استفاده از الگوریتم بهینه سازی ازدحام ذرات برای حل مسئله برنامه ریزی خطی سطح دوطرفه

Bi-level linear programming is a technique for modeling decentralized decision. It consists of the upper-level and lower-level objectives. This paper attempts to develop an efficient method based on particle swarm optimization (PSO) algorithm with swarm intelligence. The performance of the proposed method is ascertained by comparing the results with genetic algorithm (GA) using four problems in the literature and an example of supply chain model. The results illustrate that the PSO algorithm outperforms GA in accuracy.

Multi-level programming techniques are developed to solve decentralized planning problems with multiple decision makers in a hierarchical organization. The bi-level linear programming problem (BLPP) is a special case of multi-level linear programming problems with a two-level structure [1] and [2]. Most of the mathematical programming models deal with a single decision maker and a single objective function and are used for centralized planning systems. The BLPP on the other hand is developed for decentralized planning systems in which the upper level is termed as the leader and the lower level pertains to the objective of the follower. In the BLPPs, each decision maker tries to optimize its own objective function without considering the objective of the other party, but the decision of each party affects the objective value of the other party as well as the decision space. There already have been some methods for solving BLPPs, like methods based on vertex enumeration and meta-heuristics. In this study, an attempt is made to employ particle swarm optimization (PSO) algorithm for solving BLPPs due to its promising performance in optimization problems. Four problems taken from the literature are adopted to test the proposed algorithm’s performance. The experimental results indicate that PSO algorithm outperforms genetic algorithm (GA) in accuracy and has better stability. In addition, an example of supply chain model also reveals that PSO algorithm is a suitable approach for solving BLPPs. The rest of this paper is organized as follows. Section 2 provides basic concept of BLPPs and PSO algorithm. The proposed PSO algorithm for solving BLPPs will be presented in Section 3, while Section 4 makes a thorough discussion on computational experiences. Finally, the concluding remarks are made in Section 5.

This study has proposed a PSO-based method for BLPPs. The experimental results of four problems taken from the literature illustrate that the proposed PSO algorithm outperforms GA for most of the problems. Besides, PSO algorithm has smaller standard deviation. This implies that PSO algorithm has better stability compared to GA. In addition, the computational time of PSO algorithm is also smaller than that of GA. However, the current algorithm is only suitable for BLPPs. It is promising to extent the proposed method for multi-level linear programming problems in the future. Besides, it is feasible to gather virtues of PSO algorithm and GA to ascend learning efficiency.