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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
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Transcript of A Generic Bed Planning Model - TSpace Repository: Home · PDF fileA 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
Page | ii
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 | iii
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 | iv
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
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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 | vi
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 | vii
List of Tables
4.1: Average of expected patient demand for beds for a typical week, by patient-
day 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 | 1
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 | 2
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 | 3
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 life-
threatening 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 | 4
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 | 5
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 | 6
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 | 7
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 | 8
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 | 9
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 | 10
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 | 11
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 | 12
surgeries cannot be scheduled simultaneously. Bekker and Koeleman [38] combine time-
dependent 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 | 13
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 | 14
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 | 15
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 | 16
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 | 17
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 | 18
information, such as assigned operating room and the main surgeon should also be used to
further classify elective patients.
Start
Define patient
groups
Create arrival
distribution for
each shift of the
week and each
patient group
i = 1
Begin trial i
j = 1
For shift j
k = 1
Generate n patient
arrivals for patient
group k at shift j
Calculate number
of inpatients for
current and
subsequent shifts
Done all
groups?Done all shifts? Done all trials?
k = k +1
j = j +1
i = i +1
Calculate mean and standard
deviation of patient demand
for each patient group at each
shift of the week
End
yes
no
no
no
yes
yes
Simulation Design
1
2
3
4
5
Figure 4.4: Simulation design
Page | 19
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 | 20
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 | 21
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 | 22
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.
Figure 4.5: Expected patient demand for beds at acute wards
231 234
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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 Patient-
day 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 | 24
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.
Figure 4.6: Bed capacity by target occupancy level for acute wards
231 234 232 235
241 240 243
248 247 250
254 255 257 261 260 259
254 247 246
240 234
290
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Expected PDB Bed capacity with 85% occupancy level
Page | 25
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 48th
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.
Figure 4.7: Bed capacity by probability of bed blocking for acute wards
231 234 232 235
241 240 243
248 247 250
254 255 257 261 260 259
254 247 246
240 234
254 260
256 258
270 270 270
281
273 278
281 281 285 284
288
280 276
271 266
259 254
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Expected PDB Bed capacity with 5% chance of bed blocking
Page | 26
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 | 27
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 | 28
Service Received
Daily Average of The Expected PDB Across the Week
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 | 29
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 | 30
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
58 56
49 48
61
55 53
67
61 60
73
68 67
81
75 74
81
73 72 69
62 70
0
10
20
30
40
50
60
70
80
90
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Expect PDB Number of budgetted surgical beds
Page | 31
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
Wednesday 215 283 503
421
991 394 185 599 599
Thursday 103 503
539
340 240 384 828
991
421
991
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 | 32
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.
59.7
66.4
74.2 78.8
87.7 88.5
74
62.5
68.7
74.8 78.6
85.5 85.4
73.9
0
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Weekday day shift
orginal operating schedule revised operating schedule
Page | 33
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 | 34
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:
Dit the expected PDB for block i at the day t starting from the surgery date
for convenience, Di(t-7) = Dit for t = 0, 1, … , 6
Sim 1 when block i is performed by surgeon m, 0 otherwise
Bij 1 when block i can be assigned to day j, 0 otherwise
Rik 1 when block i can be assigned to operating room k, 0 otherwise
Then, Xijk can be defined as a decision variable representing whether or not block i is
assigned to day j in operating room k. Xijk = 1 if block i is at day j in operating room k,
otherwise 0. Define Yij to be a decision variable representing whether or not block i is
assigned to day j. Yij = 1 if block i is located at day j, otherwise 0. Thus,
Define Fij to be the expected patient demand for beds from block i on day j.
Page | 35
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 | 36
In the above model, constraints (1) and (2) calculate dummy variables Yij and Fij
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 Xijk and Yij.
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 | 37
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 | 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
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
ID
Block
Assignment
Expected Patient Demand for Beds t Days After Surgery
Surgeon t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6
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 | 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
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
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
66.3
74.2
78.8
87.9 88.5
73.9
65.1
73.4
76.2
80.3 80.4 80.8
73.2
64.3
72.5
77.2
80.6 80.1 81
73.7
50
55
60
65
70
75
80
85
90
95
Sun Mon Tue Wed Thu Fri Sat
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
Sunday
Monday
503
234
234
206
103
215
340
421
405
185
141
405
991
556
771
991
Tuesday
215
793
234
539
206
103
340
240
720
828
901
185
556
599
851
991
Wednesday
215
283
283
103 503
421
294 394
185
720
599
991 599
Thursday
103
215 503
539
958
240
340
384
901
828
411
991
715
991
851
Friday
958
503
206
234
294
340
793
206
185
141
411
384 715
715
771
Saturday
Table 6.4: Optimal operating room schedule from MIP
1 2 3 4 5 6 7 8
Sunday
Monday 503 234
103
421
340
240 405
141
185 991
771
991
Tuesday
215
539
234
206
206
103
340
793
720
185
901
828 556
851
991
Wednesday 215
283
503
503
294
421
103 394
185
720
599
715 599
Thursday
103
958
503
283
539
215
240
340 384
828
901
991
715
991
599
Friday
958
206
206
234
294
503
793
340
185
141 411
715
771
715
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 | 41
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.
