Operations Management under Consumer Choice Models with Multiple Purchases

Abstract

This paper investigates the effects of multiple purchases that arise in the retailing of consumer goods, in which the product choice and consumer surplus depend not only on what to purchase but also on how many units to purchase. We incorporate the multiple purchases into consumer choice behavior and study a series of associated operational problems. Most of the discrete choice models in the current literature often assume that a customer chooses exactly one unit of a product. The assumption of “one purchase” is too restrictive in some practical scenarios (e.g., consumer goods) because customers often purchase multiple units of a product. We take the widely-used multinomial logit (MNL) model as a showcase and incorporate the effects of multiple purchases into the classic discrete choice model. In the new choice framework, consumers first form a consideration set, then select one product from consideration set and determine the purchase quantity of the selected product. In the absence of fixed cost, we characterize the structure of the optimal policy for the assortment optimization problem; whereas in the presence of product-differentiated fixed costs, the assortment problem becomes NP-complete, so we propose an efficient heuristic. We further develop a polynomial time algorithm for the assortment problem with identical fixed cost for each product. For the joint assortment and pricing problem, we show it can be decoupled into multiple multi-product pricing problems with different assortment sizes, each of which can be transformed into a single-variable problem. For the price competition problem, we characterize the existence and uniqueness of the Nash equilibrium. We combine the alternating optimization algorithm with the expectation maximization algorithm to overcome the non-concavity and missing data issues in estimation. An empirical study on JD.com data shows that incorporating the effects of multiple purchases into discrete choice models can improve model fitting and prediction accuracy, while ignoring the effects of multiple purchases may lead substantial losses.