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Dealing with Uncertain Surgery Times in Operating Room Scheduling
Macarena Azar, Rodrigo A. Carrasco∗
School of Engineering and Sciences, Universidad Adolfo Ib´a˜nez,
Diagonal Las Torres 2640, Santiago, Chile
Susana Mondschein
Industrial Engineering Department, Universidad de Chile,
Instituto Sistemas Complejos de Ingenier´ıa,
Avda. Rep´ublica 701, Santiago, Chile
Abstract
The operating theater is one of the most expensive units in the hospital, representing up to 40% of
the total expenses. Because of its importance, the operating room scheduling problem has been
addressed from many different perspectives since the early 1960s. One of the main difficulties that
has reduced the applicability of the current results is the high variability in surgery duration, making
schedule recommendations hard to implement.
In this work, we propose a time-indexed scheduling formulation to solve the operational problem.
Our main contribution is that we propose the use of chance constraints related to the surgery
duration’s probability distribution for each surgeon to improve the scheduling performance. We
show how to implement these chance constraints as linear ones in our time-indexed formulation,
enhancing the performance of the resulting schedules significantly.
Through data analysis of real historical instances, we develop specific constraints that improve the
schedule, reducing the need for overtime without affecting the utilization significantly. Furthermore,
these constraints give the operating room manager the possibility of balancing overtime and
utilization through a tunning parameter in our formulation. Finally, through simulations and the
use of real instances, we report the performance for four different metrics, showing the importance
of using historical data to get the right balance between the utilization and overtime.
Keywords: Scheduling, OR in health services, Operating Room Scheduling, Scheduling under
Uncertainty
∗Corresponding author
Email addresses: mazar@alumnos.uai.cl (Macarena Azar), rax@uai.cl (Rodrigo A. Carrasco),
susana.mondschein@uchile.cl (Susana Mondschein)
Final version available at https://doi.org/10.1016/j.ejor.2021.09.010
2021. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
http://creativecommons.
org/licenses/by-nc-nd/4.0/
Preprint submitted to EJOR, v.3.3(210916234605) September 16, 2021
1. Introduction
The operating theater is one of the most expensive units within a hospital, representing up to
40% of the total expenses [
6
]. As a result, there is a vast literature addressing operating room (OR)
management at the strategic, tactical, and operational levels (e.g., see [
16
] and references therein).
One of the most important trade-offs, usually highlighted when managing ORs, is the one between
the cost of overtime when ORs are used after regular hours versus the opportunity cost for under
utilization, when there are idle OR hours [8].
Although researchers have addressed several dimensions of this problem, only a few actual
implementations have been described in the literature. One of the main reasons identified for the
lack of real applications is that the resulting schedules are hard to operationalize by OR managers,
in part due to the high variability of surgery duration [11].
Our work has been motivated by the scheduling problem faced by the Instituto de Neurocirug´ıa Dr.
Ra´ul Asenjo, in Santiago, Chile. This hospital is a specialty facility for patients with neurosurgical
needs, from around the country. Thus, in contrast of what happens at a typical hospital, there are
no competing specialties for the use of OR hours. We remark, however, that if this was not the
case, then the master surgery scheduling, when OR block hours are assigned to different specialties,
would be an input to consider at the daily scheduling process, and incorporated in the mathematical
models when specifying the OR available hours. This study focuses at the operational level where
the detailed daily scheduling takes place: date, time, OR, and, in the more general case, physician
are assigned to each patient. The current practice for the scheduling process is as follows: on
Thursdays, a medical team revises the waiting list and schedules patients for the following week,
considering the patients’ waiting times and trying to maximize the use of the scarce OR operating
hours. We notice that the majority of patients have already an assigned surgeon to perform the
procedure. However, there are simpler surgeries, such as carpal tunnel syndrome, where a physician
is assigned according to time availability. The scheduling process is manual and relies heavily on the
personal experience of the scheduling team. The latter provides an opportunity not only to optimize
the scheduling procedure in terms of performance measures but also to relief professional resources.
The medical center, which is one of the most important for neurosurgeries in the country, has
four operating rooms plus one emergency OR, where more than 110 different types of surgeries are
performed.The hospital has provided detailed historical data of their four ORs. These ORs are
dedicated exclusively to elective surgeries, having a separate OR for emergency ones.
The data shared by the hospital, of all their elective surgical procedures from 2015 and 2016,
corroborates what is shown in the literature: for a given diagnostic and procedure, surgeries may
vary significantly in their duration (also known as case length). As an example, Figure 1shows the
duration, in minutes, of the 20% most common procedures at the hospital, indicating that for the
same surgery code (i.e., the same type of operation), surgeries can have a wide range of duration.
The selected 23 surgeries account for more than 85% of the total procedures done in the Institute,
and more than 88% of the time used in the operating rooms. The box plot in Figure 1highlights
the median for each kind of surgery, with the box covering the upper and lower quartile. The notch
2
in the box represents the confidence interval for the median estimation, whereas the whiskers show
the whole range of the data for that specific type of surgery.
Although the high variability indicated in Figure 1could be explained by the difference in the
performance of the different surgeons, Figure 2shows that this result also holds when considering
surgery times for each physician separately. In this figure, we showcase the lengths of the operation
for all the physicians that performed one specific surgery in the data time-frame: nucleus pulposus
hernia. We continue to observe the high variability in surgery times for each physician ID. We
remark that the source of this variability could be found on the physicians’ personal features such
as experience or training in that specific procedure, as well as patients’ characteristics such as age
or comorbidities. It is also important to note that a physician does not have the same performance
across different types of surgeries. That is, one physician can be among the fastest for one specific
type of procedure, whereas she can be far from that group for a different kind of surgery.
Our Contributions.
In this work, we present a novel approach on how to incorporate surgery
time variability into the OR scheduling problem at the operational level. Our work presents several
contributions to address the operating room scheduling problem.
First, we present a time-indexed formulation for the OR scheduling problem. Because of large
differences in the duration of similar surgeries between the surgeons, our general model allows the
decision-maker to select the surgeon for each procedure. Hence, in the more flexible version of
the model, the recommendation given by our model will not only be at what time to schedule the
operation for each patient and in which operating room, but also the physician that should perform
the surgery.
The main methodological contribution of our work is how we improve the proposed model by
including surgery variability. We do this by developing specific chance constraints using historical
data to guide our development. These new constraints allow the decision-maker to control the
probability of having overtime in each OR in the resulting schedule.
