Seminar 1
Michael Samudra
Contents
The name of the game
Problem: Patients are served late 5
Task 9
Improvement strategies
Simulation results 12
The Gasthuisberg setting 22
Model 40
Implementation 45
References 51
Literature Review: OR planning 52
Literature Review: Patient scheduling 87
Perspectives 109
Conclusion 111
Appendix 114
1
1. Introduction
In most hospitals there are patients who receive surgery later than
medically advised. In Belgium’s largest hospital, Gasthuisberg, this
is the case for approximately every third patient. At hospitals such
as Gasthuisberg this could be solved by simply adding to their
capacities, i.e., opening a new OR (operating room) and hiring the
necessary additional personnel. Unfortunately, in today’s economic
environment this is rarely an option.
Serving patients late is a problem because their health condition
can potentially quickly worsen which exposes them to an increased
health risk. This needs to be avoided, primarily from a humanitarian
standpoint, but there is also a hidden cost perspective as a patient
in a worsened health state is likely to require larger amounts of
resources and thus to cost more money.
In order to improve the current situation, the lateness of patients
has to be, firstly, quantified and, secondly, the responsible
mechanism, the patient scheduling process, has to be understood.
We analyzed the percentage of patients being served in time at
Gasthuisberg. At the hospital, an elective patient is associated with
one of five due time (DT) intervals within which the patient has to
be served. We also analyzed the lateness of patients across
disciplines, using all data from 2012 and 22 ORs. We tried to
understand many of the different aspects related to the scheduling
process, which knowledge we then included into a simulation model.
We investigated from the data: patient arrival patterns, the relation
between estimated and realized surgery durations, rescheduling
mechanisms and the allocation patterns of emergencies.
3
We also used the model to investigate the effects of switching from
the current scheduling practice of assigning surgeries directly to
slots (OR and day) to a two-step procedure, where patients are
scheduled to a surgery week first and only in a second step to slots.
Our results suggest that in case of the two-step procedure it is very
important to allow patients with shorter DTs to break into the already
fixed weekly schedule. Additionally, it is important that in the second
step of the scheduling procedure, in the within week scheduling, the
DT is considered. We conclude that improving the patient
scheduling process can help to decrease the amount of patients
served too late. As a next step, we try to develop a sound
scheduling schema, which allows to further decrease the number of
patients served too late.
4
2. The name of the game
2.1. Problem: Patients are served
late
We are focusing on one particular problem: every third patient
receives surgery later than the medically advised time limit. This is
a serious problem as the health condition of patients that are not
served on time can become worse. This is undesirable, primarily for
humanitarian reasons, but it is also important to keep in mind that
a patient in a worsened health state is likely to require more hospital
resources and therefore ultimately will cost more money.
Figure 1: The first axis shows the offset of the number
of days patients were served respective to their DT.
Most patients (11%) are served 6-7 days before their
DT.
Whether a patient received surgery too late or is still on time is
dependent on the urgency status, that is, the DT category of the
patient. The DT is assigned to the patients by the physician in
charge. If the patient is an elective patient, then the DT category
will define a time interval measured in weeks within which surgery
is advised. For instance, it is a Monday and the surgeon decides
that a patient requires surgery with urgency category DT 4 (1 week),
then it means that the patient should undergo surgery the latest next
week Monday.
5
Non-elective
Elective
GYN
Tx
ABD
CAH
NCH
ONC
RHK
THO
TRH
URO
VAT
MKA
NKO
Category
1
2
3
4
5
6
7
8
Meaning
Now!
Up to 6 hours
Today
1 week
1 – 2 weeks
2 – 4 weeks
4 – 8 weeks
8 weeks -
Closed set
[0, 0]
[0, 5h]
[0, 23h]
[1d, 7d]
[8d, 14d]
[15d, 28d]
[29d, 56d]
[57d, 112d]
Color
GYNAECOLOGIE EN VERLOSKUNDE
Gynecology and obstetrics
HK ABD TRANSPLANTATIECHIRURGIE
Abdominal transplant surgery
HK ABDOMINALE HEELKUNDE
Abdominal surgery
HK CARDIALE HEELKUNDE
Cardiac surgery
HK NEUROCHIRURGIE
Neurosurgery
HK ONCOLOGISCHE HEELKUNDE
General medical oncological
HK PLAST RECONSTR ESTH CHIR
Plastic, reconstructive and cosmetic surgery
The DT categories can vary from 1-8 where category 1-3 are
assigned to non-elective patients and category 4-8 to electives. The
first three non-elective categories have to be scheduled within a
time interval less than or equal to 1 day. The shortness of this
interval leaves no room for planning and thus all three non-elective
categories are excluded from the scheduling process. In other
words, we only schedule elective patients (DT category 4-8). Nonelectives are nevertheless considered in the simulation model as
we in detail model their arrival to the ORs and thus implicitly test for
their impact on the elective schedule.
HK THORAXHEELKUNDE
Thoracic surgery
HK TRAUMATOLOGIE
Traumatology
HK UROLOGIE
Urology
HK VAATHEELKUNDE
Vascular surgery
MOND, KAAK, AANGEZ.CHIRURGIE
Oral and maxillofacial surgery
NKO, GELAATS- EN HALSCHIR
Head and neck surgery
Another property of the DT is that it is defined as an interval,
suggesting that it is best for the patient to get surgery only after a
certain reference period. It might seem unreasonable to let patients
wait unnecessarily, but it can be the case that the patient or surgeon
needs some time to prepare for the surgery. For example, a patient
with DT category 5 (1-2 weeks) should be able to wait without
complications for 1 week, but it is highly advisable to serve the
patient in the 2nd week the latest. From a scheduling algorithmic
perspective the end time of the interval is of greater importance.
Table 1: http://www.uzleuven.be/en/departments
6
Figure 2: For each discipline a different distribution of
DT categories is observed.
Interestingly, the distribution or mix of DT categories varies largely
across different medical disciplines. Some disciplines contain
patients from only 1 or 2 DT categories and some from 3 or 4 DT
categories. For Urology, for example, the vast majority of patients
is associated to DT 8, i.e., none of the urology patients is urgent. A
patient category with a more balanced spectrum is Oral and
maxillofacial surgery (MKA). One could wonder if Urology and MKA
patients should be handled by the same scheduling mechanism: for
urology a first-come, first-served (FCFS) approach might suffice,
whereas the same for MKA might result in many late treatments.
Surprisingly, looking at the ratio of late treatments, it appears that
DT category 4 is served the most efficiently (Figure 3). Most of DT
category 4 patients are served within their DT (1 week). This is less
true for DT category 5 patients who have to wait relatively long and
are in almost half the cases served late, that is, after 2 weeks. DT
category 6 patients have variable waiting times, but as the peak
between -28 and -21 indicates are often served immediately.
Figure 3: The amount of patients served per DT
category. DT category 4 patients are in 84.16% of the
cases served within their DT.
So far the lateness of patients was investigated for the whole
inpatient population irrespective of the disciplines. But, as Figure 4
shows, two patients with the same DT but different discipline will
have different probabilities of being served in time.
Whether patients of a certain discipline get surgery within their DT
does not only depend on the discipline’s DT distribution (Figure 2),
but also on the way how the patient scheduling process is handled
within the discipline. A discipline that performs very well is RHK,
where most patients are served in time despite the large number of
high urgencies (DT 4).
7
Figure 4: The amount of patients served late for each DT category and discipline. Care has to be taken when interpreting the results: for
example, the graph shows that 100% of DT 4 gynecology (GYN) patients are served within their DT, but as shown by Figure 2, there are
only a few patients from that DT category and thus it is of little importance that those patients are served in time.
8
2.2. Task
We are pursuing two targets. Firstly, we want to develop a
scheduling procedure that allows patients to be served in time, i.e.,
within their DT. Secondly, keeping in mind the first objective, we
want to investigate the implications of switching from the current
one-step scheduling practice of scheduling patients directly to slots
to a new two-step strategy where patients are first scheduled to a
week and only in the second step to a slot.
Our first goal is to increase the amount of patients that get timely
access to the OR, i.e., serve more patients within their DT. This
goal can be achieved in three ways. Firstly, the simplest way is to
increase the capacity on the supply side, for example, by opening
a new OR and hiring the necessary personnel. The problem is that
increasing existing OR capacities requires additional financial
resources which in the current economic environment are not
necessarily available. Moreover, in case the financial resources are
available, there are other departments of the hospital that could
need the resources as well. The second way how to schedule more
patients within their DT is to allow for more flexibility, for example,
by use of open scheduling. Open scheduling allows disciplines to
occupy any OR at any given time of the weekday, i.e., there is no
Master Surgery Schedule (MSS). As a consequence, open
scheduling, using the pooling effect, would allow to deal more
efficiently with occasional short term capacity shortages affecting
single disciplines. Despite some of the benefits of open scheduling
it is often avoided in practice, to one part, as it requires much more
effort to create a good OR plan and, on the other part, as the OR
plan is less repetitive surgeons cannot not select fixed weekly days
for consultation or teaching. The third and most cost effective way
to improve patient access to the OR is the application of operations
research methods.
9
Our second goal is to investigate the implications of switching from
the current one-step scheduling strategy, where patients are
scheduled directly to slots, to a two-step strategy where patients are
first assigned to a future week (to-week scheduling) and, in a
second step, to the actual surgery weekday and OR (within-week
scheduling). It fits within the future plans of Gasthuisberg to do this
switch from the one-step strategy to the two-step strategy. One of
the main benefits of changing to the second, two-step strategy is
that many of the scheduling decisions that had to be done in
advance using the one-step strategy, can be done later in the
second step using the two-step strategy.
Patients arrive on a Tuesday. If a week is selected that
is only partially within the DT then in the within-week
scheduling step it has to be made sure that the DT is
obeyed.
At Gasthuisberg patients are normally told about their
surgery date 3 weeks in advance. In other words, in
case patients do not fit the schedule for the next 3
weeks they are put on a waiting list. This should
normally only happen to patients with DT 6, 7 or 8.
Patients with DT 4 and 5 should be scheduled within 1
and 2 weeks respectively and thus should preferable
not be put on the waiting list.
Aligning the goal to schedule patients within their DT with the goal
of switching to the two-step procedure results in a problem: in the
two-step procedure, the second step will be largely dependent on
the first step. In case in the first step (to-week scheduling) a week
is selected that is only partly within the patients DT, then in the
second step (within-week scheduling) it has to be made sure that
the DT criterion is obeyed. In other words, depending on the week
selected in the first step, it is often the case that in the second step
only the early days of that week are valid.
Surgeons schedule their patients individually, i.e., surgeons are not
necessarily coordinating amongst each other or with other
departments. This means that a surgeon will find for a patient an
appropriate slot according to the criteria that for the individual
surgeon are important. This also means that different surgeons,
based on their past experience, might be differently efficient at
scheduling their patients. Moreover, surgeons at Gasthuisberg as
well as generally in Flanders are typically creating their patient
schedules by hand [4]. As schedules are not created centrally nor
by algorithms, it is likely that different disciplines create patient
schedules of varying quality. The current situation can be improved
in two ways. Firstly, by learning, i.e., showing surgeons the
implications of certain scheduling decisions, and secondly, by
algorithmic life support: an algorithm could either suggest several
schedules to the surgeons from which they can select the one they
prefer, or in case surgeons themselves create their schedules, the
10
algorithm could compute some of the implications of the manually
created schedule.
In order to be able to use a uniform scheduling framework, we focus
our attention on Gasthuisberg’s 22 inpatient ORs. Those inpatient
ORs are part of one of 5 kerns each denoted by a letter from A to
E. Kerns A-D are located closer to one another, whereas kern E is
positioned a little further away. Each kern is generally only used by
a restricted set of disciplines. ORT (orthopedics) serves some of its
patients in kern C and the rest of it in other facilities.
Inpatient department [8]
Kerns are fairly similar: kerns A-D contain 4 ORs each and kern E
contains 6 ORs. In a kern there is a floor in the middle and in the
center there is a room with a computer. It also contains small
elevators connecting it with the sterilization department. The floor
itself contains the facilities for washing hands.
Layout of a kern [8]. Each corner is occupied by one
of the four ORs.
11
3. Improvement strategies
3.1. Simulation results
We use DT driven scheduling decisions both to test one-step and
two-step scheduling strategies. The current practice at Gasthuisberg
is to schedule patients in one step, which means that patients are
scheduled directly to slots. Using the DT we will test four different
one-step scheduling strategies. Following that section, we will
introduce the two-step scheduling system and, in that context, test
three different basic scheduling mechanisms.
3.1.1. One-step strategy
In this section we test four different one-step assignment policies.
The first policy is FCFS which does not differentiate between
patients. The other three policies are based on the DT. We will
show that FCFS outperforms the three DT based policies.
At Gasthuisberg the planner (typically the surgeon) schedules
patients in one step. Finding an appropriate slot happens manually,
that is, the planner finds a slot without algorithmic support. There
are several criteria the planner can and often will consider [2].
Firstly, the planner will only consider slots that have enough free
capacity to accommodate the new surgery. Unlike for example in
the Erasmus Medical enter in the Netherlands where slack time is
used to ensure that the probability of overtime is 30% [5], in
12
Gasthuisberg the slots can be fully utilized. Therefore, the sum of
expected surgery durations assigned to a slot can sum up to the
total capacity of the slot (9 hours). As surgery durations are highly
variable, ORs will regularly run into overtime.
Slots can also be selected based on the preferences of surgeons.
One such preference is to have only one difficult surgery (e.g., hip
replacement) in a slot. Similarly, also the number of children can be
restricted. This is done as patients before their surgery are not
allowed to eat for a certain amount of time, which is more difficult
to do for children. It is therefore best to serve one child in a slot first
in the morning.
Figure 5: Different one-step policies result in
remarkably simlar OR related performanc measures.
The only exception is undertime, which using FCFS is
lower than for all other policies. Performances are in
percentages, for example, an averege of 1 hour
overtime for a 9 hour slot equals 11.1 % (1/9).
One of the most important criteria driving the patient scheduling
process is the patient’s urgency status. The urgency status
translates to the DT. In case the surgery is not urgent, patients can
request and propose surgery dates that would suit them the best,
e.g., when they have holiday.
The first policy we test is FCFS which assigns surgeries to the
earliest slot that has enough free capacity left. A slot has enough
free capacity to accommodate an additional surgery if adding the
surgery does not yield the slot overbooked, that is, the sum of the
estimated surgery durations of all patients served in the slot remains
less than the slot’s total capacity.
The other three assignment policies are based on the DT. The idea
behind the policies is to postpone less urgent surgeries, thereby
creating short-term buffer capacity usable by more urgent patients.
The first policy assigns patients into the closer end of the patients
DT interval. This is similar to the FCFS strategy with the restriction
that patients can only be served after a certain reference period.
The second policy pushes patients into the end of their DT, that is,
the patients are served as late as possible while still within their DT.
If there is no such slot, then a date after the patients’ DT is chosen,
in which case the patient is served late. The third policy schedules
patients as close to the middle of the DT as possible, that is, it
minimizes the temporal distance between the selected slot and (DT
end – DT start) / 2.
13
As Figure 6 shows, the four policies largely affect patient waiting
time. Using FCFS (most left figure) the average waiting time will
generally be low for patients from all DT categories. The figure also
shows that when applying any DT based policy many patients will
experience a different waiting time than what was targeted by the
policy, i.e., some dots are between the layers. More results can be
found on www.econ.kuleuven.be/healthcare/seminar_1.
Figure 6: Waiting time of neurosurgery patients. Each data point is a patient where the color represents the patient’s DT. First axis:
day the patient arrived. Second axis: the amount of time the patient waited to get surgery. The colored horizontal lines denote time
limits of DT categories 4 to 7. The most right figure shows the historical data for neurosurgery patients from the year 2012.
14
The results on the left suggest that FCFS outperforms all three DT
based policies. The least effective strategy is to schedule patients
towards the end of their DT. Interesting to note is the fact that the
shape of the distribution of the center policy shows the greatest
resemblance to the one experienced in reality (Figure 1).
The different policies were compared using both patient and OR
related performance criteria. Patient related performances are, for
example, expected patient waiting time and percentage of patients
being scheduled within the DT. Measures were taken jointly for all
disciplines together (in order to show how the OR complex as a unit
performs), but also for each discipline separately (in order to confirm
that the results are valid for all disciplines). OR related performance
measures are: utilization, overtime and undertime (Figure 5).
FCFS also benefits DT category 4 patients. One might think that
FCFS performs well because less urgent patients profit on cost of
more urgent ones. This would make sense as FCFS assigns
patients to slots regardless of their DT. Figure 49 (Appendix) shows
that this is not the case and that FCFS benefits all DT categories.
It is surprising that FCFS outperforms the three DT based policies.
The main reason is that the DT based polices fail their goal to
provide the necessary buffer capacity for DT 4 (and to some extent
DT 5). This can be seen in Figure 6. In the figure the large peaks
are the bottlenecks, that is, temporal capacity shortages leading to
excessive patient waiting times. As can be seen from the figure, in
case there is a bottleneck using FCFS, it will also appear in any of
the other policies. Difficult to spot on the images is that the DT
based policies shift the appearance of the bottlenecks some weeks
into the future. To summarize, DT based polices will on the one
hand occasionally waste OR capacity as patients are postponed.
On the other hand, DT based polices do not provide the necessary
buffer capacity required by urgent DT categories. The results do not
suggest that the DT cannot be used to guide patient scheduling
decisions, but it is probably counterproductive to use them in the
rigid way we define them. In the next section, we will incorporate
the DT into the scheduling process in a more flexible way.
15
3.1.2. Two-step strategy
In Gasthuisberg, as in most hospitals, patients are assigned to slots
directly. This means that a patient will be directly planned for
surgery to an OR and a date. At Gasthuisberg, they intend to switch
from this one-step procedure to a two-step procedure. Instead of
assigning patients directly to slots, they will be assigned to a week
first. This means that a second step is required, where for all the
patients assigned to a given week a suitable OR and weekday is
found. The advantage of the two-step procedure comes from the
fact that the second part of the procedure, the within week
scheduling, can be done just before the start of the week (e.g.,
Thursday or Friday before the scheduled week). In other words, a
large part of scheduling related decisions can be postponed to the
second step.
We use a simulation model to test the two-step procedure, focusing
primarily on the effect it has on the amount of patients scheduled
after their respective DT, i.e., too late. Our results suggest that in
case of the two-step procedure it is very important to allow higher
urgency patients to break into the already fixed weekly schedules.
Additionally, it is important that the second step, the within-week
scheduling, is guided by the patient’s DT category. Interestingly, we
found that reserving a constant amount of capacity for high urgency
patients is from a whole system perspective less beneficial.
We tested three mechanisms: push, protection levels and DT driven
within-week scheduling.
Push allows patients to be assigned to a slot that is in
the same week as they arrived.
Using the push mechanism, for example, a patient that arrives
Wednesday can be directly assigned to a slot on Thursday or Friday
in the current week. Using the two-step scheduling strategy this
would normally not be possible as the first step of the procedure,
the to-week step, requires patients to be assigned to a complete
week. Earliest such week follows the week in that the patient
arrived, i.e., is registered for surgery.
16
If patients are pushed into the current week, then they skip both
steps of the scheduling procedure. In reality, if the health condition
of the patient allows, it’s better to give them some time to prepare
for their surgery. The ‘Push (4, 5)’ mechanism excludes DT 6, 7 and
8 patients from the push mechanism as it only allows to push urgent
DT categories 4 and 5.
Protection Levels are used to reserve a certain amount
of weekly capacity for each DT category.
A general patient planning related problem is that the exact time
when a future urgent patient arrives is unknown in advance.
Consequently, there is a problem if a larger number of DT 4 and 5
patients arrive while all ORs are fully booked for the future days or
weeks. In order to mitigate this problem, some OR capacity is
reserved for urgent patients, i.e., protected from less urgent ones.
Protection levels are not defined for DT category 8 as it contains
the least urgent patients.
The amount of weekly capacity protected for each DT category
corresponds to the amount of weekly capacity needed for that DT
category. This is the multiplication of the expected number of weekly
patients (Figure 2) and their estimated average surgery duration.
Protection levels are not defined for non-electives, but as will be
shown in the next section, are implicitly accounted for by the MSS.
Learning from the results in the previous section, we decided to
generally avoid postponing surgeries. We assume therefore that the
results that were valid for the one-step strategy are to some extent
also valid for the two-step strategy. Consequently, we keep
ourselves mostly to the FCFS ideology. Therefore patients are only
postponed if we have reason to believe that there is a shortage of
OR capacity which is expected to affect more urgent patients.
