You are an analyst for a 1200 room hotel. Your job is to maximize the hotels revenue. Its late Sunday night and youre trying to find the correct rate to set for next Friday night, for which you are already at 75% occupancy. There wont be any amount of cancellations of note, because customers arent allowed to cancel without charge within 2 weeks of booking.

If your rate is too low, your remaining inventory will fill up with low-rate bookings, hurting revenue. If your rate is too high, you wont sell enough of your inventory, hurting revenue. The revenue-maximizing rate is somewhere between too low and too high.

Youve analyzed a history of booking inquiries from several similar Fridays in the companys recent history. Keep in mind, booking inquiries dont always turn into bookings; many customers decline because the rate is too high. We call this lost business. You compile an average of booking inquiries. You expect them to flow into the call center all this coming week as follows:

Monday	Tuesday	Wednesday	Thursday	Friday
50	70	170	300	550

The interpretation of this table is as follows. On Monday, 50 people will call in and request a Friday booking. On Tuesday, 70 will call in to request a Friday booking. Then 550 people will request a same-day booking on Friday.

You analyze your lost business data and reveal that:

The highest rate at which no one will decline to book is $25.

The lowest rate at which everyone will decline to book is $800.

The proportion of booking requests that turn into bookings can be adequately modeled by a half-parabola, concave up, with a vertex at rate = $800:

Some people will not show up for their reservation, so you need to overbook the hotel by a certain amount. However, you do not want to overbook too much. Having a customer show up with no room to give them is a customer service nightmare for a casino hotel. Company policy does not permit you to risk this happening more than twice in a year.

History shows average no show to be equal to 4% of bookings with a standard deviation equal to 0.416667% of bookings. History also shows no show to have a normal distribution.

For your assignment, please do the following:

Calculate the highest amount you can overbook the hotel within company policy.

With this overbooking amount you calculated, find the rate for Friday that maximizes hotel revenue.

For now, you do not need to generate simulations. For example, you may treat the table of booking inquiries as absolutely certain. We will relax this assumption in another assignment.

This is all the data we had, it does not have be a perfect simulation, but a fair prediction about the two questions.