Operating Theatre Planning and Scheduling

Abstract

In this chapter we present a number of approaches to operating theatre planning and scheduling. We organize these approaches hierarchically which serves to illustrate the breadth of problems confronted by researchers. At each hierarchicalplanning level we describe common problems, solution approaches and results from studies at partner hospitals.

Full text

Chapter 5 Operating Theatre Planning and
Scheduling
Erwin W. Hans, Peter T. Vanberkel
Center for Healthcare Operations Improvement & Research, dep. Operational Methods for
Production & Logistics, University of Twente, Enschede
Abstract In this chapter we present a number of approaches to operating theatre
planning and scheduling. We organize these approaches hierarchically which
serves to illustrate the breadth of problems confronted by researchers. At each hi-
erarchical planning level we describe common problems, solution approaches and
results from studies at partner hospitals.
Acknowledgments The hospitals Erasmus MC and Netherlands Cancer Institute, and all co-
authors involved in the various research projects described here: Boucherie RJ, Harten WH van,
Houdenhoven M van, Hulshof PJH, Hurink JL, Kazemier G, Lans M van der, Lent WAM van,
Oostrum JM van, Wullink G.
5.1 Introduction
Within the OR/OM healthcare literature, operating theatre planning and schedul-
ing is one of the most popular topics. This is not surprising, as many patients in a
hospital undergo surgical intervention in their care pathway. For a hospital, the
operating theatre accounts for more than 40% of its revenues and a similar large
part of its costs (HFMA 2005). An efficient operating theatre department thus sig-
nificantly contributes to an efficient healthcare delivery system as a whole.
An extensive overview and taxonomy of the operating theatre planning and
scheduling literature is given by Cardoen et al. (2010). They conclude that the ma-
jority of the research is directed at planning and scheduling of elective patients at
an operational level of control, and take a deterministic approach. Furthermore,
they observe that only half of the literature contributions consider up- or down-
stream hospital resources, and few papers report about implementation in practice.
This appears to be a common problem in OR/OM healthcare literature (Brailsford
et al. 2009). An up-to-date online bibliography of the operating theatre manage-
ment literature is maintained by Dexter (2011), and a structured literature review
of operations research in the management of operating theatres is given by Guerri-
ero and Guido (2011).

2
In this chapter we address operating theatre planning problems on three hierar-
chical managerial levels: strategic, tactical and offline operational planning, as in-
troduced in Section 5.2.
The remainder of this chapter addresses recent work in each of these three lev-
els of control. Section 5.2 outlines the planning and control functions on the
aforementioned hierarchical levels in an operating theatre department. Section 5.3
addresses the strategic problem of determining the target utilization of an operat-
ing theatre department. Section 5.4 addresses the strategic problem of determining
the number of surgical teams required during the night to deal with emergency
cases. Section 5.5 addresses the strategic decision whether to use emergency oper-
ating theatres. Section 5.6 addresses the tactical problem of determining a master
surgery schedule (a day-to-day allocation of operating theatres to surgical special-
ties) that levels the workload in subsequent departments (wards). Section 5.7 ad-
dresses the offline operational problem of scheduling elective surgeries with sto-
chastic durations, and sequencing them in order to reduce access time of
emergency surgeries. We will use a wide array of OR techniques, including dis-
crete event and Monte Carlo simulation, statistical modeling and meta-heuristics.
5.2 A hierarchy of resource planning and control in operating
theatres
Competitive manufacturing companies make planning and control decisions in a
hierarchical manner (Zijm 2000). For example, the long term decision of what
products to manufacture is at the top of the hierarchy and the real time decision of
whether to discard a specific part due to its quality is at the bottom of the hierar-
chy. In general the reliance of one decision on another defines the hierarchy.
Many planning and control frameworks classify decisions into the three hierar-
chical levels strategic, tactical, and operational, as suggested by Anthony (1965).
Similar hierarchical planning and control frameworks have been proposed for
healthcare (see Hans et al. 2011). Hans et al. refine the classical hierarchy by split-
ting the operational level into an offline and online operational level, where the
former is the in advance short term decision making, and the latter the monitoring
and control of the process in real time. In the remainder of this section we outline
the main operating theatre planning and control functions on these four hierar-
chical levels.
5.2.1 Strategic planning and control
To reach organizational goals, the strategic level addresses the dimensioning of
core OT resources, such as the number of OTs, the amount of personnel, instru-
ments (e.g. X-ray machines), etc. It also involves case mix planning, i.e. the selec-

3
tion of surgery types, and the determination of the desired patient type volumes
(Vissers et al. 2001). Agreements are made with surgical services / specialties
concerning their annual patient volumes and assigned OT time. The dimensioning
of subsequent departments’ resources (e.g. ward beds) is also done (Vanberkel and
Blake 2007). Strategic planning is typically based on historical data and/or fore-
casts. The planning horizon is typically long term, e.g. a year or more.
5.2.2 Tactical planning and control
The tactical level addresses resource usage over a medium term, typically with a
planning horizon of several weeks (Blake and Donald 2002, Wachtel and Dexter
2008). The actual aggregate patient demand (e.g. waiting lists, appointment re-
quests for surgery) is used as input. In this stage, the weekly OT time is divided
over specialties or surgeons, and patient types are assigned to days. For the divi-
sion of OT time, two approaches exist (Denton et al. 2010). When a closed block
planning approach is used, each specialty will receive a number of OT blocks
(usually OT-days). In an (uncommon) open block planning approach, OT time is
assigned following the arrival of requests for OT time by surgeons.
On the tactical level, the surgery sequence is usually not of concern. In-
stead, on this level it is verified whether the planned elective surgeries cause re-
source conflicts for the OT, for subsequent departments (ICU, wards), or for re-
quired instruments with limited availability (e.g. X-ray machines). The design of a
Master Surgical Schedule is a tactical planning problem.
5.2.3 Operational planning and control (offline)
The offline operational level addresses scheduling of specific patients to resources
(and as a consequence, the sequencing of activities) and typically involves a plan-
ning horizon of a week. It encompasses the rostering of OT-personnel, and reserv-
ing resources for add-on surgeries (Dexter et al. 1999). In addition, it addresses the
sequencing of surgeries (Denton et al. 2007), to prevent critical resource conflicts,
e.g. regarding X-ray machines, instrument sets, surgeons, etc. When there are no
dedicated emergency OTs, the sequencing of the elective surgeries can also aid in
spreading the planned starting times of elective surgeries (which are “break-in
moments” for emergency surgeries) in order to reduce the emergency surgery
waiting time (Wullink et al. 2007).