Patient can only be scheduled for a certain week if the
weeks usable OR capacity for the patient’s DT
category allows it. This is the case if the free OR
capacity minus the sum of protected capacities for
more urgent DT categories is larger than the surgery’s
estimated duration.
Protection levels protect some amount of capacity for each DT
class. Consequently, a patient is only postponed if the expected
capacity requirements for a more urgent patient classes make that
necessary. Protection levels are nested and therefore capacity
reserved for a given DT category can always be used by a more
urgent DT category (e.g., capacity reserved for patients of DT
category 6 can always be used by a DT 4 or 5 patient. No capacity
is protected for DT category 8.
17
We define protection levels in two ways: FCFS (PL basic) and FCFS
(PL DT). Using the FCFS (PL basic) approach scheduled patients
use capacity that is reserved for their DT category. Only if this
capacity is insufficient, then unprotected capacity or, in case this is
still not enough, capacity protected for less urgent DT categories is
used.
The FCFS (PL DT) approach differs from the FCFS (PL basic)
approach in that patients can even if there is capacity protected for
their DT category use unprotected capacity or capacity allocated to
less urgent DT categories. This happens in case they are scheduled
to a date that is later than their DT. In other words, capacity
protected for a DT category can only be claimed by patients that
are scheduled within their DT. Consequently the FCFS (PL DT)
strategy is more restrictive to low urgency patients than the FCFS
(PL basic) strategy.
Wfit creates a balanced within week schedule.
wfit
Lastly, we also tried to get a preliminary understanding of the
importance of the within week scheduling step. We tested two
heuristics: ‘wfit’ and ‘wfit (DT)’. The former evens out slot occupancy
whereas the latter additionally considers the DT of patients.
The name wfit is an abbreviation of the phrase worst fit from the
memory management literature and is the opposite of best-fit. The
aim of the heuristic is to create a schedule with slots that have a
similar utilization. Practically this translates into a strategy where a
surgery is always assigned to the slot with the most leftover
capacity. As a consequence, patients are first always assigned to
empty slots. After each slot is occupied by one patient the slot is
chosen with the most remaining capacity, that is, the one that was
previously assigned the shortest surgery. The ‘wfit’ algorithm starts
with the longest surgery and ends with the shortest. Once all
surgeries are scheduled, they are randomly permutated slot wise.
18
wfit (DT)
Figure 7: The colors represent the
last day patients may get surgery
without exceeding their DT.
An extension of ‘wfit’ is ‘wfit (DT)’ where also the DT of patients is
considered. The algorithm starts by assigning patients to one of 6
groups. The 1th group contains those patients that are already late
(dark red). The 2nd, 3rd, 4th and 5th group contain patients that have
to be served the latest by Monday, Tuesday, Wednesday and
Thursday respectively. The 6th group contains those patients who
are within their DT even if scheduled for Friday. Within each group
surgeries are processed from long to short. The algorithm first
schedules group 1. The patients in the 1th group are scheduled for
Monday slots first. A patient that is assigned to a slot is removed
from its group. At this stage, slots are not allowed to be overbooked,
that is, the sum of the estimated surgery durations assigned to a
slot has to be smaller than or equal to 9 hours. If all patients in the
1th group are scheduled or, alternatively, all Monday slots are fully
booked then the algorithm enters the second stage. In the second
stage the remaining patients from the 1th group are merged with the
patients from the 2nd group. The patients in the newly formed group
are scheduled for both Monday and Tuesday slots. As before, slots
are not allowed to be overbooked. The same procedure continues
until patients from all 6 groups are included. At this stage patients
are scheduled to days Monday until Friday. In case there are any
unscheduled patients left the slot overbooking becomes allowed and
the remaining patients are distributed using the basic ‘wfit’
algorithm.
19
Figure 8 shows how the three different mechanisms affect patient
waiting time. As the ‘Push’ mechanism schedules patients into the
week of their arrival, evidently some patients will have a very short
waiting time. This can be seen by the missing strip on the bottom
of the graphs in rows 2 and 3 of the figure. In the 3rd row the ‘Push’
mechanism is only applied to DT category 4 (yellow) and 5 (green).
It is also easy to spot the effect of protection levels (columns 2, 3
and 5, 6) which produces “layered” patient waiting times. It is more
difficult to eyeball the effect of DT driven within scheduling. What
would be seen on a higher resolution image is that for each line
representing a DT category, the points with corresponding color
would show a higher density under than above the line.
wfit
No PL
PL DT
wfit (DT)
PL basic
No PL
PL DT
PL basic
No Push
Push all
Push (4, 5)
Figure 8: Neurosurgery. First axis: patient arrival, second axis: patient waiting time. Color of the dots (patients) and lines matches DT.
20
The results suggest that in order to maximize the number of patients
served within their DT it is essential to use both ‘Push’ and ‘wfit
(DT)’ mechanisms while it does not bring any benefit to use
protection levels. The results were obtained using a full factorial
design, therefore all 18 possible combinations of different
mechanisms were tested. Detailed results can be found on
www.econ.kuleuven.be/healthcare/seminar_1.
Figure 9
The results for the one-step and two-step methods can
be compared against each other as the same
environment (e.g., same seed) is used in the simulation
model. In the one-step strategy the estimated surgery
duration of patients assigned to a slot needs to be
smaller than the slot capacity, i.e., patients are not
planned into overtime. Therefore, if the capacities
provided by the MSS are too little, the system might
turn unstable, i.e., patient waiting times start to
continuously increase. It is easy to see that this
phenomenon is less a problem using the two-step
procedure where patients have to fit the cumulative
weekly capacity of slots and where in the within-week
scheduling step planning into overtime is admissible.
Testing how the system reacts if capacities on the
supply side are decreased or, equally, on the demand
side increased is therefore only possible using the twostep method.
It is surprising that applying protection levels does not bring any real
benefits (Figure 9). This can be seen as, one the one hand, the
average improvement is not significant (below 1%) and, on the other
hand, as Figure 9 shows, batch means (dots) are largely
overlapping. Positive is that at least patients from DT category 4
seem at least minimally to profit from protection levels, though batch
means are also this time strongly overlapping (Appendix, Figure 51).
This does not mean that protection levels as a mechanism are
inadequate, but it merely points to the fact that they might need to
be used differently (e.g., time dynamic protection levels, see
Chapter 7).
Contrary to protection levels, large improvements can be achieved
using DT driven within week scheduling. In Figure 9 right graph,
‘wfit’ scenarios are consistently outperformed by ‘wfit (DT)’
scenarios (triples above them). The improvements can mostly be
attributed to better scheduled DT 4 and DT 5 patients (Appendix,
Figure 51).
Besides DT driven within week scheduling it is also important to
apply the ‘Push’ mechanism. This is shown by Figure 9 where the
12 upper scenarios using ‘Push’ outperform the respective lower 6
scenarios. Interestingly, the performance gains can almost entirely
be realized by using ‘Push (4, 5)’ and therefore only applying the
mechanism to patients with DT 4 and 5. Using ‘Push (4, 5)’,
additionally, the number of patients who are pushed into the
schedule will reduce from a daily average of 15.6 to 5.4 (Appendix,
Figure 52).
21
3.2. The Gasthuisberg setting
The people at Gasthuisberg we cooperate with are:
Frank Rademakers
Nancy Vansteenkiste
Pierre Luysmans
Christian Lamote
Philip Monnens
Jo Vandersmissen
Guido De Voldere
This section contains our understanding of the patient scheduling
process at Gasthuisberg. The information to build up this
understanding comes from two sources. Firstly, from a very detailed
data set covering patient and scheduling related information.
Secondly, from the experts at Gasthuisberg who explained to us
those aspects of the scheduling mechanism that are not in the data.
3.2.1. Patient arrival pattern
The arrival time of a patient is defined as the time when the surgeon
in charge makes the decision that a patient needs surgery. For
elective patients this will usually happen during consultation, i.e.,
working hours on weekdays. For non-electives it can happen at any
hour of the day.
We tried two approaches to understand the patient arrival process.
The first approach involves the statistical analysis of inter-arrival
times, i.e., the mean length and the variance of time intervals
measured between two consecutive arrivals. In the second
approach we are only interested in the expected number of arrivals
for each weekday.
In order to model inter-arrival times we used a Maximum Likelihood
Estimator (MLE) and fitted different distributions to the data. We
found that the exponential distribution generally fits arrival patterns
better than other distributions. Nevertheless as Figure 10 (red
continuous line) shows, the results are not satisfactory. One way to
improve the quality of the fit is to remove outliers from the data set.
Outliers are very large values, i.e., very long inter-arrival times. The
reason why some of the inter-arrival times are longer than what you
would find in reality has its roots in the property of the data set. The
data covers only those arrivals from 2010 to 2012 that received
22
surgery in the year 2012. The patients that received surgery before
2012 or after 2012 are excluded. In other words, our data according
to surgery dates covers a 1 year period while the data according to
arrival times covers a 2-3 year period. As a result, inter-arrival times
will tend to be longer and especially at the end part of the spectrum
we will find values that are unrealistically large. Those entries should
ideally be identified and removed. Since it is impossible to identify
the outliers, we chose to remove the largest 5% (red dashed line).
Figure 10: The histogram is based on 3 hour bins where each column shows how often patients arrive within 3, 6, 9 …60 hours. In
all elective graphs there is a peak at around 24 hours, representing those patients that arrived on consecutive days. This peak is one
reason why it is difficult to achieve a good fit with any standard probability distribution function.
23
As we chose to use an exponential distribution, we can make use
of the fact that its parameter can easily be calculated directly
(inverse of the mean). Calculating the parameter directly, we are
still left with the problem of outliers. The quality of fit improves but
is for electives still not satisfactory.
As non-elective patients are generally served quickly, the mismatch
between the length of the interval covered by surgery dates and the
length of the interval covered by arrival dates is much smaller than
for electives. Consequently, non-elective inter-arrival times will not
contain outliers.
The inter-arrival time of non-electives can be modeled
with an exponential distribution. The parameter of the
distribution can either be approximated with a MLE or
calculated directly. In case a MLE is used, it is
advisable to remove outliers from the dataset first.
An alternative way to understand patient arrival patterns are arrival
rates. Using arrival rates, only the total number of patients arriving
on a given weekday (Mon-Sun) is important whereas the exact
arrival hour is not. As elective patients are registered during
consulting sessions, it is not a surprise that most patients arrive on
weekdays (Figure 11). A benefit of using arrival rates in comparison
to inter-arrival times is that there are no outliers, i.e., the mismatch
between the intervals covered by the surgery dates and by arrival
dates does not influence the collected statistics.
Customer arrival rates are generally modeled using a Poisson
distribution, that is, it is assumed that the mean arrival rate equals
the variance. Investigating the patient arrival rate for different
weekdays, we noticed that for most disciplines the mean arrival rate
is significantly lower than the variance and thus they are not equal
(Figure 12). As the arrival rate does not follow a Poisson distribution,
it can be modeled, for example, by a discrete distribution.
So far we analyzed the arrival patterns for each discipline
separately, but we did not distinguish between urgency categories.
Unfortunately, this is difficult to do. As Figure 2 shows, the DT
categories are for most disciplines not well balanced, i.e., at least
some of the DT categories will have a very few patients.
Consequently, the sample size for those discipline - DT category
pairs will be small. This is even the case for ABD which is the
discipline with the largest total sample size and with a fairly well
balanced distribution of DT categories (Appendix, Figure 53).
24
Figure 11: The height of the column represents the amount of patients expected to arrive on that given weekday. For example, twice
as many gynecology (GYN) patients arrive on a usual Monday than on a Wednesday.
Figure 12: Each column represents one day of the year 2012 (52 columns for each weekday). The height of the columns
represents the number of ABD arrivals for the given weekday. The average arrival rate is the highest for Wednesday (9)
but the maximum number of patients arrived on a Friday (21). On the graph there are some zero entries which means
that there was no consulting on that day, e.g., because it was a holiday. The statistics for other disciplines can be found
on www.econ.kuleuven.be/healthcare/seminar_1.
25
3.2.2. Master Surgery Schedule
Week 1 / Week 2
Room MON TUE
WED
THU
FRI
Kern A
A1
A2
A3
A4
URO
NKO
GYN
GYN
URO
NKO
GYN
GYN
URO
NKO
GYN
URO
URO
NKO
GYN
URO
URO
NKO
GYN
NKO/GYN
Kern B
B1
B2
B3
B4
ABD
ABD
Tx
ONC
ABD
ABD
Tx
Tx/ABD
ONC
ABD
Tx
ONC
ABD
ABD
Tx
ABD
ABD
ABD
Tx
ABD
Kern C
C1
C2
C3
C4
THO
TRH
TRH
TRH
THO
THO
TRH
ORT/[]
THO
THO
TRH
TRH
THO
THO
TRH
[]/ORT
THO
TRH
TRH
ONC
NCH
RHK
MKA
NCH
NCH
RHK
RHK/MKA
NCH
RHK
MKA/[]
NCH
NCH
RHK
CAH
CAH
CAH
CAH
VAT
VAT
CAH
CAH
VAT
CAH
CAH
VAT
VAT
CAH
CAH
VAT
VAT
CAH
CAH
Kern D
D1
NCH
D2
RHK
D3
D4
RHK
Kern E
E1
E2
E3
E4
E5
E6
VAT
CAH
CAH
CAH
Table 2: MSS used at Gasthuisberg.
NCH
An easy but admittedly oversimplified way of structuring the OR
planning process is to distinguish between the following three levels:
strategic, tactical and operational. At the strategic level, a certain
amount of OR capacity is allocated to each discipline (e.g., urology).
This is the same as creating the case mix as the hospital effectively
decides on the number of future patients it wants to serve from each
discipline. On the tactical level, the MSS is created (Table 2), i.e.,
a 1 or 2 week cyclic plan is constructed that allocates to each OR
slot (weekday and OR) a corresponding discipline. On the
operational level, patients are assigned to OR slots. A more detailed
schema is described in [3] and in Chapter 5.
The basic OR to discipline allocation schema is defined by the MSS
template (Table 2). In reality, smaller changes can occur to the
weekly MSS. This is, even though often on a short notice, known in
advance and can happen if a given discipline either has not enough
patients to fill up a slot or if a needed surgeon is unavailable (e.g.
goes to a conference). Consequently, a slot can remain empty or it
can be occupied by a discipline other than predefined in the
template. The MSS including those prearranged changes forms the
planned MSS.
It can happen that some of the prearranged changes are not logged.
Unlogged changes are difficult to track. One way of doing it is to
label slots based on the discipline of the first elective patient that is
scheduled and served. This way, each slot that is open can be
associated to a discipline. A slot is regarded to be open if any
elective patient gets surgery within the slot time (7:45-16:45), i.e.,
starts or ends within slot time or started before and ended after the
slot. This relabeled MSS forms the realized MSS. The weekly
capacities allocated to disciplines by the MSS template, the planned
MSS and the realized MSS are visualized in Figure 13.
Interesting to note is the fact that for most disciplines the mismatch
between the MSS template and the planned MSS is only minimal
(Figure 13). There are some exceptions. One such discipline is
26
Patients need less OR capacity if estimated surgery
durations are taken as basis. More details on surgery
durations can be found in the next section.
thoracic surgery (THO) where relative to the template +7.2 hours
are planned but -7.6 hours seems to get realized. Other disciplines
where less is used than planned are: transplantation (TX),
cardiology (CAH) and vascular surgery (VAT). The transplantation
OR is generally regarded to be the OR that is used to accommodate
non-electives, thus it is no surprise that it is often empty or actually
occupied by non-electives. For cardiology another phenomenon
plays a role: the length of surgeries is usually long and difficult to
predict, thus some extra buffer is allocated to them.
From Figure 13 it seems that both the template and the planned
MSS is over-capacitated and therefore, at least in theory, less OR
capacity would be enough to serve the patients. In other words, the
hospital plans in some amount of buffer capacity. There are two
reasons for this. Firstly, it is used to compensate for uncertainty in
surgery durations. This is especially important in a setting where
surgery durations are systematically underestimated. A second
reason are the emergency patients [1]: more details on this topic
can be found in Section 3.2.4.
Figure 13: The amount of weekly OR capacity required by patients is shown in color. The MSS template, the planned MSS and the
realized MSS are usually similar.
27
3.2.3. Surgery duration
As Figure 13 shows, surgery durations are generally
underestimated. A more detail picture of the link between estimated
surgery duration, realized surgery durations and the DT of patients
will be given in this chapter.
Realized and estimated surgery durations are dependent variables.
The surgery duration of a patient represents the time between the
moments the patient is rolled into the OR and the time when the
patient leaves the OR. Cleaning time is not included whereas setup
time might or might not be included, depending on whether the
patient is already present in the OR. Normally patient specific setup
time is included. The estimated surgery duration is suggested to the
planner (e.g., surgeon) based on the mean of realized surgery
durations of previous similar OR sessions. The planner can accept
this value or overrule it.
Surgery durations are for most disciplines
systematically underestimated (Figure 15), that is,
estimated surgery durations are generally shorter than
realized surgery durations. If for a surgery the estimate
would turn out to be correct, in the figure the surgery
dot would be on the horizontal black line. On the figure
we see that most of the dots are above that black line,
which means that surgery durations generally take
longer than previously estimated. This happens
because surgeons try to plan as many patients into
their own slots as they can without exceeding the
capacity of their slots. Underestimating patients’
surgery durations gives them a tool to legally overfill
their slots. The same phenomenon can be seen in
Figure 13. Disciplines that only slightly underestimate
their surgery durations are head and neck surgery
(NKO), vascular surgery (VAT) and cardiology (CAH).
We assume that surgery durations are independent of the DT. More
generally, we assume that patients from different DT categories only
differ in their urgency status and the rest of their attributes are
statistically the same. This is a strong assumption but a necessary
one. In Figure 2 we see that for most disciplines the mix of DT
categories is unequal. Therefore, if we divide patients according to
their DT categories, our sample size would become for some of the
DT categories very small.
In Figure 14 we see an indication that the assumption that surgery
durations are independent of their DT categories is mostly true. The
same can be seen for neurosurgery in Figure 16, where the
marginal distributions of surgery durations are similar for all DT
categories. One of the exceptions is abdominal surgery (Figure 17),
where we see that DT 4 patients (in yellow) have a significantly
lower mean surgery duration than patients of other DT categories.
28
Figure 14: The surgery durations are independent of the DT. Notable exceptions are ABD (DT 4 are shorter), THO (DT 4 are shorter)
and CAH (DT 6 are longer).
Figure 15: Surgery durations are systematically underestimated (many of the dots are above the black line).
29
Figure 16: The distribution of both estimated and realized surgery durations follows a bell curve. For some DT categories the shape
takes the form of a multimodal distribution (distinct peaks), as interventions of the same type usually take similarly long. Results for
other disciplines can be found on www.econ.kuleuven.be/healthcare/seminar_1.
Figure 17: Estimated surgeries durations are suggested to the planner based on similar previous OR sessions. As the figure shows,
estimated surgeries are usually a multiple of 30 minutes (stripes in the figure). The figure also shows that DT 4 (yellow) abdominal
surgeries (ABD) are generally shorter.
30
3.2.4. Non-elective allocation schema
Every week over 60 non-electives require surgery at Gasthuisberg.
In this section, we will show how those non-electives are allocated
to ORs.
It is more likely that a non-elective patient arrives during slot time
than during the weekend or the night (from 22:00 – 06:00). Slot time
in a broad sense is the time period when OR slots can theoretically
be open, that is, weekdays from 06:00 to 22:00. The night is defined
as the time interval between 22:00-06:00. If non-elective patients
were to arrive with the same probability around the clock, then
theoretically 2/7 (~29%) of all non-elective patients should arrive on
the weekend. For most disciplines, we see that this number is in
reality significantly lower (Figure 18, red) with exceptions: for
thoracic surgery (THO), traumatology (TRH) and Oral and
maxillofacial surgery (MKA). The fact that fewer non-electives arrive
on weekends can also be seen in Figure 11, last image. The same
observation is true for night arrivals. Theoretically, 5/7 * 8/24 (~24%)
of patients should arrive in a week night whereas in practice this
value is significantly lower (Figure 18, blue).
Non-electives are during slot time mostly assigned to an OR which
at the moment of their arrival is allocated to their discipline. For
example, a patient with a heart attack is probable to be assigned to
an OR that at the moment the patients arrives is serving cardiology.
