Probabilistically optimized Airline Overbooking Strategies 325 Bump Threshold model Using a 0.05 bump threshold, we compute an optimal number of passengers to book on Flight 502. Given the Airbus A319 capacity of 134 passengers and a passenger arrival probability of p= 88, the optimal number of tickets to sell to guarantee that bumping occurs less than 5% of the time is B=145, or 107% of flight capacity Linear Compensation plan Table 1 shows th rofit for various linear bump functions Table 1 Linear bump functions compared Bump cost Optimal #f Expected profit Per passenger to book per flig 316 $17817 400 151 $16799 800 151 $16,692 150 16601 1000 $16526 If Frontier were to compensate bumped passengers less than the cost of airfare, bumping passengers would always cost less than revenue gained from ticket sales. Thus, assuming it could sell as many tickets as it wanted, Frontier would realize an unbounded profit on each flight! Obviously, the linear com- pensation plan is not realistic in this regime, and we must wait for subsequent models to see increased real-world applicability. These results agree with the result of using a simple bump threshold above and indicate an average profit of approximately $17,000. In comparison with using no overbooking strategy at all, Frontier gains additional profit of $4,000 per flight! The actual dynamics of the problem may be seen in Figure 4, where compet- ng effects form an optimal number of tickets to sell(B)when Frontier assumes a sizeable enough compensation average. We can also see the unbounded profit available in the unrealistic regime Nonlinear Compensation Plan Numerical results for the more realistic nonlinear model paint a more rea- sonable picture Table 2 recommends booking limmits similar to(though slightly higher than) previous models. The dynamics may be seen in the Figure 5Probabilistically Optimized Airline Overbooking Strategies 325 Bump Threshold Model Using a 0.05 bump threshold, we compute an optimal number of passengers to book on Flight 502. Given the Airbus A319 capacity of 134 passengers and a passenger arrival probability of p = .88, the optimal number of tickets to sell to guarantee that bumping occurs less than 5% of the time is B = 145, or 107% of flight capacity. Linear Compensation Plan Table 1 shows the expected profit for various linear bump functions. Table 1. Linear bump functions compared. Bump cost Optimal # Expected profit per passenger to book per flight 200 ∞ ∞ 316 162 $17,817 400 156 $17,394 500 153 $17,121 600 152 $16,940 700 151 $16,799 800 151 $16,692 900 150 $16,601 1000 150 $16,526 If Frontier were to compensate bumped passengers less than the cost of airfare, bumping passengers would always cost less than revenue gained from ticket sales. Thus, assuming it could sell as many tickets as it wanted, Frontier would realize an unbounded profit on each flight! Obviously, the linear com￾pensation plan is not realistic in this regime, and we must wait for subsequent models to see increased real-world applicability. These results agree with the result of using a simple bump threshold above and indicate an average profit of approximately $17,000. In comparison with using no overbooking strategy at all, Frontier gains additional profit of $4,000 per flight! The actual dynamics of the problem may be seen in Figure 4, where compet￾ing effects form an optimal number of tickets to sell (B) when Frontier assumes a sizeable enough compensation average. We can also see the unbounded profit available in the unrealistic regime. Nonlinear Compensation Plan Numerical results for the more realistic nonlinear model paint a more rea￾sonable picture. Table 2 recommends booking limmits similar to (though slightly higher than) previous models. The dynamics may be seen in the Figure 5