Further investigation to dual fuzzy matrix equation 1

Abstract

The paper considers the general dual fuzzy matrix equation $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ in a complete matrix method. We firstly introduce the fuzzy matrix and its operation with crisp number. Then we convert the dual fuzzy matrix equation to a model which is a crisp function matrix equation. The fuzzy approximate solution of dual fuzzy matrix equation is obtained by solving the model. The existence condition of the strong fuzzy solution is also discussed. In addition, we investigate the LR dual fuzzy matrix equation by the same way. Finally, some examples are given to illustrate our proposed method.

Keywords

Fuzzy numbers fuzzy numbers matrix dual fuzzy matrix equations fuzzy approximate solutions

1 Introduction

In the mathematical modelling of physics, engineering computation and statistical analysis, it is the linear systems that has mature theory and easy computational property. However, the uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented by fuzzy numbers. So the investigation of theory and computing method for fuzzy linear systems, plays an important role in the fuzzy mathematics and its applications. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [37], Dubois et al. [18] and Nahmias [31]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [33], Goetschell et al. [21] and Wu Congxin et al. [35, 36].

Since Friedman et al. [19] proposed a general model for solving an n × n fuzzy linear systems $A \tilde{x} = \tilde{b}$ by an embedding approach in 1998, many a work has been done about some specific fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), dual full fuzzy systems linear systems (DFFLS) and general dual fuzzy linear systems (GDFLS) see [1–6 , 38]. Recent years new approaches and theory for fuzzy linear systems emerge in endlessly [7 , 22]. Considering research of the dual fuzzy linear system, we need to go back to the past decades. In 2000, Ma et al. [27] firstly discussion a class of dual fuzzy linear system by using an embedding approach. They remarked that the system A₁x = A₂x + b is not equivalent to the system (A₁ - A₂) x = b, since for an arbitrary fuzzy number $\tilde{u}$ there does not exist an element $\tilde{v}$ such that $\tilde{u} + \tilde{v} = \tilde{0}$ . Later, Wang et al. [34] presented an iterative algorithm for solving dual linear system of the form X = AX + U, where A is real n × n matrix, the unknown vector X and the constant U are vectors consisting of fuzzy numbers. Also, Muzziloi et al. [30] considered fuzzy linear systems of the form A₁x + b₁ = A₂x + b₂ with A₁, A₂ square matrices of fuzzy coefficients and b₁, b₂ fuzzy number vectors. In 2008, Abbasbandy et al. [2] proposed a numerical method for finding the minimal solution of the m × n general dual fuzzy linear system $A \tilde{x} + \tilde{F} = B \tilde{x} + \tilde{C}$ based on pseudo-inverse calculation.

Although matrix equations which are accompany with fuzzy numbers have wide use in control theory and control engineering, few work has been done in the past. In 2009, Allahviranloo et al. [10] firstly discussed the fuzzy matrix equations (FLME) of the form $A \tilde{X} B = \tilde{C}$ . By means of the parametric form of the fuzzy numbers, they derived a necessary and sufficient conditions for the existence condition of fuzzy solutions and designed a numerical procedure for calculating fuzzy solutions. In 2011, Gong et al. [23] studied least squares solutions of the inconsistent fuzzy matrix equation $A \tilde{X} = \tilde{B}$ by the same way. In 2012, Guo et al. [25] proposed a computing method of fuzzy symmetric solutions to the fuzzy matrix equations $\tilde{X} A = \tilde{B}$ .

Meanwhile, once the uncertain elements of fuzzy systems were denoted by the parametric form of fuzzy numbers. Thus it may lead to two defects. One is that the extended linear equations always contains parameter r, 0 ≤ r ≤ 1, which makes their computation inconvenient. The other is that sometimes the weak fuzzy solution of fuzzy linear systems did not exist [8]. To overcome above two defects and handle the full fuzzy linear systems (FFLS), D. Dubois and H. Prade [18] introduced the LR fuzzy number. In 2006, Dehgham et al. [15] discussed computational methods for fully fuzzy linear systems $\tilde{A} \tilde{x} = \tilde{b}$ whose coefficient matrix and the right-hand side vector are LR fuzzy numbers. In the past ten years, LR fuzzy linear systems have been paid more attention by some researchers. In 2013, Guo et al. [26] proposed a computing method for the fuzzy Sylvester matrix equations $A \tilde{X} + \tilde{X} B = \tilde{C}$ with LR fuzzy numbers. In 2012, M. Otadi et al. [28] proposed a method for solving fully fuzzy matrix equations $\tilde{A} \otimes \tilde{X} = \tilde{B}$ . In 2014, Gong and Guo et al. [24] studied the general LR dual fuzzy linear matrix systems $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ . Later, K. Dookhitram et al. [17] discussed fully fuzzy Sylvester matrix equations in 2015.

In this paper, we investigate the general dual fuzzy matrix equation $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ in a complete matrix method. At first, we introduce the fuzzy matrix and its operation with crisp number. Then we convert the dual fuzzy matrix equation to a model which is a crisp function matrix equation. The fuzzy approximate solution of dual fuzzy matrix equation is obtained by solving the model. The existence condition of the strong fuzzy solution is also discussed. As a simple application, we study the LR dual fuzzy matrix equation by the same way. Finally, we give some illustrating examples.

2 Preliminaries

There are several definitions for the concept of fuzzy numbers (see [18 , 37]).

Definition 2.1.A fuzzy number is a fuzzy set like u : R → I = [0, 1] which satisfies:

u is upper semicontinuous,

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

u is normal, i.e., there exists x₀ ∈ R such that u (x₀) =1,

suppu = {x ∈ R ∣ u (x) >0} is the support of the u, and its closure cl(suppu) is compact.

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

Definition 2.2. A fuzzy number 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 $x = (\underline{x} (r), \bar{x} (r))$ , $y = (\underline{y} (r), \bar{y} (r)) \in E^{1}$ , 0 ≤ r ≤ 1 and k ∈ R. Then

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

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

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

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

Definition 2.4.A fuzzy number $\tilde{M}$ is said to be a LR fuzzy number if $μ_{\tilde{M}} (x) = {\begin{matrix} L (\frac{m - x}{α}), & x \leq m, & α > 0, \\ R (\frac{x - m}{β}), & x \geq m, & β > 0, \end{matrix}$ where m, α and β are called the mean value, left and right spreads of $\tilde{M}$ , respectively. The function L (·), which is called left shape function satisfies:

L (x) = L (- x),

L (0) =1 and L (1) =0,

L (x) is non increasing on [0, ∞).