4
5.2.4 Operational planning and control (online)
The online operational level addresses the monitoring and control of the day-to-
day activities in the OT. Obviously at this level of control, all uncertainty materi-
alizes and has to be dealt with. If necessary, surgeries are rescheduled, or even
cancelled (Dexter et al. 2004, McIntosh et al. 2006). This is usually done by a day
coordinator in the OT department. Emergency surgeries, which have to be dealt
with as soon as possible, are scheduled, and emergency OT teams may have to be
assembled and dispatched to the first available OT. If there are emergency OTs,
these emergency surgeries are dispatched to these OTs. If there are no such OTs,
they are scheduled within the elective surgical schedule.
In summary, strategic planning typically addresses capacity dimensioning de-
cisions, considering a long planning horizon of multiple years. Tactical planning
addresses the aggregate capacity allocation to patient types, on an intermediate
horizon of weeks or months. Offline operational planning addresses the in-
advance detailed capacity allocation to elective patients, with a short planning
horizon of days and up to a few weeks. Online operational planning addresses the
monitoring and control of the process during execution, and encompasses for ex-
ample reacting to unforeseen events.
5.3 Strategic: the problem with using target OT utilization levels
Utilization of operating theatres is high on the agenda of hospital managers and
researchers and is often used as a measure of efficiency, both introspectively as
well as in benchmarking against other OT departments. As a result, much effort is
spent trying to maximize OT utilization and sometimes, without understanding the
factors affecting it. Using straightforward statistical analysis we show how the tar-
get OT utilization of a hospital depends on the patient mix and the hospital’s will-
ingness to accept overtime. This work is described in detail in Houdenhoven et al.
(2007). Similarly, the erroneousness of target ward occupancies is studied by Har-
per and Shahani (2002) and discussed by Green in Hall (2006).
5.3.1 General model
There are various ways to compute the utilization rate. We define the OT utiliza-
tion as the expected total surgery duration (including changeover/cleaning time)
divided by the amount of time allotted (see Figure 5.1):

 (5.1)

5
Our approach can be easily extended to deal with more extensive definitions of
OT utilization.
Fig. 5.1 Timeline for surgical cases
The amount of allotted time is computed as follows:
5.2
where the slack time (reserved capacity) is determined in such a way that a certain
frequency of overtime is achieved. This is a managerial choice: slack time reduces
cancellations and/or costly overtime, but also reduces OT utilization. The frequen-
cy of overtime depends on the distribution of the total surgery duration and can be
computed according to:

5.3
Now more formally, let  and  denote the average and standard deviation of
elective surgical case durations of type s, and let  denote the number of cases
completed in one block. A type s may correspond with the surgeries of for exam-
ple a surgical specialty or a specific surgeon. Likewise, ,  and  denote the
same for emergency cases. All these parameters are based on historical data. It fol-
lows that the total expected duration of all elective cases in one OT block is
 and the standard deviation of the total duration of these  cases equals
. Accordingly:
 (5.4)
.. (5.5)
The accepted risk (or frequency) of overtime is denoted by . Now we can com-
plete equation (5.2). The amount of allotted time required to achieve an overtime
frequency of  can be computed as follows:
1
st
electivecase 2
nd
electivecase
expectedtotalsurgeryduration
allottedtime
slack
caseduration caseduration caseduration
8:00 15:30
3
rd
electivecase

6 5.6
where  is a function yielding probability . The outcome of this function de-
pends on the distribution of the surgery duration. Using  in this way allows
the approach to be independent of the surgery duration distribution, i.e. function
 can be changed to reflect various distributions. Using equations (5.4) and
(5.6) we can complete formula (5.1) for the expected OT utilization as a function
of the frequency of overtime as follows:
 
 (5.7)
5.3.2 General results
We use formula (5.7) for the expected OT utilization to illustrate the relationship
between OT utilization, patient mix and overtime frequency. In a theoretical sce-
nario where there is no surgery duration variability (i.e., 0), the ex-
pected OT utilization is obviously 100%.
As a case study we consider Erasmus Medical Center in Rotterdam, the
Netherlands. OT management in this hospital accepts a 30% risk of overtime. For
simplicity, they assume that the total surgery duration follows a normal distribu-
tion ~,. Using straightforward sta-
tistical analysis we can show that 0.5 when the acceptable frequency of
overtime is 30%.This is shown as follows. Let X be the total surgery duration,
then:
30%⟺
0.7⟺

0.7⟺
0.7
where ~0,1. It follows that 0.5.
We use 2 years of historical data from the aforementioned hospital. We
consider three different surgical specialties (i.e. three different patient mixes) and
for each we show the trade-off between expected OT utilization and overtime fre-
quency. This is illustrated in Figure 5.2.

7
Fig. 5.2 Trade-off between overtime probability and expected utilization
The calculated expected OT utilization can also be regarded as a target utilization,
or benchmark. Figure 5.2 shows that a single OT utilization target will result in
different overtime frequencies for each specialty. For example, a target utilization
of 80% will result in an overtime frequency of approximately 12% for ophthal-
mology but an overtime frequency of approximately 35% for ENT. In general, a
low risk of overtime and a complex patient mix will result in a low utilization rate.
If the accepted risk of overtime is higher and the patient mix less complex, then a
higher utilization can be achieved. Given that overtime is expensive (and perhaps
limited by collective bargaining agreements), this example illustrates the inade-
quacy of a single target OT utilization as a performance metric. It also illustrates
the importance of taking case mix characteristics into account when comparing
utilization figures between different OT departments.
5.4 Strategic: on-call or in-house nurses for overnight coverage
for emergency cases?
Treating emergency patients is a common activity for most hospitals. Likewise,
the operating theatre must be available to provide emergency operations 24 hours
per day. The night shift (e.g., from 11:00 pm to 7:30 am) is typically the most ex-
pensive shift to staff due to collective labor agreements and the inconvenient
hours. Determining minimum cost staffing levels that provide adequate coverage
to meet emergency demand is a strategic problem. In this section we describe a
30%
40%
50%
60%
70%
80%
90%
100%
0,1%
10,0%
20,0%
30,0%
40,0%
50,0%
Ear‐Nose‐Throat
Ophthalmology
Orthopedics
Overtime probability(%)
Expectedutilization(%)

