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Integrating the Number and Location of Retail Outlets on a Line with Replenishment Decisions

Joseph L. Rotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada
Faculty of Industrial Engineering and Management, Technion–Israel Institute of Technology, Technion City, 32000 Haifa, Israel
hussein.naseraldin{at}rotman.utoronto.ca
yale{at}technion.ac.il

We research the management approach that quantitatively combines decisions that affect different planning horizons—namely, the strategic and operational ones—and simultaneously derive the optimal values of these decisions. The system we investigate comprises retail outlets and customers in an infinite-horizon setting. Both retail outlets and customers are located on a finite homogenous line segment. The total demand posed by customers is normally distributed with known mean and variance. To optimally design and operate such a system, we need to determine the optimal values of the number of retail outlets, the location of each retail outlet, and the replenishment inventory levels maintained at each retail outlet. We analyze the system from an expected cost point of view, considering the fixed costs of operating the retail outlets, the expected holding and shortage costs, and the expected delivery costs. We show that all decisions can be represented as a function of the number of retail outlets. Moreover, we show that the system's expected cost function is quasi-convex in the number of retail outlets. We compare our model to a model that does not integrate these decisions at once. We show the advantage of our approach on both the solution and objective spaces. We propose an exact quantification of this advantage in terms of the cost and problem parameters. In addition, we point out several managerial insights.

Key Words: location-inventory model; implicit function theorem; risk pooling; infinite-horizon stochastic inventory; subadditive function; quasi-convex
History: Received: January 31, 2005;