Heuristics for improving the solution of p-median location problem with Erlenkotter approach

This paper deals with the problem of designing the optimal structure of a public service system. The problem can be often formulated as a p-median location problem. Solving the p-median location problem in the public service system design is NP-hard problem. Real instances of the problem are characterized by big number of possible service center locations, which can take the value of several hundreds or thousands. The optimal solution can be obtained by the universal IP solvers only for smaller instances of the problem. The universal IP solvers are very time-consuming and often fail when a large instance is solved. Our approach to the problem is based on the Erlenkotter procedure and the Lagrangean relaxation. The Erlenkotter procedure solves the uncapacitated facility location problem. Using the Lagrangean relaxation allow to solve the p-median location problem. The suggested approach finds the optimal solution in the most studied instances. The feasibility and the quality of the resulting solutions of the suggested approach depends on the convenient setting of the Lagrangean multiplier. The convenient value of the multiplier can be obtained by a bisection algorithm. The resulting multiplier cannot guarantee the determination of the optimal solution, but it ensures the lower bound and the near-to-optimal solution. If our approach does not obtain the optimal solution, then it improves the near-to-optimal solution by some heuristic. We designed some heuristics for improving the obtained solution and choose the best improving heuristic. The improving heuristics are verified on comparison the resulting solution obtained by the improving heuristics and the optimal solution obtained by the universal IP solver XPRESS-IVE in the computational time and the quality of solutions.