Solution of Fuzzy Matrix Equation System

Abstract

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: (1)𝑢(𝑟) is a bounded left continuous nondecreasing function over [0,1], (2)𝑢(𝑟) is a bounded left continuous nonincreasing function over [0,1], and (3)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] (a)𝑥=𝑦 if and only if 𝑥(𝑟)=𝑦(𝑟) and 𝑥(𝑟)=𝑦(𝑟),(b)𝑥+𝑦=(𝑥(𝑟)+𝑦(𝑟),𝑥(𝑟)+𝑦(𝑟)), and(c)𝑘𝑥=^{(𝑘𝑥,𝑘𝑥),𝑘≥0,}_{(𝑘𝑥,𝑘𝑥),𝑘<0.}

Definition 2.1. The 𝑛×𝑛 linear system is as follows: 𝑎₁₁𝑥₁+𝑎₁₂𝑥₂+⋯+𝑎_1𝑛𝑥_𝑛=𝑦₁,𝑎₂₁𝑥₁+𝑎₂₂𝑥₂+⋯+𝑎_2𝑛𝑥_𝑛=𝑦₂,⋮𝑎_𝑛1𝑥₁+𝑎_𝑛2𝑥₂+⋯+𝑎_𝑛𝑛𝑥_𝑛=𝑦_𝑛, (2.1) where the given matrix of coefficients 𝐴=(𝑎_𝑖𝑗), 1≤𝑖, 𝑗≤𝑛 is a real 𝑛×𝑛 matrix, the given 𝑦_𝑖∈𝐸, 1≤𝑖≤𝑛, with the unknowns 𝑥_𝑗∈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 [21]). A fuzzy number vector (𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑡 given by 𝑥_𝑗=𝑥_𝑗(𝑟),𝑥_𝑗(𝑟);1≤𝑗≤𝑛,0≤𝑟≤1 (2.2) is called a solution of the fuzzy linear system (2.1) if ^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑦_𝑖,_𝑛𝑗=1𝑎_𝑖𝑗𝑥_𝑗=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑦_𝑖. (2.3) If, for a particular 𝑖,𝑎_𝑖𝑗>0, for all 𝑗, we simply get ^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑦_𝑖,^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑦_𝑖. (2.4)
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: 𝑆𝑋=𝑌, (2.5) where 𝑠_𝑖𝑗 are determined as follows: 𝑎_𝑖𝑗≥0⇒𝑠_𝑖𝑗=𝑎_𝑖𝑗,𝑠_{𝑖+𝑛,𝑗+𝑛}=𝑎_𝑖𝑗,𝑎_𝑖𝑗<0⇒𝑠_{𝑖,𝑗+𝑛}=−𝑎_𝑖𝑗,𝑠_{𝑖+𝑛,𝑗}=−𝑎_𝑖𝑗, (2.6) and any 𝑠_𝑖𝑗 which is not determined by (2.1) is zero and 𝑥𝑋=¹⋮𝑥_𝑛−𝑥₁⋮−𝑥_𝑛𝑦,𝑌=¹⋮𝑦_𝑛−𝑦₁⋮−𝑦_𝑛. (2.7) The structure of 𝑆 implies that 𝑠_𝑖𝑗≥0, 1≤𝑖, 𝑗≤2𝑛 and that 𝑆=𝐵𝐶𝐶𝐵, (2.8) 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 [21]). The inverse of nonnegative matrix 𝑆=𝐵𝐶𝐶𝐵 (2.9) is 𝑆⁻¹=𝐷𝐸𝐸𝐷, (2.10) where 1𝐷=2(𝐵+𝐶)⁻¹+(𝐵−𝐶)⁻¹1,𝐸=2(𝐵+𝐶)⁻¹−(𝐵−𝐶)⁻¹. (2.11)

Corollary 2.4 (see [30]). The solution of (2.5) is obtained by 𝑋=𝑆⁻¹𝑌. (2.12)

3. Fuzzy Matrix Equation System

A matrix system such as 𝑎₁₁𝑎₁₂⋯𝑎_1𝑛𝑎₂₁𝑎₂₂⋯𝑎_2𝑛𝑎⋮⋮⋮⋮_𝑛1𝑎_𝑛2⋯𝑎_𝑛𝑛𝑥₁₁𝑥₁₂⋯𝑥_1𝑛𝑥₂₁𝑥₂₂⋯𝑥_2𝑛𝑥⋮⋮⋮⋮_𝑛1𝑥_𝑛2⋯𝑥_𝑛𝑛=𝑦₁₁𝑦₁₂⋯𝑦_1𝑛𝑦₂₁𝑦₂₂⋯𝑦_2𝑛𝑦⋮⋮⋮⋮_𝑛1𝑦_𝑛2⋯𝑦_𝑛𝑛, (3.1) where 𝑎_𝑖𝑗, 1≤𝑖, 𝑗≤𝑛, are real numbers, the elements 𝑦_𝑖𝑗 in the right-hand matrix are fuzzy numbers, and the unknown elements 𝑥_𝑖𝑗 are ones, is called a fuzzy matrix equation system (FMES).

