COMBININGCONTINUOUS AND DISCRETE MET-HEURISTIC APPROACHES FOR SOLVING NON-LINEAR TRI–LEVEL PROGRAMMING PROBLEM

The multi-level programming problems (MLPP) have received much interest from researchers because of their application in several areas such as economic, traffic, finance, management, transportation and so on. Among these, the tri-level programming problem (TLPP) is an appropriate tool to model these real problems. It has been proven that the general TLPP is an NP-hard problem, so it is a practical and complicated problem therefore solving this problem would be significant. The literature shows several algorithms to solve different forms of the bi-level programming problems (BLPP), but only one attempt for using linear TLPP model and any non-linear TLPP. The most important part in this paper is combining particle swarm optimization (PSO), which is a continuous approach, with a proposed modified genetic algorithm (MGA), which is a discrete algorithm, using a heuristic function and constructing an effective hybrid approaches (PSOMGA). Also we attempt to develop two applied problems which would be modeled to the non-linear TLPP, then by using the Karush-Kuhn-Tucker conditions the TLPP is converted to a non-smooth single level problem, and then it is smoothed by a new heuristic method. The smoothed problem is solved using PSOMGA which is a fast approximate method for solving the non-linear TLPP. The presented approach achieves an efficient and feasible solution in an appropriate time which has been evaluated by solving the problems.