Solving fuzzy matrix equation of the form X ˜ A = B ˜

Abstract

In paper a general fuzzy matrix equation $\tilde{X} A = \tilde{B}$ is investigated by using the embedding approach. A numerical procedure for calculating the solution is designed and a sufficient condition for the existence of fuzzy solution is derived. Two examples are given to illustrate the proposed method.

Keywords

Fuzzy numbers matrix analysis fuzzy matrix systems fuzzy approximate solutions

1 Introduction

Systems of simultaneous matrix equations play a major role in various areas such as mathematics, physics, statistics, engineering and social sciences. In many linear systems, some of the system parameters are vague or imprecise, and fuzzy mathematics is a better tool than crisp mathematics for modeling these problems, and hence solving a fuzzy linear system [18] or a fuzzy differential equation [30] is becoming more important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [33], Dubois et al. [17] and Nahmias [26]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [30], Goetschell et al. [20] and Wu Congxin et al. [31, 32].

Since Friedman et al. [18] proposed a general model for solving an n × n fuzzy linear systems whose coefficients matrix is crisp and the right-hand side is an arbitrary fuzzy numbers vector by an embedding approach in 1998, lots of works have been done about how to deal with some fuzzy linear systems with more advanced forms such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), fully fuzzy linear systems (FFLS), dual full fuzzy systems linear systems (DFFLS) and general dual fuzzy linear systems (GDFLS) [1–7 , 34]. However, for a fuzzy linear matrix equation which always has a wide use in control theory and control engineering, few work has been done in the past decades. In 2009, Allahviranloo et al. [8] discussed the fuzzy linear matrix equations (FLME) of the form $A \tilde{X} B = \tilde{C}$ . By means of the parametric form of the fuzzy number, they derived necessary and sufficient conditions for the existence condition of fuzzy solutions and designed a numerical procedure for calculating the solutions of the original system. In 2011, Gong et al. [21] investigated a class of fuzzy matrix equations $A \tilde{X} = \tilde{B}$ by the same way and studied least squares solutions of the inconsistent fuzzy matrix equation $A \tilde{x} = \tilde{B}$ by using generalized inverses. In 2013, Guo et al. [23, 24] proposed a computing method of fuzzy symmetric solutions to fuzzy matrix equations $A \tilde{X} = \tilde{B}$ and discussed the fuzzy Sylvester matrix equations $A \tilde{X} + \tilde{X} B = \tilde{C}$ based on LR fuzzy numbers. Recently, they studied the general dual fuzzy linear matrix systems $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ according to arithmetic operations of LR fuzzy numbers [22].

In this paper we propose a general model for solving the fuzzy matrix equation $\tilde{X} A = \tilde{B}$ where A is a crisp n × p matrix and $\tilde{B}$ is an m × p arbitrary fuzzy numbers matrix respectively. We extend the fuzzy matrix equation into a crisp system of linear equations. The fuzzy minimal solution of the fuzzy matrix equation is derived from solving the crisp function system. Moreover, the existence condition of the strong fuzzy minimal solution is discussed. Finally, some examples are given to illustrate our method. The structure of this paper is organized as follows:

In Section 2, we recall the fuzzy number and present the concept of the fuzzy matrix equation. In Section 3, the model to the fuzzy matrix equation is proposed in detail and the fuzzy minimal solution of the fuzzy linear matrix equation is obtained. Some examples are given to illustrate our method in Section 4 and the conclusion is drawn in Section 5.

2 Preliminaries

2.1 The fuzzy number

There are several definitions for the concept of fuzzy numbers (see [17 , 33]).

Definition 2.1. A fuzzy number is a fuzzy set like $\tilde{u} : R \to I = [0, 1]$ which satisfies:

$\tilde{u}$ is upper semicontinuous,

$\tilde{u}$ is fuzzy convex, i.e., $\tilde{u} (λ x + (1 - λ) y) \geq min {\tilde{u} (x), \tilde{u} (y)}$ for all x, y ∈ R, λ ∈ [0, 1],

$\tilde{u}$ is normal, i.e., there exists x₀ ∈ R such that $\tilde{u} (x_{0}) = 1$ ,

supp $\tilde{u} = {x \in R ∣ \tilde{u} (x) > 0}$ is the support of the $\tilde{u}$ , and its closure cl(supp $\tilde{u}$ ) is compact.