As an additional contribution, we do several experimental studies of our approach. Through
Figure 1: Case length of the 20% most common surgeries at the Instituto de Neurocirug´ıa
3
Figure 2: Case length of one type of surgery by physician
simulations, we show the value added by including historical data into the scheduling problem:
first by separating the performance of different physicians, and later by adding chance constraints
that use this data to control the probability of having overtime in each OR. Our experiments
show that our approach significantly improves the performance of the resulting schedules. We also
validate our methodology on historical data provided by the hospital, showing that even when the
physician cannot be selected (i.e., the surgeon is predetermined for each patient), using these chance
constraints can reduce overtime without sacrificing OR utilization as much.
The rest of the paper is organized as follows. First, in Section 2, we present the literature review
focused on how researchers have dealt with surgery duration variability. Then, Section 3presents
an initial, deterministic scheduling model, which we further improve in Section 4to include the
variability in surgery duration. In Section 5, we show experimental results, highlighting how the
performance of the schedules improve when new data is added, and our methodology is used. Section
6studies the model adding the flexibility of assigning physicians to patients at the scheduling stage.
Finally, Section 7describes our conclusions and future lines of research.
2. Literature Review
The operating room scheduling problem has been studied since the early 1960s. As mentioned
before, there are a plethora of papers presenting different approaches and methodologies to address
this problem. Surveys in [
5
] and [
11
] give comprehensive reviews of such methods for all different
decision levels: strategic, tactical, and operational. In our work, we focus on the latter, and
specifically, in the case where the randomness of surgery times has a significant impact on the
efficiency of the system.
In the particular case of Chile, during the early 2000s, the government introduced a profound
health reform, aiming for a more equitable system, with a law enacted in August of 2004 [
15
]. This
reform includes a set of guarantees regarding access, quality, opportunity, and financial protection
for all citizens when diagnosed with a pathology within a set previously defined. In particular, the
4
opportunity guarantee stipulates a maximum number of waiting days before the patient receives
appropriate treatment. If the opportunity is not satisfied, the insurer has to identify an alternative
provider that can fulfill it. The financial protection guarantee remains for the patient, and therefore,
the potential cost difference must be incurred by the original health provider: the original hospital
that made the diagnosis. This change has added significant stress on the scheduling process, which is
addressed in [
2
]. In that paper, a general framework to assist the OR planning process is developed,
solving the advance and allocation scheduling problems, considering the particular cost structure
currently in place in Chile. In this current work, we continue the work of [
2
] with the next logical
step at the operational decision scheduling level, helping the OR managers to assign the specific
time-slots, ORs, and surgeons to patients daily.
Several approaches have been used to deal with the uncertainty in surgery duration (see a
broader discussion on this topic in [
2
,
19
], and the references therein). At the operational level,
several stochastic optimization models have been developed to incorporate this uncertainty. For
example, in [
13
], the schedule provided by the medical center for one week is used as a base solution,
which is then improved by using a local search heuristic that aims to increase the capacity utilization
and reduce the risk of overtime. In a different approach, in [
20
], the authors use an approximation
method to add probabilistic capacity constraints. However, their methodology requires a large
dataset to tune it. In [
3
], a two-stage stochastic MIP is formulated to assign a patient and a block
time to an operating room. The formulation also allows deciding the number of operating rooms
that should be opened. Although these papers deal directly with the variability of surgery times,
the distribution only depends on the diagnosis and not on the physician performing the surgery. A
more recent approach, described in [
19
], used a Markov Decision Process to model the uncertainty
in surgery duration.
Another recent approach used to deal with uncertainty has been to add chance constraints to the
scheduling optimization model. In a recent result in [
23
], the authors develop a MIP with additional
chance constraints which help the OR manager limit the hospital ward usage for each day of the
planning horizon. Unlike this approach, in our setting we are interested in controlling the overtime
spent by the physicians.
Predictive analytics has also been used to reduce the uncertainty in surgery duration. In this
setting, the prediction of surgical procedure times has been addressed by many researchers (see,
for example, [
7
] and reference therein). In [
7
], the authors mention that the main two factors,
besides the diagnosis, to predict the perioperative times are the primary surgeon and the type of
anesthetic. Although other factors are reported to be significant to make a more accurate prediction,
the authors conclude that these additional factors are not cost-effective. Similar conclusions are
obtained in [
4
] and [
21
]. In all these papers, the authors mention that a more accurate prediction of
the perioperative time should be used to improve OR management. Still, recent surveys do not
mention any work using this information. To highlight this, in [
18
], the authors point out that
most studies that assume certain variability of surgery times, use the historical data to calibrate
the stochastic optimization models or heuristics, instead of using historical data to reduce the
5
uncertainty in the prediction of the corresponding surgery duration. Finally, in [
5
] and [
11
], the
authors suggest that future research should use the statistical methods available in the literature to
reduce the uncertainty of perioperative times and case duration and incorporate this estimation in
stochastic programming models for OR management.
To the best of our knowledge, the only work that schedules ORs using surgeons’ information to
predict surgery duration is [
10
]. In their work, they compare three classic data mining techniques to
estimate surgery duration and propose an integer optimization model to schedule surgeries in one
OR for one week. The authors assume a deterministic perioperative time, which in turn depends on
the surgeon assigned to the operation.
The previously cited results leave one important open question: how can we incorporate surgery
duration’s variability, at the surgeon level, in a cost-effective way that can help the OR manager
balance overtime costs with OR utilization? In this work, we evaluate the performance improvement
achieved by adding more historical data into the model and refining the way uncertainty is modeled.
Unlike previous research results, we deal with variability at the physician’s level, including the
possibility of deciding on the physician during the decision process. We do this by using a time-
indexed scheduling formulation with distribution-related chance constrains to improve the scheduling
of the operating theatre, using the information of surgery times at the surgeon level.
3. Baseline Model: Deterministic Formulation
In this section, we describe a deterministic mathematical model for computing solutions for the
OR scheduling problem, considering average surgery duration. In the subsequent sections, we use
this model as a baseline to illustrate the improvements when explicitly considering the random
nature of surgery times, which we add as chance constraints to this model. As described in [
2
],
the model presented in this section was used to operationalize the results of the aggregate weekly
schedule, also known as the advanced schedule, which was developed in that article.
3.1. Problem Description
We consider a list of patients to be scheduled for surgery during the incoming week given by the
advanced schedule model. We notice, that the latter model provides a list of patients that should
be operated within the upcoming week. Therefore, the scheduling model decides, for each patient,
the day of the week, operating room, starting time and, in the most flexible case, a surgeon for
the procedure. In this paper, we consider only elective surgeries, and therefore, some procedures
can be postponed, with the corresponding consequences on the performance indicators, such as,
for example, average waiting times. The model described in what follows, solves a problem for a
single day, and thus, for day
t
of the week (
t
= 1
,
2
, . . .
