A Continuum Approximation Approach to the Dynamic Facility Location Problem in a Growing Market

This paper proposes a continuum approximation (CA) model to solve the dynamic facility location problem for a large-scale growing market. The objective is to determine the optimal facility location and deployment time that minimize the costs for facility construction and customer service in a planning horizon. To overcome computational challenges, the CA model determines the optimal facility density in the spatiotemporal continuum. Then we propose a tube model to discretize the resulting continuous facility density function into a set of time-varying facility location trajectories. To enforce consistency in facility location over time, an iterative regulation procedure based on a penalty method is applied. We present convergence properties of the proposed procedure and further derive conditions under which the CA approach and the tube model yield tight approximation error bounds. We conduct a series of numerical experiments to illustrate the applicability and computational performance (e.g., accuracy and convergence) of the proposed modeling framework, first via comparison with discrete model counterparts using hypothetical data, and then via application to an empirical case for the state of Illinois. Our results show that the proposed method effectively solves the dynamic facility location problem to reasonable accuracy. Various managerial insights are also drawn.