Convexity Results for Stochastic Inventory Networks

In this paper, we establish the convexity of important cost functions in a general class of multi-echelon inventory models. In particular, we first study an assembly system with a single finished product managed using an echelon order-up-to policy. We show that the shortage penalty cost over any horizon is jointly convex with respect to the base-stock levels and capacity levels. Our second result pertains to an arbitrary inventory network, with multiple components, products, production stages and distribution locations, managed optimally. We show that the cost-to-go function of the dynamic program is jointly convex in the inventory state vector and the capacity vector for both the backorder and lost sales models. These convexity properties have implications for developing algorithms for making optimal inventory and capacity decisions in such systems.