Near-Optimal Mixed ( s,S ) Policy for a Multiwarehouse, Multistore Inventory System with Lost Sales and Fixed Cost

We consider a firm managing a multiperiod, multiwarehouse, multistore (MWMS) inventory problem with fixed ordering cost at each store over a finite time horizon. The warehouses are endowed with initial inventories at the start of the horizon, and the stores are periodically replenished from the warehouses. The decisions are the order quantities from each store at each period. The optimal policy for this problem is complex and computationally intractable. We construct a mixed ( s , S ) policy based on the optimal solutions of a Lagrangian relaxation. Under this policy, each store makes use of at most two ( s , S ) policies; one is applied during the first phase of the selling horizon, and the second is applied in the remaining periods. We prove that this policy is near optimal as the length of the time horizon grows. In contrast to the existing works on the MWMS problem without fixed cost for which near-optimal policies can be developed using an optimal Lagrangian solution, with fixed cost, it is crucial to adopt a mixture of Lagrangian solutions, and simply applying a pure optimal Lagrangian solution can be highly suboptimal.