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E-mail: eliana.gonzalez@javeriana.edu.co (E. M. González-Neira)
2018 Growing Science Ltd.
doi: 10.5267/j.ijiec.2018.1.002
International Journal of Industrial Engineering Computations 9 (2018) ***–***
Contents lists available at GrowingScience
International Journal of Industrial Engineering Computations
homepage: www.GrowingScience.com/ijiec
A simulation-optimization approach for the surgery scheduling problem: a case
study considering stochastic surgical times
Diana Marcela Díaz-Lópeza, Nicolás Andrés López-Valenciaa, Eliana María González-Neirab*,
David Barrerab, Daniel R. Suárezc, Martha Patricia Caro-Gutiérrezb and Carlos Sefair (MD)d
aResearch Assistant, Department of Industrial Engineering, School of Engineering, Pontificia Universidad Javeriana, Carrera 7 # 40-62,
Bogotá, Colombia
bAssistant Professor, Department of Industrial Engineering, School of Engineering, Pontificia Universidad Javeriana, Carrera 7 # 40-62,
Bogotá, Colombia
cAssociate Professor, Department of Industrial Engineering, School of Engineering, Pontificia Universidad Javeriana, Carrera 7 # 40-62,
Bogotá, Colombia
d Surgeon, Director of Department of Surgery, Hospital Universitario Mayor - Mederi, Carrera 54 # 65-28, Bogotá, Colombia
C H R O N I C L E A B S T R A C T
Article history:
Received September 20 2017
Received in Revised Format
December 25 2017
Accepted January 24 2018
Available onli ne
January 24 2018
This work studies the scheduling of elective procedures, with stochastic durations, in surgery
rooms. Given a set of rooms with limited availability and a set of procedures, it must be decided
in which room and when each procedure will be performed. The problem’s objectives are to
maximize the use of the operating rooms and to minimize the delays in starting the scheduled
surgeries. A simulation-optimization approach is proposed. First, procedures’ durations are
modeled as random variables and a set of test percentiles (i.e. it is assumed that all surgeries will
last as many minutes as the 75th percentile of its probability density function) is selected.
Subsequently, using these durations as a parameter, a greedy randomized adaptive search
procedure (GRASP) is run. Consequently, as many solutions as selected test percentiles are
generated. Finally, a Monte Carlo simulation is used to estimate three indicators: i) rooms
utilization, ii) percentage of surgeries that had delays, and iii) average delay time of scheduled
surgeries. The technique was tested by solving the elective procedures scheduling problem in a
high-complexity hospital in Bogota. This hospital has 19 operating rooms and 35,000 surgeries
performed annually. Currently, the scheduling process is manual. The simulation-optimization
proposed approach allowed to determine the relation between utilization rate and delays in the
service. As the occupation percentage increases, delay times also augment, implying a reduction
of the service level. An average reduction of 5% in delay times entails a reduction between 3%
and 9% of operating room occupancy.
© 2018 Growing Science Ltd. All rights reserved
Keywords:
Surgery scheduling problem
GRASP
Combined simulation and
optimization techni ques
1. Introduction
Hospital management has changed from a resources optimization approach to a balance between
efficiency and quality (Brailsford & Vissers, 2011). Consequently, the decision-making process has
become increasingly complex (Hulshof et al., 2012) and the use of operations research has grown steadily
in recent years (Dobrzykowski et al., 2014). The hypothesis is that the appropriate allocation of resources,
in support processes, positively influences the quality of service (Van De Klundert et al., 2008). It is also
2
expected that a proper scheduling of resources involved with the patient (human, equipment and
facilities) leads to the improvement of critical variables such as cost, coverage, and response times
(Cardoen et al., 2010; Elkhuizen et al., 2007; Liu et al., 2011; Velasco et al., 2012). Despite the growth
of research in this area, there is still a gap regarding the actual practices of hospitals (Brailsford & Vissers,
2011; van Sambeek et al., 2011). One of the main challenges for the research community is to achieve
collaborative spaces that allow a greater impact of developments in real systems (Pitt, Monks et al., 2015;
Velasco et al., 2012).
