) = b , that is, C ⊗ b = b is also solvable.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

Lemma 4.5 Let us consider the system of IFLEs A ⊗ x = b and x′( A; b) be its tolerable solution. If there exists an IFM D such that, D ⊗ x = b is solvable with ρ ( A, D ) = δ , then there also exists an IFM C such that, A → C ← b and ρ ( A, C ) ≤ δ with C ⊗ x = b is solvable.

Proof: We can choose the IFM C in three different way.

Case-1: If 〈biµ , biν 〉 ≤ 〈 aiµ , aiν 〉 ≤ 〈 d iµ , d iν 〉 or 〈biµ , biν 〉 ≥ 〈 aiµ , aiν 〉 ≥ 〈 d iµ , d iν 〉 , we set

cij

= 〈 cijµ , cijν 〉 = 〈 max{biµ , aijµ − (d ijµ − aijµ )}, min{biν , aijν + (aijν − d ijν )}〉 = 〈 max{biµ , (2aijµ − d ijµ )}, min{biν , (2aijν − d ijν )}

cij

or = 〈 cijµ , cijν 〉 = 〈 min{biµ , aijµ + (aijµ − d ijµ )}, max{biν , aijν − (d ijν − aijν )}〉 = 〈 min{biµ , (2aijµ − d ijµ )}, max{biν , (2aijν − d ijν )}

respectively.

Case-2: If 〈 aiµ , aiν 〉 ≤ 〈 d iµ , d iν 〉 ≤ 〈biµ , biν 〉 or 〈 aiµ , aiν 〉 ≥ 〈 d iµ , d iν 〉 ≥ 〈biµ , biν 〉 , then take

cij = d ij .

Case-3: If 〈 aiµ , aiν 〉 ≤ 〈biµ , biν 〉 ≤ 〈 d iµ , d iν 〉 or 〈 aiµ , aiν 〉 ≥ 〈biµ , biν 〉 ≥ 〈 d iµ , d iν 〉 , then take

cij = bij .

Now from the construction of C by the above three cases, it is obvious that ρ ( A; C ) ≤ δ and A → C ← b . More over, D → C ← b , hence by Lemma 4.4, C ⊗ x = b is solvable.

Definition 4.6 For a given IFM A ∈ F ( m × n) and the intuitionistic fuzzy vector b ∈ F (n) we denote the IFM D ∈ F ( m × n) by ( A, ∆ → b) such that for each i ∈ M and j ∈ N ,

〈 min{aijµ + ∆ µ , biµ }, max{aijν − ∆ν , biν }  d ij = 〈 d ijµ , d ijν 〉 = 〈 max{aijµ − ∆ µ , biµ }, min{aijν + ∆ν , biν }  

if aij < bi   if aij ≥ bi  .  

It is obvious that, A → ( A, ∆ → b) ← b for any non-negative ∆ = 〈 ∆ µ , ∆ν 〉 . More over as ∆ increases, we finally arrive at a matrix D such that d ij = bi for all i ∈ M , j ∈ N , which satisfy the condition , D ⊗ x′( D; b) = b . So computation of the IFM D is an iterative process, which can be describe by the following algorithm.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

Algorithm MATRIX begin k = 0 ; ∆ k = 〈 0,0〉 ; A(∆ k ) = A ; compute x′( A; b) ; If A ⊗ x′( A; b) ≠ b then

repeat ∆ k +1

= 〈 ∆ k +1µ , ∆ k +1ν 〉 = 〈 ∆ kµ + min {| A(δ k ) ijµ − biµ |; A(δ k ) ijµ ≠ biµ }, i, j

∆ kν + min {| A(δ k ) ijν − biν |; A(δ k ) ijν ≠ biν }〉 i, j

k = k + 1; A(∆ k ) = ( A; δ k → b) until A(δ k ) ⊗ x′( A(δ k ); b) = b ; output: A(δ k ); ∆ k end MATRIX. The above algorithm can be illustrate by the following example.

Let us consider the system of IFLEs A ⊗ x = b where,

〈 0.3,0.5〉 〈 0.6,0.4〉 〈 0.7,0.2〉 〈 0.4,0.5〉 〈 0.6,0.3〉 〈 0.2,0.6〉 〈 0.9,0.1〉 〈 0.1,0.8〉 A=  〈 0.3,0.5〉 〈 0.8,0.1〉 〈 0.5,0.5〉 〈 0.4,0.4〉  〈 0.5,0.3〉 〈 0.7,0.1〉 〈 0.3,0.4〉 〈 0.7,0.1〉 and b = [〈 0.4,0.4〉 , 〈 0.9,0.1〉 , 〈 0.3,0.5〉 , 〈 0.5,0.4〉 ]T .

〈 0.2,0.5〉  〈0.6,0.2〉  〈0.2,0.6〉   〈0.7,0.2〉 

The corresponding tolerable solution will be

x′( A; b) = [〈 0.5,0.4〉 , 〈 0.3,0.5〉 , 〈 0.3,0.5〉 , 〈 0.3,0.5〉 , 〈 0.5,0.4〉 ]T but A ⊗ x′( A; b) ≤ b so the system is unsolvable.

Now by the above algorithm, in the first iteration,

〈 0.4,0.4〉 〈 0.5,0.4〉 〈 0.6,0.3〉 〈 0.4,0.4〉 〈 0.3,0.4〉  〈 0.7,0.2〉 〈 0.3,0.5〉 〈 0.9,0.1〉 〈 0.2,0.7〉 〈 0.7,0.1〉   ∆1 = 〈0.1,0.1〉 , A(∆1 ) =  〈 0.3,0.5〉 〈 0.7,0.2〉 〈 0.4,0.5〉 〈 0.3,0.5〉 〈 0.3,0.5〉    〈 0.5,0.4〉 〈 0.6,0.2〉 〈 0.4,0.4〉 〈 0.6,0.2〉 〈 0.6,0.3〉  and x′( A(∆1 ); b) = [〈1,0〉 , 〈 0.3,0.5〉 , 〈 0.3,0.5〉 , 〈 0.5,0.4〉 , 〈 0.5,0.4〉 ]T .

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.3, July 2014

Here also, A ⊗ x′( A(∆1 ); b) ≤ b .

In the second iteration,

〈 0.4,0.4〉  〈 0.9,0.1〉 ∆ 2 = 〈0.2,0.2〉 , A(∆ 2 ) =  〈 0.3,0.5〉  〈 0.5,0.4〉

〈 0.4,0.4〉 〈 0.4,0.4〉 〈 0.4,0.4〉 〈 0.4,0.4〉  〈 0.5,0.3〉 〈 0.9,0.1〉 〈 0.4,0.5〉 〈 0.9,0.1〉  〈 0.5,0.4〉 〈 0.3,0.5〉 〈 0.3,0.5〉 〈 0.3,0.5〉   〈 0.5,0.4〉 〈 0.5,0.4〉 〈 0.5,0.4〉 〈 0.5,0.4〉 

and x′( A( ∆ 2 ); b) = [〈1,0〉 , 〈 0.3,0.5〉 , 〈1,0〉 , 〈1,0〉 , 〈1,0〉 ]T .

In this case, A ⊗ x′( A(∆ 2 ); b) = b . So D = A(∆ 2 ) is the Chebychev best approximation of the coefficient IFM A of the given system and x′( A(∆ 2 ); b) 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.

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