Question

Q4 Hotel California has 160 rooms. The hotel has an ample low-fare demand at the room rate of $200 per night, but the demand from the high-fare class which pays $450 per night on average, is uncertain. Table below shows the number of high-fare rooms that were booked during the past 30 days. How many rooms should Hotel California protect for high-fare customers to maximize its expected revenue? (6 points)

Number of high-fare rooms	Frequency
0	5
1	3
2	6
3	8
4	4
5	1
6	1
7	2
	Total=30

Answer 1

Booking limit is number of rooms that are available for the booking to the low-fare customers.

Protection level (Q) is the number of rooms that are reserved for the high-fare customers.

Booking Limit = Total number of rooms - Protection Level

Booking Limit = 160 - Q

We have,

Low rate = r_L = $200/night

High rate = r_H = $450/night

If D is the demand of the high-rate customers, then the ideal case will be to protect Q = D rooms for the high-rate customers

Overage Cost

If D < Q, then we are protecting too many rooms and will lead to the loss of the low-rate customers as we could have sold those protected but empty rooms that are not filled to the low-rate customers i.e. at least at a lower rate.

So, Overage Cost C_o = r_L

Underage Cost If D > Q, then we are protecting fewer rooms and it will lead to a loss of the high-rate customers and we could have sold the additional D - Q rooms at a higher rate as against the lower rate.

So, Underage Cost C_u = r_H - r_L

Optimum Fare Protection Level F(Q) = Critical Ratio = C_u / (C_u + C_o) = (r_H - r_L) / r_H

Critical Ratio = (r_H - r_L) / r_H

Critical Ratio = (450 - 200) / 450

Critical Ratio = 0.556

For a cumulative probability of 0.733, which is nearly the critical ratio, the corresponding protection level of 3 rooms will help Hotel Californiafor maximizing its revenue.

Because the critical ratio is much higher than 0.467 i.e. for 2 rooms, we prefer 3 rooms.

Answer: 3 rooms