288 The UMAP Journal 23.3(2002) Revenue vs. Tickets sold 24000 23500 23000 21000 17o 190 200 210 Figure 1. Revenue R vs overbooking strategy B for C= 150, k= l, p=0.85, and T=$140 Limitations The single-plane model fails to account for bumped passengers general dissatisfaction and propen sity to switch airlines assumes a simple constant-cost compensation function for bumped passen ignores the distinction between voluntary and involuntary bumping assumes that all tickets are identical-that is, everyone flies coach assumes that all b tickets that the airline is willing to sell are actually sold Even so, the model successfully analyzes revenue as a function of over- booking strategy plane capacity, the probability that ticket-holders become contenders, and compensation cost. Later, we develop a more complete model. The Complicating Factors First, though, we use the basic model to make preliminary predictions fo the optimal overbooking strategy in light of market changes due to the com- plicating factors post-September 11
288 The UMAP Journal 23.3 (2002) Revenue vs. Tickets Sold 21000 21500 22000 22500 23000 23500 24000 Revenue 150 160 170 180 190 200 210 Tickets (#) Figure 1. Revenue R vs. overbooking strategy B for C = 150, k = 1, p = 0.85, and T = $140. Limitations The single-plane model • fails to account for bumped passengers’ general dissatisfaction and propensity to switch airlines; • assumes a simple constant-cost compensation function for bumped passengers; • ignores the distinction between voluntary and involuntary bumping; • assumes that all tickets are identical—that is, everyone flies coach; • assumes that all B tickets that the airline is willing to sell are actually sold. Even so, the model successfully analyzes revenue as a function of overbooking strategy, plane capacity, the probability that ticket-holders become contenders, and compensation cost. Later, we develop a more complete model. The Complicating Factors First, though, we use the basic model to make preliminary predictions for the optimal overbooking strategy in light of market changes due to the complicating factors post-September 11
Optimal Overbooking 289 Of the four complicating factors, only two are directly relevant to this model the security factor and the fear factor. The primary effect of the security factor is to decrease the probability p of a ticketholder reaching the gate on time and becoming a contender. On the other hand, the primary effect of the fear factor is that a greater proportion of those who fly do so out of necessity; since such passengers are more likely to arrive for their flights than more casual flyers, the fear factor tends to increase p Figure 2 plots the optimal overbooking strategy Bopt VS. p for fixed k=1 and C= 150 Optimal Ticket sales vs. Show-up Probability Probability(p) Figure 2. Optimal overbooking strategy vs arrival probability p It is difficult to assess the precise change in p resulting from the securit and fear factors. However, airlines can determine this empirically by gathering statistics on their flights, then use our graph or computer program to determine a new optimal overbooking strategy. One-Plane model: Multifare extension Introduction and motivation Most airlines sell tickets in different fare classes(most commonly first class and coach). We extend the basic One-Plane Model to account for multiple fare classes Development For simplicity, we consider a two-fare system, with C1 first-class seats and C2 coach seats. We assume that a first-class ticket costs Ti= $280 and that
Optimal Overbooking 289 Of the four complicating factors, only two are directly relevant to this model: the security factor and the fear factor. The primary effect of the security factor is to decrease the probability p of a ticketholder reaching the gate on time and becoming a contender. On the other hand, the primary effect of the fear factor is that a greater proportion of those who fly do so out of necessity; since such passengers are more likely to arrive for their flights than more casual flyers, the fear factor tends to increase p. Figure 2 plots the optimal overbooking strategy Bopt vs. p for fixed k = 1 and C = 150. Optimal Ticket Sales vs. Show-up Probability 150 160 170 180 B_optimal 0.8 0.85 0.9 0.95 1 Probability (p) Figure 2. Optimal overbooking strategy vs. arrival probability p. It is difficult to assess the precise change in p resulting from the security and fear factors. However, airlines can determine this empirically by gathering statistics on their flights, then use our graph or computer program to determine a new optimal overbooking strategy. One-Plane Model: Multifare Extension Introduction and Motivation Most airlines sell tickets in different fare classes (most commonly first class and coach). We extend the basic One-Plane Model to account for multiple fare classes. Development For simplicity, we consider a two-fare system, with C1 first-class seats and C2 coach seats. We assume that a first-class ticket costs T1 = $280 and that a