Constrained Assortment Optimization for the Nested Logit Model

We study assortment optimization problems where customer choices are governed by the nested logit model and there are constraints on the set of products offered in each nest. Under the nested logit model, the products are organized in nests. Each product in each nest has a fixed revenue associated with it. The goal is to find a feasible set of products, i.e. a feasible assortment, to maximize the expected revenue per customer. We consider cardinality and space constraints on the offered assortment, which respectively limit the number of products and the total space consumption of the products offered in each nest. We show that the optimal assortment under cardinality constraints can be obtained efficiently by solving a linear program. The assortment optimization problem under space constraints is NP-hard. We show how to obtain an assortment with a performance guarantee of two under space constraints. This assortment also provides a performance guarantee of 1/(1 − ε) when the space requirement of each product is at most a fraction ε of the space availability in each nest. Building on our results for constrained assortment optimization, we show that we can efficiently solve joint assortment optimization and pricing problems under the nested logit model, where we choose the assortment of products to offer to customers, as well as the prices of the offered products. Discrete choice models have long been used to describe how customers choose among a set of products that differ in attributes such as price and quality. Specifically, discrete choice models represent the demand for a particular product through the attributes of all products that are in the offered assortment, capturing substitution possibilities and complementary relationships between the products. To pursue this thought, different discrete choice models have been proposed in the literature. Some of these models are based on axioms as in Luce (1959), resulting in the basic attraction model, whereas some others are based on random utility theory as in McFadden (1974), resulting in the multinomial logit model. A popular extension to the multinomial logit model is the nested logit model introduced by Williams (1977). Under the nested logit model, the products are organized in nests. The choice process of a customer proceeds in such a way that the customer first selects a nest, and then a product within the selected nest. In this paper, we study constrained assortment optimization problems when customers choose according to the nested logit model. There is a fixed revenue contribution associated with each product. The goal is to find an assortment of products to offer so as to maximize the expected revenue per customer subject to a constraint on the assortment offered in each nest. We consider two types of constraints, which we refer to as cardinality and space constraints. Cardinality constraints limit the number of products in the assortment offered in each nest. We show that the optimal assortment under cardinality constraints can be obtained by solving a linear program. Under space constraints, each product occupies a certain amount of space and we limit the total space consumption of the products offered in each nest. The assortment optimization problem under space constraints is NP-hard, but we show that we can solve a tractable linear program to obtain an assortment with a certain performance guarantee. These results establish that we can obtain provably good assortments under cardinality or space constraints. In addition, we consider joint assortment optimization and pricing problems under the nested logit model. In the joint assortment optimization and pricing problem, the goal is to decide which assortment of products to offer and set the prices of the offered products. Customers choose among the offered products according to the nested logit model and the price of a product affects its attractiveness in the sense that if we set the price of a product higher, then it becomes less attractive to customers. Building on our results for constrained assortment optimization problems, we show that an optimal solution to the joint assortment optimization and pricing problem can be obtained efficiently by solving a linear program. Therefore, our results are not only useful for solving constrained assortment optimization problems, but they are also useful for pricing. Main Contributions. In assortment optimization problems, we consider a setting with m nests, each including n products that we can offer to customers. Under cardinality constraints, we show that we can solve a linear program with 1+m decision variables and O(mn2) constraints to obtain the optimal assortment. Under space constraints, we show that we can solve a linear program of the same size to obtain an assortment whose expected revenue deviates from the optimal expected revenue by at most a factor of two. Also, if each product consumes at most a fraction ε ∈ [0, 1)