Systematic OR Block Allocation at a Large Academic Medical Center
Online Companion
I. Simplified example of the effects of block reallocation on inpatient unit census
Assume we have a system with a single OR that operates Monday through Friday. We present a simplified example
of the impact of a change in the OR block schedule on the maximum of the average weekday census. Each series
represents the expected bed-day impact of a surgical block that operates on a given day of the week. The
permutation recommends that Block 1 changes from Wednesday to Monday, Block 2 from Tuesday to Friday, Block
3 from Monday to Thursday, Block 4 from Thursday to Wednesday, and Block 5 from Friday to Tuesday (3b). Series
6 denotes the maximum weekly bed occupancy.
Figure 1. Simplified example of effects an OR block permutation on the average bed occupancy given each block’s
expected length of stay (LOS) profile.
Systematic OR Block Allocation at a Large Academic Medical Center
II. Integer Programming Formulation
In this section we fully specify the integer programming model we used to derive the new surgical schedule. We
first specify the notation used. Important sets referenced by the model are:
days in the week starting from Sunday
weeks in the month
ORs
OR-day pairs corresponding to surgical blocks
pairs of blocks corresponding to allowed moves
surgical floor units
surgeons
services
set of block-week pairs owned by surgeon
for
The parameters are:
number of patients expected to occupy a bed on floor
having surgery days ago (mod 7) in
block
the current peak average bed census for floor
number of weeks service
owns block
lower bound on number of blocks allocated to service
upper bound on number of blocks allocated to service
each day
each day
The decision variables are:
peak bed census on floor
resulting from permuted schedule
Systematic OR Block Allocation at a Large Academic Medical Center
The general form of the integer program is given by:
The objective (1) maximizes the reduction in the peak average bed census across the floors. Summing the
reductions across floors avoids improving the situation in one floor unit at the expense of another. Constraint (2)
ensures that the solution is a bonafide permutation of the schedule. In constraint (3) we link the
decision
variables so that they are at least as great as the census on each day, and thus correspond to the peak census of
each floor. The expression on the left-hand side calculates the expected census by finding the contribution of each
block on its new day to the day in question. The model guarantees that surgeons are not overbooked in constraint
(4). Note that the same surgeon is not necessarily working in an OR each week of the month, which is why it is
necessary to go down to the level of day of week and week of month. Finally, we need to have each service have a
relatively balanced access to ORs throughout the week, which is handled by constraint (5), with specified upper
and lower bounds for the number of blocks. The actual integer program contains many additional minor
constraints like surgeon-specific availability and particular linked blocks, but these are omitted here for clarity.
Systematic OR Block Allocation at a Large Academic Medical Center
III. Numerical Results
In this section we provide additional numerical details that complement the Results section of the manuscript.
Additional Average Midnight Census Details
Table andTable 2 display detailed census information corresponding to the surgical units census for patient groups
A and B, respectively. The presented performance metrics are: daily average statistics, weekday average, the
maximum of the average weekly census, and the maximum inter-day census differences. The definition of the
performance metrics can be found in the Metrics subsection of the manuscript.
We recall that Group A includes all patients in surgical units except observation patients and those who had
surgery in any of the OR blocks added after January 2012. Thus, Table describes Figure 4a of the manuscript (first
and last columns), and provides supplementary material regarding the model predictions (middle columns).
Table 1: Group A midnight census statistics for the different time frames and scenarios of study: actual data, first
model prediction, model prediction after negotiations, actual data after implementation.
Apr-10Mar-11
(actual)
Apr-10Mar-11
(1st model pred.)
Apr-10Mar-11
(model pred.
after negs.)
Jan-12Dec-12
(actual)
Sunday
272.40
277.37
275.90
265.11
Monday
299.63
298.74
296.33
291.85
Tuesday
304.58
301.75
301.85
300.96
Wednesday
313.10
302.85
309.35
303.06
Thursday
307.98
299.92
301.54
296.65
Friday
296.02
301.82
300.08
286.62
Saturday
275.73
283.69
281.35
270.46
Average Weekday Census
304.26
301.02
301.83
295.83
Average Census Peak
313.10
302.85
309.35
303.06
Maximum Inter-day
Census Differences
11.96
3.01
7.81
10.03
Thu → Fri
Mon → Tue
Mon → Tue
Thu → Fri
Metric
Average
Midnight
Census
On the other hand, Group B represents all patients in the surgical units. As such, the first three columns of Table 2
correspond to Figures 3a and 3b of the manuscript. The first and last columns describe Figure 4b.
