OPTIMAL POLICY FOR TWO-STAGE BROWNIAN INVENTORY SYSTEMS UNDER THE LONG-RUN AVERAGE COST CRITERION

A two-stage Brownian inventory system with fixed order costs at both stages under the long-run average cost criterion is investigated. We first develop some structural properties for an optimal policy such as nested and zero-inventory-ordering features when the upper stage has a zero leadtime. We decompose the two-stage system into two independent single-stage systems by utilizing the induced-penalty cost. The decomposition helps us to obtain a lower-bound of the optimal cost for the two-stage inventory system. Leveraging the nested and zero-inventory-ordering properties, by the lower-bound approach, the problem of obtaining the optimal policy is transformed to solving a constrained quasi-convex optimization problem. The parameters characterizing the optimal policy are further derived by solving the KKT conditions of the associated Lagrangian relaxation problem.