A Generic Bed Planning Model

A Generic Bed Planning Model by Tian Mu Liu A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Mechanical and Industrial Engineering University of Toronto Copyright by Tian Mu Liu 2012

A Generic Bed Planning Model Tian Mu Liu Master of Applied Science Graduate Department of Mechanical and Industrial Engineering University of Toronto 2012 Abstract In April 2008, the Ontario government announced its top two healthcare priorities for the next 4 years, one of which is reducing wait time in emergency rooms. To study the wait time in emergency rooms or any other departments in a hospital, one must investigate resource planning, scheduling, and utilization within the hospital. This thesis provides hospitals with a set of simulation and optimization tools to help identify areas of improvement, particularly when there are a number of alternatives under consideration. A simulation tool (a Monte Carlo simulation model) estimates patient demand for beds in a hospital during a typical week. Two optimization tools (an integer programming mathematical model and a heuristics model) demonstrate opportunities for smoothing the patient demand for beds by adjusting the operating room schedule. Page ii

Acknowledgements I would like to thank Dr. Michael Carter, my thesis supervisor, for his patience, guidance and insight during my time as a master s student. I would also like to thank the members of my lab at the Centre for Research in Healthcare Engineering, particularly Daphne Sniekers for her insights on data interpretation from hospitals, as well as Matthew Nelson for his suggestions on the usability and applicability of the simulation model. I further acknowledge the members of PricewaterhouseCoopers, particularly Robert Varga and Laura Van de Bogart, for their assistance in coordinating hospital data requests and meetings in the early stage of this research and for their assistance along the way. In addition, I would like to thank all the representatives from the participating hospitals: William Osler Health System, Hamilton Health Sciences, and Regina General Hospital for sharing their knowledge and for their time. Page iii

Table of Contents 1 Introduction... 1 2 Background and Problem Analysis... 3 2.1 Bed Management... 4 2.2 Operating Room Scheduling... 5 2.3 Research Objectives... 6 3 Literature Review... 7 3.1 Stochastic Models for Bed Capacity Planning... 7 3.2 Simulation Models... 8 3.3 Discrete Event Simulation vs. Monte Carlo Simulation... 10 3.4 Operating Room Scheduling... 10 4 Bed Planning Model...13 4.1 Model Design... 13 4.2 Input Data... 15 4.3 Simulation Design... 17 4.3.1 Process 1: Define Patient Groups... 17 4.3.2 Process 2: Create Patient Arrival Distribution for Each Shift of the Week and Each Patient Group... 19 4.3.3 Process 3: Generate n Patient Arrivals for Patient Group k at Shift j... 20 4.3.4 Process 4: Calculate Number of Inpatients for Current and Subsequent Shifts... 21 4.3.5 Process 5: Calculate Mean and Standard Deviation of Patient Demand for Beds for Each Patient Group at Each Shift... 21 4.4 Model Output... 22 4.5 Model Validation... 23 4.6 Bed Capacity Planning... 24 5 Scenario Planning...27 5.1 Case Study 1: Benchmark against CIHI Expected Patient LOS... 27 5.2 Case Study 2: Revising Operating Room Schedule... 28 6 Optimizing the Operating Room Schedule...33 6.1 Input Parameters... 33 6.2 Mixed Integer Programming (MIP) Approach... 34 6.3 2-Opt Heuristic Approach... 37 Page iv

6.4 Numerical Experiments... 37 7 Application...43 8 Conclusion...45 9 Future Research...47 Bibliography...48 A Bed Planning Model Outputs...53 B AMPL Code for MIP Model...58 C Numerical Experiments...60 D Graphical User Interface...62 Page v

List of Figures 4.1: Sample inpatient admission data...16 4.2: Sample actual surgical service data...16 4.3: Sample operating room schedule...17 4.4: Simulation design...18 4.5: Expected patient demand for beds at acute wards...22 4.6: Bed capacity by target occupancy level for acute wards...24 4.7: Bed capacity by probability of bed blocking for acute wards...25 5.1: Expected acute and ALC patient demand for beds based on the sample operating room schedule...30 5.2: The expected PDB based on the original and the revised operating room schedule...32 6.1: Experimental results for MIP and 2-opt approach...39 6.2: Decrease in patient demand for beds from each step in 2-opt heuristic...41 D.1: Main graphic user interface...62 D.2: Simulation interface for emergent/urgent inpatients...64 D.3: Simulation interface for patient demand for beds...65 D.4: Simulation interface for revising operating room schedule...66 Page vi

List of Tables 4.1: Average of expected patient demand for beds for a typical week, by patientday analysis and by simulation...23 5.1: Average of the expected PDB calculated based on actual and CIHI LOS...28 5.2: A sample original operating room schedule...29 5.3: Revised operating room schedule after 3 swaps...31 6.1: A sample original operating room schedule...38 6.2: A sample of expected PDB and surgeon assigned to each block...38 6.3: A sample of operating room restriction for each block...39 6.4: Optimal operating room schedule from MIP...40 6.5: Near-optimal operating room schedule from 2-opt heuristic...40 6.6: Top five swaps for the first step in 2-opt heuristic...42 A.1: Patient demand for beds at emergency department...54 A.2: Patient demand for beds at special-care unit...55 A.3: Patient demand for beds at acute wards...56 A.4: Patient demand for beds at alternative-level-of-care...57 C.1: Expected demand for beds and surgeon assigned to each block...60 C.2: Operating room restriction for each block...61 Page vii

Chapter 1 Introduction Emergency department overcrowding has been a longstanding problem in Ontario [1]. In many cases, overcrowding is due to lack of beds in the downstream departments, such as acute wards and the lack of beds generally occurs at peaks in patient demand for beds. Hospital administrators often respond to peaks in demand by opening extra beds. But with growing demand for healthcare resources, pressure on efficient usage of available bed capacity is increasing. Peaks in bed demand are due to variability in admissions and lengths-of-stay. The particular area with which this thesis is concerned is variability in elective admissions. It can be reduced by creating a balanced operating room schedule, which levels the patient demand for beds throughout a week. With a balanced schedule, peak traffic is leveled across the week, hence, reducing overcrowding without turning away any patients or increasing bed capacity. For these reasons, we build a set of simulation and optimization tools to estimate patient demand for beds in a hospital during a typical week. And then, we demonstrate opportunities for smoothing the expected patient demand for beds by adjusting the operating room schedule while preserving the equipment and staff restrictions. The remainder of the thesis is organized as follows. We start by giving a detailed description of the problem and an overview of the necessary background information to understand this problem in Chapter 2. In Chapter 3, we review the relevant literature in four areas: (1) Page 1

stochastic models for bed capacity planning; (2) application and development of simulation models; (3) differences between discrete event simulation and Monte Carlo simulation; and (4) scheduling techniques that have been used in the operating room setting. We believe that Monte Carlo simulation is a fast and easy approach for bed capacity planning. In Chapter 4, 5 and 6, we detail both the high-level and specific designs of our simulation and optimization tools. We conclude with Chapter 7 where we discuss the contribution of this paper and describe the main practical insight that can be derived. Page 2

Chapter 2 Background and Problem Analysis The Canadian healthcare system is funded through a combination of premiums and taxes, which varies from province to province [3]. For example, each province has its own insurance program, which acts as a single source payor for both hospital and physician services. The insurance programs fund hospitals in advance and pay physicians for the services they provide. The difference in funding methods for hospitals and physicians is designed to promote cost containment while protecting the physician-patient relationship [4]. In the physician-patient relationship, the physician acts the patient s agent, to determine the treatment that is best for the patient [5]. Health Canada, a federal department, publishes surveys of the healthcare system in Canada based on Canadians' first-hand experiences of the healthcare system. Although lifethreatening cases are dealt with immediately, some services are non-urgent and patients are seen at the next-available appointment in their local chosen facility [8]. A study by the Commonwealth Fund found that 57% of Canadians reported waiting 30 days (4 weeks) or more to see a specialist [9]. In April 2008, the Ontario government announced its top two healthcare priorities for the next four years: reducing wait time in emergency rooms and improving access to family health care [10]. To study wait times in emergency rooms or any other departments in a hospital, one should consider resource planning, scheduling, and utilization within the hospital. Page 3

2.1 Bed Management Bed management is the allocation and provision of beds. Beds in specialist wards are a scarce resource [11]. The bed in this context represents not simply a place for the patient to sleep, but the services that go with being cared for by the medical facility, such as admission processing, physician time, nursing care, necessary diagnostic work, appropriate treatment, and so forth. It includes all the resources (e.g., physicians, nurses, medical equipments and supplies) that are needed to provide care for the patients. As such, bed management is an essential part of resource planning in a hospital. Hospitals cannot force a patient to leave if they cannot find a place to provide safe and sufficient care. Beds may be unavailable for new, acutely sick patients because of the continued presence of the previous patients. This shortage of beds is sometimes known as a bed blocking. It is one of the primary reasons for cancellations of admissions for planned (elective) surgery, admission to inappropriate wards (medical vs. surgical, male vs. female, etc), delay in admitting emergency patients (long wait time at emergency department), and transfers of patients between wards [12]. Hospital capacity decisions have traditionally been made, both at the government and institutional levels, based on target occupancy levels the average percentage of occupied beds. The number of beds needed at a hospital can be calculated from expected patient demand for beds and target occupancy level, such that number of beds needed is equal to expected demand divided by target occupancy level. The most commonly used occupancy target has been 85% [13]. Lower occupancy levels are often viewed as indicative of operational inefficiency. Higher occupancy levels, on the other hand, result in a higher chance of bed blocking. Page 4

