Improved Approximation Schemes for MNL-Driven Sequential Assortment Optimization

In this paper, we consider assortment optimization under the sequential Multinomial-Logit (MNL) choice model, recently proposed by Lui et. al. (2019) to capture a multitude of applications, ranging from appointment scheduling in hospitals, restaurants, and fitness centers to product recommendation in e-commerce settings. In this problem, the purchasing dynamics of customers unfold sequentially over T stages. Within each stage, the retailer selects an assortment of products to make available for purchase, with the intent of maximizing expected revenue. However, motivated by practical applications, the caveat is that each product can be offered in at most one stage. Moving from one stage to the next, the customer either purchases one of the currently offered products according to MNL preferences and leaves the system, or decides not to make any purchase at that time. In the former scenario, the retailer gains a product-associated revenue; in the latter scenario, the customer progresses to the next stage, or eventually leaves the system once all T stages have been traversed. We focus our attention on the most general formulation of this problem, in which purchasing decisions are governed by a stage-dependent MNL choice model, reflecting the notion that customers' preferences might change from stage to stage due to updated perceptions, patience waning over time, etc. Concurrently, we consider a more structured formulation, in which purchasing decisions are stage-invariant, utilizing a single MNL model across all stages. Our main contribution comes in the form of a strongly polynomial-time approximation scheme (PTAS) for both formulations of the sequential assortment problem in their utmost generality. Additionally, we provide evidence for the practical relevance of these theoretical findings through extensive numerical experiments. Finally, we present a case study in which the benefits of offering assortments in stages are explored and exploited within the context of dynamically scheduling restaurant reservations. For this purpose, we make use of historical reservation data from a fine-dining Michelin starred restaurant to build a realistic simulation of their booking process. Ultimately, we discover that offering assortments of potential booking times in two or three stages can lead to revenue improvements of up to 10%.