On the handling of inconsistent systems based on max-product and max-Lukasiewicz compositions

In this article, we study the inconsistency of systems of fuzzy relational equations based either on the max-product composition or on the max-Lukasiewicz composition. Given an inconsistent system, we compute by an explicit analytical formula the Chebyshev distance (defined by L-infinity norm) between the right-hand side vector of the inconsistent system and the set of right-hand side vectors of consistent systems defined with the same matrix: that of the inconsistent system. The main result that allows us to obtain these formulas is that the Chebyshev distance is the smallest positive value that satisfies a vector inequality, whatever the t-norm used in the composition. We then give explicit analytical formulas for computing the greatest Chebyshev approximation of the right-hand side vector of an inconsistent system and its greatest approximate solution. The analytical formula of the Chebyshev distance associated with max-Lukasiewicz systems allows us to show that the fuzzy matrices which are invertible for max-Lukasiewicz composition are the permutation matrices.