276 The UMAP Journal 23.3(2002) Expected Revenue of a Flight Let a flight have capacity of c and we book b passengers. Let r be the poten- tial revenue from a passenger and p the potential penalty cost of a passenger bumped. Finally, let z be the percentage of ticket holders who show up for the flight. The revenue generated by the flight is if rb< c revenue(al cr-(ab-c)p, if xb>c. The percentage a of passengers who show up follows some probability distri- bution with density function f(a)and an appropriate mean(in our case, 0.9) We find the value of b that maximizes the expected revenue for b passengers expected- revenue(b) f(ar). revenue(a, b)dr Repeat this process for all flights and you have a complete recommendation for an overbooking policy. Examining Compensation Policies We can adjust our model even further by examining the effects of different compensation policies. Airlines have several forms of compensation at their disposal, from food to hotel stays to vouchers. The cost of the compensation policy is the penalty paid to a bumped passenger(p in our formulas above) By rerunning our expected revenue calculations for each compensation policy, we can see how each policy affects the maximum expected revenue of a flight Key Overbooking Flights An airline can determine from historical data the"key"overbooking flights, the ones most likely to require overbooking. It can then use a compensation policy that concentrates on maximizing expected revenue for those flights From Theory to Reality: Vanguard airlines We illustrate our ideas by a case study of Vanguard Airlines, using publicly available information below [Vanguard Airlines 2001]. We assume that the January 2001 through September 2001 statistics provide an accurate picture of he airline RASM= $0.073/seat-mile RPM=817 330 passenger-miles276 The UMAP Journal 23.3 (2002) Expected Revenue of a Flight Let a flight have capacity of c and we book b passengers. Let r be the poten￾tial revenue from a passenger and p the potential penalty cost of a passenger bumped . Finally, let x be the percentage of ticket holders who show up for the flight. The revenue generated by the flight is revenue(x, b) = xbr, if xb ≤ c; cr − (xb − c)p, if xb > c. The percentage x of passengers who show up follows some probability distri￾bution with density function f(x) and an appropriate mean (in our case, 0.9). We find the value of b that maximizes the expected revenue for b passengers: expected revenue(b) = 1 0 f(x) · revenue(x, b) dx Repeat this process for all flights and you have a complete recommendation for an overbooking policy. Examining Compensation Policies We can adjust our model even further by examining the effects of different compensation policies. Airlines have several forms of compensation at their disposal, from food to hotel stays to vouchers. The cost of the compensation policy is the penalty paid to a bumped passenger (p in our formulas above). By rerunning our expected revenue calculations for each compensation policy, we can see how each policy affects the maximum expected revenue of a flight. Key Overbooking Flights An airline can determine from historical data the “key” overbooking flights, the ones most likely to require overbooking. It can then use a compensation policy that concentrates on maximizing expected revenue for those flights. From Theory to Reality: Vanguard Airlines We illustrate our ideas by a case study of Vanguard Airlines, using publicly available information below [Vanguard Airlines 2001]. We assume that the January 2001 through September 2001 statistics provide an accurate picture of the airline: • RASM = $ 0.073/seat-mile. • RPM = 817,330 passenger-miles.