Fuzzy Approximate Solution of Positive Fully Fuzzy Linear Matrix Equations

Abstract

The fuzzy matrix equations 𝐴⊗𝑋⊗𝐵=𝐶 in which 𝐴, 𝐵, and 𝐶 are 𝑚×𝑚, 𝑛×𝑛, and 𝑚×𝑛 nonnegative LR fuzzy numbers matrices, respectively, are investigated. The fuzzy matrix systems is extended into three crisp systems of linear matrix equations according to arithmetic operations of LR fuzzy numbers. Based on pseudoinverse of matrix, the fuzzy approximate solution of original fuzzy systems is obtained by solving the crisp linear matrix systems. In addition, the existence condition of nonnegative fuzzy solution is discussed. Two examples are calculated to illustrate the proposed method.

1. Introduction

Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modeling, which can formulate uncertainty in actual environment. In many matrix equations, some or all 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 matrix equation is becoming more important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [1, 2], Dubois and Prade [3], and Nahmias [4]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [5], Goetschel, Jr. and Voxman [6], and Wu and Ma [7, 8].

In the past decades, many researchers have studied the fuzzy linear equations such as fuzzy linear systems (FLS), dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), fully fuzzy linear systems (FFLS), dual fully fuzzy linear systems (DFFLS), and general dual fuzzy linear systems (GDFLS). These works were performed mainly by Friedman et al. [9, 10], Allahviranloo et al. [11–17], Abbasbandy et al. [18–21], Wang and Zheng [22, 23], and Dehghan et al. [24, 25]. The general method they applied is the fuzzy linear equations were converted to a crisp function system of linear equations with high order according to the embedding principles and algebraic operations of fuzzy numbers. Then the fuzzy solution of the original fuzzy linear systems was derived from solving the crisp function linear systems. However, for a fuzzy matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past. In 2009, Allahviranloo et al. [26] discussed the fuzzy linear matrix equations (FLME) of the form 𝐴𝑋𝐵=𝐶 in which the matrices 𝐴 and 𝐵 are known 𝑚×𝑚 and 𝑛×𝑛 real matrices, respectively; 𝐶 is a given 𝑚×𝑛 fuzzy matrix. By using the parametric form of 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 fuzzy matrix equations. In 2011, Guo et al. [27–29] investigated a class of fuzzy matrix equations 𝐴𝑥=𝐵 by means of the block Gaussian elimination method and studied the least squares solutions of the inconsistent fuzzy matrix equation 𝐴𝑥=𝐵 by using generalized inverses of the matrix, and discussed fuzzy symmetric solutions of fuzzy matrix equations 𝐴𝑋=𝐵. What is more, there are two shortcomings in the above fuzzy systems. The first is that in these fuzzy linear systems and fuzzy matrix systems the fuzzy elements were denoted by triangular fuzzy numbers, so the extended model equations always contain parameter 𝑟, 0≤𝑟≤1 which makes their computation especially numerical implementation inconvenient in some sense. The other one is that the weak fuzzy solution of fuzzy linear systems 𝐴𝑥=𝑏̃ does not exist sometimes see; [30].

To make the multiplication of fuzzy numbers easy and handle the fully fuzzy systems, Dubois and Prade [3] introduced the LR fuzzy number in 1978. We know that triangular fuzzy numbers and trapezoidal fuzzy numbers [31] are just specious cases of LR fuzzy numbers. In 2006, Dehghan et al. [25] discussed firstly computational methods for fully fuzzy linear systems 𝐴𝑥=𝑏̃ whose coefficient matrix and the right-hand side vector are LR fuzzy numbers. In 2012, Allahviranloo et al. studied LR fuzzy linear systems [32] by the linear programming with equality constraints. Otadi and Mosleh considered the nonnegative fuzzy solution [33] of fully fuzzy matrix equations 𝐴𝑋=𝐵 by employing linear programming with equality constraints at the same year. Recently, Guo and Shang [34] investigated the fuzzy approximate solution of LR fuzzy Sylvester matrix equations 𝐴𝑋+𝑋𝐵=𝐶. In this paper we propose a general model for solving the fuzzy linear matrix equation 𝐴⊗𝑋⊗𝐵=𝐶, where 𝐴,  𝐵, and 𝐶 are 𝑚×𝑚, 𝑛×𝑛, and 𝑚×𝑛 nonnegative LR fuzzy numbers matrices, respectively. The model is proposed in this way; that is, we extend the fuzzy linear matrix system into a system of linear matrix equations according to arithmetic operations of LR fuzzy numbers. The LR fuzzy solution of the original matrix equation is derived from solving crisp systems of linear matrix equations. The structure of this paper is organized as follows.

