INVERSE PROBLEM IN FUZZY RELATIONAL EQUATIONS

The paper deals with the problem of handling fuzzy relational equations of the form X · R = Y in the case of an empty solution set with respect to X, X(R, Y) = θ. Discussing the essence of the structure of the fuzzy relation R with respect to provided Y, algorithms have been proposed which enable us to modify the original equation (i.e. the fuzzy set Y) in a way leading to its genuine solution or at least an approximate one. It is explained how these modifications relax a set of constraints needed to be satisfied for any solvable equation. A grade of the modification of the original problem is expressed in terms of distortion of the fuzzy set Y which forms a cornerstone of the proposed procedures. Numerical examples are provided. Some aspects related to the area of diagnostic models in which these equations are studied are also treated. © 1990.