Question

The Baruch Star Hotel in Miami, FL is considering doing overbooking in order to deal with the constant problem they have with no-shows. The table given below presents the number of no-shows and the probability of each occurring. # of No- Probability of No- Shows Shows occurring (d) P(d) 0 0.07 0.07 2 0.10 3 0.02 4 0.12 5 0.01 0.04 0.19 8 0.16 9 0.22 6 a) What would be your recommendation for overbooking if the average rate per room per night is $145 and the cost of not honoring a reservation is $205? b) What is the expected loss for your overbooking choice?

Answer 1

a)

Underbooking cost, Cu = $ 145

Overbooking cost, Co = $ 205

Critical ratio = Cu/(Cu+Co)

= 145/(145+205)

= 0.4143

Cumulative probability distribution

# of no shows	Probability	Cumulative probability
0	0.07	0.07
1	0.07	0.14
2	0.1	0.24
3	0.02	0.26
4	0.12	0.38
5	0.01	0.39
6	0.04	0.43
7	0.19	0.62
8	0.16	0.78
9	0.22	1

In the above table, look for cumulative probability just greater than 0.4143

That value of cumulative probability is 0.43 and corresponding # of no shows is 6

Therefore, Recommendation overbooking is:  6

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b)

Loss occurs when # of no shows is greater than overbooking, i.e. for 7, 8, 9 no shows

Expected # of no shows greater than overbooking = (7-6)*0.19+(8-6)*0.16+(9-6)*0.22

= 1.17

Expected loss = 1.17*Cu

= 1.17*145

= $ 169.65