Algorithms For Calculating Exactly One Minimal Solution Of A Finite Fuzzy Relation Equation With Max-min Composition

This paper deals with finding one minimal solution of a finite fuzzy relation equation with max-min composition assigned on [0,1]. First, it gives an algorithm to find one minimal solution of the fuzzy relation equation. The algorithm which uses the maximum solution of the fuzzy relation equation as an initial vector can find exactly one minimal so-lution of the fuzzy relation equation. We finally analyze the computational complexity of the algorithm and supply a numerical example to illustrate the algorithm. Then, this paper introduces a concept of an ε-minimal solution of the fuzzy relation equation. It proposes an algorithm to find one e-minimal solution by using fuzzy neural network who learns starting from the maximum solution of the fuzzy relation equation, and it shows the convergence of the algorithm and illustrates the algorithm by a numerical example.