Let E¹ be the set of all fuzzy numbers on R.

Definition 2.2. A fuzzy number $\tilde{u}$ in parametric form is a pair $(\underline{u}, \bar{u})$ of functions $\underline{u} (r)$ , $\bar{u} (r)$ , 0 ≤ r ≤ 1, which satisfies the requirements:

$\underline{u} (r)$ is a bounded monotonic increasing left continuous function,

$\bar{u} (r)$ is a bounded monotonic decreasing left continuous function,

$\underline{u} (r) \leq \bar{u} (r)$ , 0 ≤ r ≤ 1.

A crisp number x is simply represented by $(\underline{u} (r), \bar{u} (r)) = (x, x)$ , 0 ≤ r ≤ 1. By appropriate definitions the fuzzy number space ${(\underline{u} (r), \bar{u} (r))}$ becomes a convex cone E¹ which could be embedded isomorphically and isometrically into a Banach space.

Definition 2.3. Let $\tilde{x} = (\underline{x} (r), \bar{x} (r))$ , $\tilde{y} = (\underline{y} (r), \bar{y} (r)) \in E^{1}$ , 0 ≤ r ≤ 1 and k ∈ R. Then

$\tilde{x} = \tilde{y}$ iff $\underline{x} (r) = \underline{y} (r)$ and $\bar{x} (r) = \bar{y} (r)$ ,

$\tilde{x} + \tilde{y} = (\underline{x} (r) + \underline{y} (r), \bar{x} (r) + \bar{y} (r))$ ,

$\tilde{x} - \tilde{y} = (\underline{x} (r) - \bar{y} (r), \bar{x} (r) - \underline{y} (r))$ ,

$k \tilde{x} = {\begin{matrix} (k \underline{x} (r), k \bar{x} (r)), & k \geq 0, \\ (k \bar{x} (r), k \underline{x} (r)), & k < 0 . \end{matrix}$

2.2 Fuzzy matrix equations

Definition 2.6. A matrix $\tilde{A} = ({\tilde{a}}_{ij})$ is called a fuzzy matrix, if each element ${\tilde{a}}_{ij}$ of $\tilde{A}$ is a fuzzy number. Let $\tilde{A} = ({\tilde{a}}_{ij}) = ({\underline{a}}_{ij} (r), {\bar{a}}_{ij} (r)), i, j = 1, 2, \dots, n,$ the fuzzy matrix $\tilde{A}$ can be represented by $\tilde{A} = (\underline{A} (r), \bar{A} (r))$ where $\underline{A} (r) = ({\underline{a}}_{ij} (r)), \bar{A} (r) = ({\bar{a}}_{ij} (r)), i, j = 1, 2, \dots, n$ .

Definition 2.7. Let $\tilde{A} = ({\tilde{a}}_{ij})$ be a m × n fuzzy matrix and B = b_ij be a n × p crisp matrix. The size of the product of two matrices is m × p and is written as follows: $\tilde{A} B = \tilde{C} = ({\tilde{c}}_{ij})$ where ${\tilde{c}}_{ij} = \sum_{k = 1, . . ., n}^{+} {\tilde{a}}_{ik} \times b_{kj}$ .

Definition 2.8. The matrix system

$\begin{matrix} (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{1 n} \\ {\tilde{x}}_{21} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{2 n} \\ \dots & \dots & \dots & \dots \\ {\tilde{x}}_{m 1} & {\tilde{x}}_{m 2} & \dots & {\tilde{x}}_{mn} \end{matrix}) (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 p} \\ a_{21} & a_{12} & \dots & a_{2 p} \\ \dots & \dots & \dots & \dots \\ a_{n 1} & a_{n 2} & \dots & a_{np} \end{matrix}) \\ = (\begin{matrix} {\tilde{b}}_{11} & {\tilde{b}}_{12} & \dots & {\tilde{b}}_{1 p} \\ {\tilde{b}}_{21} & {\tilde{b}}_{12} & \dots & {\tilde{b}}_{2 p} \\ \dots & \dots & \dots & \dots \\ {\tilde{b}}_{m 1} & {\tilde{b}}_{m 2} & \dots & {\tilde{b}}_{mp} \end{matrix}), \end{matrix}$ (2.1) where a_ij, 1 ≤ i ≤ n, 1 ≤ j ≤ p are crisp numbers and ${\tilde{b}}_{ij}, 1 \leq i \leq m, 1 \leq j \leq p$ are fuzzy numbers, is called a fuzzy matrix equations (FMEs).