The definition of a right shape function R (·) is similar to that of L (·).

Clearly, two LR fuzzy numbers $\tilde{M} = (m, α, β)_{LR}$ and $\tilde{N} = (n, γ, δ)_{LR}$ are said to be equal, if and only if m = n, α = γ and β = δ. Also, $\tilde{M} = (m, α, β)_{LR}$ is positive (negative) if and only if m - α > 0(m + β < 0).

Definition 2.5. For arbitrary LR fuzzy numbers $\tilde{M} = (m, α, β)_{LR}$ and $\tilde{N} = (n, γ, δ)_{LR}$ , we have

Addition $\begin{matrix} \tilde{M} \oplus \tilde{N} & = & (m, α, β)_{LR} \oplus (n, γ, δ)_{LR} \\ = & (m + n, α + γ, β + δ)_{LR} . \end{matrix}$

Subtraction $\begin{matrix} \tilde{M} - \tilde{N} & = & (m, α, β)_{LR} - (n, γ, δ)_{LR} \\ = & (m - n, α - δ, β - γ)_{LR} . \end{matrix}$

Scalar multiplication $\begin{matrix} λ \otimes \tilde{M} & = & λ \otimes (m, α, β)_{LR} \\ ≅ & {\begin{matrix} (λ m, λ α, λ β)_{LR}, λ \geq 0, \\ (λ m, - λ β, - λ α)_{RL}, λ < 0 . \end{matrix} \end{matrix}$

2.1 The dual fuzzy matrix equation

Definition 2.6. The linear matrix system

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

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

A fuzzy numbers matrix $\tilde{X} = ({\tilde{x}}_{ij}) = ({\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r)), 1 \leq i \leq n, 1 \leq j \leq p, 0 \leq r \leq 1$ is said to the solution of the general dual fuzzy matrix equation (2.1) if $\tilde{X}$ satisfies the Equation (2.2), i.e., $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ .

3 Solving dual fuzzy matrix equations

In this section we investigate the dual fuzzy matrix equation (2.2). Firstly, we propose a model for solving the dual fuzzy matrix equation, i.e., convert it to a crisp function matrix equation. Then we define the fuzzy solution and give its solution representation to the dual fuzzy matrix equation. The existence condition of the strong fuzzy solution to the dual fuzzy matrix equation is also discussed. At last, the LR dual fuzzy matrix equation is treated in similar way.

3.1 The model

Definition 3.1. 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 $\underline{A} (r) = ({\underline{a}}_{ij} (r)), \bar{A} (r) = ({\bar{a}}_{ij} (r)), i, j = 1, 2, \dots, n$ .

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

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

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

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

By using arithmetic operations of fuzzy numbers matrix with parametric form, we extend the dual fuzzy matrix equation (2.2) into a crisp function matrix equation.

Theorem 3.1. The dual fuzzy matrix equation (2.2) can be extended into a crisp function matrix equation as follows:

(S - T) X (r) = D (r) - B (r),

(3.1)

where

$\begin{matrix} S & = & (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}), T = (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}), \\ X (r) & = & (\begin{matrix} \underline{X} (r) \\ - \bar{X} (r) \end{matrix}), B (r) = (\begin{matrix} \underline{B} (r) \\ - \bar{B} (r) \end{matrix}), \\ D (r) & = & (\begin{matrix} \underline{D} (r) \\ - \bar{D} (r) \end{matrix}), \end{matrix}$ (3.2)

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 $a_{ij}^{+} = 0, 1 \leq i, j \leq n$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i, j \leq n$ . The elements $c_{ij}^{+}$ of matrix C⁺ and $c_{ij}^{-}$ of matrix C^- are determined in the same way.

Proof. Let the fuzzy matrix $\tilde{B}$ with $\tilde{B} = [\underline{B} (r), \bar{B} (r)]$ and $\tilde{D} = [\underline{D} (r), \bar{D} (r)]$ . We suppose the unknown fuzzy matrix denoted by $\tilde{X} = [\underline{X} (r), \bar{X} (r)]$ . We also suppose A = A⁺ + A^- and C = C⁺ + C^- 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 $a_{ij}^{-} = 0, 1 \leq i \leq n, 1 \leq j \leq m$ . And the elements $c_{ij}^{+}$ of matrix C⁺ and $c_{ij}^{-}$ of matrix C^- are determined in a same way.

For fuzzy matrix equation $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ , we can express it as

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

Since $k {\tilde{x}}_{ij} = {\begin{matrix} (k {\underline{x}}_{ij} (r), k {\bar{x}}_{ij} (r)), & k \geq 0, \\ (k {\bar{x}}_{ij} (r), k {\underline{x}}_{ij} (r)), & k < 0 \end{matrix},$ we have $A \tilde{X} = {\begin{matrix} (A \underline{X} (r), A \bar{X} (r)), & A \geq O, \\ (A \bar{X} (r), A \underline{X} (r)), & A < O . \end{matrix}$

So the Equation (3.3) be rewritten is $\begin{matrix} A^{+} [\underline{X} (r), \bar{X} (r)] + A^{-} [\underline{X} (r), \bar{X} (r)] + [\underline{B} (r), \bar{B} (r)] \\ = C^{+} [\underline{X} (r), \bar{X} (r)] + C^{-} [\underline{X} (r), \bar{X} (r)] + [\underline{D} (r), \bar{D} (r)], \end{matrix}$ i.e., $\begin{matrix} [A^{+} \underline{X} (r), A^{+} \bar{X} (r)] + [A^{-} \bar{X} (r), A^{-} \underline{X} (r)] \\ + [\underline{B} (r), \bar{B} (r)] = [C^{+} \underline{X} (r), C^{+} \bar{X} (r)] \\ + [C^{-} \bar{X} (r), C^{-} \underline{X} (r)] + [\underline{D} (r), \bar{D} (r)] . \end{matrix}$

Thus we get ${\begin{matrix} A^{+} \underline{X} (r) + A^{-} \bar{X} (r) + \underline{B} (r) \\ = C^{+} \underline{X} (r) + C^{-} \bar{X} (r) + \underline{D} (r), \\ A^{+} \bar{X} (r) + A^{-} \underline{X} (r) + \bar{B} (r) \\ = C^{+} \bar{X} (r) + C^{-} \underline{X} (r) + \bar{D} (r) . \end{matrix}$ (3.4) or ${\begin{matrix} A^{+} \underline{X} (r) - A^{-} (- \bar{X} (r)) + \underline{B} (r) \\ = C^{+} \underline{X} (r) - C^{-} (- \bar{X} (r)) + \underline{D} (r), \\ A^{+} (- \bar{X} (r)) - A^{-} \underline{X} (r) - \bar{B} (r) \\ = C^{+} (- \bar{X} (r)) - C^{-} \underline{X} (r) - \bar{D} (r) . \end{matrix}$