8
case study to determine appropriate night shift staffing levels at Erasmus Medical
Center. The outcomes of the study were successfully implemented.
5.4.1 General problem formulation
Covering the night shift is usually accomplished by using in-hospital and on-call
nurses. The in-hospital nurses are stationed in the hospital while waiting for emer-
gency cases. The on-call nurses wait at their homes for emergency cases (typically
there is a requirement that they can be present in the hospital within e.g. 30
minutes of being requested to do so) and are typically cheaper than in-hospital
teams. In general, A single nurse can complete exactly one case at a time but can
complete any number of cases in series until the end of the shift. The decision re-
quired in this problem is to determine how many in-hospital and on-call nurses are
necessary to meet the demand for emergency cases. Timeliness is of the essence
here, as these emergency cases may be very urgent.
When the first emergency case presents for surgery during a night shift,
in-house nurses respond. Depending on the hospital policy and the total number of
in-house nurses, an on-call nurse may be called in. In other words, some hospitals
may wait until 1, 2, … or all in-house nurses are busy before calling in a nurse
from home, while other hospitals may wait until all in-house nurses are busy and
an emergency case is present. For each subsequent emergency case this process is
repeated. Note that nurses are available to complete multiple surgeries per night
and are available again after completing a surgery. Finally surgeries cannot be
preempted. In this subsection we assume the hospital’s policy for calling an on-
call nurse is fixed, although determining this policy is, in and of itself, and inter-
esting research question.
There are generally two types of emergency cases: those that need to be
started immediately and those that can be delayed before being started. The former
we refer to as emergent cases and the latter as urgent cases. The acceptable delay,
or safety interval, for starting an urgent case varies: “for example a facility may
consider it imperative for a patient with a ruptured abdominal aortic aneurysm to
be operated on within 30 min of arrival, while a patient with an amputated finger
should be operated on within 90 min of arrival, and a patient with a perforated
gastric ulcer should be operated on within 3 h of arrival” (Oostrum et al. 2008).
By incorporating the acceptable delays for urgent cases it is possible to
postpone urgent case demand to a later cheaper shift, and/or postpone the case un-
til busy in-house nurses are free. To examine these possibilities in detail, Oostrum
et al. (2008) use a discrete-event simulation and a case study at Erasmus Medical
Center Rotterdam (Erasmus MC). To illustrate the benefits of postponing surger-
ies, the authors compare results with surgery postponements with the approach of
Dexter and O’Neill (2001) where surgery postponements are not used. In the fol-
lowing subsection we provide an overview of the results.

9
5.4.2 General results
Current practice at Erasmus MC had a team composition of 9 in-house nurses and
2 on-call nurses. Using the approach of Dexter and O’Neill, a team of 8 in-house
nurses and 2 on-call nurses was determined to be appropriate. A number of other
team compositions were considered, ranging from a total of 11 nurses to a total of
6. Each team composition represented a what-if scenario in the simulation model.
The simulation model was used to determine the number of surgeries cases start-
ing later than required.
To compute the cost of each team composition, observe that – since the
number of working hours does not depend on the team composition – we only
need to look at the cost of idle staff. Under Dutch law, the costs for nurses who
wait during the night shift are 107.5% of the regular hourly daytime wage for in-
house nurses, and 106% for nurses on call. We thus compute the cost of waiting
nurses in each team composition as follows:
numberofin‐housenurses1.075
numberofon‐callnurses1.06hourlywage
Figure 5.3 displays the cost of waiting and percentage of surgeries starting late for
the considered team compositions, where we assumed for simplicity that a regular
hour’s wage is 1. It shows that current practice of 9 in-house and 2 on-call nurses
performs the best. However, the waiting cost can be decreased by approximately
18.5% by switching to a team composition of 5 in-house and 4 on-call nurses, at
the expense of a 2% increase of late starts.
Fig. 5.3 Cost of waiting and late surgery starts for various team compositions
6
7
8
9
10
11
12
10% 12% 14% 16% 18% 20% 22% 24%
Costofwaiting(hourlywage=1)
Percentageofsurgeriesstartinglate
CurrentPractice:9in‐house,2on‐call
Dexter &O'Neill:8in‐house,2on‐call
7in‐house,2on‐call
5in‐house,4on‐call
4in‐house,4on‐call
4in‐house,3on‐call
4in‐house,
2on‐call

10
For policy making, managers can use results like these to see the relative perfor-
mance cost associated with each staff assignment. The decision autonomy remains
with the policy makers and they are left to determine if cheaper staffing levels jus-
tify the a decrease in performance.
For more extensive results, we refer to (Oostrum et al. 2008),where the
authors present the distribution of cases starting later than required, surgical spe-
cialty specific results, results for multiple nurse types, and an extensive sensitivity
analyses. The sensitivity analyses showed that the approach can be generalized for
use in other centers.
Oostrum et al. (2008) report that heavy involvement of clinical staff in
this project was essential for the following reasons. Staff assessed the safety inter-
vals for urgent patients to ensure changes did not negatively affect patient’s safety.
They validated the discrete event simulation model, and suggested various scenar-
ios for sensitivity analyses. The visualizations provided by the computer simula-
tion aided to convince them of the final conclusions. As a result, despite the nega-
tive impact on their salary, the staff accepted the adjustment of the team
composition to 5 in-house and 4 on-call nurses. For Erasmus MC this intervention
resulted in an annual cost saving of 275,000 Euro.
5.5 Strategic: Emergency operating theatres or not?
During regular working hours, most hospitals either perform emergency opera-
tions in dedicated emergency OTs, or in regular elective patient OTs. For the se-
cond option a certain amount of slack is scheduled in order to fit in emergency
cases without causing excessive cancellations of elective cases. The choice to use
Policy 1 (reserving capacity in dedicated emergency OTs) or Policy 2 (reserving
capacity in multiple regular emergency OTs) is the strategic decision addressed in
this section. The difference between these two policies is illustrated graphically in
Figure 5.4.

11
Fig. 5.4 Cost of waiting and late surgery starts for various team compositions
The flow of patients is summarized as follows: “Emergency patients arriving at a
hospital that has adopted the first policy, will be operated immediately if the dedi-
cated OT is empty and will have to queue otherwise, whereas patients arriving at a
hospital that has adopted the second policy can be operated once one of the ongo-
ing elective cases has ended. Other planned cases will then be postponed to allow
the emergency operation” (Wullink et al. 2007).
Policy 1 has the advantage that the first emergency case of the day can
begin without delay, but all following cases may be subject to delay. Furthermore
this policy means only the emergency OTs needed to be equipped to deal with
emergency cases. Finally, as a result of emergency surgeries elective surgeries
will experience no delay (Bhattacharyya et al. 2006, Ferrand et al. 2010) and elec-
tive OTs will experience no overtime (Wixted et al. 2008).
Policy 2 cannot guarantee any emergency case will begin without delay,
but since emergency cases can be completed in more OTs, an opening (i.e. a case
finishing) for the subsequent cases may a happen sooner than in policy 1. The
benefits from this policy essentially result from flexibility. To ensure this flexibil-
ity (and the corresponding benefits) multiple (or all) of the OTs must be equipped
to deal with emergency cases.
5.5.1 General problem description
The decision that is required is to determine how to reserve OT capacity for emer-
gency cases, i.e. according to policy 1 or policy 2. There are advantages and dis-
advantages of both policies introduced above. Due to the stochastic nature of
emergency cases (arrivals and surgery durations) choosing the best policy is not
Policy 1:
Emergency OTs Reserved
OT time for
emergency
surgery
Scheduled
elective
surgeries
Policy 2:
Elective OTs
OT1 OT2 OT3 OT4 OT5 OT6 OT7 OT8
OT1 OT2 OT3 OT4 OT5 OT6 OT7 OT8

