An Approximative Lexicographic Min-Max Approach to the Discrete Facility Location Problem

We propose a new approximative approach to the discrete facility location problem that provides solutions close to the lexicographic minimax optimum. The lexicographic minimax optimum is concept that allows to find equitable location of facilities. Our main contribution is the approximation approach, which is based on the rules allowing: (i) to take into account the multiplicities assigned to different customers; (ii) to detect whether for a given distance active customers can reach higher, equal or smaller distance to the closest located facility; and (iii) to use methods customized for solving the p-median problem. Customized methods can handle larger problems than state-of-the-art general purpose integer programming solvers. We use the resulting algorithm to perform extensive study using the well-known benchmarks and benchmarks derived from the real-world road network data. We demonstrate that our algorithm allows to solve larger problems than existing algorithms and provides high-quality solutions. The algorithm found the optimal solution for all tested benchmarks, where we could compare the results with the exact algorithm.