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

3.2 Solving the model

In order to solve the dual fuzzy matrix equation (2.2), we need to consider crisp function matrix equations (3.1). Since Equation (3.1) is matrix equation, its solving is relatively easy. For instance, when matrix G = S - T is invertible, the solutions of the Equation (3.5) is expressed by $X (r) = (S - T)^{- 1} (D (r) - B (r)),$ (3.5) i.e.,

$\begin{matrix} (\begin{matrix} \underline{X} (r) \\ - \bar{X} (r) \end{matrix}) \\ = {((\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) - (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}))}^{- 1} \\ ((\begin{matrix} \underline{D} (r) \\ - \bar{D} (r) \end{matrix}) - (\begin{matrix} \underline{B} (r) \\ - \bar{B} (r) \end{matrix})) \\ = {(\begin{matrix} A^{+} - C^{+} & - A^{-} + C^{-} \\ - A^{-} + C^{-} & A^{+} - C^{+} \end{matrix})}^{- 1} \\ (\begin{matrix} \underline{D} (r) - \underline{B} (r) \\ - \bar{D} (r) + \bar{B} (r) \end{matrix}) . \end{matrix}$ (3.6)

Theorem 3.2.[24] Let G belong to R^m×n and H belong to R^n×p. Then the minimal solution X^* of the matrix equation GX = H is expressed by $X^{*} = G^{†} H .$ where G^† is the Moore-Penrose generalized inverse of matrix G.

From Theorem 3.2, the minimal solutions of the crisp function matrix equations (3.1) is $X (r) = (S - T)^{†} (D (r) - B (r)),$ (3.7) i.e.,

$\begin{matrix} (\begin{matrix} \underline{X} (r) \\ - \bar{X} (r) \end{matrix}) & = & {(\begin{matrix} A^{+} - C^{+} & - A^{-} + C^{-} \\ - A^{-} + C^{-} & A^{+} - C^{+} \end{matrix})}^{†} \\ (\begin{matrix} \underline{D} (r) - \underline{B} (r) \\ - \bar{D} (r) + \bar{B} (r) \end{matrix}) . \end{matrix}$ (3.8)

However, the solution matrix may still not be an appropriate fuzzy numbers matrix. Restricting the discussion to triangular fuzzy numbers, i.e., $({\underline{b}}_{ij} (r), {\bar{b}}_{ij} (r)), ({\underline{d}}_{ij} (r), {\bar{d}}_{ij} (r)), 1 \leq i \leq m, 1 \leq j \leq p$ and consequently ${\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), 1 \leq i \leq n, 1 \leq j \leq p$ are all linear functions of r, and having calculated X (r) which solves (3.8), we define the fuzzy minimal solution to the dual fuzzy matrix systems (2.2) as follows.

Definition 3.3. Let $X (r) = ({\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r)), 1 \leq i \leq n, 1 \leq j \leq p$ be the minimal solution of (3.1). The fuzzy number matrix $U = {({\underline{u}}_{ij} (r), {\bar{u}}_{ij} (r)), 1 \leq i \leq n, 1 \leq j \leq p$ defined by ${\begin{matrix} {\underline{u}}_{ij} (r) = min {{\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), {\underline{x}}_{ij} (1), {\bar{x}}_{ij} (1)}, \\ {\bar{u}}_{ij} (r) = max {{\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), {\underline{x}}_{ij} (1), {\bar{x}}_{ij} (1)}, \\ 1 \leq i \leq n, 1 \leq j \leq p . \end{matrix}$ (3.9) is called the fuzzy minimal solution of the function matrix Equation (3.1). If ${\underline{x}}_{ij} (r), {\bar{x}}_{ij} (r), 1 \leq i \leq n, 1 \leq j \leq p$ 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 dual fuzzy matrix systems (2.2). Otherwise, U is called a weak fuzzy minimal solution.

3.3 A sufficient condition for strong fuzzy solution

To illustrate the expression (3.8) to be a appropriate fuzzy solution matrix, we now discuss the generalized inverses of matrix $S - T = (\begin{matrix} A^{+} - C^{+} & - A^{-} + C^{-} \\ - A^{-} + C^{-} & A^{+} - C^{+} \end{matrix})$ in a special structure.

Lemma 3.1.[14] Let $S - T = (\begin{matrix} A^{+} - C^{+} & - A^{-} + C^{-} \\ - A^{-} + C^{-} & A^{+} - C^{+} \end{matrix})$ be a non negative matrix, i.e., S - T ≥ O. Then the matrix $\begin{matrix} (S - T)^{†} & = & (\begin{matrix} E & F \\ F & E \end{matrix}) \\ = & \frac{1}{2} (\begin{matrix} (A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†} & (A - C)^{†} - (∣ A ∣ - ∣ C ∣)^{†} \\ (A - C)^{†} - (∣ A ∣ - ∣ C ∣)^{†} & (A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†} \end{matrix}), \end{matrix}$ (3.10)in which ∣A∣ is the absolute values matrix of matrix A.

The key points to make the solution matrix being a strong fuzzy solution is that (S - T) ^† (D (r) - B (r)) is fuzzy matrix, i.e., each element in which is a triangular fuzzy number. By the analysis, it is equivalent to the condition (S - T) ^† ≥ O and D (r) ≥ B (r), 0 ≤ r ≤ 1.

Theorem 3.3. If S - T is a non negative matrix, ((A - C) ^† + (∣ A ∣ - ∣ C ∣) ^†) ≥ O, ((A - C) ^† + (∣ A ∣ + ∣ C ∣) ^†) ≥ O and D (r) ≥ B (r), 0 ≤ r ≤ 1 the dual fuzzy matrix equation (2.2) has a strong fuzzy minimal solution as follows: $\tilde{X} = [\underline{X} (r), \bar{X} (r)],$ where ${\begin{matrix} \underline{X} (r) = E (\underline{D} (r) - \underline{B} (r)) - F (\bar{D} (r) - \bar{B}) (r)), \\ \bar{X} (r) = - F (\underline{D} (r) - \underline{B} (r)) + E (\bar{D} (r) - \bar{B}) (r)), \\ E = \frac{1}{2} ((A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†}), \\ F = \frac{1}{2} ((A - C)^{†} + (∣ A ∣ + ∣ C ∣)^{†}) . \end{matrix}$ (3.11)

Proof. Let $S^{†} = (\begin{matrix} E & F \\ F & E \end{matrix}) = \frac{1}{2} (\begin{matrix} (A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†} & (A - C)^{†} - (∣ A ∣ - ∣ C ∣)^{†} \\ (A - C)^{†} - (∣ A ∣ - ∣ C ∣)^{†} & (A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†} \end{matrix}) .$

We know the condition that (S - T) ⁺ ≥ 0 is equivalent to E ≥ O, F ≥ O.