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

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

Theorem 3.1. Suppose that the inverse of matrix 𝐴 exists and 𝑥_𝑗=(𝑥_𝑗1,𝑥_𝑗2,…,𝑥_𝑗𝑛)^𝑇 is a solution of this equation. Then 𝑥_𝑗+𝑥_𝑗=(𝑥_𝑗1+𝑥_𝑗1,𝑥_𝑗2+𝑥_𝑗2,…,𝑥_𝑗𝑛+𝑥_𝑗𝑛)^𝑇 is the solution of the following systems: 𝐴𝑥_𝑗+𝑥_𝑗=𝑦_𝑗+𝑦_𝑗,𝑗=1,2,…,𝑛, (3.5) where 𝑦_𝑗+𝑦_𝑗=(𝑦_𝑗1+𝑦_𝑗1,𝑦_𝑗2+𝑦_𝑗2,…,𝑦_𝑗𝑛+𝑦_𝑗𝑛)^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: 𝑎₁₁𝑥_𝑗1+𝑥_𝑗1+⋯+𝑎_1𝑛𝑥_𝑗𝑛+𝑥_𝑗𝑛=𝑦_𝑗1+𝑦_𝑗1,𝑎₂₁𝑥_𝑗1+𝑥_𝑗1+⋯+𝑎_2𝑛𝑥_𝑗𝑛+𝑥_𝑗𝑛=𝑦_𝑗2+𝑦_𝑗2,⋮𝑎_𝑛1𝑥_𝑗1+𝑥_𝑗1+⋯+𝑎_𝑛𝑛𝑥_𝑗𝑛+𝑥_𝑗𝑛=𝑦_𝑗𝑛+𝑦_𝑗𝑛,𝑗=1,2,…,𝑛. (3.6) Using matrix notation, we have 𝐴𝑋+𝑋=𝑌+𝑌. (3.7) Suppose that the solution of (3.7) is as 𝑑_𝑗=𝑑_𝑗1𝑑_𝑗2⋮𝑑_𝑗𝑛=𝑥_𝑗+𝑥_𝑗=𝑥_𝑗1+𝑥_𝑗1𝑥_𝑗2+𝑥_𝑗2⋮𝑥_𝑗𝑛+𝑥_𝑗𝑛,𝑗=1,2,…,𝑛. (3.8) 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: 𝐵𝑋(𝑟)−𝐶𝑋(𝑟)=𝑌(𝑟),𝐵𝑋(𝑟)−𝐶𝑋(𝑟)=𝑌(𝑟). (3.9) 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 (𝐵+𝐶)𝑋(𝑟)=𝑌(𝑟)()+𝐶𝐷,𝐵+𝐶𝑋(𝑟)=𝑌(𝑟)+𝐶𝐷, (3.10)(3.11) therefore, we have 𝑋(𝑟)=(𝐵+𝐶)⁻¹𝑌(𝑟)+𝐶𝐷,𝑋(𝑟)=(𝐵+𝐶)⁻¹𝑌(𝑟)+𝐶𝐷. (3.12) 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 𝑋=𝑥₁,𝑥₂,…,𝑥_𝑛,−𝑥₁,−𝑥₂,…,−𝑥_𝑛^𝑇=𝑆⁻¹𝑌, (3.13) (the proposed method in Friedman et al. [21]) and 𝑥_𝑗=𝑥_𝑗1,𝑥_𝑗2,…,𝑥_𝑗𝑛,𝑥_𝑗1,𝑥_𝑗2,…,𝑥_𝑗𝑛^𝑇, (3.14) from (3.7)–(3.10), respectively. Then 𝐺≤𝑔 and 𝑔−𝐺=𝑛².

Proof. According to Section 2, we have 𝑆⁻¹=𝐷𝐸𝐸𝐷, (3.15) where 1𝐷=2(𝐵+𝐶)⁻¹+(𝐵−𝐶)⁻¹1,𝐸=2(𝐵+𝐶)⁻¹−(𝐵−𝐶)⁻¹. (3.16) 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 ℎ(𝑆)=ℎ(𝐵+𝐶)+ℎ(𝐵−𝐶)=2ℎ(𝐴), (3.17) and hence 𝑔=2ℎ(𝐴)+4𝑛². (3.18) For computing 𝑥_𝑗+𝑥_𝑗=(𝑥_𝑗1+𝑥_𝑗1,𝑥_𝑗2+𝑥_𝑗2,…,𝑥_𝑗𝑛+𝑥_𝑗𝑛)^𝑇 from (3.7) and 𝑥_𝑗=(𝑥_𝑗1,𝑥_𝑗2,…,𝑥_𝑗𝑛)^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 𝐺=2ℎ(𝐴)+3𝑛², (3.19) 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: 2−1𝑥11¹¹𝑥₁₂𝑥₂₁𝑥₂₂=()()3𝑟−3,3−3𝑟)(4𝑟−4,6−6𝑟2𝑟+1,5−2𝑟)(3𝑟,7−4𝑟. (3.20) By using (3.7) and (3.10), we have 𝑥₁₁(𝑟)+𝑥₁₁(𝑟)𝑥₁₂(𝑟)+𝑥₁₂(𝑟)𝑥₂₁(𝑟)+𝑥₂₁(𝑟)𝑥₂₂(𝑟)+𝑥₂₂(𝑟)=,𝑥23−𝑟44₁₁(𝑟)𝑥₁₂(𝑟)𝑥₂₁(𝑟)𝑥₂₂(𝑟)=,𝑟𝑟1+𝑟2𝑟 (3.21) and hence 𝑥₁₁(𝑟)𝑥₁₂(𝑟)𝑥₂₁(𝑟)𝑥₂₂(𝑟)=2−𝑟3−2𝑟3−𝑟4−2𝑟. (3.22) 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.