Airline Overbooking Problem

You are taking reservations for an airline flight. This particular flight uses an aircraft with 50 first-class seats and 190 economy-class seats.

First-class tickets on the flight cost $600, with demand to purchase them distributed like a Poisson random variable with mean 50. Each passenger who buys a first-class ticket has a 93% chance of showing up for the flight. If a first-class passenger does not show up, he or she can return their unused ticket for a full refund. Any first class passengers who show up for the flight with tickets but are denied boarding are entitled to a full refund plus a $500 inconvenience penalty.

Economy tickets cost $300. Demand for them is Poisson distributed with a mean of 200, and is independent of the demand for first-class tickets. Each ticket holder has a 96% chance of showing up for the flight, and “no shows” are not entitled to any refund. If an economy ticket holder shows up and is denied a seat, however, they get a full refund plus a $200 penalty. If there are free seats in first class and economy is full, economy ticket holders can be seated in first class.

The airline allows itself to sell somewhat more tickets than it has seats. This is a common practice called “overbooking”.  The firm is considering the 18 possible polices obtained through all possible combinations of

• Allowing overbooking of up to 0, 5, or 10 first-class seats
• Allowing overbooking of up to 0, 5, 10, 15, 20, or 25 economy seats

Which option gives the highest average profit?  What are the average numbers of first-class and economy passengers denied seating under this policy.  If no overbooking of first class is allowed, what is the best policy?