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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.

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