Adaptive Lagrangian Policies for a Multiwarehouse, Multistore Inventory System with Lost Sales

We consider the inventory control problem of a multiwarehouse, multistore system over a time horizon when the warehouses receive no external replenishment. This problem is prevalent in retail settings, and it is referred to in the work of [ Jackson PL (1988) Stock allocation in a two-echelon distribution system or “what to do until your ship comes in.” Management Sci. 34(7):880–895] as the problem of “what to do until your (external) shipment comes in.” The warehouses are stocked with initial inventories, and the stores are dynamically replenished from the warehouses in each period of the planning horizon. Excess demand in each period at a store is lost. The optimal policy for this problem is complex and state dependent, and because of the curse of dimensionality, computing the optimal policy using standard dynamic programming is numerically intractable. Static Lagrangian base-stock (LaBS) policies have been developed for this problem [Miao S, Jasin S, Chao X (2022) Asymptotically optimal Lagrangian policies for one-warehouse multi-store system with lost sales. Oper. Res. 70(1):141–159] and shown to be asymptotically optimal. In this paper, we develop adaptive policies that dynamically adjust the control parameters of a vanilla static LaBS policy using realized historical demands. We show, both theoretically and numerically, that adaptive policies significantly improve the performance of the LaBS policy, with the magnitude of improvement characterized by the number of policy adjustments. In particular, when the number of adjustments is a logarithm of the length of time horizon, the policy is rate optimal in the sense that the rate of the loss (in terms of the dependency on the length of the time horizon) matches that of the theoretical lower bound. Among other insights, our results also highlight the benefit of incorporating the “pooling effect” in designing a dynamic adjustment scheme.