In this context, the scheduling of operating rooms is particularly relevant. Surgical services account for
40% of total costs of hospitals and are an important part of their income (Denton et al., 2007).
Consequently, this issue has been addressed from different approaches (Cardoen et al., 2010; Hasvold &
Scholl, 2011). In particular, the surgery scheduling problem (SSP) aims to define the time and room
where a set of surgeries must be performed (Abdelrasol et al., 2014). Several authors have demonstrated
the positive impact of the use of formal methodologies and tools to make this decision (Díaz-López et
al., 2015; Ghazalbash et al., 2012; Litvak, 2010; Shylo et al., 2013). However, in a regional context, only
a few approaches have been made regarding this problem (Alonso et al., 2013; Ceballos-Acevedo et al.,
2014; Díaz-López et al., 2015; Velásquez-Restrepo et al., 2012). Moreover, Velasco et al. (2012)
concluded that there was a field of interest for the academic community in the study of problems in real
environments.
Regarding this, Abdelrasol et al. (2014) reviewed the literature and identified two lines of work that will
be of interest to hospitals and researchers in the coming years. The authors conclude that the development
of efficient algorithms and the inclusion of the variability of the surgical times should be addressed in
order to close the gap between management practices and research. Assuming a deterministic surgical
time, as is usual in research, makes the implementation of the results difficult (Velásquez-Restrepo et al.,
2012). The same challenge has been addressed in other classical problems (Holm et al. 2013; Juan et al.,
2014; Sarin et al., 2013). Consequently, the use of hybrid simulation-optimization techniques has
flourished (Juan et al., 2015).
Due to the complexity of relations to be modeled, it is common to use discrete event simulation for
operating rooms scheduling (Aringhieri et al., 2015; Baesler et al., 2015; M’Hallah & Al-Roomi, 2014).
However, optimization could overcome the limitations of simulation when selecting the evaluated
scenarios (Zhang & Xie, 2015). According to Baesler et al. (2015), most of the work in this area is
concentrated in the simulation phase and uses dispatching rules for the optimization problem. This
situation implies an enhancement field since, in scheduling problems, the use of more elaborate
optimization techniques has shown a positive impact on the quality of the solutions (Molina-Sánchez &
González-Neira, 2016). In fact, after a revision of 216 papers in literature, published between 2000 to
2014, Samudra et al. (2016) suggest the usage of simulation-optimization methods to solve the complex
surgery scheduling problems.
Recently, some studies have explored this approach using search techniques for the optimization phase.
For example, Testi et al. (2007) proposed a hierarchical approach of three stages weekly scheduling. The
objectives were improve throughput, overtime and reduce waiting list. In the first phase the number of
sessions to assign to each specialty is solved ass a bin packing problem. The second phase consists in the
assignment of specialties to the operating rooms. And, in the final step, the simulation procedure is
implemented to define the best sequence. Saadouli et al. (2015) studied the problem of scheduling
surgeries within a planning horizon of one day. The authors solve the optimization problem using exact
methods and discrete events simulation is used to evaluate the solution quality. Baesler et al. (2015) use
simulated annealing at the optimization phase. The authors solve instances of up to 170 patients, and
found that the use of more sophisticated search techniques, other than the dispatching rules, can deliver
up to 18% of improvement in utilization levels. Wang et al. (2014) group surgeries to schedule each kind
of them in a different operating room to reduce the flow variability and waiting time. They present an
alternate simulation-optimization structure where the results of simulation are the inputs of the
D. M. Díaz-López et al. / International Journal of Industrial Engineering Computations 9 (201 8)
3
optimization model and its results are used for the next simulation, and so on, until meeting the stopping
criteria. With this approach authors reduced the waiting time in 89% and the overtime in 46%. Beaulieu
et al. (2012) proposed a methodology with four phases to deal with surgery scheduling and rescheduling.
