A model is proposed and evaluated using an Excel/VBA simulation to schedule full time and part time nurses in outpatient wards in face of probabilistic patient arrivals.

1. Workforce scheduling in healthcare facilities
Project for graduate course on stochastic modeling and simulation

2. Scheduling of Nurses
Executive Summary
This report outlines the analysis carried out to prepare a work schedule for nurses in an
upcoming medical facility in College station, Texas.
Scheduling involves the traditional tradeoff between quality and cost. We have arrived with a
variable quality level throughout the day so as to provide the best possible service at a
reasonable cost. Based on sample means and standard deviations provided for individual time-
slots in a day, statistical distributions have been arrived at to forecast the number of patients
arriving at the facility in a given hour. A variable quality level, varying from 80% to 95% has
been set for each hour, arrived at by comparing the individual means and standard deviations.
The hours for which the mean to standard ratio was the highest were given a 95% quality level
while the ones with the aforementioned ratio less than 1 were given an 80% quality level.
Scheduling of nurses has been done considering both fixed and irregular shift hours. Based on
available data and our understanding of the problem, following assumptions and constraints
have been arrived at:
• The two wards in the facility have different functionality and thereby, distinct requirement
for nurses.
• Permanent nurses, who are skilled and scarce, are limited in number. To arrive at a
definite solution for the problem, the number of available permanent nurses has been set
at 30. Temporary nurses can be sourced from the local nursing schools and are readily
available without any restrictions.
• Permanent nurses work in 8 hour shifts and can work in either of the two wards.
Temporary nurses can work any number of contiguous hours less than 8 and can only
work in a specific ward (depending upon their skill sets).
• The wages have been set at $25 an hour for the permanent nurses and $10 an hour for
the temporary nurses.
• 5% of the days are allowed for vacation, sickness, etc., for each nurse.
Scheduling has been done heuristically, by considering various possible shifts and trying
various combinations of permanent and temporary nurses and an optimal solution (providing a
reasonable quality of service at the minimum possible cost) has been proposed.
The final schedule has been tested using a simulation model and the results show that the
facility can provide at least a 95% service quality level on 80% of the days and at least a 90%
service quality level on all the days. The cost estimate for the proposed schedule is $10500 per
day.
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3. Scheduling of Nurses
The solution is divided into 3 major steps.
1. Data analysis: approximating the expected number of patients in a particular hour
2. Solution: scheduling of nurses
3. Simulation: model validation and calculation of efficiency of proposed solution.
1. Data analysis: expected number of Patients
Towards finding the approximate number of patients, the possible distributions which can fit the
given mean and standard deviations were found. Four distributions found to satisfy the
conditions. a) Gamma, b) lognormal, c)Pareto and d)Weibull. The corresponding parameters
for the distributions were calculated.
95% confidence levels for the distributions were found to be very high above the mean owing to
the high variance. To reduce the possible under utilization in the majority of the days, decision
was taken to lower the confidence level for high variance hours.
Decision: The decisions we made at this point was to give different service levels during
different hours of operation. 80% service level (manpower to cater if no. of patients
derived considering 80% confidence interval) during late-night hours to upto 95% during
the peak hours.
Reasons: Considering 95% during the odd hours which had so much variability took the
staffing required to very high values which implied under utilization.
Thumb rule: If the ratio of mean to variance is less than 1, the value for 80% confidence
interval considered. If ratio between 1 – 1.5, then 85%, 90% if the value is between 1.5 -
2 and 95% for the hours when the ratio was above 2.
Validity: Close to 80 percent of the total patients visiting the facility visit during the hours
when the ratio is greater than 2 and so the hospital can claim 92% service level overall
with this structure. For marginal increase in the service levels during off peak hours we
need to staff very high than with the current structure.
Another decision was to staff for the highest value of patients among the four distributions. This
decision will further increase the service level.
The following table for the number of patients in wards A and B were obtained.
