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Resolution of a system of the MAX-product fuzzy relation equations using $$L \circ U$$-factorization. (English) Zbl 1284.65045
Summary: In this paper, the $$LU$$-factorization is extended to the fuzzy square matrix with respect to the max-product composition operator called $$L \circ U$$-factorization. Equivalently, we will find two fuzzy (lower and upper) triangular matrices $$L$$ and $$U$$ for a fuzzy square matrix $$A$$ such that $$A= L \circ U$$, where “$$\circ$$” is the max-product composition. An algorithm is presented to find the matrices $$L$$ and $$U$$. Furthermore, some necessary and sufficient conditions are proposed for the existence and uniqueness of the $$L \circ U$$-factorization for a given fuzzy square matrix $$A$$. An algorithm is also proposed to find the solution set of a square system of Fuzzy Relation Equations (FRE) using the $$L \circ U$$-factorization. The algorithm finds the solution set without finding its minimal solutions and maximum solution. It is shown that the two algorithms have a polynomial-time complexity as $$O(n^3)$$. Since the determination of the minimal solutions is an NP-hard problem, the algorithm can be very important from the practical point of view.

##### MSC:
 65F05 Direct numerical methods for linear systems and matrix inversion 03E72 Theory of fuzzy sets, etc. 15B15 Fuzzy matrices
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