Discrete-continuous model for facility location problem with capacity-cost relation constraints

The facility location problem (FLP) involves optimally locating a set of facilities that must satisfy the demands of a group of customers distributed in a planar area. Traditionally, various FLP models have been applied to find new facilities by considering already-existing facilities, and choosing new locations among a given set of discrete candidates. These are unreasonable because a limited set of candidates may not include the optimal locations. In this study, we developed a mathematical model based on mixed-integer linear programming (MILP) for FLP. In this model, the locations of existing facilities are known and categorized as discrete. On the other hand, the locations of new facilities to be found are categorized as continuous and discrete. In addition, the optimal number of new facilities is determined by a judgment formula. The cost of facilities is also considered, including fixed opening costs and variable capacity-related costs. To make the model closer to reality, the entire area is divided into several zones with varying costs. We developed a greedy multiphase location method for the proposed MILP (MPL-MILP) that efficiently determines the exact locations of new facilities in a stepwise manner. For large-scale problems, we developed a two-variable-based dynamic iterative partial optimization (TVB-DIPO) to obtain near-optimal solutions within a given time limit. Computational experiments were conducted to test the performance of the proposed MPL-MILP and TVB-DIPO. The results indicate that MPL-MILP is suitable for small- and medium-sized problems with exact solutions, while TVB-DIPO is more suitable for large-scale problems with higher solution efficiencies.