Even though this bed capacity planning method is popular at the government and institutional levels, it has many problems. Garling noted that this 85% target occupancy level is based on a theoretical stochastic model derived from a highly simplified view of the dynamics of queues, which ignores a variety of dynamic behavioral responses to work pressure in the real world [14]. Goronescu et al exemplified a better approach to determining the number of beds that a hospital unit should have. They showed that the optimal number of beds depends on the relative cost that is incurred when a patient is blocked compared with that of maintaining an empty bed. The optimal utilization at which the unit should be maintained also depends on this relative cost [15]. Hospital executives and government officials need to be aware of the trade-off between utilization and the ability to provide an appropriate bed in a timely fashion. Green introduced factors such as nursing unit sizes, the variability and time-dependent patterns of demands for beds, and bed allocation policies to determine appropriate bed capacity [16]. Due to the controversy surrounding target occupancy level analysis, we define the patient demand for beds (PDB) as the standard unit of analysis. Given the value of PDB, hospital managers and researchers can plan for bed capacity with the methods that they prefer. 2.2 Operating Room Scheduling The planning and scheduling of operating room time is known as operating room scheduling. Typically, a multiple stage process is used [17]. Stage 1 starts with the long-term allocation of operating room time to the surgical specialties, such as the number of surgery hours per year. This allocation is a strategic decision that reflects patient demand patterns and the priorities defined by hospital management. In stage 2, the master surgical schedule is Page 5

developed from this strategic decision. This is a cyclic operating room schedule for a shorter time horizon, which divides operating room time (aggregated into blocks) amongst the specialties. The specific assignment of patients to blocks with the master surgical schedule is commonly referred to as Stage 3. Stage 4 addresses the monitoring and control of the operating room activities on the day of surgery. 2.3 Research Objectives This thesis provides hospitals with a set of simulation and optimization tools to help identify areas of improvement, particularly when there are a number of alternatives under consideration. The simulation tool (a Monte Carlo simulation model) estimates patient demand for beds in hospital during a typical week. The optimization tools (an integer programming mathematical model and a heuristics model) demonstrate opportunities for smoothing the patient demand for beds by adjusting operating room schedule. Using these quantitative decision support tools, hospital management could reduce the overall cost of healthcare system redesign. Page 6

Chapter 3 Literature Review Faced with diminishing government subsidies, competition, and the increasing influence of managed care, hospitals are under enormous pressure to cut costs. In this environment, it is more important than ever for hospital managers to identify ways to deploy their resources more effectively [18]. This chapter presents an overview of the work done on the bed capacity planning problem. It further reviews the use of stochastic modeling and, more specifically, of Monte Carlo simulation. Finally, it details operating room scheduling and gives some insight into scheduling techniques by providing a review of the relevant literature. 3.1 Stochastic Models for Bed Capacity Planning A number of researchers have investigated patient demand and bed capacity planning at a specific department within a hospital. McClain has developed a stochastic model to forecast the allocation of non-obstetric patient-days to the obstetric unit and to predict the effect of such allocations on demand for obstetric beds [19]. Dexter and Macario have modeled the distribution of patients at an obstetrical unit as a Poisson distribution and minimized the number of staffed beds subject to remaining below a specified probability of patient overflow [20]. Harris has developed a simulation model to aid decision making in the area of operating theatre time tables and the resultant hospital bed requirements [21]. Page 7

Furthermore, many authors have created models for the entire hospital, while capturing the inherent variability in patient arrival and length-of-stay [22, 23]. They have demonstrated that managing capacities based on simple deterministic spreadsheet calculations typically do not provide the appropriate information, and result in underestimating true bed requirements. However, they ignore the patient demand for beds at each department within a hospital, such as emergency rooms, intensive care units, and acute care units. To calculate the patient demand at each department, Gorunescu et al. and Harrison have used compartment models, in which a facility is subdivided into categories of patients with different transition rates to model patient flow through wards [15, 24]. 3.2 Simulation Models Today s healthcare providers recognize the importance of implementing simulation to support quality learning outcomes [25]. It has been applied to practically every topic in healthcare, such as space considerations, physiology, crisis management, critical care, and general surgery [26]. In comparison to analytical models, more procedural details can be included in a computer simulation model [27, 28]. Linear or nonlinear programming models, queuing models and Markov chains often rely on closed-form mathematical solutions [29]. They are more sensitive to the size, complexity and level-of-detail required by the system under study. Simulation models, on the other hand, are much less sensitive to these parameters [29]. However, simulation may be more difficult to use for several reasons [19]. First, the added complexity of constructing a more realistic model requires considerable institution-specific data that may be costly to collect. Second, computer programming is usually expensive and Page 8

time-consuming. Third, forecasts of parameters used in such models are often subject to significant error, which may negate gains in accuracy achieved through simulation. Sinreich and Marmor incorporate three principles to minimize the short-comings of simulation, and to increase management s involvement and confidence in their model [29]: 1. The simulation tool has to be general and flexible enough to model different possible hospital settings. 2. The simulation tool has to be intuitive and simple to use. This way, managers, hospital engineers, and other nonprofessional simulation modelers can run the simulation tool with very little effort. 3. The simulation tool has to include reasonable default values for many of the system parameters. This will reduce the need for comprehensive, costly, and time-consuming time and motion studies, which are usually among the first steps taken when building any simulation model. Sinreich and Marmor satisfy the first principle by testing their model against five hospital data sets. They address the second principle by designing a user-friendly interface that mirrors a unified patient process chart, which managers are familiar with. To comply with the third principle, default values are used in the simulation and can be easily accessed through the model s interface. In this thesis, our approach follows these three principles closely to offset the difficulties of developing simulation models. Page 9

3.3 Discrete Event Simulation vs. Monte Carlo Simulation Monte Carlo simulation is a recognized approach in healthcare, but it is not used as extensively as the discrete event simulation [30]. In discrete event simulation, the operation of a system is represented as a chronological sequence of events. Each event occurs at an instant in time and marks a change of state in the system. For example, Zhu has developed a discrete event simulation to reflect the complex patient flow of the ICU system and to determine the proper ICU bed capacity which strikes a balance between service level and cost effectiveness [31]. On the other hand, Monte Carlo simulation samples probability distribution for each system variable to produce hundreds or thousands of possible outcomes. Compared to discrete event simulation, Monte Carlo simulation is much more simplistic, as it does not deal with events or time. It therefore cannot be used to investigate wait time in a system. Since we do not investigate wait time in this thesis, Monte Carlo simulation is a fast and easy approach to achieve our goal. 3.4 Operating Room Scheduling Operating room scheduling has received quite some attention in the literature. One of the early works on operating room scheduling is done by Blake and Donald [7]. They use integer programming to model the nurse manager s schedule development process. The model minimizes the shortfalls from target hours allocated to each department. The solutions are bounded by limits on the number of rooms that can be assigned to any department, equipment restrictions, surgeon availability, assumed patient volumes, and by terms of the nurses collective agreement. The benefit of the model is its ability to produce a relatively Page 10

unbiased, equitable schedule through a consistent process, thus reducing conflict both amongst surgeons and between surgeons and the nurse manager. Belien and Demeulemeester [32] use a nonlinear integer programming model to construct master surgical schedules. They try to level the bed usage by finding the best allocation of blocks to surgical disciplines. They view the number of patients admitted on a day and length-of-stay for each operated patient as stochastic variables with a distribution depending on the specialty that used the operating room. Van Oostrum et al. [33] find the optimal master surgical schedule, in which they schedule all regularly performed surgeries on a specific day in the planning cycle. Their objective function is a combination of operating time usage and the maximum number of beds needed on every day. They treat the length-of-stay as deterministic, with the length depending on the type of surgery performed. Vanberkel et al. [34] study the effect of a given surgical schedule on the usage of beds, taking emergency arrivals and different ward types into account as well. However, they do not use an optimization algorithm and only try to improve step-by-step by trial and error. Their approach has been applied in practice with good results [7, 32, 33, 34]. Gallivan and Utley [35] present a generic model for determining the distribution of bed occupation for a given cyclic admission schedule. They give an example of how these results could be used in an optimization context. However, they restrict themselves to a single ward. Denton et al. [36] and Jebali et al. [37] demonstrate mathematical models to allocate surgeries to operating rooms (ORs). The objective of the model is to minimize total cost of operating ORs. However, Denton et al. ignore the upstream (intake) and downstream (recovery) resources required to support surgery, under the assumption that ORs tend to be the bottleneck in the overall process. Also, their model is missing constraints that certain Page 11

surgeries cannot be scheduled simultaneously. Bekker and Koeleman [38] combine timedependent analysis with a quadratic programming model to determine admission quota for scheduled admissions and to analyze the impact of variability in scheduled admissions on the required bed capacity. They derive three generic practical insights that apply to almost all hospital situations: 1. Reducing the variation in length-of-stay leads to less variable bed occupancy only for stable arrival processes. 2. Scheduling patients with a longer expected length-of-stay on Fridays can help to minimize unused capacity in the absence of scheduled admissions during weekend. 3. More admissions should be scheduled on Mondays compared to the other days of the week in absence of scheduled admissions during weekend. These approaches [35, 36, 37, 38] provide great insights but fail to demonstrate real life implementation. Page 12