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4. Scheduling of Nurses
Ward A
Confidence
Time Mean SD Patients Mean/SD level
00:00-01:00 9.7 20.3 13 0.48 80%
01:00-02:00 8.1 12.9 12 0.63 80%
02:00-03:00 6.1 10.1 9 0.60 80%
03:00-04:00 7.2 8.6 11 0.84 80%
04:00-05:00 9.7 10.3 14 0.94 80%
05:00-06:00 12.9 13.1 19 0.98 80%
06:00-07:00 23.4 17.9 41 1.31 85%
07:00-08:00 56.1 12.8 79 4.38 95%
08:00-09:00 61.6 13.2 85 4.67 95%
09:00-10:00 59.1 14.9 86 3.97 95%
10:00-11:00 46.4 20.4 85 2.27 95%
11:00-12:00 65.8 10.9 85 6.04 95%
12:00-13:00 71.4 9.4 88 7.60 95%
13:00-14:00 60.6 15.2 88 3.99 95%
14:00-15:00 48.3 19.8 85 2.44 95%
15:00-16:00 57 10.3 75 5.53 95%
16:00-17:00 84.7 5.2 94 16.29 95%
17:00-18:00 64.8 8.2 79 7.90 95%
18:00-19:00 39.5 11 59 3.69 95%
19:00-20:00 22.9 18 41 1.27 85%
20:00-21:00 20 14 34 1.42 85%
21:00-22:00 15.1 9 27 1.72 90%
22:00-23:00 12.5 12 23 1.05 85%
23:00-00:00 10.9 15 18 0.72 80%
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5. Scheduling of Nurses
Ward B
Confidence
Time Mean SD Patients Mean/SD level
00:00-01:00 4.1 7.2 6 0.57 80%
01:00-02:00 3.1 5.6 5 0.55 80%
02:00-03:00 6.9 10.7 10 0.64 80%
03:00-04:00 7.4 9.9 11 0.75 80%
04:00-05:00 8.1 12.1 12 0.67 80%
05:00-06:00 14.7 10.3 23 1.43 85%
06:00-07:00 20.6 15.2 35 1.36 85%
07:00-08:00 30.2 17.3 54 1.75 90%
08:00-09:00 33.6 16.1 64 2.09 95%
09:00-10:00 42.8 12.6 66 3.40 95%
10:00-11:00 31.7 20.3 59 1.56 90%
11:00-12:00 29.8 24.9 53 1.20 85%
12:00-13:00 51.2 14.2 77 3.61 95%
13:00-14:00 66.1 12.4 88 5.33 95%
14:00-15:00 24.1 16.8 34 1.43 85%
15:00-16:00 29.3 19.9 41 1.47 85%
16:00-17:00 38.5 13.6 64 2.83 95%
17:00-18:00 44.2 14.5 71 3.05 95%
18:00-19:00 21 10.2 40 2.06 95%
19:00-20:00 18.3 8.1 34 2.26 95%
20:00-21:00 17.1 7.1 30 2.41 95%
21:00-22:00 16.9 9.3 30 1.82 90%
22:00-23:00 14.2 6.1 26 2.33 95%
23:00-00:00 13.1 4.1 21 3.20 95%
Sample calculation for various distribution and the various confidence intervals. (Ward A)
Gamma (%) Lognormal (%) Weibull (%) Pareto (%)
Time 95 90 85 80 95 90 85 80 95 90 85 80 95 90 85 80
18:00-19:00 59 59 57 58
19:00-20:00 40 37 41 29
20:00-21:00 34 32 34 25
21:00-22:00 27 26 27 22
22:00-23:00 23 21 23 16
23:00-00:00 18 15 17 12
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6. Scheduling of Nurses
2. Solution
2.1) Constraints
2.1.1) Wage structure
Following wage structure was assumed for the solution:
• Hourly pay of a temporary nurse: $10
• Hourly pay of a permanent nurse: $25
For the two wards, the following chart outlines the ratio of work done by the temporary
nurses and permanent nurses. This formed the basis for deciding a wage structure for
our solution.
