Abstract
In this paper we propose some new methods to solve the polynomial form of composite fuzzy relational equations represented in the form of ∪i=1I(A(i)_??_X_??_B(i))=C or ∩i=1I(A(i)ΔX-ΔB(i))=C for the unknown fuzzy relation X, where A(i), B(i), C are the given fuzzy relations, and _??_, Δ denote the operations of max-min, min-max compositions, respectively.
Several methods to solve the monomial form of composite fuzzy relational equations, which in general represented in the form of A*X=B, were formerly presented by E. Sanchez in 1976 and Y. Tsukamoto in 1979, and have been fully giving the availabilities to the various fields of practical applications, such as the fuzzy medical diagnosis, fuzzy failure diagnosis, fuzzy control problem and fuzzy information processing. However, when needed to approach to the more complex fuzzy system and to represent the system more in detail, we must face with the polynomial form of composite fuzzy relational equations representing the system. Then it arises that the former methods for the monomial form can not be directly applied to the method of solving the polynomial form of composite fuzzy relational equations.
In this paper we show the methods to resolve the polynomial form into the monomial form by the equivalent simplification based on the direct product operations of fuzzy relations and the equivalent decomposition, and to obtain the solutions to the polynomial form of composite fuzzy relational equations on the basis of Sanchez's method of inverse operations to the monomial form. Several numerical examples are presented in order to show the effectiveness of the methods of solving th polynomial form of composite fuzzy relational equations.