Chapter 4 Bed Planning Model This thesis provides hospitals with a set of simulation and optimization tools to help identify process improvements, particularly when there are a number of alternatives under consideration. Developing these tools can be seen as a three-stage process. In the first stage, we build a simulation tool (a Monte Carlo simulation model) to estimate the patient demand for beds in a hospital during a typical week. In the second stage, we apply the model to various real-life scenarios to identify areas of improvement. The third stage involves developing optimization tools (an integer programming mathematical model and a heuristic model) to demonstrate opportunities for smoothing the expected patient demand for beds by adjusting operating room schedule. In this chapter, we describe the design, the assumptions, and the outputs of the Monte Carlo simulation model. We will refer it as the bed planning model. The front end of the model is built using Excel UserForm and the back end coding is done in VBA. 4.1 Model Design The purpose of the bed planning model is to estimate the patient demand for beds in a hospital during a typical week. It is designed with following features: 1. The results of the simulation tool are based on patient traffic in a typical busy week, which means doctors are working full time (omitting vacation) and all licensed/certified beds are open (staffed). Page 13

2. Within a week, there are 21 shifts (3 shifts a day for 7 days) in which patients could arrive and stay. Each shift is 8 hours long. The start time of each shift is arbitrary (left for the users to decide), but shifts must directly follow each other (sequential without gaps). We define the first shift of the day as the night shift, which is followed by the day shift, and finally the evening shift. We also assume elective surgeries start in the day shift. 3. The patient demand for beds (PDB) in a shift equals the number of inpatients in that shift. To calculate number of inpatients, we assume all patients depart and arrive at the beginning of the shift. Since a bed is available if a patient leaves mid-shift, we do not count this patient as an inpatient in his/her last shift. When patients arrive in the middle of a shift, we assume they enter at the beginning. 4. The PDB is separated into departments in a hospital, such as PDB in the emergency room, the ICU, the acute ward, the ALC, etc. The simulation model must allow the users to define the number of departments in the hospital and the role of each department. 5. The PDB in each department is also separated into patient groups. Patient groups are classified by user defined patient categories. For example, users can classify patient groups by hospital programs (surgical procedure or medical procedure), by specialties (cardiology, oncology, etc.), or even by both hospital programs and specialties (surgical cardiology, medical cardiology, surgical oncology, medical oncology, etc). 6. A distinction is made between emergent/urgent patients and elective patients. Elective patient arrivals depend on the operating room schedule, while emergent/urgent patient Page 14

arrivals are based on emergency department arrival patterns. The simulation model must account for this while generating patient arrivals. 7. Patient arrivals are randomly generated from patient arrival distributions. Patient arrival distributions are constructed from patient historical records. 8. Due to the stochastic nature of simulation, PDB is a random variable. The simulation model needs to run multiple trials to estimate the mean and the variability of PDB. 4.2 Input Data To simulate emergent/urgent PDB, inpatient admission data is required. It includes admission time, length-of-stay (LOS) at each department, and service and patient categories for at least six months. However, to simulate elective PDB, three sets of input data are needed: 1. Inpatient admission data (admission time, surgeon, LOS at each department in the order of the visit) with service and patient categories for at least six months 2. Actual surgical service data (surgery date, surgery duration, main surgeon, patient type) for the same time period as the inpatient admission data 3. A typical operating room schedule A sample of inpatient admission data is shown in Figure 4.1. Admission time should include both date and time for the model to identify the shift in which the patients were admitted. LOS is measured in hours. Some examples of patient categories are service received, admission method (elective or emergent/urgent) and ward name. Doctors can be represented by names or identification numbers to protect their identity. Page 15

Figure 4.1: Sample inpatient admission data A sample of actual surgical service data is shown in Figure 4.2. Because the model assumes elective patients arrive during the day shift, only the date of surgery is required as the model no longer needs the time stamp to determine the shift in which the patients were admitted. Surgery duration is measured in minutes. Surgeon name must be in the same format as inpatient admission data. Since same day surgery does not result in bed use, the model needs patient type information to distinguish between same day surgery and inpatient surgery. We do not remove same day surgery from actual surgical service data because they are used to determine if a surgeon is given a full day operation or a half day operation. Figure 4.2: Sample actual surgical service data Page 16

A typical operating room schedule is shown in Figure 4.3. The first row of the table represents the ID of the operating room. Surgeon name must be in the same format as inpatient admission data. If a surgeon is assigned to both morning and afternoon operation at the same day of the week in an operating room, he/she is given a full day operation. Figure 4.3: Sample operating room schedule 4.3 Simulation Design The bed planning model simulates patient arrivals and patient stays in a hospital, as shown in Figure 4.4. We describe the detailed design of each numbered process. 4.3.1 Process 1: Define Patient Groups Patient groups are classified by arbitrary patient categories. For example, users can classify patient groups by hospital programs (surgical procedure or medical procedure), by specialties (cardiology, oncology, etc.), or even by both hospital programs and specialties (surgical cardiology, medical cardiology, surgical oncology, medical oncology, etc). Surgical Page 17

information, such as assigned operating room and the main surgeon should also be used to further classify elective patients. Start Simulation Design 1 Define patient groups 2 Create arrival distribution for each shift of the week and each patient group i = 1 i = i +1 Begin trial i j = 1 j = j +1 For shift j no k = 1 k = k +1 no 3 Generate n patient arrivals for patient group k at shift j no 4 Calculate number of inpatients for current and subsequent shifts Done all groups? yes Done all shifts? yes Done all trials? yes 5 Calculate mean and standard deviation of patient demand for each patient group at each shift of the week End Figure 4.4: Simulation design Page 18

4.3.2 Process 2: Create Patient Arrival Distribution for Each Shift of the Week and Each Patient Group Three sets of data are required to generate the arrival distribution of emergent/urgent patients and elective patients, described in the previous section. A distinction is made between emergent/urgent patients and elective patients. For emergent/urgent patients, we create an arrival distribution for each combination of patient group and arrival shift. For elective patients, we create an arrival distribution for each surgeon. Note that, within a week, there are 21 shifts (3 shifts a day for 7 days) in which patients could arrive and stay. Each shift is 8 hours long. The start time of each shift is arbitrary (left for the users to decide), but shifts must directly follow each other (sequential without gaps). We define the first shift of the day as the night shift, which is followed by the day shift, and finally the evening shift. To generate an emergent/urgent patient arrival distribution for each combination of patient group and arrival shift, the model separates inpatient admission data by patient group and arrival shift. Then, for each portion of the data, the model creates a frequency table for number of patient arrivals. The frequency is the number of times that the given number of patient arrivals has happened on the same shift. Finally, from each frequency table, the model normalizes and creates an empirical distribution for the number of patient arrivals. For example, there are 27 stroke patient arrivals on Monday day shift for the past 32 weeks. From these records, the model finds 1 occurrence of 3 stroke patient arrivals on the same shift, 5 occurrences of 2 arrivals, 14 occurrences of 1 arrival and 12 occurrences of no arrivals. Finally, the model creates an empirical arrival distribution for stroke patients on Monday day shift, such that, probability of no arrivals is 12/32 or 37.5%, probability of 1 arrival is 14/32 Page 19

or 43.75%, probability of 2 arrivals is 5/32 or 15.625%, and probability of 3 arrivals is 1/32 or 3.125%. To generate the elective patient arrival distribution for each surgeon, the model divides actual surgical service data by surgeon. Then, for each portion of the data, the model creates two frequency tables for number of patient arrivals. The first frequency table stores the number of times that a given number of patient arrivals have happened if the surgeon is given a half day operation (morning or afternoon operation). The second table stores the number of times that a given number of patient arrivals have happened if the surgeon is given a full day operation (the surgeon has booked the whole day). Finally, from each frequency table, the model normalizes and creates an empirical distribution for the number of patient arrivals. 4.3.3 Process 3: Generate n Patient Arrivals for Patient Group k at Shift j To generate emergent/urgent patients, the model uses the emergent/urgent patient arrival distributions that have been generated in process 2 for each combination of patient group and arrival shift. Given the patient arrival distribution for patient group k at shift j, a random number is used to determine number of arrivals. For n number of patient arrivals, n patient records are randomly drawn from the inpatient admission data for patient group k and shift j. To generate elective patients, the model uses elective patient arrival distributions that are generated in process 2 for each surgeon and a typical operating room schedule that consists of blocks that are assigned to a surgeon. A block is described by four parameters: the name of the operating room (room 1, room 2, etc), the day of the week (Monday, Tuesday, etc), the assigned shifts and the assigned surgeon. We assume surgeons do elective surgeries in the day shift (the second shift of the day). For all surgeons who are working in shift j, the model uses the patient arrival distribution for a surgeon and a random number to determine a Page 20

