Hotel manager Basil Fawlty and his resourceful assistant Manuel run a 26 room hotel in a quaint little town called Tourkey on the eastern sea coast. A combination of Mr. Fawlty's genial attitude and the absence of a respectable inn in the nearby vicinity imply that Mr. Fawlty enjoys ample demand at his low fare of $159 per night.

Manuel notes that some customers will walk into the inn requesting a room for that evening and they are willing to pay a high fare of $325 per night. Manuel knows this demand is variable. (In reality, this demand is Poisson distributed with mean 7.5). He suggests some rooms should be kept unsold to the low fare customers, so that they can serve the high fare customers.

a. To maximize profits with Manuel's plan, what is the booking limit that should be set for low fare customers?

b. On average, how many empty rooms will the hotel have under Manuel's plan, if the protection level is 9 rooms?

c. Basil scoffs at the idea: "Empty rooms! A bird in hand is better than two in the bush. This brilliant idea of yours, Manuel, might work in Barcelona but definitely not at Tourkey. Let me run the hotel my way. We will sell to everyone who reserves in advance and ignore the walk-in demand". If Basil has his way, what will be the hotel's expected revenue?

d. Manuel replies: "If you are going to forgo the opportunity to sell to last minute customers, let's at least accept more than 26 reservations for the evening." Checking the data, Manuel observes that the number of "no-shows" is Poisson distributed with mean 2.75. (Recall, a "no-show" is when a customer makes a reservation but doesn't show up to use the room that evening.) Manuel also notes that a $100 non-refundable deposit is required with all reservations. However, if the hotel does not have a room for a reservation holder, then they need to book that person in a B&B in the nearest town. They decide that in those cases they would refund the customer's deposit, and pay for the customer's stay in the B&B, which is $450. The customer would not be happy, but they are getting a free night, so Manuel figures that there would be no loss of good will. Finally, if they have an empty room due to a no-show, they also figure that they would not be able to fill the room with a last minute customer. What is the critical ratio they should use to choose an overbooking quantity to maximize revenue?