Tractable Open Loop Policies for Joint Overbooking and Capacity Control Over a Single Flight Leg with Multiple Fare Classes

In this paper, we consider the joint overbooking and capacity control problem over a single flight leg with multiple fare classes. The objective is to maximize the net expected revenue, which is given by the difference between the expected revenue from the accepted requests and the expected penalty cost from the denied reservations. We study a class of open loop policies that accept the requests for each fare class with a fixed acceptance probability. In this case, the challenge becomes that of finding a set of acceptance probabilities that maximize the net expected revenue. We derive a simple expression that can be used to compute the optimal acceptance probabilities, despite the fact that the problem of finding the optimal acceptance probabilities is a high dimensional optimization problem. We show that the optimal acceptance probabilities randomize the acceptance decisions for at most one fare class, indicating that the randomized nature of our open loop policies is not a huge practical concern. We bound the performance loss of our open loop policies when compared with the optimal policy. Computational experiments demonstrate that open loop policies perform remarkably well, providing net expected revenues within two percent of the optimal on average. Revenue management operations of an airline deal with the problem of allocating the limited seats available on the flight legs among different fare classes. Perhaps, the two most important ingredients of these operations are overbooking and capacity control. Overbooking decides how much the physically available seats on the flight legs should be oversold given that not all reservations show up at the departure time, either due to prior cancellations or last minute no shows. On the other hand, capacity control decides which fare classes should be left open for purchase by the passengers, trying to keep a balance between not selling too many seats to low fare class passengers and making sure that the flight legs do not depart with too many empty seats. Naturally, the overbooking and capacity control ingredients have effects on each other. Overbooking determines the working capacities available on the flight legs, which influence the lowest fare class that should be left open for purchase, whereas capacity control determines the fare class mix of the passengers, which influences the penalty cost of denying boarding to a reservation and the likelihood that a reservation shows up at the departure time. In this paper, we study a joint overbooking and capacity control model over a single flight leg. In our problem setting, the requests for different fare classes arrive according to a Poisson process. We need to decide whether to accept or reject each arriving request. An accepted request provides a revenue and becomes a reservation, whereas a rejected request leaves the system. At the departure time of the flight leg, a portion of the reservations show up. If we cannot accommodate all of these reservations on the flight leg, then we incur a penalty cost for the denied reservations. The objective is to maximize the net expected revenue, which is the difference between the expected revenue from the accepted requests and the expected penalty cost from the denied reservations. A dynamic programming model for this problem can be computationally difficult since it requires keeping track of the number of accepted requests for each fare class, resulting in a high dimensional state variable. Instead, we work with a class of open loop policies that accept a request for each fare class with a fixed acceptance probability. In this case, the goal is to find a set of acceptance probabilities that maximize the net expected revenue. We make contributions along the dimensions of computational tractability of our model, structure of the optimal acceptance probabilities and performance of open loop policies. First, we note that there is one acceptance probability for each fare class and the problem of finding a set of acceptance probabilities that maximize the net expected revenue is a high dimensional optimization problem. We show that it is possible to come up with an essentially closed form expression for the optimal acceptance probabilities. This expression requires only a few lookups to the cumulative distribution function of the Poisson distribution. Besides computing the optimal acceptance probabilities in a tractable fashion, this closed form expression suggests that fare to show up probability ratio should be used as a metric to prioritize the fare classes when deciding which requests to accept. Second, the class of policies that we use are randomized policies as they employ probabilistic decision rules. Therefore, a possible cause for concern may be that such randomized policies, while acceptable in theory, may not be ideal for practical implementation. We alleviate this concern by showing that the optimal acceptance probabilities randomize the acceptance decisions for at most one fare class. Third, we develop a performance bound for our class of open loop policies. This performance bound allows us to identify the crucial problem parameters that affect the performance of open loop policies. Fourth, we point out several extensions of our model that can be important from a practical perspective. In particular, our initial development