We first show a sufficient and necessary condition that a solution of fuzzy relation inequalities with addition-min composition is a minimal one. We then prove that for every solution of the fuzzy relation inequalities there exists a minimal solution that is less than or equal to the solution in a very different way. We finally give an algorithm to find a minimal solution for a given solution, which is illustrated by numerical examples.

We first show a sufficient and necessary condition that a solution of fuzzy relation inequalities with addition-min composition is a minimal one. We then prove that for every solution of the fuzzy relation inequalities there exists a minimal solution that is less than or equal to the solution in a very different way. We finally give an algorithm to find a minimal solution for a given solution, which is illustrated by numerical examples.

This paper investigates minimal solutions of fuzzy relation inequalities with addition-min composition. It first shows the conditions that an element is a minimal solution of the inequalities, and presents the conditions that the inequalities have a unique minimal solution. It then proves that every solution of the inequalities has a minimal one and proposes an algorithm to searching for a minimal solution with computational complexity O (n 2) where n is the number of unknown variables of the inequalities. This paper finally describes all minimal solutions of the inequalities.