The main is to develop a method to solve an arbitrary fuzzy matrix equation system by using the embedding approach. Considering the existing solution to n×n fuzzy matrix equation system is done. To illustrate the proposed model a numerical example is given, and obtained results are discussed.

1. Introduction

The concept of fuzzy numbers and fuzzy arithmetic operations was first introduced by Zadeh [1], Dubois, and Prade [2]. We refer the reader to [3] for more information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to study a variety of problems including fuzzy metric spaces [4], fuzzy differential equations [5], fuzzy linear systems [6–8], and particle physics [9, 10].

One of the major applications of fuzzy number arithmetic is treating fuzzy linear systems [11–20], several problems in various areas such as economics, engineering, and physics boil down to the solution of a linear system of equations. Friedman et al. [21] introduced a general model for solving a fuzzy n×n linear system whose coefficient matrix is crisp, and the right-hand side column is an arbitrary fuzzy number vector. They used the parametric form of fuzzy numbers and replaced the original fuzzy n×n linear system by a crisp 2n×2n linear system and studied duality in fuzzy linear systems Ax=Bx+y where A and B are real n×n matrix, the unknown vector x is vector consisting of n fuzzy numbers, and the constant y is vector consisting of n fuzzy numbers, in [22]. In [6–8, 23, 24] the authors presented conjugate gradient, LU decomposition method for solving general fuzzy linear systems, or symmetric fuzzy linear systems. Also, Abbasbandy et al. [25] investigated the existence of a minimal solution of general dual fuzzy linear equation system of the form Ax+f=Bx+c, where A and B are real m×n matrices, the unknown vector x is vector consisting of n fuzzy numbers, and the constants f and c are vectors consisting of m fuzzy numbers.

In this paper, we give a new method for solving a n×n fuzzy matrix equation system whose coefficients matrix is crisp, and the right-hand side matrix is an arbitrary fuzzy number matrix by using the embedding method given in Cong-Xin and Min [26] and replace the original n×n fuzzy linear system by two n×n crisp linear systems. It is clear that, in large systems, solving n×n linear system is better than solving 2n×2n linear system. Since perturbation analysis is very important in numerical methods. Recently, Ezzati [27] presented the perturbation analysis for n×n fuzzy linear systems. Now, according to the presented method in this paper, we can investigate perturbation analysis in two crisp matrix equation systems instead of 2n×2n linear system as the authors of Ezzati [27] and Wang et al. [28].

2. Preliminaries

Parametric form of an arbitrary fuzzy number is given in [29] as follows. A fuzzy number u in parametric form is a pair (u_,u¯) of functions u_(r),u¯(r),0≤r≤1, which satisfy the following requirements:

u_(r) is a bounded left continuous nondecreasing function over [0,1],

u¯(r) is a bounded left continuous nonincreasing function over [0,1], and

u_(r)≤u¯(r),0≤r≤1.

The set of all these fuzzy numbers is denoted by E which is a complete metric space with Hausdorff distance. A crisp number α is simply represented by u_(r)=u¯(r)=α,0≤r≤1.

For arbitrary fuzzy numbers x=(x_(r),x¯(r)),y=(y_(r),y¯(r)), and real number k, we may define the addition and the scalar multiplication of fuzzy numbers by using the extension principle as [29]

x=y if and only if x_(r)=y_(r) and x¯(r)=y¯(r),

x+y=(x_(r)+y_(r),x¯(r)+y¯(r)), and

kx={(kx_,kx¯),k≥0,(kx¯,kx_),k<0.

Definition 2.1.

The n×n linear system is as follows:
(2.1)a11x1+a12x2+⋯+a1nxn=y1,a21x1+a22x2+⋯+a2nxn=y2,⋮an1x1+an2x2+⋯+annxn=yn,
where the given matrix of coefficients A=(aij), 1≤i, j≤n is a real n×n matrix, the given yi∈E, 1≤i≤n, with the unknowns xj∈E, 1≤j≤n is called a fuzzy linear system (FLS). The operations in (2.1) is described in next section.

Here, a numerical method for finding solution [21] of a fuzzy n×n linear system is given.

Definition 2.2 (see [<xref ref-type="bibr" rid="B22">21</xref>]).

A fuzzy number vector (x1,x2,…,xn)t given by
(2.2)xj=(x_j(r),x¯j(r));1≤j≤n,0≤r≤1
is called a solution of the fuzzy linear system (2.1) if
(2.3)∑j=1naijxj_=∑j=1naijxj_=y_i,∑j=1naijxj¯=∑j=1naijxj-=y¯i.
If, for a particular i,aij>0, for all j, we simply get
(2.4)∑j=1naijx_j=y_i,∑j=1naijx¯j=y¯i.

Finally, we conclude this section by a reviewing on the proposed method for solving fuzzy linear system [21].

The authors [21] wrote the linear system of (2.1) as follows:
(2.5)SX=Y,
where sij are determined as follows:
(2.6)aij≥0⇒sij=aij,si+n,j+n=aij,aij<0⇒si,j+n=-aij,si+n,j=-aij,
and any sij which is not determined by (2.1) is zero and
(2.7)X=[x_1⋮x_n-x¯1⋮-x¯n],Y=[y_1⋮y_n-y¯1⋮-y¯n].
The structure of S implies that sij≥0, 1≤i, j≤2n and that
(2.8)S=(BCCB),
where B contains the positive entries of A, and C contains the absolute values of the negative entries of A, that is, A=B-C.

Theorem 2.3 (see [<xref ref-type="bibr" rid="B22">21</xref>]).

The inverse of nonnegative matrix
(2.9)S=(BCCB)
is
(2.10)S-1=(DEED),
where
(2.11)D=12[(B+C)-1+(B-C)-1],E=12[(B+C)-1-(B-C)-1].

