Question

Operating Room

The hospital has to allocate a certain amount of operating room (OR) time to specific cardiac
procedures. Since the actual procedure time in the OR is random and will - in the best of all cases
- vary around the expected procedure time, some procedures will exceed the forecasted durations
while others will be completed ahead of schedule. If the hospital reserves too much time to a case,
the OR is likely to incur excessive idle time. If, however, the hospital reserves too little time to a
case, the hospital is likely to face schedule over-runs and decreased service quality.

Case	Reserved Time (min)	Actual Time (min)	A/F Ratio
A	90	122	1.35
B	100	83	.83
C	120	121	1.01
D	150	145	.97
E	180	209	1.16

There is a new case, say, F. The reservation system predicts the operating time to be 110 minutes.
From past experience, the doctors think that the reservation system has a certain percentage of
predicting errors (see the table for the data from previous 5 cases).

The doctors wonder if Newsvendor model would help in their planning.

a) Construct an empirical distribution (discrete distribution) for operating time of case F based
on the historical data.

b) If the doctors want to make sure that the reserved time is enough for the procedure with at
least 80% of the chance, how much time should be reserved for case F? (Hint: First decide
what corresponds to “demand” and what corresponds to “order quantity” in this OR context.
Then apply the appropriate service level formula.)

c) Given the reserved time in part b), what is the expected overtime for case F? (Hint: overtime
happens when the procedure time exceeds the reservation duration.)

d) Given the reserved time in part b), what is the expected idle time for case F? (Hint: idle time
happens when the procedure time is less than the reservation duration.)

Answer 1

(a)

Reserved Time (min) - F	A/F Ratio	Actual Time (min) = F x (A/F)	Probability	Cumulative Probability
110	0.83	91.3	0.2	0.2
110	0.97	106.7	0.2	0.4
110	1.01	111.1	0.2	0.6
110	1.16	127.6	0.2	0.8
110	1.35	148.5	0.2	1.0

(b)

The Procedure time is equivalent to demand and the Reserved time is equivalent to the Order quantity.

At 80% service level the prediction should be 127.6 (using the cumulative distribution function from the above table).

(c) & (d)

Actual Time (min)	Probability	Reserved time (min)	Overtime (min)	Idle time (min)
91.3	0.2	127.6	0	36.3
106.7	0.2	127.6	0	20.9
111.1	0.2	127.6	0	16.5
127.6	0.2	127.6	0	0
148.5	0.2	127.6	20.9	0
Expected values =			4.18	14.74