On upper bounds for assortment optimization under the mixture of multinomial logit models

The assortment optimization problem under the mixture of multinomial logit models is NPcomplete and there are different approximation methods to obtain upper bounds on the optimal expected revenue. In this paper, we analytically compare the upper bounds obtained by the different approximation methods. We propose a new, tractable approach to construct an upper bound on the optimal expected revenue and show that it obtains the tightest bound among the existing tractable approaches in the literature to obtain upper bounds. Assortment optimization has important applications in retailing and revenue management and has received much attention lately. In the assortment problem, we have a firm that is interested in maximizing revenues by selling products to customers, where each product has a revenue associated with it and customers choose among the offered products according to a given discrete choice model. The goal therefore is to figure out the set of products, or the assortment, that maximizes the expected revenue obtained from a customer. While there are a large number of discrete choice models that can be used to describe customer choice behavior, the multinomial logit model and its variants have been a popular choice in the assortment optimization literature. In this paper, we consider the assortment problem under a mixture of multinomial logit models. In this model, we have multiple customers segments and an arriving customer belongs to a particular segment with a given probability and chooses among the offered products according to the multinomial logit model. The parameters of the multinomial logit model are allowed to depend on the segment to which the customer belongs. The assortment optimization problem under the mixture of multinomial logit models is NPcomplete; see Bront, Mendez-Diaz and Vulcano (2009) and Rusmevichientong, Shmoys, Tong and Topaloglu (2013). On the other hand, McFadden and Train (2000) show that the mixture of multinomial logits is a rich choice model that can approximate any random utility choice model arbitrarily closely. So there has been considerable interest in the assortment problem under the mixture of multinomial logit models and there is a growing literature that focuses on developing approximation methods that generate assortments with provable performance guarantees; see for example Rusmevichientong et al. (2013) and Mittal and Schulz (2013). ∗Indian School of Business, Hyderabad, 500032, India, email: sumit [email protected]