5), the patients that are considered for this
allocation are the weekly list minus those already assigned on days 1 to t−1.
Since some surgeries require special conditions, such as leaded OR for x-ray based operations,
some patients could be limited to have their surgeries in a subset of specially upgraded ORs.
Similarly, only a subset of surgeons is qualified for performing specific procedures.
6
Operating rooms are available for
T
regular hours each day, and any procedure that extends
beyond these hours incurs in overtime costs. We consider surgery duration as random variables
that depend on the pathology and the surgeon in charge. The deterministic model proposed in
subsection 3.2 utilizes the average length as an approximation of the surgery time.
Several competing goals can be used to evaluate and compare two OR schedules. The most
common ones are the maximization of OR utilization and the minimization of OR overtime. These
metrics are generally described as very relevant in the literature and coincidentally the ones required
by the OR manager at the Instituto de Neurocirug´ıa.
These goals become particularly challenging when considering that procedure times are, in
nature, random variables, as shown in Figures 1and 2; and highly dependent on the patient and
surgeon attributes. Therefore, as we show in Section 5, the sole use of average surgery times in the
scheduling process might lead to weak OR schedules, with high overtime or low utilization rates.
3.2. Mathematical Formulation
In this subsection, we present a time-indexed integer-optimization model for the allocation of
patients to physicians, ORs, and surgery starting times. In this allocation, we consider average
surgery times for each surgeon and type of procedure.
We would like to highlight that all mathematical formulations, from here on, consider the more
general case where patients can be assigned optimally to physicians at the daily OR scheduling process.
We remark, however, that this assignment considers, through the corresponding mathematical
constraints, that not all physicians are trained to perform each type of surgery. Therefore, for each
type of surgery, a set of qualified physicians is defined. We are aware that, in practice, this might
not be possible and in many cases the pair patient–physician is already determined when performing
the scheduling procedure. Our mathematical formulations easily capture these cases, by restricting
the set of physicians that can operate on a patient to the already assigned professional. To reflect
the fact that the latter situation is commonly found in practice, in the Experimental Results Section
5, we first study this case to subsequently study the impact of relaxing this constraint on the metrics
considered.
For formulation purposes, we divide the total amount of time the OR is operative,
T
, in steps of
time of length ∆
t
hours. We define
T
=
T/
∆
t
, as the total number of time-steps where surgeries
can be scheduled. We introduce the following notation:
Parameters
T: Total number of time-steps of length ∆thours, when the ORs are operative.
I: Total number of patients, where for each patient i∈ {1, . . . , I}, we have:
–Ki
: Set of physicians that can operate patient
i
. Notice that when the pair patient–
physician is predetermined, then for patient
i
, the set
Ki
only contains the assigned
physician.
7
–Ji: Set of operating rooms where patient i’s surgery can be performed.
–Wi: Patient i’s waiting time at the moment of scheduling.
K
: Total number of surgeons, where each surgeon
k∈ {
1
, . . . , K}
, has [
ak, bk
] as the range of
time-steps where she is available.
J: Total number of available ORs.
Random Variables
Slik
: random variable for the surgery duration when patient
i
with pathology
li
is operated by
surgeon k, with plik=Elk[Slik].
Decision Variables
xijkt
= 1 if patient
i
is assigned to OR
j
, surgeon
k
, and the surgery starts at time-step
t
; and
0 otherwise.
We defined the objective function in collaboration with the OR manager at the Instituto de
Neurocirug´ıa. Since the hospital’s objective is to operate as many patients as possible, we use the
maximization of patient throughput as our objective function, prioritizing those patients that have
waited the most. Thus, our objective is to maximize the daily number of patients served, weighted
by their corresponding waiting times. In what follows, we describe the deterministic mathematical
model for this problem.
(BCP) max
x
I
X
i=1
J
X
j=1
K
X
k=1
T
X
t=0
Wixijkt ,(1)
subject to the following operational constraints:
1. At each time-step t, OR jhas at most one surgery scheduled:
I
X
i=1
K
X
k=1
t
X
u=max{0,t−plik+1}
xijku ,≤1,∀j, ∀t. (2)
2. At each time-step t, surgeon kperforms at most one surgery:
I
X
i=1
J
X
j=1
t
X
u=max{0,t−plik+1}
xijku ,≤1,∀k, ∀t. (3)
3. Each patient ican be assigned at most to one OR, surgeon, and time-step starting time:
X
k∈KiX
j∈Ji
T
X
t=0
xijkt ≤1,∀i. (4)
8
4.
For each surgeon
k
, her assigned surgeries can only be scheduled within her available time-steps
at the hospital:
xijkt = 0,∀i, ∀j, ∀k, if t<ak,∀k, (5)
xijkt = 0,∀i, ∀j, ∀k, if t+plik−1> bk,∀k. (6)
5. All surgeries have to be scheduled within the hospital operating hours:
xijkt = 0,∀i, ∀j, ∀k, if t+plik−1> T, ∀t. (7)
6. Binary decision variables:
xijkt ∈ {0,1},∀i, ∀j, ∀k, ∀t. (8)
By solving the problem (1) – (8), which we denote as (
BCP
), the OR manager gets a decision
support tool to assign patients to OR, surgeons, and starting surgery times. We notice that in
constraints (2) and (3) we consider as case length duration the average surgery duration by type of
procedure and surgeon in charge. We remark that the most common approach in the literature is to
consider this time independent of who is performing the surgery. In Section 5.3, we present the
improvement achieved by using this distinction.
Due to the deterministic nature of this problem, its solution does not guarantee that there
will not be overtime when implementing it in a real-life instance, due to the inherent variability
of surgery times. Therefore, to deal with this shortcoming, in Section 4we include probabilistic
restrictions for the daily OR utilization time in the form of chance constraints.
4. Incorporating Surgery Times’ Variability Through Chance Constraints
In this section, we use chance constraints to restrict the probability of using the OR rooms over
the available regular time
T
, and therefore, to incur in overtime. For this purpose, we impose the
condition that the probability that the total stochastic surgery time allocated to operating room
j
surpasses the regular operating hours, must be less than or equal to a tuning parameter
. This
parameter is determined by the decision maker. Thus, at a given day, the total stochastic surgery
time allocated to operating room j, (Sj), can be written as:
Sj=X
ikt
Slikxikjt ⇒El[Sj] = X
ikt
plikxikjt ,(9)
and,
P(Sj≥T)≤⇔P(Sj−E[Sj]≥T−E[Sj]) ≤, ∀j. (10)
9
We propose two different sets of linear sets of constraints, which can be used to substitute
constraints (10), assuming that surgery times have either a Uniform or a Gamma probability
distributions. The rationale for using these specific distributions comes from the data analysis of the
surgery times at the Instituto de Neurocirug´ıa. We analyzed surgery times for all elective procedures
done between 2015 and 2016, using the Kolmogorov-Smirnov test, grouping them by each surgeon
and procedure type. We found that 27% of the surgery times have uniform distributions, whereas
44% of them have Gamma distributions, with the remaining 29% being Lognormal. The linear
transformations for (10), which we will present later, require the use of the Laplace Transform of
the moment generating function of the underlying distribution. Since the lognormal distribution
does not have an analytical moment generating function, we focus our approach on the Uniform
and Gamma distributions.