In Section 2, we recall the LR fuzzy numbers and present the concept of fully fuzzy linear matrix equation. The computing model to the positive fully fuzzy linear matrix equation is proposed in detail and the fuzzy approximate solution of the fuzzy linear matrix equation is obtained by using pseudo-inverse in Section 3. Some examples are given to illustrate our method in Section 4 and the conclusion is drawn in Section 5.

2. Preliminaries

2.1. The LR Fuzzy Number

Definition 1 (see [1]). A fuzzy number is a fuzzy set like 𝑢∶𝑅→𝐼=[0,1] which satisfies the following.(1)𝑢is upper semicontinuous,(2)𝑢 is fuzzy convex, that is, 𝑢(𝜆𝑥+(1−𝜆)𝑦)≥min{𝑢(𝑥),𝑢(𝑦)} for all 𝑥,𝑦∈𝑅, 𝜆∈[0,1],(3)𝑢 is normal, that is, there exists 𝑥₀∈𝑅 such that 𝑢(𝑥₀)=1,(4)supp 𝑢={𝑥∈𝑅∣𝑢(𝑥)>0} is the support of the 𝑢, and its closure cl(supp 𝑢) is compact.
Let 𝐸¹ be the set of all fuzzy numbers on 𝑅.

Definition 2 (see [3]). A fuzzy number 𝑀 is said to be an LR fuzzy number if 𝜇_𝑀(𝑥)=𝐿𝑚−𝑥𝛼𝑅,𝑥≤𝑚,𝛼>0,𝑥−𝑚𝛽,𝑥≥𝑚,𝛽>0, (1) where 𝑚, 𝛼, and 𝛽 are called the mean value, left, and right spreads of 𝑀, respectively. The function 𝐿(⋅), which is called left shape function satisfies(1)𝐿(𝑥)=𝐿(−𝑥), (2)𝐿(0)=1 and 𝐿(1)=0,(3)𝐿(𝑥) is nonincreasing on [0,∞).
The definition of a right shape function 𝑅(⋅) is similar to that of 𝐿(⋅).
Clearly, 𝑀=(𝑚,𝛼,𝛽)_LR is positive (negative) if and only if 𝑚−𝛼>0  (𝑚+𝛽<0).
Also, two LR fuzzy numbers 𝑀=(𝑚,𝛼,𝛽)_LR and 𝑁=(𝑛,𝛾,𝛿)_LR are said to be equal, if and only if 𝑚=𝑛, 𝛼=𝛾, and 𝛽=𝛿.

Definition 3 (see [5]). For arbitrary LR fuzzy numbers 𝑀=(𝑚,𝛼,𝛽)_LR and 𝑁=(𝑛,𝛾,𝛿)_LR, we have the following.(1)Addition: 𝑀⊕𝑁=𝑚,𝛼,𝛽_LR⊕𝑛,𝛾,𝛿_LR=𝑚+𝑛,𝛼+𝛾,𝛽+𝛿_LR. (2)(2)Multiplication:(i)If 𝑀>0 and 𝑁>0, then 𝑀⊗𝑁=𝑚,𝛼,𝛽_LR⊗𝑛,𝛾,𝛿_LR≅𝑚𝑛,𝑚𝛾+𝑛𝛼,𝑚𝛿+𝑛𝛽_LR. (3)(ii)if 𝑀<0 and 𝑁>0, then 𝑀⊗𝑁=𝑚,𝛼,𝛽_RL⊗𝑛,𝛾,𝛿_LR≅𝑚𝑛,𝑛𝛼−𝑚𝛿,𝑛𝛽−𝑚𝛾_RL, (4)(iii)if 𝑀<0 and 𝑁<0, then 𝑀⊗𝑁=𝑚,𝛼,𝛽_RL⊗(𝑛,𝛾,𝛿)_LR≅𝑚𝑛,−𝑚𝛿−𝑛𝛽,−𝑚𝛾−𝑛𝛼_RL, (5)(3) Scalar multiplication: 𝜆×𝑀=𝜆×𝑚,𝛼,𝛽_LR=𝜆𝑚,𝜆𝛼,𝜆𝛽_LR,𝜆≥0,𝜆𝑚,−𝜆𝛽,−𝜆𝛼_RL,𝜆<0. (6)

