EXPRESS: Joint Assortment and Price Optimization with Multiple Purchases

In this article, we incorporate the effects of multi-unit purchases into consumer choice models and investigate the associated joint assortment and price optimization problems. Under the proposed two-stage choice framework, consumers first form a consideration set, and then select a product with the optimal quantity from the consideration set to maximize their utility. For the joint assortment and pricing problem with homogeneous consumers, we find that a multi-purchase-willingness-ordered assortment is optimal under certain technical conditions, and propose a polynomial-time algorithm to find the optimal assortment and prices for general cases. We further show that the optimal prices and assortment size are not monotone in multi-purchase preference and multi-purchase resistance. For the joint assortment and pricing problem with heterogeneous consumers, we first prove that this problem is NP-hard. We then consider a scenario with discrete prices and develop a fully polynomial-time approximation scheme by considering a finite number of consumer segments and nested consideration sets. We conduct an empirical study on the JD.com dataset and show that the new choice framework can improve model fitting and prediction accuracy compared with existing choice models with multiple purchases. We reveal that our model is especially suitable for datasets exhibiting a strong diminishing marginal effect for product variety. We further extend our analysis to price competition and establish the existence and uniqueness of a Nash equilibrium. For the joint optimization with fixed costs, we develop an algorithm that can find a solution guaranteeing at least half the optimal expected profit.