A small hotel has 50 rooms that rent for $105 per night and cost $45 to clean and prepare each night they are used. All rentals are by reservation and there is a 10 percent chance that an individual reservation will not show up. If a customer arrives at the hotel with a reservation and there is no room available due to overbooking, the hotel will rebate the cost of the room and pay on average $150 to put the customer up at another hotel. The hotel would like to determine its overbooking limit: the number of rooms to accept reservations for each night in excess of capacity. Its objective is to maximize expected profits.
a. What is the expected profit with an overbooking limit of 50?
b. What is the optimal overbooking limit?
c. Using the optimal overbooking limit, what is the expected number of customers turned away?
d. How sensitive is the optimal overbooking limit to rental prices in the range from $90 to $125 per night?