We also studied the use of Hoeffding’s inequality to develop more general chance constraints,
which do not require any underlying probability distribution requirements other than its support
limits. Our results were not encouraging, obtaining extremely conservative schedules, even for large
values of
. This limitation, together with the added complexity of solving a MIP for this model
(see the Appendix for the description of this model), made this methodology neither practical nor
useful. These poor results have led us not to include these results in this study. In what follows, we
describe the two different set of change constraints we developed, depending on the assumption on
the probability distributions of surgery times.
4.1. Surgery Times with Uniform Probability Distributions
We propose using Markov’s inequality [
17
] to derive chance constraints for the specific setting of
surgery times having an underlying uniform distribution:
P(Sj≥T)≤e−sT
M
Y
m=1
φm(s),∀j, (11)
where
φm
is the Laplace Transform of the moment generating function of the
m
-th random variable
summing in Sj.
In our setting, each random variable represents the surgery time of patient
i
when the procedure
is performed by surgeon
k
, which we assume to have a uniform distribution in [
αlik, βlik
]. Hence, we
use
φlik
(
s
) as the Laplace Transform of the moment generating function when patient
i
is operated
by physician kin a specific OR j.
Using (11) and our decision variables, chance constraints (10) are satisfied when imposing the
following inequalities for each OR j:
I
X
i=1
K
X
k=1
T
X
t=1
xijkt ln (φlik(s)) −sT ≤ln(),∀j,
10
which, when the random variables have uniform distributions, can be rewritten as follows:
I
X
i=1
K
X
k=1
T
X
t=1
xijkt ln esβlik−esαlik
s(βlik−αlik)!−sT ≤ln(),∀j. (12)
Note that constraints (12) are linear as a function of the decision variables
x
. Furthermore,
since (11) it is satisfied for all
s >
0, we would like to select
s
such that we have the best inner
approximation of the feasible set. Additionally, since doing no surgeries in an OR should be a
feasible solution (i.e.
x
= 0), from (12) we obtain that
s≥ln
(
)
/T
. The value of
s
that achieves
the goal of getting the best inner approximation depends on the surgeries assigned to the OR; thus,
we cannot choose one
s
beforehand. Hence, to solve this problem, we pre-build a set of possible
values of sand select among them in the optimization model.
Let
{su}
, with
u
=
{
1
, . . . , U }
, be a set of positive real values of
s
. We can then approximate
constraints (12) by:
min
u(I
X
i=1
K
X
k=1
T
X
t=1
xijkt ln esuβlik−esuαlik
su(βlik−αlik)!−suT)≤ln(),∀j, (13)
which, in turn, can be rewriten as a set of linear constraints, using auxiliary variables. Then, for
each OR j, we have
zj≤ln(),(14)
I
X
i=1
K
X
k=1
T
X
t=1
xijkt ln esuβlik−esuαlik
su(βlik−αlik)!−suT−Myuj ≤zj,∀u, (15)
U
X
u=1
yuj ≤U−1,(16)
yuj ∈ {0,1},∀u, (17)
where Mis an adequately large number.
We can now write the new chance constrained problem with (1) – (8), and (14) – (17) for each
OR j. We denote this problem as (UCCM).
4.2. Surgery Times with Gamma Probability Distributions
Similarly to the previous case, in this subsection, we use the particularities of the gamma
distribution to derive a mathematically tractable expression for the chance constraints. Using
equation (12) and the moment generating function for the Gamma distribution of parameters
κ
and
θ, we can rewrite constraints (10) as
I
X
i=1
K
X
k=1
T
X
t=1
−xijkt κlikln (1 −θliks)−sT ≤ln(),∀j. (18)
11
Note that constraints (18) are linear as functions of the decision variabes. Thus, we use the same
technique used for the uniform setting to determine the best value of swithin a set, to compute a
good inner approximation of the previous constraint. It is important to note that, just like before,
s
needs to comply with
s≥ln
(
)
/T
. In this case, we also need that 1
−θliks >
0
,∀i, k
to have a
feasible solution in (18) when no patients are scheduled on an OR, giving us an upper bound on the
possible values of s.
min
u(I
X
i=1
K
X
k=1
T
X
t=1
−xijkt κlikln (1 −θliksu)−suT)≤ln(),∀j. (19)
These constraints can be replaced by the following set of linear constraints, for each OR j:
zj≤ln(),(20)
I
X
i=1
K
X
k=1
T
X
t=1
−xijkt κlikln (1 −θliksu)−suT−Myuj ≤zj,∀u, (21)
U
X
u=1
yuj ≤U−1,(22)
yu∈ {0,1},∀u, (23)
where Magain is an adequately large number.
Therefore, the chance constrained optimization problem is given by (1)–(8), and (20) – (23)
for each OR
j
. We denote this problem as (
GCCM
). In the following section, we evaluate the
performance of the models described above.
Finally, it is important to note that this approach does not protect against the impact of duration
uncertainty with respect to the intervals [
ak, bk
] for each surgeon
k
, and it only helps balancing
overtime and utilization within each OR.
5. Experimental Results
In this section, we evaluate the performance of the various models presented in Sections 3and 4.
For this purpose, we use simulations based on historical data from the Instituto de Neurocirug´ıa for
the probability distributions of surgeries’ duration and the procedures’ composition of waiting lists.
We also use historical instances from the hospital for validation purposes.
As mentioned in Section 3.2, through this Section, we analyze the setting most commonly found
in practice, where physicians are previously assigned to patients, and therefore, the pair patient–
physician is known when the OR scheduling takes place. This simplification is easily implemented
by setting Kito the specific physician assigned to patient ifor each constraint (4)
12
5.1. Experimental Setting and Performance Metrics
For performance evaluation, we used the following experimental setup: we simulated 100 different
queues of 20 patients each (
I
= 20), all of which are available for surgery in a single day. The
composition of the procedures required by the patients in the queue is statistically similar to the
composition observed in the daily data available from the Instituto de Neurocirug´ıa.