From Figure 18 we see that the only exception to this rule is
oncology which often serves its non-electives in the abdominal or
the transplantation OR. One might think that this will only apply to
non-electives that are not very urgent (DT 2 and 3) and that the
most urgent non-electives (DT 1) are served immediately and
therefore enter the first OR that becomes free. This is indeed true
for MKA and to some extent for ONC, but for all other disciplines
not (Appendix, Figure 55).
31
As a significant part of non-electives are allocated to ORs used by
the elective program, it becomes necessary to recalcuate the MSS
capacity calculations found in Figure 13. The result is shown in
Figure 19. The calculations were done the following way: nonelective cardiology patients need 16.2 hours of OR time a week.
From Figure 18 we know that only 67% will interfere with the elective
cardiology program, i.e., arrive during slot hours and be served in
an OR reserved for cardiology. As a result, in Figure 19 the red bars
(DT 1-3) associated to cardiology will represent a length of 10.9
hours (0.67 * 16.2). Non-electives entering an OR other than their
own discipline are assumed to be random and not included into the
capacity calculations. This is an assumption valid for all ORs except
B3 that despite being assigned to transplantation surgery (TX) is
generally regarded to be the “emergency OR”.
32
Figure 18: Red columns are generally lower than the red line which means that less non-electives arrive on a day of the weekend than
on a day during the week. The same logic applies to night arrivals (in blue).
Figure 19: Where the lower column is larger than the capacity of the planned MSS are disciplines that are physically unable to serve
all their patients in their planned slot time. Therefore, even if scheduled optimally, they will run into overtime. At Gasthuisberg the
impact of overtime is reduced by the funneling mechanism (Section 3.2.5).
33
3.2.5. Funnel
Not all ORs are planned to close at the same time but depending
on the hour of the day some can remain open for longer: 8 (18:00
- 19:00), 4 (19:00 - 20:00) and 2 (20:00 – next morning). This means
that gradually more and more ORs are closing towards the evening,
therefrom the analogy of a “funnel” (Figure 20). This funneling
mechanism helps the hospital to reduce overtime costs that would
otherwise certainly occur (Figure 19 - some disciplines will
unavoidably run into overtime).
The amount of surgeries planned into an OR corresponds to 9 hours
(7:45-16:45). An exception to this rule is cardiology, where for each
day 1 or 2 surgeries are planned. As can be seen from Figure 15,
cardiology surgeries tend to be long, therefore planning 2 cardiology
surgeries into one OR almost certainly means that the OR goes into
overtime (cardiology ORs have in average 1.2h overtime).
One might wonder how it is decided what ORs to leave open, thus
how the funnel is formed. First of all, ORs that finish in time do not
join the funnel. It is the ORs that experience some sort of
complication with one or more of their surgeries that might go into
overtime and thus join the funnel. An OR can always be stopped
from going into overtime by rescheduling its last surgery (cancel or
reassign the surgery to another OR). To make those rescheduling
decisions is difficult as it requires to balance between many
competing criteria such as: patient satisfaction (not too many
patients can be canceled), surgeon’s preference and OR staff
requirements (nurses and anesthesiologist). At Gasthuisberg the
rescheduling decisions are made by the head anesthesiologist and
the head nurse. The anesthesiologist is responsible for decisions
that affect the OR complex for the next 24 hours (OR
reassignments) and is therefore in charge of the funnel, whereas
the head nurse makes decisions that affect the OR complex on the
longer run (cancelations).
One might get the impression that non-electives are served as last
ones (Figure 21), i.e., after the elective schedule. This can be
34
explained, on the one hand, by the fact that the number of ORs that
can accommodate them is limited (Figure 18), i.e., it is unlikely that
an OR just turns free. On the other hand, surgeons like to make
sure that all of their electives scheduled for the day get served. In
case they leave a non-elective patient as the last one of the day
their schedule is “safe” as the head anesthesiologist will not force
them to cancel that last non-elective case.
If it would be the case that non-electives are scheduled last on a
day (Figure 21), one might wonder whether there is a difference
between DT 2 and 3, i.e., non-electives who have to be served
within 6 hours and within 24 hours. As shown by Figure 22, there
is a difference between the two: for DT 2 patients there is a 44.5%
chance to be served within 3 hours of arrival, whereas for DT 3
patients this is only an 18.1% chance. For both DT 2 and 3 patients
it is true that they rarely enter an OR immediately (within half an
hour). By 48 hours (Figure 23, right) we see that most patients get
served for all three DT categories. Non-electives that arrive during
the weekend or on a holiday are excluded in Figure 22 and Figure
23, but are included in the Appendix in Figure 56 and Figure 57.
DT categories 1-3 largely determine a patient’s expected waiting
time in the first 6 hours (Figure 23, left). As the graph shows, the
longer patients wait, the less it matters whether they are assigned
to DT 1 or DT 2. By 6 hours, the chance of being served is 85% for
DT 1, 76% for DT 2 and only 40.2% for DT 3 patients.
35
Figure 20 (day 205, 2012): The number of ORs that can remain open after 6:45 depends on the funnel (dashed line). The number of
open ORs relates to nurse requirements: after 20:00, all surgeries could have be done in two ORs (E1, C3) despite the fact that
physically five ORs were open (E1, C3, B2, D2 and D3). This means that only 2 OR teams were available. The staffing requirements
for anesthesiologists follows a similar but slightly shifted funnel. The two ORs are open after 20:00 on weekdays and during the whole
weekend mainly to serve non-electives.
Figure 21 (day 332, 2012): All schedules for the year 2012 can be found on www.econ.kuleuven.be/healthcare/seminar_1.
36
Figure 22: Non-elective waiting time on weekdays. On the left side each bin is of size 30 min, e.g., within 30 minutes 19.8%, whereas
after 30 minutes but before 60 minutes around 22% of DT 1 patients are served. On the right side the bin size is changed to 3 hours.
Figure 23: Cumulative function of non-elective waiting time on weekdays. The bin size is 30 min on the left and 3 hours on the right.
For example, for DT 1, within 30 min 19.8%, within 60 min 42% ... within 6 hours 85% of patients are served. Interesting to note is the
fact that after 12 hours the probability of being served is higher for DT 2 than for DT 1.
37
3.2.6. Rescheduling
Rescheduling is used to decrease the load on ORs that otherwise
would go into overtime. Generally, in order to serve all the planned
surgeries, the available capacity of an OR is enough. There are two
major reasons why the capacity of an OR might turn out to be
insufficient. Firstly, if during a surgery a complication occurs then
the surgery duration can be larger than planned (Figure 15).
Secondly, non-elective arrivals might interfere with the already fixed
daily schedule (Figure 21).
We distinguish between two types of rescheduling actions: OR
reassignment and canceling. If a surgery is OR reassigned then the
surgery will still be performed on the same day as planned and only
the OR is changed (Figure 24). In case a surgery is canceled then
the surgery is scheduled to the next slot available for the discipline
even in case that slot is already fully booked.
It is always preferred to perform a surgery on the planned surgery
day and therefore cancelation is only used if there are no other
options left. This is the case as it can become frustrating for a
patient to physically and mentally prepare for a surgery that in the
end is not carried out. Consequently, cancelations are not
happening often: On average 1.4 surgeries are canceled a day while
on the other hand 5.5 surgeries are OR reassigned a day. Examples
of both OR reassignments (in blue) and cancelations (in black) are
shown in Figure 20 and Figure 21.
As Figure 25 shows, rescheduling activities are mostly concentrated
in the morning and in the afternoon hours. Interestingly, the shapes
of both the histograms of OR reassignments and canceling are
similar with the difference being that the latter is shifted by 1-2 hours
to the right, i.e., canceling decisions are generally made later in the
day. Unexpectedly there is increased rescheduling activity
observable in the early morning. This can be explained by the fact
that sometimes complete ORs are switched, which in the system is
registered as many individual OR reassignments.
38
Figure 24: OR reassignment schema. (1) It is preferred to OR reassign a patient to a slot with the same discipline. (2) Alternatively a
surgery can be OR reassigned to another discipline within the kern. (3) Least favorable is to move a surgery within kern A, B, C and
D. CAH and NCH can only be OR reassigned to slots of their own respective disciplines.
Figure 25: The bin size represents the amount of rescheduling activity that is expected for the given hour of day. The figure shows, for
example, that most OR reassignments happen between 13:00 and 14:00.
39
3.3. Model
Within the simulation model, we implemented most of the patient
and hospital specific mechanisms described in the previous section.
In this section we will introduce three of the mechanisms in more
detail. The first one relates to the relationship between real and
estimated surgery durations. The second one relates to the way
how rescheduling was modeled while the third one describes the
way how the MSS is constructed and used in the simulation model.
3.3.1. Rescheduling model
As described in Section 3.2.5, OR reassignment and canceling is
used to decrease the load of ORs that otherwise would go into
overtime. A surgery that is canceled is assigned to the slot closest
in date. If on the closest date several slots of a discipline are
available, the one with the lowest occupancy is chosen. A surgery
that is canceled once is in the replanned OR always served first,
ensuring that it is not canceled again. Neurosurgery cancelations
are modeled slightly differently as in their case it is essential that
the surgery is performed by the same surgeon. The surgeon’s next
slot is one week later on the same day.
Rescheduling actions are based on the estimated closing time of
ORs. This estimate is the sum of two components: firstly, the sum
of the estimated surgery durations still waiting and, secondly, the
amount of time the current surgery is still expected to need. This
last component is regarded to be zero in case the surgery already
takes longer than its estimate. This is not a correct estimate, but is
likely to be close to the value that is used in reality. The correct
estimate would be the expected value of the conditional probability
distribution of the surgery length given the amount of time the
patient is already in the OR.
40
Figure 26: The expected closing time of Slot ‘A’ can
fall into any of the three intervals: Green – OR closes
in time, nothing has to be done; Yellow – OR goes
overtime, if possible move last surgery, else keep it;
Red – OR goes heavily into overtime, if possible move
last surgery, else cancel it.
The search procedure for an OR that can accept a
moved surgery follows the hierarchy explained in
Figure 24. An OR can only accept a moved surgery if,
including the new surgery, the OR is still estimated to
close before 16:45. Cancelation becomes an option for
those ORs that are estimated to close after 18:00, but
also in this case first it is always attempted to reassign
its last surgery to another OR.
As shown by Figure 28, rescheduling decisions are made
continuously from the morning until the evening. Consequently,
instead of rescheduling patients on one certain hour of the day, in
our model we allow rescheduling interventions to happen on an
hourly basis starting from 9:00 to 19:00. At each intervention point,
we collect the ORs that are believed to run into overtime. For those
ORs, it is attempted to move their last surgery to another less
occupied OR.
There is one problem with the method explained in Figure 26,
namely, that if implemented, a disproportionally large number of
surgeries will be rescheduled already early in the morning. This
happens in case the first surgery is taking longer than expected.
In reality, whether we believe that an OR goes overtime or not will
depend on the degree of trust we put into our estimate. The later
we are in the day, the more surgeries have been realized and
consequently the better becomes our estimate. This relationship
between the current time and the estimated OR closing time is
captured by the formulas in Figure 27. In the formulas, the degree
of trust we have in our estimates is modeled as a linear function of
time (in blue). Put differently, we normalize the time of the day, that
is, we map the time of the day onto the unit interval. Similarly we
also normalize the estimated OR closing time. Estimates can be
normalized in the two ways represented by the functions in yellow
and red. The yellow function is used to check whether an OR fulfils
the criteria to move its last surgery to another OR, that is, the OR
is believed to go into overtime to a degree that justifies an OR
reassignment. Similarly, the red function is used to test for
cancelation. Important to note is that OR reassignments will never
happen if the estimated OR closing time is before 16:45, as the
yellow function will be on a constant zero. For the same reason we
will never cancel a surgery from an OR that is estimated to close
before 18:00. Applying this model in the simulation yields the results
shown in Figure 28.
41
Figure 27: The last surgery of OR A is reassigned to another OR if, for example, it is 12:00 (blue function takes the value 1/3) and the
estimated closing time of OR A is 18:00 (yellow function takes the value 1). The multiplication of 1/3 and 1 results in 1/3 which is larger
than the threshold of 0.3. This would not be the case if the intervention point would be checked an hour earlier at 11:00. A similar logic
applies to cancelations. The thresholds ‘0.3’ and ‘0.5’ were chosen on a ‘try and error’ basis, trying to fit the histograms in Figure 28.
Noteworthy is the fact that the blue function reaches its maximum at 18.00 which means that after that point in time, we have full
confidence in the estimates.
Figure 28: The rescheduling statistics for Gasthuisberg are in blue and the statistics produced by our model are in green.
42
3.3.2. Copula
The connection between realized and estimated surgery durations
can be captured using a copula. Despite the fact that the theoretical
background behind copulas is complex, they turn out to be relatively
easy to use in practice [9]. The idea is the following: transform the
durations (dots in Figure 16) into the unit space (e.g., using a kernel
estimator of the cumulative distribution function). In the second step,
measure the dependence between realized and estimated surgery
durations (e.g., using maximum likelihood, estimate the linear
correlation matrix and the degrees of freedom of the copula).
Knowing the dependence between the variables it is easy to
generate an arbitrary number of pairs of realized and estimated
surgery durations in the unit space. Scaling them back into their
original space yields the final random estimated and realized
surgery duration pairs.
The surgery durations generated with the copula method have the
same pattern as the original, real data.
43
3.3.3. The applied Master Surgery Schedule
In the simulation model we use a one week MSS that merges the
two week cycle of the MSS (Table 2) used at Gasthuisberg. This
can easily be done as the two weeks are almost identical. While
creating the MSS applied in the simulation model, we were trying to
find a trade-off between the MSS template, the planned MSS and
the realized MSS.
A practical problem we encountered in our simulation model
concerns the disciplines cardiology and neurosurgery. The two
disciplines can, in case they need to OR reassign, only do this to
slots of their own disciplines. As a consequence, surgeries not fitting
the slots will often be canceled. In case this happens, it is possible
that chains of cancelations appear in the simulated schedule. This
is to be avoided as, firstly, it is unrealistic because in reality this
could be handled (e.g., by allowing more overtime or opening
additional slots) and, secondly, it yields wrong results as many
patients might be canceled to a slot that is not within their DT
anymore.
Figure 29: In the simulation
model we use a MSS that
constitutes a trade-off between
the MSS template, the planned
MSS and the realized MSS.
44
3.4. Implementation
This section contains some of the technicalities related to the
simulation model and the data processing step.
3.4.1. Simulation model
Using the understanding of the Gasthuisberg patient scheduling
setting, we needed a way to test and analyze newly developed
scheduling policies. Because of the complexity of the setting we
chose to use simulation. The simulation model was created based
on the hospital’s data. Models of mechanisms that were not covered
by the data were created on the basis of the explanations of our
contacts at Gasthuisberg.
Simulation models are often created using either custom healthcare
related simulation software or from scratch using a general purpose
language. Both methods have their benefits but also their
drawbacks. The first option is quick to implement but there are
mechanisms that are difficult to model within the software.
Additionally, it is usually a standalone software and thus it is tedious
and often impossible to properly integrate it with other
environments. The second option, using a general purpose
language, has the drawback of being time consuming to implement.
Advantages are: flexibility, speed and high integrate-ability.
We chose to create a simulation model that is based on a general
purpose language but is seamlessly integrated with a custom
discrete event simulation (DES) environment. We are using Matlab
to code routines and Simulink’s SimEvents toolbox for the DES
framework.
45
An advantage of using Matlab together with Simulink is that
components from each environment are easily integrated.
Practically, this means that we are able to call the Simulink DES
model from Matlab while within Simulink we are able to use Matlab
code.
SimEvents is used to simulate the patient service process. This
involves timing patient arrivals: for electives calling appropriate
scheduling modules and for non-electives calling the appropriate
non-elective to OR allocation schema (Figure 18 or appendix Figure
54). Also hospital mechanisms are implemented in SimEvents such
as the surgery process in the OR and patient rescheduling actions.
One of the drawbacks of SimEvents, in comparison to other DES
environments, is its rudimentary nature, making it difficult to directly
implement more complex mechanisms. This is to some extent
compensated for by the fact that different models from other
Simulink environments (e.g., state machines) or Matlab code can
be mixed into SimEvents. One of the strong sides of SimEvents is
that an entity (patient) can enter an attribute function block. The
strength of this block comes from the fact that within the block it is,
on the one hand, possible to arbitrarily change the entities’ attributes
(e.g., scheduled OR) and, on the other hand, it allows to import
object handles. This means that patient attributes can freely be
processed within the simulation model using any class function
created in Matlab. This practically means that an entity can be
processed in an arbitrary way. This results in a highly flexible and
capable environment.
Matlab is used to create patient arrivals (Section 3.2.1) with their
attributes (e.g., Section 3.2.3) and to model scheduling
mechanisms. Matlab is a general purpose language and as such
usable for many different tasks. For example, Matlab code is used
in the simulation model to achieve parts of the funneling behavior,
i.e., from 22:00 – 7:45 only 2 ORs are allowed to be open in parallel.
Matlab code is also used to process and visualize simulation results.
46
Using a simulation model, and for that matter any model, one might
wonder whether the results are correct. We will briefly elaborate on
points regarding model verification, validity and credibility.
The first point is model verification which is often defined as
“ensuring that the computer program of the computerized model and
its implementation are correct” [6]. The verification of our model
happened on several levels. For example, we visualized and
inspected the patient schedules (e.g., Figure 20), checked for all
created entities the time they were in the system and whether they
all left the system. We checked whether simplified inputs result in
the correct output, e.g., used the same value for estimated and
realized surgery durations and checked whether scheduling routines
show the intended behavior.
The second issue is model validation which is the “substantiation
that a computerized model within its domain of applicability
possesses a satisfactory range of accuracy consistent with the
intended application of the model” [7]. The intended application of
our model is only to show how different scheduling policies and
methods compare against one another. That is, we do not try to
predict the real performances of different scheduling policies in case
they are implemented by the hospital. This would not be possible
as the setting is too complex, e.g., some of the post-surgery units
cannot be modeled as they serve patients from a variety of different
facilities within the hospital but also from outside the hospital.
The third issue is model credibility which is concerned with
“developing in potential users the confidence they require in order
to use a model and in the information derived from that model” [6].
We believe that our model is credible to the people interested in our
results at Gasthuisberg. We created the model based on the data
of Gasthuisberg and through several meetings we confirmed that
we have the right understanding of the data. Also the model of
surgery scheduling related processes are both based on the
hospital’s data and on the managerial insights and expertise of the
people at the hospital.
47
3.4.2. Data processing
In order to test patient scheduling policies that are applicable to
Gasthuisberg, we needed to understand their scheduling setting.
Therefore, firstly, we analyzed the properties of the patients getting
surgery at their hospital and, secondly, examined the service
process of their ORs. In order to achieve the aforementioned tasks
we used the hospitals historical data.
The hospital provided us with patient scheduling records of the
complete year 2012 and helped us to correctly interpret those
records. The data was grouped into three datasets: Patient
trajectory, OR information and Patient planning related data. The
first data set, Patient trajectory, contains 468.599 entries and 23
attributes. The dataset contains, on the one hand, the date and time
patients were transferred to different facilities in the hospital and, on
the other hand, general information concerning the patient such as
arrival date, ID of surgeon in charge and whether the patient arrived
as an emergency. The second dataset, OR information, contains
17.310 entries and 14 attributes. The dataset includes surgery
specific information, such as the IDs of the planned surgical
interventions, estimated surgery duration and the DT of the patient.
The last dataset, Patient planning, contains 28.514 entries and 19
attributes. The data contains surgery scheduling related records,
such as the surgery’s planned date and OR. A new entry is created
in the table anytime the planner changes the scheduling information
of a patient.
Before merging the three datasets, each of them was individually
preprocessed. Preprocessing involved, for example, combining
entries in a dataset that relate to the same surgery. The datasets
were merged in a way that all the information attached to one
surgery became easily accessible and processable. Some entries
were lacking important attributes, such as the arrival data and were
therefore removed from the dataset. After removing all uncompleted
entries 15596 patients remained from 13 disciplines.
48
As surgeries can be replanned several times, the Patient planning
table will generally contain more than one entry. Patient trajectory
and OR information related datasets are merged using the attribute
‘aanwezigheidKey’. Patient planning related data is included using
attributes ‘okBonNr’ and ‘OpnamevoorstelNr’.
Further post-processing steps were made to make the dataset
easier to interpret. One of the steps includes the interpretation of
the patient planning points based on the type of rescheduling action
they represent (Figure 30).
