Optimal control of a divergent N-echelon inventory system

Consider a divergent multi-echelon inventory system, e.g. 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 at each facility for holding inventory. Also, linear penalty costs are incurred at the most downstream facilities for backorders. 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 rationing functions at each facility, can be determined by decomposition of the system. This decomposition result reduces complex multi-dimensional control problems to 'simple' one-dimensional problems, which closely resemble the classical newsboy problem achieving a proper trade-off between the consequences of too much stock and too little inventory. Keywords: multi-echelon, optimal control, inventory, allocation, rationing, divergent