Optimal control of a divergent multi-echelon inventory system

Consider a divergent multi-echelon inventory system, such as a distribution system or a production system. At every facility in the system orders are placed (or production is initiated) periodically. The order arrives after a fixed lead time. At the end of each period linear costs are incurred penalty costs are incurred at the most downstream facilities for back-orders. The objective is to minimize the expected holding and penalty costs per period. We prove that under the balance assumption it is cost optimal to control every facility by an order-up-to-policy. The optimal replenishment policy, i.e. the order-up-to-level and the allocation functions at each facility, can be determined by system decomposition. This decomposition reduces complex multi-dimensional control problems to simple one-dimensional problems.