Assortment Optimization Under Variants of the Nested Logit Model

We study a class of assortment optimization problems where customers choose among the offered products according to the nested logit model. There is a fixed revenue associated with each product. The objective is to find an assortment of products to offer so as to maximize the expected revenue per customer. We show that the problem is polynomially solvable when the nest dissimilarity parameters of the choice model are less than one and the customers always make a purchase within the selected nest. Relaxing either of these assumptions renders the problem NP-hard. To deal with the NP-hard cases, we develop parsimonious collections of candidate assortments with worst-case performance guarantees. We also formulate a convex program whose optimal objective value is an upper bound on the optimal expected revenue. Thus, we can compare the expected revenue provided by an assortment with the upper bound on the optimal expected revenue to get a feel for the optimality gap of the assortment. By using this approach, our computational experiments test the performance of the parsimonious collections of candidate assortments that we develop. Discrete choice models have been used for nearly half a century to understand how customers select among a group of products that vary in terms of price and quality. Of particular interest is how demand for the different products changes as the offer set changes in composition, quality or price. To advance this agenda, researchers have developed discrete choice models based on axioms as in Luce (1959), resulting in the basic attraction model, and based on random utility theory as in McFadden (1974), resulting in the celebrated multinomial logit model. Important extensions include the nested attraction model of which the nested logit model, introduced byWilliams (1977), is a special case. Justifications and extensions for the nested logit model are provided in McFadden (1980) and Borsch-Supan (1990). Under the nested logit model, customers first select a nest, and then, a product within the selected nest. The nested logit model was developed primarily to avoid the independence of irrelevant alternatives property suffered by the multinomial logit model; see Ben-Akiva and Lerman (1997). In this paper, we study a class of assortment optimization problems where the choices of the customers are governed by the nested logit model. Under this model, customers first select a nest, and then, a product within the nest. We assume that there is a revenue associated with each product and the objective is to find a set of products, or an assortment, to offer that maximizes the expected revenue per customer. This assortment optimization problem is combinatorial in nature and the number of possible assortments can be very large, particularly when there are many potential products to offer. In airline and hotel revenue management settings, for example, the number of products can easily exceed 30 or 40, yielding 230 or 240 possible assortments. Therefore, it is important to classify when the problem is polynomially solvable. When not, it is important to find solution methods with worst-case performance guarantees. One of our main contributions is to classify the complexity of the assortment problem for nested attraction models. We do this along two dimensions. The first dimension is the magnitude of the nest dissimilarity parameters, which characterize the degree of dissimilarity of the products within a nest. The second dimension is the presence or absence of the no purchase alternative within a nest. We show that the only polynomially solvable case is when the nest dissimilarity parameters are less than one and the no purchase alternative is only available at the time of selecting a nest. If the nest dissimilarity parameters exceed one or the customers can choose a no purchase option after selecting a nest, then we show that the problem is NP-hard. For the NP-hard cases, we develop tractable methods to obtain assortments with worst-case performance guarantees. Research on pricing in the context of the multinomial logit and nested logit models has been fairly active. In that setting, the problem is to choose a set of prices for the products, where the prices of all products jointly determine the probability that a customer purchases a particular product. The objective is to maximize the expected revenue per customer. For the pricing problem, Hanson and Martin (1996) notice that the expected revenue function fails to be concave in prices for the multinomial logit model, but significant progress was made by formulating the pricing problem in terms of market shares, as this results in a concave expected revenue function; see Song and Xue (2007) and Dong et al. (2009). Li and Huh (2011) extend the concavity result to the nested logit model by assuming that the price sensitivities of the products are constant within each nest and the nest dissimilarity parameters are all between zero and one. They show that the expected revenue maximization problem can be reduced to optimizing over a single variable. Gallego and Wang (2011) relax both of the assumptions in Li and Huh (2011)