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
1
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
0.2
0.4
0.6
0.8
1
1.2
1.4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
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 length-of-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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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
Emer
gen
cy D
epa
rtm
ent
Me
an o
f P
atie
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ds
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aye
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igh
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00
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7.2
5.4
78
6.8
6.7
86.
47.
28.
46.
46.
97.
96.
57.
38.
16.
26.
47
5.5
66.
9
Su
rge
ry0.
70.
71.
41.
41.
11.
41.
30.
81.
41.
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81.
41.
91.
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7
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Tab
le A
.1:
Pat
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dem
and f
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bed
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em
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t
Page | 55
Spec
ial C
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itM
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0.5
0.5
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13
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3.1
3.2
3.1
3.2
32.
92.
92.
93.
23
3.1
3.3
3.2
3.2
3.2
3.3
3.2
3.1
Su
rge
ry2.
42.
42.
32.
62.
72.
62.
52.
62.
62.
72.
72.
72.
72.
52.
62.
52.
52.
42.
32.
32.
32.
5
GI M
ed
icin
e0.
60.
60.
60.
60.
60.
50.
60.
60.
60.
60.
60.
60.
50.
50.
50.
50.
50.
50.
50.
50.
50.
6
On
colo
gy0.
80.
80.
80.
80.
80.
80.
90.
91
0.9
11
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.8
0.9
He
mat
olo
gy1.
31.
21.
21.
11.
21.
21.
21.
21.
21.
11.
21.
21.
21.
21.
31.
31.
21.
31.
31.
21.
21.
2
Ort
ho
pe
dic
s1.
11.
41.
41.
41.
91.
91.
92
21.
81.
81.
71.
41.
41.
41.
41.
31.
21.
21.
11.
11.
5
Ort
ho
pe
dic
On
colo
gy0.
60.
50.
40.
50.
50.
40.
40.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
5
V-M
ed
icin
e0
00
00
00
00
00
00
00
00
00
00
0
Gyn
eco
logy
On
colo
gy0.
20.
20.
20
00
0.1
0.1
0.1
0.1
00
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
Sat
Sun
Mo
nTu
eW
ed
Thu
Fri
Sat
Sun
Mo
nTu
eW
ed
Thu
Fri
Tab
le A
.2:
Pat
ient
dem
and f
or
bed
s at
spec
ial-
care
unit
Page | 56
Acu
te W
ard
sM
ean
of
Pat
ien
t D
em
and
fo
r B
ed
s
Pat
ien
ts c
ate
gori
zed
by
serv
ice
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gA
vera
ge
GI S
urg
ery
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
Me
dic
ine
6868
6767
6767
6868
6868
6868
6969
7070
7069
6969
6868
Su
rge
ry56
5655
5859
5861
6160
6365
6667
6968
6865
6463
6159
62
GI M
ed
icin
e4.
94.
64.
44.
54.
44.
64.
54.
74.
94.
64.
54.
54.
54.
54.
64.
64.
84.
84.
84.
64.
64.
6
On
colo
gy19
1819
2019
2020
2021
2121
2121
2121
2120
2020
1919
20
He
mat
olo
gy26
2627
2626
2727
2728
2827
2828
2829
2928
2827
2727
27
Ort
ho
pe
dic
s31
3533
3239
3636
4039
3841
4041
4340
4038
3534
3330
37
Ort
ho
pe
dic
On
colo
gy1.
51.
51.
51.
41.
51.
31.
21.
61.
61.
61.
61.
61.
71.
91.
91.
91.
81.
81.
71.
71.
61.
6
V-M
ed
icin
e24
2424
2423
2424
2424
2424
2424
2424
2423
2323
2323
24
Gyn
eco
logy
On
colo
gy0.
20.
20.
20.
30.
20.
20.
20.
20.
20.
20.
20.
20.
30.
20.
20.
20.
20.
20.
20.
20.
20.
2
SUM
231
234
232
235
241
240
243
248
247
250
254
255
257
261
260
259
254
247
246
240
234
246
Stan
dar
d D
evi
atio
n o
f P
atie
nt
De
man
d f
or
Be
ds
Pat
ien
ts c
ate
gori
zed
by
serv
ice
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gA
vera
ge
GI S
urg
ery
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
Me
dic
ine
88
88
88
88
88
88
77
88
88
77
88
Su
rge
ry8
88
99
99
88
99
99
99
98
88
88
8
GI M
ed
icin
e2.
11.
92
2.1
22.
22.
12
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
On
colo
gy4
44
44
44
44
44
44
44
44
44
44
4
He
mat
olo
gy5
55
55
55
55
55
55
55
55
55
55
5
Ort
ho
pe
dic
s6
66
66
66
66
66
66
66
66
66
56
6
Ort
ho
pe
dic
On
colo
gy1.