In the first phase authors proposed a MILP model for the assignment of the surgeries to a given day. In
the second step, diverse strategies are implemented to schedule the surgeries in that day. These
assignments are evaluated in the third stage with the use of simulation. Then, in the fourth phase decisions
of rescheduling are taken if it is required. In this work, a four steps approach is proposed. First, surgery
cases are assigned to a given day. Second, they are scheduled according to different strategies, which are
evaluated through a simulation tool in the third step. If needed, feedback and rescheduling occur in the
fourth step. Authors evaluated the performance of their approach with randomly generated instances
based on real data of a large hospital in Montreal. Duma and Aringhieri (2015) designed an online
solution method to adjust a patient schedule that have been affected by a delay during its execution.
Possible modifications are rescheduling the surgery or assignment of overtime for the surgery. The
authors also developed an offline simulation-optimization model to evaluate the impact of the online
procedure. Aringhieri and Duma (2015) developed a simulation-optimization model to Schedule
surgeries. The method includes a greedy algorithm to construct the solution and a local search that
improve it. Authors center their attention on the analysis of a surgical clinical path of a patient to optimize
the use of resources and assess the effect of the optimization in different indexes focused both on patients
and facility. Molina-Pariente et al. (2016) consider stochastic surgery times and uncertain arrivals of
surgeons and emergency surgeries to minimize under and overtime costs of operating the rooms and costs
of exceed the capacity constraints. Authors proposed an iterated greedy local search hybridized with a
Monte Carlo simulation and compare the results obtained with this method with the deterministic solution
of the problem, demonstrating the cost reduction when solving the stochastic problem. Landa et al. (2016)
solve the advance and allocation scheduling problems. In the first one authors assigned a date for surgery
and an operating room block for the surgeries. They determined the sequence of the surgeries in each
operating room in a specific day. They developed a hybrid algorithm that combines Monte Carlo
simulation with neighborhood search methods to obtain solutions that maximizes the utilization rate
while patient cancellations are reduced. Addis et al. (2016) schedule also elective procedures using the
block scheduling strategy and considering stochastic surgery times and stochastic arrivals of new
patients. Therefore, authors propose a rolling horizon method for patients selection and assignment where
an integer linear programming model (ILP) is used iteratively, minimizing waiting time and tardiness of
assignment of patients, the uncertainties that are generated between one period to another are recovered
at the next iteration of the ILP model. Finally, Ozcan et al. (2017) applied a simulation-optimization
approach, in which simulated annealing metaheuristic deals with the optimization issue, to allocate
resources for thyroid surgical treatment without ignoring the other types of surgeries. The implemented
approach allows managers to involve hospital and patients objectives by obtaining the opportunity costs
of prioritizing one objective to another.
In this context, it is necessary to continue exploring other optimization techniques to improve the quality
of the solutions and reduce computational time—a factor that can hamper implementation of
sophisticated optimization techniques (Baesler et al., 2015). Additionally, research is focused on
optimizing objective functions related to efficiency. However, in environments of high variability, it is
interesting to quantify the impact of high rates of utilization in the service level (Holm et al., 2013).
Consequently, this paper proposes a simulation-optimization technique to solve the stochastic version of
the surgery scheduling problem.
Instead of generating one solution for each instance, the proposed technique generates a set of solutions
so that the hospital can decide which one to implement. In general, the proposed technique implies that
the surgical time can be modeled by probability density functions. Then, these functions are used to
generate a set of realizations corresponding to different percentiles of surgical time. Subsequently, for
each percentile, the deterministic version of the problem is solved using GRASP. Thus, there are as many
solutions for one instance as test percentiles chosen. Finally, through Monte Carlo simulation, the
4
confidence intervals are calculated for three indicators: i) utilization percentage of rooms, ii) surgeries
percentage that are delayed, and iii) average delay time in surgery. In this manner, the proposed technique
quantifies the impact of high levels of utilization in the service level.
