Balancing retailer inventories

This paper examines the optimality of inventory ‘balancing’ in a one-warehouse N-retailer distribution system facing stochastic demand for a single product over T successive-time intervals. In particular, we consider the division of predetermined quantities of warehouse stock among retailers in each interval. Balancing attempts to bring the retailer-inventories to the same (normalized with respect to first-interval demand) net inventory level. When the demand distribution over the T periods is symmetric with respect to all pairs of retailers, and retailer cast over the T-intervals is a convex function of the T shipments, we show that balancing divisions are optimal. The required convexity condition is shown to be satisfied for some familiar cost functions when shortages are backordered and lost. However for other cost functions, the convexity condition is satisfied under backordering, but not under lost sales. We also consider the nonidentical-retailer (demands are not symmetric) case and prove examples to show that balancing is generally not optimal, even for relatively common, simple cost functions.