41.
51.
51.
41.
51.
31.
31.
41.
41.
41.
41.
41.
41.
61.
51.
51.
41.
41.
41.
41.
31.
4
V-M
ed
icin
e5
55
55
55
55
55
55
55
55
55
55
5
Gyn
eco
logy
On
colo
gy0.
50.
40.
40.
50.
50.
50.
50.
40.
40.
40.
40.
40.
50.
50.
50.
50.
50.
50.
50.
50.
50.
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
Sat
Sun
Mo
nTu
eW
ed
Thu
Fri
Sat
Sun
Mo
nTu
eW
ed
Thu
Fri
Tab
le A
.3:
Pat
ient
dem
and f
or
bed
s at
acu
te w
ards
Page | 57
Alt
ern
ati
ve L
evel
of
Ca
re U
nit
Me
an o
f P
atie
nt
De
man
d f
or
Be
ds
Pat
ien
ts c
ate
gori
zed
by
serv
ice
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gA
vera
ge
GI S
urg
ery
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
Me
dic
ine
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
Su
rge
ry2.
52.
52.
52.
52.
62.
52.
62.
72.
72.
82.
82.
82.
72.
72.
72.
62.
62.
72.
62.
52.
52.
6
GI M
ed
icin
e0.
30.
30.
30.
30.
40.
30.
30.
30.
30.
30.
30.
30.
30.
30.
30.
40.
40.
40.
40.
40.
40.
3
On
colo
gy4.
84.
84.
84.
84.
94.
74.
84.
84.
84.
84.
84.
84.
84.
94.
85
55.
15
4.9
4.9
4.9
He
mat
olo
gy0.
80.
80.
70.
70.
70.
70.
60.
60.
60.
70.
60.
70.
60.
70.
70.
60.
60.
70.
60.
60.
60.
7
Ort
ho
pe
dic
s12
.913
12.9
12.9
12.9
1313
13.1
13.1
13.2
13.1
13.1
1313
.113
.213
.213
.113
.313
.313
.313
.513
.1
Ort
ho
pe
dic
On
colo
gy0.
40.
40.
40.
40.
40.
50.
50.
50.
50.
50.
50.
50.
50.
50.
40.
40.
40.
40.
40.
40.
40.
5
V-M
ed
icin
e0
00
00
00
00
00
00
00
00
00
00
0
Gyn
eco
logy
On
colo
gy0
00
00
00
00
00
00
00
00
00
00
0
SUM
41.2
41.5
41.1
4140
.740
.740
.740
.740
.640
.740
.740
.640
.540
.840
.841
40.9
41.3
41.2
40.7
41.1
40.9
Stan
dar
d D
evi
atio
n o
f P
atie
nt
De
man
d f
or
Be
ds
Pat
ien
ts c
ate
gori
zed
by
serv
ice
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gn
igh
td
aye
ven
ing
nig
ht
day
eve
nin
gA
vera
ge
GI S
urg
ery
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
Me
dic
ine
4.7
4.7
4.8
4.6
4.5
4.4
4.2
4.2
4.3
4.1
44
44.
14.
24.
44.
64.
54.
64.
64.
54.
4
Su
rge
ry1.
51.
51.
51.
51.
51.
51.
51.
61.
61.
61.
71.
71.
61.
61.
61.
61.
61.
61.
61.
51.
51.
6
GI M
ed
icin
e0.
50.
50.
60.
60.
60.
60.
60.
60.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
50.
5
On
colo
gy2.
32.
32.
32.
32.
52.
42.
42.
32.
32.
42.
32.
22.
22.
22.
12.
22.
32.
32.
22.
22.
22.
3
He
mat
olo
gy0.
80.
80.
80.
80.
80.
80.
80.
70.
70.
80.
80.
80.
80.
80.
70.
70.
70.
70.
70.
70.
70.
8
Ort
ho
pe
dic
s3.
63.
73.
63.
63.
43.
43.
43.
43.
53.
43.
53.
43.
33.
43.
43.
43.
33.
33.
33.
43.
43.
4
Ort
ho
pe
dic
On
colo
gy0.
60.
60.
60.
70.
70.
70.
70.
70.
70.
70.
70.
70.
60.
60.
60.
60.
60.
60.
60.
60.
60.
7
V-M
ed
icin
e0
00
00
00
00
00
00
00
00
00
00
0
Gyn
eco
logy
On
colo
gy0
00
00
00
00
00
00
00
00
00
00
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
Sat
Sun
Mo
nTu
eW
ed
Thu
Fri
Sat
Sun
Mo
nTu
eW
ed
Thu
Fri
Tab
le A
.4:
Pat
ient
dem
and f
or
bed
s at
alt
ernat
ive-
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
Appendix C
Numerical Experiments
Block
ID
Block
Assignment
Expected Patient Demand for Beds t Days Since Surgery
Surgeon t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6
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 | 61
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
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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.
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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.
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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.
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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.
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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.