2. Methods
The problem studied is the SSP for elective procedures. This is due to some hospitals has separate rooms
for elective surgeries than those for emergency and urgency surgical procedures. The set of surgeries to
be performed and the availability of a set of homogeneous operating rooms are known. In this context,
two decisions must be made: i) the allocation of the procedures to the rooms, and ii) the starting time of
each procedure. The duration of surgeries is not known with certainty; however, it can be modeled by
using a random variable. The aim is to maximize the occupation of operating rooms. Likewise, the impact
of the chosen strategy by means of two indicators of service quality must be quantified: i) the percentage
of surgeries that had delays, and ii) the average delay time of scheduled surgeries.
A methodology that combines simulation and optimization techniques to attack this problem was
designed. Figueira and Almada-Lobo (2014) have proposed a classification of hybridization approaches
of these two methods, depending on their role in solving the problem. Specifically, from the point of
view of hierarchical structure, the simulation-optimization approach selected for this research is
sequential simulation-optimization (SSO) and in particular the evaluation procedure named Solution
Completion by Simulation (SCS). The SSO consists in that optimization and simulation modules run
sequentially one after another. The SCS allows to complete the initial solution obtained in the
optimization phase by giving more accurate values for the different variables and objective function
through simulation (Figueira & Almada-Lobo, 2014). Thence, we resolved the deterministic SSP at the
optimization phase using GRASP metaheuristic due to the NP-hardness of the problem (Brucker, 2007;
Pinedo, 2012). Posteriorly, we applied a Monte Carlo simulation to validate the results and calculate the
expected value of the indicators. A graphical representation of the methodology used in this study, based
on the proposed algorithm in (Juan et al., 2015), is shown in Fig. 1.
Fig. 1. Simulation-Optimization approach (Adapted from (Juan et al., 2015))
Sta
rt
Stochastic Problem
Deterministic Problem
Surgical time
realization
Simplification process
Generate a GRASP
solution
More iteration needed?
Performance analysis
End
No
Yes
Simulation Process
D. M. Díaz-López et al. / International Journal of Industrial Engineering Computations 9 (201 8)
5
Initially, it was assumed that all procedures would last as many minutes as the 50th percentile of its
probability density function. Taking the durations as known, with the value of 50t h percentile, GRASP
was used to schedule the operating rooms. The resulting schedule was analyzed by using a Monte Carlo
simulation model, taking the corresponding probability distribution for each surgery. It allows us to
estimate the expected values of utilization and quality of service indicators. Finally, this process was
repeated by increasing 5% in the percentile used for surgeries’ durations up to the 95th percentile. Thus,
a total of ten different scheduling scenarios were considered. The details of the phases of optimization
and simulation will be discussed next.
2.1 Optimization phase
For this stage we firstly proposed the mathematical formulation of the problem and secondly solved it
with a simulation-optimization procedure in order to evaluate the performance of the solution method
under uncertainty.
2.1.1 Mathematical formulation
The mixed integer linear programming MILP proposed model is:
Sets:
∶ Surgeries
: Operating rooms
: Days of a week
: Position of the sequence in which a surgery is scheduled in a specific room and day
Parameters:
: duration [hours] of surgery
:maximum scheduling time [hours] for a day
: big positive number
Decision variables:
,,,: Binary variable that takes value 1 if surgery is sequenced immediately after surgery on day
at room
: continuous variable for the finishing time of surgery
Objective function:
:
,
,
,
∗
∈
∈
∈
∈
(1)
Subject to:
,
,
,
∈
≤
1
;
∀
∈
,
∀
∈
,
∀
∈
(2)
6
,
,
,
∈
∈
∈
≤
1
;
∀
∈
(3)
,
,
,
∈
≥
,
,
,
∈
;
∀
∈
,
∀
∈
,
∀
∈
,
<
|
|
(4)
+
1
−
,
,
,
≥
+
−
1
−
,
,
,
;
∀
∈
,
∀
ℎ
∈
,
ℎ
≠
,
∀
∈
,
∀
∈
,
∀
∈
,
<
|
|
(5)
≥
,
,
,
∈
∈
∈
;
∀
∈
(6)
≤
,
,
,
∈
∈
∈
;
∀
∈
(7)
,
,
,
∈
{
0
,
1
}
;
∀
∈
,
∀
∈
,
∀
∈
,
∀
∈
(8)
Eq. (1) calculates the utilization rate in hours. Constraints set (2) ensures that in each room at each day
in each position of the sequence at most only one surgery is scheduled. Constraints set (3) guarantees
that every surgery is scheduled at most one time in the scheduling horizon. Constraints set (4) makes that
at each room on each day the surgeries be sequenced one next to another one. Constraints set (5) controls
that surgeries do not overlap and calculates the finalization time of every scheduled surgery and together
with constraint set (6) allows to calculate the completion time of a scheduled surgery. Constraints set (7)
states that the completion time of a surgery must not exceed the maximum scheduling time and sets in
zero the completion time of an unscheduled surgery. Constraints set (9) define the domain of the binary
decision variables.
