Competitive Facility Location with Market Expansion and Customer-centric Objective

TL;DR

This paper introduces a novel approach to competitive facility location considering market expansion and customer satisfaction, using advanced nonlinear optimization techniques and approximation algorithms to improve solution quality and computational efficiency.

Contribution

It presents a new nonlinear formulation with market expansion, develops an inner-approximation method for solving MINLPs as MILPs, and extends the approach to non-concave market functions.

Findings

Concave market expansion functions allow for a (1-1/e) approximation via greedy algorithms.

Inner-approximation method provides lower bounds and arbitrary precision solutions for MINLPs.

Experiments show the efficiency and effectiveness of the proposed methods.

Abstract

We study a competitive facility location problem, where customer behavior is modeled and predicted using a discrete choice random utility model. The goal is to strategically place new facilities to maximize the overall captured customer demand in a competitive marketplace. In this work, we introduce two novel considerations. First, the total customer demand in the market is not fixed but is modeled as an increasing function of the customers' total utilities. Second, we incorporate a new term into the objective function, aiming to balance the firm's benefits and customer satisfaction. Our new formulation exhibits a highly nonlinear structure and is not directly solved by existing approaches. To address this, we first demonstrate that, under a concave market expansion function, the objective function is concave and submodular, allowing for a $(1-1/e)$ approximation solution by a simple polynomial-time greedy algorithm. We then develop a new method, called Inner-approximation, which enables us to approximate the mixed-integer nonlinear problem (MINLP), with arbitrary precision, by an MILP without introducing additional integer variables. We further demonstrate that our inner-approximation method consistently yields lower approximations than the outer-approximation methods typically used in the literature. Moreover, we extend our settings by considering a\textit{ general (non-concave)} market-expansion function and show that the Inner-approximation mechanism enables us to approximate the resulting MINLP, with arbitrary precision, by an MILP. To further enhance this MILP, we show how to significantly reduce the number of additional binary variables by leveraging concave areas of the objective function. Extensive experiments demonstrate the efficiency of our approaches.