Using matrix notation, we have $\tilde{X} A = \tilde{B} .$ (2.2)

A fuzzy numbers matrix $\begin{matrix} \tilde{X} & = & ({\tilde{x}}_{ij}) = ({\bar{x}}_{ij} (r), {\underline{x}}_{ij} (r)), \\ 1 \leq i \leq m, 1 \leq j \leq n, 0 \leq r \leq 1 \end{matrix}$ is called a solution of the fuzzy matrix Equation (2.1) if $\tilde{X}$ satisfies $\tilde{X} A = \tilde{B} .$

3 Solving fuzzy matrix equation

In this section we investigate the fuzzy linear matrix Equations (2.6). Firstly, we set up a computing model for solving FMEs. Then we define the LR fuzzy solution of FMEs and obtain its solution representation by means of generalized inverses of matrices. Finally, we give a sufficient condition for strong fuzzy approximate solution to the original fuzzy matrix equation.

Definition 3.1. A fuzzy number matrix $\tilde{A}$ in parametric form is a pair $(\underline{A}, \bar{A})$ of functions $\underline{A} (r)$ , $\bar{A} (r)$ , 0 ≤ r ≤ 1, which satisfies the requirements:

$\underline{A} (r)$ is a bounded monotonic increasing left continuous function matrix,

$\bar{A} (r)$ is a bounded monotonic decreasing left continuous function matrix,

$\underline{A} (r) \leq \bar{A} (r)$ , 0 ≤ r ≤ 1.

Definition 3.2. Let $\tilde{A} = (\underline{A} (r), \bar{A} (r))$ , $B = (\underline{B} (r), \bar{B} (r)) \in E^{m \times n}$ , 0 ≤ r ≤ 1 be two fuzzy matrices and k ∈ R. Then

$\tilde{A} = \tilde{B}$ iff $\underline{A} (r) = \underline{B} (r)$ and $\bar{A} (r) = \bar{B} (r)$ ,

$\tilde{A} + \tilde{B} = (\underline{A} (r) + \underline{B} (r), \bar{A} (r) + \bar{B} (r))$ ,

$\tilde{A} - \tilde{B} = (\underline{A} (r) - \bar{B} (r), \bar{A} (r) - \underline{B} (r))$ ,

$k \tilde{A} = {\begin{matrix} (k \underline{A} (r), k \bar{A} (r)), & k \geq 0, \\ (k \bar{A} (r), k \underline{A} (r)), & k < 0 . \end{matrix}$

Theorem 3.1. The fuzzy matrix equation (2.2) can be extended to a crisp function linear matrix system as follows $X (r) S = B (r),$ (3.1) where $\begin{matrix} \tilde{X} & = & [\underline{X} (r), - \bar{X} (r)], S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), \\ B (r) = [\underline{B} (r), - \bar{B} (r)], \end{matrix}$ in which the elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined by the following way:

if $a_{ij} \geq 0, a_{ij}^{+} = a_{ij}$ else $a_{ij}^{+} = 0, 1 \leq i \leq n, 1 \leq j \leq m$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i \leq n, 1 \leq j \leq m$ .

Proof. We denote the right fuzzy matrix $\tilde{B}$ with $\tilde{B} = [\underline{B} (r), \bar{B} (r)] = ([{\underline{b}}_{ij} (r), {\bar{b}}_{ij} (r)])_{m \times m}, 0 \leq r \leq 1$ and the unknown fuzzy matrix denoted by $\tilde{X} = [\underline{X} (r), \bar{X} (r)] = ([{\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r)])_{m \times n}$ . We also suppose A = A⁺ + A^- in which the elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined by the following way:

if $a_{ij} \geq 0, a_{ij}^{+} = a_{ij}$ else $a_{ij}^{+} = 0, 1 \leq i \leq n, 1 \leq j \leq m$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i \leq n, 1 \leq j \leq m$ .

