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Fuzzy symmetric solutions of fuzzy linear systems

机译：模糊线性系统的模糊对称解

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摘要

In this paper, we propose a simple and practical method to solve a fuzzy linear system AX=b, where X and b are fuzzy triangular vectors with non-zero spreads and matrix A is nonsingular with real coefficients. The aim of this paper is twofold. First, we obtain the crisp solution of a fuzzy linear system. To this end, we solve the 1-cut of a fuzzy linear system. Second, we allocate some unknown symmetric spreads to any rows of fuzzy linear system in 1-cut position. Thus, a fuzzy linear system in 1-cut will be transformed to a system of interval equations. The symmetric spreads of each element of a fuzzy vector solution are derived by solving such an interval system. Moreover, based on the obtained symmetric spreads we derive three types of solutions. However, one of the mentioned spreads has pessimistic/optimistic attitude that is determined with a decision maker. It seems that such a solution is a connection solution between Tolerable Solution Set (TSS) and Controllable Solution Set (CSS). Also, we derive the maximal solution and the minimal solution of an original fuzzy linear system which are placed in TSS and CSS, respectively. Finally, some numerical examples are given to illustrate the efficiency and ability of the proposed method.
机译：在本文中，我们提出了一种简单实用的方法来求解模糊线性系统AX = b，其中X和b是具有非零扩展的模糊三角向量，而矩阵A是具有实系数的非奇异向量。本文的目的是双重的。首先，我们获得了模糊线性系统的清晰解。为此，我们解决了模糊线性系统的1-cut问题。其次，我们将一些未知的对称展布分配给1割位置的模糊线性系统的任何行。因此，将一切模糊线性系统转换为区间方程组。通过求解这种间隔系统，可以得出模糊矢量解的每个元素的对称分布。此外，基于获得的对称扩展，我们导出了三种类型的解。但是，上述价差之一具有由决策者确定的悲观/乐观态度。看来这种解决方案是可容忍解决方案集（TSS）和可控解决方案集（CSS）之间的连接解决方​​案。同样，我们推导了分别放置在TSS和CSS中的原始模糊线性系统的最大解和最小解。最后，通过数值算例说明了该方法的有效性。

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