For each instance, we considered a hospital with the same setting presented at the Instituto:
four ORs (
J
= 4) and 8 hours of OR availability, discretized into time-steps of 15 minutes (
T
= 32).
We consider ten surgeons (
K
= 10) from the Instituto with their respective time distributions for
the different types of surgeries. Using these parameters, each of the 100 queues was used as an
input for the different proposed models, each resulting in an optimal schedule.
Since a crucial factor in the quality of the resulting OR schedules is the surgery duration’s
randomness, we simulate multiple realizations of surgery times for each procedure, for each optimal
schedule computed. Therefore, once a model determines an optimal schedule, we simulated 10,000
realizations of that schedule, with random surgery times.
For each patient
i
, we simulated her case length using the hospital’s historical data of procedures
of type
li
. We indicate when we use a time depending on the physician performing the surgery.
As mentioned previously, we considered the three probability distributions identified in the data:
Uniform, Gamma, and Lognormal, and used historical data to tune the distribution parameters, such
that the simulated distribution has the same mean and variance. The rationale behind the decision
to use these distributions lies in our analysis of historical data using the Kolmogorov-Smirnov test,
which showed that these three distributions were the best to represent the case duration of the
surgeries at the hospital. Unless otherwise stated, we always simulate the case length considering
the identified probability distribution for the patient–physician pair.
We used four performance metrics to evaluate the quality of the solutions:
Utilization (UT): the average percentage of regular hours that ORs are occupied in the
T
time-steps available at each day.
Overtime (OT): the average amount of overtime (hours) per OR required to complete all
surgeries in the resulting schedule.
ORs with overtime (OROT): the average fraction of ORs that used overtime to complete all
scheduled surgeries.
Patients operated (PAT): daily average number of operated patients.
All models and simulations were implemented in Python 3.7.9 with Gurobi 9.1.1 as a modeling
toolbox and optimization engine [
12
] and ran in an Intel i7-6700 CPU with 16 GB of RAM. In all
the instances solved in this section, the MIP Gap tolerance for Gurobi was set to 0.1%, with a time
limit of 600 seconds.
13
(a) Utilization (UT) (b) Overtime (OT)
(c) ORs with Overtime (OROT) (d) Patients (PAT)
Figure 3: Base Case (BCS) Performance – Empirical Procedure Times
5.2. Baseline Model: Deterministic Formulation
Most operational scheduling models in the literature use average values for surgery duration,
without distinguishing among physicians performing the corresponding procedure [
5
,
9
]. Given that
this approach is widely used in the literature, we consider it as our starting point, which we name
(
BCS
) or Base Case with average surgery times, against which we compare the results obtained
with the alternative approaches proposed in this study. We remark, however, that unlike in the
(
BCP
) model we evaluate later, in (
BCS
)surgery duration in equations (2), (3), (6), and (7) use
the average duration among all physicians for a specific surgery, and therefore, they only depend on
the type of procedure. Thus, (BCS) considers plik=pli∀k.
Figure 3shows the performance analysis results for the (
BCS
) model when surgery lengths are
simulated with their corresponding underlying distribution. The parameters for each procedure
were determined using the data from the hospital mentioned above. In each subfigure, we show a
histogram of the 1,000 realizations of each of the 100 queues used as input for the (
BCS
) model, for
that corresponding performance metric. We highlight the average value (solid line) for each metric
and add the 99
.
5% confidence interval for the average estimation (dotted lines around). Since we
14
have a large number of simulations, the confidence interval’s width is extremely small, landing over
the mean in most plots.
Figure 3a shows that the model results in a schedule with high utilization of the ORs, with
an average of 87.4%, with 2% of the realizations using 100% of the OR capacity. However, this
high utilization comes at a price, as shown in Figure 3b, where on average 0.8 hours of overtime
are required to complete the scheduled surgeries, with instances in which more than 10 hours were
needed. Overall, 88.2% of the ORs required overtime to complete all scheduled surgeries, as shown
in Figure 3c. Finally, Figure 3d shows that, on average, 15.4 patients are operated daily.
It is important to highlight that, although the (
BCS
) model uses the average surgery times
across all physicians as input, the simulations to evaluate the performance of the resulting schedule
use the actual parameters for the procedure for the specific surgeon that is scheduled to do it.
Also, notice that in Figure 3, the 99
.
5% confidence intervals for the mean are narrow and barely
discernible to the mean line. This observation holds for all the studies made in this and subsequent
subsections.
5.3. Reducing Uncertainty: Considering Surgery Times by Procedure and Physician
The intrinsic randomness of surgery times leads to high overtimes in the Base case (
BCS
). Thus,
when the actual surgery time is longer than the average, the effect would be, in most cases, to
postpone subsequent procedures. However, when the actual time is shorter than expected, following
surgeries, in general, cannot be moved to earlier times due to operational restrictions, such as patient
preparation or availability of medical staff. Therefore, in this subsection, we analyze the advantages
of reducing uncertainty by considering random surgery times for each pair procedure – surgeon in
charge. This case corresponds to the deterministic model described in Section 3, namely (
BCP
), in
which plik=Eli,k[Slik].
Figure 4shows a comparison between the performance of the (
BCS
) and (
BCP
) models for
the four performance metrics. As before, in the simulations, we use the empirical distributions
for the surgery duration. In Figure 4a we observe that by considering that case lengths are the
average for a patient – physician pair and not just surgery type, utilization is slightly improved to
89.4%. Similarly, overtime is also slightly reduced from 0.80 to 0.74 hours on average, with a smaller
variability as shown in Figure 4b, thanks to having a better estimation of case length duration. In
the case of ORs having overtime, Figure 4c shows that this increases slightly to 46.1%, whereas in
Figure 4d we observe that the number of patients is increased to 16.1 on average. These results are
promising and only require distinguishing surgery duration by each surgeon, which, as discussed in
the Introduction, are significantly different in practice. Our results validate the hypothesis stated
in [
5
] and [
11
], that the use of surgeon performance in case length estimation could improve the
resulting schedules. Furthermore, we quantify this improvement by showing that overtime can be
reduced by around 8%. In contrast, the number of patients is increased by almost 4.7%, with minor
changes in the overall ORs utilization rates.
15
5.4. Variability Control: Applying Chance Constraints
In this subsection, we analyze the performance of the resulting schedules when chance constraints
are used. This analysis focuses on uniform and gamma chance constraints, corresponding to the
models (UCCM) and (GCCM), respectively.
Since the (
BCP
) model improved the performance metrics, and the following approach uses
data by each physician, we compare these new models’ performance with the (
BCP
) results instead
of the (
BCS
) one. To do so, we recomputed optimal schedules using the (
BCP
) setting and used
that as a baseline model for comparison purposes.
