Joint Assortment Optimization and Customization Under a Mixture of Multinomial Logit Models: On the Value of Personalized Assortments

We consider a joint assortment optimization and customization problem under a mixture of multinomial logit models. In this problem, a firm faces customers of different types, each making a choice within an offered assortment according to the multinomial logit model with different parameters. The problem takes place in two stages. In the first stage, the firm picks an assortment of products to carry the subject to a cardinality constraint. In the second stage, a customer of a certain type arrives into the system. Observing the type of the customer, the firm customizes the assortment that it carries by, possibly, dropping products from the assortment. The goal of the firm is to find an assortment of products to carry and a customized assortment to offer to each customer type that can arrive in the second stage to maximize the expected revenue from a customer visit. The problem arises, for example, in online platforms, where retailers commit to a selection of products before the start of the selling season; but they can potentially customize the displayed assortment for each customer type. We refer to this problem as the Customized Assortment Problem ( CAP ). Letting m be the number of customer types, we show that the optimal expected revenue of ( CAP ) can be Ω ( m ) times greater than the optimal expected revenue of the corresponding model without customization and this bound is tight. We establish that ( CAP ) is NP-hard to approximate within a factor better than 1 − 1 / e , so we focus on providing an approximation framework for ( CAP ). As our main technical contribution, we design a novel algorithm, which we refer to as Augmented Greedy; building on it, we give a Ω ( 1 / log m ) -approximation algorithm to ( CAP ). Also, we present a fully polynomial-time approximation scheme for ( CAP ) when the number of customer types is constant. Considering the case where we have a cardinally constraint on the assortment offered to each customer type in the second stage of ( CAP ), we give a Ω ( 1 / m log m ) -approximation algorithm. In our computational experiments, we demonstrate the value of customization by using a data set from Expedia and check the practical performance of our approximation algorithm.