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

This paper proposes a continuum approximation model framework to solve a 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. To overcome computational challenge, a continuous approximation model is developed to find the optimal facility density in the spatiotemporal continuum. Then we propose a tube model to discretize the resulted continuous facility density function into a set of time-varying facility location trajectories. To enforce consistency of facility location over time, an iterative regulation procedure based on a penalty method is applied. We present convergence properties of the proposed iterative regulation procedure and further derive conditions under which the proposed continuous approximation approach and the tube model provide tight approximation error bounds. A series of numerical experiments, including comparison with discrete model counterparts, are conducted to illustrate the performance (accuracy and convergence) of the proposed modeling framework. Our results show that the proposed method solves the dynamic facility location problem effectively to reasonable accuracy.