A straightforward approach for solving dual fuzzy linear systems

Abstract

The main purpose of this paper is to solve a dual fuzzy linear system algebraically. In considered systems, the coefficient matrices are crisp-valued matrices and the left and right hand sides vectors are fuzzy number-valued vectors. Two types of solutions are defined and the relationship between them is investigated. Finally, based on the obtained results, a simple method is presented to obtain a unique algebraic solution of a dual fuzzy linear system. The main advantage of the proposed method over existing methods is that it does not need to convert a dual fuzzy linear system to two crisp linear systems. Also, a new necessary and sufficient condition for the existence of unique algebraic solution of a dual fuzzy linear system is presented. To illustrate our method, two numerical examples and an applied example in economy are given.

Introduction

Fuzzy set-based mathematics are widely used in many scientific fields such as physics, transportation planning [22], optimization [27], business, finance, management [9], current flow and control theory [11]. In most of the times, more parameters or variables in the under study models are real numbers. But, in practice, these parameters or variables may take on the uncertain values. One approach to quantify and cope with uncertainty is to use fuzzy numbers instead of the real numbers.

Throughout this paper, we denote the matrices by bold capital letters and the vectors by non-bold capital letters. Moreover, we place a bar over a symbol if it represents a fuzzy quantity.

A well known class of problems involving fuzzy sets is the systems of fuzzy linear equations, or in particular, fuzzy linear systems. In 1998, a general method for solving the fuzzy linear system $\mathbf{\text{A}}\tilde{X}=\tilde{B}$, where A is a crisp-valued matrix and $\tilde{B}$ is a fuzzy number-valued vector, is introduced by Fridman et al. [5], [14]. After that, different methods and approaches were applied to solve this type of fuzzy linear system [4], [6], [7], [15], [16], [20].

One of the important topics in fuzzy mathematics that has been rapidly growing in recent years is the fuzzy linear systems in dual form. This type of system is called dual fuzzy linear system and it has many applications in various branches of science such as economics, finance, engineering and physics [19]. In 2000, Ma et al. [18] investigated the existence of a solution for the dual fuzzy linear system $\mathbf{A}\tilde{\mathbf{X}}=\mathbf{B}\tilde{\mathbf{X}}+\tilde{\mathbf{Y}}$, where A and B are two crisp-valued matrices and $\tilde{Y}$ is a fuzzy number-valued vector. Also, they showed that the system $\mathbf{A}\tilde{\mathbf{X}}=\mathbf{B}\tilde{\mathbf{X}}+\tilde{\mathbf{Y}}$ can not be replaced by the system $\left(\mathbf{A}-\mathbf{B}\right)\tilde{X}=\tilde{Y}$, because for an arbitrary fuzzy number $\tilde{x}$ there exists no element $\tilde{y}$ such that $\tilde{x}+\tilde{y}=0$. In 2001, Wang et al. [25] considered the dual fuzzy linear systems of the form $\tilde{X}=\mathbf{\text{A}}\tilde{X}+\tilde{Y}$ and presented iteration algorithms for solving them. Muzzioli and Reynaerts in 2006 [19], investigated those dual fuzzy linear systems where all the coefficients matrices and the left and right hand sides vectors are fuzzy number-valued vectors. In 2008, Abbasbandy et al. [2] presented two necessary and sufficient conditions for the existence of a minimal solution of non-square dual fuzzy linear systems of the form $\mathbf{A}\tilde{\mathbf{X}}+\tilde{\mathbf{Y}}=\mathbf{B}\tilde{\mathbf{X}}+\tilde{\mathbf{Z}}$. In the same year, Ezzati [12] proposed a method for solving a non-square symmetric dual fuzzy linear system as $\mathbf{A}\tilde{\mathbf{X}}=\mathbf{\text{B}}\tilde{\mathbf{X}}+\tilde{\mathbf{Y}}$. In 2009, Tian et al. [24] improved the obtained results by Wang et al. [25] and then presented a necessary and sufficient condition to convergence of their method. Also, Sun and Guo [23], introduced a general model for solving non-square dual fuzzy linear systems of the form $\mathbf{\text{A}}\tilde{X}+\tilde{Y}=\mathbf{\text{B}}\tilde{X}+\tilde{Z}$ with non-full rank matrices A and B. In 2012, Fariborzi Araghi and Hosseinzadeh [13] applied a special algorithm of the class of ABS algorithms called Huang algorithm to solve the non-square dual fuzzy linear system $\mathbf{\text{A}}\tilde{X}+\tilde{Y}=\mathbf{\text{B}}\tilde{X}+\tilde{Z}$. A new model for solving the dual fuzzy linear system $\mathbf{\text{A}}\tilde{X}=\mathbf{\text{B}}\tilde{X}+\tilde{Y}$ is presented by Otadi in 2013 [21]. Recently, Allahviranloo [3] and Chehlabi [10] considered the dual fuzzy linear systems in complex form and introduced practical and simple approaches for solving them.

In this paper, we focus on the dual fuzzy linear systems of the form $\mathbf{\text{A}}\tilde{X}+\tilde{Y}=\mathbf{\text{B}}\tilde{X}+\tilde{Z}$, where the coefficient matrices A and B are crisp-valued matrices and the left and right hand sides columns $\tilde{Y}$ and $\tilde{Z}$ are fuzzy number-valued vectors. Two types of solutions are defined for these systems, called extended solution and algebraic solution. The relationship between these solutions is investigated and finally a direct approach is presented to obtain the algebraic solution using the extended solution. Unlike the existing methods (see [2], [12], [13], [18], [21], [23]), based on the proposed method, we don't need to convert a dual fuzzy linear system to two crisp linear systems that is main advantage of our method. Also, a new necessary and sufficient condition for the existence of unique algebraic solution of a dual fuzzy linear system is presented.

The structure of paper is organized as follows. In Section 2, the fundamental concepts of fuzzy mathematics and definition of dual fuzzy linear systems are given. Then, two types of solutions for a dual fuzzy linear system are presented and the relationship between them is investigated. In Section 3, our method is explained by presenting a theorem. In Section 4, several numerical examples are presented to illustrate the proposed method. Conclusions are brought in Section 5.