Corollary 2.4 (see [<xref ref-type="bibr" rid="B26">30</xref>]).

The solution of (2.5) is obtained by
(2.12)X=S-1Y.

3. Fuzzy Matrix Equation System

A matrix system such as
(3.1)(a11a12⋯a1na21a22⋯a2n⋮⋮⋮⋮an1an2⋯ann)(x11x12⋯x1nx21x22⋯x2n⋮⋮⋮⋮xn1xn2⋯xnn)=(y11y12⋯y1ny21y22⋯y2n⋮⋮⋮⋮yn1yn2⋯ynn),
where aij, 1≤i, j≤n, are real numbers, the elements yij in the right-hand matrix are fuzzy numbers, and the unknown elements xij are ones, is called a fuzzy matrix equation system (FMES).

Using matrix notation, we have
(3.2)AX=Y.
A fuzzy number matrix
(3.3)X=(x1,…,xj,…,xn)
is called a solution of the fuzzy matrix system (2.1) if
(3.4)Axj=yj,1≤j≤n.

In this section, we propose a new method for solving FMES.

Theorem 3.1.

Suppose that the inverse of matrix A exists and xj=(xj1,xj2,…,xjn)T is a solution of this equation. Then xj_+xj¯=(xj_1+xj¯1,xj_2+xj¯2,…,xj_n+xj¯n)T is the solution of the following systems:
(3.5)A(xj_+xj¯)=yj_+yj¯,j=1,2,…,n,
where yj_+yj¯=(yj_1+yj¯1,yj_2+yj¯2,…,yj_n+yj¯n)T,j=1,2,…,n.

Proof.

It is the same as the proof of Theorem 3 in [27].

For solving (3.2), we first solve the following system:
(3.6)a11(xj_1+xj¯1)+⋯+a1n(xj_n+xj¯n)=(yj_1+yj¯1),a21(xj_1+xj¯1)+⋯+a2n(xj_n+xj¯n)=(yj_2+yj¯2),⋮an1(xj_1+xj¯1)+⋯+ann(xj_n+xj¯n)=(yj_n+yj¯n),j=1,2,…,n.
Using matrix notation, we have
(3.7)A(X_+X¯)=(Y_+Y¯).
Suppose that the solution of (3.7) is as
(3.8)dj=[dj1dj2⋮djn]=xj_+xj¯=[xj_1+xj¯1xj_2+xj¯2⋮xj_n+xj¯n],j=1,2,…,n.
Let matrices B and C have defined as Section 2. Now using matrix notation for (3.7), we get in parametric form (B-C)(X_(r)+X¯(r))=(Y_(r)+Y¯(r)). We can write this system as follows:
(3.9)BX_(r)-CX¯(r)=Y_(r),BX¯(r)-CX_(r)=Y¯(r).
By substituting X¯(r)=D-X_(r) and X_(r)=D-X¯(r) in the first and second equation of above system, respectively, we have
(3.10)(B+C)X_(r)=Y_(r)+CD,(3.11)(B+C)X¯(r)=Y¯(r)+CD,
therefore, we have
(3.12)X_(r)=(B+C)-1(Y_(r)+CD),X¯(r)=(B+C)-1(Y¯(r)+CD).
Therefore, we can solve fuzzy matrix equation system (3.2) by solving (3.7)–(3.10).

Theorem 3.2.

Let in (3.3) j=1, also g and G are the number of multiplication operations that are required to calculate
(3.13)X=(x_1,x_2,…,x_n,-x¯1,-x¯2,…,-x¯n)T=S-1Y,
(the proposed method in Friedman et al. [21]) and
(3.14)xj=(xj_1,xj_2,…,xj_n,xj¯1,xj¯2,…,xj¯n)T,
from (3.7)–(3.10), respectively. Then G≤g and g-G=n2.

Proof.

According to Section 2, we have
(3.15)S-1=(DEED),
where
(3.16)D=12[(B+C)-1+(B-C)-1],E=12[(B+C)-1-(B-C)-1].
Therefore, for determining S-1, we need to compute (B+C)-1 and (B-C)-1. Now, assume that M is n×n matrix and denote by h(M) the number of multiplication operations that are required to calculate M-1. It is clear that
(3.17)h(S)=h(B+C)+h(B-C)=2h(A),
and hence
(3.18)g=2h(A)+4n2.
For computing xj_+xj¯=(xj_1+xj¯1,xj_2+xj¯2,…,xj_n+xj¯n)T from (3.7) and xj_=(xj_1,xj_2,…,xj_n)T from (3.10) the number of multiplication operations is h(A)+n2 and h(B+C)+2n2, respectively. Clearly h(B+C)=h(A), so
(3.19)G=2h(A)+3n2,
and hence g-G=n2. This proves theorem.

Remark 3.3.

In (3.3) if j=1, then this paper is similar to [27].

Example 3.4.

Consider the 2×2 fuzzy matrix equation system as follows:
(3.20)(2-111)(x11x12x21x22)=((3r-3,3-3r)(4r-4,6-6r)(2r+1,5-2r)(3r,7-4r)).
By using (3.7) and (3.10), we have
(3.21)(x11_(r)+x11¯(r)x12_(r)+x12¯(r)x21_(r)+x21¯(r)x22_(r)+x22¯(r))=(23-r44),(x11_(r)x12_(r)x21_(r)x22_(r))=(rr1+r2r),
and hence
(3.22)(x11¯(r)x12¯(r)x21¯(r)x22¯(r))=(2-r3-2r3-r4-2r).
Obviously, x11,x12,x21 and x22, are fuzzy numbers.

4. Conclusions

In this paper, we propose a general model for solving fuzzy matrix equation system. The original system with matrix coefficient A is replaced by two n×n crisp matrix equation systems.

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