number of patients to arrive for that surgeon. If a surgeon generated n patients, then n patient records from his/her portion of inpatient admission data are randomly drawn to represent them. By collecting patients from all surgeons at shift j, the model now has all patient arrivals at shift j, each associated with a patient record. 4.3.4 Process 4: Calculate Number of Inpatients for Current and Subsequent Shifts From the previous process, the model has a collection of patient arrivals and a real patient record to represent each of them. Since each patient record comes with patient length-of-stay at each department, the model can determine the location of every patient at current and subsequent shifts. It can then calculate the number of inpatients in each department at current and subsequent shifts. 4.3.5 Process 5: Calculate Mean and Standard Deviation of Patient Demand for Beds for Each Patient Group at Each Shift For each simulation run, there are number of trials. For each trial run, there are number of weeks as a warm up period to reach a steady state system. The number of trials and the length of warm up period are defined by the user. Result collection period is a week (21 shifts), which means only the last week of each simulation trial is used to calculate the number of inpatients in each department at each shift of the week for each patient group. Finally, the model uses results from all of the trials to calculate mean and standard deviation of patient demand for beds (PDB) in each department at each shift of the week for each patient group. Page 21

night day evening night day evening night day evening night day evening night day evening night day evening night day evening 4.4 Model Output The bed planning model outputs the mean and the standard deviation of PDB in each department at each shift of a week for each patient group. Tables A.1 A.4 (see Appendix A) show the mean and the standard deviation of PDB in emergency departments, special care units, acute wards and alternative-level-of-care (ALC) units at each shift. The patients are categorized by the main service that they received. The simulation runs for 50 trials/iterations with 20 weeks of warm up period. The night shift (first shift of a day) starts at 9PM and ends at 5AM. The operating room opens 8 hours for operations. These parameters are user defined and should reflect the actual hospital settings. For example, hospitals with long LOS patients should have longer warm up period to reach steady state. 270 260 250 240 230 231 234 232 235 241 240 243 248 247 250 254 255 257 261 260 259 254 247 246 240 234 220 210 200 Sun Mon Tue Wed Thu Fri Sat Figure 4.5: Expected patient demand for beds at acute wards Page 22

The model outputs can be represented graphically, as shown in Figure 4.5 for acute wards. The expected (mean) PDB is represented by solid bars. The results are based on historical patient records that are provided by Hamilton Health Sciences. 4.5 Model Validation This section describes the validation of the bed planning model. We validate our approach by applying patient-day analysis on the bed planning model s input data from Hamilton Health Sciences. Patient-day is a unit in a system of accounting used by healthcare facilities and healthcare planners. Each represents a unit of time during which the services of the institution or facility are used by a patient; thus 50 patients in a hospital for 1 day would represent 50 patient-days. In a time period, the average number of patients per day is equal to the total patient-days divided by the total number of days. We calculate the average number of patients per day using patient-day analysis and compare it with the results (average of expected PDB for a typical week) from the bed planning model, shown in Table 4.1. The difference between the averages is insignificant (less than 2%) and hence the bed planning model is valid. Average of Expected PDB Throughout the Week by Patientday Analysis by Bed Planning Model ED 9.64 9.7 SCU 23.54 23.2 acute wards 247.82 246 ALC 41.49 40.9 Table 4.1: Average of expected patient demand for beds for a typical week, by patient-day analysis and by simulation Page 23

night day evening night day evening night day evening night day evening night day evening night day evening night day evening 4.6 Bed Capacity Planning Hospital bed capacity decisions have traditionally been made based on target occupancy levels the average percentage of occupied beds. Historically, the most commonly used occupancy target has been 85%. Given an occupancy target, bed capacity is equal to expected PDB divided by target occupancy level. We use target occupancy level method to estimate bed capacity for acute wards at Hamilton Health Sciences, shown in Figure 4.6. The average of expected PDB throughout the week is 246. To achieve the standard 85% occupancy level, 290 beds are needed. However, on Thursday day shift, the expected occupancy level is actually 90% based on 261 expected PDB. On Sunday night shift, the expected occupancy level drops to 80% based on 231 expected PDB. 300 290 280 290 270 260 250 240 230 231 234 232 235 241 240 243 248 247 250 254 255 257 261 260 259 254 247 246 240 234 220 210 200 Sun Mon Tue Wed Thu Fri Sat Expected PDB Bed capacity with 85% occupancy level Figure 4.6: Bed capacity by target occupancy level for acute wards Page 24

night day evening night day evening night day evening night day evening night day evening night day evening night day evening We can also estimate bed capacity given an acceptable probability of bed blocking, such as 5%. A 5% chance of bed blocking means that patient demand for beds is satisfied 95% of the time. Given a bed blocking chance of p and N simulation trials, bed capacity must satisfy PDB for at least (1-p)N trials. For example, for a simulation with 50 trials and 5% bed blocking, bed capacity must satisfy PDB for 48 out of 50 trials. In this case, we would set bed capacity equal to the PDB of the 48 th smallest (third largest) trial. We use probability of bed blocking method to estimate bed capacity for acute wards at Hamilton Health Sciences, shown in Figure 4.7. To achieve the maximum of 5% bed blocking for each and every shift of the week, 288 beds are needed. With this method, we do not have to maintain a constant bed capacity throughout the week. For example, Hamilton could reduce its bed capacity to 266 beds on the weekends. 300 290 280 270 260 250 240 230 220 210 200 285 284 288 281 281 281 278 280 276 273 270 270 270 260 261 241 243 248 247 250 254 260 255 257 259 254 256 258 254 231 234 232 235 240 271 266 259 254 247 246 240 234 Sun Mon Tue Wed Thu Fri Sat Expected PDB Bed capacity with 5% chance of bed blocking Figure 4.7: Bed capacity by probability of bed blocking for acute wards Page 25

Due to its simplicity in calculation, occupancy target analysis has been a well-known measure for determining bed requirements at the individual hospital and even hospital unit level. On the other hand, probability of bed blocking is one of the indicators of healthcare accessibility. Planning bed capacity based on this probability provides quantifiable measurement of system performance. In this thesis, we define the patient demand for beds (PDB) as the standard unit of analysis. The bed capacity can be easily calculated by either occupancy targets analysis or by the probability of bed blocking method. The users of the bed planning model can decide which method to use, given the PDB from each simulation trial. Page 26

Chapter 5 Scenario Planning The purpose of scenario planning is to identify problems in the existing system and to study the effect of various solutions. In this chapter, we evaluate what-if scenarios with the bed planning model to identify areas of improvement in the existing hospital settings. Specifically, we consider scenarios, such as changes in length-of-stay using the CIHI benchmark and revisions in the operating room schedule. 5.1 Case Study 1: Benchmark against CIHI Expected Patient LOS The Canadian Institute for Health Information (CIHI) collects and analyzes information on health and healthcare in Canada and makes it publicly available. The expected (50 percentile) CIHI LOS is the average acute LOS in hospital for patients with the same case mix group, age category, comorbidity level, and intervention factors. In this case study, we will identify the strengths and weaknesses of Hamilton Health Sciences by benchmarking current patient length-of-stay (LOS) against expected patient LOS from CIHI. We categorized patients by the main service they received because it is a major factor in predicting the nursing time that they will need. We ran two separate simulations: one using the actual patient LOS from Hamilton, and the other one uses the expected LOS from CIHI, while all other input data and parameters stay the same. We compared the results from the two simulation runs in Table 5.1. For most of the services, there is no change to PDB. However, the current patient demand for Orthopedics beds is much higher than the demand based on CIHI LOS. Page 27

