Assortment Optimization with Multi-Item Basket Purchase under 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 assortment optimization 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. Under MVMNL, products are clustered into different groups or categories, and a customer can simultaneously purchase from as many groups as possible, where at most one product gets selected from each group. We first show that the traditional revenue-ordered assortment may not be optimal. Nonetheless, we show that under some mild condition, 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 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