12
immediately obvious. To compare the policies we suggest evaluating the follow-
ing metrics:
- emergency surgery waiting time: the total delay, or the delay past what is
allowed to receive emergency surgery.
- elective surgery waiting time: the difference between the planned and ac-
tual starting time of an elective surgery.
- OT overtime: the time used for surgical procedures after the regular block
time has ended.
- OT utilization: the ratio between the total used operating time for elective
procedures and the available regular time.
The following instance parameters are taken into account: elective surgery volume
and duration characteristics, emergency surgery arrival and duration characteris-
tics.
5.5.2 General results
We summarize a case study (presented in detail in Wullink et al. 2007) where dis-
crete event simulation was used to prospectively evaluate both policies. The case
study was used to support decision making at Erasmus MC. When applying Policy
2, the hospital decided that all of their 12 OTs would be equipped to handle emer-
gency cases. In policy 1, with emergency capacity allocated to 1 dedicated emer-
gency OT, the remaining free OT time is allocated exclusively to elective OTs. In
policy 2, with emergency time allocated to each elective OT, the reserved OT time
is distributed evenly over all elective OTs. Figure 5.5 and Table 5.1 summarize the
results from the discrete event simulation.
Table 5.1 Summary of simulation results for policy 1 and 2
Policy 1 Policy 2
Total overtime per day 10.6 8.4
Mean number of OTs with overtime per day 3.6 3.8
Mean emergency patient’s waiting time 74 (±4.4) 8 (±0.5)
OT utilization 74% 77%
From Table 5.1 it is clear that policy 2 outperforms policy 1 on all given out-
comes.
Under Policy 1, all emergency patients were operated on within 7 hours
with a mean waiting time of 74 (±4.4) min. Under Policy 2, all emergency patients
were operated upon within 80 min with a mean waiting time of 8 (±0.5) min. OT
utilization for policy 1 was 74% and 77% for policy 2. Policy 1 resulted in 10.6
hours of overtime on average per day and policy 2 resulted in 8.4. Policy 2, with
emergency capacity allocated to all elective OTs, thus substantially outperforms
policy 1, on all outcomes measured.

13
Fig. 5.5 Cumulative percentage of emergency patients in policy 1 and 2 (simulation results)
Table 5.2 summarizes the results of additional simulation experiments in which
we vary the number of emergency OTs (0, 1, 2, or 3) as well as the number of
elective OTs used for emergency surgeries (0, 5, 10, or 15). We use the case mix
of the previous experiment, but resize the problem to 15 elective OTs (instead of
12). Furthermore, approximately 10% of the surgeries are emergency surgeries.
The results show that policy 2 (dealing with emergencies in (some) elective OTs)
results in improved emergency waiting performances, at the expense of increased
waiting time of the elective surgeries. A mixed policy combines the advantages of
both policies – the table can be used as a guideline to make a trade-off.
Table 5.2 Simulation results: (1) average and (2) maximum emergency surgery waiting
time (minutes), (3) percentage of emergency surgeries that has to wait, (4) average elective
surgery waiting time (minutes)
Number of emergency OTs
Elective OTs
used for
emergency
0 1 2 3
0 -
21.9 2.4 0.5
3026 949 292
4.4% 2.4% 1.1%
12.6 12.6 12.6
5
1.3 0.6 0.3 0.1
204.9 152.7 113.1 83.3
4.5% 2.9% 1.5% 0.7%
32.3 21.2 14.2 11.4
10 0.5 0.3 0.1 0.1
0%
20%
40%
60%
80%
100%
10
40
70
100
130
160
190
220
250
280
310
340
370
400
Cumulative % of urgent surgery
Waiting time (minutes)
Policy 1
Policy 2

14
94.2 76.3 63.8 50.2
4.2% 2.6% 1.3% 0.6%
22.2 16.0 12.1 10.3
15
0.3 0.2 0.1 0.0
60.3 52.3 43.3 36.0
4.0% 2.5% 1.2% 0.5%
18.6 14.9 11.6 10.0
5.6 Tactical: designing a master surgical schedule to level ward
usage
Managers are inclined to solve problems at the moment they occur (i.e., on the op-
erational level). In Hans et al. (2011) we refer to this phenomenon as the “real-
time hype” of managers. For healthcare managers, while inundated with opera-
tional problems, the universal panacea for all productivity related problems is
“more capacity”. It is thereby often overlooked to tactically allocate and reorgan-
ize the available resources, which may turn out to be even more effective, and will
certainly be cheaper. However, due to its longer (intermediate) planning horizon,
tactical planning is less tangible and inherently more abstract than operational
planning. In the majority of our healthcare process optimization research projects
we find that the tactical planning level is typically not formalized and overlooked.
Tactical planning decisions are rather a result of historical development (“This
year’s tactical plan is last year’s tactical plan”), than a result of periodic planning.
We also find they have often evolved to hard constraints for operational planning
(“We don’t do orthopedic patients on Wednesday afternoons. Why? Well, we just
don’t!”).
This is also typical for the tactical planning of OTs, the block planning or
Master Surgical Scheduling (MSS) problem, which concerns the weekly allocation
of OT-days to surgical specialties (or surgeons). To a surgeon: “operating theatre
6 on Monday is her/his OT”. Re-allocating OT-days may however lead to a more
stable workload in subsequent departments (wards, ICU), and even reduce the re-
quired capacity of these departments. In this section we present a model to analyze
and improve the impact of the MSS on the resource usage in subsequent depart-
ments.
5.6.1 General problem description
Tactical OT planning, typically involves the assignment of OT capacity to aggre-
gate patient groups (i.e. patient cohorts) for a fixed planning horizon. This as-
signment should reflect the strategic goals of the hospital. For example, consider a

p
lanni
n
1000
o
special
t
geries
p
ning h
o
which
p
lishe
d
days d
u
to plan
ule als
o
how
m
when
c
ning h
o
availa
b
case
m
tients)
t
Q). Fo
r
assign
e
descri
b
block

MSS (
i
in Fig
u
that m
u
The M
S
LOS o
f
how p
a
n
g horizon o
f
o
rthopedic su
r
t
y should be
p
er month.
The tactica
l
o
rizon such
t
operational l
e
d
with a MSS.
u
ring the pla
n
their other f
u
o
allows the
O
m
any pieces o
f
c
an OT maint
e
When an
M
o
rizon are kn
o
b
le, etc. Wha
t
m
ix is expecte
d
t
o OT time is
A MSS rep
r
r
each day 
∈
e
d to one of
t
b
ed by the as

