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：产品组合优化因其实际重要性在过去几十年中得到了积极探索。尽管已有大量文献涉及优化算法和潜在评分估计，但最优产品组合的不确定性量化仍需探索，且具有重要的实际意义。决策者可能不需要估计和恢复完整的最优推荐集，而只关心测试最优产品组合是否满足某个给定属性，例如是否应将若干感兴趣的产品纳入最优组合，或最优组合应包含多少类产品。本文提出了一种新颖的推断框架来测试这类属性。我们考虑广泛采用的多项Logit模型，其中假设每位顾客会以与产品相关潜在偏好评分成比例的概率购买推荐产品中的某一项。我们将一般性最优组合属性的推断简化为对边际收益差距符号变化点检测的不确定性量化。我们证明了边际收益差距估计量的渐近正态性，并通过该估计量构建最大统计量来检测符号变化点。利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。我们还进行了数值实验来评估我们方法的性能。