Randomization Approaches for Network Revenue Management with Customer Choice Behavior

In this paper, we present new approximation methods for the network revenue management problem with customer choice behavior. Our methods are sampling-based and so we require only minimal assumptions regarding the underlying customer choice model. The starting point for our methods is a dynamic program that allows randomization. An attractive feature of this dynamic program is that the size of its action space is linear in the number of itineraries, as opposed to exponential. It turns out that this dynamic program has a structure that is similar to the dynamic program for the network revenue management problem under the so called independent demand setting. Our approximation methods exploit this similarity and build on ideas developed for the independent demand setting. We present two approximation methods. The first one is based on relaxing the flight leg capacity constraints using Lagrange multipliers, whereas the second method involves solving a perfect hindsight relaxation problem. We show that both methods yield upper bounds on the optimal expected total revenue. Computational experiments indicate that our methods can generate tighter upper bounds and higher expected revenues when compared with the standard deterministic linear program that appears in the literature. Network revenue management with customer choice behavior is well-studied and has many applications in the airline, hotel and car rental industries. In the context of airlines, a representative example, it involves controlling the sale of itineraries over a flight network. Customers arrive over the booking period to purchase itineraries. The airline has to decide which itineraries to make available for sale at each point in time taking into account the remaining capacities on the flight legs. This is a crucial decision to make since the customer’s purchasing decision is influenced by the set of itineraries that are offered. Depending on the offer set, the customer may purchase one of the offered itineraries, or may not purchase anything and simply leave. The airline’s goal is to determine the set of itineraries to offer at each point in time that maximizes the expected total revenues over the booking period. The airline’s decision problem can be formulated as a dynamic program. However, computing the value functions and the optimal policy quickly become intractable and one has to resort to approximation methods. Many of the approximation methods for the network revenue management problem with customer choice build on methods developed for network revenue management under the assumption that the customer’s purchasing decision is not influenced by the set of offered itineraries. This is the so called independent demand setting, where we assume that customers arrive with the intention of purchasing a fixed itinerary. If the itinerary is available, they make the purchase. Otherwise, they leave without making any purchase. Even with the independent demand assumption, the network revenue management problem becomes intractable as the size of the state space increases exponentially with the number of flight legs. Consequently, the approximation methods for the network revenue management problem with independent demand have mainly been concerned with reducing the dimensionality of the state space. Incorporating customer choice behavior adds another layer of complexity since the size of the action space also increases exponentially with the number of itineraries. This is because of the combinatorial nature of the problem of deciding which subset of itineraries to offer for sale from the set of all possible itineraries. So, while many of the approximation methods for the network revenue management problem with customer choice are able to handle the dimensionality of the state space quite well, they are less effective in dealing with the complexity of the action space. As a result, the tractability of many of the existing methods depends on the underlying model of customer choice. It is usually assumed that the customer choices are governed by the multinomial logit model and that the consideration sets, the sets of itineraries of interest to the different customer segments, are disjoint. In this paper, we propose new approximation methods that remain tractable for a large class of choice models. We assume that a customer’s choice decision is governed by a simple utility maximization principle. That is, a customer has a utility for purchasing each of the itineraries and to not purchasing anything. Of the available alternatives, the customer chooses the one with the highest utility. The starting point for our methods is a dynamic program that allows randomization. We generate a sample path of customer arrivals along with their utilities for the different itineraries and formulate a dynamic program in order to compute the optimal offer sets. We show that it is possible to reformulate this problem as a dynamic program where the number of decision variables is linear in the number of itineraries. As a result, the size of the action space becomes manageable. In fact, the resulting formulation is similar to the dynamic programming formulation of the network revenue management problem with independent demand. Consequently, we use ideas from the independent demand setting to reduce the