Finding Minimal Solution to Generalized Min-Max Programming Problem With Addition-Min Composition

Abstract

In order to describe the data transmission mechanism in the BitTorrent-like peer-to-peer file-sharing system, the addition-min fuzzy relation inequality system is introduced. At the same time, in order to distinguish the different importance of different terminals, the generalized min-max programming problem under the addition-min fuzzy relation inequality is proposed and studied. On the premise of ensuring that the quality level of data transmission meets all users, the minimal optimal solution can better avoid network congestion and improve the stability of data transmission. In this paper, we present how to find a minimal optimal solution to generalized min-max programming problems with addition-min composition. Firstly, some properties of the maximum optimal solution for the generalized min-max programming problem with addition-min combination are given. Then, by using the maximum optimal solution, the minimal optimal solution of the original problem is obtained by finding the minimal solution of the simplified constraint inequality. Moreover, convexity of the optimal solution set and the number of minimal optimal solutions are discussed. Finally, we propose an algorithm for finding the minimal optimal solution of the generalized min-max programming problem. Some examples are given to illustrate the feasibility and efficiency of the algorithm.

Keywords

• Generalized min-max programming problem
• fuzzy relation inequalities
• addition-min composition
• minimal solution