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Deriving minimal solutions for fuzzy relation equations with max-product composition. (English) Zbl 1151.03345
Summary: 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 Syst. 153, 261–273 (2005; Zbl 1073.03538)]. 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.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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