2.2. The LR Fuzzy Matrix

Definition 4. A matrix 𝐴=(𝑎_𝑖𝑗) is called an LR fuzzy matrix, if each element 𝑎_𝑖𝑗 of 𝐴 is an LR fuzzy number.
𝐴 will be positive (negative) and denoted by 𝐴>0(𝐴<0) if each element 𝑎_𝑖𝑗 of 𝐴 is positive (negative), where 𝑎_𝑖𝑗=(𝑎_𝑖𝑗,𝑎^𝑙_𝑖𝑗,𝑎^𝑟_𝑖𝑗)_LR. Up to the rest of this paper, we use positive LR fuzzy numbers and formulas given in Definition 3. For example, we represent 𝑚×𝑛 LR fuzzy matrix 𝐴=(𝑎_𝑖𝑗), that 𝑎_𝑖𝑗=(𝑎_𝑖𝑗,𝛼_𝑖𝑗,𝛽_𝑖𝑗)_LR with new notation 𝐴=(𝐴,𝑀,𝑁), where 𝐴=(𝑎_𝑖𝑗), 𝑀=(𝛼_𝑖𝑗) and 𝑁=(𝛽_𝑖𝑗) are three 𝑚×𝑛 crisp matrices. In particular, an 𝑛 dimensions LR fuzzy numbers vector 𝑥 can be denoted by (𝑥,𝑥^𝑙,𝑥^𝑟), where 𝑥=(𝑥_𝑖), 𝑥^𝑙=(𝑥^𝑙_𝑖) and 𝑥^𝑟=(𝑥^𝑟_𝑖) are three 𝑛 dimensions crisp vectors.

Definition 5. Let 𝐴=(𝑎_𝑖𝑗) and 𝐵=(𝑏̃_𝑖𝑗) be two 𝑚×𝑛 and 𝑛×𝑝 fuzzy matrices; we define 𝐴⊗𝐵=𝐶=(𝑐̃_𝑖𝑗) which is an 𝑚×𝑝 fuzzy matrix, where 𝑐̃_𝑖𝑗=^⊕_{𝑘=1,2,…,𝑛}𝑎_𝑖𝑘⊗𝑏̃_𝑘𝑗. (7)

2.3. The Fully Fuzzy Linear Matrix Equation

Definition 6. The matrix system: 𝑎₁₁𝑎₁₂⋯𝑎_1𝑚𝑎₂₁𝑎₂₂⋯𝑎_2𝑚𝑎⋮⋮⋮⋮_𝑚1𝑎_𝑚2⋯𝑎_𝑚𝑚⊗𝑥₁₁𝑥₁₂⋯𝑥_1𝑛𝑥₂₁𝑥₂₂⋯𝑥_2𝑛𝑥⋮⋮⋮⋮_𝑚1𝑥_𝑚2⋯𝑥_𝑚𝑛⊗𝑏̃₁₁𝑏̃₁₂⋯𝑏̃_1𝑛𝑏̃₂₁𝑏̃₂₂⋯𝑏̃_2𝑛𝑏̃⋮⋮⋮⋮_𝑛1𝑏̃_𝑛2⋯𝑏̃_𝑛𝑛=𝑐̃₁₁𝑐̃₁₂⋯𝑐̃_1𝑛𝑐̃₂₁𝑐̃₂₂⋯𝑐̃_2𝑛𝑐̃⋮⋮⋮⋮_𝑚1𝑐̃_𝑚2⋯𝑐̃_𝑚𝑛, (8) where 𝑎_𝑖𝑗, 1≤𝑖, 𝑗≤𝑚, 𝑏̃_𝑖𝑗, 1≤𝑖, 𝑗≤𝑛, and 𝑐̃_𝑖𝑗, 1≤𝑖≤𝑚, 1≤𝑗≤𝑛 are LR fuzzy numbers, is called an LR fully fuzzy linear matrix equation (FFLME).
Using matrix notation, we have 𝐴⊗𝑋⊗𝐵=𝐶. (9)
A fuzzy numbers matrix 𝑋=(𝑥_𝑖𝑗,𝑦_𝑖𝑗,𝑧_𝑖𝑗)_LR,1≤𝑖≤𝑚,1≤𝑗≤𝑛 (10) is called an LR fuzzy approximate solution of fully fuzzy linear matrix equation (8) if 𝑋 satisfies (9).
Up to the rest of this paper, we will discuss the nonnegative solution 𝑋=(𝑋,𝑌,𝑍)≥0 of FFLME 𝐴⊗𝑋⊗𝐵=𝐶, where 𝐴=(𝐴,𝑀,𝑁)≥0, 𝐵=(𝐵,𝐸,𝐹)≥0, and 𝐶=(𝐶,𝐺,𝐻)≥0.

