An efficient branch-and-cut approach for the sequential competitive facility location problem under partially binary rule

We investigate the sequential competitive facility location problem (SCFLP) under partially binary rule where two companies sequentially open a limited number of facilities to maximize their market shares, requiring customers to patronize, for each company, the facility with the highest utility. The SCFLP is a bilevel mixed integer nonlinear programming (MINLP) problem and can be rewritten as a single-level MINLP problem, where each nonlinear constraint corresponds to a hypograph of a multiple ratio function characterizing the leader's market share for a fixed follower's location choice. By establishing the submodularity of the multiple ratio functions, we characterize the mixed 0-1 set induced by each hypograph using submodular inequalities and extend a state-of-the-art branch-and-cut (B&C) algorithm to the considered SCFLP. To address the challenge of poor linear programming (LP) relaxation of the underlying formulation, we develop two new mixed integer linear programming (MILP) formulations for the SCFLP as well as efficient B&C algorithms based on them. The first MILP formulation is based on a class of improved submodular inequalities, which include the classic submodular inequalities as special cases, and together with the trivial inequalities characterize the convex hull of the mixed 0-1 set. The second one is an extended formulation of the first one that provides the same LP relaxation bound. We also develop efficient algorithms for the separations of the exponential families of the inequalities in the MILP formulations. Extensive computational experiments show that the proposed B&C algorithms significantly outperform an adapted state-of-the-art B&C algorithm and a sophisticated heuristic algorithm in the literature. Moreover, the proposed B&C algorithms can find optimal solutions for SCFLP instances with up to 1000 customers and facilities within a two-hour time limit.