Daily Average of The Expected PDB Across the Week Service Received based on actual LOS based on CIHI LOS GI Surgery 2 3 Medicine 105 101 Surgery 72 73 GI Medicine 6 6 Oncology 26 26 Hematology 30 31 Orthopedics 53 39 Orthopedic Oncology 2 2 Vascular Medicine 23 10 Gynecology Oncology 0 0 Total 320 291 Table 5.1: Average of the expected PDB calculated based on actual and CIHI LOS Further investigation has shown that, on average, 13 out of the 53 Orthopedics patients (or 25%) require alternative-level-of-care (ALC). ALC patients, those who have healthcare needs that could be better addressed in other settings, are staying in acute care hospitals for prolonged and often excessive periods of time; the largest proportion of ALC days is for those waiting for long-term care homes placement [39]. By placing ALC patients in less costly long-term care homes, ALC days in acute care facilities would be reduced. We recommend Hamilton to work with its Local Health Integration Network to reduce the number of ALC patients. 5.2 Case Study 2: Revising Operating Room Schedule One of the most expensive resources in a hospital is the operating room department. Since the majority of elective admissions involve surgery [40], optimal utilization of operating room capacity is of paramount importance. Most surgeries are scheduled during weekdays plus a few emergencies on evenings and weekends. In the absence of elective admissions Page 28

during the weekend, elective PDB usually peaks on Thursday and Friday. Shifting the PDB to earlier days in the week would lower the peak and reduce the number of beds needed without changing patient volume. When the peak exceeds the actual bed capacity, reducing the peak also reduces cancellations of elective surgeries, which is a major cause of inefficient use of operating room time and a waste of recourses [41]. In this case study, we level the expected PDB throughout a week by modifying the operating room schedule. We revise the operating schedule by assigning blocks to different operating rooms and days of the week. A sample operating room schedule from Hamilton Health Sciences is shown in Table 5.2. Each block is given a full day shift. The blocks on Sunday and Saturday are empty because there are no elective patient admissions during the weekend. The surgeon names are represented by their identification numbers to protect their identity. 1 2 3 4 5 6 7 8 Sunday Monday 503 234 103 340 405 141 991 771 Tuesday 215 234 206 340 720 901 556 851 Wednesday 215 283 503 421 394 185 599 599 Thursday 103 503 539 240 384 828 991 991 Friday 958 206 294 793 185 411 715 715 Saturday Table 5.2: A sample original operating room schedule The PDB based on this sample operating room schedule is shown in Figure 5.1, which shows how PDB fluctuates over the course of a week. The expected acute and ALC PDB varies from 48 on the Monday night shift to 81 on the Friday day shift. However, Hamilton only had 70 budgeted surgical beds at the time. This fluctuation in the PDB leads to bed blocking of surgical wards on Wednesday to Saturday. The excess demand for beds is soaked up by medical wards if possible. Otherwise, some elective surgeries are cancelled. Page 29

night day evening night day evening night day evening night day evening night day evening night day evening night day evening 90 80 70 60 50 58 56 70 49 48 61 55 53 67 61 60 73 68 67 81 75 74 81 73 72 69 62 40 30 20 10 0 Sun Mon Tue Wed Thu Fri Sat Expect PDB Number of budgetted surgical beds Figure 5.1: Expected acute and ALC patient demand for beds based on the sample operating room schedule We felt that by revising operating room schedule to shift elective PDB to earlier days of the week, we could reduce the variation in PDB. To illustrate, we revised the sample operating room schedule manually (trial-and-error) by reassigning six blocks, as shown in Table 5.3. We will discuss ways to revise the sample operating room schedule automatically in the next chapter. We assume the lengths-of-stay of all patients remains the same after the blocks are reassigned. There are rules that must be followed when revising the operating room schedule: 1. One block per day of the week per operating room 2. Two blocks with same surgeon cannot be assigned to the same day of the week Page 30

3. Surgeons cannot be assigned to the weekend 1 2 3 4 5 6 7 8 Sunday Monday 503 234 103 340 405 141 991 771 Tuesday 215 991 234 206 340 539 720 901 556 851 421 Wednesday 215 283 503 991 394 185 599 599 539 991 991 Thursday 103 503 340 240 384 828 421 215 Friday 958 206 294 793 185 411 715 715 Saturday Table 5.3: Revised operating room schedule after 3 swaps With the revised operating room schedule, we compared the expected PDB between the original and revised operating room schedule, as shown in Figure 5.2. Only the demand during the day shift is shown because it is generally much higher than the other shifts of the day, and we are only interested in the peak number of beds when planning for bed capacity. In Figure 5.2, the expected PDB includes SCU, acute and ALC patients. From Figure 5.2, the peak of the expected PDB has decreased from 88.5 to 85.5. The revised schedule has therefore freed 3 beds. The results are promising but there are many problems with revising the operating room schedule manually. First of all, it is time-consuming due to the large number of possible permutations of blocks. If we save time by not iterating through all possible permutations, then we may miss good answers. Furthermore, surgeons have individual preferences on which day of the week they can work. This will result in back-and-forth negotiations between surgeons and the nurse manager that is even more time-consuming and may lead to conflict among staff. Page 31

Expected patient demand for beds 100 90 80 70 60 62.5 59.7 68.7 66.4 74.2 74.8 78.8 78.6 87.7 88.5 85.5 85.4 74 73.9 50 40 30 20 10 0 Sun Mon Tue Wed Thu Fri Sat Weekday day shift orginal operating schedule revised operating schedule Figure 5.2: The expected PDB based on the original and the revised operating room schedule In spite of these challenges, we believe Hamilton can revise its operating room schedule, because more elective services could be provided with a more balanced schedule. However, we need a better process to improve the schedule. We address this issue in the next chapter, where we develop optimization models to automatically generate optimal and near-optimal operating room schedules. Page 32

Chapter 6 Optimizing the Operating Room Schedule The purpose of this chapter is to propose and compare optimization models for building operating room schedules. We aim to level the patient demand for beds (PDB) throughout a typical week. The optimization models take results from the bed planning model in Chapter 5 as input parameters. We assume the length-of-stay of all patients remain the same after the blocks are reassigned. This assumption might not be true as there are more patient discharges on Friday and fewer on weekend. We keep this assumption for simplicity. 6.1 Input Parameters Our first input parameter is the expected PDB on each day of the week by each block. The second parameter is surgeon information. In order to prevent assigning two blocks with the same surgeon to the same day of the week, the models also need to know the surgeon that each block is assigned to. The third parameter is a list of infeasible days of the week for each block. This parameter adds flexibility to the model in order to deal with surgeons who are not available on certain days of the week. Page 33

Sometimes, a surgery cannot be performed in a certain operating room due to specialty equipment or room size restriction. The fourth parameter represents operating room availability for each block. 6.2 Mixed Integer Programming (MIP) Approach The following set notation is used in the mathematical model: i operating room block (i = 1, 2,, I = number of blocks) j the day of the week (j = 1 for Sunday, 2 for Monday,, 7 for Saturday) k operation room (k = 1, 2,, K = number of operations rooms) t days relative to the surgery date (t = -7, -6,, 0,, 6) m surgeon (m = 1, 2,, M = number of surgeons) The following parameter notation is used in the mathematical model: D it S im B ij R ik the expected PDB for block i at the day t starting from the surgery date for convenience, D i(t-7) = D it for t = 0, 1,, 6 1 when block i is performed by surgeon m, 0 otherwise 1 when block i can be assigned to day j, 0 otherwise 1 when block i can be assigned to operating room k, 0 otherwise Then, X ijk can be defined as a decision variable representing whether or not block i is assigned to day j in operating room k. X ijk = 1 if block i is at day j in operating room k, otherwise 0. Define Y ij to be a decision variable representing whether or not block i is assigned to day j. Y ij = 1 if block i is located at day j, otherwise 0. Thus, Define F ij to be the expected patient demand for beds from block i on day j. Page 34

To generalize, Define Z to be the peak expected PDB throughout the week. Then, We define the mixed integer programming portion of the problem of allocating blocks as follows: subject to Page 35

In the above model, constraints (1) and (2) calculate dummy variables Y ij and F ij respectively. Since Z is minimized in the objective function, constraint (3) is sufficient to calculate Z (the peak expected PDB throughout the week) by restricting it to greater than or equal to the PDB of each day of the week. Constraint (4) is designed to ensure that there is a max of one block per day of the week per operating room. Constraint (5) is designed to ensure that each block is assigned. Constraint (6) restricts blocks to the days of the week that are available to them. This constraint allows surgeons to resolve conflicts with surgery times, if necessary. Constraint (7) ensures that blocks with the same surgeon cannot be assigned to the same day of the week. Constraint (8) is an arbitrary bound on which day of the week a block can be assigned to. In our model, we limit our blocks to weekdays (Monday to Friday). Constraint (9) restricts blocks to operating rooms that are available to them, such as rooms that have the equipment needed for the surgery or rooms that are large enough for the procedure. Constraints (10) and (11) are binary constraints on model variables X ijk and Y ij. When run, this model provides the allocation of blocks to an operating room schedule that minimizes the peak expected PDB throughout a typical week. The model s bounds ensure that the resulting schedule is feasible. The front end of this model is coded in AMPL (an algebraic modeling language for linear and nonlinear optimization problems). Less than a Page 36

second is required to generate an optimal solution using the Gurobi solver (a mathematical programming solver). However, Gurobi and AMPL licensing and training are expensive. A sample set of code for AMPL is shown in Appendix B. 6.3 2-Opt Heuristic Approach A near-optimal and feasible operating room schedule can be generated quickly by an r-opt algorithm, which means the exchanges of r blocks are tested until there is no feasible exchange that improves the current solution; this solution is said to be r-optimal. Since the number of operations increases rapidly with increases in r, r = 2 and r = 3 are most commonly used. A 2-opt heuristic considers pair-wise block swaps, starting from an initial operating room schedule. Our starting point is the sample operating schedule that the hospital has been using. Each pair-wise swap requires the 2-opt to reduce the peak expected PDB throughout the week. The 2-opt algorithm considers all possible swaps in the current solution and chooses the best one to take. It does this while preserving the operating room schedule restrictions, as described in section 6.1. It then repeats this process using the new operating room schedule generated from the previous step, until it cannot find a better solution. At this point, it is assumed that the (local) optimum has been reached. The front end of this model is Excel, and the heuristic is coded in Excel Visual Basic Application. 6.4 Numerical Experiments 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 37

