Competitive facility location problem with foresight considering discrete-nature attractiveness for facilities: Model and solution

Location of distribution and collection/inspection centers for an entrant firm in (a) a closed-loop supply chain from the set of their corresponding candidate points considering the already-exist firm's facilities location as shown in (b), and its future reactions. • Developing a leader–follower game in a closed-loop supply chain for the facility location. • Proposing a novel reformulation procedure to handle discrete decisions in the follower’s optimization problem. • Developing an efficient global optimization strategy for the new model using the improved branch-and-refine algorithm. • Illustrating some applications of the model and solution method. • Using some numerical examples to provide additional managerial insights. This paper addresses a bi-level mixed-integer nonlinear programming (MINLP) model for the competitive facility location problem in a closed-loop supply chain (CLSC), in which a firm (i.e., leader) aims at entering a market by locating new distribution and collection facilities, where a competitor (i.e., follower) already exists. The goal is to find the location and attractiveness of each facility going to be established by the leader who seeks to maximize its profit while also taking the follower’s response into account. The attractiveness of each facility is a function of integer variables related to the facility’s characteristics. Customer behavior is considered to be probabilistic based on the Huff gravity-based rule. To globally optimize the model, a procedure that handles the discrete decisions of the follower’s problem is proposed. Afterward, by replacing the inner level convex program with its corresponding Karush–Kuhn–Tucker (KKT) conditions, the bi-level MINLP is converted into a single-level MINLP model, optimized by an improved branch-and-refine algorithm. Numerical experiments on randomly generated instances are conducted to illustrate the model’s applicability. Moreover, through a computational analysis of the proposed model, the amount of gain the leader makes and the follower loses due to foresight in the competition are calculated.