Minimax programming problems subject to addition-Łukasiewicz fuzzy relational inequalities and their optimal solutions

This article focuses on minimax programming problems subject to addition-Łukasiewicz fuzzy relational inequalities. We first establish two necessary and sufficient conditions that a solution of the fuzzy relational inequalities is a minimal one and explore the existence condition of the unique minimal solution. We also supply an algorithm to search for minimal solutions of the fuzzy relational inequalities starting from a given solution. We then apply minimal solutions of the fuzzy relational inequalities to the minimax programming problems for searching optimal solutions. We provide two algorithms to solve a kind of single variable optimization problems, and obtain the greatest optimal solution. The algorithm for finding minimal solutions of a given solution are also used for searching minimal optimal solutions.