Since $(\tilde{D} - \tilde{B}) = [\underline{D} (r) - \underline{B} (r), \bar{D} (r) - \bar{B} (r)]$ , $(\underline{D} (r) - \underline{B} (r))$ is a bounded monotonic increasing left continuous function matrix and $(\bar{D} (r) - \bar{B} (r))$ a bounded monotonic decreasing left continuous function matrix with $(\underline{D} (r) - \underline{B} (r)) \leq (\bar{D} (r) - \bar{B} (r)), 0 \leq r \leq 1$ by assumption in Theorem 3.3.

According to Equation (3.8), we have $\begin{matrix} (\begin{matrix} \underline{X} (r) \\ - \bar{X} (r) \end{matrix}) & = & {(\begin{matrix} A^{+} - C^{+} & - A^{-} + C^{-} \\ - A^{-} + C^{-} & A^{+} - C^{+} \end{matrix})}^{†} \\ (\begin{matrix} \underline{D} (r) - \underline{B} (r) \\ - (\bar{D} (r) - \bar{B}) (r) \end{matrix}), \end{matrix}$

i.e., $\begin{matrix} \underline{X} (r) & = & E (\underline{D} (r) - \underline{B} (r)) - F (\bar{D} (r) - \bar{B}) (r)), \\ - \bar{X} (r) & = & F (\underline{D} (r) - \underline{B} (r)) - E (\bar{D} (r) - \bar{B}) (r)) . \end{matrix}$

Now that E ≥ 0, F ≥ 0 and $(\underline{D} (r) - \underline{B} (r)), - (\bar{D} (r) - \bar{B}) (r))$ are bounded monotonic increasing left continuous function vectors, we know that $\underline{X} (r)$ is a bounded monotonic increasing left continuous function vector and $\bar{X} (r)$ is a bounded monotonic decreasing left continuous function vector. And $\begin{matrix} \bar{X} (r) - \underline{X} (r) \\ = E ((\bar{D} (r) - \bar{B} (r)) - (\underline{D} (r) - \underline{B} (r))) \\ + F ((\bar{D} (r) - \bar{B} (r)) - (\underline{D} (r) - \underline{B} (r))) \\ = (E + F) (((\bar{D} (r) - \bar{B} (r)) - (\underline{D} (r) - \underline{B} (r))) \geq O . \end{matrix}$

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

Example 3.1. Consider the dual fuzzy linear system

$\begin{matrix} (\begin{matrix} 2 & 1 \\ 1 & - 1 \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) + (\begin{matrix} 2 r, 3 - r \\ 1 + r, 3 - r \end{matrix}) \\ = (\begin{matrix} 1 & 1 \\ - 1 & 0 \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) + (\begin{matrix} 1 + 3 r, 6 - 2 r \\ 3 + 2 r, 7 - 2 r \end{matrix}) . \end{matrix}$

By the Theorem 3.1, $\begin{matrix} S - T & = & (\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 0 & - 1 & 1 \\ 0 & 0 & 1 & 0 \\ - 1 & 1 & 1 & 0 \end{matrix}), \\ \tilde{d} - \tilde{b} & = & (\begin{matrix} 1 + r, 3 - r \\ 2 + r, 4 - r \end{matrix}) . \end{matrix}$

Matrix S - T is invertible and from the Equation (3.8), we have $\begin{matrix} (\begin{matrix} \underline{x} (r) \\ - \bar{x} (r) \end{matrix}) & = & {(\begin{matrix} A^{+} - C^{+} & - A^{-} + C^{-} \\ - A^{-} + C^{-} & A^{+} - C^{+} \end{matrix})}^{- 1} (\begin{matrix} \underline{d} (r) - \underline{b} (r) \\ - \bar{d} (r) + \bar{b} (r) \end{matrix}) \\ = & (\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 0 & - 1 & 1 \\ 0 & 0 & 1 & 0 \\ - 1 & 1 & 1 & 0 \end{matrix}) (\begin{matrix} 1 + r \\ 2 + r \\ - 3 + r \\ - 4 + r \end{matrix}) = (\begin{matrix} 1 + r \\ r \\ - 3 + r \\ - 2 + r \end{matrix}) . \end{matrix}$

It means ${\tilde{x}}_{1} = (1 + r, 3 - r), {\tilde{x}}_{2} = (r, 2 - r) .$

Since it is a appropriate fuzzy numbers vector, the solution of the dual fuzzy matrix system we obtained is a strong minimal fuzzy solution given by $(\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) = (\begin{matrix} 1 + r, 3 - r \\ r, 2 - r \end{matrix}) .$

3.4 LR dual fuzzy matrix system

Now we discuss LR dual fuzzy matrix equations

$\begin{matrix} (\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{12} & \dots & a_{2 n} \\ ⃜ & ⃜ & ⃜ & ⃜ \\ a_{m 1} & a_{m 2} & \dots & a_{mn} \end{matrix}) (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{1 p} \\ {\tilde{x}}_{21} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{2 p} \\ ⃜ & ⃜ & ⃜ & ⃜ \\ {\tilde{x}}_{n 1} & {\tilde{x}}_{n 2} & \dots & {\tilde{x}}_{np} \end{matrix}) + (\begin{matrix} {\tilde{b}}_{11} & {\tilde{b}}_{12} & \dots & {\tilde{b}}_{1 p} \\ {\tilde{b}}_{21} & {\tilde{b}}_{12} & \dots & {\tilde{b}}_{2 p} \\ ⃜ & ⃜ & ⃜ & ⃜ \\ {\tilde{b}}_{m 1} & {\tilde{b}}_{m 2} & \dots & {\tilde{b}}_{mp} \end{matrix}) \\ = (\begin{matrix} c_{11} & c_{12} & \dots & c_{1 n} \\ c_{21} & c_{12} & \dots & a_{2 n} \\ ⃜ & ⃜ & ⃜ & ⃜ \\ c_{m 1} & a_{m 2} & \dots & c_{mn} \end{matrix}) (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{1 p} \\ {\tilde{x}}_{21} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{2 p} \\ ⃜ & ⃜ & ⃜ & ⃜ \\ {\tilde{x}}_{n 1} & {\tilde{x}}_{n 2} & \dots & {\tilde{x}}_{np} \end{matrix}) + (\begin{matrix} {\tilde{d}}_{11} & {\tilde{d}}_{12} & \dots & {\tilde{b}}_{1 p} \\ {\tilde{d}}_{21} & {\tilde{d}}_{12} & \dots & {\tilde{d}}_{2 p} \\ ⃜ & ⃜ & ⃜ & ⃜ \\ {\tilde{d}}_{m 1} & {\tilde{d}}_{m 2} & \dots & {\tilde{d}}_{mp} \end{matrix}), \end{matrix}$ (3.12) where a_ij, c_ij, 1 ≤ i ≤ m, 1 ≤ j ≤ n are crisp numbers and elements ${\tilde{b}}_{ij}, {\tilde{d}}_{ij}, 1 \leq i \leq m, 1 \leq j \leq p$ are LR fuzzy numbers.

