Bounding Optimal Expected Revenues for Assortment Optimization under Mixtures of Multinomial Logits

We consider assortment problems under a mixture of multinomial logit models. There is a fixed revenue associated with each product. There are multiple customer types. Customers of different types choose according to different multinomial logit models whose parameters depend on the type of the customer. The goal is to find a set of products to offer so as to maximize the expected revenue obtained over all customer types. This assortment problem under the multinomial logit model with multiple customer types is NP-complete. Although there are heuristics to find good assortments, it is difficult to verify the optimality gap of the heuristics. In this paper, motivated by the difficulty of finding optimal solutions and verifying the optimality gap of heuristics, we develop an approach to construct an upper bound on the optimal expected revenue. Our approach can quickly provide upper bounds and these upper bounds are remarkably tight. In our computational experiments, over a large set of problem instances, the upper bounds provided by our approach deviate from the optimal expected revenues by only 0.11% on average and by only 0.83% in the worst case. By using our upper bounds, we are able to verify the optimality gaps of a greedy heuristic accurately, even when optimal solutions are not available. Customer choice models are becoming increasingly popular for modeling demand in modern revenue management systems. In particular, traditional models of demand assume that each customer arrives into the system with the intention of purchasing a fixed product. If this product is available for purchase, then the customer makes a purchase. Otherwise, the customer simply leaves the system. However, modern revenue management systems are able to offer a variety of products to customers, possibly by exploiting the availability of online sales channels. Often times, there are multiple offered products that satisfy the needs of a customer, in which case, the customer makes a choice among the offered products. Due to the choice process, the demand for a particular product depends on what other products are offered. Thus, customer choice models emerge as a useful tool for capturing the dependencies between the demands for the offered products. In this paper, we study assortment problems that capture the customer choice process of the kind mentioned above. In our problem setting, a firm wants to find a set of products to offer to its customers. There is a fixed revenue associated with each product. An arriving customer may be one of multiple customer types. The firm does not known the type of an arriving customer, but it has access to the probability that an arriving customer is of a particular type. Customers choose among the offered products according to the multinomial logit model and customers of different types choose according to different multinomial logit models whose parameters depend on the type of the customer. This choice model is known as the mixture of multinomial logit models. The goal of the firm is to find a set or an assortment of products to offer to its customers so as to maximize the expected revenue obtained from each customer. Bront et al. (2009) show that this assortment problem is NP-complete, give a mixed integer programming formulation to obtain the optimal solution and provide computational experiments that demonstrate that a greedy heuristic performs quite well when compared with the optimal solutions obtained through the mixed integer programming formulation. However, one shortcoming of using a heuristic is that we use a heuristic simply due to the fact that we cannot obtain the optimal solution and there is no immediate way of being confident that the solution provided by a heuristic is actually a good one. In this paper, motivated by the difficulty of obtaining optimal solutions and evaluating the quality of the solutions provided by a heuristic, we develop a method to obtain upper bounds on the optimal expected revenue in our assortment problem. Thus, we can check the gap between the expected revenue from the solution provided by a heuristic and the upper bound on the optimal expected revenue to assess the optimality gap of the heuristic. Our method for obtaining an upper bound on the optimal expected revenue has two crucial pieces. First, a natural approach for obtaining an upper bound on the optimal expected revenue is to assume that the firm knows the type of an arriving customer. In this case, we can focus on each customer type one by one and separately find an assortment that maximizes the expected revenue from each customer type. This approach essentially allows us to offer different assortments of products to customers of different types, whereas our assortment problem requires that we find a single assortment to offer to all customer types. Talluri and van Ryzin (2004) show that if we focus on one customer type at a time, then the assortment that maximizes the expected revenue