Workforce scheduling in healthcare facilities




   Project for graduate course on stochastic modeling and simulation
Scheduling of Nurses


Executive Summary

This report outlines the analysis carried out to prepare a work schedule for nur...
Scheduling of Nurses


The solution is divided into 3 major steps.

   1. Data analysis: approximating the expected number...
Scheduling of Nurses


Ward A
                                                 Confidence
Time          Mean   SD     Pati...
Scheduling of Nurses


            Ward B
                                                                          Confid...
Scheduling of Nurses




2. Solution

2.1) Constraints

2.1.1) Wage structure

Following wage structure was assumed for th...
Scheduling of Nurses


2.2) Solution methodology

   •    Interpolated tables were generated based on the data provided, u...
Scheduling of Nurses




2.3) Proposed schedule

                                      WARD A
         Projected          ...
Scheduling of Nurses


                                         WARD B
    Projected                  Shift (Permanent nur...
Scheduling of Nurses


3) Simulation: model validation

3.1) Simulation set-up

The model was validated using a spreadshee...
Scheduling of Nurses




                Appendix 1

     Interpolated tables: Ward A
PERMANENT       TEMPORARY
No. of Max...
Scheduling of Nurses




     Interpolated tables: Ward B
PERMANENT       TEMPORARY
No. of Max      No. of Max
Nurses pati...
Scheduling of Nurses


                35       65
                36       66
                37       67
               ...
Scheduling of Nurses


                                            Appendix 2

Simulation results

Five runs were executed...
Scheduling of Nurses




Run 2

           80th percentile                         4.11
           90th percentile        ...
Scheduling of Nurses




Run 4

                    80th percentile                     4.37
                    90th perc...
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Excel/VBA model for nurse scheduling in outpatient wards

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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.

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Excel/VBA model for nurse scheduling in outpatient wards

  1. 1. Workforce scheduling in healthcare facilities Project for graduate course on stochastic modeling and simulation
  2. 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. 2
  3. 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. 3
  4. 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% 4
  5. 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 5
  6. 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. 6
  7. 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. 7
  8. 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 8
  9. 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 9
  10. 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. 10
  11. 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 11
  12. 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 12
  13. 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 13
  14. 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 14
  15. 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 15
  16. 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 16

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