2.1.2 Simulation-optimization procedure
As it was mentioned in Methods section GRASP metaheuristic was selected for the optimization phase
of the simulation-optimization procedure. GRASP is an iterative multi-start metaheuristic, designed for
solving combinatorial problems. Each iteration of the metaheuristic consists of two steps: construction
and local search (Resende & Ribeiro, 2010). For construction, the surgeries were organized according to
a greedy function. Subsequently, a list of candidates (i.e. surgeries that can be scheduled) was
constructed, and one of them was randomly selected. This process was repeated until all the surgeries
were scheduled. Later in the local search phase, the initial solution is taken and all exchanges of surgeries
among different days and rooms were evaluated. Finally, it was assessed whether, after all exchanges of
local search, it was possible to schedule some of the surgeries that had not initially been sequenced in the
construction phase. Fig. 2 presents a graphical representation of the GRASP algorithm proposed. To
conclude on quality, the metaheuristic’s solutions were compared with exact solutions of an equivalent
linear programming model.
2.2 Simulation phase
In problems where the parameters have random behavior, it is expected that the performance measures
have it also (Li et al., 2012). Consequently, the Monte Carlo simulation was used to validate the results
and to estimate the risk (Bayer, 2014; Tako et al., 2014). To calculate the required number of iterations
of the simulation, the proposed algorithm in Framinan and Perez-Gonzalez (2015) was used. An
amplitude interval for indicators of ±2.5% around the mean was established as a convergence threshold
to be reached. Thus, the simulation stopped when the indicator with the highest variance reached such
accuracy. Three indicators were calculated: occupation rate (14), waiting time (15) and waiting rate (16).
The occupation rate shows the proportion of available time that the rooms are used for surgeries. The
D. M. Díaz-López et al. / International Journal of Industrial Engineering Computations 9 (201 8)
7
waiting time represents the average time in minutes that a patient has to wait if his corresponding surgery
is starting late. The waiting time is the proportion of surgeries that begin after its schedule hour.
Ocupation
rate
:
(14)
Waiting
time
:
(15)
Waiting
rate
:
ℎ
(16)
3. Case Study
The proposed methodology was tested to schedule surgeries for one week at the Méderi University
Hospital in Bogotá, Colombia. The surgery service at the hospital has 19 operating rooms and performs
approximately 35,000 procedures a year. According to a previous study, the use of formal tools for
scheduling rooms has high potential for impact at this hospital (Estupiñán et al., 2016). The authors found
that by shifting from manual to algorithmic programming with the use of dispatching rules, the
occupation indicator can be improved considerably. These results confirm the hypothesis that a low-
quality scheduling can explain poor performances in service indicators (Litvak, 2010; van Sambeek et
al., 2011; Velasco et al., 2012).
Currently, the scheduling process is made manually and there are no specialty blocks, this implies an
open surgery scheduling. Historical data corresponding to all surgeries performed during May 2014 were
analyzed. According to the method proposed by Baesler et al. (2015), by using tests of homogeneity and
goodness of fit, the probability density function that best describes the duration of surgeries was
determined. In total, 26 probability density functions for surgeries were generated, which explain 89%
of the occupation of operating rooms. The goodness of fit tests was run on Expertfit (Flexim 7).