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 Sunday Monday 503 234 103 340 405 141 991 771 Tuesday 215 234 206 340 720 901 556 851 Wednesday 215 283 503 421 394 185 599 599 Thursday 103 503 539 240 384 828 991 991 Friday 958 206 294 793 185 411 715 715 Saturday 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 Block Expected Patient Demand for Beds t Days After Surgery ID Assignment t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 Surgeon 1 Mon OR1 2.7 2.6 2.4 2.1 1.5 1 0.6 503 6 Mon OR6 3.9 3.7 3.3 3.1 2.9 2.8 2.2 141 15 Tue OR8 2.9 2.6 2.5 2.4 2.1 1.8 1.6 851 16 Wed OR1 4.2 3.8 3.6 3.5 2.8 2.2 1.6 215 19 Wed OR4 2.2 2 1.8 1.4 0.7 0.5 0.4 421 39 Fri OR8 1.3 0.9 0.8 0.6 0.3 0.1 0.1 715 40 Tue OR4 2.4 2.3 2.2 2.1 1.7 1 0.6 340 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 38

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 2 Mon OR2 Orth 1, 2, 3, 4 3 Mon OR3 Orth 1, 2, 3, 4 4 Mon OR4 Orth 1, 2, 3, 4 5 Mon OR5 Gen 5, 6 6 Mon OR6 Gen 5, 6 7 Mon OR7 Urol 7, 8 8 Mon OR8 Urol 7, 8 Table 6.3: A sample of operating room restriction for each block 95 90 85 80 75 70 65 60 55 50 Expected Day Shift Patient Demand for Beds 87.9 88.5 80.3 80.6 80.4 80.1 80.8 81 78.8 76.2 77.2 73.4 74.2 73.9 72.5 73.2 73.7 65.1 66.3 64.3 59.8 Sun Mon Tue Wed Thu Fri Sat original schedule MIP optimal schedule 2-opt local optimal schedule 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 Page 39

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. Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2 3 4 5 6 7 8 503 234 234 206 103 215 215 234 206 793 539 103 215 283 283 103 503 103 539 215 503 958 958 206 294 503 234 340 340 421 405 185 340 720 240 828 421 294 394 240 384 340 901 793 185 206 141 Table 6.4: Optimal operating room schedule from MIP 141 405 991 556 771 991 901 556 851 185 599 991 185 599 720 991 599 828 991 991 411 715 851 411 715 384 715 771 Sunday Monday 503 234 215 234 Tuesday 539 206 283 Wednesday 215 503 103 503 Thursday 958 283 958 206 Friday 206 234 1 2 3 4 5 6 7 8 103 421 206 103 503 294 539 215 294 503 340 240 405 340 720 793 185 421 103 394 240 340 384 793 340 185 141 411 141 185 991 771 991 901 851 828 556 991 185 599 720 715 599 828 991 991 901 715 599 715 715 771 851 Saturday 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 40

benchmark tool to show the potential cost savings of an optimal schedule, but it would be difficult to implement in practice. On the other hand, the 2-opt heuristic is capable of generating a near-optimal solution in steps (swaps). This allows the users to evaluate the benefit at each step and perhaps only performs swaps with a significant improvement. The reduction in the peak expected PDB from each swap is presented in Figure 6.2. There is a 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. 1.4 1.2 1 1.2 1 0.8 0.6 0.4 0.2 0.7 0.6 0.5 0.6 0.3 0.5 0.3 0.2 0.4 0.3 0.2 0.1 0.1 0.1 0.3 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 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 Page 41

step if the best swap is questioned by the stakeholders and the cost associated with choosing these lesser options. Expected Day Shift Patient Demand for Beds Top Ranked Available Swaps 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 42

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 43

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 44

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 length-of-stay against expected patient length-ofstay 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-levelof-care are staying for prolonged periods of time while waiting for rehab, home care, longterm 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 45

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 46

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 47

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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 53

Emergency Department Mean of Patient Demand for Beds Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 0 0 0.1 0.1 0 0.1 0.1 0 0.1 0 0 0 0.1 0 0 0 0 0 0 0 0 0 Medicine 7.2 5.4 7 8 6.8 6.7 8 6.4 7.2 8.4 6.4 6.9 7.9 6.5 7.3 8.1 6.2 6.4 7 5.5 6 6.9 Surgery 0.7 0.7 1.4 1.4 1.1 1.4 1.3 0.8 1.4 1.2 0.8 1.4 1.9 1.1 1.4 1.4 0.9 0.9 1.1 0.6 0.9 1.1 GI Medicine 0.3 0.2 0.3 0.2 0.2 0.3 0.4 0.3 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.5 0.4 0.4 0.3 0.2 0.3 0.3 Oncology 0.4 0.3 0.8 0.7 0.4 0.6 0.4 0.3 0.9 0.6 0.1 0.6 0.5 0.3 0.5 0.4 0.4 0.5 0.5 0.5 0.8 0.5 Hematology 0.2 0.2 0.3 0.3 0.3 0.4 0.3 0.2 0.4 0.3 0.2 0.4 0.3 0.2 0.4 0.6 0.4 0.4 0.4 0.3 0.2 0.3 Orthopedics 0.3 0.3 0.3 0.5 0.2 0.4 0.7 0.4 0.6 0.6 0.3 0.6 0.4 0.3 0.5 0.4 0.2 0.3 0.4 0.2 0.2 0.4 Orthopedic Oncology 0 0 0 0 0 0 0 0 0 0 0 0.1 0.1 0 0 0 0 0 0 0 0 0 V-Medicine 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Gynecology Oncology 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SUM 9.1 7.3 10.3 11.3 8.9 10 11.2 8.4 10.8 11.3 8 10.5 11.7 8.9 10.5 11.5 8.5 9 9.7 7.3 8.6 9.7 Standard Deviation of Patient Demand for Beds Sun Mon Tue Wed Thu Fri Sun Mon Tue Wed Thu Fri Sat Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 0 0.2 0.3 0.3 0.1 0.2 0.3 0.2 0.2 0 0 0.2 0.2 0.1 0.2 0.2 0 0.2 0 0 0.2 0.2 Medicine 2.4 2.4 2.6 2.5 2.6 2.4 2.6 2.4 2.7 2.9 2.5 2.4 2.9 3 2.7 2.9 2.2 2.3 2.7 2.2 2.2 2.5 Surgery 0.8 1 1.6 1.5 1.1 1.5 1.4 1.1 1.5 1.2 0.8 1.3 1.4 0.9 1.2 1.1 1 1 1.2 0.8 1.1 1.2 GI Medicine 0.5 0.5 0.6 0.5 0.4 0.5 0.6 0.6 0.5 0.4 0.4 0.6 0.7 0.6 0.6 0.6 0.6 0.6 0.5 0.4 0.6 0.5 Oncology 0.6 0.5 1 1 0.8 0.9 0.6 0.6 1 0.8 0.3 0.7 0.8 0.6 0.7 0.6 0.6 0.6 0.8 0.6 1 0.7 Hematology 0.6 0.5 0.6 0.6 0.5 0.6 0.7 0.4 0.6 0.5 0.5 0.8 0.6 0.4 0.6 0.8 0.8 0.6 0.8 0.6 0.5 0.6 Orthopedics 0.5 0.6 0.6 0.8 0.5 0.7 0.8 0.5 0.7 0.7 0.5 0.8 0.7 0.6 0.7 0.7 0.5 0.5 0.6 0.4 0.5 0.6 Orthopedic Oncology 0 0 0.1 0 0 0 0.1 0.3 0.2 0 0.1 0.3 0.2 0 0 0 0 0.1 0 0.1 0.1 0.1 V-Medicine 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Gynecology Oncology 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SUM 2.8 2.8 3.4 3.3 3 3.1 3.2 2.8 3.4 3.4 2.8 3.1 3.6 3.3 3.3 3.4 2.8 2.8 3.3 2.6 2.8 3.1 Table A.1: Patient demand for beds at emergency department Sat Page 54

