Deriving minimal solutions for fuzzy relation equations with max-product composition

This work considers fuzzy relation equations with max-product composition. The critical problem in solving such equations is to determine the minimal solutions when an equation is solvable. However, this problem is NP-hard and difficult to solve [A.V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy Sets and Systems 153 (2005) 261–273]. This work first examines the attributes of a solvable equation and characteristics of minimal solutions, then reduces the equation to an irreducible form, and converts the problem into a covering problem, for which minimal solutions are correspondingly determined. Furthermore, for theoretical and practical applications, this work presents a novel method for obtaining minimal solutions. The proposed method easily derives a minimal solution, and obtains other minimal solutions from this predecessor using a back-tracking step. The proposed method is compared with an existing algorithm, and some applications are described in detail.