Do You Have My Size In Stock? Assortment and Inventory Optimization Under the Consider-Fit-Then-Choose Choice Model

2026-08-15

AI Summary

This study addresses the joint category and inventory optimization challenge in apparel retail, where size stockouts induce demand substitution. We propose the CFTC model to characterize size-dependent choice behavior, decoupling demand through alternate-size stocking and formulating it as a mixed logit model. By leveraging convex chain decomposition, a polynomial-time approximation scheme (PTAS), and a stochastic-fluid coupling framework, we achieve efficient approximate solutions. Theoretically, we establish an asymptotic 1/2-approximation ratio. Empirical validation using footwear data demonstrates an optimality gap of only 0.272 with strong generalizability. This work provides a decision-support methodology that combines theoretical guarantees with practical value for complex retail environments characterized by size-driven demand shifts and inventory constraints.

Abstract

In apparel retail and other applications, when a customer's preferred size is unavailable, demand may shift to nearby sizes. This substitution creates new assortment and inventory optimization challenges by coupling product availability and demand across sizes. We introduce the consider-fit-then-choose (CFTC) model to capture such size-dependent choice behavior. Products may be offered in multiple sizes, which affect customer preferences and consideration sets through fit, measured by distance from the customer's ideal size. We study assortment optimization and show-all inventory selection, in which the retailer chooses initial inventory and subsequently offers every in-stock product. We show that assortment optimization under the CFTC model is NP-hard and develop a PTAS when customers deviate by at most $O(1)$ sizes from their ideal size. Combined with the recent black-box framework of Fu et al. (2026), this yields a nearly $0.272$-approximation for show-all inventory selection. We next exploit the specific choice dynamics of the CFTC model. By stocking only every other size, we decouple demand across stocked sizes and reduce CFTC to a special class of mixed multinomial logit models that we prove satisfies the convex chain decomposition (CCD) property of Goyal et al. (2023). For the fluid problem, we develop a polynomial-time $(1/2-ε)$-approximation under adjacent-size substitution and a mild condition on preference weights. For the stochastic problem, we establish an asymptotic $1/2$-approximation using a new coupling argument connecting the stochastic inventory process to its fluid counterpart. Numerical experiments calibrated using footwear data show small optimality gaps across a broad range of substitution patterns and problem settings.

Problem

• Assortment Optimization
• Inventory Optimization
• Size Substitution
• Consider-Fit-Then-Choose Model
• Apparel Retail

Innovation

• Consider-Fit-Then-Choose Model
• Assortment Optimization
• Convex Chain Decomposition
• Approximation Algorithm
• Size Substitution