Distributionally Robust Discrete Choice Model and Assortment Optimization

Abstract

We consider an assortment optimization problem where the retailer needs to choose a set of products to offer to customers from the range of available products. By choosing the assortment, which will affect customers’ purchase choice, the retailer aims to obtain a high expected revenue. We study the robust setting in the sense that we do not have exact knowledge of the distribution of customers’ utility from each product. Specifically, we introduce two distributionally robust assortment formulations, robust assortment revenue optimization and robust assortment revenue satisficing. While the former uses a pre-specified ambiguity set to characterize the scope of the probability distributions of customers’ utilities, the latter uses a target-driven approach to take all probability distributions into account. By using the multinomial logit model as the reference choice model for both formulations, we show that the optimal assortments exhibit a revenue-ordered property, i.e., products in the optimal assortment have higher revenue than those not in the assortment. We derive the worst-case distribution, construct worst-case choice model based on the worst-case distribution and provide insights on the effects of distributionally robust setting. When the assortment optimization problems have a cardinality constraint, we develop efficient methods to find optimal solutions. Theoretically, we show that by comparison with the revenue optimization approach, the revenue satisficing approach can achieve the target revenue with a higher probability and has a lower computational complexity. We also provide computational studies to demonstrate that the revenue satisficing approach can outperform the benchmark approaches in terms of achieving target revenues. To describe our modeling power, we extend distributionally robust framework under nested logit choice model.

Keywords: Assortment optimization, robust optimization, robust satisficing, data-driven optimization