On the handling of inconsistent systems based on min-implication compositions

In this article, we study the inconsistency of systems of fuzzy relational equations based on min-implication compositions, where the implications are residual implicators associated with continuous t-norms. The study focuses on three min-implication compositions: min-Gödel, min-Goguen, and min-Łukasiewicz. To address the inconsistency of a min-implication system, we study how to construct consistent systems close to the inconsistent system by minimally perturbing the right-hand side vector of the inconsistent system. The solutions of the obtained consistent systems are considered as approximate solutions of the inconsistent system. Closeness is measured with the Chebyshev distance, i.e. the distance induced by the L-infinity norm, between the right-hand side vector of the considered inconsistent system and the set of right-hand side vectors of the consistent systems that are defined using the same matrix: the matrix of the inconsistent system. We give explicit analytical formulas for computing the Chebyshev distances for min-Gödel, min-Goguen, and min-Łukasiewicz systems.