For fuzzy matrix equation $\tilde{X} A = \tilde{B}$ , we can express it as $[\underline{X} (r), \bar{X} (r)] (A^{+} + A^{-}) = [\underline{B} (r), \bar{B} (r)] .$ (3.2)

Since ${\tilde{x}}_{ij} k = {\begin{matrix} (k {\underline{x}}_{ij} (r), k {\bar{x}}_{ij} (r)), & k \geq 0, \\ (k {\bar{x}}_{ij} (r), k {\underline{x}}_{ij} (r)), & k < 0 \end{matrix},$ we have $\tilde{X} A = {\begin{matrix} (\underline{X} (r) A, \bar{X} (r)) A, & A \geq 0, \\ (\bar{X} (r) A, \underline{X} (r)) A, & A < 0 . \end{matrix}$ So the Equations (3.2) can be rewritten as

$\begin{matrix} [\underline{X} (r), \bar{X} (r)] A^{+} + [\underline{X} (r), \bar{X} (r)] A^{-} \\ = [\underline{X} (r) A^{+}, \bar{X} (r) A^{+}] + [\bar{X} (r) A^{-}, \underline{X} (r) A^{-}] \\ [\underline{X} (r) A^{+} + \bar{X} (r) A^{-}, \bar{X} (r) A^{+} + \underline{X} (r) A^{-}] \\ = [\underline{B} (r), \bar{B} (r)] . \end{matrix}$ Thus we get ${\begin{matrix} \underline{X} (r) A^{+} + \bar{X} (r) A^{-} = \underline{B} (r), \\ \bar{X} (r) A^{+} + \underline{X} (r) A^{-} = \bar{B} (r) . \end{matrix}$ (3.3)

Denoting in matrix form, we have $[\underline{X} (r), - \bar{X} (r)] (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) = [\underline{B} (r), - \bar{B} (r)] . □$

By the approach of Freidman [18], the following results are apparent.

Theorem 3.2. The matrix S is nonsingular if and only if the matrices A⁺ + A^- and A⁺ - A^- are both nonsingular. If S is nonsingular, then

$S^{- 1} = \frac{1}{2} (\begin{matrix} (A^{+} - A^{-})^{- 1} + (A^{+} + A^{-})^{- 1} & (A^{+} - A^{-})^{- 1} - (A^{+} + A^{-})^{- 1} \\ (A^{+} - A^{-})^{- 1} - (A^{+} + A^{-})^{- 1} & (A^{+} - A^{-})^{- 1} + (A^{+} + A^{-})^{- 1} \end{matrix})$ (3.4)

where (A⁺ + A^-) ^-1, (A⁺ - A^-) ^-1 are inverse matrices of matrices A⁺ + A^- and A⁺ - A^-, respectively. In this case, the solution of model Equation (3.1) is $[\underline{X} (r), - \bar{X} (r)] = [\underline{B} (r), - \bar{B} (r)] {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{- 1} .$ (3.5)

Theorem 3.3. Let S belong to R^n×p and C belong to R^m×p. Then the minimal solution X^* of the matrix equation XS = C is expressed by $X^{*} = {CS}^{†} .$

Proof. We write the matrix equation XS = C in block form of matrix as follows: $X_{i} S = C_{i}, i = 1, 2, \dots, m,$ where X_i, B_i are the ith row of matrices X, S, respectively.

By the matrix theory, we know that the matrix equation XS = C is consistent if and only if each linear equation xS = C_i, i = 1, 2, …, m is consistent, and the matrix equation XS = C is inconsistent if and only if that at least one of linear equations xS = C_i, i = 1, 2, …, m is inconsistent.

When XS = C is consistent, the result that X = CS^† is its minimal solution is straightforward.

