A Stochastic Approximation Algorithm to Compute Bid Prices for Joint Capacity Allocation and Overbooking over an Airline Network

In this paper, we develop a stochastic approximation algorithm to find good bid price policies for the joint capacity allocation and overbooking problem over an airline network. Our approach is based on visualizing the total expected profit as a function of the bid prices and searching for a good set of bid prices by using the stochastic gradients of the total expected profit function. We show that the total expected profit function that we use is differentiable with respect to the bid prices and derive a simple expression that can be used to compute its stochastic gradients. We show that the iterates of our stochastic approximation algorithm converge to a stationary point of the total expected profit function with probability 1. Our computational experiments indicate that the bid prices computed by our approach perform significantly better than those computed by standard benchmark strategies and the performance of our approach is relatively insensitive to the frequency with which we recompute the bid prices over the planning horizon. The notion of bid prices forms a powerful tool for finding good policies for network revenue management problems. The fundamental idea is to associate a bid price with each flight leg that captures the opportunity cost of a seat. In this case, an itinerary request is accepted only if the revenue from the itinerary request exceeds the sum of the bid prices associated with the flight legs that are in the requested itinerary; see Williamson (1992) and Talluri and van Ryzin (1998). It is known that the optimal policies are not necessarily characterized by bid prices, but the intuitive appeal and ease of implementation of bid price policies make them a popular choice in practice. Bid prices are traditionally computed by solving a deterministic linear program. This deterministic linear program can be visualized as an approximation to the network revenue management problem that is formulated under the assumption that all random quantities are known in advance and they take on their expected values. In the deterministic linear program, there exists one capacity availability constraint for each flight leg and the right side of this constraint is the total capacity available on the flight leg. Therefore, the optimal values of the dual variables associated with the capacity availability constraints are used as bid prices. This approach for computing bid prices has seen acceptance from both academic community and industry, but it is inherently a deterministic approximation to a problem that actually takes places under uncertainty and it does not capture the temporal dynamics of the network revenue management problem accurately. In this paper, we consider the problem of finding good bid price policies for making the capacity allocation and overbooking decisions over an airline network. We have a set of flight legs that can be used to satisfy the itinerary requests that arrive randomly over time. Whenever an itinerary request arrives, we need to decide whether to accept or reject this itinerary request. An accepted itinerary request generates a revenue and becomes a reservation. At the departure time of the flight legs, a portion of the reservations show up and we need to decide which reservations should be allowed boarding. The objective is to maximize the total expected profit, which is the difference between the total expected revenue obtained by accepting the itinerary requests and the total expected penalty cost incurred by denying boarding to the reservations. Our approach in this paper is based on visualizing the total expected profit as a function of the bid prices and searching for a good set of bid prices by using the stochastic gradients of the total expected profit function in a stochastic approximation algorithm. Since the stochastic gradients of the total expected profit function depend on the realizations of the itinerary requests and the show up decisions of the reservations, the hope is that our approach captures the stochastic aspects of the problem more accurately than the deterministic linear program. Focusing on a class of policies that are characterized by a small number of parameters and using stochastic approximation algorithms to search for a good set of values for the parameters is a common approach in stochastic optimization. However, the nondifferentiable nature of the bid price policies creates problems when we use this idea in the network revenue management setting. In particular, if we perturb the bid price associated with a flight leg by an infinitesimal amount, then the cardinality of the subset of the itineraries for which are willing to accept a request either does not change at all or changes by a discrete amount. Therefore, an infinitesimal change in the bid prices either does not change the total expected profit at all or changes the total expected profit by a discrete amount. This implies that