In order to parametrize GRASP, random instances were generated for the parameterization of the
metaheuristic through a 3 factorial design. Factors were greedy parameter of GRASP (with levels 0.2,
0.25 and 0.3), way in surgeries are sorted (ascending order of duration, ascending order of variability,
descending order of duration) and number of iterations of GRASP (100, 500 and 900). According to
ANOVA, the interaction of the three factors is statistically significant (p- value < 0.0001 for a 95%
confidence). Additionally, a Tukey test was performed—to determine the best treatment. It was found
that the best combination of parameters is as follows: sorting surgeries in ascending order according to
their duration, selecting the 20% of non-scheduled procedures for the list of candidates, and running 900
iterations. Assumptions of normality (p -value> 0.05 in the Kolmogorov-Smirnov test) and homogeneity
of variance (p- value = 0.056 for the Levene test) were tested.
Expected results of the case study are divided into two categories:
i) The performance GRASP-based approach results were measured with the outcomes obtained
by a mixed integer linear programming model that was run for six hours.
ii) The real instance obtained with the hospital database was run with the simulation-
optimization approach to establish utilization and service levels for ten different scenarios.
8
Fig. 2. Proposed GRASP algorithm
4. Results and discussion
4.1 GRASP performance in comparison with MILP model
A first aspect of interest is the comparison of GRASP against the results obtained with an integer
programming model. The problem of scheduling operating rooms is highly related to the scheduling
problem for parallel machines; therefore, it can be formulated in that way (Pinedo, 2012). However, since
the objective of the mathematical model is to determine the quality of the solutions given by the
metaheuristic, a simplification of the formulation through a multiple knapsack problem approach is
proposed (Chen et al., 2016).
To compare the results, ten deterministic small instances were generated, one for each percentile of
surgical time, by using the probability distributions previously fitted with data of original surgeries at the
Hospital. The random generated instances consisted in 27 types of surgeries, 10 operating rooms and a
total of 63 surgeries to be scheduled in only one day. Those instances were tested with the MILP model
and with GRASP metaheuristic. To define these ten instances, many tests were carried out with instances
of different sizes. The process of creating them begun with big instances similar to the Hospital ones,
D. M. Díaz-López et al. / International Journal of Industrial Engineering Computations 9 (201 8)
9
and we were reducing the size of them by decrementing the number of surgeries, days and rooms until
we achieve to solved it in GAMS. With big and small instances, we obtained always an out of memory
response. The reason for creating only small instances is that the big and medium instances designed,
were tested with GAMS and the response was always out of memory. GRASP was executed 10 times
(with 10 different seeds) for each Scenario with the parameters selected in section 3, and the outcomes
of worst, best and average results of occupation rate and execution time were recorded (Tables Table 1
and Table 2 respectively). The model was solved using the solver CPLEX 12.7 of GAMS and the GRASP
approach was coded using Visual Basic for Applications of Excel.
Table 1
Occupation rate of MILP and GRASP, and GAP of GRASP for each scenario
Percentile (Scenario) Occupation rate % Ocupation rate GAP %
(with respect to MILP)
MILP GRASP
M inimum Average Maximum Standard
Deviation
Minimum Average Maximum
50 99.61 94.56 95.70 97.91 1.16 5.05 3.91 1.70
55 99.61 93.45 96.22 97.34 1.17 6.16 3.39 2.27
60 99.71 94.52 95.93 97.12 0.76 5.19 3.78 2.59
65 98.85 94.90 95.94 97.29 0.75 3.95 2.91 1.56
70 96.76 93.29 95.39 96.76 1.29 3.47 1.37 0.00
75 99.83 91.83 94.23 95.75 1.12 8.00 5.60 4.08
80 99.38 92.24 94.29 95.60 0.97 7.14 5.09 3.78
85 99.53 91.65 94.56 96.92 1.75 7.88 4.97 2.61
90 98.90 92.12 93.67 96.55 1.39 6.78 5.23 2.35
95 98.58 88.04 92.21 94.84 2.26 10.54 6.37 3.74
It was found that the use of the algorithm involves an average reduction of 4.26% in the utilization
compared to the solutions found with the model. However, the computation time savings is on average
around 93%. Furthermore, because the comparison is being done with a simplified model, this
underestimates the computational saving of the algorithm. The result is important for two reasons. On
one hand, although there is recent evidence of good performance using GRASP in other operating room
scheduling problems (Cartes Rubilar & Medina Duran, 2016), to the best of our knowledge there are no
previous studies that test the performance of GRASP in the surgery scheduling problem. The reduction
in the found objective function is good, compared to what was reported by other authors in production
scheduling problems (Molina-Sánchez & González-Neira, 2016). On the other hand, to solve stochastic
problems it is vital to find efficient algorithms from a computational time point of view (Figueira &
Almada-Lobo, 2014). Since deterministic solutions present difficulties in implementation, in real
systems, it is often preferable to find good solutions in reasonable computational times (Sarin et al.,
2013).