Some planning points are regarded to be invalid. It can happen that
planners after making a planning decision for a surgery change their
mind and decide to plan the surgery differently: in this case we are
only interested in the planners’ final decision. Therefore, if there are
two entries which are made within 30 minutes then only the latter is
considered. Furthermore, as Figure 30 shows, a surgery can be
replanned before the surgery day in which case however it’s not
regarded to be an OR reassignment or cancelation. The statistics
related to the time of the day when surgeries are OR reassigned or
cancelled can be found in Section 3.2.5.
49
Figure 30: Planning decisions that were made for one patient. This patient arrived the 1 th of December at 17:03 and was planned
into OR C3. The patient was OR reassigned to E2 at 11:07 on the planned surgery date. As E2 was apparently overfilled, the
patient was canceled at 20:37 and finally served 2 days later.
50
4. References
1. Adan I, Bekkers J, Dellaert N, Jeunet J, Vissers J (2011) Improving operational effectiveness of tactical master
plans for emergency and elective patients under stochastic demand and capacitated resources. Eur J Oper Res In
Press, Corrected Proof. doi:DOI: 10.1016/j.ejor.2011.02.025
2. Cardoen B, Demeulemeester E, Belien J (2009) Optimizing a multiple objective surgical case sequencing problem.
Int J Prod Econ 119 (2):354-366. doi:10.1016/j.ijpe.2009.03.009
3. Cardoen B, Demeulemeester E, Belien J (2010) Operating room planning and scheduling: A literature review. Eur
J Oper Res 201 (3):921-932. doi:10.1016/j.ejor.2009.04.011
4. Cardoen B, Demeulemeester E, Van der Hoeven J (2010) On the use of planning models in the operating theatre:
Results of a survey in flanders. The International Journal of Health Planning and Management 25 (4):400-414.
doi:10.1002/hpm.1027
5. Hans E, Wullink G, van Houdenhoven M, Kazemier G (2008) Robust surgery loading. Eur J Oper Res 185 (3):10381050. doi:10.1016/j.ejor.2006.08.022
6. Sargent RG (2005) Verification and validation of simulation models. Paper presented at the Proceedings of the
37th conference on Winter simulation, Orlando, Florida,
7. Schlesinger S, Crosby R, Cagne R, Innis G, Lalwani CS, Loch J, Sylvester R, Wright R, Kheir N, Bartos D (1979)
{terminology for model credibility}. SIMULATION 32 (3):103-104. doi:citeulike-article-id:4813312
8. Schoenmaeker AD, Kimpen S Onthaalbrochure e905 & e906.
9. Trivedi P, Zimmer D (2006) Copula modeling: An introduction for practitioners. Foundations and Trends® in
Econometrics 1 (1):1-111. doi:citeulike-article-id:6425795
51
5. Literature Review: OR planning
Abstract
Operating room (OR) planning and scheduling decisions involve the coordination of patients, medical staff and hospital facilities. The
patients arriving to the hospital are assigned to a surgery date and a surgery time slot. At the time of surgery, a suitable OR, the
attending surgeon, supporting anesthesiologists, nurses and, after the surgery, room in secondary facilities such as post anesthesia
care unit (PACU), intensive care unit (ICU) and ward need to be available. In order to deal with the complexity and the variety of
problems faced in OR scheduling, it is useful to involve methods from operations research. In this chapter, we review the recent
literature on the application of operations research to OR planning and scheduling. We start by discussing the impact of planning and
scheduling of the ORs on the overall performance of a hospital. Next, we discuss the criteria for included publications and summarize
the structure of Cardoen et al. [29] that served as the guideline for organization of this chapter. In the remainder of the chapter, we
describe the evolution of the literature over the last ten years with regards to the patient type, the different performance measures, the
decision that has to be made, the incorporation of uncertainty, the operations research methodology and the applicability of the research.
Moreover, each of these evolutions will be demonstrated with a short review of some relevant papers. The chapter ends with conclusions
and a discussion of interesting topics for further research.
5.1. Introduction
Healthcare has a heavy financial burden for governments within the European Union as well as oversees. While growing economies
and newly emerging technologies could lead us to believe that supporting our respective national healthcare systems might get less
expensive over time, data show the contrary is true. For example, within the U.S. the NHE (National Health Expenditure) as a share
of the Gross Domestic Product (GDP) increased from 15.6% in 2004 up to 16.2% in 2009 [112]. The data suggest that increasing
health expenditures are an ongoing trend with an estimated annual growth rate of 6.3%. The NHE share of the GDP in the U.S. is
thereby being projected to hit 19.6% by the year 2019 [31].
52
Similarly on the European continent, even though differences exist across member states, healthcare spending as a share of the GDP
is increasing, where countries that are hit hard by the global recession are most affected. For example, in Ireland, the percentage of
GDP devoted to healthcare increased from 7.6% in 2004 to 9.5% in 2009. In the UK during the same time interval, a rise from 8% to
9.8% was experienced [112].
A large amount, estimated at 31% of the spending on healthcare, pertains to hospitals [32], which are consequently being pressured
to reduce costs. Hospitals are expensive from the patient perspective as well. For instance, Milliman Medical Index (MMI) estimates
that, for a typical family of four, 48% of the family health care spending involves hospital costs [104].
Within the hospital, special attention is given to ORs as they represent the largest costs and provide the largest revenues [1]. It comes
as no surprise that the body of literature dealing with topics related to OR efficiency and profitability is steadily increasing. Out of the
many aspects, we focus our attention on planning and scheduling problems and do not include topics related to business process reengineering, the impact of introducing new technologies or facility design.
Our work is the natural continuation of the literature review carried out by Cardoen et al. [29]. In this chapter we complement the work
by adding the recent body of literature (2007-2010) and including a more in-depth analysis of the trends being followed by the research
community. Trends are investigated on the interval starting from 2000 and ending at 2010, while detailed descriptions are only provided
for the most recent contributions being published after 2006 and which were not included in [29].
In the past 60 years, a large body of literature on the management of ORs has evolved. Magerlein and Martin [95] distinguish between
advance scheduling and allocation scheduling as they provide us with a review on surgical demand scheduling. Advance scheduling
is the process of fixing a surgery date for a patient, whereas allocation scheduling determines the OR and the starting time of the
procedure on the specific day of surgery. Blake and Carter [17] elaborate on this taxonomy in their literature review and add the domain
of external resource scheduling, which they define as the process of identifying and reserving all resources external to the surgical
suite necessary to ensure appropriate care for a patient before and after surgery. Przasnyski [125] structures the literature on OR
scheduling based on general areas of concern, such as cost containment or scheduling of specific resources.
The more recent review by Gupta and Denton [67] focuses on appointment scheduling from a general perspective and describes three
commonly encountered health care scheduling environments, namely primary care, specialty care, and elective surgery (nonemergencies). In primary care usually a primary care physician, such as a general practitioner or a family physician, acts as the principle
point of consultation. Specialty care is focused on a specific and often complex diagnosis and treatment, whereas elective surgery is
focused on a specific procedure. They discuss various factors, which affect the performance of the appointment system, including
arrival and service time variability, patient and provider preferences and performance measures such as patient waiting time, OR idle
time and overtime. According to these factors, the existing literature is classified into three groups. The difference between Gupta and
Denton’s review and our review is that our focus is limited to scheduling problems which arise in close relationship to the OR.
Consequently, we include elective surgery scheduling and exclude considerations related to primary and specialty care.
In the literature review of Guerriero and Guido [64], a selected number of articles are categorized according to the commonly used
three hierarchical decision levels: strategic, tactical and operational. Strategic decisions involve defining both number and types of
surgeries to be performed, and hence affect the OR function in the long term. The tactical level usually involves the construction of a
53
cyclic schedule, which assigns time blocks to surgeons or surgeon groups. The final, operational level does not influence the number
and type of performed procedures, but deals mostly with daily staffing and surgery scheduling decisions. The three hierarchical levels
give the OR planning problem structure. Nonetheless, in our literature review we chose not to use the three hierarchical levels but
rather to define descriptive fields. The justification for our decision can be found in Section 2. Other reviews, in which OR management
is covered as a part of global health care services, can be found in [21,124,135,165].
We searched the databases Pubmed and Web of Knowledge1 for relevant manuscripts, which are written in English and appeared in
2000 or afterwards. Search phrases included combinations of the following words: operating, surgery, case, room, theatre(er),
scheduling, planning and sequencing. We searched in both titles and abstracts and in addition checked the complete reference list of
any found article. As the search process happened in an unbiased way, we believe to have arrived at a set of articles, which objectively
represents most of the articles in the field. At the end of the search procedure, a set of 181 articles was identified of which 136 [2,3,516,18-20,22-25,28,27,33-41,43,45,46,48-57,59-63,65,66,68-73,58,74-84,86-94,96-100,105-111,113-115,117-123,126-132,136141,143,142,144-148,150-157,161-164,166-168] were found to be technically oriented. We define an article as “technical” if it contains
an algorithmic description of a method directly related to patient scheduling. Some articles missing this algorithmic component instead
provide managerial insights. Those types of articles are classified as “managerial” and excluded from the classification process itself.
However, these are mentioned in the text where they seem appropriate. The quantitative descriptions provided in later sections, which
try to give insights into the dynamics of the trends followed by the research community, are exclusively based on the technical
contributions. The distribution of included articles according to their year of publication is shown in Figure 31.
1
Includes: Web of Science, Current Contents Connect and Inspec
54
35
30
25
Both technical and
managerial
20
Technical
15
Managerial
10
5
0
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Figure 31: The distribution of included articles according to their year of publication. Figures appearing in the text are based on technical
contributions only.
The structure we use is meant to balance between simplicity and expressiveness. We provide a simplified, but in our belief for the
majority of the readers, sufficiently accurate way to identify and select articles they are interested in. In the detailed tables, all researched
manuscripts are listed and categorized. Pooling these tables over the several fields should enable the reader to reconstruct the content
of specific papers. They furthermore act as a reference tool to obtain the subset of papers that correspond to a certain characteristic.
To search for literature in a more direct way the database containing the detailed classification of each analyzed article is made
available at www.econ.kuleuven.be/healthcare/review2011 in the form of an Excel [Microsoft, Redmond, WA] spreadsheet.
5.2. Organization of the review
As in [64], many authors differentiate between strategic (long term), tactical (medium term) and operational (short term) approaches,
and situate their planning or scheduling problem accordingly. With respect to the operational level, a further distinction can be made
between offline (i.e. before schedule execution) and online (i.e. during schedule execution) approaches. The boundaries between these
major categories can vary considerably for different settings and hence are often perceived as vague and interrelated [134].
Furthermore, this categorization seems to lack an adequate level of detail. Other taxonomies, for instance, are structured and
categorized on a specific characteristic of the paper, such as the use of the solution or evaluation technique. However, when a
55
researcher is interested in finding papers on OR utilization, a taxonomy based on solution technique does not seem very helpful.
Therefore, we propose a literature review that is structured using descriptive fields. As in Cardoen et al. [29] each field analyzes the
manuscripts from a different perspective, which can be either problem or technically oriented. In particular, we distinguish between six
fields:
 Patient characteristics (Section 3): reviewing the literature according to the elective (inpatient or outpatient) or non-elective
(urgency or emergency) status of the patient.
 Performance measures (Section 4): discussion of the performance criteria such as utilization, waiting time, preferences,
throughput, financial value, makespan, and patient deferral.
 Decision delineation (Section 5): indicating what type of decision has to be made (date, time, room or capacity) and whether
this decision applies to a medical discipline, a surgeon or a patient (type).
 Uncertainty (Section 6): indicating to what extent researchers incorporate arrival or duration uncertainty (stochastic versus
deterministic approaches).
 Research methodology (Section 7): providing information on the type of analysis that is performed and the solution or
evaluation technique that is applied.
 Applicability of research (Section 8): information on the testing (data) of research and its implementation in practice.
Each section clarifies the terminology if needed and includes a brief discussion based on a selection of appropriate manuscripts.
Furthermore, a detailed table is included in which all relevant manuscripts are listed and categorized. Plots are provided to point out
some of the trends followed by the research community. It should be noted that, if not stated otherwise, the percentages are calculated
in relation to the total amount of technical papers. Also note that some categories are not interpretable for some methods and even
though rare, some articles contain more than one single method. As a consequence, the sum of mutually exclusive categories does
not necessarily add up to 100%.
5.3. Patient characteristics
Two major patient classes are considered in the literature, namely elective patients and non-elective patients. The former class
represents patients for whom the surgery can be planned in advance, whereas the latter class groups patients for whom a surgery is
unexpected and hence needs to be planned on short notice. A non-elective surgery is considered an emergency if it has to be performed
immediately and an urgency if it can be postponed for a short time (i.e. days).
56
As shown in Figure 32 and Table 3, the literature on elective patient scheduling is vast compared to the non-elective counterpart.
Although many researchers do not indicate what type of elective patients they are considering, some distinguish between inpatients
and outpatients. Inpatients refer to hospitalized patients who have to stay overnight, whereas outpatients typically enter and leave the
hospital on the same day. As pointed out by [103], in reality an ongoing shift of services from the inpatient to the outpatient setting is
present, which is reflected in a higher growth rate of the latter. Moreover, according to the MMI, outpatient costs increase with a yearly
value of 10%, of which 90% are attributable to increasing prices of existing and more expensive emerging services [104]. Despite the
increasing importance of outpatient care in general, the share of outpatient related academic articles as shown by Figure 32 remains
constant in time.
100%
90%
80%
70%
60%
Elective
50%
Elective Inpatient
40%
Elective Outpatient
Non-elective
30%
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 32: The majority of contributions relate to the elective patient. 2 Surprising is that, contrary to what might be expected, the share
of outpatient related articles is not increasing.
2
As some articles deal with both elective and non-elective patients, the sum of the two will add up to a value higher than 100%.
57
Elective
inpatient
2,8,13,14,16,23,25,36,48,62,67,72,74,84,94,108,109,118,123,136,137,139,140,143,144,148,154,156,167
outpatient
8,15,23,25,27,28,40,43,46,48,52,61,67,74,75,76,84,108,109,123,126,129,139,140,148,154,157,167
not specified
3,5,6,7,9,11,22,24,34,35,37,38,39,45,49,50,54,55,56,57,58,59,60,65,68,69,70,71,73,77,80,81,82,83,86,87,88,89,9
0,91,92,96,99,100,105,106,107,111,113,114,115,117,119,120,121,122,127,128,130,132,138,141,142,145,146,150
,151,152,153,
155,161,162,163,164,166,168
Non-elective
urgent
22,54,65,67,96,109,110,123,168
emergent
9,25,60,67,71,72,82,87,88,89,90,92,100,106,108,113,122,123,139,141,142,150,154,163,167
not specified
83,84,114,151,162
Table 3: It is not always specified what type of patients is considered and, especially for the elective patient case, not always clear
whether an inpatient or outpatient setting is used.
Scientific contributions on outpatients often (22 out of 37 articles) investigate ORs in an integrated way, i.e., a preoperative or
postoperative unit is taken into account. For example, Huschka et al. [76] use both an intake and a recovery area as part of a simulation
model of an outpatient procedure center. Several daily scheduling and sequencing heuristics are applied and tested on their impact on
patient waiting time and the amount of OR overtime. The authors found that defining the order of surgeries has less influence on patient
waiting time and OR overtime as the arrival time schedules. Additionally, the importance of a proper daily surgery mix is stressed.
Other methods focus on assigning patients to days and do not define the exact procedure starting times. Lamiri et al. [88] consider
several stochastic optimization methods to plan elective surgeries in case OR capacity is shared by both elective and emergency
patients and present a solution method combining Monte Carlo sampling and mixed integer programming. Several heuristics were also
tested, from which the most efficient one proved to be tabu search. In their problem setting, it is surprising that the quality of heuristic
solutions degrades as the variability on the amount of emergency arrivals decreases, i.e., the stochastic problem is easier than the
deterministic one. Planning for stochastic emergency arrivals poses a problem because it leaves an uncertain amount of capacity left
for elective patients. Ferrand et al. [61] investigate whether eliminating this source of uncertainty by channeling emergency arrivals to
dedicated emergency ORs is beneficial. This requires however that a constant number of ORs be reserved for emergencies, and
therefore leaves less free capacity for elective patients. Nevertheless, based on their results using a simulation model, they find elective
patients benefit from this, whereas emergency arrivals have an increased waiting time.
A scenario where a hospital dedicates all of its ORs to emergency services is the case of a disaster. As a consequence, all elective
surgeries are cancelled while resources are redirected to provide quick care to non-electives. This type of non-elective patient is an
urgency, as quick but not necessarily immediate care is required. Nouaouri et al. [110] sequence a large number of patients resulting
58
from a disaster, with the objective of maximizing patient throughput. Their approach identifies patients which cannot be served by the
given hospital and therefore have to be transported to another one.
Zonderland et al. [168] refers to patients who have to be treated within 1-2 weeks as semi-urgencies and uses queuing theory to
analyze the trade-off between reserving too much OR time for their arrival (which results in unused OR time) and excessive cancellations
of elective surgeries. Additional complexity results from the fact that cancelled elective patients turn into semi-urgent patients, which
consequently need to be served within the following two weeks. An insight gained by the authors is that focusing only on the average
behavior of the system can result in undesired system outcomes, i.e., shortage of capacity, which ultimately leads to the cancellation
of many elective patients.
5.4. Performance measures
Different performance measures emphasize different priorities and will favor the interests of some stakeholders over others. A hospital
administrator could, for example, be interested in achieving high utilization levels and low costs. Medical staff, on the other hand, might
care less about cost factors and rather aim to achieve low overtime. The patient as the client of the hospital might care little about the
above factors and only desires high quality service and short waiting times. Many authors in the scientific community try to find a
compromise between the interests of different stakeholders and simultaneously enforce several kinds of performance criteria. As a
matter of fact, as shown by Figure 33, a gradually decreasing number of authors restrict themselves to only one single performance
measure.
59
100%
90%
80%
70%
60%
50%
Multiple performance criteria
40%
Single performance criteria
30%
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 33: It is increasingly popular to use multiple instead of single performance criteria. The criteria are not interpretable for all
articles, their sum is thus lower than 100%.
The major performance measures we distinguish can be found in Table 4: utilization, waiting time, leveling, preference, throughput,
financial measures, makespan, and patient deferral. As shown by Figure 34, patient waiting time is a frequently used performance
measure, which is understandable as one of the major problems in general health care consists of long waiting lists but also extensive
waits on the day of surgery. Wachtel and Dexter [159,160] investigated increases in waiting time on the day of surgery, for both surgeon
and patient, caused by tardiness from scheduled start times. They concluded that the total duration of preceding cases is an important
predictor of tardiness, i.e., the tardiness per case grew larger as the day progressed. If case durations are systematically
underestimated, tardiness results. The estimated amount of under-utilized (over-utilized) time, which was to be expected at the end of
the workday, caused average tardiness to increase (decrease). A reduction of tardiness can be achieved by modifying the OR schedule
to incorporate corrections for both lateness of first cases of the day and case duration bias.
60
Utilization
underutilization
/undertime
OR
2,19,20,36,50,55,56,57,58,59,68,73,78,79,80,90,91,100,111,113,115,140,142,148,152,156,161,164,167,16
8
Ward
156
ICU
2,36,156
PACU
2,36
overutilization
/overtime
OR
2,11,22,25,33,36,37,38,39,40,41,50,51,54,55,56,57,58,59,60,64,65,69,73,76,77,78,79,80,87,88,89,90,91,9
6,98,
100,105,106,111,113,115,119,121,122,123,127,128,132,137,140,141,142,145,148,150,151,156,161,163,1
64
Ward
54,156
ICU
2,36,115,156
PACU
2,27,28,36
general
3,8,9,12,22,23,25,33,34,48,50,54,60,61,69,71,72,91,111,122,132,139,142,145,148,150,151,163
Waiting time
patient
3,9,25,33,36,37,38,40,54,60,61,64,65,67,76,78,79,81,91,93,106,107,109,111,118,120,121,122,132,136,13
9,141,
142,143,144,148,150,154,157,163,167
surgeon
11,37,38,91,157
Leveling
OR
15,98,99,111
Ward
13,14,16,68,94,130,140,153
PACU
15,27,28,75,96,97,131,139,152
61
Patient volume
111,140,142
Preference
16,18,27,28,34,35,52,62,83,88,94,105,106,107,114,115,118,138,140,143,144,145,146,155,162,166
Throughput
3,8,9,12,71,100,110,126,130,132,142,145,154
Financial
11,18,24,39,43,45,46,48,49,65,84,93,100,108,138
Makespan
5,6,7,35,57,58,59,73,74,75,86,96,123,129,137,147
Patient deferral
22,25,36,54,71,72,81,82,92,120,121,122,132,139,145,168
Other
2,9,10,12,33,35,36,61,68,77,83,87,89,90,93,96,99,100,117,121,123,127,128,136,141,142,153,155
Table 4: The main performance criteria are: Utilization, Waiting time, Leveling, Preference (e.g., priority scooring), Financial (e.g.,
minimization of surgery costs), Makespan (completion time), Patient deferral, and Other (e.g., number of required porter teams).
