The Zigus hotel routinely experiences no-shows (people making reservations and not showing up) during the peak season when the hotel is always full. No-shows follows the probability distribution given in the table below:

No-shows (x)  Probability

0                    0.1

1                    0.13

2                    0.31

3                    0.16

4                    0.21

5                    0.09

In order to lessen the number of vacant rooms, the hotel uses an overbooking policy through which it accepts three more reservations than the number of rooms available. On a day when the hotel experiences fewer than three no-shows, there are not enough rooms for those who have reservations. The hotel's policy is to send these guests to a closely located competing hotel, at a cost, supported by the Zigus hotel, of $125 per person. If the number of no-shows is more than three, the hotel has vacant rooms, resulting in an opportunity cost of $50 per person.

Required:

Question 1) Simulate 3000 days of operation to calculate the hotel's average daily cost due to overbooking and opportunity costs.

Question 2) Determine the optimal number (0, 1, 2, 3, 4, 5, 6) of rooms that the Zigus hotel should overbook in order to minimize average daily costs.

Solve the given numerical problem and illustrate step by step calculation.