 where ∈
i
.e. before sp
e
u
re 5.6 where
u
ltiple blocks
Fig.
5
S
S is defined
f
any patien
t
.
a
tients overla
p
f
one month
a
r
ge
r
ies over t
h
assigned eno
u
l
plan is used
t
hat long te
r
m
e
vel planning
The MSS de
f
n
ning horizon
u
nctions, suc
h
O
T departme
n
f
equipment a
r
e
nance happe
n
M
SS is being
d
o
wn, i.e. whi
c
is known is
d
. Hence the
a
the primary
fa
r
esents a rep
e
∈
1,2,…,
t
he available
s
signment of
1,2,…, a
n
e
cialties have
b
each cell rep
r
are assigned
t
5
.6 Example e
m
for period Q
a
Figure 5.7 d
i
p
.
a
nd a hospita
l
h
e next six
m
u
gh OT capa
c
to organize
c
m
goals are
m
can be base
d
f
ines which s
u
. Such a sch
e
h
as outpatien
t
n
t to make th
e
r
e needed ea
c
n
, etc.
developed, n
o
ch patients
w
that a certai
n
a
ssignment o
f
f
actor conside
r
e
titive pattern
in the MSS e
a
surgical spec
i
a surgical sp
n
d ∈1,2,
…
b
een assigne
d
r
esents an op
e
t
o a single sp
e
m
pty Master Su
r
and executed
i
splays how t
h
l
with a strat
e
m
onths, then t
h
c
ity to compl
e
c
apacity over
m
et and to c
r
d
. In the OT
t
u
rgical speci
a
e
dule allows t
h
t
clinics, edu
c
e
ir own plan
n
c
h day, what
a
o
t all of the
d
w
ill show up,
w
n
volume of
p
f
patient coho
r
ed when des
i
over a certai
n
a
ch of the I a
v
i
alties. More
ecialty j to
e
…
,. Using t
h
d
operating th
e
e
rating theatr
e
e
cialty on the
r
gery Schedule
repeatedly.
L
h
e multiple
M
e
gic goal to
c
h
e orthopedic
e
te 1000/6 ≈
an intermedi
a
r
eate a struct
u
t
his is usuall
y
l
ties operate
o
h
e surgical s
p
c
ation, etc. T
h
ing decisions
,
a
re the staffi
n
d
etails about
t
w
hich doctor
s
p
atients with
a
r
ts (not indiv
i
i
gning a MSS
.
n
number of
d
v
ailable OTs
h
p
recisely, th
e
e
ach operatin
g
h
is notation,
a
e
atre blocks)
i
e
block. It is
c
s
ame day.
(MSS)
et M
b
e the
m
M
SS cycles re
15
c
omplete
surgical
167 sur-
a
te plan-
u
re from
y
accom-
o
n which
p
ecialties
h
e sched-
, such as
n
g levels,
t
he plan-
s
will be
a certain
i
dual pa-
.
d
ays (say
h
as to be
e
MSS is
g
theatre
a
n empty
i
s shown
common
m
aximum
e
pea
t
and

16
In this
s
the mo
d
termedi
a
forman
c
MSS, c
o
inpatie
n
p
ropos
a
tic to fi
n
sion.
of the
M
distribu
t
termine
recover
i
ing inp
a
has the
p
ute th
e
for eac
h
dischar
g
experi
m
b
er of
e
distribu
t
Thus o
n
charges
tomorr
o
b
er of
a
each da
y
tion for
for the
d
Finally,
The for
m
the M
S
1,2,…
,
gical s
p
crete di
Fig. 5.7 I
l
s
ection we d
e
d
el is to make
a
te term plan
n
c
e goals are
a
o
mpute a nu
m
n
ts. This ap
p
r
o
a
ls. Adopting
n
d the best
M
The aim is t
M
SS. We do t
h
t
ions. We th
e
the number
o
i
ng, we predi
c
a
tient care, di
s
Consider a
s
option of sta
y
e
probability
o
h
day of the
p
g
e probabiliti
e
m
ents have tw
o
e
xperiments
r
t
ion (assumi
n
n
any day an
d
and consequ
o
w. For a giv
e
a
dmission to
y
) for each p
a
the admissio
n
d
ischarge rat
e
using discre
t
m
al model de
Assume tha
t
S
S be define
d
,
 indexes t
h
p
ecialty j
b
e c
h
stribution for
l
lustration of t
h
e
scribe a mo
d
a cyclical as
s
n
ing horizon,
a
chieved. W
e
m
ber of work
l
o
ach does not
t
his approac
h
M
SS proposal
o determine t
h
is by model
i
e
n add these
d
o
f patients re
c
c
t a number o
s
charges and
s
s
ingle patient
y
ing or bein
g
o
f being disc
h
p
atient’s LOS.
e
s. From pro
b
o
(and only t
w
esulting in e
a
n
g experimen
t
d
for each pat
i
e
ntly we kno
w
e
n MSS and
u
the ward (i.
e
a
tient cohort.
T
n
rate and usi
n
e
and we can
e
t
e convolutio
n
s
cription foll
o
t
each surgica
l
d
such that
h
e OTs and 
h
aracterized
b
the number
o
h
e overlap bet
w
d
el by Vanbe
r
s
ignment of
O
such that str
a
e
allow for a
l
oad metrics
a
find the opti
m
h
to be used i
n
is of course
a
t
he number o
f
i
ng the recov
e
d
iscrete distri
b
c
overing. On
c
o
f workload
m
s
pecialized in
p
t
who is reco
v
g
discharged.
h
arged and c
o
.
Now consid
e
b
ability theo
r
w
o) outcome
s
a
ch outcome
t
s are indepe
n
i
ent cohort,
w
w
the numbe
r
u
sing historic
e
. the numbe
r
Thus for eac
h
n
g a binomial
e
asily compu
t
n
s we can co
m
o
ws.
l
specialty re
p
 is an o
p
∈1,2,…,

b
y two param
o
f surgeries c
a
w
een multiple
M
r
kel et al. (20
O
T time to pat
i
a
tegic “produ
c
stochastic
L
a
ssociated wi
t
m
al MSS but
r
n
conjunction
a
natural and
f
patients in r
e
e
ring patient
c
b
utions (with
c
e we know t
h
m
etrics includi
n
p
atient care.
v
ering from
s
From histori
c
nversely the
p
e
r a cohort o
f
r
y it is know
n
s
, then the pr
o
can be comp
u
n
dent and id
e
w
e can comp
u
r
of patients
w
r
ecords, we c
r
of complet
e
h
patient coho
r
distribution
w
t
e the ward o
c
m
pute the ov
e
p
resents a sin
g
p
erating thea
t

the days in
a
eters  and

a
rried out in
o
M
SS cycles
1
1). The obj
e
i
ent cohorts f
o
c
tion levels”
a
OS and, for
h
recovering
r
ather evalua
t
with a searc
h
very plausibl
e
e
covery as a
f
c
ohorts with
b
convolution
s
h
e number of
n
g admission
s
s
urgery and e
a
c
al data we c
a
p
robability o
f
f
patients wit
h
n
that when
m
o
bability for t
h
u
ted with a
b
e
ntically dist
r
t
e the numbe
r
w
ho will rem
a
a
n compu
t
e t
h
e
d inpatient s
u
r
t we have a
d
w
e have a dist
r
c
cupancy dist
r
e
rall ward occ
g
le patient co
h
t
re block w
h
a
cycle. Let e
a


, where 
i
o
ne OT bloc
k
e
ctive of
o
r an in-
a
nd per-
a given
surgical
t
es MSS
h
heuris-
e exten-
f
unction
b
inomial
s
) to de-
patients
s, ongo-
ach day
a
n com-
f
staying
h
similar
m
ultiple
h
e num-
b
inomial
r
ibuted).
r
of dis-
a
in until
h
e num-
urgeries
d
istribu-
t
ribution
r
ibution.
c
upancy.
h
ort. Let
h
ere ∈
ach sur-
is a dis-
k
and 


