A Dynamic Programming Decomposition Method for Making Overbooking Decisions Over an Airline Network

In this paper, we develop a revenue management model to jointly make the capacity allocation and overbooking decisions over an airline network. Our approach begins with the dynamic programming formulation of the capacity allocation and overbooking problem and uses an approximation strategy to decompose the dynamic programming formulation by the flight legs. This decomposition idea opens up the possibility of obtaining approximate solutions by concentrating on one flight leg at a time, but the capacity allocation and overbooking problem that takes place over a single flight leg still turns out to be intractable. We use a state aggregation approach to obtain high quality solutions to the single leg problem. Overall, our model constructs separable approximations to the value functions, which can be used to make the capacity allocation and overbooking decisions for the whole airline network. Computational experiments indicate that our model performs significantly better than a variety of benchmark strategies from the literature. Capacity allocation and overbooking are two main ingredients of network revenue management. In particular, capacity allocation deals with the question of which itineraries to keep open for purchase and which itineraries to close as the remaining capacities on the flight legs are depleted over time with the customer purchases. Overbooking deals with the question of to what extent the sales should exceed the physically available capacity on the flight legs, given that not everyone with a reservation ends up showing up at the departure time. The capacity allocation and overbooking decisions are inherently connected. What fare classes to make available for purchase depends on how many seats in excess of the physically available capacity the airline is willing to sell. On the other hand, how much to overbook depends on what itineraries the airline keeps open and the probability that a customer who purchases a reservation for one of the open itineraries shows up at the departure time. In this paper, we propose a revenue management model that makes the joint capacity allocation and overbooking decisions over an airline network. Our approach formulates the problem as a dynamic program and uses an approximation strategy to decompose the dynamic programming formulation by the flight legs. This decomposition idea opens up the possibility of obtaining approximate solutions by concentrating on one flight leg at a time, though the capacity allocation and overbooking problem that takes place over a single flight leg still happens to be intractable. In particular, the state variable in the dynamic programming formulation of the single leg problem involves a large number of dimensions in practical applications. We overcome this difficulty by using state aggregation to obtain high quality solutions to the single leg capacity allocation and overbooking problem. Ultimately, our model provides separable approximations to the value functions, which can be used to construct a capacity allocation and overbooking policy for the whole airline network. Our work in this paper draws on two streams of literature. The first stream of literature is the work on dynamic programming decomposition methods in network revenue management. Dynamic programming decomposition methods date back to Belobaba (1987) and they are approximate methods aimed at decomposing the network revenue management problem by the flight legs. The basic idea is to associate a displacement adjusted fare with each itinerary over each flight leg, which is different from the actual fare that the airline charges. The displacement adjusted fares immediately allow us to solve a sequence of single leg revenue management problems. In the single leg revenue management problem that takes place over a particular flight leg, we only concentrate on the remaining capacity on this flight leg and assume that the fares associated with the itineraries are equal to the displacement adjusted fares over this flight leg. Once we have solved the single leg problem over each flight leg, we add up the value functions obtained for different flight legs to obtain separable approximations to the value functions. In this paper, we use a similar idea to decompose the capacity allocation and overbooking problem. The important distinction of our paper is that we explicitly deal with overbooking, whereas the earlier decomposition methods work exclusively under the assumption that overbooking is not possible and all reservations show up at the departure time. Our extension to overbooking is nontrivial and has important practical implications as overbooking plays a major role in airline operations. The second stream of literature that we draw on is the work on solving the capacity allocation and overbooking problem over a single flight leg. This stream of literature becomes especially useful