The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing methods using the embedding approach for calculating the solution. The main aim in this paper is to design a class of mixed type splitting iterative methods for solving FLS. Furthermore, convergence analysis of the method is proved. Numerical example is illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm. 1. Introduction Solving fuzzy systems has been considered by many researchers, for example, [1–9] and the references therein. In [1, 2] Kandel et al. applied the embedding method for fuzzy linear system (hereafter denoted by FLS) and replaced the FLS by a crisp linear system. This model has been modified later by some other researchers; see [10–16] and the references therein. Here, based on mixed type splitting, we introduce a new iterative method to FLS. The mixed type splitting iterative method [17, 18] is given for the linear system of equations , where is positive real. Cheng et al. in [19] presented a class of the mixed type splitting iterative methods based on [17, 18] and some convergence conditions were given. They also proposed some sufficient and necessary conditions of convergence when coefficient matrix of the linear system is certain matrices such as -matrix. In this paper, the mixed type splitting iterative method for FLS will be established, which is a generalization of mixed type splitting iterative method for linear system. Some sufficient conditions for convergence of the mixed type splitting iterative method will be considered. Moreover, we will discuss a comparison theorem, which describes the influences of the parameters on the convergence rates of the new methods. 2. Preliminaries In this section we provide some basic notations and definitions of fuzzy number and fuzzy linear system. Definition 1. An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions , , which satisfy the following requirements (see [2, 3]). (i) is a bounded monotonic increasing left continuous function over .(ii) is a bounded monotonic decreasing left continuous function over .(iii), . A crisp number can be simply expressed as , . The addition and scalar multiplication of fuzzy numbers and can be described as follows:(i) if and only if and ;(ii); (iii)for all ; Definition 2. Consider the linear system of the following equations: where the coefficient matrix , , , is a crisp matrix and , , is called a FLS. Definition 3. A fuzzy number vector , given by the parametric form , , , is called a