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

We study a joint inventory placement and online fulfillment model. In the beginning, the inventory is distributed to different warehouses. At each subsequent period, an order arrives from one of the demand regions, and the decision maker makes an irrevocable decision: whether to accept or reject the order and, if accepted, from which warehouse to fulfill it. To study this problem, we introduce the notion of joint (placement and fulfillment) regret, the regret of a given inventory placement and fulfillment policy. We analyze the joint regret of two state-of-the-art fulfillment policies: probabilistic fulfillment and score-based fulfillment. We prove that the joint regret of probabilistic fulfillment under any inventory placement scales with the square root of the time horizon, which is significantly worse than the regret it achieves under fixed inventory placement when the corresponding fluid solution is nondegenerate. In contrast, the joint regret of score-based fulfillment at the offline inventory placement is independent of the time horizon and is upper bounded by a number that scales polynomially with 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.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2024.0983.