Models for Evaluating Airline Overbooking 305 able information on the actual requests for reservations how much revenue might have been gained from overbooking, compared to how much actually was? This provides a very tangible measure of overbooking performance but very little insight into the reasons for results. The enormous number of factors affecting the design and evaluation of an overbooking policy force us to make simplifying assumptions to construct models that meet both of these goals Assumptions Fleet-wide revenues can be near-optimized one leg at a time Maximizing revenue involves complicated interactions between flights. For instance, a passenger purchasing a cheap ticket on a flight into a major hub might actually be worth more to the airline than a business-class passenger, on account of connecting flights. We assume that such effects can be compen- sated for by placing passengers into fare classes based on revenue potential rather than on the fare for any given leg. This assumption effectively reduces the network problem to a single-leg optimization problem Airlines set fares optimally Revenue maximization depends strongly on the prices of various classes of tickets. To avoid getting into the economics of price competition and revenue maximization to sermng optimal fare-class(over)booking limits es supply/demand, we assume that airlines set prices optimally. This redu Historical demand data are available and applicable The model needs to estimate future demand for tickets on any given flight We assume that historical data are available on the number of tickets sold any given number of days t before a flights departure. In some respects, this assumption is unrealistic because of the problem of data censorship-that is the failure of airlines to record requests beyond the booking limit for a fare class [ Belobaba 1989. On the other hand, statistical methods can be used to reconstruct this information [Boeing Commercial Airline Company 1982, 7-16;Swan1990] Low-fare passengers tend to book before high-fare ones. Discount tickets are often sold under advance purchase restrictions, for the precise reason that it enables price discrimination. Because of restrictions like these, and because travelers who plan ahead search for cheap tickets, low-fare passengers tend to book before high-fare ones Predicting Overbooking effectiveness Disentangling the effects of overbooking, seat allocation, pricing schemes, and external factors on revenues of an airline is extremely complicated To
Models for Evaluating Airline Overbooking 305 able information on the actual requests for reservations, how much revenue might have been gained from overbooking, compared to how much actually was? This provides a very tangible measure of overbooking performance but very little insight into the reasons for results. The enormous number of factors affecting the design and evaluation of an overbooking policy force us to make simplifying assumptions to construct models that meet both of these goals. Assumptions • Fleet-wide revenues can be near-optimized one leg at a time. Maximizing revenue involves complicated interactions between flights. For instance, a passenger purchasing a cheap ticket on a flight into a major hub might actually be worth more to the airline than a business-class passenger, on account of connecting flights. We assume that such effects can be compensated for by placing passengers into fare classes based on revenue potential rather than on the fare for any given leg. This assumption effectively reduces the network problem to a single-leg optimization problem. • Airlines set fares optimally. Revenue maximization depends strongly on the prices of various classes of tickets. To avoid getting into the economics of price competition and supply/demand, we assume that airlines set prices optimally. This reduces revenue maximization to setting optimal fare-class (over)booking limits. • Historical demand data are available and applicable. The model needs to estimate future demand for tickets on any given flight. We assume that historical data are available on the number of tickets sold any given number of days t before a flight’s departure. In some respects, this assumption is unrealistic because of the problem of data censorship—that is, the failure of airlines to record requests beyond the booking limit for a fare class [Belobaba 1989]. On the other hand, statistical methods can be used to reconstruct this information [Boeing Commercial Airline Company 1982, 7–16; Swan 1990]. • Low-fare passengers tend to book before high-fare ones. Discount tickets are often sold under advance purchase restrictions, for the precise reason that it enables price discrimination. Because of restrictions like these, and because travelers who plan ahead search for cheap tickets, low-fare passengers tend to book before high-fare ones. Predicting Overbooking Effectiveness Disentangling the effects of overbooking, seat allocation, pricing schemes, and external factors on revenues of an airline is extremely complicated. To
