Problem definition: We study a joint product framing and order fulfillment problem with both inventory and cardinality constraints faced by an e-commerce retailer. There is a finite selling horizon and no replenishment opportunity. In each period, the retailer needs to decide how to “frame” (i.e., display, rank, price) each product on his or her website as well as how to fulfill a new demand. Academic/practical relevance: E-commerce retail is known to suffer from thin profit margins. Using the data from a major U.S. retailer, we show that jointly planning product framing and order fulfillment can have a significant impact on online retailers’ profitability. This is a technically challenging problem as it involves both inventory and cardinality constraints. In this paper, we make progress toward resolving this challenge. Methodology: We use techniques such as randomized algorithms and graph-based algorithms to provide a tractable solution heuristic that we analyze through asymptotic analysis. Results: Our proposed randomized heuristic policy is based on the solution of a deterministic approximation to the stochastic control problem. The key challenge is in constructing a randomization scheme that is easy to implement and that guarantees the resulting policy is asymptotically optimal. We propose a novel two-step randomization scheme based on the idea of matrix decomposition and a rescaling argument. Managerial implications: Our numerical tests show that the proposed policy is very close to optimal, can be applied to large-scale problems in practice, and highlights the value of jointly optimizing product framing and order fulfillment decisions. When inventory across the network is imbalanced, the widespread practice of planning product framing without considering its impact on fulfillment can result in high shipping costs, regardless of the fulfillment policy used. Our proposed policy significantly reduces shipping costs by using product framing to manage demand so that it occurs close to the location of the inventory.
Hybrid quantum–classical optimization with cardinality constraints and applications to finance