Definition 3.4. A matrix $\tilde{A} = ({\tilde{a}}_{ij})$ is called a LR fuzzy matrix, if each element ${\tilde{a}}_{ij}$ of $\tilde{A}$ is a LR fuzzy number. For example, we represent m × n LR fuzzy matrix $\tilde{A} = ({\tilde{a}}_{ij})$ , that ${\tilde{a}}_{ij} = (a_{ij}, α_{ij}, β_{ij})_{LR}$ with new notation $\tilde{A} = (A, M, N)$ , where A = (a_ij), M = (α_ij) and N = (β_ij) are three m × n crisp matrices.

Definition 3.5. For two LR fuzzy numbers matrices $\tilde{A} = (A, A^{l}, A^{r})$ and $\tilde{B} = (B, B^{l}, B^{r})$ , we have

Addition $\begin{matrix} \tilde{A} + \tilde{B} & = & (A, A^{l}, A^{r}) + (B, B^{l}, B^{r}) \\ = & (A + B, A^{l} + B^{l}, A^{r} + B^{r}) . \end{matrix}$

Subtraction $\begin{matrix} \tilde{A} - \tilde{B} & = & (A, A^{l}, A^{r}) - (B, B^{l}, B^{r}) \\ = & (A + B, A^{l} - B^{r}, A^{r} - B^{l}) . \end{matrix}$

Scalar multiplication $\begin{matrix} λ \tilde{A} & = & λ (A, A^{l}, A^{r}) \\ ≅ & {\begin{matrix} (λ A, λ A^{l}, λ A^{r}), λ \geq 0, \\ (λ A, - λ A^{r}, - λ A^{l}), λ < 0 . \end{matrix} \end{matrix}$

Theorem 3.4.The LR dual fuzzy matrix system $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ can be extended into the following system of linear matrix equations ${\begin{matrix} AX + B = CX + D, \\ (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) + (\begin{matrix} B^{l} \\ B^{r} \end{matrix}) \\ = (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) + (\begin{matrix} D^{l} \\ D^{r} \end{matrix}), \end{matrix}$ (3.13) where $\tilde{X} = (X, X^{l}, X^{r}) .$ And 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 $a_{ij}^{+} = 0, 1 \leq i \leq n, 1 \leq j \leq p$ ; if $a_{ij} < 0, a_{ij}^{-} = a_{ij}$ else $a_{ij}^{-} = 0, 1 \leq i \leq n, 1 \leq j \leq p$ . The elements $c_{ij}^{+}$ of matrix C⁺ and $c_{ij}^{-}$ of matrix C^- are determined in the same way.

Proof. We denote the fuzzy matrix $\tilde{B}$ with $\tilde{B} = (B, B^{l}, B^{r})$ and $\tilde{D}$ with $\tilde{D} = (D, D^{l}, D^{r})$ and the unknown fuzzy matrix $\tilde{X}$ by $\tilde{X} = (X, X^{l}, X^{r})$ . We also suppose A = A⁺ + A^- and C = C⁺ + C^- in which the elements $c_{ij}^{+}$ of matrix C⁺ and $c_{ij}^{-}$ of matrix C⁺ are determined by the following way:

if $c_{ij} \geq 0, c_{ij}^{+} = c_{ij}$ else $c_{ij}^{+} = 0, 1 \leq i \leq m, 1 \leq j \leq n$ ; if $c_{ij} < 0, c_{ij}^{-} = c_{ij}$ else $c_{ij}^{-} = 0, 1 \leq i \leq m, 1 \leq j \leq n$ . The elements $a_{ij}^{+}$ of matrix A⁺ and $a_{ij}^{-}$ of matrix A^- are determined in the same way.

For fuzzy matrix equation $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ , we can express it as

$\begin{matrix} (A^{+} + A^{-}) (X, X^{l}, X^{r}) + (B, B^{l}, B^{r}) \\ = (C^{+} + C^{-}) (X, X^{l}, X^{r}) + (D, D^{l}, D^{r}) . \end{matrix}$ (3.14)

Since $k {\tilde{x}}_{j} = {\begin{matrix} ({kx}_{j}, {kx}_{j}^{l}, {kx}_{j}^{r}), & k \geq 0, \\ ({kx}_{j}, - {kx}_{j}^{r}, - {kx}_{j}^{l}), & k < 0 \end{matrix},$

we have $A \tilde{x} = {\begin{matrix} (Ax, {Ax}^{l}, {Ax}^{r}), & A \geq 0, \\ (Ax, - {Ax}^{r}, - {Ax}^{l}), & A < 0 \end{matrix}$

So the Equation (3.11) be rewritten is $\begin{matrix} A^{+} (X, X^{l}, X^{r}) + A^{-} (X, X^{l}, X^{r}) + (B, B^{l}, B^{r}) \\ = C^{+} (X, X^{l}, X^{r}) + C^{-} (X, X^{l}, X^{r}) + (D, D^{l}, D^{r}) \end{matrix}$ i.e., $\begin{matrix} (A^{+} X, A^{+} X^{l}, A^{+} X^{r}) \\ + (A^{-} X, - A^{-} X^{r}, - A^{-} X^{l}) + (B, B^{l}, B^{r}) \\ = (C^{+} X, C^{+} X^{l}, C^{+} X^{r}) \\ + (C^{-} X, - C^{-} X^{r}, - C^{-} X^{l}) + (D, D^{l}, D^{r}), \\ (A^{+} X + A^{-} X, A^{+} X^{l} - A^{-} X^{r}, A^{+} X^{r} - A^{-} X^{l}) \\ + (B, B^{l}, B^{r}) \\ = (C^{+} X + C^{-} X, C^{+} X^{l} - C^{-} X^{r}, \\ C^{+} X^{r} - C^{-} X^{l}) + (D, D^{l}, D^{r}) . \end{matrix}$

