We evaluate our optimization models with input parameters representing elective patients at Hamilton Health Sciences. The first input parameter, the expected PDB by each block for
Page | 38 each day of the week, is calculated by the bed planning model using a sample operating room schedule from the previous year, shown in Table 6.1.
1 2 3 4 5 6 7 8
Table 6.1: A sample original operating room schedule
The value of the first input parameter (the expected PDB on each day of the week for each block) is shown in Table C.1. A sample of Table C.1 is shown below in Table 6.2.
Block ID
Block Assignment
Expected Patient Demand for Beds t Days After Surgery
Surgeon
Table 6.2: A sample of expected PDB and surgeon assigned to each block
In this experiment, we did not restrict any block to an operating room or day of the week. As a result, the third parameter (list of infeasible days of the week for each block) is an empty list. For our fourth parameter (operating room availability for each block), we have determined the operating room settings from staff at Hamilton. Out of the eight operating rooms, the first four are reserved to Orthopedics surgeries. Operating rooms 5 and 6 are used for general surgeries, and operating rooms 7 and 8 are dedicated to Urology. Based on the sample operating room schedule, we determine the service that each block provides. Blocks
Page | 39 can only be assigned to operating rooms with the same service type, as shown in Table C.3.
A sample of Table C.2 is shown below in Table 6.3.
Block ID Block Info Procedure Available Operating Rooms 1 Mon OR1 Orth 1, 2, 3, 4
Table 6.3: A sample of operating room restriction for each block
Figure 6.1: Experimental results for MIP and 2-opt approach
Figure 6.1 presents the computational results from the optimization models. It displays the expected day shift PDB based on the original operating room schedule from the previous year, the optimal operating room schedule generated by the mixed integer programming model (shown in Table 6.4), and the local optimal operating room schedule generated by the
59.8
Expected Day Shift Patient Demand for Beds
original schedule MIP optimal schedule 2-opt local optimal schedule
Page | 40 2-opt heuristic (shown in Table 6.5). The optimal schedule results in a peak expected PDB of 80.8, followed closely by the 2-opt solution with peak expected PDB of 81. These two schedules reduce the peak demand by about 8. Since we simply shift the demand on Thursday or Friday to other days of the week, the average of expected PDB stays the same.
1 2 3 4 5 6 7 8
Table 6.4: Optimal operating room schedule from MIP
1 2 3 4 5 6 7 8
Table 6.5: Near-optimal operating room schedule from 2-opt heuristic
The decrease in peak expected PDB is not without penalties. The optimal schedule moves 35 out 40 blocks in the original schedule and the 2-opt schedule moves 30 blocks. We believe that such dramatic changes to the existing schedule would encounter considerable resistance from the surgeons involved. We believe the mixed integer programming model is useful as a
Page | 41 diminishing return on bed saving while revising the operating room schedule. For example, the first two steps have provided the greatest decrease in PDB, 1.2 and 1 respectively. 4 blocks are moved from these 2 swaps, resulting in 2.2 out of the 7.5 total potential reductions by 2-opt heuristic (almost 30%). The later steps provide considerably less improvement.
Figure 6.2: Decrease in patient demand for beds from each step in 2-opt heuristic
Furthermore, managers have to negotiate with the surgeons on how much change to implement and who is affected. Therefore, the cost and the benefit for each feasible swap at each step should be transparent to all parties. The 2-opt heuristic assesses alternatives at each step and presents the degree of improvement. For example, the top five swaps for the first step in 2-opt heuristic is shown in Table 6.6. This table provides four alternatives at the first
1.2
Page | 42 step if the best swap is questioned by the stakeholders and the cost associated with choosing these lesser options.