3. Method for Solving FFLME

First, we extend the fuzzy linear matrix system (9) into three systems of linear matrix equations according to the LR fuzzy number and its arithmetic operations.

Theorem 7. The fuzzy linear matrix system (9) can be extended into the following model: 𝐴𝑋𝐵=𝐶,𝐴𝑋𝐸+𝐴𝑌𝐵+𝑀𝑋𝐵=𝐺,𝐴𝑋𝐹+𝐴𝑍𝐵+𝑁𝑋𝐵=𝐻. (11)

Proof. We denote 𝐴=(𝐴,𝑀,𝑁), 𝐵=(𝐵,𝐸,𝐹) and 𝐶=(𝐶,𝐺,𝐻), and assume 𝑋=(𝑋,𝑌,𝑍)≥0, then 𝐴𝑋()⊗()⊗()=()⊗()=()=(),𝐵=𝐴,𝑀,𝑁𝑋,𝑌,𝑍𝐵,𝐸,𝐹𝐴𝑋,𝐴𝑌+𝑀𝑋,𝐴𝑍+𝑁𝑋𝐵,𝐸,𝐹𝐴𝑋𝐵,𝐴𝑋𝐸+𝐴𝑌𝐵+𝑀𝑋𝐵,𝐴𝑋𝐹+𝐴𝑍𝐵+𝑁𝑋𝐵𝐶,𝐺,𝐻 (12) according to multiplication of nonnegative LR fuzzy numbers of Definition 2. Thus we obtain a model for solving FFLME (9) as follows: 𝐴𝑋𝐵=𝐶,𝐴𝑋𝐸+𝐴𝑌𝐵+𝑀𝑋𝐵=𝐺,𝐴𝑋𝐹+𝐴𝑍𝐵+𝑁𝑋𝐵=𝐻. (13)

Secondly, in order to solve the fuzzy linear matrix equation (9), we need to consider the crisp systems of linear matrix equation (11). Supposing 𝐴 and 𝐵 are nonsingular crisp matrices, we have 𝑋=𝐴⁻¹𝐶𝐵⁻¹,𝑌=𝐴⁻¹𝐺−𝑋𝐸𝐵⁻¹−𝐴⁻¹𝑀𝑋,𝑍=𝐴⁻¹𝐻−𝑋𝐹𝐵⁻¹−𝐴⁻¹𝑁𝑋, (14) that is, 𝑋=𝐴⁻¹𝐶𝐵⁻¹,𝑌=𝐴⁻¹𝐺−𝐴⁻¹𝐶𝐵⁻¹𝐸𝐵⁻¹−𝐴⁻¹𝑀𝐴⁻¹𝐶𝐵⁻¹,𝑍=𝐴⁻¹𝐻−𝐴⁻¹𝐶𝐵⁻¹𝐹𝐵⁻¹−𝐴⁻¹𝑁𝐴⁻¹𝐶𝐵⁻¹. (15)

Definition 8. Let 𝑋=(𝑋,𝑌,𝑍) be an LR fuzzy matrix. If (𝑋,𝑌,𝑍) is an exact solution of (11) such that 𝑋≥0, 𝑌≥0, 𝑍≥0, and 𝑋−𝑌≥0; we call 𝑋=(𝑋,𝑌,𝑍) a nonnegative LR fuzzy approximate solution of (9).

