On Size Substitution and Its Role in Assortment and Inventory Planning

Problem definition: How should (apparel) retailers manage product sizes? For example, if most customers wearing a given shoe size, such as 9.5, are willing to accept a half-size up or down, is it necessary for a retailer to carry that size at all? Additionally, although identical products in different sizes are treated as distinct stock-keeping units in inventory management, they are often aggregated for assortment and strategic planning. However, there is no theoretical justification for this approach. In this paper, we address the fundamental questions about size management that have remained largely unexplored in the operations literature. Methodology/results: We propose a choice model where each customer forms a consideration set based on the in-stock availability of products of her best-fit size and adjacent sizes. Using a real-world data set from a large footwear retailer, we show that nearly 25% of the unmet demand caused by stockouts spills over to adjacent sizes. We further solve the assortment and inventory optimization problems under the proposed choice model. Our findings demonstrate that the optimal assortment remains unchanged, regardless of the likelihood that customers might purchase adjacent sizes. We utilize this finding and further show that inventory policies that ignore size substitution can be (asymptotically) optimal when the demand rate is high or the selling horizon is long. We also propose a mixed-integer program to determine inventory levels that account for size substitution and achieve higher profits in low-demand settings. Managerial implications: We show that the prevalent size-aggregation approach employed in apparel retail operations is sensible in high-demand settings, such as e-commerce. In contrast, when the expected demand over the selling horizon is low, size substitution can be relevant and should be considered in stocking decisions. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0674 .