Top Ranked Available Swaps
Expected Day Shift Patient Demand for Beds Sun Mon Tue Wed Thu Fri Sat Peak
no swap 59.8 66.3 74.2 78.8 87.9 88.5 73.9 88.5
OR 7 on Thursday with OR 8 on Tuesday 60.1 66.5 75.4 79.6 86.6 87.3 73.9 87.3 OR 8 on Tuesday with OR 8 on Thursday 60.1 66.8 75.5 79.5 86.8 87.3 73.4 87.3 OR 1 on Tuesday with OR 3 on Thursday 61.1 67.6 73.8 78 87.2 87.5 74.2 87.5 OR 3 on Thursday with OR 4 on Tuesday 60.8 67.3 74.3 78.4 87 87.5 74.1 87.5 OR 3 on Tuesday with OR 3 on Thursday 61.3 67.4 73.7 77.8 86.9 87.6 74.7 87.6
Table 6.6: Top five swaps for the first step in 2-opt heuristic
In this section, we presented the numerical results from the mixed integer programming model and 2-opt heuristic. Both approaches generate excellent operating room schedule based on the data provided by Hamilton Health Sciences. The mixed integer programming model generates an optimal schedule resulting in a peak expected PDB of 80.8, followed closely by the 2-opt solution with peak expected PDB of 81. However, the optimal schedule moves 35 out 40 blocks in the original schedule. Realistically speaking, such dramatic change to the existing schedule will meet resistance by the stakeholders involved. As a result, the numerical results from the mixed integer programming model could only be used as a benchmark for the 2-opt solution. On the other hand, 2-opt heuristic allows the users to improve the operating room schedule incrementally by showing the trade-off of each feasible swap at each step.
Page | 43
Chapter 7
Application
In this chapter, we describe the work we have done for hospitals other than Hamilton Health Sciences. First of all, we were asked to investigate patient demand for beds in each department at William Osler Health Centre (Brampton Civic and Etobicoke General) to derive bed capacity for the upcoming fiscal year. In addition to that, we determined that there is a potential for large bed saving if patients in alternative-level-of-care, were discharged earlier. We also analyzed the hospital bed requirements using the CIHI (Canadian Institute for Health Information) 25 percentile benchmark for ward length-of-stay. Again, this provided a significant reduction in bed capacity required.
Furthermore, we worked closely with Regina General Hospital to design dedicated wards capacity and to balance elective patient demand for beds. Allocating proper ward capacity reduces off-service patient placements, thereby improving quality of care. Lowering peak patient demand for beds frees up beds when they are most needed and potentially reduces patient wait time and recovery time. In this case, the peak patient demand for beds occurs on Thursday and Friday. We established two policies with regard to balancing elective patient demand for beds:
1. If possible, move surgical procedures that generate long length-of-stay inpatients to Friday. This will maximize bed utilization on the weekends and the early weekdays.
2. If possible, move surgical procedures that generate a lot of inpatients to Monday and Tuesday. This will shift patient demand for beds to earlier weekdays.
Page | 44 All in all, our tools were not designed specifically for just one hospital setting; instead we built them for generally purpose, in which any hospital can be represented with appropriate parameters and inputs. See Appendix D for a demonstration of the tools.
Page | 45
Chapter 8
Conclusion
This thesis detailed the development, validation, and results of a set of simulation and optimization tools. The bed planning (simulation) model estimates patient demand for beds in a hospital during a typical week. The bed capacity can be calculated from patient demand for beds by either the occupancy target level analysis or the probability of bed blocking method.
Our simulation model imitates an existing hospital and then manipulates it by adjusting parameters used to build the model. The parameters we tested include patient length-of-stay and operating room schedule. We identified one of the improvement opportunities of a hospital by benchmarking current patient stay against expected patient length-of-stay from CIHI. The simulation results showed that there are more Orthopedics bed days than expected. We believe it was the result of Orthopedics patients who require alternative-level-of-care are staying for prolonged periods of time while waiting for rehab, home care, long-term care home, or placement. We want to stress that the bed planning model is generic and can easily be implemented in any hospital. As a simulation, the bed planning model has great potential in decision making during the evaluation of alternatives. If a manager could simulate alternatives and predict their outcomes at this point in the decision process, he or she could eliminate much of the guesswork from decision making.
Finally, we were able to smooth the expected patient demand for beds (PDB) throughout the weekdays by modifying the operating room schedule, which reduced the maximum number
Page | 46 of beds needed without affecting patient volume. Our approach for revising operating room schedule included mixed integer programming, and a 2-opt heuristic. Both the mixed integer programming model and the 2-opt heuristic lowered the peak elective PDB by about 8%.
However, the decrease in peak elective PDB is not without penalties. At least 75% of the blocks in the final operating room schedule are moved. In practice, such dramatic change to an existing schedule would be unpopular by the surgical team. We believe the mixed integer programming model is useful as a benchmark tool to show the potential cost savings of an optimal schedule, but it is hard to implement in practice. 2-opt heuristic not only generates a near-optimal solution, it also shows the trade-off of each feasible swap at each step. Since 2-opt heuristic enables scenario planning for the hospital administration to test alternative operating room schedules quickly, it is useful when there are significant differences of opinion over the relative merits of the different courses of action within a hospital.
