An airline is taking reservations for a direct flight from New York to Chicago. The aircraft used for this flight has 150 seats in total. The company is practicing a flat pricing policy for this route with the price of $300/ticket. Suppose the operating cost is zero dollars per seat. The airline customers have been known to cancel their bookings near a flight date. To address this problem of cancellation, the airline implements an over-booking method. A passenger is bumped if he/she already booked and paid for a ticket but can't get a seat on the airplane because the actual number of passengers showing up is larger than the seat capacity of the airplane. In this case, the airline has to use a backup plan that costs the company $500 per a bumped passenger.

a. Suppose the airline implements a booking policy under which a customer does not pay/deposit any amount of money when booking (so there is no penalty for cancellation). What is the critical ratio (i.e. the probability that the number of cancellations is less than or equal to the overbooking level that the firm should target at)?

b. Suppose the airline implements a booking policy under which a customer does not pay/deposit any amount of money when booking (so there is no penalty for cancellation). From the past data, the airline estimates that that the number of cancellations follows a normal distribution with a mean of 40 seats and a standard deviation of 30 seats. Find the optimal number of overbookings that maximizes the expected revenue from this flight for the company.

c. What is the optimal total number of bookings to maximize the expected revenue under this booking policy?