Theorem 9. Let 𝐴=(𝐴,𝑀,𝑁), 𝐵=(𝐵,𝐸,𝐹), and 𝐶=(𝐶,𝐺,𝐻) be three nonnegative fuzzy matrices, respectively, and let 𝐴 and 𝐵 be the product of a permutation matrix by a diagonal matrix with positive diagonal entries. Moreover, let 𝐺𝐵≥𝐶𝐵⁻¹𝐸+𝑀𝐴⁻¹𝐶, 𝐻𝐵≥𝐶𝐵⁻¹𝐹+𝑁𝐴⁻¹𝐶, and 𝐶+𝐶𝐵⁻¹𝐸+𝑀𝐴⁻¹𝐶≥𝐺𝐵. Then the systems 𝐴⊗𝑋⊗𝐵=𝐶 has nonnegative fuzzy solutions.

Proof. Our hypotheses on 𝐴 and 𝐵 imply that 𝐴⁻¹ and 𝐵⁻¹ exist and they are nonnegative matrices. Thus 𝑋=𝐴⁻¹𝐶𝐵⁻¹≥0.
On the other hand, because 𝐺𝐵≥𝐶𝐵⁻¹𝐸+𝑀𝐴⁻¹𝐶 and 𝐻𝐵≥𝐶𝐵⁻¹𝐹+𝑁𝐴⁻¹𝐶, so with 𝑌=𝐴⁻¹(𝐺𝐵−𝐶𝐵⁻¹𝐸−𝑀𝐴⁻¹𝐶)𝐵⁻¹ and 𝑍=𝐴⁻¹(𝐻𝐵−𝐶𝐵⁻¹𝐹−𝑁𝐴⁻¹𝐶)𝐵⁻¹, we have 𝑌≥0 and 𝑍≥0. Thus 𝑋=(𝑋,𝑌,𝑍) is a fuzzy matrix which satisfies 𝐴⊗𝑋⊗𝐵=𝐶. Since 𝑋−𝑌=𝐴⁻¹(𝐶−𝐺𝐵+𝐶𝐵⁻¹𝐸+𝑀𝐴⁻¹𝐶)𝐵⁻¹,  the positivity property of 𝑋 can be obtained from the condition 𝐶+𝐶𝐵⁻¹𝐸+𝑀𝐴⁻¹𝐶≥𝐺𝐵.

When 𝐴 or 𝐵 is a singular crisp matrix, the following result is obvious.

Theorem 10 (see [35]). For linear matrix equations 𝐴𝑋𝐵=𝐶, where 𝐴∈𝑅^𝑚×𝑚, 𝐵∈𝑅^𝑛×𝑛, and 𝐶∈𝑅^𝑚×𝑛. Then 𝑋=𝐴^†𝐶𝐵^† (16) is its minimal norm least squares solution.
By the pseudoinverse of matrices, we solve model (11) and obtain its minimal norm least squares solution as follows: 𝑋=𝐴^†𝐶𝐵^†,𝑌=𝐴^†𝐺−𝑋𝐸𝐵^†−𝐴^†𝑀𝑋,𝑍=𝐴^†𝐻−𝑋𝐹𝐵^†−𝐴^†𝑁𝑋, (17) that is, 𝑋=𝐴^†𝐶𝐵^†,𝑌=𝐴^†𝐺−𝐴^†𝐶𝐵^†𝐸𝐵^†−𝐴^†𝑀𝐴^†𝐶𝐵^†,𝑍=𝐴^†𝐻−𝐴^†𝐶𝐵^†𝐹𝐵^†−𝐴^†𝑁𝐴^†𝐶𝐵^†. (18)

Definition 11. Let 𝑋=(𝑋,𝑌,𝑍) be an LR fuzzy matrix. If (𝑋,𝑌,𝑍) is a minimal norm least squares solution of (11) such that 𝑋≥0, 𝑌≥0, 𝑍≥0, and 𝑋−𝑌≥0, we call 𝑋=(𝑋,𝑌,𝑍) a nonnegative LR fuzzy minimal norm least squares solution of (9).
At last, we give a sufficient condition for nonnegative fuzzy minimal norm least squares solution of FFLME (9) in the same way.