When XS = C is inconsistent, the expression $∥ C - X^{*} S ∥_{F}^{2} = min ∥ C - XS ∥_{F}^{2}$ holds is equivalent to the following conditions: $∥ C_{i} - X_{i}^{*} S ∥_{F}^{2} = min ∥ C_{i} - X_{i}^{*} S ∥_{F}^{2}, i = 1, \dots, m,$ where $X^{*} = (\begin{matrix} X_{1}^{*} \\ \dots \\ X_{m}^{*} \end{matrix})$ is the least squares solution of the matrix equation XS = C and $X_{i}^{*}$ is the least squares solution of linear equation xS = C_i, i = 1, 2, …, m and $\begin{matrix} ∥ C - XS ∥_{F}^{2} & = & \sum_{i = 1}^{m} ∥ C_{i} - X_{i} S ∥_{F}^{2} \\ = & \sum_{i = 1}^{m} \sum_{j = 1}^{p} ∥ c_{ij} - \sum_{k = 1}^{n} x_{ik} s_{kj} ∥_{F}^{2} . \end{matrix}$

From above analysis, we know that $X_{i}^{*}$ is the minimal solution of inconsistent linear equation xS = C_i, i = 1, 2, …, m is equivalent to that $X^{*} = (\begin{matrix} X_{1}^{*} \\ \dots \\ X_{m}^{*} \end{matrix})$ is the minimal solution of the inconsistent matrix equation XS = C. So we know that the minimal solution of the matrix equation XS = C is expressed uniformly by $X^{*} = {CS}^{†} . □$

In order to solve the fuzzy matrix Equation (2.2), we need to consider the systems of linear Equations (3.1). It seems that we have obtained the minimal solution of the function linear system (3.1) as $X (r) = B (r) S^{†},$ i.e. $[\underline{X} (r), - \bar{X} (r)] = [\underline{B} (r), - \bar{B} (r)] {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†},$ (3.6) where S^† is the Moore-Penrose generalized inverse [11] of matrix S.

However, the solution matrix may still not be an appropriate fuzzy numbers matrix. Restricting the discussion to triangular fuzzy numbers, i.e., ${\underline{b}}_{ij} (r), {\bar{b}}_{ij} (r), 1 \leq i, j \leq m$ and consequently ${\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), 1 \leq i \leq m, 1 \leq j \leq n$ are all linear functions of r, and having calculated X (r) which solves (3.6), we define the fuzzy minimal solution to the fuzzy linear systems (2.2) as follows.

Definition 3.3. Let $X (r) = ({\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r)), 1 \leq i \leq n, 1 \leq j \leq m$ be the minimal solution of (3.6). The fuzzy number matrix $U = {({\underline{u}}_{ij} (r), {\bar{u}}_{ij} (r)), 1 \leq i \leq m, 1 \leq j \leq n$ defined by ${\begin{matrix} {\underline{u}}_{ij} (r) = min {{\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), {\underline{x}}_{ij} (1), {\bar{x}}_{ij} (1)}, \\ {\bar{u}}_{ij} (r) = max {{\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), {\underline{x}}_{ij} (1), {\bar{x}}_{ij} (1)}, \\ 1 \leq i \leq m, 1 \leq j \leq n . \end{matrix}$ (3.7) is called the fuzzy minimal solution of the function matrix Equation (3.1). If ${\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), 1 \leq i \leq m, 1 \leq j \leq n$ are all fuzzy numbers then ${\underline{u}}_{ij} (r) = {\underline{x}}_{ij} (r), {\bar{u}}_{ij} (r) = {\bar{x}}_{ij} (r), 1 \leq i \leq m, 1 \leq j \leq n$ and U is called a strong fuzzy minimal solution of the fuzzy linear systems (2.2). Otherwise, U is called a weak fuzzy minimal solution.

Lemma 3.1. [11] Let $S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) .$ Then the matrix

$S^{†} = \frac{1}{2} (\begin{matrix} (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} \\ (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} \end{matrix})$ 3.8 is the Moore-Penrose inverse of the matrix S, where(A⁺ + A^-) ^†, (A⁺ - A^-) ^† are Moore-Penrose inverses of matrices A⁺ + A^- and A⁺ - A^-, respectively.

