Assortment Optimization under the Multinomial Logit Model with Nested Consideration Sets

We study assortment optimization problems when customers choose under the multinomial logit model with nested considerations sets. In this choice model, there are multiple customer types and a customer of a particular type is interested in purchasing only a particular subset of products. We use the term consideration set to refer to the subset of products that a customer of a particular type is interested in purchasing. The consideration sets of customers of different types are nested in the sense that the consideration set of one customer type is included in the consideration set of another. The choice process for customers of different types is governed by the same multinomial logit model except for the fact that customers of different types have different consideration sets. Given that customers choose according to the multinomial logit model with nested consideration sets and there is a revenue associated with each product, the assortment optimization problem finds a set of products to offer to the customers to maximize the expected revenue obtained from a customer. We show that the assortment optimization problem is NP-complete. Motivated by this complexity result, we give a fully polynomial time approximation scheme for the problem. Our approach exploits the fact that the consideration sets are nested and its running time is polynomial in both the number of products and customer types. We extend our approach to the case where each product occupies a certain amount of space and there is a constraint on the total space consumption of the offered products. Incorporating customer choice behavior into revenue management models has been seeing considerable attention. In traditional revenue management models, it is common to represent the demand for a product by using an exogenous random variable. However, this way of modeling the demand becomes inadequate when there are multiple products that can serve the needs of a customer and the customers choose among the offered products. When the customers choose among the offered products, the demand for a particular product depends on what other products are offered, creating complex interactions between the demands for the products. Capturing the fact that the customers choose among the offered products is becoming more important than ever, as online sales channels bring a variety of products to the customers among which the customers make a choice. When modeling the customer choice process, a critical tradeoff is that a sophisticated choice model may capture the customer choice process more faithfully, but solving operational problems under a more sophisticated choice model may become intractable. In this paper, we consider assortment optimization problems when customers choose among the offered products according to the multinomial logit model with nested consideration sets. In the assortment optimization problem, a firm has access to a set of products among which it picks an assortment of products to offer to its customers. There is a revenue associated with each product. Customers choose among the offered assortment of products according to a particular choice model. The goal of the firm is to pick an assortment of products to offer to maximize the expected revenue obtained from a customer. We capture the choice process of the customers by using the multinomial logit model with nested consideration sets. In this choice model, there are multiple customer types. A customer of a particular type is interested in purchasing only a particular subset of products. We use the term consideration set to refer to the subset of products that a customer of a particular type is interested in purchasing. The consideration sets of customers of different types are nested, which is to say that we can index the customer types such that the consideration set of a customer type with a smaller index is included in the consideration set of a customer type with a larger index. This consideration set structure arises when different customers are interested in products that satisfy different desired thresholds in a particular attribute, such as price or quality. The choice process for customers of different types is governed by the same multinomial logit model except for the fact that customers of different types have different consideration sets. When making a purchase, a customer observes which of the products in her consideration set are actually included in the offered assortment and she makes a choice only among these products according to the multinomial logit model. Main Contributions. We show that the problem of finding the assortment of products that maximizes the expected revenue obtained from a customer is NP-complete. Motivated by this complexity result, we give a fully polynomial time approximation scheme (FPTAS) for the assortment optimization problem. To our knowledge, our FPTAS is the first algorithm that is directed at solving assortment optimization problems with multiple consideration sets, while making sure that the running time scales polynomially with the number of products, the number of customer types and the approximation guarantee. Our FPTAS is based on a dynamic programming