Theorem 12. Let 𝐴^† and 𝐵^† be nonnegative matrices. Moreover, let 𝐺𝐵≥𝐶𝐵^†𝐸+𝑀𝐴^†𝐶, 𝐻𝐵≥𝐶𝐵^†𝐹+𝑁𝐴^†𝐶, and 𝐶+𝐶𝐵^†𝐸+𝑀𝐴^†𝐶≥𝐺𝐵. Then the systems 𝐴⊗𝑋⊗𝐵=𝐶 has nonnegative fuzzy minimal norm least squares solutions.

Proof. Since 𝐴^† and 𝐵^† are nonnegative matrices, we have 𝑋=𝐴^†𝐶𝐵^†≥0.
Now that 𝐺𝐵≥𝐶𝐵^†𝐸+𝑀𝐴^†𝐶 and 𝐻𝐵≥𝐶𝐵^†𝐹+𝑁𝐴^†𝐶, therefore, with 𝑌=𝐴^†(𝐺𝐵−𝐶𝐵^†𝐸−𝑀𝐴^†𝐶)𝐵^† and 𝑍=𝐴^†(𝐻𝐵−𝐶𝐵^†𝐹−𝑁𝐴^†𝐶)𝐵^†, we have 𝑌≥0 and 𝑍≥0. Thus 𝑋=(𝑋,𝑌,𝑍) is a fuzzy matrix which satisfies 𝐴⊗𝑋⊗𝐵=𝐶. Since 𝑋−𝑌=𝐴^†(𝐶−𝐺𝐵+𝐶𝐵^†𝐸+𝑀𝐴^†𝐶)𝐵^†, the nonnegativity property of 𝑋 can be obtained from the condition 𝐶+𝐶𝐵^†𝐸+𝑀𝐴^†𝐶≥𝐺𝐵.

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

Theorem 13 (see [36]). The inverse of a nonnegative matrix 𝐴 is nonnegative if and only if 𝐴 is a generalized permutation matrix.

Theorem 14 (see [37]). Let 𝐴 be an 𝑚×𝑚 nonnegative matrix with rank 𝑟. Then the following assertions are equivalent:(a)𝐴^†≥0. (b)There exists a permutation matrix 𝑃, such that 𝑃𝐴 has the form 𝑆𝑃𝐴=¹𝑆₂⋮𝑆_𝑟𝑂, (19) where each 𝑆_𝑖 has rank 1 and the rows of 𝑆_𝑖 are orthogonal to the rows of 𝑆_𝑗, whenever 𝑖≠𝑗, the zero matrix may be absent.(c)𝐴^†=^{𝐾𝑃⊤𝐾𝑄⊤}_{𝐾𝑃⊤𝐾𝑃⊤} for some positive diagonal matrix 𝐾. In this case, (𝑃+𝑄)^†=𝐾(𝑃+𝑄)^⊤,(𝑃−𝑄)^†=𝐾(𝑃−𝑄)^⊤. (20)

4. Numerical Examples

Example 15. Consider the fully fuzzy linear matrix system (2,1,1)_LR(1,0,1)_LR(1,0,0)_LR(2,1,0)_LR⊗𝑥₁₁𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃⊗(1,0,1)_LR(2,1,1)_LR(1,1,0)_LR(2,1,0)_LR(1,0,0)_LR(2,1,1)_LR(1,0,0)_LR(2,1,1)_LR(1,0,1)_LR=(17,12,23)_LR(22,22,29)_LR(17,16,28)_LR(19,15,13)_LR(23,23,16)_LR(19,16,17)_LR. (21)
By Theorem 7, the model of the above fuzzy linear matrix system is made of the following three crisp systems of linear matrix equations 21𝑥12¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=,𝑥1212121211722171923192112¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃+𝑦0111010102112¹¹𝑦₁₂𝑦₁₃𝑦₂₁𝑦₂₂𝑦₂₃+𝑥1212121211001¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=,𝑥1212121211222161523202112¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃+𝑧1100010112112¹¹𝑧₁₂𝑧₁₃𝑧₂₁𝑧₂₂𝑧₂₃+𝑥1212121211100¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=.121212121232928131617 (22)
Now that the matrix 𝐵 is singular, according to formula (18), the solutions of the above three systems of linear matrix equations are as follows: 𝑋=𝐴^†𝐶𝐵^†=2112^†172217192319121212121^†=,1.50001.00001.50001.50002.00001.5000𝑌=𝐴^†𝐺−𝑋𝐸𝐵^†−𝐴^†𝑀𝑋=2112^†122216192319−𝑋011101010121212121^†−2112^†𝑋=,10011.03330.66670.87251.12501.65531.2578𝑍=𝐴^†𝐻−𝑋𝐹𝐵^†−𝐴^†𝑁𝑋=2112^†232928131617−𝑋110001011121212121^†−2112^†𝑋=.11008.333311.66679.33331.41671.33332.4167 (23)
By Definition 11, we know that the original fuzzy linear matrix equations have a nonnegative LR fuzzy solution