The key points to make the solution matrix being a strong fuzzy solution is that B (r) S^† is fuzzy matrix, i.e., each element in which is a triangular fuzzy number. By the following analysis, we know that it is equivalent to the condition S^† ≥ O.

Theorem 3.4. If $\begin{matrix} (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†}) \\ \geq O, (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†}) \geq O, \end{matrix}$ the fuzzy matrix equation (2.2) has a strong fuzzy minimal solution as follows: $\tilde{X} = [\underline{X} (r), \bar{X} (r)],$ where ${\begin{matrix} \underline{X} (r) = \underline{B} (r) E - \bar{B} (r) F, \\ \bar{X} (r) = - \underline{B} (r) F + \bar{B} (r) E, \\ E = \frac{1}{2} ((A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†}), \\ F = \frac{1}{2} ((A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†}) . \end{matrix}$ (3.9)

Proof. Let $S^{†} = (\begin{matrix} E & F \\ F & E \end{matrix}) = \frac{1}{2} (\begin{matrix} (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} \\ (A^{+} - A^{-})^{†} - (A^{+} + A^{-})^{†} & (A^{+} - A^{-})^{†} + (A^{+} + A^{-})^{†} \end{matrix}) .$ We know the condition that S⁺ ≥ 0 is equivalent to E ≥ O, F ≥ O.

Since $\tilde{B} = [\underline{B} (r), \bar{B} (r)]$ , $\underline{B} (r)$ is a bounded monotonic increasing left continuous function matrix and $\bar{B} (r)$ is a bounded monotonic decreasing left continuous function matrix with $\underline{B} (r) \leq \bar{B} (r), 0 \leq r \leq 1$ by Definition 3.1.

According to Equations (3.6), we have $\begin{matrix} [\underline{X} (r), - \bar{X} (r)] & = & [\underline{B} (r), - \bar{B} (r)] {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†} \\ = & [\underline{B} (r), - \bar{B} (r)] (\begin{matrix} E & F \\ F & E \end{matrix}), \end{matrix}$

i.e., $\underline{X} (r) = \underline{B} (r) E - \bar{B} (r) F, - \bar{X} (r) = \underline{B} (r) F - \bar{B} (r) E .$

Now that E ≥ 0, F ≥ 0 and $\underline{B} (r), - \bar{B} (r)$ are bounded monotonic increasing left continuous function matrices, we know that $\underline{X} (r)$ is a bounded monotonic increasing left continuous function matrix and $\bar{X} (r)$ is a bounded monotonic decreasing left continuous function matrix. And

$\begin{matrix} \bar{X} (r) - \underline{X} (r) = (\bar{B} (r) - \underline{B} (r)) E + (\bar{B} (r) - \underline{B} (r)) F \\ = (\bar{B} (r) - \underline{B} (r)) (E + F) = (\bar{B} (r) - \underline{B} (r)) (A^{+} - A^{-})^{†} \geq O . \end{matrix}$

Thus the fuzzy matrix equation (2.2) has a strong fuzzy minimal solution. □

The following Theorems give some results for such S^-1 and S^† to be nonnegative. As usual, (.) ^⊤ denotes the transpose of a matrix (.).

Theorem 3.5. [29] The inverse S^-1 of a nonnegative matrix S is nonnegative if and only if S is a generalized permutation matrix.

Theorem 3.6. [12] Let S be an 2n × 2m nonnegative matrix with rank r. Then the following assertions are equivalent:

S^† ≥ 0 .

There exists a permutation matrix P, such that PS has the form

$PS = (\begin{matrix} T_{1} \\ T_{2} \\ ⋮ \\ T_{r} \\ O \end{matrix}),$ where each T_i has rank 1 and the rows of T_i are orthogonal to the rows of T_j, whenever i ≠ j, the zero matrix may be absent.

