Adaptive Algorithms for Multi-Warehouse Multi-Store Inventory System with Lost Sales and Fixed Replenishment Cost

Abstract

We consider a two-echelon inventory control problem for a Multi-Warehouse Multi-Store (MWMS) system with lost sales and fixed ordering cost at each store. We focus on a time horizon with no external replenishment, or the problem of “what to do until your (external) shipment comes in. The warehouses are stocked with inventories at the beginning of the horizon, and the stores dynamically replenish from the warehouses in each period. The delivery lead time is zero from a warehouse to any of the store it serves (e.g., each store is within 24 hrs driving distance from its servicing warehouses). This is a prevalent problem in many retail settings. The optimal policy for this problem is complex and state-dependent, and due to the well-known curse of dimensionality, computing the optimal policy using dynamic program is numerically intractable. In this paper, we develop provably near-optimal algorithms by integrating and extending three key ideas/approaches: (i) Lagrangian relaxation of the original complex dynamic inventory control problem, (ii) approximating the finite-horizon cost of an inventory problem with its infinite-horizon average cost, and (iii) dynamic re-adjustments of both replenishment and allocation policies. We first present a Lagrangian-based algorithm that is open-loop and non-adaptive, and discuss its performance in terms of loss rate (optimality gap) between its expected total cost and that of a true optimal policy. Then, we develop several easy-to-compute adaptive algorithms that dynamically re-adjust the policy parameters after observing sales at each store. We highlight the importance of proper policy adjustment to realize the benefit of pooling effect, which typically exists in the setting where one warehouse replenishes multiple stores. Our results show that adaptive readjustments significantly improve the performance of the base algorithm both theoretically and numerically.