Assortment Optimization with Multi-Item Basket Purchase under the Multivariate MNL Model

Assortment selection is one of the most important decisions faced by retailers. Most existing papers in the literature assume that customers select at most one item out of the offered assortment. While this is valid in some cases, it contradicts practical observations in many shopping experiences, both in online and brick-and-mortar retail, where customers may buy a basket of products instead of a single item. In this paper we incorporate customers' multi-item purchase behavior into the assortment optimization problem. We consider both uncapacitated and capacitated assortment problems under the so-called Multivariate MNL (MVMNL) model, which is one of the most popular multivariate choice models used in the marketing and empirical literature. We first show that the traditional revenue-ordered assortment may not be optimal. Nonetheless, we show that under some mild conditions, a certain variant of this property holds (in the uncapacitated assortment problem) under the MVMNL model---that is, the optimal assortment consists of revenue-ordered local assortments in each group. Finding the optimal assortment is still computationally expensive as the revenue thresholds for different groups cannot be computed separately. We show that the optimization problem under MVMNL is NP-complete even in the setting where there is no interaction among the product categories. Motivated by this result, we develop FPTAS for several variants of (capacitated and uncapacitated) assortment problems under MVMNL. Our analysis reveals that disregarding customers' multi-item purchase behavior in assortment decisions can indeed have a significant negative impact on a retailer's profitability, demonstrating its practical importance in retail. In particular, we show that our proposed algorithm can improve a retailer's expected total revenues (compared to some benchmark policies that do not properly take into account the impact of customer's multi-item choice behavior in assortment decisions) by around 5-7% for the uncapacitated problems, and around 10-54% for the capacitated problems, both of which are quite significant.