Solution of Non-Square Fuzzy Linear Systems

TL;DR

This paper develops a comprehensive geometric approach to solve fuzzy linear systems with non-square matrices, providing explicit solutions for all cases and introducing a novel method for general systems.

Contribution

It introduces a unified geometric framework for solving fuzzy linear systems with non-square matrices, including new methods for overdetermined and underdetermined cases.

Findings

Solution set for square systems is a parallelepiped.

Overdetermined systems form convex polyhedra with a new computation method.

General solutions are derived for non-full rank coefficient matrices.

Abstract

In this paper, a linear system of equations with crisp coefficients and fuzzy right-hand sides is investigated. All possible cases pertaining to the number of variables, n, and the number of equations, m, are dealt with. A solution is sought not as a fuzzy vector, as usual, but as a fuzzy set of vectors. Each vector in the solution set solves the given fuzzy linear system with a certain possibility. Assuming that the coefficient matrix is a full rank matrix, three cases are considered: For m = n (square system), the solution set is shown to be a parallelepiped in coordinate space and is expressed by an explicit formula. For m > n (overdetermined system), the solution set is proved to be a convex polyhedron and a novel geometric method is proposed to compute it. For m < n (underdetermined system), by determining the contribution of free variables, general solution is computed. From the results of three cases mentioned above, a method is proposed to handle the general case, in which the coefficient matrix is not necessarily a full rank matrix. Comprehensive examples are provided and investigated in depth to illustrate each case and suggested method.