Document Type : Original Research


1 Department of Mathematics, Payam Noor University of Tehran, Tehran, Iran

2 Department of Industry, University of Kurdistan, Sanandaj, Iran


In the recent years, the multi-level programming problems specially the bi-level and tri-level programming problems (TLPP) are interested by many researchers and these problems, particularly TLPP, are known as an appropriate tool to solve the real problems in several areas of optimization such as economic, traffic, finance, management, transportation, computer science and so on. Also, it has been proven that the general bi-level and TLPP are NP-hard problems. The literature shows it has been proposed a few attempts for solving using TLPP. In this paper, we attempt to propose a new function for smoothing the tri-level programming problem after using Karush-Kuhn-Tucker condition, also we develop two effective approaches, one based on Genetic algorithm, which it is an approximate approach, and the other based on the hybrid algorithm by combining the proposed method based on penalty function and the line search algorithm for solving the linear TLPP. In both of these approaches, by using the Karush-Kuhn-Tucker condition the TLPP is converted to a non-smooth single problem, and then it is smoothed by proposed functions. Finally, the smoothed problem is solved using both of the proposed approaches. The presented approaches achieve an efficient and feasible solution in an appropriate time which has been evaluated by comparing to references and test problems.


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