Thus we get ${\begin{matrix} (A^{+} + A^{-}) X + B = (C^{+} + C^{-}) X + D, \\ A^{+} X^{l} - A^{-} X^{r} + B^{l} = C^{+} X^{l} - C^{-} X^{r} + D^{l}, \\ A^{+} X^{r} - A^{-} X^{l} + B^{r} = C^{+} X^{r} - C^{-} X^{l} + D^{r} . \end{matrix}$ (3.15) or ${\begin{matrix} AX + B = CX + D, \\ (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) + (\begin{matrix} B^{l} \\ B^{r} \end{matrix}) \\ = (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) + (\begin{matrix} D^{l} \\ D^{r} \end{matrix}) . □ \end{matrix}$

In order to solve the LR dual fuzzy matrix equation (2.6), we need to consider the system of linear equations (3.13). It seems that we have obtained the minimal solution of the linear matrix system (3.13) as $X = (A - C)^{†} (D - B),$ (3.16) $(\begin{matrix} X^{l} \\ X^{r} \end{matrix}) = {((\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) - (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}))}^{†} ((\begin{matrix} D^{l} \\ D^{r} \end{matrix}) - (\begin{matrix} B^{l} \\ B^{r} \end{matrix})) .$ (3.17) where (S - T) ^† is the Moore-Penrose generalized inverse of matrix S - T.

It seems that we obtained the fuzzy matrix $\tilde{X}$ as the above expression (3.16) and (3.17). However, the solution matrix may still not be an appropriate LR fuzzy numbers vector except for X^l ≥ O, X^r ≥ O. So we give the definition of LR fuzzy solution to the equation (2.2) as follows:

Definition 3.5. Let $\tilde{X} = (X, X^{l}, X^{r})$ . If (X, X^l, X^r) is the minimal solution of Equation (3.13) such that X^l ≥ O, X^r ≥ O, we call $\tilde{X} = (X, X^{l}, X^{r})$ is a strong LR fuzzy minimal solution of dual fuzzy matrix equation (2.2). Otherwise, the $\tilde{X} = (X, X^{l}, X^{r})$ is said to a weak LR fuzzy fuzzy minimal solution of dual fuzzy matrix equation (2.2) given by $\tilde{X} = {\tilde{x}}_{ij}$ where

$\begin{matrix} {\tilde{x}}_{ij} \\ = {\begin{matrix} (x_{ij}, x_{ij}^{l}, x_{ij}^{r}), & x_{ij}^{l} > 0, & x_{ij}^{r} > 0, \\ (x_{ij}, 0, max {- x_{ij}^{l}, x_{ij}^{r}}), & x_{ij}^{l} < 0, & x_{ij}^{r} > 0, \\ (x_{ij}, max {x_{ij}^{l}, - x_{ij}^{r}}, 0,), & x_{ij}^{l} > 0, & x_{ij}^{r} < 0, \\ (x_{ij}, - x_{j}^{l}, - x_{ij}^{r}), & x_{ij}^{l} < 0, & x_{ij}^{r} < 0 . \end{matrix} \\ i = 1, \dots, n, j = 1, \dots, p . \end{matrix}$ (3.18)

Example 3.2. Consider the dual fuzzy matrix system $\begin{matrix} (\begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix}) & = & (\begin{matrix} 1 & - 1 \\ 1 & 3 \end{matrix}) (\begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix}) \\ + (\begin{matrix} (1, 2, 1)_{LR} & (2, 1, 1)_{LR} \\ (4, 1, 1)_{LR} & (3, 1, 2)_{LR} \end{matrix}) . \end{matrix}$

By the Theorem 3.4, the model we proposed for treating the above system is ${\begin{matrix} I_{2} X = CX + D, \\ (\begin{matrix} I_{2} & O \\ O & I_{2} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) \\ = (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) + (\begin{matrix} D^{l} \\ D^{r} \end{matrix}), \end{matrix}$

where $I_{2} - C = (\begin{matrix} 0 & 1 \\ - 1 & - 2 \end{matrix})$

and

$I_{4} - T = (\begin{matrix} 0 & 0 & 0 & - 1 \\ - 1 & - 2 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & - 2 \end{matrix}) .$

Since that I₂ - C and I₄ - T are both nonsingular, from the Equations (3.16) and (3.17) we obtain the mean value X, and the left spread X^l and the right spread X^r of solution $\begin{matrix} X & = & (I_{2} - C)^{- 1} D \\ = & (\begin{matrix} - 2.000 & - 1.0000 \\ 1.0000 & 0.0000 \end{matrix}) (\begin{matrix} 1 & 2 \\ 4 & 3 \end{matrix}) \\ = & (\begin{matrix} - 6.000 & - 7.000 \\ 1.000 & 2.000 \end{matrix}) \end{matrix}$ and $\begin{matrix} (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) \\ = (\begin{matrix} x_{11}^{l} & x_{12}^{l} \\ x_{21}^{l} & x_{22}^{l} \\ x_{11}^{r} & x_{12}^{r} \\ x_{21}^{r} & x_{22}^{r} \end{matrix}) = (I_{4} - T)^{- 1} Y \\ = (\begin{matrix} 0 & - 1.0000 & 2.0000 & 0 \\ 0 & - 0.0000 & - 1.0000 & 0 \\ 2.0000 & 0 & 0 & - 1.0000 \\ - 1.0000 & 0 & 0 & - 0.0000 \end{matrix}) \\ (\begin{matrix} 2 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 2 \end{matrix}) \\ = (\begin{matrix} 1.0000 & 1.0000 \\ - 1.0000 & - 1.0000 \\ 3.0000 & 0.000 \\ - 2.0000 & - 1.0000 \end{matrix}) . \end{matrix}$

By the Definition 3.1, the solution of the original dual fuzzy system is $\tilde{x} = (\begin{matrix} (- 6.0000, 1.0000, 3.0000)_{LR} & (- 7.0000, 1.0000, 0.0000)_{LR} \\ (1.0000, 2.0000, 1.0000)_{LR} & (2.0000, 1.0000, 1.0000)_{LR} \end{matrix}),$ and it admits a weak minimal LR fuzzysolution.

The key point to make the solution matrix be a strong LR fuzzy solution is that $\tilde{X} = (X, X^{l}, X^{r})$ is LR fuzzy matrix, i.e., each element in which is a LR fuzzy number. By the the following analysis, we know that it is equivalent to the condition (S - T) ^† ≥ O and $((\begin{matrix} D^{l} \\ D^{r} \end{matrix}) - (\begin{matrix} B^{l} \\ B^{r} \end{matrix})) \geq O$ .

