-Matrices in Fuzzy Linear Systems

We consider a class of fuzzy linear system of equations and demonstrate some of the existing challenges. Furthermore, we explain the efficiency of this model when the coefficient matrix is an -matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis. 1. Introduction In the field of scientific and technical computation, various equations which describe realistic problems like natural phenomena or engineering problems such as computational fluid dynamics, finite differences methods, finite element methods, statistics, time/frequency domain circuit simulation, dynamic and static modeling of chemical processes, cryptography, magnetohydrodynamics, electrical power systems, differential equations, quantum mechanics, structural mechanics (buildings, ships, aircraft, and human body parts…), heat transfer, MRI reconstructions, vibroacoustics, linear and nonlinear optimization, financial portfolios, semiconductor process simulation, economic modeling, oil reservoir modeling, astrophysics, crack propagation, Google page rank, Gene page rank, 3D computer vision, cell phone tower placement, tomography, model reduction, nanotechnology, acoustic radiation, density functional theory, quadratic assignment, elastic properties of crystals, natural language processing, DNA electrophoresis, and so forth must be solved numerically. These problems can lead to solving a system of linear equations. There are many methods for solving linear systems; see [1–7] and the references therein. Nevertheless, when coefficients of a system are ambiguous and there is some inexplicit information about the exact amount of parameters, one can solve a linear equation system by fuzzy logic. In 1965 [8], fuzzy logic was proposed by Zadeh and, following his work, many papers and books were published in fuzzy system theory. In particular, the solutions of fuzzy linear systems have been considered by many researchers, for example, [8–14]. Some investigations on the numerical solution of the fuzzy linear systems have also been reported; see [15–23] and references therein. Most of these studies use the same expansion of [12], where the coefficient matrix is crisp and the right-hand side is an arbitrary fuzzy number vector. Friedman et al. [12], using the embedding method, presented a general model for solving an fuzzy linear system and replaced the fuzzy linear system by crisp linear system. They studied also the uniqueness of the fuzzy solution for this model [12]. Dehghan and Hashemi [16] investigated the existence of a solution for this model under the