Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.

翻译：产品组合优化由于其实践重要性在过去几十年中得到了广泛研究。尽管已有大量文献探讨优化算法和潜在得分估计，但针对最优产品组合的不确定性量化仍需进一步探索，且具有重要的实践意义。决策者可能不需要完全估计和恢复完整的最优产品集合，而仅需检验最优产品组合是否满足特定性质，例如是否应将某些感兴趣的产品纳入最优集合，或最优集合应包含多少类产品。本文提出了一种新颖的推断框架来检验此类性质。我们采用广泛使用的多项Logit（MNL）模型，该模型假设每位客户会以与产品潜在偏好得分成比例的概率购买所提供的商品。我们将一般性最优产品组合性质的推断问题转化为对边际收益差额符号变化点检测的不确定性量化。我们证明了边际收益差额估计量的渐近正态性，并基于这些估计量构造了最大统计量以检测符号变化点。通过使用乘子自助法近似最大统计量的分布，我们提出了一种有效的检验程序。数值实验验证了所提方法的性能。