Page | 47
Chapter 9
Future Research
Future research would involve extending the bed planning model to be more generic. First, the existing model should be applied to more hospital sites to ensure that its assumptions hold and improvements can be found for those sites as well. Secondly, if one or more assumptions are violated in the future application, the bed planning model should be modified to remain generic to any hospital settings.
Page | 48
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Page | 53
Appendix A
Bed Planning Model Outputs
The bed planning model outputs mean and standard deviation of patient demand on beds in each department at each of 21 shifts for each patient group. Table A.1 – A.4 shows the mean and the standard deviation of patient demand on beds in emergency department, special care unit, acute wards and alternative level of care unit at each shift. The patients are categorized by the main service that they received.
Page | 54
Emergency Department Mean of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery000.10.100.10.100.10000.1000000000 Medicine7.25.4786.86.786.47.28.46.46.97.96.57.38.16.26.475.566.9 Surgery0.70.71.41.41.11.41.30.81.41.20.81.41.91.11.41.40.90.91.10.60.91.1 GI Medicine0.30.20.30.20.20.30.40.30.20.20.20.40.40.40.40.50.40.40.30.20.30.3 Oncology0.40.30.80.70.40.60.40.30.90.60.10.60.50.30.50.40.40.50.50.50.80.5 Hematology0.20.20.30.30.30.40.30.20.40.30.20.40.30.20.40.60.40.40.40.30.20.3 Orthopedics0.30.30.30.50.20.40.70.40.60.60.30.60.40.30.50.40.20.30.40.20.20.4 Orthopedic Oncology000000000000.10.1000000000 V-Medicine0000000000000000000000 Gynecology Oncology0000000000000000000000 SUM9.17.310.311.38.91011.28.410.811.3810.511.78.910.511.58.599.77.38.69.7 Standard Deviation of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery00.20.30.30.10.20.30.20.2000.20.20.10.20.200.2000.20.2 Medicine2.42.42.62.52.62.42.62.42.72.92.52.42.932.72.92.22.32.72.22.22.5 Surgery0.811.61.51.11.51.41.11.51.20.81.31.40.91.21.1111.20.81.11.2 GI Medicine0.50.50.60.50.40.50.60.60.50.40.40.60.70.60.60.60.60.60.50.40.60.5 Oncology0.60.5110.80.90.60.610.80.30.70.80.60.70.60.60.60.80.610.7 Hematology0.60.50.60.60.50.60.70.40.60.50.50.80.60.40.60.80.80.60.80.60.50.6 Orthopedics0.50.60.60.80.50.70.80.50.70.70.50.80.70.60.70.70.50.50.60.40.50.6 Orthopedic Oncology000.10000.10.30.200.10.30.200000.100.10.10.1 V-Medicine0000000000000000000000 Gynecology Oncology0000000000000000000000 SUM2.82.83.43.333.13.22.83.43.42.83.13.63.33.33.42.82.83.32.62.83.1
Sat SunMonTueWedThuFriSat
SunMonTueWedThuFri Table A.1: Patient demand for beds at emergency department
Page | 55
Special Care Unit Mean of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery0.20.10.10.10.60.60.40.40.40.40.60.60.60.50.40.40.30.30.30.20.20.4 Medicine111111.111.411.11110.911.11111.111.11111.210.810.810.810.510.310.510.610.710.9 Surgery5.55.75.67.17.16.97.47.67.47.47.57.37.27.67.57.16.86.465.85.66.8 GI Medicine0.30.40.40.40.40.40.40.40.40.40.30.40.30.30.30.30.30.40.30.30.30.3 Oncology0.60.60.70.70.70.70.70.70.80.80.80.80.80.80.90.80.80.80.80.80.70.7 Hematology1.71.71.61.61.61.61.61.51.61.41.51.61.61.61.61.61.61.61.71.71.71.6 Orthopedics1.32.1222.82.72.83.12.92.72.72.52.32.32.221.81.81.51.21.22.2 Orthopedic Oncology0.30.20.20.20.20.20.20.30.30.30.30.30.30.30.30.30.30.20.20.20.20.3 V-Medicine0000000000000000000000 Gynecology Oncology0.10.10.10000000000000000000 