$S^{†} = (\begin{matrix} {GC}^{⊤} & {GD}^{⊤} \\ {GD}^{⊤} & {GC}^{⊤} \end{matrix})$ for some positive diagonal matrix G. In this case, $\begin{matrix} (C + D)^{†} & = & G (C + D)^{⊤}, (C - D)^{†} \\ = & G (C - D)^{⊤} . \end{matrix}$

4 Numerical examples

Example 4.1. Consider the following fuzzy matrix system $\begin{matrix} (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \end{matrix}) (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}) \\ = (\begin{matrix} (r, 2 - r) & (1 + r, 3 - r) \\ (0, 1 - r) & (2 r, 2 - r) \end{matrix}) . \end{matrix}$

Let $\begin{matrix} \tilde{X} & = & [\underline{X} (r), \bar{X} (r)] \\ = & [(\begin{matrix} {\underline{x}}_{11} (r) & {\underline{x}}_{12} (r) \\ {\underline{x}}_{21} (r) & {\underline{x}}_{22} (r) \end{matrix}), (\begin{matrix} {\bar{x}}_{11} (r) & {\bar{x}}_{12} (r) \\ {\bar{x}}_{21} (r) & {\bar{x}}_{22} (r) \end{matrix})], \end{matrix}$ $A = A^{+} + A^{-} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) + (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ and $\tilde{B} = [\underline{B} (r), \bar{B} (r)] = [(\begin{matrix} r & 1 + r \\ 0 & 2 r \end{matrix}), (\begin{matrix} 2 - r & 3 - r \\ 1 - r & 4 - r \end{matrix})] .$

By the Theorem 3.1, the original fuzzy matrix equation is equivalent to the following linear system $X (r) S = B (r),$ where $\begin{matrix} \tilde{X} & = & [\underline{X} (r), - \bar{X} (r)], S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), \\ B (r) = [\underline{B} (r), - \bar{B} (r)], \end{matrix}$ in which $S = (\begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}) .$

The matrix S is obviously not invertible, its Moore-Penrose inverse is $S^{†} = \frac{1}{4} (\begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}) \geq O .$

From Equations (3.5), the minimal solution of model is as follows:

$\begin{matrix} [\underline{X} (r), - \bar{X} (r)] = [\underline{B} (r), - \bar{B} (r)] {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†} \\ = \frac{1}{4} [(\begin{matrix} - 3 + 2 r & - 1 + 2 r \\ - 4 + r & - 1 + 3 r \end{matrix}), (\begin{matrix} - 1 + 2 r & - 3 + 2 r \\ - 1 + 3 r & - 4 + r \end{matrix})] \end{matrix}$

i.e., $\begin{matrix} (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \end{matrix}) \\ = \frac{1}{4} (\begin{matrix} (- 3 + 2 r, 1 - 2 r) & (- 1 + 2 r, 3 - 2 r) \\ (- 4 + r, 1 - 3 r) & (- 1 + 3 r, 4 - r) \end{matrix}) . \end{matrix}$

Since $\tilde{X}$ is an appropriate triangular fuzzy numbers matrix, we obtained the solution of the fuzzy matrix system is $\begin{matrix} \tilde{U} = (\begin{matrix} {\tilde{u}}_{11} & {\tilde{u}}_{12} \\ {\tilde{u}}_{21} & {\tilde{u}}_{22} \end{matrix}) \\ = \frac{1}{4} (\begin{matrix} (- 3 + 2 r, 1 - 2 r) & (- 1 + 2 r, 3 - 2 r) \\ (- 4 + r, 1 - 3 r) & (- 1 + 3 r, 4 - r) \end{matrix}), \end{matrix}$ it admits a strong fuzzy solution matrix by Definition 3.3.

Example 4.2. Consider the following fuzzy matrix system $\begin{matrix} (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \end{matrix}) (\begin{matrix} 1 & 1 & 2 \\ 1 & - 1 & 1 \end{matrix}) \\ = (\begin{matrix} (r, 2 - r) & (- 1 + r, 1 - r) & (1 + r, 3 - r) \\ (0, 1 - r) & (1 + r, 1 - r) & (2 r, 4 - r) \end{matrix}) . \end{matrix}$