Figure 34 shows that patient throughput is rarely used as a performance measure, and patient or surgeon preferences are increasingly
gaining popularity. Preference most often covers some qualitative aspect, whereas throughput is associated with volume. Noteworthy
is the fact that both in general healthcare [149] and in the operations research literature value or quality based approaches seem to
be getting increasingly important. For example, the preferences of cataract surgery patients of one surgeon are investigated by Dexter
et al. [42]. The surgeon’s patients place a high value on receiving care on the day chosen by them, at a single site, during a single
visit, and in the morning. Preferences can also be embodied in patient priorities, i.e., preference to perform surgery first on patients
who have a more acute condition. Priorities are most often defined at the level of a patient group. Testi et al. [144,146] define a model
where the position of a patient on a waiting list is defined by a priority scoring algorithm, which considers both patient urgency
(progression of disease, pain or dysfunction and disability) and time spent on the surgical waiting list. It is easy to see that priority
scoring minimizes the total weighted waiting time of all patients, i.e., using an algorithm where patient priorities are uniform, we will
minimize the average patient waiting time.
62
100%
90%
80%
70%
60%
Overtime
50%
Patient waiting time
40%
Preference
Throughput
30%
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 34: Some selected performance criteria. Overtime, despite used slightly less in 2010, it is still the most frequently used
performance measure. From 2008 on, preference related measures seem to become increasingly popular.
Including patient priorities drives OR scheduling in a more patient oriented direction. Min and Yih [105] go a step further and explicitly
incorporate an additional payment factor, defined as the cost of overtime. In their model, if many high priority patients are on a waiting
list, ORs will be kept open longer in order to avoid high postponement costs, i.e., the surgery postponement costs are balanced versus
OR overtime costs. The authors establish that patient prioritization is only useful if the difference between the cost coefficients
associated to different priority classes is high, as otherwise a similar schedule can be obtained by using average postponement costs.
Additionally, the relative cost ratio between the cost of patient postponement and OR overtime should not be low, as a low ratio would
imply high overtime costs and therefore prioritization would only marginally affect the surgery schedule.
Purely financial objectives are rarely used in the literature. In Stanciu and Vargas [138], protection levels, the amount of OR time
reserved in a partitioned fashion for each individual patient class, are used to determine which patients to accept and which to postpone
during the planning period under study. A patient class is a combination of the patient reimbursement level and the type of surgery. A
patient class enjoys higher priority if its expected revenue per unit surgery time is higher. The goal is to maximize expected revenues
incurred by the surgical unit. Patients, given their class, are accepted or postponed to a later date when the protection level for their
class can accommodate them. The central question becomes how many requests to accept from low revenue patients, and how much
capacity to reserve for future high revenue patients. Financial considerations are also expressed by Wachtel and Dexter [158], who
argue that if additional OR capacity is expanded, it should be assigned to those subspecialties that have the greatest contribution
margin per OR hour (revenue minus variable cost), that have the potential for growth, and that have minimal need for a limited resource
such as ICU beds.
63
Figure 34 also reveals the fact that minimizing overtime is a popular performance measure. This is understandable as overtime results
both in the dissatisfaction of the surgical staff and high costs for the hospital (as higher wages typically apply for the time beyond the
normal working hours). Reducing overtime is consequently highly beneficial and often desirable to practitioners. Dexter and Macario
[47] establish that a correction of systematically underestimated lengths of case durations would not markedly reduce overutilization of
ORs. Tancrez et al. [141] proposed an analytical approach that takes into account both stochastic operating times and random arrivals
of emergency patients. They showed how the probability of overtime in the OR changes as a function of the total number of scheduled
operations per day. As shown in Table 4, we relate underutilization to undertime and overutilization to overtime, although they do not
necessarily represent the same concept. Utilization actually refers to the workload of a resource, whereas undertime or overtime
includes some timing aspect. Hence, it is possible to have an underutilized set of ORs, although with overtime in some of the ORs. In
some manuscripts it is unclear which view is applied. Therefore, we group underutilization with undertime and similarly overutilization
with overtime.
Adan et al. [2] formulated an optimization problem which minimizes the deviation from a targeted utilization level of the OR, the ICU,
the medium care unit and nursing hours. The deviation is measured as the sum of overutilization and underutilization. They recommend
using stochastic time durations as well as a broad perspective on supporting resources such as the ICU and the wards. Pandit and
Dexter [116] defined rules to determine whether an OR should be staffed for 8 or 10 hours. They concluded that in case the average
OR time is less than 8 h 25 min, 8 h staffing should be planned, while in case the average OR time is over 8 h 50 min, 10 h of staffing
is needed. For averages in between, the full analysis of McIntosh et al. [102] should be performed.
Augusto et al. [7] minimized the sum of the makespan (completion time) of patients undergoing surgery. Makespan in general defines
the time span between the entrance of the first patient and the finishing time of the last patient. Since minimizing the makespan often
results in a dense schedule, deviations from the plan can result in complications requiring adjustments to the schedule. An example is
the arrival of an emergency patient to the hospital which results in a dense schedule with no free ORs available. In a case like this, a
likely scenario includes the deferral of an elective patient, who will have to be reassigned to another surgery slot at a later point in
time. Zonderland et al. [168], used queuing theory to investigate the trade-off between cancellations of elective surgeries due to semiurgent surgeries and unused OR time. They provide a decision support tool, which assists the scheduling process of elective and semiurgent cases. General reasons for patient deferrals in one specific hospital are discussed by Argo et al. [4].
In addition to emergency patient arrivals, other reasons can cause cancelation of elective surgeries. For example, a fully occupied
PACU would prohibit patients who have already completed surgery from leaving the OR. A blocked OR will delay the service of
preceding elective surgeries, which as a final measure may have to be cancelled. This situation can be avoided if the OR schedule is
constructed in a way that resource utilization is leveled. Similar approaches may be used for other resources. For instance, Ma and
Demeulemeester [94] maximize the availability of the number of expected spare beds and investigate bed occupancy levels at wards,
whereas Van Houdenhoven et al. [152] target the ICU.
Some of the articles in the literature used methods which have not been covered in the previous paragraphs. Does et al. [53] used Six
Sigma to decrease the tardiness of surgeries, which are performed first on a day. Applied to two hospitals in the Netherlands, substantial
savings were achieved and the number of surgeries was increased by 10% without requiring additional resources. Epstein and Dexter
64
[44] introduced a method through which analysts can screen for the economic impact of improving first-case starts. In [2,77] nursing
hours are considered, while [128] considers the number of open ORs, and [141] the number of disruptions.
5.5. Decision delineation
A variety of planning and scheduling decisions are studied in the literature. In Table 5, we provide a matrix that indicates what type of
decisions are examined, such as the assignment of a date (e.g., on Friday, February 25), a time indication (e.g., at 2 p.m.), an OR
(e.g., OR 1, OR of type B) or the allocation of capacity (e.g., one hour of OR time). The manuscripts are further categorized according
to the decision level they address, i.e., to whom the particular decisions apply. We distinguish between the discipline (e.g., pediatrics),
the surgeon and the patient level. Figure 35 shows that a large part of the literature aims at the patient level, whereas the discipline
as well as the surgeon level receives less attention.
100%
90%
80%
70%
60%
Patient
50%
Discipline
40%
Surgeon
Other
30%
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 35: In the literature, most decision problems relate to the patient level. A typical problem setting, for example, consists of finding
the optimal patient to day / OR assignment.
The discipline level unites contributions in which decisions are taken for a medical specialty or department as a whole. This frequently
involves the construction of a cyclic timetable, which aims at minimizing the underallocation of a specialties’ OR time with respect to
its predetermined target time. On a lower level, the target is a surgeon group or a single surgeon. In Denton et al. [39], surgeries
65
consecutively carried out by one surgeon define a surgery block, which are subsequently assigned to ORs. The problem is formulated
as a stochastic optimization model, which balances the cost of opening an OR with the cost of overtime.
Discipline Level Surgeon level Patient level
Other
Date
54,143,156
13,19,20,24,33,49,65,13
14,15,16,25,74,80,120,1
2,25,33,34,36,48,49,54,55,56,57,58,59,64,65,68,69,72,73,
42
0,132,
78,79,80,81,82,83,87,88,89,90,94,105,106,107,111,115,
144,145,167
118,119,120,121,123,127,128,132,136,140,143,144,145,
146,152,153,155,161,164,166
Time
13,33,49,65,71,132,145
11,14,15,16,25,375,6,7,11,25,27,28,33,37,38,40,41,49,57,58,59,65,70,71,73,
11,12
75,76,77,78,79,83,86,91,96,97,99,110,123,127,128,132,
137,145,147,150
Room
19,20,33,62,
130,132,144,
145,167
11,143
11,15,16,39,74,80,120,1
11,23,27,28,33,34,35,41,50,51,55,56,57,58,59,60,64,68,69,
42
70,73,76,77,78,79,80,83,86,87,90,96,98,99,106,107,110,
111,115,118,120,121,123,127,128,129,132,137,143,144,
145,152,153,161,163,164
Capacity 22,24,33,49,65,67,71,13
11,18,25,37,43,45,46,80,
2,11,22,25,33,36,37,49,54,64,65,68,71,72,80,94,105,108,
11,12,52,54,61,92,93,
84,120
114,120,121,132,138,141,145,168
109,117,122,126,131,154
0,132,
139,145,167
Other
151
120
3,7,35,54,111,120,136,143,157,162,168
52,54,143,148
Table 5: Type and level of decisions. For example, articles dealing with the sequencing problem are found in column 3 and row 2.
Articles dealing with the problem generally referred to as the assignment step are found in column 3 and row 1. Defining patient
capacity requirement for a given day of the week are articles found in column 3 and both row 1 and row 4.
On the patient level, the decision variables are formulated on the individual patient or patient type. Fei et al. [59] describe a two stage
approach, where in the first stage patients are assigned to days and rooms, and the exact daily sequence of surgeries (timing aspect)
is set in the second stage. This is a common way of patient scheduling, as the assignment of the day and the room of a given surgery
is more easily planned ahead than the exact time, which is often fixed close to the actual surgery date.
Date, room, time and capacity questions as shown in Table 5 are answered on all three decision levels.
Figure 36 shows that the share of manuscripts dealing with decisions on exact times is decreasing, whereas date, room or capacity
problems are continuously addressed in the literature. A capacity problem is discussed by Masursky et al. [101] who forecasted longterm anesthesia and OR workload. They concluded that forecasting future workload should be based on historical and current workload
related data and advise against using local population statistics. The problem of forecasting workload is addressed by Gupta et al. [68]
as well. In his case study, simulation is used to answer capacity related questions. They concluded that a one-time infusion of capacity
66
in the hope to clear backlogs will fail to reduce waiting times permanently, while targeting extra capacity to highest urgency categories
reduces all-over waiting times including those of low urgency patients. In situations where arrival rates increased, even if only within a
specific urgency class, waiting times increased dramatically and failed to return to the baseline for a long time.
100%
90%
80%
70%
Assignment of date
60%
Assignment of time
50%
Assignment of room
40%
Assignment of capacity
30%
Other
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 36: Solution approaches related to the assignment of dates and rooms are increasingly popular in the literature, whereas the
time assignment step (e.g., sequencing) is slightly less popular than it used to be.
We added both a row and a column (other) to Table 5 to provide entries for manuscripts that study OR planning and scheduling
problems in a way that is not well captured by the main matrix. Manuscripts that are categorized in this column or row examine for
instance, capacity considerations with regards to beds [92,131], OR to ward [143] and patient to week assignments [168] or different
timing aspects, such as the amount of recovery time spent within the OR [7].
As OR planning and scheduling decisions affect facilities throughout the entire hospital, it seems to be useful to incorporate facilities,
such as the ICU or PACU, in the decision process in an attempt to improve their combined performance. If not, we believe that
improving the OR schedule may worsen the efficiency of those related facilities.
Figure 37 shows that the ratio of manuscripts between 2005 and 2010, which deal with the OR in an integrated way, and those which
deal with the OR in an isolated way, are oscillating around the 50% mark. This is surprising as we would expect to observe an
increasing interest in integrated approaches. Whether a manuscript uses an integrated or an isolated approach can be looked up in
Table 6.
67
100%
90%
80%
70%
60%
50%
OR is isolated
40%
OR is integrated
30%
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 37: Every second article deals with the OR planning and scheduling process in an integrated way.
Isolated OR
5,10,11,19,20,22,33,34,37,38,39,41,45,48,49,50,51,52,55,56,57,58,59,60,64,65,67,68,69,73,74,83,84,87,88,89,9
0,91,98,99,105,107,110,111,113,114,117,118,119,120,122,127,128,132,138,141,146,147,148,150,151,157,161,1
62,163,164,
166,168
Integrated OR
2,3,6,7,8,9,12,13,14,15,16,18,23,24,25,27,28,36,40,43,46,54,58,61,62,65,68,70,71,72,73,75,76,77,78,79,82,86,9
4,96,97,100,106,108,109,115,121,123,126,130,131,136,137,139,140,142,143,145,152,153,154,155,156,167
Table 6: In an integrated OR, supporting facilities such as the ICU, PACU, and wards are considered.
The problem of the congested PACU, which previously had been scarcely addressed in the literature, received more attention between
2008 and 2010. In this problem, patients are not allowed to enter the fully occupied PACU and are therefore forced to start their
recovery phase in the OR itself, keeping it blocked. Iser et al. [77] used a simulation model to tackle the problem and compare overtime
occurring in the OR with PACU specific performance measures. Augusto et al. [7] showed the benefit of using a mathematical model
to plan ahead the exact amount of recovery time a patient will spend within the OR. As it is typical for highly utilized systems, there is
a sensitive relationship between overall case volume, capacity (of the PACU) and the effect on waiting time (to enter the PACU). This
relationship is described in more detail by Schonmeyr et al. [131] using queuing theory.
68
Besides the PACU, a downstream facility, which could affect the function of the OR is the ICU. Kolker [82] reduced diversion of an ICU
to an acceptable level by defining the maximum number of elective surgeries per day that are allowed to be scheduled along with the
competing demand from emergency arrivals. Litvak et al. [92] went a step further and tackled the ICU capacity problem in a cooperative
framework. In their model, several hospitals of a region jointly reserve a small number of beds in order to accommodate emergency
patients and achieve an improved service level for all patients.
5.6. Uncertainty
One of the major problems associated with the development of accurate OR planning and scheduling strategies is the uncertainty
inherent to surgical services. Deterministic planning and scheduling approaches ignore uncertainty, whereas stochastic approaches
explicitly try to incorporate it. In Table 7, we list the relevant manuscripts classified according to the type of uncertainty they incorporate.
100%
90%
80%
Deterministic
70%
60%
Stochastic (arrival + durration
+ other)
50%
Stochastic arrival
40%
Stochastic durration
30%
20%
Stochastic other
10%
0%
2005
2006
2007
2008
2009
2010
Figure 38: Uncertainty incorporation. Implicit in the figure is the fact that if duration uncertainty is accounted for, it is often the case that
arrival uncertainty is considered as well.
69
Deterministic
6,7,10,14,15,18,19,20,24,27,28,33,34,35,43,46,52,55,56,57,58,59,64,68,70,73,75,77,78,79,80,83,84,86,97,99,107,1
08,110,
111,114,115,118,120,121,123,127,128,130,136,137,140,143,144,145,146,150,152,155,156,164,166,167
Stochastic
Arrival
3,9,13,16,22,23,25,36,48,49,54,62,65,67,71,72,74,81,82,87,89,90,93,94,100,105,109,117,121,122,126,132,138,139,
141,
142,145,150,154,162,163,167,168
Duration
2,3,5,8,9,11,12,13,16,22,25,36,37,38,39,40,41,49,50,54,62,65,67,68,69,71,72,76,81,87,88,89,91,93,94,96,97,98,100
,105,
106,109,113,117,119,122,126,131,132,139,141,142,145,148,150,151,153,154,157,161,162,163,167
Other
3,5,9,25,45,46,72,92,94,106,126,162
Table 7: Methods frequently take stochasticity into account. Most common forms are duration and arrival uncertainty.
As shown in Figure 38, stochasticity in the form of uncertain patient arrival and surgery duration is frequently used in the OR literature.
If we narrow the literature to recent contributions, which explicitly incorporate non-elective patients, we see that over 80% of methods
incorporate some sort of uncertainty. Non-elective patient arrivals are in most cases impossible to predict in advance and additionally
occupy a random amount of OR time, which often leaves OR managers with no option but to keep a safety margin to accommodate
them [141]. In contrast, the arrival of elective patients contains a lower amount of uncertainty, and as shown in Figure 39, is frequently
considered as deterministic in the literature.
70
100%
90%
80%
Non-elective patient and
stochastic duration
70%
60%
Non-elective patient and
stochastic arrival
50%
Elective patient and
stochastic duration
40%
30%
Elective patient and
stochastic arrival
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 39: Stochasticity in the elective and non-elective patient setting.3 If uncertainty is considered in an elective or non-elective patient
setting, stochastic aspects are in either case slightly more often applied to surgery durations than to patient arrival times.
Duration uncertainty is a central element in Denton et al. [39] as well as in Batun et al. [11]. In Denton et al. [39] decisions include the
number of ORs to open and surgery block to OR assignments, whereas in Batun et al. [11] this is supplemented by patient sequencing
and setting surgeon start times. Both methods aim at minimizing OR opening and OR overtime costs, where Batun et al. [11] additionally
consider surgeon idle times. The functional difference between their methods lies in the way surgery to OR assignments are carried
out. In Denton et al. [39], the common practice of assigning a surgery block to a single surgeon (block scheduling) is followed, whereas
Batun et al. [11] consider the scenario of pooled ORs, and therefore surgeons are allowed to switch between ORs. OR pooling allows
carrying out surgeries in parallel as the main surgeon only needs to be present during the critical part of the surgery and can move to
the next patient before the close-up of the patient.
A timing aspect, which is different from the actual surgery duration but is characterized by large variations, is the patient length of stay
(LOS) in e.g. PACU, ICU or ward. The variability of patient time spent in the ward is considered by Ma and Demeulemeester [94] in
which the rate of patient misplacements caused by bed shortages is minimized.
It should be clear that operations research techniques are able to deal with stochasticity, especially simulation techniques (included in
66% of the stochastic literature) and analytical procedures (included in 22% of the stochastic literature), and that an adequate planning
and scheduling approach may lower the negative impact of uncertainty. Mostly, studies assumed a certain level of variability, based on
analyzing historical data, and used this information as input for models. However, only limited attention is paid to the reduction of
variability within the individual processes. As an example, consider the estimation of surgery durations. Instead of the immediate
determination of the distribution of a surgery duration, one should examine whether the population of patients for which the durations
3
Percentages are calculated in ratio to the total number of articles dealing with the respective patient type.
71
are taken into account is truly homogenous. If not, separating the patient population may result in a decreased variability even before
the planning and scheduling phase is executed. As the estimation of surgery durations exceeds the scope of this literature review, we
do not elaborate further on this issue.
5.7. Operations Research Methodology
The literature on OR planning and scheduling exhibits a wide range of methodologies that fit within the domain of operations research
and that combine a certain type of analysis with some solution or evaluation technique. Table 8 provides an overview of the ways in
which OR planning and scheduling problems are analyzed. The table shows that mathematical programming and heuristics are
frequently applied, generally to patient sequencing or assignment type of problems (e.g., patient to surgeon/OR assignment). These
types of problems are combinatorial optimization problems.
In some approaches the impact of specific changes to the problem setting is examined. We refer to this type of analysis as scenario
analysis since multiple scenarios, settings or options are compared to each other with respect to the performance criteria.