𝑥𝑋=¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=(1.5000,1.0333,8.3333)_LR(1.0000,0.6667,11.6667)_LR1.5000,0.8725,9.3333_LR(1.5000,1.1250,1.4167)_LR(2.0000,1.6553,1.3333)_LR1.5000,1.2578,2.4167_LR, ()()(24)

since 𝑋≥0,  𝑌≥0, 𝑍≥0, and 𝑋−𝑌≥0.

Example 16. Consider the following fuzzy matrix system: (1,1,0)_LR(2,0,1)_LR(2,0,1)_LR(1,0,0)_LR𝑥₁₁𝑥₁₂𝑥₂₁𝑥₂₂(1,0,1)_LR(2,1,0)_LR(2,1,1)_LR(3,1,2)_LR=(15,14,18)_LR(25,24,24)_LR(12,10,12)_LR(20,17,15)_LR. (25)
By Theorem 7, the model of the above fuzzy linear matrix system is made of following three crisp systems of linear matrix equations: 21𝑥12¹¹𝑥₁₂𝑥₂₁𝑥₂₂=,𝑥1223152510202112¹¹𝑥₁₂𝑥₂₁𝑥₂₂+01𝑦112112¹¹𝑦₁₂𝑦₂₁𝑦₂₂+𝑥12231000¹¹𝑥₁₂𝑥₂₁𝑥₂₂=,𝑥1223142410172112¹¹𝑥₁₂𝑥₂₁𝑥₂₂+10𝑧122112¹¹𝑧₁₂𝑧₂₁𝑧₂₂+𝑥12230110¹¹𝑥₁₂𝑥₂₁𝑥₂₂=.122318241215 (26)
By the same way, we obtain that the solutions of the above three systems of linear matrix equations are as follows: 𝑋=𝐴⁻¹𝐶𝐵⁻¹=,1122𝑌=𝐴⁻¹𝐺−𝑋𝐸𝐵⁻¹−𝐴⁻¹𝑀𝑋=,0101𝑍=𝐴⁻¹𝐻−𝑋𝐹𝐵⁻¹−𝐴⁻¹𝑁𝑋=.0010 (27)
Since 𝑋≥0, 𝑌≥0, 𝑍≥0, and 𝑋−𝑌≥0, we know that the original fuzzy linear matrix equations have a nonnegative LR fuzzy solution given by 𝑥𝑋=¹¹𝑥₁₂𝑥₂₁𝑥₂₂=(1.000,1.000,0.000)_LR(1.000,1.000,0.000)_LR(2.000,0.000,1.000)_LR(2.000,1.000,0.000)_LR. (28)

5. Conclusion

In this work we presented a model for solving fuzzy linear matrix equations 𝐴⊗𝑋⊗𝐵=𝐶 in which 𝐴 and 𝐵 are 𝑚×𝑚 and 𝑛×𝑛 fuzzy matrices, respectively, and 𝐶 is an 𝑚×𝑛 arbitrary LR fuzzy numbers matrix. The model was made of three crisp systems of linear equations which determined the mean value and the left and right spreads of the solution. The LR fuzzy approximate solution of the fuzzy linear matrix equation was derived from solving the crisp systems of linear matrix equations. In addition, the existence condition of strong LR fuzzy solution was studied. Numerical examples showed that our method is feasible to solve this type of fuzzy matrix equations. Based on LR fuzzy numbers and their operations, we can investigate all kinds of fully fuzzy matrix equations in future.

Acknowledgments

The work is supported by the Natural Scientific Funds of China (no. 71061013) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-1120).