5.4.1. Uniform-Based Chance Constraints
The (
UCCM
) model was tested with the same experimental setting used in Sections 5.2 and 5.3.
As indicated before, we used the empirical case length distributions to simulate surgery durations.
We also used the solution of the (
BCP
) model for each instance, to compute a set of values of
s
that are useful for implementing equation (13). Figure 5shows the resulting performance metric’s
improvements compared to the (
BCP
) model when using Uniform-based chance constraints with
(a) Utilization (UT) (b) Overtime (OT)
(c) ORs with Overtime (OROT) (d) Patients (PAT)
Figure 4: (BCP) Performance – Empirical Procedure Times
16
(a) Utilization (UT) (b) Overtime (OT)
(c) ORs with Overtime (OROT) (d) Patients (PAT)
Figure 5: (UCCM) Performance – Empirical Procedure Times for = 0.8
= 0
.
8, which in our view, has a good balance between overtime (significantly reduced) and patients
operated (similar to the (BCS) results). We show the results for various values of later.
For the specific case when
= 0
.
8, Figure 5a shows that OR utilization drops to 71.0% compared
to the 89.2% of the recomputed (
BCP
) case, but with a significant improvement in overtime, which
is reduced from 0.65 to 0.18 hours, a reduction of over 72%, as shown in Figure 5b. We also observe
a substantial reduction of ORs that have overtime, from 47.0% to 15.4%, and a reduction in the daily
average of the number of patients attended from 18.2 to 15.1, see Figures 5c and 5d respectively.
Note that, although the overtime is significantly reduced, that number of patients is very similar to
results obtained with the (BCS) approach.
As expected, the performance metrics of the (
UCCM
) model depend on the selected value of
. Therefore,
is a tuning parameter for the OR manager to determine the trade-off’s relative
importance between utilization and overtime. By choosing different values of
, we built an efficient
frontier that shows how these two metrics interact when using this approach, see Figure 6. As the
value of
increases, the expected overtime worsens, whereas the utilization rate improves. Since
each point only shows the average value of the corresponding performance metric, we have added
17
Figure 6: Efficient Frontier for the (UCCM) Model – Empirical Procedure Times
the associated histogram for selected values of
on each axis. The color of the histogram is related
to the same color in the efficient frontier, with red for
= 0
.
4, green for
= 0
.
8, and blue for
= 1
.
0.
In the Figure, and as a reference, we have also included the result for the (BCP) model.
Using Figure 6, the OR manager can decide the correct value of
, according to her balance
requirements between overtime and utilization of the ORs. This value is selected according to
the hospital’s mission, the relation they have with the surgeons, overtime costs, and the director’s
utilization goals, among other factors.
5.4.2. Gamma-Based Chance Constraints
In this subsection, we present the results for the (
GCCM
) model. We use the same experimental
setting used for the (UCCM) model to make them comparable.
Figure 7shows a summary of the results when using this type of chance constraints, using
= 0
.
9. As shown in Figure 7a, utilization drops significantly, to just 36.7% for this setting,
with that reduction allowing an important improvement in overtime. Figure 7b shows that the
average overtime has a significant reduction of almost 90% (0.068 hours), with only 4.6% of the
ORs having overtime at some point, as indicated in Figure 7c. In terms of patients, using model
(GCCM) results in significant reduction as well, with only 8.9 patients on average.
18
(a) Utilization (UT) (b) Overtime (OT)
(c) ORs with Overtime (OROT) (d) Patients (PAT)
Figure 7: (GCCM) Performance – Empirical Procedure Times for = 0.9
Similarly to the (
UCCM
) model,
is a tuning parameter that relates overtime and utilization
of the ORs. Hence, we can evaluate the model’s performance for several different values to build
an efficient exchange frontier between OT and UT, as shown in Figure 8. It is important to note
that since 1
−θliks >
0
,∀i, k
, and
s≥ln
(
)
/T
, we have a limit to the smallest value of
for our
specific instance. Given the Instituto’s data, this limit is
= 0
.
84, and thus our frontier will be
computed from that value onwards. As with the uniform case, the frontier’s shape is similar, with
an increase in utilization as
increases, with overtime increasing as well. In this Figure, we have
also included the histograms of simulations for specific values of
. In this case, we present
= 0
.
88
in red,
= 0
.
96 in green, and
= 1
.
0 in blue. In this case, the chance constraints’ performance is
much worse than in the uniform once, but we will see in the following Section that the model is
useful for a setting with added flexibility.
5.5. Performance Validation
To conclude this Section, we use the hospital’s historical instances to run the models and
compare their performances against those obtained with the actual schedules at the Instituto de
Neurocirug´ıa, validating that our approach adds value to the current scheduling process. Since the
19
Figure 8: Efficient Frontier for the (GCCM) Model – Empirical Procedure Times
(
UCCM
) model has shown the best performance in the previous experiments, we used it for the
experiments shown in what follows.
From the database of more than 700 days of surgeries from 2015 to 2016, we selected all the
days with more than ten elective surgeries in the four ORs available at the hospital. This selection
resulted in 138 days to evaluate our methodology. Considering fewer than ten daily surgeries, it
resulted in easy to schedule instances without much overtime, and thus there was no advantage in
using additional data.
Figure 9a shows an example of an actual day from those selected instances, where each block
represents the actual duration of each surgery as recorded in the hospital’s data. Above each block,
we indicate the patient and surgeon ID numbers according to the hospital’s data. We also use
different colors and hatching pattern combinations to differentiate each surgeon. We highlight that
this is a real instance that occurred and not a simulated realization.
Using these 138 instances, we simulated 10,000 realizations of the random surgery duration
using the probability distributions identified in the historical data for that procedure – physician
pair. Figure 9b shows a histogram of the OR utilization for all 10,000 realizations, with an average
of 73.7%. We also include the 99.5% confidence interval for the mean, shown in dashed lines around
the mean. As before, the confidence interval is minimal. Figure 9c shows the overtime needed to
20
(a) Schedule Example – Historical
(b) Utilization (UT) (c) Overtime (OT)
Figure 9: Example – Real Instance with Empirical Procedure Times
complete the whole schedule for each of the 10,000 realizations, with a daily average of 3.8 additional
hours to perform all procedures.
Next, to test our methodology, we used the same 138 instances as an input for the (
UCCM
) model.
For each case, we predefined the patient – surgeon pair according to the historical data for that
instance and computed an optimal schedule using our model. We used the past average duration
of each procedure
li
by each surgeon
k
for estimating
plik
, and the minimum and maximum case
lengths for
αlik
and
βlik
. We highlight that we did not use the actual realization time for each
surgery when solving the optimization problem, but the historical averages as required by our model.