72
Analytical procedure
22,37,59,62,65,88,89,92,93,105,113,114,131,141,151,157,162,168
Mathematical Programming
Linear Programming
7,37,43,46,84,108,119
Goal Programming
2,18,36,115,140
Integer Programming
19,20,27,33,52,110,130,142,143,144,145,153
Mixed integer programming
11,13,16,39,68,78,79,80,87,88,89,106,107,118,120,121,123,128,167
Column generation
55,57,58,59,68,73,86,87,90,152,153
Branch-and-price
14,28,56
Dynamic programming
6,7,14,28,56,73,90,105,164
Other
6,7,13,16,45,70,99,119
Dedicated branch-and-bound
27,39,111,156
Scenario analysis
3,5,7,8,9,12,15,18,22,23,25,34,35,36,37,38,39,40,41,43,46,48,49,50,51,54,55,57,58,59,61,62,67,
69,71,72,74,75,76,78,81,82,84,86,91,92,93,94,96,97,98,100,106,107,108,109,111,113,115,117,1
18,121,122,126,128,130,131,132,136,138,139,141,142,145,146,148,150,151,152,154,156,157,1
62,163,167
Simulation
Discrete-event
3,5,8,9,12,22,23,25,36,48,49,50,54,58,60,61,67,71,72,74,76,77,81,82,92,93,94,96,97,98,100,106
,109,121,122,126,132,139,142,145,148,150,154,163,167
Monte-Carlo
22,40,46,69,87,88,89,91,111,117
Heuristics
improvement heuristics
Simulated annealing
13,16,40,69,88,150
Tabu search
35,73,75,88
Genetic algorithm
34,58,72,73,127,128,136,137,147,161,166
other
19,20,38,69,87,88,90,98,106,138,150,164
constructive heuristics
5,6,13,16,33,37,38,39,49,50,51,55,58,59,64,69,76,77,83,86,87,88,90,126,142,143,150
73
Table 8: Different solution techniques are used in the literature: Analytical procedures (e.g., queuing theory or new vendor model),
mathematical programming, dedicated branch-and-bound, scenario analysis (or sensitivity analysis), simulation, and heuristics.
As is shown by Figure 40, performing scenario analysis is popular, especially in the discrete-event simulation (DES) modeling literature.
Scenario analysis can be done by plotting the efficiency frontier formed by respective scenarios’ (possibly multidimensional) performance
scores. This helps to identify and distinguish between advantageous and disadvantageous scenarios. The performance criteria most
frequently used in the DES modeling literature are patient waiting time and different kinds of utilization measures such as the overtime
related to the OR, ward, ICU or PACU.
100%
90%
80%
70%
Analytical
60%
Mathematical programming
50%
Scenario or sensitivity analysis
40%
Discrete-event simulation
30%
Heuristics
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 40: Only selected solution techniques are shown. Most articles include a scenario or sensitivity analysis.
In Section 4, we expressed our surprise about the lack of an increasing use of integrated approaches. In the DES modeling literature,
however, the proportion of integrated approaches does increase, as OR supporting facilities such as the PACU or ward are increasingly
taken into account. We think that modeling the OR in an integrated way is an important step towards the construction of more realistic
and applicable models.
An integrated DES model is introduced by Steins et al. [139], in which pre-operative care as well as a PACU are considered. The
arrival of case types, the surgery time and the LOS in the PACU are represented as probabilistic distributions. The patients in their
model are differentiated according to their urgency status, i.e., whether a patient is elective or non-elective. It is true in general that
DES modeling approaches take explicit account of non-electives.
In the literature, the analytical approach is less often encountered than DES models. Aside from their differences on the methodological
side, both DES and analytical methods are often related to a similar problem setting. In case of a stochastic environment where capacity
questions have to be answered and non-electives possibly play a role, both of the approaches are useful. Tancrez et al. [141] define
74
the amount of OR capacity, which is needed to accommodate for non-elective patients in a Markovian model setting. Simulation is
used to show that the assumptions required to build the Markov chain have a minor influence on their final analytical results. In their
work, the stochasticity in OR capacity is the consequence of randomly arriving non-elective patients occupying an uncertain amount of
OR time. Also without non-elective patient arrivals it is difficult to predict the required OR capacity on a day, as surgery durations are
unknown in advance and can vary considerably in length. In Olivares et al. [113], the decision making process of reserving OR capacity
is investigated using the newsvendor model. In the analytical approach, an estimate is given of the cost placed by the hospital on
having idle capacity and the cost of a schedule overrun. Their results reveal that the hospital under study places more emphasis on
the tangible costs of having idle capacity than on the costs of a schedule overrun and long working hours for the staff.
As shown by Figure 40, MPs are popular. As opposed to DES and analytical models, MPs, such as mixed integer programs, deal with
combinatorial optimization problems. In the majority of cases (>60%), the objective function of the optimization problem includes
under/overtime or under/overutilization. Those performance criteria are rarely used by themselves but are usually part of a multiple
objective formulation. The use of multiple objectives in MPs, as is the case in general, is increasingly popular. In 2010, less than 20%
of mathematical formulations found in the literature still restrict themselves to a single objective. Their popularity can be explained in
two ways. First, the development of better solvers makes it increasingly practical to use them. Second, defining multiple objectives
allows capturing stakeholder preferences more realistically.
Despite the increasing complexity of the objective functions of MPs, there are no indications that the same would be true in respect to
their constraints. In other words, the variety of constraints used in MPs seems to be constant. The most frequently used constraints
are resource related, which in many cases relate to the OR (under-, regular or overtime) or medical personnel. Regularly applied
constraints, which do not focus on a given resource, are priority constraints (a high priority patient always needs to be served before
a low priority patient), demand related constraints (a given specialty needs to be given a certain amount of OR time) and release
related (a patient belonging to a given category needs to be served before a given deadline). As in Min and Yih [106], the decisions in
most of the mathematical programs apply to the elective patient. In their work, a stochastic mixed integer programming model is
proposed and solved by a sampling based approach. The surgery durations, the LOS, the availability of a downstream facility (ICU)
and new demand are assumed to be random with known distributions.
In some cases mathematical programs are too difficult to solve within a reasonable time limit and therefore heuristics are proposed. In
Fei et al. [60] a column generation based heuristic is used to solve the patient scheduling problem. In their setting, a column corresponds
to a feasible plan, in other words, the assignment of surgical cases to an OR. Roland et al. [128] propose a method, which includes
the assignment of cases to ORs, planning days and operating time periods. The NP-hard problem is tackled by means of a genetic
algorithm. Similarly to mathematical programming, also heuristics are in most of the cases used for scheduling tasks involving the
elective patient. Noteworthy is that in 2006 all heuristic methods found in the literature were time assignment problems whereas in
2010, as a result of a gradual decrease, this was true for only 20% of the articles as date and room assignment problems become
more popular.
75
5.8. Applicability of Research in Practice
Many researchers provide a thorough testing phase in which they illustrate the applicability of their research. Whether applicability
points at computational efficiency or at showing to what extent objectives may be realized, a substantial amount of data is desired.
From Figure 41 and Table 9, we notice that most of this data is on real health care practices. This evolution is noteworthy and results
from the improved hospital information systems from which data can be easily extracted. Unfortunately, a single testing of procedures
or tools based on real data does not imply that they finally get implemented in practice. Lagergren [85] indicates that the lack of
implementation in the health services seems to have improved considerably. Figure 41 shows, however, that only a very small share
of the articles report on actual implementation. An exception to this is Wachtel and Dexter [157] who introduce a website, which is
used by the hospital under study to decide on the exact times patients have to arrive to their surgery appointment. The problem tackled
by the authors arises from the fact that a case is often started earlier than scheduled, but it cannot be known in advance if it will happen
or not. Patient availability must therefore be balanced against patient waiting times and fasting times. Daily applicability is entailed by
their method. However, there are problems, which have to be solved on a less frequent basis. An example is the application of a case
mix model that is applied every year, clearly resulting in a different degree of implementation. A clear comparison of manuscripts on
this aspect is hence not straightforward. Even if the implementation of research can be assumed, authors often provide little detail
about the process of implementation. Therefore, we encourage the provision of additional information on the behavioral factors that
coincide with the actual implementation. Identifying the causes of failure, or the reasons that lead to success, may be of great value to
the research community [26].
100%
90%
80%
70%
60%
No testing
50%
Theoretic data
40%
Based on real data
Implemented and applied
30%
20%
10%
0%
2005
2006
2007
2008
2009
2010
Figure 41: Even though most data used in the literature is based on real data, this does not mean that the methods are applied in
reality.
76
In many contributions a problem is solved and applied to the problem setting specific to one single hospital and it is unclear whether
or to what extent a method is applicable to another setting. In order to justify the generality of their modeling assumptions Schoenmeyr
et al. [131] surveyed several hospitals. Introducing generalizable methods makes it easier to spread and implement good working
operations research practices to more than one hospital.
Only limited research has been done to study which planning and scheduling expertise is currently in use in hospitals. Using a survey,
Sieber and Leibundgut [133] reported that the state of OR management in Switzerland is far from excellent. A similar more recent
exercise for Flemish (Belgium) hospitals is described in Cardoen et al. [30]. It seems contradictory that so little research is effectively
applied in a domain as practical as OR planning and scheduling.
No testing
65,161,162
Data for testing
theoretic
6,7,13,14,37,49,51,54,55,56,64,73,77,78,79,83,86,87,88,89,90,91,94,96,98,110,111,120,123,136,137,138,143,
164
based on real data
2,3,5,8,9,10,11,12,15,16,22,23,24,25,27,28,33,34,35,36,38,39,40,41,43,45,46,48,50,51,52,57,58,59,60,61,62,6
7,68,69,70,71,72,74,75,76,80,81,82,84,92,93,97,99,100,105,106,107,108,109,111,113,114,115,117,118,119,1
20,121,122,123,126,127,128,130,131,132,139,140,141,144,145,146,147,148,150,151,152,153,154,155,156,15
7,163,166,167,168
Implemented and applied18,19,20,60,70,128,141,144
Table 9: Both theoretic and real data are frequently used for testing purposes.
5.9. Opportunities for Future Research
Our review suggests that methods introduced in the literature are rarely implemented at hospitals and, if implemented, details usually
remain unpublished. Both the problem of low success rates of implementations and the lack of reports could be mitigated by actively
involving surgeons, head nurses and IT personnel as coauthors. In order to avoid developing scheduling software that is only specified
to the needs of one hospital, it might be wise that projects cover several hospitals. Including more than one hospital in a study provides,
besides generalizability, also other opportunities: resource pooling on the level of emergency ORs, anesthesia rooms, equipment or
even nursing staff can lead to an integrated approach, which profits all participating hospitals. Integration is also an important concept
within the hospital itself. Considering its importance, it appears there is an opportunity to study the role of supporting facilities such as
77
the anesthesia department, PACU, ICU and/or wards in an integrated way with the OR. Similarly, important as integrality is the
incorporation of uncertainty. First and most prominently, uncertainty is accounted for in respect to patient surgery times, which are
unknown until surgeries are actually realized. Second, it is unknown whether the operating room will be available at the planned surgery
start as an emergency could occupy it. Third, while booking a surgery into a certain slot, it is unknown whether the slot might be
needed to allocate a future more urgent patient. Even though we see that uncertainty is frequently incorporated, the question arises
whether it should be a prerequisite for an algorithm to take aspects of uncertainty into account, i.e., is a strictly deterministic scheduling
approach able to provide the robustness required in reality? These and other open questions present many opportunities for future
research related to OR planning and scheduling.
5.10.
Conclusion
In this chapter we have studied and described recent trends in the field of OR planning and scheduling. Based on the data, we found
that most attention is given to elective patients, and even though often not stated explicitly, it is in many cases implied to be an inpatient
setting. Less frequently occurring in the literature are methods which consider outpatients. This is surprising as outpatient care is
gaining in importance and therefore we would expect to observe an increasing amount of literature dealing in this area. We also
observed a gradual shift from determining the exact time of a surgery to problems related to date and/or room assignments. With
respect to the performance measures considered, we found that overtime is the most popular measure and that preference related
criteria are gaining popularity. Noteworthy is the fact that preferences are increasingly used in multi-criteria settings. Mathematical
programming, DES and Heuristics are the techniques most frequently used. It is also true that the majority of articles present results
based on real data. However, it is important to note that this does not imply that the
5.11.
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6. Literature Review: Patient
scheduling
ABSTRACT
Hospitals are under growing pressure to cut costs, meanwhile sustaining the level of service provided to the patients. One part of
the solution to this difficult task is to improve the efficiency of the operating rooms by using a sound patient scheduling framework.
This can be done by identifying and applying good operations research methods from the healthcare literature. This literature review
tries to help the reader to achieve this goal, firstly by pointing out relevant articles and, secondly by identifying and describing some
of the major research groups in the field. Each research group is approaching the patient planning problem in a different way as,
for example, some groups look at timing decisions such as finding an appropriate surgery date (e.g., Dec. 16) or time (e.g., 11:30)
for a patient, whereas some include decisions regarding the surgery location, that is, determining an appropriate operating room.
We found that seven major research groups are focusing on patient scheduling related problems and that their findings can be
highly relevant to achieve a better and more efficiently used operating room. Reading about the research groups hopefully helps
the reader, on the one hand, to get an overview over the patient planning literature and, on the other hand, to identify useful
methods.
6.1. Introduction
Hospitals are under increased pressure to cut costs meanwhile requested to keep or even improve their level of service. Given this
difficult task, many managers are trying to realize cost savings at one particular part of the hospital, namely the operating rooms
(OR). Tools from the operations research literature can help hospital managers to achieve this difficult task and to increase the
efficiency of the OR’s by improving the way patient planning and scheduling is done.
This review focuses on those topics from the surgery planning and scheduling literature that are primarily about the patient. Problems
of this kind involve timing decisions such as finding an appropriate surgery date (e.g., Dec. 16 2012) and time (e.g., 11:30 AM) for
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a patient but also decisions regarding the surgery location such as determining an appropriate OR. A more general literature review
on OR scheduling can be found in Demeulemeester et al. [10] and Cardoen et al. [8].
Excluded from this review are scheduling problems related to the discipline level (e.g., pediatrics), the surgeon itself or to any other
medical entity. Not many research groups devote a substantial amount of their scientific effort on researching problems specific to
patient scheduling. Moreover, each research group uses, from the methodological perspective, a restricted set of tools and reapplies
or extends similar methods across the scientific contributions they make. It is therefore straightforward to introduce the research
groups one by one, together with their applied methodology. Discussing a research group at an earlier (or at a later) stage does
not mean that their method is less (or more) sophisticated than the one of another group, but merely reflects a sequencing choice,
which supports a logical and continuous way of describing the research field.
In hospitals the demands of both patients and medical personnel are important and have to be considered alongside cost
considerations. This means, on the one hand, that patient demands have to be satisfied both at the individual level (e.g., by
decreasing patient waiting time) and on the patient group level (e.g., considering patient priorities). Additionally, surgeon and hospital
requirements need to be met (e.g., allowing surgeons to schedule patients into preferred slots). Cost considerations, on the other
hand, translate into strategies, where the amount of salary paid to medical personnel is decreased by ensuring that an OR closes
in time and therefore no overtime costs occur. The strategy can also mean that we try to increase profitability by serving, for
example, more patients.
To cope with the many aspects related to patient scheduling, operations researchers are continuously developing new and
innovative methods or searching and defining best practices. Recent problems found to be important by the operations research
community are OR integration (section 2.1-2.2) and uncertainty (section 2.6). OR integration ensures that the created surgery
schedule is to allow for a smooth patient care in pre/post-surgery facilities such as the post anesthesia care unit (PACU). Uncertainty
is incorporated in order to achieve robust scheduling. It is considered on different levels. First and most prominently, it is accounted
for with respect to patient surgery times, which are unknown until surgeries are actually realized (section 2.6). Second, it is unknown
whether the OR will be available at the planned surgery start time as the arrival of an emergency patient could occupy it (section
2.1). Third, while booking a patient for surgery into a certain slot, it is unknown whether that slot could be needed to allocate a
future more urgent patient (section 2.5). The second and third problems are similar as in both problems future arriving patients
have to be anticipated and prepared for.
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6.2. Literature review
The following section contains a summary on the research activity observed in the field of patient scheduling. For each group only
those key articles are given, which are directly linked to patient scheduling. Some groups are more focused on patient scheduling
problems than others. The research output of the former groups will be described in more detail. Articles of a group that are highly
related are grouped together and are referred to by the same capital letter (e.g., A1, A2 and A3). Single contributions not associated
to a particular research group are discussed at the end of the text. Throughout the text, boldface and uppercase letters indicate
vectors and random variables respectively.
6.2.1. The managerial way
The first group publishes material that is highly managerial in nature and deals with applicability and implementability issues. The
group is centered around Jan Vissers who works at the Erasmus University Medical Center in the Netherlands. The people
associated to the group are:
Vissers (Jan), Bekkers (Jos) – Erasmus University Medical Centre (Rotterdam) | Adan (Ivo), Dellaert (Nico) – Eindhoven University
of Technology | Jeunet (Jully) - CNRS (Université Paris Dauphine)
Article:
 2011 | Improving operational effectiveness of tactical master plans for emergency and elective patients under stochastic
demand and capacitated resources [1]
The article describes and provides routines which can be followed step by step by practitioners. First, an approach is given to
determine the amount of optimal OR capacity dedicated to patient types (tactical). Second, rules are given to guide the process of
populating the dedicated capacities with patient instances. Third and finally, rules are given to guide the decision making process
on the day of surgery. What distinguishes their article from most others in the field is the fact that it presents a holistic approach
including many levels of the patient scheduling process. Moreover, it is also rare that authors give instructions usable on the surgery
day itself.
The authors of the article use a highly descriptive way of introducing their methods and rules. In order to give a rudimentary
understanding of the basics of their work, we will use an explanation method that is based on state vectors. The values of the state
vectors are calculable by the routines given in the article. The vectors are:
 Capacity vector c, where ci,t is the number of category ‘i’ patients to be operated on day ‘t’. The vector c is defined in
advance (weeks or months) and is based on the average number of arriving patients of that category (similar to a master
surgery schedule). It does not contain actual arriving patient instances.
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 Booking state s (operational plan), where si,t is the number of ‘i’ patients assigned to day ‘t’. The vector s is set ‘7’ or
more days prior to the surgery date and contains real patient instances. Patients in s are getting notified about their
surgery date.
 Planning vector p (executed plan), where pi,t is the number of ‘i’ patients to be certainly operated on day ‘t’. The vector is
defined in the morning of the surgery day and is therefore regarded to be fixed.
Goal programming is used to determine the value of vector c. In order to prepare for arriving non-electives (urgent or emergency
patients), slack capacity is included. Vector c provides a template guiding the process of allocating actual patients to vector s. The
allocation can happen in a strict or flexible way, depending on whether unused space is allowed to be occupied by patients from a
different patient category. There might be unused space available if, of a given patient category, a less than expected number
arrives. In order to calculate p, in the morning of the surgery day, a rule based daily scheduling algorithm is applied on s. This
allows p to incorporate information with respect to still ongoing surgeries leftover from the night shift. At this point, as daily
emergency arrivals are still not being realized, capacity for emergencies is still being reserved in the form of slack capacity. While
determining p in the morning of the surgery day, if an elective surgery is estimated not to fit, it will be late canceled (severe action)
and scheduled for day t+7. This may, in return, interfere with a patient assigned to s i,t+7 causing an early cancelation (less severe
action). Arriving emergencies are handled in the same way as electives, as also they have to fit the schedule, i.e., they are not
allowed to take the place of an elective. The difference between an emergency and an elective patient is that the emergency patient
can be allocated into slack time. If an emergency does not fit the schedule, it is deferred to another hospital. Figure 42 gives an
illustration of the whole procedure.
Figure 42: Goal programming is used to arrive to the tactical plan ‘c’. Scheduling rules are applied to derive an operational plan ‘s’.
Then, each morning, the daily scheduling algorithm is applied in order to derive the executed plan ‘p’. The same algorithm is used,
at the moment an emergency arrives, to decide whether the emergency should be accepted and scheduled or deferred to another
hospital. [2]
Whether a surgery is expected to fit a schedule depends not only on the OR occupancy but also on the bed loads in the medium
care unit (MCU) and intensive care unit (ICU). Additionally, it is required that nurses are available to support the ICU.
An evaluation of the scheduling system is given through a Discrete-event simulation (DES) study where hospital efficiency measures
are compared against patient satisfaction factors. Hospital efficiency measures the consistency in which a plan was carried out
whereas patient satisfaction relates to waiting time, elective late cancelation and non-elective deferral.
In their model hospital resources are exclusively devoted to a targeted discipline, namely cardiothoracic pathology. Hence, it would
be interesting to see the results of their method applied to the general OR setting.