Ward A Ward B
No of patients Perm. Nurses Temp Nurses Ratio No of patients Perm. Nurses Temp Nurses Ratio
0-10 2 4 2 0-10 3 7 2.3333333
10-20 4 9 2.25 10-20 7 16 2.2857143
20-30 5 11 2.2 20-30 9 20 2.2222222
30-40 6 14 2.333333 30-40 12 25 2.0833333
40-50 7 16 2.285714 40-50 13 28 2.1538462
50-60 7 18 2.571429 50-60 14 32 2.2857143
60-70 8 22 2.75 60-70 15 40 2.6666667
70-80 8 25 3.125 70-80 17 46 2.7058824
80-90 9 28 3.111111 80-90 17 51 3
90-100 10 31 3.1 90-100 19 58 3.0526316
Since each permanent nurse does 2 to 3.1 times the work of a temporary nurse it was
decided that the permanent nurses will be paid more than double the hourly pay of a
temporary nurse (including benefits).
2.1.2) Number of available permanent nurses
Since it is mentioned in the problem statement that permanent nurses are skilled and
scarce, it was decided to use this fact as a constraint. The initial analysis showed that the
number of permanent nurses required to adequately staff the facility is between fifty and fifty
five. Thus, it was decided to use approximately sixty percent of that value as the number of
available permanent nurses and a figure of 30 was arrived at.
Hence the proposed solution does not optimize the number of permanent and temporary
nurses as it has been considered a constraint. Instead, the solution is aimed at minimizing
the cost incurred by the facility by way of wages to the nurses.
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7. Scheduling of Nurses
2.2) Solution methodology
• Interpolated tables were generated based on the data provided, using commercially
available software (please refer appendix 1). Using the tables, maximum patient
handling capacity was assigned for various numbers of permanent and temporary
nurses.
• For each time slot, the number of extra permanent nurses required would translate to a
patient handling capacity; corresponding values for the number of temporary nurses
required to handle those many patients were located using the interpolated tables.
• A schedule was prepared for permanent nurses; the temporary nurses were allocated to
complement the patient handling capacity for each time slot.
• To allow for 5% missed shifts/sickness/leave, one extra temporary nurse was added to
each slot. In addition, it was ensured that there is at least one permanent nurse in each
slot. Thus, for the late night shifts, a total of 3 nurses are scheduled so that at least one
can stay in each ward in case of an absence.
Based on the schedule obtained and the assumptions regarding nurses’ wages, the
establishment will need to pay USD 10,500 per day by way of wages to the nurses.
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8. Scheduling of Nurses
2.3) Proposed schedule
WARD A
Projected Shift (Permanent nurses) Temporary
Time
patients 6AM-2PM 8AM-4PM 2PM-10PM 10PM-6AM nurses
0-1 13 1 3
1-2 12 1 2
2-3 9 1 2
3-4 11 1 2
4-5 14 1 3
5-6 19 1 6
6-7 41 4 9
7-8 79 4 17
8-9 85 4 5 1
9-10 86 4 5 1
10-11 85 4 5 1
11-12 85 4 5 1
12-13 88 4 5 1
13-14 88 4 5 1
14-15 85 5 3 1
15-16 75 5 3 1
16-17 94 3 24
17-18 79 3 19
18-19 59 3 15
19-20 41 3 10
20-21 34 3 8
21-22 27 3 5
22-23 23 1 8
23-24 18 1 5
Total 146
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9. Scheduling of Nurses
WARD B
Projected Shift (Permanent nurses) Temporar
patients 6AM-2PM 8AM-4PM 2PM-10PM 10PM-6AM y nurses
6 2 1
5 2 1
10 2 1
11 2 1
12 2 1
23 2 13
35 9 2
54 9 17
64 9 4 12
66 9 4 13
59 9 4 6
53 9 4 2
77 9 4 18
88 9 4 24
34 4 2 14
41 4 2 17
64 2 30
71 2 34
40 2 21
34 2 18
30 2 16
30 2 16
26 2 15
21 2 11
Total 304
Cost Estimate:
The estimated cost according to the schedule is:
(30 × 25 × 8) + (304 + 146) × 10 = $10500 per day
Cost of permanent Cost of temporary
nurses nurses
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10. Scheduling of Nurses
3) Simulation: model validation
3.1) Simulation set-up
The model was validated using a spreadsheet to simulate random arrivals based on the
distributions chosen for each time slot.