Finally, just as in the previous case, we simulated 10,000 realizations of the random surgery duration
using the same probability distributions as in the previous case. Figure 10 shows the results of these
experiments.
Figure 10a shows the resulting schedule of the same real instance shown in 9a. As shown in
the Figure, although we kept the same surgeon – patient pair as before, we can see in the actual
21
(a) Schedule Example - (UCCM) Result
(b) Utilization (UT) (c) Overtime (OT)
Figure 10: Example – Using the (UCCM) Model with Empirical Procedure Times
realization of this instance that there is no overtime required to do all the required procedures if
the hospital followed the proposed optimal schedule.
Following the same simulation approach described before, Figures 10b and 10c show the results
for the 10,000 realizations.
These experiments show encouraging results, with significant improvements in the performance
metrics. Figure 10b shows an increase in the utilization rate from 73.7% to 79.9%, thanks to a
better balancing of ORs; and a decrease in overtime of 52.5%, from 3.8 to 1.8 hours. These results
indicate that our approach can significantly help decision-makers, even when the surgeon performing
each surgery is already predefined. However, our approach can go even further. In some settings,
particularly with routine simpler surgeries, the physicians can be selected between the available
ones at the moment of the OR scheduling process. Our integer optimization model can handle
this specific case and choose not only how to allocate patients between ORs, but also decide which
physician should perform which surgery. In the following section, we study this additional setting.
22
(a) Utilization (UT) (b) Overtime (OT)
(c) ORs with Overtime (OROT) (d) Patients (PAT)
Figure 11: (BCP) Performance – Adding Physician Flexibility
6. Adding Flexibility: Assigning Physicians to Patients
Due to how general is our OR scheduling formulation, we can extend the study from Section 5
to the setting where physicians can be selected for each patient. This setting is also useful at the
Instituto since they can choose physicians for simple, routine surgeries. In this section, we study the
improvements that can be achieved when this flexibility is plausible.
6.1. Deterministic Formulation – Adding Flexibility
Figure 11 shows the result of comparing the (
BCS
) model with the (
BCP
) model with the
added flexibility. As Figure 11a shows, the utilization remains similar, improving only slightly to
89.2%. Likewise, the percentage of ORs with overtime remains similar on average, with a slight
increase to 47.0%, as shown in Figure 11d. On the other hand, overtime is improved almost 30%,
reducing it to 0.65 hours as Figure 11b shows. Similarly, Figure 11d shows that the number of
patients operated, increases by more than 15%, to 18.2 patients on average.
23
(a) Utilization (UT) (b) Overtime (OT)
(c) ORs with Overtime (OROT) (d) Patients (PAT)
Figure 12: (UCCM) Performance – Physician Flexibility, = 0.8
6.2. Variability Control: Applying Chance Constraints
This subsection analyzes the resulting schedules’ performance when chance constraints are used
on the extended flexibility formulation. Like before, we study the improvements achieved for the
(
UCCM
) and (
GCCM
) formulations, comparing the results to the (
BCP
) model with the added
flexibility. For this setting, we increased the MIP Gap tolerance to 1% to ensure that we could solve
all the instances for each type of chance constraint and for all the desired values of
in a reasonable
amount of time.
6.2.1. Uniform-Based Chance Constraints
For the specific case when
= 0
.
8, we show the results in Figure 12. Figure 12a shows that
OR utilization drops to 77.4%, compared to the 89.2% of the baseline case. This reduction in
utilization translates to a significant improvement in overtime, which is reduced from 0.65 to 0.24
hours, reducing over 62%, as shown in Figure 12b. We also observe a substantial reduction in the
percentage of ORs that have overtime, from 47.0% to 21.0%, and a reduction in the daily average of
the number of patients attended, from 18.2 to 16.2. See Figures 12c and 12d respectively. Note that
the number of patients is still superior to the ones obtained through the (BCS) model.
24
Figure 13: Efficient Frontier for the (UCCM) Model – Physician Flexibility Added
Similar to the previous analysis of the (
UCCM
) approach, we can use
as a tuning parameter
to balance overtime and utilization. Figure 13 shows the result for several different values of
,
highlighting some of them in the histograms on each axis. Note that, unlike the setting in which
physicians are assigned to a patient previously, adding the flexibility reduces the need for overtime
significantly, shifting the whole efficient frontier. Additionally, the variability is also reduced, as
shown in the histograms, which are tighter than in Figure 6. We note that for this setting, each of
the 900 instances were solved in average in 10.2 seconds, with only 10 instances reaching the limit
of 600 seconds.
In the Appendix, we show an additional study over this approach, for the setting where all case
lengths have an underlying uniform distribution, improving the results significantly.
6.2.2. Gamma-Based Chance Constraints
Next, we study the performance of the (
GCCM
) approach with the added flexibility. We use
the same experimental setting used for the (UCCM) model before.
Figure 14 shows a summary of the results for this setting, using
= 0
.
9. As shown in Figure
14a, utilization has a steep drop to 32.2%, whereas the average overtime has a massive reduction to
0.02 hours, which is more than 96% reduction, shown in Figure 14b. This reduction also implies a
25
(a) Utilization (UT) (b) Overtime (OT)
(c) ORs with Overtime (OROT) (d) Patients (PAT)
Figure 14: (GCCM) Performance – Physician Flexibility, = 0.9
significant reduction in the number of ORs that have overtime, to 2.5%, as well as the number of
patients operated in average, which is reduced to 9.3. See Figures 14c and 14d respectively.
Finally, in Figure 15, we show the efficient frontier obtained by computing the optimal schedule
for different values of
. Unlike the setting in which physicians are assigned to a patient previously,
in this case, the (
GCCM
) approach returns a consistent advantage over the deterministic models,
even for large values of
. The variability in overtime and utilization of the resulting schedules is
also reduced, as shown in the accompanying histograms.
7. Conclusions and Future Work
Uncertainty in surgery duration is one of the most critical challenges when computing useful,
efficient, and robust OR planning schedules. Furthermore, reducing the uncertainty is viewed by
many researchers as an essential next step to improve OR management decision support systems
[5,9,11].
In our work, we address this uncertainty by acknowledging that, according to empirical data,
surgery duration depends not only on the type of procedures but also on the physician in charge
26
Figure 15: Efficient Frontier for the (GCCM) Model – Physician Flexibility Added
of it. We show that by considering the surgeon in charge of each procedure to estimate the case
length, we reduce the uncertainty and significantly improve the schedule performance.