Special Care Unit Mean of Patient Demand for Beds Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 0.2 0.1 0.1 0.1 0.6 0.6 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.5 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.4 Medicine 11 11 11.1 11.4 11.1 11 10.9 11.1 11 11.1 11.1 11 11.2 10.8 10.8 10.8 10.5 10.3 10.5 10.6 10.7 10.9 Surgery 5.5 5.7 5.6 7.1 7.1 6.9 7.4 7.6 7.4 7.4 7.5 7.3 7.2 7.6 7.5 7.1 6.8 6.4 6 5.8 5.6 6.8 GI Medicine 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.4 0.3 0.3 0.3 0.3 0.3 0.4 0.3 0.3 0.3 0.3 Oncology 0.6 0.6 0.7 0.7 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.8 0.8 0.9 0.8 0.8 0.8 0.8 0.8 0.7 0.7 Hematology 1.7 1.7 1.6 1.6 1.6 1.6 1.6 1.5 1.6 1.4 1.5 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.7 1.7 1.7 1.6 Orthopedics 1.3 2.1 2 2 2.8 2.7 2.8 3.1 2.9 2.7 2.7 2.5 2.3 2.3 2.2 2 1.8 1.8 1.5 1.2 1.2 2.2 Orthopedic Oncology 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.3 V-Medicine 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Gynecology Oncology 0.1 0.1 0.1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SUM 21 21.9 21.8 23.5 24.4 24.1 24.5 25.1 24.7 24.4 24.9 24.5 24.4 24.3 24.1 23.3 22.5 21.8 21.4 20.8 20.6 23.2 Standard Deviation of Patient Demand for Beds Sun Mon Tue Wed Thu Fri Sun Mon Tue Wed Thu Fri Sat Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 0.4 0.4 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.5 Medicine 3 3.1 3 3.3 3.1 3.2 3.1 3.2 3 2.9 2.9 2.9 3.2 3 3.1 3.3 3.2 3.2 3.2 3.3 3.2 3.1 Surgery 2.4 2.4 2.3 2.6 2.7 2.6 2.5 2.6 2.6 2.7 2.7 2.7 2.7 2.5 2.6 2.5 2.5 2.4 2.3 2.3 2.3 2.5 GI Medicine 0.6 0.6 0.6 0.6 0.6 0.5 0.6 0.6 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 Oncology 0.8 0.8 0.8 0.8 0.8 0.8 0.9 0.9 1 0.9 1 1 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.8 0.9 Hematology 1.3 1.2 1.2 1.1 1.2 1.2 1.2 1.2 1.2 1.1 1.2 1.2 1.2 1.2 1.3 1.3 1.2 1.3 1.3 1.2 1.2 1.2 Orthopedics 1.1 1.4 1.4 1.4 1.9 1.9 1.9 2 2 1.8 1.8 1.7 1.4 1.4 1.4 1.4 1.3 1.2 1.2 1.1 1.1 1.5 Orthopedic Oncology 0.6 0.5 0.4 0.5 0.5 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 V-Medicine 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Gynecology Oncology 0.2 0.2 0.2 0 0 0 0.1 0.1 0.1 0.1 0 0 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 SUM 4.4 4.5 4.4 4.7 4.8 4.9 4.7 4.9 4.8 4.7 4.7 4.7 4.7 4.6 4.7 4.8 4.6 4.6 4.5 4.5 4.4 4.6 Table A.2: Patient demand for beds at special-care unit Sat Page 55

Acute Wards Mean of Patient Demand for Beds Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 1.7 1.7 1.7 1.7 1.8 1.7 1.8 1.7 1.6 1.6 1.6 1.6 1.7 1.8 1.7 1.7 1.7 1.8 1.7 1.6 1.7 1.7 Medicine 68 68 67 67 67 67 68 68 68 68 68 68 69 69 70 70 70 69 69 69 68 68 Surgery 56 56 55 58 59 58 61 61 60 63 65 66 67 69 68 68 65 64 63 61 59 62 GI Medicine 4.9 4.6 4.4 4.5 4.4 4.6 4.5 4.7 4.9 4.6 4.5 4.5 4.5 4.5 4.6 4.6 4.8 4.8 4.8 4.6 4.6 4.6 Oncology 19 18 19 20 19 20 20 20 21 21 21 21 21 21 21 21 20 20 20 19 19 20 Hematology 26 26 27 26 26 27 27 27 28 28 27 28 28 28 29 29 28 28 27 27 27 27 Orthopedics 31 35 33 32 39 36 36 40 39 38 41 40 41 43 40 40 38 35 34 33 30 37 Orthopedic Oncology 1.5 1.5 1.5 1.4 1.5 1.3 1.2 1.6 1.6 1.6 1.6 1.6 1.7 1.9 1.9 1.9 1.8 1.8 1.7 1.7 1.6 1.6 V-Medicine 24 24 24 24 23 24 24 24 24 24 24 24 24 24 24 24 23 23 23 23 23 24 Gynecology Oncology 0.2 0.2 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 SUM 231 234 232 235 241 240 243 248 247 250 254 255 257 261 260 259 254 247 246 240 234 246 Standard Deviation of Patient Demand for Beds Sun Mon Tue Wed Thu Fri Sun Mon Tue Wed Thu Fri Sat Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 1.4 1.4 1.4 1.4 1.3 1.4 1.4 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.2 1.2 1.3 Medicine 8 8 8 8 8 8 8 8 8 8 8 8 7 7 8 8 8 8 7 7 8 8 Surgery 8 8 8 9 9 9 9 8 8 9 9 9 9 9 9 9 8 8 8 8 8 8 GI Medicine 2.1 1.9 2 2.1 2 2.2 2.1 2 2.1 2.1 2.2 2.2 2.2 2.1 2.1 2.2 2.4 2.3 2.4 2.2 2.1 2.1 Oncology 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Hematology 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Orthopedics 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 6 6 Orthopedic Oncology 1.4 1.5 1.5 1.4 1.5 1.3 1.3 1.4 1.4 1.4 1.4 1.4 1.4 1.6 1.5 1.5 1.4 1.4 1.4 1.4 1.3 1.4 V-Medicine 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Gynecology Oncology 0.5 0.4 0.4 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 SUM 14.7 14.9 15.1 15.4 15.3 15.7 15.6 15.6 15.6 16.0 15.7 15.6 15.4 15.7 15.7 15.6 15.6 15.2 14.9 14.7 14.9 15.4 Table A.3: Patient demand for beds at acute wards Sat Page 56

Alternative Level of Care Unit Mean of Patient Demand for Beds Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.2 0.2 0.2 0.2 Medicine 19.3 19.6 19.2 19.1 18.8 18.8 18.8 18.6 18.4 18.3 18.3 18.3 18.4 18.5 18.5 18.7 18.6 18.7 18.7 18.4 18.6 18.7 Surgery 2.5 2.5 2.5 2.5 2.6 2.5 2.6 2.7 2.7 2.8 2.8 2.8 2.7 2.7 2.7 2.6 2.6 2.7 2.6 2.5 2.5 2.6 GI Medicine 0.3 0.3 0.3 0.3 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.3 Oncology 4.8 4.8 4.8 4.8 4.9 4.7 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.9 4.8 5 5 5.1 5 4.9 4.9 4.9 Hematology 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.7 0.6 0.7 0.6 0.7 0.7 0.6 0.6 0.7 0.6 0.6 0.6 0.7 Orthopedics 12.9 13 12.9 12.9 12.9 13 13 13.1 13.1 13.2 13.1 13.1 13 13.1 13.2 13.2 13.1 13.3 13.3 13.3 13.5 13.1 Orthopedic Oncology 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 V-Medicine 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Gynecology Oncology 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SUM 41.2 41.5 41.1 41 40.7 40.7 40.7 40.7 40.6 40.7 40.7 40.6 40.5 40.8 40.8 41 40.9 41.3 41.2 40.7 41.1 40.9 Standard Deviation of Patient Demand for Beds Sun Mon Tue Wed Thu Fri Sun Mon Tue Wed Thu Fri Sat Patients categorized by service night day evening night day evening night day evening night day evening night day evening night day evening night day evening Average GI Surgery 0.4 0.4 0.4 0.4 0.3 0.4 0.4 0.5 0.5 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 Medicine 4.7 4.7 4.8 4.6 4.5 4.4 4.2 4.2 4.3 4.1 4 4 4 4.1 4.2 4.4 4.6 4.5 4.6 4.6 4.5 4.4 Surgery 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.6 1.6 1.6 1.7 1.7 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.5 1.5 1.6 GI Medicine 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 Oncology 2.3 2.3 2.3 2.3 2.5 2.4 2.4 2.3 2.3 2.4 2.3 2.2 2.2 2.2 2.1 2.2 2.3 2.3 2.2 2.2 2.2 2.3 Hematology 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.7 0.8 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.8 Orthopedics 3.6 3.7 3.6 3.6 3.4 3.4 3.4 3.4 3.5 3.4 3.5 3.4 3.3 3.4 3.4 3.4 3.3 3.3 3.3 3.4 3.4 3.4 Orthopedic Oncology 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 V-Medicine 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Gynecology Oncology 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SUM 6.6 6.7 6.7 6.6 6.5 6.4 6.2 6.2 6.3 6.1 6.1 6.1 5.9 6.1 6.1 6.3 6.4 6.3 6.4 6.4 6.3 6.3 Table A.4: Patient demand for beds at alternative-level-of-care Sat Page 57

