Solving Large-scale Competitive Facility Location under Random Utility Maximization Models

This work concerns the maximum capture problem with random utilities, i.e., the problem of seeking to locate new facilities in a competitive market such that the captured demand of users is maximized, assuming that each individual chooses among all available facilities according to a random utility maximization model. The challenge when solving this problem is the nonlinearity of the objective function. Existing approaches often only use the multinomial (MNL) logit model due to its simple structure, and address this challenge by formulating the problem into Mixed-Integer Linear Programing (MILP) models, so it can be solved by a MILP solver. There are two issues associated with this approach, namely, (i) the use of the MNL model retains the well-known independence of irrelevant alternatives property that may be undesirable in the context, and (ii) the MILP model could be difficult to solve for large instances. In this paper, we address the aforementioned issues by dealing with the facility location problem under the mixed MNL (MMNL) model, which is general and fully flexible, compared to the MNL model. In addition, we do not solve equivalent MILP models, instead we propose an exact algorithm based on the outer approximation framework to quickly solve the associated 0-1 nonlinear problems. We test our algorithm using generated instances of different sizes, and show that our algorithm is much faster, compared to other existing approaches in the context. Especially, for several large-scale instances (thousands of locations and zones) under the MMNL model, our algorithm manages to find an optimal solution in few seconds, while other approaches cannot converge within a time budget of several hours.