Assortment Optimization under a Single Transition Choice Model

Abstract

In this study, we consider a new customer choice model which we call the single transition choice model. In this model, there is a universe of products and customers arrive at each product with a certain probability. If the arrived product is unavailable, then the seller can recommend a subset of available products and the customer will purchase one of the recommended products or choose not to purchase with certain transition probabilities. The distinguishing features of the model are that the seller can control which products to recommend depending on the arrived product, and each customer either purchases a product or leaves the market after one transition. We study the assortment optimization problem under this model. Particularly, we show that it is NP‐Hard even if the customer can transition from each product to at most two products. Despite the computational complexity, we provide polynomial time algorithms or approximation algorithms for several special cases, such as when the customer can only transition from each product to at most a given number of products and the size of each recommended set is bounded. Our approximation algorithms are developed by invoking the submodularity arguments, or connecting the problem with maximum constraint satisfaction problem and applying randomized rounding techniques to its semidefinite programming relaxation. We also provide a tight worst‐case performance bound for revenue‐ordered assortments. In addition, we propose a compact mixed‐integer program formulation, which is efficient for moderate size problems. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed algorithms.