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 58

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 59

Appendix C Numerical Experiments Block Block Expected Patient Demand for Beds t Days Since Surgery ID Assignment t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 Surgeon 1 Mon OR1 2.7 2.6 2.4 2.1 1.5 1 0.6 503 2 Mon OR2 4 3.6 3.4 3 2 1.4 1.1 234 3 Mon OR3 3.8 3.4 3.1 3.1 2.7 2 1.6 103 4 Mon OR4 2.6 2.4 2.3 2.2 1.5 1 1 340 5 Mon OR5 3 2.6 2.5 2.2 1.5 1 0.8 405 6 Mon OR6 3.9 3.7 3.3 3.1 2.9 2.8 2.2 141 7 Mon OR7 2.4 2 1.1 0.6 0.5 0.3 0.3 991 8 Mon OR8 1.1 1.1 0.9 0.5 0.5 0.5 0.6 771 9 Tue OR1 3.6 3.3 3.2 3.1 2.6 1.7 1.2 215 10 Tue OR2 4 3.7 3.5 3.3 2.4 1.4 0.9 234 11 Tue OR3 3 2.9 2.9 2.6 1.8 1 0.6 206 12 Tue OR5 5.3 4.8 4.4 4.3 4.1 3.7 3.1 720 13 Tue OR6 3.1 2.7 2.5 2.2 2 1.9 1.7 901 14 Tue OR7 2.3 1.9 1.6 1.3 1 0.8 0.7 556 15 Tue OR8 2.9 2.6 2.5 2.4 2.1 1.8 1.6 851 16 Wed OR1 4.2 3.8 3.6 3.5 2.8 2.2 1.6 215 17 Wed OR2 3.1 2.7 2.5 2.3 2.1 1.6 1.3 283 18 Wed OR3 2.8 2.6 2.5 2.4 1.6 0.9 0.5 503 19 Wed OR4 2.2 2 1.8 1.4 0.7 0.5 0.4 421 20 Wed OR5 1.4 1.4 0 0 0 0 0 394 21 Wed OR6 4.6 4.4 4 3.7 3.3 2.9 2.8 185 22 Wed OR7 0.9 0.4 0.2 0.1 0 0 0 599 23 Wed OR8 1.2 1 0.3 0.1 0 0 0 599 24 Thu OR1 3.7 3.3 3.1 2.7 2.5 1.8 1.1 103 25 Thu OR2 3.1 2.8 2.5 2.5 1.9 1.3 0.9 503 26 Thu OR3 1.5 1.3 0.4 0.1 0.1 0 0 539 27 Thu OR4 2.2 1.8 1.6 1.2 0.8 0.5 0.3 240 28 Thu OR5 3.2 2.8 2.8 2.3 2.1 1.9 1.6 384 29 Thu OR6 3.6 3.5 2.9 2.5 2.4 2.2 1.9 828 30 Thu OR7 2.7 2.1 1 0.7 0.6 0.4 0.3 991 31 Thu OR8 2.7 2 1.2 0.6 0.3 0.3 0.3 991 32 Fri OR1 1.6 1.3 1.4 1.3 1.1 0.9 0.5 958 33 Fri OR2 3.4 3.1 2.8 2.4 1.8 1 0.6 206 34 Fri OR3 2.3 2 1.8 1.7 1.6 1.1 1.1 294 35 Fri OR4 2.1 2 1.3 1.2 0.9 0.7 0.6 793 36 Fri OR5 4.3 3.9 3.7 3.2 3.1 2.9 2.6 185 37 Fri OR6 3.2 2.8 2.6 2.4 2.1 2 1.7 411 38 Fri OR7 1.4 1.2 0.8 0.8 0.3 0.1 0.1 715 39 Fri OR8 1.3 0.9 0.8 0.6 0.3 0.1 0.1 715 40 Tue OR4 2.4 2.3 2.2 2.1 1.7 1 0.6 340 Table C.1: Expected demand for beds and surgeon assigned to each block Page 60

Block ID Block Info Procedure Available Operating Rooms 1 Mon OR1 Orth 1, 2, 3, 4 2 Mon OR2 Orth 1, 2, 3, 4 3 Mon OR3 Orth 1, 2, 3, 4 4 Mon OR4 Orth 1, 2, 3, 4 5 Mon OR5 Gen 5, 6 6 Mon OR6 Gen 5, 6 7 Mon OR7 Urol 7, 8 8 Mon OR8 Urol 7, 8 9 Tue OR1 Orth 1, 2, 3, 4 10 Tue OR2 Orth 1, 2, 3, 4 11 Tue OR3 Orth 1, 2, 3, 4 12 Tue OR5 Gen 5, 6 13 Tue OR6 Gen 5, 6 14 Tue OR7 Urol 7, 8 15 Tue OR8 Urol 7, 8 16 Wed OR1 Orth 1, 2, 3, 4 17 Wed OR2 Orth 1, 2, 3, 4 18 Wed OR3 Orth 1, 2, 3, 4 19 Wed OR4 Orth 1, 2, 3, 4 20 Wed OR5 Gen 5, 6 21 Wed OR6 Gen 5, 6 22 Wed OR7 Urol 7, 8 23 Wed OR8 Urol 7, 8 24 Thu OR1 Orth 1, 2, 3, 4 25 Thu OR2 Orth 1, 2, 3, 4 26 Thu OR3 Orth 1, 2, 3, 4 27 Thu OR4 Orth 1, 2, 3, 4 28 Thu OR5 Gen 5, 6 29 Thu OR6 Gen 5, 6 30 Thu OR7 Urol 7, 8 31 Thu OR8 Urol 7, 8 32 Fri OR1 Orth 1, 2, 3, 4 33 Fri OR2 Orth 1, 2, 3, 4 34 Fri OR3 Orth 1, 2, 3, 4 35 Fri OR4 Orth 1, 2, 3, 4 36 Fri OR5 Gen 5, 6 37 Fri OR6 Gen 5, 6 38 Fri OR7 Urol 7, 8 39 Fri OR8 Urol 7, 8 40 Tue OR4 Orth 1, 2, 3, 4 Table C.2: Operating room restriction for each block Page 61

Appendix D Graphical User Interface In this section, we discuss the graphical user interface of our tools and the reasons we think they are generic. The bed planning model and the 2-opt heuristic are built with Excel UserForm while the mixed integer programming model is coded in AMPL. Figure D.1: Main graphical user interface Figure D.1 shows the main user interface. It guides the users through the setup process. Short descriptions are shown upfront while additional instructions and details are easily accessible. Page 62

The Simulation Settings button leads to the parameters we use for our simulations and allows other users to update with their own settings. The Simulate Emergent/Urgent Inpatient Bed Demand button leads to emergent/urgent simulation setup, shown in Figure D.2. The instruction for each control (labeled in Figure D.2) is listed below: 1. Click on Load Patients Record, a window will pop up asking the location of patient record Excel file. Then, find and select patient record file. 2. If there is only one worksheet in the patient record Excel file, Excel will automatically load all the fields in that file. Otherwise, this control allows you to choose the name of the worksheet containing patient records. 3. This control allows you to choose a valid arrival period. A valid arrival period of 2009/04/01 to 2009/12/31 means all patients who were admitted between 2009/04/01 and 2009/12/31 inclusive exist in the patient record file. 4. This control allows you to select the name of the field that stores patient s admit date time in the Columns section, and click the right arrow to move it into Admit Date and Time section. 5. This control contains the names of the fields that store patient s length-of-stay in its corresponding departments. Order the names in the order of patient flow between each department, for example, emergency department to special care units to acute wards to alternative-level-of-care. 6. This control contains the name of the fields that store patient categories that you want to use to divide patients into groups. 7. Click to start simulation. Page 63

These controls are the foundation of a generic model in which the user can customize simulation inputs and parameters to represent different hospital settings. Figure D.2: Simulation interface for emergent/urgent inpatients Furthermore, the Show Saved Results on Patient Demand for Beds button in Figure D.1 leads to patient demand for beds interface, shown in Figure D.3. This interface allows the user to format and organize simulation outputs. For example, in the Figure D.3, the user chose to exclude patient demand for beds at emergency department and SCU by deselecting those fields. Also, only patient demand for beds at JHCC site is shown by removing the highlight on MUMC sites. And finally, service patient category is activated to categorized patient demand for beds in the output panel, which displays the expected and the standard deviation of the results. Page 64

Figure D.3: Simulation interface for patient demand for beds The Integer -> Decimal, Copy to Clipboard, Graph buttons are options to display or plot customized output results. However, the Revised OR Schedule button leads to operating room scheduling interface, shown in Figure D.4. Top ranked available swaps are displayed to show the benefit of each available swap. This could be used as evidence to convince a surgeon regarding improvement opportunities worth pursuing. Also, operating schedule based on the chosen swaps is displayed to keep track of the benefit and the cost of the current progress. Page 65

Figure D.4: Simulation interface for revising operating room schedule The mixed integer programming model is not implemented in Excel VBA because Gurobi solver is a commercial product and we want to provide these tools to our users for free. However, the code for our mixed integer programming model is provided in Appendix B, in case some of our users have access to AMPL and Gurobi, and want to try it out. To conclude, we mentioned three principles to minimize the short-comings of simulation in Section 3.2 and we are confident that we have built a comprehensive simulation model that incorporates all those principles. Page 66