Theorem 3.5. If S - T is a non negativematrix, $\begin{matrix} ((A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†}) \\ \geq O, ((A - C)^{†} + (∣ A ∣ + ∣ C ∣)^{†}) \geq O \end{matrix}$ and $((\begin{matrix} D^{l} \\ D^{r} \end{matrix}) - (\begin{matrix} B^{l} \\ B^{r} \end{matrix})) \geq O,$ the fuzzy matrix equation (2.2) has a strong LR fuzzy minimal solution as follows: $\tilde{X} = (X, X^{l}, X^{r})$ where ${\begin{matrix} X = (A - C)^{†} (D - B), \\ (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) = {((\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) - (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}))}^{†} ((\begin{matrix} D^{l} \\ D^{r} \end{matrix}) - (\begin{matrix} B^{l} \\ B^{r} \end{matrix})) . \end{matrix}$

Proof. Since B^l, D^l and B^r, D^r are the left and right spreads fuzzy matrix $\tilde{B}$ and $\tilde{D}$ , B^l ≥ O, B^r ≥ O and D^l ≥ O, D^r ≥ O. By the hypothesis, $((\begin{matrix} D^{l} \\ D^{r} \end{matrix}) - (\begin{matrix} B^{l} \\ B^{r} \end{matrix})) \geq O$ .

Let $S^{†} = (\begin{matrix} E & F \\ F & E \end{matrix}) = \frac{1}{2} (\begin{matrix} (A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†} & (A - C)^{†} - (∣ A ∣ - ∣ C ∣)^{†} \\ (A - C)^{†} - (∣ A ∣ - ∣ C ∣)^{†} & (A - C)^{†} + (∣ A ∣ - ∣ C ∣)^{†} \end{matrix}) .$

We know the condition that (S - T) ⁺ ≥ 0 is equivalent to E ≥ O, F ≥ O.

Now that E ≥ O and F ≥ O, the product of two non negative matrices $\begin{matrix} (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) & = & (\begin{matrix} E & F \\ F & E \end{matrix}) (\begin{matrix} D^{l} - B^{l} \\ D^{r} - B^{r} \end{matrix}) \\ = & (E (D^{l} - B^{l}) + F (D^{r} - B^{r}), F (D^{l} - B^{l}) \\ + E (D^{r} - B^{r})) \geq O \end{matrix}$ is non negative in nature. i.e. X^l ≥ O and X^r≥ O. □

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

Theorem 3.6. [32] The inverse of a nonnegative matrix S - T is nonnegative if and only if S - T is a generalized permutation matrix.

Theorem 3.7. [14] Let S - T be a 2m × 2n nonnegative matrix with rank r. Then the following assertions are equivalent:

(S - T) ^† ≥ 0.

There exists a permutation matrix P, such that P (S - T) has the form $P (S - T) = (\begin{matrix} B_{1} \\ B_{2} \\ ⋮ \\ B_{r} \\ O \end{matrix}),$ where each B_i has rank 1 and the rows of B_i are orthogonal to the rows of B_j, whenever i ≠ j, the zero matrix may be absent.

$(S - T)^{†} = (\begin{matrix} {UZ}^{⊤} & {UW}^{⊤} \\ {UW}^{⊤} & {UZ}^{⊤} \end{matrix})$ for some positive diagonal matrix U. In this case, $\begin{matrix} (Z + W)^{†} = U (Z + W)^{⊤}, (Z - W)^{†} = U (Z - W)^{⊤} . \end{matrix}$

4 Numerical examples

Example 4.1. Consider the dual fuzzy linear system $\begin{matrix} (\begin{matrix} 2 & 1 \\ 1 & - 1 \\ 0 & 1 \end{matrix}) (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \end{matrix}) \\ + (\begin{matrix} (2 r, 3 - r) & (1 + r, 4 - r) \\ (1 + r, 3 - r) & (r, 1 - 2 r) \\ (2 + r, 4 - r) & (2 + 2 r, 4 - 3 r) \end{matrix}) \\ = (\begin{matrix} 1 & 0 \\ 1 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \end{matrix}) \\ + (\begin{matrix} (1 + 2 r, 5 - 2 r) & (2 + 3 r, 8 - 2 r) \\ (2 + 2 r, 6 - 2 r) & (2 + 3 r, 4 - 2 r) \\ (3 + 2 r, 7 - 2 r) & (5 r, 5 - 6 r) \end{matrix}) . \end{matrix}$

By the Theorem 3.1, the model for solving the above fuzzy system is $(S - T) X (r) = D (r) - B (r),$ where $S - T = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 1 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & - 1 & 1 \end{matrix})$ and $\tilde{D} - \tilde{B} = (\begin{matrix} (1 + r, 3 - r) & (1 + 2 r, 4 - r) \\ (1 + r, 3 - r) & (2 + 2 r, 3) \\ (1 + r, 3 - r) & (- 2 + 3 r, 1 - 3 r) \end{matrix}) .$

From the Equation (3.8), we have $\begin{matrix} (\begin{matrix} \underline{X} (r) \\ - \bar{X} (r) \end{matrix}) & = & {(\begin{matrix} A^{+} - C^{+} & - A^{-} + C^{-} \\ - A^{-} + C^{-} & A^{+} - C^{+} \end{matrix})}^{†} (\begin{matrix} \underline{D} (r) - \underline{B} (r) \\ - \bar{D} (r) + \bar{B} (r) \end{matrix}) \\ = & (\begin{matrix} 0.7778 & - 0.1111 & - 0.2222 & 0.2222 & 0.1111 & 0.2222 \\ 0.5556 & - 0.2222 & 0.5556 & 0.4444 & 0.2222 & 0.4444 \\ 0.2222 & 0.1111 & 0.2222 & 0.7778 & - 0.1111 & - 0.2222 \\ 0.4444 & 0.2222 & 0.4444 & 0.5556 & - 0.2222 & 0.5556 \end{matrix}) (\begin{matrix} 1 + r & 1 + 2 r \\ 1 + r & 2 + 2 r \\ 1 + r & - 2 + 3 r \\ - 3 + r & - 4 + r \\ - 3 + r & - 3 \\ - 3 + r & - 1 + 3 r \end{matrix}) \\ = & (\begin{matrix} - 1.2222 + 1.0000 r & - 0.4444 + 1.5556 r \\ - 2.4444 + 2.0000 r & - 3.8889 + 4.1111 r \\ - 0.7778 + 1.0000 r & - 2.5556 + 1.4444 r \\ - 1.5556 + 2.0000 r & - 2.1111 + 4.8889 r \end{matrix}) . \end{matrix}$

