Online Demand Fulfillment Problem with Initial Inventory Placement: A Regret Analysis

Abstract

We investigate a joint inventory placement and online fulfillment problem. At the beginning, the inventory of a single item is distributed to different warehouses. Then, at each period, an order arrives from one of the demand regions, and the decision maker makes an irrevocable decision on whether to accept or reject the order. In our model, we propose the minimum-inventory regret, a notion that includes both the selection of initial inventories and the performance of the selected fulfillment policy. We consider two state-of-the-art fulfillment policies: probabilistic fulfillment and score-based fulfillment. We prove that probabilistic fulfillment has a minimum-inventory regret that scales with the square root of the time horizon. On the other hand, we show that the score-based fulfillment policy has a minimum-inventory regret bound that is independent of the time horizon and polynomial with respect to the number of warehouses and demand regions. Our results have the following implication: the score-based fulfillment policy, when paired with offline inventory placement, outperforms probabilistic fulfillment with any inventory placement, and the performance gap increases with the time horizon.

Keywords: Online Fulfillment, inventory placement, resolving policies, minimum-inventory regret