AI 中文总结
针对服装零售中尺码缺货导致的需求转移问题，引入CFTC模型，证明其品类优化NP难并开发PTAS，结合框架得到全展示库存近似算法，还利用模型特性解耦需求并获不同问题的近似算法，实验验证效果良好。
AI 中文摘要
在服装零售及其他场景中，当顾客偏好的尺码缺货时，需求可能会转移至相邻尺码。这种替代效应通过将各尺码的产品可得性与需求耦合，带来了品类规划与库存优化的新挑战。我们引入“先考虑适配再选择”（CFTC）模型，以捕捉这种尺码依赖的选择行为。产品可提供多种尺码，尺码通过与顾客理想尺码的距离所衡量的适配性，影响顾客偏好与考虑集。我们研究品类优化与全展示库存选择问题，其中零售商需选择初始库存，随后提供所有库存内的产品。我们证明，在CFTC模型下的品类优化问题是NP难的，并在顾客与理想尺码的偏差最多为O(1)个尺码时，开发出一个多项式时间近似方案（PTAS）。结合Fu等人（2026）近期的黑箱框架，这为全展示库存选择问题提供了一个近似比约为0.272的近似算法。接下来，我们利用CFTC模型的特定选择动态：仅每隔一个尺码备货，可解耦已备货尺码间的需求，并将CFTC简化为一类特殊的混合多项logit模型，我们证明该模型满足Goyal等人（2023）提出的凸链分解（CCD）性质。对于流体问题，我们在相邻尺码替代和偏好权重的温和条件下，开发出一个多项式时间的(1/2-ε)近似算法；对于随机问题，我们通过一种新的耦合论证，将随机库存过程与其流体对应物关联，建立了一个渐近1/2近似算法。基于鞋类数据校准的数值实验表明，在广泛的替代模式和问题设置下，优化差距较小。
英文摘要
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.
