Multi-Item Order Fulfillment Revisited: LP Formulation and Prophet Inequality

In this work, we revisit the multi-item order fulfillment model introduced by [Jasin and Sinha 2015]. Specifically, we study a dynamic setting in which an e-commerce platform (or online retailer) with multiple warehouses and finite inventory is faced with the problem of fulfilling orders that may contain multiple items. The platform's goal is to minimize the expected cost incurred from the fulfillment process, subject to warehouses' inventory constraints. Unlike the classical literature on multi-item fulfillment, we propose an alternative offline formulation of the problem. In particular, in our model, the platform sequentially selects methods to fulfill the arriving orders. A method consists of a set of facilities that will determine which warehouses the items will ship from and, more importantly, whether multi-item orders will be split. Under this formulation, we design a class of dynamic policies that combine ideas from randomized fulfillment, prophet inequalities and subgradient methods for the general multi-item fulfillment model. Specifically, by establishing connections between the fulfillment and prophet inequality literature, we prove that our algorithm is both asymptotically optimal and has strong approximation guarantees in non-asymptotic settings. Our result shows that there is a simple and near-optimal procedure for solving multi-item fulfillment problems once the online retailer has enough inventory, independently of other problem parameters. To the best of our knowledge, this is the first result of this type in the context of multi-item order fulfillment. In addition, and of independent interest, our analysis also leads to new asymptotically optimal bounds for network revenue management problems.