A Tractable Revenue Management Model for Capacity Allocation and Overbooking 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. The crucial observation behind our model is that if the penalty cost of denying boarding to the reservations were given by a separable function, then the optimality equation for the joint capacity allocation and overbooking problem would decompose by the itineraries. We exploit this observation by building an approximation to the penalty cost that is separable by the numbers of reservations for different itineraries. In this case, we can obtain an approximate solution to the optimality equation by plugging the separable approximation into the boundary condition of the optimality equation. Our computational experiments compare our approach with a standard deterministic linear programming formulation, as well as a recent joint capacity allocation and overbooking model. When compared with the standard deterministic linear programming formulation, our approach can provide significant profit improvements. On the other hand, when compared with the recent joint capacity allocation and overbooking model, our approach can provide similar profit performance with substantially shorter runtimes. Capacity allocation and overbooking form two important components of network revenue management operations. Capacity allocation deals with the question of what itinerary requests should be accepted given that there is uncertainty about the future itinerary requests, whereas overbooking deals with the question of how many seats in excess of the physically available seat inventory should be sold given that not all reservations show up at the departure time. These two classes of decisions clearly interact. In particular, what itinerary requests should be accepted depends on how much we are willing to overbook, and how much we are willing to overbook depends on what itinerary requests we tend to accept and the likelihood that these accepted itinerary requests show up at the departure time. Nevertheless, despite this clear interaction, the capacity allocation and overbooking decisions are traditionally made in a sequential manner. First, an overbooking model is solved to obtain the overbooking pads that indicate how many seats in excess of the physically available seat inventory should be sold. Following this, a capacity allocation model is solved under the assumption that the capacities on the flight legs are equal to the sum of the physically available seat inventory and the overbooking pads. In this paper, we present a network revenue management model that jointly makes the capacity allocation and overbooking decisions. We begin by formulating the capacity allocation and overbooking problem as a dynamic program. In this dynamic program, the decisions that we make during the intermediate time periods of the planning horizon are related to whether we should accept or reject the itinerary requests, whereas the decisions that we make at the end of the planning horizon are related to which reservations we should deny boarding. As it is commonly the case for many practical problems, the dynamic programming formulation of the capacity allocation and overbooking problem quickly gets intractable, but this formulation allows us to observe that if the penalty cost of denying boarding to the reservations were given by a separable function, then the problem would decompose by the possible itineraries in the airline network. Therefore, our approach exploits this observation by building a separable approximation to the penalty cost of denying boarding to the reservations at the departure time. In particular, this approximation is separable by the numbers of reservations for different itineraries. In this case, we can obtain an approximate solution to the dynamic program by plugging the separable approximation into the boundary condition. To construct a separable approximation to the penalty cost, we start with a “reasonable” policy to accept and reject the itinerary requests. We simulate the behavior of this policy to get a rough estimate for the numbers of reservations that show up at the departure time. We fix the numbers of reservations that show up at the departure time at these estimates and vary the number of reservations for only one itinerary at a time. By observing how the penalty cost changes as we vary the number of reservations for only one itinerary, we construct a separable approximation to the penalty cost of denying boarding to the reservations. Although the literature on the capacity allocation problem is quite rich, there are surprisingly few papers that consider the interactions between the capacity allocation and overbooking decisions. Early models consider the capacity allocation and overbooking decisions over a single flight leg rather than an airline network. For example, the papers by Beckmann (1958), Thompson (1961) and Coughlan (1999) develop capacity allocation and overbooking models over a single flight leg that treat the demand from different fare classes as static random variables. These models essentially ignore the temporal dynamics of the arrivals of the itinerary requests and assume that the total demand from different fare classes are