Let $\begin{matrix} \tilde{X} & = & [\underline{X} (r), \bar{X} (r)] \\ = & [(\begin{matrix} {\underline{x}}_{11} (r) & {\underline{x}}_{12} (r) \\ {\underline{x}}_{21} (r) & {\underline{x}}_{22} (r) \end{matrix}), (\begin{matrix} {\bar{x}}_{11} (r) & {\bar{x}}_{12} (r) \\ {\bar{x}}_{21} (r) & {\bar{x}}_{22} (r) \end{matrix})], \\ A = A^{+} + A^{-} = (\begin{matrix} 1 & 1 & 2 \\ 1 & 0 & 1 \end{matrix}) + (\begin{matrix} 0 & 0 & 0 \\ 0 & - 1 & 0 \end{matrix}) \end{matrix}$

and $\begin{matrix} \tilde{B} & = & [\underline{B} (r), \bar{B} (r)] \\ = & [(\begin{matrix} r & - 1 + r & 1 + r \\ 0 & 1 + r & 2 r \end{matrix}), (\begin{matrix} 2 - r & 1 - r & 3 - r \\ 1 - r & 1 - r & 4 - r \end{matrix})] . \end{matrix}$

By the Theorem 3.1., the original fuzzy matrix equation is equivalent to the following linear system $X (r) S = B (r),$ where $\begin{matrix} \tilde{X} & = & [\underline{X} (r), - \bar{X} (r)], S = (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), \\ B (r) = [\underline{B} (r), - \bar{B} (r)] . \end{matrix}$

From Equations (3.5), the minimal solution of model is as follows:

$\begin{matrix} [\underline{X} (r), - \bar{X} (r)] = [\underline{B} (r), - \bar{B} (r)] {(\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix})}^{†} \\ = [(\begin{matrix} r & - 1 + r & 1 + r \\ 0 & 1 + r & 2 r \end{matrix}), (\begin{matrix} - 2 + r & - 1 + r & - 3 + r \\ - 1 + r & - 1 + r & - 4 + r \end{matrix})] {(\begin{matrix} 1 & 1 & 2 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 2 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{matrix})}^{†} \\ = [(\begin{matrix} 0.6430 & - 0.4286 + r \\ - 0.7855 + 0.8572 r & 1.3571 - 0.0715 r \end{matrix}), (\begin{matrix} - 0.6430 & - 1.5714 + r \\ - 2.7145 + 0.6428 r & 1.6429 + 0.0715 r \end{matrix})] \end{matrix}$ i.e., $\begin{matrix} (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \end{matrix}) \\ = (\begin{matrix} (0.6430, 0.6430) & (- 0.4286 + r, 1.5714 - r) \\ (- 0.7855 + 0.8572 r, 2.7145 - 0.6428 r) & (1.3571 - 0.0715 r, - 1.6429 - 0.0715 r) \end{matrix}) . \end{matrix}$

Since ${\tilde{x}}_{22} = (1.3571 - 0.0715 r, - 1.6429 - 0.0715 r)$ is not an appropriate triangular fuzzy number, the solution we obtained is a weak fuzzy solution matrix of the fuzzy matrix system as $\begin{matrix} \tilde{U} = (\begin{matrix} {\tilde{u}}_{11} & {\tilde{u}}_{12} \\ {\tilde{u}}_{21} & {\tilde{u}}_{22} \end{matrix}) \\ = (\begin{matrix} (0.6430, 0.6430) & (- 0.4286 + r, 1.5714 - r) \\ (- 0.7855 + 0.8572 r, 2.7145 - 0.6428 r) & (- 1.7180, 1.2756) \end{matrix}) . \end{matrix}$

5 Conclusion

In this work we presented a model for solving fuzzy matrix equations $\tilde{X} A = \tilde{B}$ where A is a crisp n × p matrix and $\tilde{B}$ is an m × p arbitrary fuzzy numbers matrix. The fuzzy matrix equation was converted to a linear function system. The fuzzy minimal solution of the fuzzy matrix equation is derived from solving the crisp function system. Moreover, the existence condition of the strong fuzzy minimal solution is discussed. Numerical examples showed that our method is feasible to solve this type of fuzzy matrix equations. The proposed method can be applied to some more complex fuzzy matrix equations.

Footnotes

Acknowledgement

The work is supported by the Natural Scientific Funds of PR China (No. 61262022).

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