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6.2.2. The exact way
Developing efficient patient scheduling algorithms is a challenging task as patient scheduling problems are generally combinatorially
complex and therefore difficult to solve. The combinatorial complexity of a problem depends on factors such as the problem objective
and the restricting constraints. Research groups would generally, if possible, provide exact solution approaches. Exact solutions
are especially from an academic perspective more interesting and favorable than approximate ones. The research group centered
around Erik Demeulemeester is using exact methods in the major part of their research. The people associated to the group are:
Demeulemeester (Erik), Ma (Guoxuan), Samudra (Michael) – KU Leuven | Belien (Jeroen) – Hogeschool-Universiteit Brussel |
Cardoen (Brecht) – Vlerick Leuven Gent Management School
Articles:
 A: 2009 | Optimizing a multiple objective surgical case sequencing problem [6]
 B: 2008 | Sequencing surgical cases in a day-care environment: An exact branch-and-price approach [7]
In articles A and B, patient sequences and surgery starting times are determined. Their scheduling algorithm is applied on the fixed
patient sets of each surgery block. A separate OR assignment step is therefore obsolete as a block is by definition allocated to an
OR.
Even though in A and B identical problems are solved, each of the articles describes different solution methods. In A, solutions
based on Mixed Integer Linear Programming (MILP) are proposed, whereas in B, column generation as part of an enumerative
branch-and-price framework is discussed.
The problem objectives of both A and B are:





Minimize the sum of surgery starting times carried out on children;
Minimize the sum of surgery starting times of prioritized patients;
Minimize the number of patients coming from further than 150km and scheduled for early surgery;
Minimize the number of overtime periods in the recovery areas;
Minimize the peak bed occupancy in PACU 1 and PACU 2 (patients first enter PACU 1 and are later transferred to PACU 2).
The cost coefficients associated to each objective are a product of two components. The first component is a normalization factor
mapping the coefficient to a value between 0 and 1. The second component is chosen by the hospital management and reflects
the importance of the objective relative to other objectives. Constraints in their model are:





91
Surgeons need to allocate their surgeries into the time window provided to them (by the master surgery schedule);
Each surgery has to be given a surgery time;
Patients required to undergo pre-surgical tests need to be scheduled after a reference period;
Peak occupancy in PACU 1 and PACU 2 cannot exceed the respective capacities of the units;
The number of medical equipment used in parallel cannot exceed their availability;
 The OR requires a long cleaning session after the treatment of a patient infected by the bacterium called Methicillinresistant Staphylococcus aureus (MRSA).
In A, three MILP based solution procedures are introduced. The first and the second are a basic and a preprocessed MILP. In the
preprocessed version, the implications of fixing each binary variable to 0 or 1 (probing) are investigated. Additionally, the possible
starting times of both MRSA and non-infected patients are limited. In the 3rd, the iterated MILP, the starting times of a particular set
of surgeries is iteratively fixed.
In B, a column (representing a surgery sequence) oriented reformulation of the MILP is given. This allows to enforce most constraints
within the column, i.e., independently of other columns. The constraints which cannot be enforced within a column are the ones
related to shared resources (beds, medical instruments).
Figure 43 gives an outline of the column generation approach. In the model, the restricted master problem is solved with an initial
set of existing columns. Through a process called pricing, new columns with reduced costs are iteratively identified and added to
the set of already selected columns. The process terminates when no more columns price out negatively (in which case the relaxed
problem is solved to optimality).
Figure 43: In the column generation approach, iteratively newly generated columns (surgery sequences) are added to the problem.
In the restricted master problem, the decision variable (selecting a column) is relaxed. In order to arrive to an integer solution, the
column generation approach is embedded into an enumerative branch-and-bound framework. The authors propose several
branching strategies and argue that it is more effective to branch on the individual surgeries than on the decision variables
themselves.
If the MILP or column generation based procedures are given a time limit (e.g., 5 minutes), we are likely to arrive at a sub-optimal
solution. In many cases though, the algorithm terminates within the time limit and hence provides the optimal solution (iterated
MILP is an exception as a probability based selection procedure is used).
6.2.3. Heuristics
92
Approximate solutions are used if it is difficult to find an efficient and exact algorithm to solve a particular problem. The group
centered around Nadine Meskens seems to prefer solution methods of exact nature but turns to heuristics if they seem necessary.
An example is their column generation approach. While the column generation of Demeulemeester’s group is ensured to be optimal
by a branch-and-bound framework, Meskens’ group uses a column generation based heuristic. The people associated to the group
are:
Meskens (Nadine), Fei (Hongying) – Louvain School of Management, Catholic University of Mons | Chu (Chengbin) – Ecole Central
Paris | Combes (Catherine) – University of Lyon
Articles:
 A: 2009 | Solving a tactical operating room planning problem by a column-generation-based heuristic procedure with four
criteria [15]
 B: 2009 | The endoscopy scheduling problem: A case study with two specialized operating rooms [17]
 C: 2010 | A planning and scheduling problem for an operating theatre using an open scheduling strategy [16]
 D: 2010 | Using constraint programming to schedule an operating theatre [20]
Articles A, B and C are related and are hence discussed together. In A, the date and OR assignment problem is solved in an open
scheduling problem setting. In open scheduling, a surgery can be assigned to any OR-day (e.g., OR 5 next Monday) within the
planning horizon. Their objective is to minimize costs of unexploited opening hours and overtime. A column based formulation is
used, where a column represents the surgeries that are assigned to an OR-day. This way, a set of columns will represent a solution.
The decision variables select from the pool of possible columns the subset that represents the optimal solution. A problem related
to column generation based approaches is that the decision variables may not be integral, i.e., the columns are not selected
decisively. To overcome this problem a heuristics procedure is applied.
B and C extend the problem setting of A by including a sequencing step. The assignment and sequencing step are independent of
each other, i.e., the sequencing step provides no feedback to the OR-day assignment step. In B, a special case of the scheduling
problem is introduced where only 2 ORs are considered. The problem is modeled as a two-machine open-shop scheduling problem
and is solved by the Gonzalez-Sahni [5] algorithm.
A more general problem setting is used in C, where the sequencing problem is solved by a genetic algorithm. The genetic algorithm
ensures compliance with two major constraints. Firstly, it is ensured that a surgeon will perform one surgery at a time. Secondly,
recovery bed limitations are obeyed as patients are simultaneously scheduled for the OR and the recovery room. The objective of
their problem is to minimize the makespan in ORs and recovery rooms.
In D, the problem of determining good surgery sequences and starting times is solved by a constraint programming approach.
Constraints ensure that two surgeries are not carried out simultaneously in the same OR. Their model is based on a threedimensional Boolean matrix otr(o,t,r), where ‘o’ denotes the surgery, ‘t’ the time slot and ‘r’ the room. The Boolean corresponding
to a surgery ‘o’ scheduled at time ‘t’ in operating room ‘r’ takes the value 1, whereas all other variables are set to 0. Constraints
ensure that a surgery is performed continuously without stops in between. Constraints are also set for earliest availability and duedates.
93
Meskens’ group deals separately with the date and the time assignment step. In their work, an inpatient setting is presumed which
means highly variable surgery durations and frequent emergency disturbances. An interesting extension of their work could therefore
use stochastic surgery durations and include aspects of emergency patient arrivals.
6.2.4. Uncertainty
Noticeable is the way how research groups relate to approximate methods. Demeulemeester’s group generally avoids them,
Meskens’ applies them intermittently, and the group centered around Xiaolan Xie heavily, almost exclusively uses heuristics.
Whether a group uses exact or approximate methods might be, on the one hand, dependent on the group’s attitude. On the other
hand, the group might have no other option, as after a problem reaches a certain level of complexity, it can become impossible to
solve it exactly. The level of complexity when researches tend to turn to heuristics seems to be reached if uncertainty is introduced.
The people associated to the group are:
Xie (Xiaolan), Lamiri (Mehdi), Augusto (Vincent), Grimaud (Frédéric) – Ecole Nationale Supérieure des Mines de Saint Etienne
(Engineering and Health Division)
Articles:




A1: 2008 | A stochastic model for operating room planning with elective and emergency demand for surgery [24]
A2: 2009 | Optimization methods for a stochastic surgery planning problem [23]
A3: 2008 | Column generation approach to operating theater planning with elective and emergency patients [25]
A4: 2007 | Operating room planning with random surgery times [22]
 B: 2010 | Operating theatre scheduling with patient recovery in both operating rooms and recovery beds [3]
Xie’s group focuses on the date assignment step in all articles except B. In B, a deterministic time assignment problem is solved.
Additionally, the patient to OR assignment step complements the general patient to date assignment in A3 and A4.
In A1 and A2, it is unnecessary to determine the patient to OR assignment as only the cumulated capacity of ORs is considered.
Since non-electives might occupy some of the OR’s capacity, the accessible capacity left to electives is uncertain. Implicit in the
problem is therefore the determination of allocated elective OR capacity.
Formally, an assignment between elective patient ‘i’ and day ‘t’ has to be found, given that the total OR capacity is decreased by
random capacity Wt (arriving non-electives). The objective includes the minimization of patient related costs (cit ) and overtime.
Patient related costs will depend on the patient type and the assigned date. Patient costs are independent of non-elective arrivals
and are not influenced by the realizations of random capacity Wt , i.e., the stochastic problem only affects OR overtime costs. The
objective is:
+
J∗ = Minimize J(x) = ∑ ∑ cit xit + ∑ ct Ewt [(Wt + ∑ di xit − Tt ) ]
i
94
t
t
i ∈ It
(1)
where ct is the unit cost of overtime and Tt is the total capacity available on day ‘t’.
In A1 and A2, a Monte Carlo type of solution method is presented. In A1, the solution algorithm involves a MIP, while in A2 a
heuristic is used. Since a Monte Carlo solution method is inherently approximate, the solution to the MIP can only be an
approximation.
In the Monte Carlo solution, Ewt is approximated by sample averages. The method is consequently called sample average
approximation (SAA). Other names of the method are: the sample path or stochastic counterpart method. The resulting objective
function is formulated as:
+
K
1
JK∗ = Minimize JK (x) = ∑ ∑ cit xit + ∑ ct ∑
[(wtk + ∑ di xit − Tt ) ](2)
K k=1
i
t
t
i ∈ It
where wtk is the k-th sample from Wt and ‘K’ is the total sample size. The SAA method optimizes over a finite number of scenarios
where a scenario represents the outcome of a random process. In A1 and A2, the random process relates to Wt only. Other
examples of a random process in general include: elective surgery times, length of stay (LOS) in the PACU or other units. Stochastic
considerations can also be applied to hard constraints. In that case, it is ensured that the constraint holds for every scenario ‘k’.
In A2, the authors prove that their problem’s SAA convergence rate to the true optimum is exponential. The quality of the SAA
method with respect to the deterministic counterpart is investigated in A1. The authors show that for a moderate sample size K
(~20) a better result is obtained by SAA than by the deterministic method. In the deterministic model, Wt = E[Wt(fixed non-elective
OR time)] and the corresponding solution is calculated by a MIP.
In A2, several constructive and improvement heuristics as well as meta-heuristics are introduced. Depending on the heuristic,
different moves are defined. Moves are, for example, to add a surgery ‘i’ to period ‘t’ or to exchange two surgeries. The quality of
a move is determined by assessing the gained benefit according to objective (1). The authors’ results suggest that Tabu search is
the most suitable heuristic and performs best for large instance sizes. It outperforms the Mont-Carlo type of MIP solution with fixed
time budget. The best constructive heuristic is to sort surgeries in increasing order of their surgery durations. The surgeries are
added to the schedule one by one, maximizing at each step objective (1). A counterintuitive observation of the authors is that as
the variability of Wt decreases, so does the solution quality of the heuristic.
In A3, the problem setting is widened by the patient to OR assignment step. The applied solution method is column generation.
The decision variables are yip (patient ‘i’ is assigned to plan ‘p’) and ztsp (plan ‘p’ is assigned to OR ‘s’ on day ‘t’). A set of feasible
plans will give the solution. The variable λp indicates whether plan ‘p’ is selected or not. Similarly to the work of Meskens, heuristics
are used to ensure integrality of λp’s and to improve existing feasible solutions.
Overtime costs and the penalty for exceeding the OR-day availability are set by the authors to 500€/hour and 3000€/hour
respectively. The cost of underutilization is 1.75 times less than the cost of overtime. The authors set the costs as fixed and choose
values which are typically used in French hospitals.
We think that an interesting future extension of their method could investigate the case when emergencies are regarded to arrive
to the entire hospital instead of single ORs, i.e., instead of excluding a random amount of emergency capacity from each OR, a
total amount of emergency capacity could be excluded from all the ORs in a shared manner. This is similar to a queuing system,
95
where modeling one common queue for all the servers or a separate queue for each server will cause the system to behave
differently.
In A4, the problem setting is in two major points different from A3. First, not only non-elective capacity requirements are stochastic
but also elective surgery durations. Second, the objective function is a simplified version of the one in A3 as it does not include a
penalty component for capacity violations nor excessive idle time.
The time assignment problem is solved in B. In their model, an integrated approach is considered as the availability of recovery
beds and porter teams is included. Additionally, the model allows for patient recovery in the OR. In the article, the authors chose
to minimize makespan and conclude that recovery in the OR is beneficial as soon as the ratio of recovery beds to ORs is lower
than 3/2.
6.2.5. Dynamic allocation problem
The group centered around Yuehwern Yih also incorporates aspects of uncertainty with a Monte Carlo type of solution approach.
The dynamic scheduling problem, arising when future patient arrivals are taken into account, is solved by value iteration. The
people associated to the group are:
Yih (Yuehwern), Min (Daiki) – Purdue University (School of Industrial Engineering, West Lafayette)
Articles:
 A: 2010 | An elective surgery scheduling problem considering patient priority [26]
 B: 2010 | Scheduling elective surgery under uncertainty and downstream capacity constraints [27]
In A, the dynamic date assignment problem with ‘i’ different types of priority patients is solved. The described model finds the best
action vector a = (a1, a2, …, aI) given a state vector s = (s1, s2, …, sI), where ai is the number of patients of priority class ‘i’ selected
to undergo surgery in the next period and si is the number of priority class ‘i’ patients waiting for surgery. Vector a has to be chosen
carefully as if too many patients are booked and many high priority patients will arrive in the future, high overtime costs become
inevitable. Future demand is denoted by vector d = (d1, d2, …, dI). The problem objective is to minimize the amount of overtime
and patient postponement costs. The corresponding Bellman optimality equation is:
v(s) = min c(s, a) + λ ∑ p(d)v(s − a + d)
a
(3)
d
where v(s) is the value function and λϵ[0,1] the discount factor. The discount factor denotes the present value of future costs
c(s, a) and is usually chosen to be close to 1. The cost vector c(s, a) is the sum of postponement and overtime costs.
I
∞
c(s, a) = ∑ ci (si − ai ) + co ∫ (x − cap) dFa (x)
i
96
cap
(4)
where ci is the cost coefficient of priority class ‘i’, co is the unit overtime cost, cap is the capacity of ORs (without overtime) and
Fa (x) is the cumulative distribution function of total surgery durations. Figure 44 shows how the different cost factors sum up to
yield value function v(s).
Figure 44: As more and more patients are scheduled, overtime is increasingly dominating the cost function. [26]
As shown by Figure 45, the scheduling rules are monotonic. For example, the more patients in the waiting list, the more patients
will be scheduled. The same is true for priorities, i.e., a waiting list containing a larger number of high priority patients will schedule
at least as many patients as if the list would contain many low priority patients.
Figure 45: The solid line denotes the case without priorities. Different points represent scenarios consisting of different mixes of
priority classes. [26]
Their model shows how practices from the reinforcement learning literature are applicable to healthcare scheduling problems. Their
procedure, the value iteration algorithm, is one of the elementary reinforcement learning methods. We would be interested to see
whether some other methods from the field would be applicable in a similar fashion. For example, reinforcement learning algorithms
are used in nondeterministic environments where state s and action a result in an uncertain state s’. This could be used to model
patient cancelation.
97
In B, a technique resembling Xie’s A1 is introduced. They resemble as they apply the same methodology (SAA) and define
coinciding objectives (minimize patient related and overtime costs). In B, stochastic aspects related to surgery durations, LOS in
the ICU and non-elective patient arrivals are considered. The complexity of their problem setting is increased by restricting surgery
assignments to a subset of surgery blocks. This is an important consideration as many hospitals use block scheduling in reality. To
deal with this increased complexity of the problem, a stochastic programming model with recourse is defined where stochasticity is
only included in the recourse function. As stochasticity only affects overtime, the objective of the recourse function likewise includes
the minimization of overtime only.
A general problem is that the solution quality of approximation algorithms is hard to assess. In B, a method based on solution
averages is given to find the lower bound of a solution. The upper bound is found by evaluating the “true” objective value of a
suboptimal solution.
Yih’s group deals with two highly relevant parts of the patient to day assignment problem. In A, patient prioritization is considered
and capacity is reserved for future bookings. In B, uncertainty related to surgery duration, LOS and non-elective patient arrivals is
considered. Both A and B deal with different parts of the same problem, namely the robust assignment of electives to days or
blocks. It would be very interesting to see whether and how A and B could be effectively unified in one model.
6.2.6. Stochastic programming
The group centered around Brian Denton, similarly to Xie’s and Yih’s group, generally includes stochastic elements in its models.
The group often combines Monte Carlo type of solution methods and mathematical programming. Their preferred problem setting
includes the patient to day assignment step, sequencing and determining surgery starting times. The people associated to the
group are:
Denton (Brian), Viapiano (James), Erdogan (S. Ayca), Berg (Bjorn) – North Carolina State University | Gupta (Diwakar) – University
of Minnesota (Dep. Of Mechanical Engineering) | Huschka (Todd R.), Rohleder (Thomas) – Mayo Clinic | Batun (Sakine), Schaefer
(Andrew) – University of Pittsburgh (Dep. of Industrial Engineering)
Articles:






A1: 2007 | Surgical suits’ operations management [18]
B1: 2003 | A sequential bounding approach for optimal appointment scheduling [11]
B2: 2006 | Simulation of a multiple operating room surgical suite [13]
B3: 2011 | Dynamic appointment scheduling with uncertain demand [14]
C1: 2007 | Optimization of surgery sequencing and scheduling decisions under uncertainty [12]
C2: 2010 | Operating room pooling and parallel surgery processing under uncertainty [4]
Throughout their articles, the group consistently uses similarly defined performance measures and solves similarly formulated
decision problems. Therefore first both their preferred objective function and their regularly used decision variables will be introduced
and the exact details of each single article will only be shown afterwards.
98
Their universally used decision variable ‘xi’ denotes the time allowance for patient ‘i’, i.e., the amount of OR time reserved for the
patient. The surgery starting time of a patient is the sum of the time allowances assigned to all preceding patients. The variable
‘Zi ′ represents the actual surgery time of patient ‘i’. The two performance measures generally included in their models are waiting
‘W’ and facility overtime ‘L’. Waiting time targets either the surgeon or the patient. The third performance measure, idle time ‘S’, is
used in some of their articles and is usually defined with respect to the facility. Given patient ‘i’, the three measures are defined as:
Wi = max(Wi−1 + Zi−1 − xi−1 , 0)
Si = max(−Wi−1 − Zi−1 + xi−1 , 0)
n−1
L = (−Wn − Zn − ∑ xi + d)
i=1
where ‘n’ is the total number of scheduled patients and ‘d’ is the facility’s total capacity. Given cost coefficients c w, cs and c l , the
objective function can then be formulated as:
n
min {∑(ciw E[Wi ] + cis E[Si ]) + c l E[L]}
x
(5)
i=1
It is presumed that the surgeries are already sequenced in all articles of type B. In articles of type C also sequencing decisions are
included.
The group models surgery start times, as in railway scheduling [33], in a fashion that surgeries never start earlier than planned (in
case a previous surgery requires less time). This is the case as the performance measures are recursive in the variable ′Wi ′, a
variable which is always bigger than or equal to 0. This models a setting where, for example, each surgery is performed by a
different surgeon who is only available at the exact starting time of the surgery.
Denton’s group formulates their problems as two-stage stochastic programs. In stochastic programming, “the decision maker takes
some action in the first stage, after which a random event occurs affecting the outcome of the first-stage decision. A recourse
decision can then be made in the second stage that compensates for any bad effects that might have been experienced as a result
of the first-stage decision. The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions
(a decision rule) defining which second-stage action should be taken in response to each random outcome” (Wikipedia - stochastic
programming). The first stage decisions are ′xi ’, whereas the second stage decisions correspond to the calculated amount of
waiting, idling and overtime.
