Presolving and cutting planes for the generalized maximal covering location problem

Abstract

This paper considers the generalized maximal covering location problem (GMCLP) which establishes a fixed number of facilities to maximize the weighted sum of the covered customers, allowing customers' weights to be positive or negative. The GMCLP can be modeled as a mixed integer programming (MIP) formulation and solved by off-the-shelf MIP solvers. However, due to the large problem size and particularly, poor linear programming (LP) relaxation, the GMCLP is extremely difficult to solve by state-of-the-art MIP solvers. To improve the computational performance of MIP-based approaches for solving GMCLPs, we propose customized presolving and cutting plane techniques, which are the isomorphic aggregation, dominance reduction, and two-customer inequalities. The isomorphic aggregation and dominance reduction can not only reduce the problem size but also strengthen the LP relaxation of the MIP formulation of the GMCLP. The two-customer inequalities can be embedded into a branch-and-cut framework to further strengthen the LP relaxation of the MIP formulation on the fly. By extensive computational experiments, we show that all three proposed techniques can substantially improve the capability of MIP solvers in solving GMCLPs. In particular, for a testbed of 40 instances with identical numbers of customers and facilities in the literature, the proposed techniques enable to provide optimal solutions for 13 previously unsolved benchmark instances; for a testbed of 56 instances where the number of customers is much larger than the number of facilities, the proposed techniques can turn most of them from intractable to easily solvable.