Capacity Constraints Across Nests in Assortment Optimization Under the Nested Logit Model

We consider assortment optimization problems when customers choose according to the nested logit model and there is a capacity constraint limiting the total capacity consumption of all products offered in all nests. When each product consumes one unit of capacity, our capacity constraint limits the cardinality of the offered assortment. For the cardinality constrained case, we develop an efficient algorithm to compute the optimal assortment. When the capacity consumption of each product is arbitrary, we give an algorithm to obtain a 4-approximate solution. Furthermore, we develop a convex program that computes an upper bound on the optimal expected revenue for an individual problem instance. In our numerical experiments, we consider problem instances involving products with arbitrary capacity consumptions. Comparing the expected revenues from the assortments obtained by our 4-approximation algorithm with the upper bounds on the optimal expected revenues, our numerical results indicate that the 4-approximation algorithm performs quite well, yielding about 2% optimality gap on average. A conventional approach to modeling demand in revenue management is to assume that each customer arrives into the system with the intention of purchasing a fixed product. If this product is available for sale, then the customer purchases it. Otherwise, the customer leaves the system without making a purchase. In reality, however, there may be multiple products that can potentially serve the needs of a customer, in which case, customers may make a choice between the products and may substitute a product for another one when their favorite product is not available. This kind of a choice process creates interactions between the demand for the different products, inflating the demand for an available product when some other product is not available so that customers satisfy their needs by substituting for the available product. A common question that arises in this setting is what products to make available to customers so as to maximize the expected revenue, given that customers choose and substitute according to a particular choice model. In this paper, we consider assortment optimization problems when customers choose according to the nested logit model and there is limited capacity for the products in the offered assortment. We consider a setting where we need to decide which assortment of products to offer. Each arriving customer chooses among the offered products according to the nested logit model. Under the nested logit model, the products are organized in nests. Each customer, after viewing the offered assortment, decides either to make a purchase within one of the nests or to leave the system without purchasing anything. If a nest is chosen, then the customer purchases one of the products within the chosen nest. There is a capacity constraint limiting the total capacity consumption of the products in the offered assortment. The goal is to choose an assortment of products to offer so as to maximize the expected revenue obtained from each customer. We consider two types of capacity constraints. In the first type of constraints, each product occupies one unit of space, in which case, the capacity constraint limits the total number of products in the offered assortment. We refer to this type of a capacity constraint as a cardinality constraint. In the second type of constraints, the capacity consumption of a product is arbitrary, possibly reflecting the space or capital requirement of a product. We refer to this type of a capacity constraint as a space constraint. Under a cardinality constraint, we show that we can compute the optimal assortment in a tractable fashion. As far as we are aware, the assortment problem was not known to be tractable when customers choose according to the nested logit model and there is a cardinality constraint limiting the total number of products in the offered assortment. This paper gives the first exact solution method for this problem. On the other hand, we give a 4-approximation algorithm under a space constraint, providing an assortment whose expected revenue deviates from the optimal expected revenue by at most a factor of four. The running time of this algorithm scales gracefully with the number of products and the number of nests. To our knowledge, this paper gives the first algorithm for the assortment problem that scales gracefully with the number of nests and the number of products, when there is a capacity constraint on the space consumption of all offered products and customers choose according to the nested logit model. In addition to giving algorithms to solve the assortment problem, we give a tractable convex program that computes an upper bound on the optimal expected revenue. By comparing the expected revenues of the assortments obtained