Until now, the common properties of the group articles were described. Next, a short description on article A1 is given. The article
describes three models, of which two will be mentioned together with their given preliminary solution approaches.
The first of the two models describes the elective surgery booking problem of prioritized patients. It is solved, as in Yih, with the
value iteration algorithm. While in Yih the question over the amount of OR capacity to be reserved for the next time period is being
answered, in A1, the (preliminary) answer to a different question is given. The question at hand is: how much downstream capacity
should we reserve on each future day ‘t’ for priority class ‘i’, given stochastic future demand?
The second of the two problems deals with the surgery sequencing problem. The objective of the introduced model is defined by
(5). The sequencing problem is computationally expensive since the set of possible sequences grows factorially in the number of
99
surgeries. Additionally, the objective value does not depend on the sequence only, but also on the surgery allowances. In order to
gain insights into the problem structure, the author radically simplifies the modeling assumptions. In the article, solution guidelines
based on stochastic ordering are being formulated, suggesting that surgeries should be sequenced in the order of increasing
variance of their surgery duration.
Articles B1, B2 and B3 exclude the sequencing step and contain methods to determine OR time allowances. In B3, the sequencing
step is obsolete as the problem setting excludes patient priorities. The patients are therefore scheduled on a FCFS (first-come,
first-served) basis. In the articles of type B, the objective function (5) is used, with the exception of B3, where idle time is excluded.
In B1, time allowances are determined based on objective (5). In the article the stochastic linear programming formulation is
introduced which will reappear in many of their later works. As it is common for the group, the authors try to exploit the models’
structural properties in an effort to increase its solvability. The optimal solution is derived using a variation of the L-shaped algorithm
[9]. A finding discussed in the article is that the optimal solution will exhibit a dome shape. In other words, assuming i.i.d. surgery
times and equal waiting and idling costs, allowances will initially increase and towards the end of the surgery day decrease again.
Also in B2, time allowances are determined with objective (5). In the article, a Simulated Annealing (SA) approach using Monte
Carlo sampling is introduced. The SA procedure will in each step evaluate a new schedule on the basis of the Monte Carlo
procedure given sample size ‘K’ (similarly to (2)). A new schedule is generated by perturbating the current schedule.
In B3, as in previous B articles, time allowances are determined. The objective is defined by (5), but without considering idle time.
The article, as A1 and Yih’s A1, deals with the dynamic appointment scheduling problem. The major difference in respect to Yih’s
A1 considers the task description and the methodology. The task description is different from Yih’s A1: a set of differently prioritized
patients is considered whereas in B3 patients are equally prioritized, i.e., patients are considered in sequence of their arrival.
Additionally, in Yih’s A1, patients are assigned to a virtual one period overall capacity (sum across ORs), while in B3 surgery
starting times are determined for one OR-day. The two models are also from a methodological perspective different. Yih applies
the value iteration algorithm, whereas in B3 a multi-stage stochastic linear program is used.
In Figure 46, the modeling idea followed in B3 is exemplified by a simple case. In the example, the first patient is scheduled and
two additional patients may arrive. Patients are scheduled with imperfect knowledge about both the number of future arriving
patients and their respective surgery durations. Each new request is therefore treated as an additional stage in the stochastic
program.
Figure 46: Multi-stage stochastic linear program. Outcomes and probabilities in case 1 patient is scheduled and 2 additional ones
may arrive. [14]
100
The C type of articles are different from the B type of articles in one major point. Whereas in B allowances were determined and a
surgery sequence was assumed to be given, in C also the sequencing step is carried out.
In C1, patient sequences and allowances are determined using objective (5). The model is represented as a two-stage stochastic
program. As the model is highly complex and an exact solution approach would be computationally expensive, an interchange
heuristic is proposed. The heuristic, at each iteration step, performs a randomly generated pairwise interchange and checks the
resulting solution quality. The quality of a solution is evaluated by a traditional two-stage stochastic model with patient sequences
fixed. In addition to the interchange heuristics some easy to implement constructive heuristics are presented. The most effective
constructive heuristic sequences surgeries in increasing order of the variance of their surgery duration. This is a strategy that is
also praised in many of their other articles.
In C2, ideas from C1 are further developed. The problems tackled in C2 involve:
 The number of ORs to open;
 Surgery to OR assignment;
 The sequence of surgeries;
 The start time for each surgeon in the morning.
The objective function is similar to (5), but instead of including waiting time a factor representing OR opening costs is added. In the
article, the authors exploit the fact that a surgery is dividable into 3 parts: preincision, incision and postincision. Surgeons are only
required to be present at the incision part of the surgery and are able to freely switch between ORs. The authors refer to this
situation as parallel surgery processing, since surgeries assigned to the same surgeon might be carried out in parallel (as long as
the incision parts are not overlapping). In order to make effective use of parallel surgery processing, it is necessary to handle ORs
as a pooled resource. This means that an open scheduling environment is assumed, i.e., not block scheduling.
Important is the fact that the length of each of the three parts of the surgeries is stochastic. The problem is once again tackled by
a two-stage stochastic model, where the first stage variables are the basic decision variables. Second stage decisions are
completion times, surgeon idle times and the overtime in each OR.
As the problem is computationally difficult to solve, the authors restrict the search space and define antisymmetry constraints with
respect to OR orders. Induced feasibility constraints stemming from the problem structure are used in a standard L-shape and an
L-shaped-based branch-and-cut algorithm. The theoretical cost reductions that can be achieved by using their method range from
~21% to ~59%.
Denton’s group is specialized in optimization methods where stochasticity plays an important role. Since stochastic optimization
problems are highly complex, the group generally searches for ways to exploit the structural properties of their problem instances.
Besides mathematical programming, they often turn to either constructive or improvement heuristics.
6.2.7. Robust schedule
101
The group centered around Erwin Hans approaches stochasticity in a less traditional way than previous groups and uses solution
methods, which are based on common sense. The group deals with two main problems. Firstly, the maximization of OR usage
given a fixed probability of overtime is solved. Secondly, they reduce the maximum waiting time of urgent arrivals. The people
associated to the group are:
Hans (Erwin), van der Lans (Marieke) – University of Twente | Wullink (Gerhard), van Houdenhoven (Mark), Kazemier (Geert) –
Erasmus Medical Centre
Articles:
 A: 2006 | Anticipating urgent surgery in operating room departments [34]
 B: 2008 | Robust surgery loading [19]
In B, the authors consider the robust surgery loading problem. Given a predetermined schedule (surgery to OR-days), the authors
discuss different reassignment strategies aiming at the maximization of OR utilization and the minimization of the risk of overtime.
The risk of overtime in a given OR depends on the variation of the surgery durations in the OR and on the amount of slack time
(buffer). In other words, the lower the total variance of the surgery durations, the less slack time is required. Figure 47 shows how
the portfolio effect can reduce the variance of the summed surgery durations in a 2 OR setting. In the figure (left), the standard
deviation of the summed surgery durations equals 2 ∗ √502 + 102 = 102 time units. In the second scenario (right), the
amount is reduced to √502 + 502 + √102 + 102 = 84.9 time units. The shown reduction in variance allows to use
less slack time and therefore a larger amount of OR time can be effectively used by surgeons.
Figure 47: Decreasing the variance of summed surgery time using the portfolio effect. [34]
In A, the focus is on reducing the waiting time of urgent patients. In order to achieve this, possible urgent surgery entry points,
referred to as break-in-moments (BIMs), are distributed evenly during the day. In Figure 48, BIMs are shown for a 2 OR case. The
break-in-interval (BII) represents the time intervals between BIMs. The optimization problem boils down to minimizing the maximum
BII.
102
Figure 48: Break-in-moments and break-in-intervals in a 2 OR setting. [34]
Different heuristics are introduced, which try to approximate the case where BIIs are of even length (ideal case). The authors
introduce different strategies with respect to the way how slack time is divided among ORs. They conclude that distributing slack
over all the ORs while performing BIM optimization performs best.
We think that using BIM optimization in a setting with a high number of ORs might have a lessened impact since if there are many
ORs also BIMs will occur more frequently. This is however not really a problem as we know from practice that in large hospitals
emergency patients are often only allowed to enter ORs allocated to their own discipline. BIM optimization can then simply be
restricted to ORs of the same discipline.
6.2.8. Summarizing table
In Table 10 and Table 11 the groups are compared according to a few selected attributes. The tables are not meant to give detailed
information over the groups, but to merely provide a way to easily classify and compare the different research groups. Therefore
attributes are selected only if they are valid for at least half of the articles of the given research group (e.g., if in 2 out of 4 articles
considerations over non-elective arrivals are incorporated then the keyword ‘Non-elective’ is going to be included into the column
‘Patient type’ of that group).
103
Group
Patient type
Patient assigned to
Uncertainty
Supporting facilities
Date
LOS
ICU, MCU
2.1
The
way
managerial
Inpatient,
Non-elective
2.2
The exact way
Outpatient
Time, Room
Pacu
Elective
Date, Time, Room
Pacu
Elective,
Non-elective
Date, Room
Emergency capacity
Elective,
Non-elective
Date, Room
Surgery
LOS in ICU
Elective
Time
Surgery duration
Elective,
Non-elective
Date, Time, Room
Surgery duration
2.3
Heuristics
2.4
Uncertainty
2.5
Dynamic
allocation problem
ICU
duration,
2.6
Stochastic
programming
2.7
Robust schedule
Table 10: Some groups frequently incorporate considerations over non-electives whereas other groups focus on electives only. The
table also shows that the patient to date assignment problem is solved most often.
104
Group
Constraints
Performance
measure
Solution technique
2.1
The
way
managerial
Patient
waiting
time,
Deferral, Cancelation
programming, DES
Utilization,
OR time, ICU/MCU Goalbeds,
ICU nurse hours
2.2
The exact way
Prioritized
patient
starting Pacutime,
Leveling bed usage in Medical
Pacu,equipment
Pacu overtime
Branch-and-Price,
capacity,
Column generation
MIP,
OR utilization, Makespan
Medical personnel
Constructive
Column generation
heuristic,
Patient
OR overtime
waiting
Slot time,
time
MIP, SAA
Patient
OR overtime
waiting
Slot time,
time
Dynamic
MIP, SAA
programming,
Patient
waiting
OR overtime, OR idle time
time,
Linear
SAA
Programming,
2.3
Heuristics
2.4
Uncertainty
2.5
Dynamic
allocation problem
2.6
Stochastic
programming
2.7
Robust schedule
OR overtime
Patient waiting time, OR overtime,
OR
utilization
Constructive
annealing
heuristic,
Simulated
Table 11: Most groups include patient waiting time as one of their performance criteria. The table also shows that mathematical
programming is a popularly used method among the research groups.
6.2.9. Single Contributions
The following section contains articles that are not published by any of the previously introduced research groups. In Jebali et al.
[21] a MIP is introduced to solve the patient planning problem in two steps. In other words, patients are assigned to ORs first and
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are then sequenced. Again a MIP is used by Pham and Klinkert [29]. In their model, the availability of resources is stressed and
modes are defined. A mode is a subset of resources that are occupied by a given patient. The usability of their method is limited
by the capabilities of general purpose MILP solvers. To overcome complexity problems Roland et al. [31] turn to genetic algorithms.
They define a 4-dimensional decision variable denoting for each surgery the operating room, the date and the time of the procedure.
The same decision problem is solved heuristically by Riise and Burke [30]. The authors focus on the properties of the heuristics
and describe the search space defined by the neighborhood structure of a relocate and two-exchange operator. The objective value
is the weighted sum of patient waiting time (days), overtime and children waiting time in the morning of their surgery. Assigning
different cost coefficients results in different fitness surfaces. Using search space analysis, properties of the fitness space are
described with respect to ruggedness and the distance properties of local optima.
An approach based on the combination of simulation and optimization is presented by Persson and Persson [28]. In their approach,
if requested by the patient, their surgery has to be carried out within a time window of 90 days. This affects the schedule and
increases the mean waiting time of medium priority patients.
Patient priorities are also considered in the revenue maximization problem of Stanciu and Vargas [32]. The focus of their work is
on capacity decisions pertaining to patient classes. A patient class is a combination of the patient reimbursement level and type of
surgery. A patient class enjoys higher priority if its expected revenue per unit surgery time is higher. The problem shares some
similarities with the airlines resource scheduling problem. The difference is that passenger requests are discrete (one seat), while
surgery patient requests are continuous and random (surgery time).
6.3. Conclusion
In this review we have focused on the healthcare literature over the surgery planning and scheduling problem of patients. The
publications of seven research groups were used to describe the field. We saw that the seven research groups are approaching
the planning problem in different ways. For example, some groups give very general, almost holistic models, meant to guide
managerial decision making, whereas others are focusing on smaller problems and their solutions. Moreover, some groups give
priority to exact methods whereas for other groups different objectives are important such as incorporating the stochastic aspects
of the problem. Those different goals require the use of different operations research methods. Having different research groups
dealing with the same problem leads thus to a large diversity of solution methods and helps to get a better and more diverse
understanding of the surgery planning and scheduling problem.
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7. Perspectives
The remainder of this project will be focused on improving our
understanding of the two step scheduling procedure. This
includes finding appropriate protection levels for each urgency
class and patient discipline. In other words, for each urgency
class we want to find the appropriate amount of reserved weekly
capacity so that, in total, most of the patients are served within
their DT. It is a general problem in hospitals that the exact time
future high urgency patients will be arriving is unknown in
advance. This makes it necessary to reserve some amount of
buffer capacity for them. As seen from the revenue management
literature in the airline industry, the optimal strategy could be
one where for the coming weeks less capacity is being reserved
(protected) than for weeks further away. This can be intuitively
understood in the following way. The further away a week is in
time, the more uncertainty there is about the amount of capacity
we will require to use from that week. As a consequence, we
will protect a larger amount of capacity for high urgency patients.
For weeks closer in time, we have more certainty about the
amount of capacity we will need for the high urgency class and
therefore we let a larger amount of lower urgency class patients
occupy that space, i.e., we protect less. This means that the
protection levels have to be dynamic. Using static protection
levels, many patients of lower urgency categories are served
late. The solution approach we are currently pursuing is based
on Markov Decision Processes (MDP). One of the largest
problems we encountered is that the search space even for a
simplified problem setting is very large and thus an approximate
solution needs to be found. At this point in time, we are not yet
sure whether an MDP solution is possible. Alternatively, a
heuristic could be used that is called expected marginal seat
revenue (EMSR). The problem with the heuristic approach is
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that it requires us to attach a value to each patient urgency
class. This is difficult since a more urgent patient is not more
valuable per se, but simply needs to be served faster.
Another part of the problem, for which we currently only have a
simple heuristic, is the within week scheduling. This is the
problem of assigning the patients scheduled for a particular
week to an exact OR, to a weekday (Monday-Friday) and a time,
i.e., for each patient an appropriate slot and for each OR an
appropriate sequence of surgeries. The exact components to
include into the scheduling algorithm are still to be selected. The
components will be selected in a way that the resulting problem,
firstly, is from an operations research perspective challenging to
solve while, secondly, is of practical use. In order to ensure that
the problem is practically usable, we will rely on the insights of
our collaborators at Gasthuisberg. Depending on the selected
components it could be possible that a mathematical
programming based optimal solution can be found. If this is not
the case, then a branch-and-bound or a branch-and-price based
solution approach will be proposed.
Besides finding appropriate protection levels and solving the
within week scheduling problem, one of the important targets for
us is to help to explore with Gasthuisberg the practical
consequences of switching from the direct slot scheduling to the
two-step scheduling procedure.
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8. Conclusion
The first stage of this project involved understanding the
literature and thus the research done so far. We did this by
analyzing 181 relevant articles and categorized them based on
several descriptive fields. The exact categorization can be found
at: http://www.econ.kuleuven.be/healthcare/review2011. We
found that most attention is given to elective patients, and even
though often not stated explicitly, it is in many cases implied to
be an inpatient setting. Less frequently occurring in the literature
are methods that consider outpatients. This is surprising as
outpatient care is gaining in importance and therefore we would
expect to observe an increasing amount of literature concerning
this area. We also observed a gradual shift from determining
the exact time of a surgery to problems related to date and/or
room assignments. With respect to the performance measures
considered, we found that overtime is the most popular measure
and that preference-related criteria are gaining popularity.
Noteworthy is the fact that preferences are increasingly used in
multi-criteria settings. Mathematical programming, discrete
event simulation and heuristics are the techniques most
frequently used. It is also true that the majority of articles present
results based on real data. However, it is important to note that
this does not imply that the methods are applied in practice. The
applicability aspect of methods is one of the major points that
we emphasize in our own research.
After learning about the current research, we were exploring
some of the practical aspects of the patient scheduling problem.
This required us to analyze many of the different patient related
attributes of the 13 patient disciplines served in the 22 inpatient
ORs at Gasthuisberg. Those attributes relate, for example, to
the weekday dependent arrival patterns, to urgency categories
as well as to both estimated and realized surgery durations.
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Analyzing the distribution of estimated and realized surgery
durations, we noticed that for many disciplines the length of the
surgery durations is consistently underestimated. This can be
explained by the fact that surgeons try to plan as many patients
into their own slot time as they can while not being allowed to
exceed the capacity of those slots. Underestimating the patient’s
surgery durations gives them a tool to legally overfill their slots.
We captured this relationship between estimated and realized
surgery durations by modeling the marginal distribution of the
two components and a copula connecting them.
Naturally, overfilled ORs will result in a larger probability of the
OR to run into actual overtime. To avoid excessive overtime,
rescheduling actions are taken in practice, i.e., moving a patient
from one OR to another or canceling a patient. In order to arrive
to a realistic simulation model, we also included rescheduling
actions. The rescheduling model is created using the
managerial insights of the head nurse of the hospital. The
implementation of the rescheduling method into the simulation
model was followed by an investigation on whether the
distribution of the number of performed simulated rescheduling
actions for each hour of the day matches the one observed in
reality. We found that the most realistic rescheduling model
takes into consideration several factors such as the hour of day,
the patient discipline and the OR kern. Similarly, also the
allocation of emergency arrivals to ORs is modeled in detail. An
example is the emergency discipline to OR discipline time
dependent allocation schema, e.g., oncology emergencies are
during the day allocated to oncology slots if available, otherwise
to abdominal or transplantation slots, whereas in the nights or
weekends to OR ‘B2’ or ‘B3’.
Finally, the simulation model is used to investigate the impact
of different scheduling policies. The main base for these policies
is a concept called the DT. The DT of a patient is a time interval
within which surgery should be performed or we risk the
worsening of the patient’s health condition. The exact waiting
time within the interval is of lesser importance as the objective
is to simply serve as many patients as possible within their
respective DT. We investigated the effects of pushing patients
with a longer DT interval into the future in order to serve a larger
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part of patients with a shorter DT in time. Our simulation results
suggest that this strategy might be too rigid as it does not allow
to effectively distribute high peaks of demand. We are currently
working on a more selective strategy.
In most hospitals, as in Gasthuisberg, patients are assigned to
slots directly. This means that a patient who is planned for
surgery will be directly planned to an OR and a day. This system
of direct slot assignment is going to be changed in the near
future to a two-step procedure. Instead of assigning patients
directly to slots, they will be assigned to a week first. This means
that a second step is required, where for all the patients
assigned to a given week a suitable OR and weekday is
selected. The advantage of the two-step procedure comes from
the fact that the second part of the procedure, the within week
scheduling part, can be done just before the start of that given
week. We use the simulation model to test for the implications
of switching to this two-step procedure, focusing primarily on the
effect it has on the amount of patients scheduled after their
respective DT, i.e., too late. Our current results suggest that in
case of the two-step procedure it is very important to allow
patients with higher urgencies to break into the already fixed
weekly schedules. Additionally, it is important that the second
step, the within week scheduling, is guided by the patients
urgency category. Interestingly, we found that reserving a
constant amount of capacity for high urgency patients is from a
whole system perspective less beneficial.
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9. Appendix
Figure 49: One-step strategies. FCFS benefits all DT categories.
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Figure 50
Figure 51: DT 6 and 7 patients receive surgery mostly in time.
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Figure 52
Figure 53. The arrival rates for ABD for each weekday and DT category
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Figure 54: Non-electives will also during the weekend or the night enter a predetermined set of ORs.
Figure 55: Also the most urgent non-elective category (DT 1) is during slot hours assigned to ORs of their own discipline. Exceptions
are oncology (ONC) and Oral and maxillofacial surgery (MKA) patients.
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Figure 56
Figure 57
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