17
the probability that a patient, who is still in the ward on day n, is to be discharged
that day (0,1,…,, where  denotes the maximum LOS for specialty j).
Using  and 
 as model inputs, for a given MSS the probability distri-
bution for the number of recovering patients on each day q can be computed. The
required number of beds is computed with the following three steps. Step 1 com-
putes the distribution of recovering patients from a single OT block of a specialty
j; i.e. we essentially pre-calculate the distribution of recovering patients expected
from an OT block of a specialty. In Step 2, we consider a given MSS and use the
result from Step 1 to compute the distribution of recovering patients given a single
cycle of the MSS. Finally in Step 3 we incorporate recurring MSSs and compute
the probability distribution of recovering patients on each day q.
Step 1: For each specialty j we use the binomial distribution to compute the num-
ber of beds required from the day of surgery 1 until . Since we know
the probability distribution for the number of patients having surgery (), which
equates to the number of beds needed on day 0, we can use the binomial dis-
tribution to iteratively compute the probability of needing beds on all days 0.
Formally, the distribution for the number of recovering patients on day n is recur-
sively computed by:




0




 1

.
Step 2: We calculate for each OT block  the impact this OT block has on the
number of recovering patients in the hospital on days q, q+1, …. If j denotes the
specialty assigned to OT block , then let 
, be the distribution for the number
of recovering patients of OT block  on day 1,2,…,,1,…. It follows
that: 
,


where 0 means 
,01. Let  be a discrete distribution for the total number
of recovering patients on day m resulting from a single MSS cycle. Since recover-
ing patients do not interfere with each other we can simply iteratively add the dis-
tributions of all the OT blocks corresponding to the day m to get . Adding two
independent discrete distributions is done by discrete convolutions which we indi-
cated by “∗”. For example, let A and B be two independent discrete distributions.
Then ∗, which is computed by:




18
where τ is equal to the largest x value with a positive probability that can result
from ∗ (e.g. if the maximum value of A is 3 and the maximum value of B is 4,
then when convoluted the maximum value of the resulting distribution is 7, there-
fore in this example 7). Using this notation,  is computed by:


,∗

,∗…∗

,∗

,∗…∗

,
Step 3: We now consider a series of MSSs to compute the steady-state probability
distribution of recovering patients. The cyclic structure of the MSS implies that
patients receiving surgery during one cycle may overlap with patients from the
next cycle. In the case of a small Q patients from many different cycles may over-
lap.
In Step 2 we have computed  for a single MSS in isolation. Let M be the last
day where there is still a positive probability that a recovering patient is present in
. To calculate the overall distribution of recovering patients when the MSS is
repeatedly executed we must take into account / consecutive MSSs. Let 
denote the probability distribution of recovering patients on day q of the MSS cy-
cle, resulting from the consecutive MSSs. Since the MSS does not change from
cycle to cycle,  is the same for all MSS cycles. Such a result, where the proba-
bilities of various states remain constant over time, is referred to as a steady-state
result. Using discrete convolutions,  is computed by:
∗∗∗…∗/
From this result a number of workload metrics can be derived. To determine the
demand for ward beds from the variable  consider the following example. Let
the staffing policy of the hospital be such that they staff for the 90th percentile of
demand and let  denote the 90th percentile of demand on day q. It follows that
 is also the number of staffed beds needed on day q, and is computed from 
as follows: |0.9
In practice, patients tend to be segregated into different wards depending
on the type of surgery they received. To incorporate this segregation into the mod-
el and to consequently have recovering patient distributions for each ward, a mi-
nor modification needs to be made to the model. Let  be the set of specialties j
whose patients are admitted to ward k. Then in Step 2 we only have to consider
those OT blocks assigned to a specialty in  and continue with the calculations.
Ward occupancy alone does not fully account for the workload associated
with caring for recovering patients. During patient admissions and discharges the
nursing workload can increase. From the model the probability distribution for
daily admissions and discharges can be computed. To compute the admission rate,

19
set 1 for all j and repeat the steps above. The resulting  will denote the
admissions on day q.
The discharge rate is the rate at which patients leave the ward and can be
computed by adding an additional calculation in Step 1. Let 
 be a discrete dis-
tribution for the number of discharges from specialty j on day n which is comput-
ed as follows:
ℙ



 
1
ℙ

Finally, after computing 
, one can set 

 and continue with Step 2. By
doing so, the resulting  will denote the distribution for daily discharges for
each day q of the MSS.
The inherent assumption of the described method is that all patients with
a patient cohort have equal probability of being discharged and that it is independ-
ent of other patients, i.e. it is assumed that patients are identically distributed and
independent. The independence assumption implies that the amount of time one
patient is in the hospital does not influence the amount of time another patient is in
the hospital. This seems like a natural assumption in most cases and appropriate so
long as surgeries are rarely cancelled due to a bed shortage (cancellations due to
bed shortages creates a dependency). The identically distributed requirement
means that we must compute the number of beds needed tomorrow (and the num-
ber of case completed in one OT block), for all identically distributed cohorts of
patients separately. In other words, the parameters of the binomial distribution
must reflect all of the patients in a given cohort.
5.6.2 General results
The model was applied at the Netherlands Cancer Institute - Antoni van Leeuwen-
hoek Hospital (NKI-AVL) to support the design of a new MSS. Selected results
from Vanberkel et al. (2011-2) are summarized in this subsection.
Management at NKI-AVL strive to staff enough beds such that for 90%
of the week days there is sufficient coverage. This implies that on 10% of the days
they will be required to call in additional staff. Using the model a number of MSS
proposals were evaluated and eventually staff choose an MSS that the model pre-
dicted would lead to a balanced ward occupancy.
An unbalanced ward occupancy makes staff scheduling, and ward opera-
tions, difficult. Early in the week, beds would be underutilized whereas later in the
week, beds would become highly utilized and the risk of a shortage would in-
crease. Such peaks and valleys represent variation in the system which possibly
could be eliminated with a different MSS. This variation leads to significant prob-
lems, particularly as the wards approach peak capacity. For example, when inpa-

