An active-set approach to finding a minimal-optimal solution to the min-max programming problem with addition-min fuzzy relational inequalities

In the literature, a BitTorrent-like peer-to-peer (BT-P2P) file-sharing system has been modeled as a system of fuzzy relational inequalities (FRI) with addition-min composition. And a min-max programming problem has been proposed to study its system congestion. From a cost-saving viewpoint, optimal solutions to the min-max programming problem may not be the minimal-optimal solution. The “minimal” solution gives better cost performance while the “optimal” solution gives the least system congestion. Such a drawback has been studied in the literature. In this paper, we propose a simple active-set approach to finding a cost-saving optimal solution, i.e. a minimal-optimal solution to the min-max programming problem. The complexity of our approach is O(m²n), where m is the number of decision variables and n is the number of constraints. Numerical examples are given to illustrate our procedures. Since our active-set approach depends on the given order of decision variables, by using a different sequence of decision variables, it may be able to find other minimal-optimal solutions. Therefore, our method may be able to provide more choices for the manager to use in decision-making.