It means $\begin{matrix} (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \end{matrix}) = (\begin{matrix} (- 1.2222 + 1.0000 r, 0.7778 - 1.0000 r) & (- 0.4444 + 1.5556 r, 2.5556 - 1.4444 r) \\ (- 2.4444 + 2.0000 r, 1.5556 - 2.0000 r) & (- 3.8889 + 4.1111 r, 2.1111 - 4.8889 r) \end{matrix}) . \end{matrix}$

Since it is a appropriate fuzzy numbers matrix, the solution of the dual fuzzy matrix system we obtained is a strong minimal fuzzy solution given by $\begin{matrix} \tilde{U} & = & (\begin{matrix} {\tilde{u}}_{11} & {\tilde{u}}_{12} \\ {\tilde{u}}_{21} & {\tilde{u}}_{22} \end{matrix}) \\ = & (\begin{matrix} (- 1.2222 + 1.0000 r, 0.7778 - 1.0000 r) & (- 0.4444 + 1.5556 r, 2.5556 - 1.4444 r) \\ (- 2.4444 + 2.0000 r, 1.5556 - 2.0000 r) & (- 3.8889 + 4.1111 r, 2.1111 - 4.8889 r) \end{matrix}) . \end{matrix}$

Example 4.2. Consider the LR dual fuzzy matrix system $\begin{matrix} (\begin{matrix} 2 & 0 & 1 \\ 0 & 1 & 1 \end{matrix}) (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \\ {\tilde{x}}_{31} & {\tilde{x}}_{32} \end{matrix}) \\ + (\begin{matrix} (2, 2, 1)_{LR} & (1, 1, 2)_{LR} \\ (2, 1, 1)_{LR} & (3, 2, 1)_{LR} \end{matrix}) \\ = (\begin{matrix} 1 & 0 & 1 \\ 0 & - 1 & 0 \end{matrix}) (\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \\ {\tilde{x}}_{31} & {\tilde{x}}_{32} \end{matrix}) \\ + (\begin{matrix} (4, 3, 2)_{LR} & (3, 2, 3)_{LR} \\ (5, 2, 2)_{LR} & (4, 2, 2)_{LR} \end{matrix}) . \end{matrix}$

By the Theorem 3.4, the model we proposed for treating the above system is

${\begin{matrix} AX + B = CX + D, \\ (\begin{matrix} A^{+} & - A^{-} \\ - A^{-} & A^{+} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) + (\begin{matrix} B^{l} \\ B^{r} \end{matrix}) = (\begin{matrix} C^{+} & - C^{-} \\ - C^{-} & C^{+} \end{matrix}) (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) + (\begin{matrix} D^{l} \\ D^{r} \end{matrix}), \end{matrix}$

where $A - C = (\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \end{matrix})$

and

$S - T = (\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 1 \end{matrix}) .$

From the Equations (3.16) and (3.17), the mean value X, and the left spread X^l and the right spread X^r of solution are derived from

$\begin{matrix} X & = & (\begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \end{matrix}) = (A - C)^{†} (D - B) \\ = & (\begin{matrix} 1.0000 & 0 \\ 0 & 0.4000 \\ 0 & 0.2000 \end{matrix}) (\begin{matrix} 2 & 2 \\ 3 & 1 \end{matrix}) = (\begin{matrix} 2.0000 & 2.0000 \\ 1.2000 & 0.4000 \\ 0.6000 & 0.2000 \end{matrix}) \end{matrix}$ and $\begin{matrix} (\begin{matrix} X^{l} \\ X^{r} \end{matrix}) = (S - T)^{†} ((\begin{matrix} D^{l} \\ D^{r} \end{matrix}) - (\begin{matrix} B^{l} \\ B^{r} \end{matrix})) \\ = (\begin{matrix} 1.0000 & 0 & 0 & 0 \\ 0 & 0.2000 & 0 & - 0.2000 \\ 0 & 0.6000 & 0 & 0.4000 \\ 0 & 0 & 1.0000 & 0 \\ 0 & - 0.2000 & 0 & 0.2000 \\ 0 & 0.4000 & 0 & 0.6000 \end{matrix}) (\begin{matrix} 1 & 1 \\ 1 & 0 \\ 1 & 1 \\ 1 & 1 \end{matrix}) = (\begin{matrix} 1.0000 & 1.0000 \\ 0.0000 & - 0.2000 \\ 1.0000 & 0.40000 \\ 1.0000 & 1.0000 \\ 0.0000 & 0.2000 \\ 1.0000 & 0.60000 \end{matrix}) . \end{matrix}$

It means $(\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} \\ {\tilde{x}}_{31} & {\tilde{x}}_{32} \end{matrix}) = (\begin{matrix} (2.0000, 1.0000, 1.0000)_{LR} & (2.0000, 1.0000, 1.0000)_{LR} \\ (1.2000, 0.0000, 0.0000)_{LR} & (0.4000, - 0.200, 0.2000)_{LR} \\ (0.6000, 1.0000, 1.0000)_{LR} & (0.2000, 0.4000, 0.6000)_{LR} \end{matrix}) .$

Since the element $x_{22}^{l}$ in ${\tilde{x}}^{22}$ is not positive, the solution of the dual fuzzy matrix system we obtained is a weak minimal LR fuzzy solution given by

$\tilde{X} = (\begin{matrix} (2.0000, 1.0000, 1.0000)_{LR} & (2.0000, 1.0000, 1.0000)_{LR} \\ (1.2000, 0.0000, 0.0000)_{LR} & (0.4000, 0.0000, 0.2000)_{LR} \\ (0.6000, 1.0000, 1.0000)_{LR} & (0.2000, 0.4000, 0.6000)_{LR} \end{matrix}) .$

5 Conclusion

In this work, we proposed a matrix method for solving the general dual fuzzy matrix equation $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ in which A, C are m × n crisp matrices and $\tilde{B}, \tilde{D}$ are n × p fuzzy numbers matrices. We directly converted the dual fuzzy matrix equation to a crisp function matrix equation based on fuzzy numbers matrix and its operation. The existence condition of strong fuzzy solution was studied in detail. Moreover, the LR dual fuzzy matrix equation was deal with in similar way. Some numerical examples were given to show that our method was feasible to solve the dual fuzzy matrix equation. The proposed method can be applied to all kind s of fuzzy linear systems.

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