20
tient w
a
ten scr
a
ditional
the elec
ed time
b
le, it
o
ward c
a
Either
w
care. A
l
out an
e
minimi
z
over a
3
used fo
r
fit tests
model.
cle) we
r
the sev
e
ward o
c
Fig. 5
.
5.6.3
D
The m
a
ward st
a
a soluti
o
several
ly predi
strictio
n
a
rds reach th
e
a
mble to try a
n
staff are call
e
tive surgery i
for surgeons
o
ften means a
a
pable of cari
n
w
ay, extra wo
r
l
though com
p
e
xorbitant am
o
z
e occurrence
After imple
m
3
3 week peri
o
r
each day of
, these obse
r
Six of the se
v
r
e found to b
e
e
nth day, this
c
cupancy wit
h
.
8 Comparison
D
iscussion
a
in benefit of
a
ff, thereby p
r
o
n. Staff was
modification
s
ct model out
p
In this proj
n
s as unchang
e
ir peak capac
n
d make a be
d
e
d in (or in r
a
s cancelled),
w
and extra an
x
patient was t
r
n
g for the pa
t
r
k is required
p
letely elimin
a
o
unt of resou
r
s
.
m
enting the
n
o
d. From thes
e
the MSS cyc
l
v
ed distribut
i
v
en projected
e
a good fit fo
was true at
a
h
the observe
d
of the projecte
d
using the m
o
r
oviding a pl
a
quick to em
b
s
to the origin
a
p
ut for a give
n
ect we treat
e
e
able. It is po
c
ity and a pat
i
d
available. I
f
a
re cases whe
n
w
hich causes
x
iety for pati
e
t
ransferred fr
o
t
ient but not
t
by ward staf
f
a
ting such pr
o
r
ces, sound p
n
ew MSS, t
h
e observatio
n
l
e were deriv
e
i
ons were co
m
distributions
o
r the observe
d
a
level α0.2
.
d
ward occup
a
d and observe
d
o
del was to b
a
tform from
w
b
race the mo
d
al MSS, at w
h
n
modificatio
n
e
d the equip
m
ssible that fu
r
i
ent admissio
n
f
one cannot b
n
additional
s
extra work f
o
e
nts. When a
b
o
m one ward
t
t
he designate
d
f
and there is
a
o
blems is lik
e
l
anning ahea
d
h
e ward occ
u
n
s, probability
e
d. Using Ch
i
m
pared to th
(one for eac
h
d
data at a le
v
.
Figure 5.8 c
a
ncy during th
d
(90th percent
i
e
able to qua
w
hich they co
u
d
el output, pa
r
h
ich point the
y
n
.
m
ent and p
h
r
ther improve
m
n
is pending,
s
e made avail
a
s
taff cannot b
e
o
r OR planne
r
b
ed was mad
e
t
o another (o
f
d
one) or dis
c
a
disruption i
n
e
ly not possib
d
of time ma
y
u
pancy was
o
distributions
i
-square good
n
o
se projecte
d
h
day of the
M
v
el α0.05,
w
o
mpare the p
r
e
33 week pe
r
le) ward occup
a
n
tify the con
c
u
ld begin to n
r
ticularly afte
r
y
were able t
o
h
ysician sche
d
m
ents in the
w
staff of-
a
ble, ad-
e
found,
r
s, wast-
e
availa-
f
ten to a
c
harged.
n
patient
b
le wi
t
h-
y
help to
o
bserved
of beds
d
ness-of-
d
by the
M
SS cy-
w
hile for
rojected
r
iod.
ancies
c
erns of
n
egotiate
r seeing
o
rough-
d
ule
r
e-
w
ard oc-

21
cupancy could have been achieved if these restrictions were relaxed. In this way
the model can be used to illustrate the benefits of buying an extra piece of equip-
ment or of changing physicians’ schedules. An additional restriction, which if re-
laxed may have allowed further improvements, is the assignment of wards to sur-
gical specialties. In other words, in addition to changing when a specialty
operates, it may prove advantageous to change which ward the patients are admit-
ted to after surgery. Finally, we chose the best MSS from those created through
swapping OR block and surgical specialty assignments. It is possible that a search
heuristic may have found a better MSS, although it would have required the many
surgical department restrictions to be modeled and the more complex model may
not have garnered the same level of staff understanding and support.
Oostrum et al. (2008-2) propose another approach, where the MSS is
planned in more detail: here it comprises of a cyclical schedule of frequently oc-
curring elective surgery types. The resulting combinatorial optimization problem
is to determine a MSS that balances OT utilization and ward occupancy. By
scheduling surgery types, the surgeon/surgical specialty can assign a patient’s
name at a later time, without affecting the performance of the MSS. The model
considers stochastic OT capacity constraints and empirical LOS distributions. As
the resulting problem is NP-hard, heuristics are provided. For a review on the suit-
ability and managerial implication of this particular MSS approach see Oostrum et
al. (2010).
5.7 Operational: elective surgery scheduling and sequencing
Operational planning and scheduling of operating theatres is arguably one of the
most popular topics in the healthcare operations research literature. The literature
reviews of Cardoen et al. (2010) and Guerriero and Guido (2011) outline many
contributions regarding the elective surgery scheduling and sequencing literature.
Cardoen et al. (2010-2) also propose a classification scheme for operating theatre
planning and scheduling problems, which contains four descriptive fields
|||. Here,  holds the patient characteristics,  the delineation of the deci-
sion,  the extent to which uncertainty is incorporated, and  the performance
measures.
In previous work (Hans et al. 2008) we demonstrated that by combining
advanced optimization techniques with extensive historical statistical records on
surgery durations, the OT department utilization can be improved significantly.
We demonstrated that, if slack time is reserved in OTs according to the method
described in Section 5.3.1 (particularly equation 5.6), the portfolio effect can be
exploited in a local search meta-heuristic as follows. By swapping surgeries be-
tween OTs (1-swap or 2-swap), the total slack time of both involved OTs is af-
fected. By clustering surgeries with similar duration variability characteristics, the
total slack time is reduced due to the portfolio effect. This principle can be used in
a local search heuristic to minimize the total slack time, and thus free OT time. A

22
result of the portfolio optimization is that the fragmentation of the free OT time is
minimized. In fact, OTs in resulting solutions are either filled to a great extent
with surgery and slack time, or are empty. As a result, OTs can be closed, or time
is freed to perform more surgeries.
In this section we discuss the optimization of the elective surgery sched-
ule, in order to minimize emergency surgery waiting time. This problem follows
from policy 2 outlined in Section 5.5 (i.e., emergency surgeries are dealt with in
elective OTs).
5.7.1 General problem description
Emergency surgery waiting time increases a patient’s risk of postopera-
tive complications and morbidity. When dealing with emergency patients accord-
ing to policy 2 (Section 5.5), waiting time will occur when all elective OTs are
busy. Typically, at the beginning of the regular working day all OTs will be busy
with long procedures, as surgeries are often scheduled according to the LPT rule.
As a result, emergency surgeries that arrive just after the start of the elective pro-
gram may have to wait a long time, as surgeries cannot be preempted. This pleads
for scheduling a short surgery at the beginning of the day, to obtain a so-called
“Break-in-Moment” (BIM), at an early time for emergency surgeries. Extending
on this idea, we may sequence the elective surgeries within their assigned OTs in
such a way, that their expected completion times, which are BIMs for emergency
surgery, are spread as evenly as possible. We do not re-assign surgeries to other
OTs, but instead only re-sequence elective surgeries within their assigned OT.
This is illustrated in Figure 5.9: the BIMs are clearly spread more evenly after re-
sequencing the surgeries.
OT1 OT2 OT3 OT1 OT2 OT3
Initialsolution AfterBIMoptimization
BIMs
Fig. 5.9 Example BIM optimization