Through experimental studies, we show that differentiating surgery duration by each physician
and procedure type helps reduce overtime while improving utilization rates and the number of
patients that can be operated on.
We further improve our approach by incorporating constraints to control the probability of
having overtime, allowing the OR manager to have a new tuning parameter that balances overtime
and utilization. We develop new linear constraints that allow us to control the probability of having
overtime while maximizing patients’ throughput. These constraints are based on the underlying
probability distributions by procedure and surgeon. Our results show significant improvements in
the performance of the resulting schedules. These improvements were consistent through different
probability distributions of surgery duration, including uniform, gamma, and lognormal.
Finally, we show that these improvements come from patients’ right assignment to ORs and
their specific order. We show this through historical instances by noting that the developed chance
constraints improve schedule performance even when physicians are predefined for each procedure,
reducing overtime significantly while enhancing OR utilization.
Case length duration studies, like [
22
], show that lognormal distributions are superior when
27
simulating surgery duration. Therefore, an interesting open question is if it is possible to develop
chance constraints specific to lognormal distributions. The approach described in Section 4 does
not allow this extension directly since lognormal distributions have no explicit moment generating
function. New methods, such as the one described in [
1
], allow for the numerical computation of
the Laplace transform of the moment generating function of the sum of lognormal distributions.
Thus, a future improvement is to use these methods to develop lognormal-based chance constraints,
which might improve our approach’s performance.
Another important future work is to incorporate surgery times case predictions to reduce
variability to improve schedule performance metrics. The use of relevant patient information, such
as gender, clinical parameters, and comorbidities, have been used in the recent literature to predict
the duration of surgeries or stays at the post-anesthesia care unit (PACU), see, for example, [
9
].
Therefore, predictive analytic tools that return estimations and confidence intervals, which could
be used as inputs for the chance constraints developed in this paper, might further enhance the
resulting schedules’ performance.
As a final comment, we would like to emphasize the importance of selecting good performance
metrics for these scheduling problems and implementing them. The metric presented in this paper,
weighted throughput, was designed in conjunction with the physicians at the hospital after testing
the performance of several different ones. However, even though it resulted in a good balance for this
hospital and the approach improves schedules when surgeons cannot be selected, in other settings,
it can prioritize faster surgeons, if available, which are not necessarily the best ones for a specific
patient. This problem can be addressed by reducing the set of feasible surgeons for a specific patient,
Ki, leaving only those who can give the correct care or have the necessary experience.
Acknowledgment
Rodrigo A. Carrasco would like to acknowledge that this work has been partially funded by
Project Anillo 1407 and ANID Projects Fondecyt 1151098 and Fondecyt 1200809. The authors
would also like to thank Professor Javiera Barrera for her insights, and the anonymous reviewers
and the editor for their comments and recommendations. They all helped us improve this work.
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Appendix A. Appendix: Experimental Results for Uniform Distribution Case Lengths
In the experimental analysis of Sections 5and 6, we showed the performance of the methodology
using the empirical distributions found at the hospital. Since the (
UCCM
) approach is explicitly
designed for surgeries with uniform distributions, in this Appendix, we show how the methodology
performs when all case lengths have a uniform distribution.
Figure A.16 shows the efficient frontier for this case. As expected, the performance is much
better than the empirical procedure times setting shown before, improving both overtime and
utilization.
Appendix B. Appendix: Chance Constraints for the General Case with no Distribu-
tion Assumptions
In this appendix, we study the development of chance constrains in a more general setting,
where no underlying distributions are known. In this more general case, we make no assumptions
regarding the probability distribution of surgery times. Accordingly, we use Hoeffding’s inequality
[14] to reformulate chance constraints (10). In the general case, Hoeffding’s inequality is given by:
P(Sj−E[Sj]≥t)≤exp −2t2
Pn
i=1(βi−αi)2,(B.1)
30
where
Sj
is the sum of
n
independent random variables, with [
αi, βi
] being the support of the
i
-th
of them. Combining inequalities (10) and (B.1), and using the decision variables of problem (1) –
(8) we obtain the following constraint:
exp −2 (T−E[Sj])2
PI
i=1 PK
k=1 PT
t=1 xijkt (βik −αik )2!≤, ∀j, (B.2)
which guarantees condition (10).
Let zijk denote if surgeon kperforms the surgery of patient iin OR j(zijk =PT
t=1 xijkt ) and
Γj(x, ) = −1
2ln()
I
X
i=1
K
X
k=1
T
X
t=1
xijkt (βik −αik )2−T2.(B.3)
Assuming that surgery times are statistically independent and using that
zijk1zij k2
= 0 if
k16
=
k2
,
Figure A.16: Efficient Frontier for the (UCCM) Model – Uniform Procedure Times
31
we rewrite inequality (B.2) as follows:
−2TE[Sj] + E[Sj]2≥Γj(x, ),∀j,
⇔ −2T
I
X
i=1
K
X
k=1
E[Sik]zij k + I
X
i=1
K
X
k=1
E[Sik]zij k !2
≥Γj(x, ),∀j,
⇔
I
X
i=1
K
X
k=1 p2
lik−2T plikzijk + 2
I
X
i=1
I
X
q=i+1
K
X
k=1
plikplqkzijk zqj k ≥Γj(x, ),∀j.
(B.4)
Note that Γ
j
(
x,
) is linear in
xijkt
, thus constraints (B.4) are non linear only as a function of
zijk
variables. To rewrite them as linear constraints, we define auxiliary variables
ykjiq
as 1 if surgeon
k
performs surgeries on patient
i
and
q
, in OR
j
, and 0 otherwise. This condition is represented by
the following inequalities:
T
X
t=1
(xijkt +xq jkt)≤1 + ykj iq ,∀k, ∀j, ∀i, ∀q6=i, (B.5)
T
X
t=1
xijkt ≥ykjiq ,∀k, ∀j, ∀i, ∀q6=i, (B.6)
T
X
t=1
xljkt ≥ykjiq ,∀k, ∀j, ∀q, ∀i6=q. (B.7)
Thus, constraints (B.4) can be rewritten by:
I
X
i=1
K
X
k=1
T
X
t=1 1
2ln()(βik −αik)2+p2
lik−2T plikxijkt +
2
I
X
i=1
I
X
q=i+1
K
X
k=1
plikplqkykjiq ≥ −T2,∀j, (B.8)
and constraints (B.5), (B.6), and (B.7).
Finally, we can write the new chance constrained problem with (1) – (8), and (B.5) – (B.8).
Although these constraints are linear and easy to implement in an integer optimization model,
the performance in our experimental setting was much worst than the other chance constrains
approaches. Thus, we limit our exposition to the theoretical construction without the experimental
results.
32
