Use the following information to determine the optimal overbooking policy for a
ID: 432941 • Letter: U
Question
Use the following information to determine the optimal overbooking policy for a motel at a rural town in South Carolina. The motel’s capacity is 30 rooms. The historic number of no-shows for a typical day, along with the probability of occurrence, is shown in the following table. The average profitability per room is $110, and the cost of lost goodwill per guest due to overbooking is approximately $60. No-Shows Probability 0 0.25 1 0.35 2 0.15 3 0.25 Calculate the total expected profit with 33 reservations (which means 3 overbookings). $3,300.00 $3,204.00 $2,897.50 $2,883.50 $2,827.00
Explanation / Answer
Profit per room = $110
Cost per loss of goodwill = $60
For an alternative of 3 overbookings, there are 4 states of nature that can exist, i.e. 0 no-shows, 1 no-shows, 2 no shows or 3 no shows
For 0 no-shows, the cost will be due to loss of goodwill of 3 customers, i.e. 3 * 60 = $180
For 1 no-shows, the cost will be due to loss of goodwill of 2 customers, i.e. 2 * 60 = $120
For 2 no-shows, the cost will be due to loss of goodwill of 1 customers, i.e. 1 * 60 = $60
For 3 no-shows, the cost will be due to loss of goodwill of 0 customers, i.e. 0 * 60 = $0
The probability of 0 no-shows = 0.25
The probability of 1 no-shows = 0.35
The probability of 2 no-shows = 0.15
The probability of 3 no-shows = 0.25
The total expected loss due to overbooking= 0.25 * 180 + 0.35 * 120 + 0.15 * 60 + 0.25 * 0 = $96
Total profit from filling all the 30 rooms = $110 * 30 = $3300
Total expected profit = Total profit from filling all the 30 rooms? - Expected loss due to overbooking policy
= 3300 - 96
= $3204
Hence the correct answer is option B.