23
The problem of sequencing elective surgeries in such a way that the BIMs are
spread as evenly as possible (or, alternatively, the break-in-intervals/BIIs are min-
imized) is in fact a new type of scheduling problem. This innovative idea was a re-
sult of a MSc thesis project at Erasmus MC (Lans et al. 2006), where it was prov-
en that the problem is NP-hard by reduction from 3-partition.
We assume, as illustrated in Figure 5.9, that surgeries are executed direct-
ly after another, i.e., there is no planned slack between surgeries. The planning
horizon is within a day, and starts on the first moment when all OTs are scheduled
to have elective surgeries. If all OTs start at the same time, then this time marks
the start of the planning horizon. It ends on the first moment when there is an OT
without a scheduled surgery, since after this moment there are infinitely many
BIMs. The objective is to lexicographically minimize the largest break-in-intervals
(BIIs). In other words, we minimize the largest BII, then the second largest (with-
out affecting the largest), etc. The reason that we do not only minimize the largest
BII is that the expected duration of the shortest surgery is a lower bound to the
longest BII. This can be seen as follows: assuming all OTs start at the same time,
placing the shortest surgery at the beginning of its OT gives a BII that cannot be
shortened.
In forthcoming work we will propose various exact and heuristic ap-
proaches for the BIM/BII optimization problem. Here we give the results of a
Simulated Annealing (SA) local search heuristic, which iteratively swaps surgeries
within their sequence. The SA method uses the following parameters: start tem-
perature 0.2, final temperature 0.0001, Markov chain length 150, decrease factor
0.8. We fix the shortest surgery at the beginning of its OT.
5.7.2 General results
We generate instances with the case mix of Section 5.5 (academic hospital Eras-
mus MC), scaled to fill 4, 8 or 12 OTs. Surgeries are scheduled “First Fit” (Hans
et al. 2008). First Fit assigns surgeries from the top of the list to the first available
OT plus an amount of slack (Section 5.3.1, equation 5.6) to achieve a 30% proba-
bility of overtime caused by surgery duration variability, until no surgery can be
found anymore that fits in the remaining OT capacity. Each instance has two vari-
ants: with full flexibility (all surgery sequences are allowed), and with reduced
flexibility (randomly, 
 of the first surgeries are fixed on this position in their
OT, and 
 of the last surgeries are fixed on this position in their OT). For exam-
ple, surgeries on children are typically done first, and surgeries after which exten-
sive OT cleaning is required are typically done last.
Table 5.3 presents the results for the SA algorithm. It compares the solu-
tions found by SA to the initial First Fit solution (which does not aim to optimize
BIM/BIIs). Particularly, it shows the frequency of the BIIs of size >15, >30, …,

24
>90 minutes. Each number is an average over 260 instances (52 weeks of 5 work-
ing days). SA solves each instance in less than 2 seconds. Clearly, the large inter-
vals are eliminated to a great extent, the more so when there are more OTs (and
thus more BIMs).
Table 5.3 Avg. frequency of break-in-interval size (initial solution  SA solu-
tion; 260 instances per parameter setting)
#
OTs Reduced
flexibility >90 min. >75 min. >60 min. >45 min. >30 min. >15 min.
4 No
1.010.29 1.510.67 2.011.50 2.722.84 4.095.52 5.517.11
-71.3% -55.6% -25.4% 4.4% 35% 29%
4 Yes
1.010.30 1.510.74 2.001.59 2.722.85 4.105.47 5.557.01
-70.3% -51% -20.5% 4.8% 33.4% 26.3%
8 No
0.480.00 0.820.01 1.210.09 1.970.46 3.843.75 6.8910.22
-100% -98.8% -92.6% -76.6% -2.3% 48.3%
8 Yes
0.470.00 0.820.02 1.230.11 1.940.56 3.823.91 6.8810.02
-100% -97.6% -91.1% -71.1% 2.4% 45.6%
12 No
0.330.00 0.690.00 0.950.02 1.470.11 3.141.35 6.9710.49
-100% -100% -97.9% -92.5% -57% 50.5%
12 Yes
0.360.00 0.700.00 0.950.02 1.460.13 3.151.58 6.9210.26
-100% -100% -97.9% -91.1% -49.8% 48.3%
The question now is what impact these optimized BIMs/BIIs have on emergency
surgery waiting time, particularly given the fact that elective surgery durations are
stochastic, and the BIMs are expected surgery completion times. Table 5.4 pre-
sents the results of a Monte Carlo simulation of 260 instances with 12 OTs and
reduced sequencing flexibility. The elective surgeries are assumed to have a
lognormal distribution. The emergency surgeries arrive according to a Poisson
process (on average 5.1 arrivals per day), and are served on a FCFS basis. Elective
surgeries are not preempted.
Table 5.4 Waiting time for the 1st, 2nd and 3rd arriving emergency patients (12
OTs, run length 780 days, max. relative error 10%, min. confidence level
90%)
1st emergency
surgery 2nd emergency
surgery 3rd emergency
surgery
Waiting
time
(minutes)
Initial
solution SA
solution Initial
solution SA
solution Initial
solution
SA
solu-
tion
< 10 28.8% 48.6% 34.9% 44.9% 40.4% 46.2%
< 20 53.0% 75.8% 56.9% 73.6% 63.0% 69.8%
< 30 70.5% 90.9% 71.8% 87.2% 76.3% 86.7%

25
We observe that the BIM/BII optimization by SA, despite the reduced flexibility,
has a significant impact on emergency surgery waiting. For example, the relative
number of first emergency patients that wait at most 10 minutes increases by 69%
from 28.8% to 48.6%. The improvement decreases with every next arriving emer-
gency patient of the day. This may be expected, as these emergency patients in-
creasingly distort the original schedule.
5.7.3 Discussion
BIM/BII optimization has a big impact on emergency surgery waiting. More re-
search is required into exact solution approaches, and perhaps applications of
BII/BIM optimization in other sectors. For healthcare, it is easy to implement: it
only requires re-sequencing of elective surgeries. As a first step, managers are ad-
vised to plan the shortest surgery at the beginning of the regular working day.
5.8 Future Directions
The operating theatre department offers challenging planning and control prob-
lems on all hierarchical levels of control. While operational planning and control
has received a lot of attention from the OR/OM in healthcare research community,
tactical planning is less exposed, and research has had less of an impact in practice
due to its inherent complexity. In our experience, decision support software tools
mostly focus on the operational planning level, whereas tools for the tactical plan-
ning level are scarce and are too simplified or limited in scope to deal with tactical
decision making. Future research therefore has to focus on the tactical level, to a
greater extent. This raises opportunities to expand the scope beyond the operating
theatre department. From our survey of healthcare models that encompass multi-
ple departments we concluded that researchers often model hospitals in a way that
reflects the limited/departmental view of healthcare managers (Vanberkel et al.
2010). The research scope should particularly include the polyclinics, where new
patients are taken in, and the wards, which are typically managed to follow the OT
department but whose workloads may be leveled significantly by tactically opti-
mizing the OT’s master surgery schedule. Ultimately, we should aim to optimize
the entire patient care pathway.
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