Assortment optimization under the multi-purchase multinomial logit choice model

In this paper, we introduce the Multi-Purchase Multinomial Logit choice model, which extends the random utility maximization framework of the classical Multinomial Logit model to a multiple-purchase setting. In this model, customers sample random utilities for each offered product as in the Multinomial Logit model. However, rather than focusing on a single product, they concurrently sample a "budget" parameter M, which indicates the maximum number of products that the customer is willing to purchase. Subsequently, the M highest utility products are purchased, out of those whose utilities exceed that of the no-purchase option. When fewer than M products satisfy the latter condition, only these products will be purchased. Our primary contribution resides in proposing the first multi-purchase choice model that can be fully operationalized. Specifically, we provide a recursive procedure to compute the choice probabilities in this model, which in turn provides a framework to study its resulting assortment problem, where the goal is to select a subset of products to make available for purchase so as to maximize expected revenue. Our main algorithmic results consist of two distinct polynomial time approximation schemes (PTAS); the first, and simpler of the two, caters to a setting where each customer may buy only a constant number of products, whereas the second more nuanced algorithm applies to our multi-purchase model in its general form. Additionally, we study the revenue potential of making assortment decisions that account for multi-purchase behavior in comparison with those that overlook this phenomenon. In particular, we relate both the structure and revenue performance of the optimal assortment under a traditional single purchase model to that of the optimal assortment in the multi-purchase setting. Finally, we complement our theoretical work with an extensive set of computational experiments, where the efficacy of our proposed PTAS is tested against natural heuristics. Ultimately, we find that our approximation scheme outperforms these approaches by 1% to 5% on average.