An efficient branch-and-cut approach for large-scale competitive facility location problems with limited choice rule

In the paper, we consider the competitive facility location problem with limited choice rule (CFLPLCR), which attempts to open a subset of facilities to maximize the net profit of a newcomer company, requiring customers to patronize only a limited number of opening facilities and an outside option. We propose an efficient branch-and-cut (B&C) approach for the CFLPLCR based on newly proposed mixed integer linear programming (MILP) formulations. Specifically, by establishing the submodularity of the probability function, we develop an MILP formulation for the CFLPLCR using the submodular inequalities. For the special case where each customer patronizes at most one open facility and the outside option, we show that the submodular inequalities can characterize the convex hull of the considered set and provide a compact MILP formulation. Moreover, for the general case, we strengthen the submodular inequalities by sequential lifting, resulting in a class of facet-defining inequalities. The proposed lifted submodular inequalities are shown to be stronger than the classic submodular inequalities, enabling to obtain another MILP formulation with a tighter linear programming (LP) relaxation. By extensive numerical experiments, we show that the proposed B&C approach outperforms the state-of-the-art generalized Benders decomposition approach by at least one order of magnitude. Furthermore, it enables to solve CFLPLCR instances with 10000 customers and 2000 facilities.