306 The UMAP Journal 23.3 (2002) isolate the effects of overbooking as much as possible, we want a simple, well- understood seat allocation model that provides an easy way to incorporate various overbooking schemes. In light of this objective, we pass up several methods for finding optimal booking limits on single-leg flights detailed in, for example, Curry [1990] and Brumelle [1993], in favor of the simpler expected marginal seat revenue(EMsr)method [Belobaba 1989] EMSR was developed as an extension of the well-known rule of thumb, popularized by Littlewood [1972], that revenues are maximized in a two-fare system by capping sales of the lower-class ticket when the revenue from selling an additional lower-class ticket is balanced by the expected revenue from selling the same seat as an upper-class ticket. In the emsr formulation any number of fare classes are permitted and the goal is"to determine how many seats not to sell in the lowest fare classes and to retain for possible sale in higher fare classes closer to departure day"[ Belobaba 1989 The only information required to calculate booking levels in the EMSR model is a probability density function for the number of requests that will arrive before the flight departs, in each fare class and as a function of time For simplicity, this distribution can be assumed to be normal, with a mean and standard deviation that change as a function of the time remaining. Thus, the only information an airline would need is a historical average and standard deviation of demand in each class as a function of time. Ideally, the informa tion would reflect previous instances of the particular flight in question. Let the mean and standard deviations in question be denoted by ui(t)and oi(t)for each fare class i= 1, 2,...,k. Then the probability that demand is greater than some specified level Si is given by Pi(Si, t) (r-Hi(t))"/oi(t).dr 2丌o;(t)Js This spill probability is the likelihood that the Si th ticket would be sold if offered in the ith category. If we further allow fi(t) to denote the expected revenue resulting from a sale to class i at a time t days prior to departure, we can define EMSRi (Si, t)=fi(t). Pi(Si, t), ply the revenue for a ticket in class i times the probability that the Si th seat will be sold. The problem, however, is to find the number of tickets S; that should be protected from the lower class j for sale to the upper class i(ignoring other classes for the moment). The optimal value for S; satisfies EMSR(S;, t)=f,(t), so that the expected marginal revenue from holding the S; th seat for class i is exactly equal to(in practice, slightly greater than) the revenue from selling it immediately to someone in the lower class j. The booking limits that should be enforced can be derived easily from the optimal S; values by letting the
306 The UMAP Journal 23.3 (2002) isolate the effects of overbooking as much as possible, we want a simple, wellunderstood seat allocation model that provides an easy way to incorporate various overbooking schemes. In light of this objective, we pass up several methods for finding optimal booking limits on single-leg flights detailed in, for example, Curry [1990] and Brumelle [1993], in favor of the simpler expected marginal seat revenue (EMSR) method [Belobaba 1989]. EMSR was developed as an extension of the well-known rule of thumb, popularized by Littlewood [1972], that revenues are maximized in a two-fare system by capping sales of the lower-class ticket when the revenue from selling an additional lower-class ticket is balanced by the expected revenue from selling the same seat as an upper-class ticket. In the EMSR formulation, any number of fare classes are permitted and the goal is “to determine how many seats not to sell in the lowest fare classes and to retain for possible sale in higher fare classes closer to departure day” [Belobaba 1989]. The only information required to calculate booking levels in the EMSR model is a probability density function for the number of requests that will arrive before the flight departs, in each fare class and as a function of time. For simplicity, this distribution can be assumed to be normal, with a mean and standard deviation that change as a function of the time remaining. Thus, the only information an airline would need is a historical average and standard deviation of demand in each class as a function of time. Ideally, the information would reflect previous instances of the particular flight in question. Let the mean and standard deviations in question be denoted by µi(t) and σi(t) for each fare class i = 1, 2,... ,k. Then the probability that demand is greater than some specified level Si is given by P¯i(Si, t) ≡ 1 √2π σi(t) ∞ Si e(r−µi(t))2/σi(t)2 dr. This spill probability is the likelihood that the Sith ticket would be sold if offered in the ith category. If we further allow fi(t) to denote the expected revenue resulting from a sale to class i at a time t days prior to departure, we can define EMSRi(Si, t) = fi(t) · P¯i(Si, t), or simply the revenue for a ticket in class i times the probability that the Sith seat will be sold. The problem, however, is to find the number of tickets Si j that should be protected from the lower class j for sale to the upper class i (ignoring other classes for the moment). The optimal value for Si j satisfies EMSRi(Si j , t) = fj (t), (1) so that the expected marginal revenue from holding the Si j th seat for class i is exactly equal to (in practice, slightly greater than) the revenue from selling it immediately to someone in the lower class j. The booking limits that should be enforced can be derived easily from the optimal Si j values by letting the