STATS Module 2 Problem 2

While vacationing in England, you decide to take an impromptu trip outside
of London to see some of the country. You phone several hotels in the area but are
told that all rooms are reserved. You and your friends decide to go anyway, vowing
that if worse comes to worse you will all sleep in the car. Having driven the 100
miles from London, you stop at the first hotel you see, Ye Olde Ox and Bow, and are
surprised to find that rooms are available. Having settled in, you go down to the
Hotel Pub for a drink and strike up a conversation with the Hotel Manager. You
inform him of your experience and he points out that it is not uncommon. The
problem is that about 20% of the people with reservations either call back and
cancel or just dont show up. He tells you that there is a great deal of uncertainty in
how long guests will stay. From experience he has found that about half of the
guests who stayed the night will stay for another night irrespective of their original
plans. He therefore adds half the number of rooms rented to the number of vacant
rooms to come up with his estimate of the number of rooms that will be available (if
there are 92 rooms rented, one half is 46 to which he adds the 8 vacant rooms to
come up with an estimated 54 vacant rooms the next night). He then will take a
maximum of 54 reservations. Of course, many of the reservations become no-shows
so that he usually has rooms available.
You point out that airlines regularly overbook flights since they have a
similar no-show problem. He has never tried this approach and is worried since
if a person with a reservation showed up, he would feel compelled to pay for a
similar room at a competitor and provide the inconvenienced party with a free
dinner and breakfast.
You wake up the next morning to a terribly rainy English day, which ruins
all of your plans. Your friends decide to see the sites of the town, which is a
Museum dedicated to Quilting. You beg off and find yourself with most of the
morning and afternoon free. Your thoughts return to the previous evenings
conversation and you think that you might be able to help the owner make a bit
more profit by changing his reservation procedure.

Using the following: Poisson and Binomial distributions
Assignment:

By using simulation techniques compare the current reservation system with
an open reservation system where anyone who calls for a reservation is given one.
Specifically determine if the open reservation system is more or less profitable than
the current system. Your simulation should be no smaller than one year in length.
Facts and Assumptions:
1) The hotel has 100 rentable rooms:
2) Assume that of the n rooms that are occupied on a given night, the number
that will vacate the next day follows the Binomial Distribution with
parameters n and p = :
3) Assume that the number of new reservations follows the Poisson
distribution with mean equal to 60:
4) If m is the number of reservations made, assume that the number of noshows will follow the Binomial Distribution with parameters m and p=. 2:
5) The profit break-even point is 50 rented rooms:
6) The net profit is $90 per room for each rented room over 50:
7) If a person with a reservation cannot be accommodated, the cost to put the
person up at another hotel plus dinner and breakfast is $270.
8) In your simulation you may assume that holidays and weekend lodging
and reservations are the same as on any other day.
Hints:
a) Assume that the hotel starts off with 92 rented rooms.
b) Use the day as your basic unit, i.e. use the previous days occupancy as the
starting point from which you will determine vacancies and, in the case of the
present reservation system, the number of reservations you will take. Then
determine how many reservations in fact show up and compute the final number of
rooms rented. In the case of the present reservation system, use this figure as the
starting point for the next day.

Is this More Or Less Profitable?
Facts:

100 Total Rooms Available, starting off with 92 rented rooms
n=?
p = .5
60 new reservations