Joint Inventory, Pricing, and Fulfillment Control for a Multi-Warehouse Multi-Store Problem : An Asymptotically Optimal Lagrangian Policy

We consider the setting of a firm operating multiple warehouses and multiple stores where each warehouse is endowed with an initial non-replenishable inventory. There is a finite selling horizon and, at the beginning of each period, the firm needs to decide for each store how many units of inventory to replenish from each warehouse together with the selling price. We assume zero delivery lead time from each warehouse to its feasible stores. At the end of each period, the firm needs to decide what to do with the realized demands. It can either fulfill them using the available inventory at the store, fulfill them using the inventory at the warehouses, or simply backlog (some of) the demand to the next period. These three options are not mutually exclusive, and the firm can use any combination of them. Computing the optimal policy is numerically intractable due to the well-known curse of dimensionality. Thus, our objective is to develop an easily implementable joint inventory, pricing, and fulfillment heuristic policy that can also be shown to be provably near-optimal. In this paper, we focus on a Lagrangian-based policy. We first study the optimal policy of a Lagrangian relaxation of the original problem. Next, we use this policy to develop our heuristic policy, which we call the Base Stock List Price with Fulfillment (BSLP-F). We show that BSLP-F is asymptotically optimal with a loss bound of order \tilde{O}(\sqrt{MNT}) where M, N, and T denote the number of warehouses, the number of stores and the length of the horizon, respectively