SUM2121.921.823.524.424.124.525.124.724.424.924.524.424.324.123.322.521.821.420.820.623.2 Standard Deviation of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery0.40.40.30.30.60.60.60.60.60.60.70.70.70.60.60.60.50.50.50.50.40.5 Medicine33.133.33.13.23.13.232.92.92.93.233.13.33.23.23.23.33.23.1 Surgery2.42.42.32.62.72.62.52.62.62.72.72.72.72.52.62.52.52.42.32.32.32.5 GI Medicine0.60.60.60.60.60.50.60.60.60.60.60.60.50.50.50.50.50.50.50.50.50.6 Oncology0.80.80.80.80.80.80.90.910.9110.90.90.90.90.90.90.90.90.80.9 Hematology1.31.21.21.11.21.21.21.21.21.11.21.21.21.21.31.31.21.31.31.21.21.2 Orthopedics1.11.41.41.41.91.91.9221.81.81.71.41.41.41.41.31.21.21.11.11.5 Orthopedic Oncology0.60.50.40.50.50.40.40.50.50.50.50.50.50.50.50.50.50.50.50.50.50.5 V-Medicine0000000000000000000000 Gynecology Oncology0.20.20.20000.10.10.10.1000.20.20.20.10.10.10.10.10.10.1 SUM4.44.54.44.74.84.94.74.94.84.74.74.74.74.64.74.84.64.64.54.54.44.6
Sat SunMonTueWedThuFriSat
SunMonTueWedThuFri Table A.2: Patient demand for beds at special-care unit
Page | 56
Acute Wards Mean of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery1.71.71.71.71.81.71.81.71.61.61.61.61.71.81.71.71.71.81.71.61.71.7 Medicine68686767676768686868686869697070706969696868 Surgery56565558595861616063656667696868656463615962 GI Medicine4.94.64.44.54.44.64.54.74.94.64.54.54.54.54.64.64.84.84.84.64.64.6 Oncology19181920192020202121212121212121202020191920 Hematology26262726262727272828272828282929282827272727 Orthopedics31353332393636403938414041434040383534333037 Orthopedic Oncology1.51.51.51.41.51.31.21.61.61.61.61.61.71.91.91.91.81.81.71.71.61.6 V-Medicine24242424232424242424242424242424232323232324 Gynecology Oncology0.20.20.20.30.20.20.20.20.20.20.20.20.30.20.20.20.20.20.20.20.20.2 SUM231234232235241240243248247250254255257261260259254247246240234246 Standard Deviation of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery1.41.41.41.41.31.41.41.21.21.21.21.21.31.31.31.31.31.31.31.21.21.3 Medicine8888888888887788887788 Surgery8889999889999999888888 GI Medicine2.11.922.122.22.122.12.12.22.22.22.12.12.22.42.32.42.22.12.1 Oncology4444444444444444444444 Hematology5555555555555555555555 Orthopedics6666666666666666666566 Orthopedic Oncology1.41.51.51.41.51.31.31.41.41.41.41.41.41.61.51.51.41.41.41.41.31.4 V-Medicine5555555555555555555555 Gynecology Oncology0.50.40.40.50.50.50.50.40.40.40.40.40.50.50.50.50.50.50.50.50.50.5 SUM14.714.915.115.415.315.715.615.615.616.015.715.615.415.715.715.615.615.214.914.714.915.4
Sat SunMonTueWedThuFriSat
SunMonTueWedThuFri Table A.3: Patient demand for beds at acute wards
Page | 57
Alternative Level of Care Unit Mean of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery0.10.10.10.10.10.10.10.20.20.20.20.20.20.20.20.20.20.10.20.20.20.2 Medicine19.319.619.219.118.818.818.818.618.418.318.318.318.418.518.518.718.618.718.718.418.618.7 Surgery2.52.52.52.52.62.52.62.72.72.82.82.82.72.72.72.62.62.72.62.52.52.6 GI Medicine0.30.30.30.30.40.30.30.30.30.30.30.30.30.30.30.40.40.40.40.40.40.3 Oncology4.84.84.84.84.94.74.84.84.84.84.84.84.84.94.8555.154.94.94.9 Hematology0.80.80.70.70.70.70.60.60.60.70.60.70.60.70.70.60.60.70.60.60.60.7 Orthopedics12.91312.912.912.9131313.113.113.213.113.11313.113.213.213.113.313.313.313.513.1 Orthopedic Oncology0.40.40.40.40.40.50.50.50.50.50.50.50.50.50.40.40.40.40.40.40.40.5 V-Medicine0000000000000000000000 Gynecology Oncology0000000000000000000000 