Simulation setup:
• Each nurse was assigned a probability of 0.05 of not showing up on her shift.
• Random numbers for each timeslot (based on the distributions used for each slot) were
generated using an add-in for MS Excel.
• The quality of service was defined as:
Number of patients who had to wait (or who missed the service), cumulated over the
entire day/Total number of arrivals
Since the service times for the nurses were not considered in the problem, an accurate
assessment was not carried out for how the waiting patients would be serviced/managed.
Considering the fact that the subsequent hours might register a fewer number of arrivals, the
patients would eventually be treated. However, since the patients were not attended to
promptly, or chose to renege seeing a large queue, it gives a broad indication of the quality level
of service at the facility.
3.2) Results
Five simulation runs were carried out, each run simulated for 250 days (i.e. 250 iterations). The
percentage of patients who were not catered to promptly was obtained from the experiment.
Results of the simulation are shown in appendix 2.
As can be observed from the results, the 80th percentile value for the simulation runs is
approximately 4.5%; i.e. on 80% of the days, 95.5% service level was achieved. Moreover, in all
the runs, the maximum percentage of patients who were not catered to is approx 11%. Hence,
we can conclude from the results that the schedule developed would enable the hospital to
maintain at least an 89% quality level on all days.
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11. Scheduling of Nurses
Appendix 1
Interpolated tables: Ward A
PERMANENT TEMPORARY
No. of Max No. of Max
Nurses patients Nurses patients
0 0 0 0
1 5 1 2
2 10 2 4
3 15 3 8
4 20 4 10
5 30 5 12
6 40 6 14
7 60 7 16
8 80 8 17
9 90 9 20
10 100 10 24
11 30
12 33
13 36
14 40
15 44
16 50
17 55
18 60
19 62
20 65
21 67
22 70
23 72
24 76
25 80
26 82
27 86
28 90
29 94
30 97
31 100
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12. Scheduling of Nurses
Interpolated tables: Ward B
PERMANENT TEMPORARY
No. of Max No. of Max
Nurses patients Nurses patients
0 0 0 0
1 5 1 3
2 8 2 5
3 10 3 6
4 12 4 8
5 14 5 9
6 17 6 9
7 20 7 10
8 26 8 11
9 30 9 12
10 33 10 12
11 35 11 13
12 40 12 14
13 50 13 15
14 60 14 17
15 70 15 18
16 78 16 20
17 90 17 23
18 95 18 25
19 100 19 28
20 30
21 32
22 34
23 36
24 38
25 40
26 43
27 47
28 50
29 53
30 56
31 58
32 60
33 62
34 63
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13. Scheduling of Nurses
35 65
36 66
37 67
38 68
39 70
TEMPORARY
No. of Nurses Max patients
40 72
41 73
42 75
43 77
44 78
45 80
46 82
47 84
48 86
49 88
50 90
51 92
52 94
53 95
54 97
55 98
56 99
57 100
58 100
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14. Scheduling of Nurses
Appendix 2
Simulation results
Five runs were executed and the results are tabulated in the subsequent pages.
On the X Axis: Percentage of patients who were not serviced/attended to promptly
On the Y axis: Frequency of occurrence in a 250 day period.
Run 1
80th percentile 3.71
90th percentile 5.36
95th percentile 6.33
30.00
25.00
20.00
15.00
10.00
5.00
0.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
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15. Scheduling of Nurses
Run 2
80th percentile 4.11
90th percentile 5.47
95th percentile 6.69
35.00
30.00
25.00
20.00
15.00
10.00
5.00
0.00
-5.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00
Run 3
80th percentile 4.20
90th percentile 5.70
95th percentile 7.11
30.00
25.00
20.00
15.00
10.00
5.00
0.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
-5.00
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16. Scheduling of Nurses
Run 4
80th percentile 4.37
90th percentile 5.79
95th percentile 6.86
30
25
20
15
10
5
0
0.00 2.00 4.00 6.00 8.00 10.00 12.00
-5
Run 5
80th percentile 4.89
90th percentile 6.25
95th percentile 7.73
30
25
20
15
10
5
0
0.00 2.00 4.00 6.00 8.00 10.00 12.00
-5
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