Two-Level Capacitated Discrete Location with Concave Costs

In this paper, we study a general class of two-level capacitated discrete location problems with concave costs. The concavity arises from the economies of scale in production, inventory, or handling at the facilities and from the consolidation of flows for transportation and transshipment on the links connecting the facilities. Given the discrete nature of the problem, it is naturally formulated as a mixed-integer nonlinear program that uses binary variables for locational decisions and continuous variables for routing flows. We present an alternative formulation that only uses continuous variables and discontinuous functions, resulting in a nonlinear program with a concave objective function. Our main goal is to computationally compare these two modeling approaches under the same solution framework. In particular, we present an exact branch-and-bound algorithm that uses (integer) linear relaxations of the proposed formulations to optimally solve large-scale instances. The algorithm is enhanced with a cost-dependent spatial branching strategy and preprocessing step to improve its convergence. Extensive computational experiments are performed to assess the performance of the exact algorithm. Based on real location data from 3,109 counties in the contiguous United States, we also present a sensitivity analysis to showcase the impact of considering concave costs in location and assignment decisions.