Solvability of system of intuitionistic fuzzy linear equations

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.


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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 ≤ .


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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 AAåå

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.


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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


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},,{ 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 ikikjkijj

jkijj

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 kkjkjkj

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 kkjkjkj

jkjkjkjkj

bbaaaaaa .

Hence, ),(<),(min),,(max ⟩⟨⟩⟨ νµννµµ kkjjkj

jjkj

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 kkjkjkj

bbaa

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

),(>),(max,),,,(min ⟩⟨⟩⟨≥⟩⟨≥⟩⟨⟩⟨ νµνµνµνµνµ kkjkjkj

jkjkjjjkjkj

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,


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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 Tb ]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,

TxxxxxxX ],,,,,[= 332211 ⟩⟨⟩⟨⟩⟨ νµνµνµ and

Tb ]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

TbAx ]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 =⊗


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),,(max00⟩⟨≠⟩⟨ νµννµµ kkjjk

jjjk

j

bbxamaxxamin for at least one 0k .

By definition of x , since ⟩⟨≤⟩⟨ νµνµ kkjj bbxx ,, for each k , so ⟩⟨≤⟩⟨ νµνµ00

,, kkjj bbxx . By

our assumption, ⟩⟨⟩⟨ νµνµ00

,<),(max kkjjkj

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, ⟩⟨⟩⟨ νµνµ0000

,>,(max kkjkjkj

bbaa for each 0k .

Hence, ⟩⟨≠⟩⟨ ),(min),,(max,0000

ννµµνµ jjkj

jjkj

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

Tb ]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[= Tx ⟩⟨⟩⟨⟩⟨ Thus, φ≠),( bAS and bxA =⊗


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 Tb ]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 TAbbAx ]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


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is defined by ⟩−−⟨ ||min|,|max=),(,,

ννµµρ ijijji

ijijji

babaBA .

The Chebychev distance of an IFM )( nmFA ×∈ and the set )( nmFS ×∈ is defined by

),(inf=),( BASASB

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.


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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.


23

Algorithm MATRIX

begin 0=k ; ⟩⟨∆ 0,0=k ; AA k =)(∆ ;

compute );( bAx′ ;

If bbAxA ≠′⊗ );( then

⟩≠−+∆

≠−+∆⟨

⟩∆∆⟨∆ +++

})(|;)({|min

},)(|;)({|min=

,=

,

,

111

ννννν

µµµµµ

νµ

δδ

δδ

iijkiijkji

k

iijkiijkji

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 Tb ]0.5,0.4,0.3,0.5,0.9,0.1,0.4,0.4[= ⟩⟨⟩⟨⟩⟨⟩⟨ .

The corresponding tolerable solution will be

TbAx ]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 TbAx ]0.5,0.4,0.5,0.4,0.3,0.5,0.3,0.5,1,0[=));(( 1 ⟩⟨⟩⟨⟩⟨⟩⟨⟩⟨∆′ .


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 TbAx ]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.

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Technology

Solvability of system of intuitionistic fuzzy linear equations