Adaptive Lagrangian Policies for Multi-Warehouse Multi-Store Inventory System with Lost Sales

We consider the inventory control problem of a Multi-Warehouse Multi-Store (MWMS) system over a time horizon that the warehouses receive no external replenishment. This problem is prevalent in retail settings, and it is referred to in Jackson (1988) as “what to do until your (external) shipment comes in". The warehouses are stocked with initial inventories and the stores dynamically replenish inventory from the warehouses in each period of the planning horizon. Excess demand in each period at any store is lost. The optimal policy for this problem is complex and state-dependent, and due to the curse of dimensionality, computing the optimal policy using standard dynamic programming is numerically intractable. Static Lagrangian Base Stock (LaBS for short) policy has been developed for this problem (Miao et al. 2022) and shown to be asymptotically optimal. In this paper, we develop adaptive policies by dynamically adjusting the control parameters of a vanilla static LaBS policy using realized historical demands. We show, both theoretically and numerically, that the adaptive policies significantly improve the performance of LaBS policy, with the magnitude of improvement characterized by the number of times the policy can be adjusted. In particular, when the number of adjustment is a logarithm of the length of time horizon, the policy is rate-optimal in the sense that the rate of the loss (against the performance benchmark in terms of the length of the time horizon) matches that of the theoretical lower bound for any policy. Our analysis highlights the importance of a careful policy adjustment scheme to enjoy the most benefit of "pooling effect"