Sequential Competitive Facility Location: Exact and Approximate Algorithms

TL;DR

This paper introduces exact and approximate algorithms for a sequential competitive facility location problem modeled as a Stackelberg game, improving solution speed and quality for large instances with probabilistic demand.

Contribution

It develops a bilevel MINLP reformulation, valid inequalities, and algorithms that enhance solution efficiency and quality for the CFLP, including extensions and practical insights.

Findings

Exact algorithm accelerates large instance solutions

Approximation algorithm finds high-quality solutions quickly

Numerical studies demonstrate improved computational performance

Abstract

We study a competitive facility location problem (CFLP), where two firms sequentially open new facilities within their budgets, in order to maximize their market shares of demand that follows a probabilistic choice model. This process is a Stackelberg game and admits a bilevel mixed-integer nonlinear program (MINLP) formulation. We derive an equivalent, single-level MINLP reformulation and exploit the problem structures to derive two valid inequalities, based on submodularity and concave overestimation, respectively. We use the two valid inequalities in a branch-and-cut algorithm to find globally optimal solutions. Then, we propose an approximation algorithm to find good-quality solutions with a constant approximation guarantee. We develop several extensions by considering general facility-opening costs, outside competitors, as well as diverse facility-planning decisions, and discuss solution approaches for each extension. We conduct numerical studies to demonstrate that the exact algorithm significantly accelerates the computation of CFLP on large-sized instances that have not been solved optimally or even heuristically by existing methods, and the approximation algorithm can quickly find high-quality solutions. We derive managerial insights based on sensitivity analysis of different settings that affect customers' probabilistic choices and the ensuing demand.