SUM41.241.541.14140.740.740.740.740.640.740.740.640.540.840.84140.941.341.240.741.140.9 Standard Deviation of Patient Demand for Beds Patients categorized by servicenightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningnightdayeveningAverage GI Surgery0.40.40.40.40.30.40.40.50.50.40.50.50.50.50.50.50.40.40.40.40.40.4 Medicine4.74.74.84.64.54.44.24.24.34.14444.14.24.44.64.54.64.64.54.4 Surgery1.51.51.51.51.51.51.51.61.61.61.71.71.61.61.61.61.61.61.61.51.51.6 GI Medicine0.50.50.60.60.60.60.60.60.50.50.50.50.50.50.50.50.50.50.50.50.50.5 Oncology2.32.32.32.32.52.42.42.32.32.42.32.22.22.22.12.22.32.32.22.22.22.3 Hematology0.80.80.80.80.80.80.80.70.70.80.80.80.80.80.70.70.70.70.70.70.70.8 Orthopedics3.63.73.63.63.43.43.43.43.53.43.53.43.33.43.43.43.33.33.33.43.43.4 Orthopedic Oncology0.60.60.60.70.70.70.70.70.70.70.70.70.60.60.60.60.60.60.60.60.60.7 V-Medicine0000000000000000000000 Gynecology Oncology0000000000000000000000 SUM6.66.76.76.66.56.46.26.26.36.16.16.15.96.16.16.36.46.36.46.46.36.3
Sat SunMonTueWedThuFriSat
SunMonTueWedThuFri Table A.4: Patient demand for beds at alternative-level-of-care
Page | 58
Appendix B
AMPL Code for MIP Model
The mixed integer programming model is built in AMPL and solved using the Gurobi solver.
When run, the model provides the allocation of blocks to operating room schedule that minimizes the peak demand for beds throughout a week. The model’s bounds ensure that the resulting schedule is feasible. ORO.run is the run file for AMPL, including calling the proper solver, initiating the model, importing the input parameter values, solving for optimization, and finally, exporting the optimal values of decision variables. ORO.mod is the AMPL model file for the MIP model described in Chapter 6.2.
ORO.run
# run command: ampl ORO.run reset;
option solver gurobi_ampl;
model ORO.mod;
data ORO.dat;
solve;
print 'solve system time, solve user time, solve time' >> ORO.out;
print _solve_system_time, _solve_user_time, _solve_time >> ORO.out;
print 'objective:' >> ORO.out;
display obj >> ORO.out;
print 'variables:' >> ORO.out;
display X >> ORO.out;
display Y >> ORO.out;
display F >> ORO.out;
Page | 59 ORO.mod
set block;
set day;
set OR;
set days_from_surgery;
set surgeon;
param Demand {block,days_from_surgery};
param SurgeonMapping {block,surgeon};
param DayRestriction {block,day};
param RoomRestriction {block,OR};
var X {block,day,OR} binary;
var Y {block,day} binary;
var F {block,day} >= 0;
minimize obj: Z;
subject to c1 {i in block, j in day}: sum {k in OR} X[i,j,k] == Y[i,j];
subject to c2 {i in block}: sum {j in day} Y[i,j] == 1;
subject to c3 {j in day, k in OR}: sum {i in block} X[i,j,k] <= 1;
subject to c4 {i in block, j in day}: F[i,j] == sum {x in 1..7} Y[i,x]*Demand[i,j-x];
subject to c5 {j in day}: sum {i in block} F[i,j] <= Z;
subject to c6 {i in block, j in day}: sum {k in OR} X[i,j,k] <= DayRestriction[i,j];
subject to c7 {m in surgeon, j in day}: sum {i in block} SurgeonMapping[i,m]*Y[i,j] <= 1;
subject to c8 {i in block}: Y[i,1] == 0;
subject to c9 {i in block}: Y[i,7] == 0;
subject to c10 {i in block, k in OR}: sum {j in day} X[i,j,k] <= RoomRestriction[i,k];
Page | 60
Expected Patient Demand for Beds t Days Since Surgery
Surgeon
Table C.1: Expected demand for beds and surgeon assigned to each block
Page | 61 Block ID Block Info Procedure Available Operating Rooms
1 Mon OR1 Orth 1, 2, 3, 4
Table C.2: Operating room restriction for each block