Question

Problem 2. (25 Points) To get full credit, please show all your work. 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- Shows (d) 0 2 3 4 5 Probability of No- Shows occurring P(d) 0.07 0.07 0.10 0.02 0.12 0.01 0.04 0.19 0.16 0.22 6 7 8 9 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)

Cost of overbooking one less than the cancellation, C_u = Cost of an empty room = 145
Cost of overbooking one more than the cancellation, C_o = Cost of a dishonored guest = 205

So,

The optimal service level = C_u / (C_o + C_u) = 145 / (205+145) = 0.414

d	P(d)	F(d)
0	0.07	0.07 < 0.414
1	0.07	0.14 < 0.414
2	0.10	0.24 < 0.414
3	0.02	0.26 < 0.414
4	0.12	0.38 < 0.414
5	0.01	0.39 < 0.414
6	0.04	0.43 < 0.414
7	0.19	0.62 > 0.414
8	0.16	0.78
9	0.22	1.00

The F(d) is >= 0.414 at d=7, so, the optimum overbooking level = 7

(b)

d	P(d)	Empty rooms	# disnonored
0	0.07	0	7
1	0.07	0	6
2	0.10	0	5
3	0.02	0	4
4	0.12	0	3
5	0.01	0	2
6	0.04	0	1
7	0.19	0	0
8	0.16	1	0
9	0.22	2	0

So, the expected number of empty rooms = 1*0.16 + 2*0.22 = 0.60

and the expected number of dishonors = 7*0.07 + 6*0.07 + 5*0.1 + 4*0.02 + 3*0.12 + 2*0.01 + 1*0.04 = 1.91

So,

The expected loss = 0.60*145 + 1.91*205 = $478.55