Combining a Continuous Search Algorithm with a Discrete Search Algorithm for Solving Non-linear Bi-level Programming Problem

The multi-level programming problems, 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 bi-level programming problem (BLPP) is an appropriate tool to model these real problems. It has been proven that the general BLPP is an NP-hard problem, so it is a practical and complicated problem therefore solving this problem would be significant. However the literature shows several algorithms to solve different forms of the bi-level programming problems (BLPP), but there is no any hybrid approach of combining of two meta-heuristic algorithms. In this paper, the authors combine 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). Using the Karush-Kuhn-Tucker conditions the BLPP is converted to a non-smooth single level problem, and then it is smoothed by a new heuristic method for using PSOMGA. The smoothed problem is solved using PSOMGA which is a fast approximate method for solving the non-linear BLPP. The presented approach achieves an efficient and feasible solution in an appropriate time, as justified by comparison with test problems.