You are in charge of the revenue management department at the Richardson$$s Days Out Hotel, which has a total of $200$ rooms. Xpedia $($a travel fare aggregator company$)$ is offering to buy your rooms for Valentine$$s Day at a rate of $$200.$ On the other hand, you can book the rooms through your own website $($which is far less popular than Xpedia$)$ at a rate of

$$400.$ The demand on your own website follows a uniform distribution between $15$ and $25,$ and all the rooms that you did not sell to Xpedia nor booked through your website result unoccupied.

a$.$ How many rooms should you sell to Xpedia? $(3$ points$)$

Suppose now that, instead of selling rooms to Xpedia, you partner with them and do all the bookings through their website. The problem is that, since Xpedia is an aggregator, many guests tend to cancel their reservations at the last minute, which results in hotel rooms being unoccupied. You see this as a missed revenue opportunity and decide to implement the practice of overbooking. The price of each room is $$200,$ and if a guest shows up and a room is not available, you have to book a room at the nearby hotel that costs you $$450.$ Based on historical data, you know that late cancellations happen according to the following distribution:b$.$ How many overbookings should you accept? $(3$ points$)$

Due to the same late cancellations problem, you decide to reduce the price of your rooms to $$150$ and request customers to pay a non$-$refundable fee of $$50$ at the moment of booking $($as a result, customers that show$-$up only pay $$100$ on sight$).$ Despite these changes, you estimate that the distribution of late cancelations will be the same as before.

c$.$ How many overbookings should you accept? $(4$ points$)$