Systematic OR Block Allocation at a Large Academic Medical Center
Table 2: Group B midnight census statistics for the different time frames and scenarios of study: actual data, first
model prediction, model prediction after negotiations, and actual data after implementation.
Apr-10Mar-11
(actual)
Apr-10Mar-11
(1st model pred.)
Apr-10Mar-11
(model pred.
after negs.)
Jan-12Dec-12
(actual)
Sunday
284.00
288.90
287.31
285.09
Monday
316.63
315.78
313.13
322.09
Tuesday
322.56
319.79
319.98
335.14
Wednesday
329.71
319.19
326.17
333.84
Thursday
325.06
316.77
318.75
328.27
Friday
312.98
318.88
316.86
319.12
Saturday
288.29
296.54
293.92
292.52
Average Weekday Census
321.39
318.08
318.98
327.69
Average Census Peak
329.71
319.79
326.17
335.14
Maximum Inter-day
Census Differences
12.08
4.01
7.42
13.05
Thu → Fri
Mon → Tue
Mon → Tue
Mon → Tue
Metric
Average
Midnight
Census
Figure Figure 2 illustrates the abrupt changes that the observation-patient midnight census suffered between the
control and the study periods. The census statistics for the model predictions were extremely similar to those of
the data of the control period; they are omitted here for intelligibility.
Figure 2: Observation patients midnight census during control and study periods.
Apr-10-Mar-11
(actual)
Jan-12-Dec-12
(actual)
Sunday
11.60
17.06
Monday
17.00
26.09
Tuesday
17.98
30.02
Wednesday
16.62
26.29
Thursday
17.08
27.94
Friday
16.96
29.13
Saturday
12.56
19.06
Average Weekday Census
17.13
27.89
Average Census Peak
17.98
30.02
Maximum Inter-day
Census Differences
1.37
3.93
Tue → Wed
Mon → Tue
Metric
Average
Midnight
Census
Systematic OR Block Allocation at a Large Academic Medical Center
Pre and Post-Surgical LOS
Table 3 introduces the breakdown of the ALOS into Pre-Surgical and Post-Surgical LOS for each surgical category.
These statistics are computed with respect to the first surgical procedure for the cases in which patients had
multiple surgeries within a single hospital admission.
Table 3: Change in average pre and post-surgical LOS. ALOS is measured for patients who stayed at least one night
in any of the surgical units. Time Frames: April 2010 to March 2011 and January 2012 to December 2012.
Patient group
Group A
Group B
Patient
Surgical Type
Average Pre-Surgical LOS
Apr 10 –
Mar 11
Jan 12 –
Dec 12
Scheduled
0.37
0.31
-16.8%
Wait list
1.32
1.16
Total
0.61
Scheduled
Average Post-Surgical LOS
∆%
Apr 10 –
Mar 11
Jan 12 –
Dec 12
∆%
**
3.87
3.79
-1.9%
-12.6%
*
5.59
5.51
-1.5%
0.53
-14.0%
**
4.30
4.23
-1.6%
*
0.36
0.27
-25.4%
**
3.75
3.44
-8.2%
**
Wait list
1.30
1.15
-12.2%
*
5.46
5.37
-1.7%
Total
0.59
0.47
-20.3%
**
4.17
3.89
-6.8%
*
**
* p-value < 0.05, ** p-value < 0.001, two-sample, one-sided Mann–Whitney–Wilcoxon test
Patient Mix Trends
In the manuscript, we described a dramatic shift in the elective census patient mix: SD-patients
increased in volume by 12% while the IN-patient volume decreased by 19% between the control and
study periods. In other words, excluding observation patients, the composition of the elective surgical
census shifted significantly to contain many more SD-patients and fewer IN-patients. Figure 3 illustrates
that this shift has occurred at a small, but significant rate since March 2010 (2.4% per year, p << 0.001).
Systematic OR Block Allocation at a Large Academic Medical Center
Figure 3. Change in percentage of scheduled Same-Day and Inpatient patients in midnight census in surgical units
over time, excluding observation patients. All regression coefficients are statistically significant (t-test, p << 0.001).
Time frame: March, 2010 to February, 2013.