Table 2
Computational time of MILP and GRASP for each scenario
Percentile (Scenario) Computational time (min) Computational time GAP (min)
(with respect to MILP)
MILP GRASP
M inimum Average Maximum Standard
Deviation Minimum Average Maximum
50 17.01 1.08 1.21 1.31 1.16 15.93 15.80 15.70
55 17.05 0.88 1.16 1.28 1.17 16.17 15.89 15.77
60 17.03 0.88 1.16 1.27 0.76 16.15 15.87 15.76
65 17.17 0.86 1.15 1.26 0.75 16.31 16.02 15.91
70 17.03 0.86 1.12 1.18 1.29 16.17 15.91 15.85
75 17.09 0.98 1.13 1.23 1.12 16.11 15.96 15.86
80 17.12 0.84 1.03 1.08 0.97 16.28 16.09 16.04
85 17.01 0.83 1.04 1.28 1.75 16.19 15.97 15.73
90 17.07 0.80 1.19 1.27 1.39 16.27 15.88 15.81
95 17.02 1.16 1.18 1.19 2.26 15.86 15.84 15.83
10
4.2 Solving hospital instance
A hospital instance, obtained from the May 2014 records, has been analyzed. Fig. 3 shows the solution
frontier of the expected value of the three indicators. On the left side, occupation and the percentage of
patients waiting are compared. On the right side, occupation and the average waiting time for patients
are plotted. Additionally, the figure shows the performance of the deterministic solution proposed by
Estupiñán et al. (2016) who used the same dataset of surgeries, allowing us to make a straightforward
comparison. It can be seen that, compared with the result of a deterministic approach, the proposed
methodology is a solution that improves the three indicators. Also, this border can be used to support the
decision-making process.
Fig. 3. Pareto frontier between waiting time, percentage of waiting patients and occupation rate
On the other hand, Fig. 4 compares the confidence intervals of the solutions obtained with the
deterministic approach proposed by Estupiñán et al. (2016) and those obtained with the simulation-
optimization methodology. It is important to note that the deterministic approach projected an 80% of
occupancy. However, with a confidence level of 95%, this indicator will be between 75.6% and 76.3%.
This is because the authors did not take into account the variability of surgical times.
Fig. 4. Confidence intervals of 95% for the indicators using the Tukey method
Compared to the deterministic approach, the proposed methodology represents two advantages: i) it
allows managing the risk associated with the indicators estimation, and ii) it allows setting realistic targets
0
20
40
60
80
100
120
140
40% 50% 60% 70% 80% 90%
Waiting minutes
Occupation rate
Sim/Opt Solutions Deterministic Solution
0%
10%
20%
30%
40%
50%
60%
40% 50% 60% 70% 80% 90%
% waiting patients
Occupation rate
Sim/Opt Solutions Deterministic Solution
75.5
75.7
75.9
76.1
76.3
76.5
Deterministic Sim/Opt
Occupation rate [%]
75.5
85.5
95.5
105.5
115.5
Deterministic Sim/Opt
Waiting time [min]
34
34.5
35
35.5
36
36.5
Deterministic Sim/Opt
Waiting rate [%]
D. M. Díaz-López et al. / International Journal of Industrial Engineering Computations 9 (201 8)
11
considering the resource constraints. On the one hand, a deviation of 5% compared to the projected
solution may be problematic in some systems. Hence, the variability of surgical times should be included.
