Resolution of a system of the max-product fuzzy relation equations using L∘U-factorization

In this paper, the LU-factorization is extended to the fuzzy square matrix with respect to the max-product composition operator called L∘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∘U, where “∘” 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∘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∘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(n3). Since the determination of the minimal solutions is an NP-hard problem, the algorithm can be very important from the practical point of view.