On the other hand, setting a target for the occupancy level without considering the reductions in the
service might lead to patient dissatisfaction. In this context, the proposed methodology delivers a set of
solutions to the hospital. From this set, the one that is more aligned with hospital objectives will be
selected.
In general, the behavior of the indicators is as expected: high levels of utilization involving low levels of
service. What is important, however, is to quantify the reductions to be made in an objective to enhance
the other, as their relationships are not linear. Thus, for example, if the hospital is interested in having an
average occupancy of 80%, it must be willing to let 44% of their patients wait for surgery delays and for
the average waiting time to be 85 minutes. Table 3 presents the confidence intervals for the percentage
of patients that have to wait, the occupation rate and the average time (in minutes) that a patient have to
wait if the waiting occurs, all of them depending on the percentile used for the duration of the surgeries.
It can be seen that for percentage of patients that have to wait and occupation rate indicators none of the
confidence intervals overlap. That is there here are statistically significant differences in these indicators
as the percentile used for surgeries duration changes. The average waiting time for patients is the
opposite. It cannot be concluded that the indicator to weaken with the increase in the occupancy. The
reduction made by the increase in occupation is mainly associated with having more patients waiting.
Table 3
Confidence intervals (95%) for the indicators
Percentile
Percentage of patients
waiting
Occupation rate
Average waiting time (min)
50
48.69%
49.01%
83.84%
83.98%
87.00
91.00
55
44.23%
44.55%
81.41%
81.57%
83.60
87.88
60
39.50%
39.82%
78.49%
78.65%
83.45
87.73
65
34.23%
34.54%
76.27%
76.44%
84.72
89.06
70
29.20%
29.49%
72.46%
72.64%
88.22
92.75
75
24.31%
24.63%
68.89%
69.09%
89.05
93.62
80
19.30%
19.60%
64.58%
64.79%
90.39
95.03
85
14.55%
14.81%
60.16%
60.36%
94.70
99.56
90
9.78%
10.03%
56.01%
56.23%
90.82
95.47
95
4.85%
5.04%
47.48%
46.70%
95.20
100.02
5. Conclusions
This work presents a simulation-optimization approach to room surgery scheduling with real data from
an important hospital in Bogota, Colombia. The Monte Carlo simulation dealt with stochastic duration
times of surgeries and GRASP metaheursitic for optimization purposes. Real data of surgeries’ duration
times of a normal month was analyzed to determine its probability distribution and simulate real durations
in the proposed scheduling method.
Results show that there is an inversely proportional relation between the service level and utilization rate.
While utilization decreases, the percentage of people that must wait increases. In fact, reductions in
utilization levels can be clustered in three groups depending on the level of service. The sacrifice, in
terms of utilization, needed to decrease delay times is explained as follows. The percentage of patients
who will not have to wait for a delay in starting their surgery is measured. The reductions of 5% in the
percentage of patients that have to wait correspond to increases of the same size in the percentile used
for the duration of the procedure. The sacrifices in occupation rate can be classified into three groups.
The first group (circle points in Fig. 4) corresponds to increments in the percentage of delayed surgeries
between 34% and 49%, which represents reductions of about less than 3% of occupation (2.52% on
average). In the second group, triangle points on the graph, there are increases of up to a 4% of the
utilization level, implying an increment in the percentage of patients that have to wait between 9% and
12
34%. Finally, the third group is the scenario where, in order to ensure that less than 5% of patients must
wait, a reduction in the utilization is around 9% (see the squares points on graph). Thus, it implies that
there is no one unique optimal solution, but there is a wide range of possibilities that the institution can
analyze to choose the one that adjusts better to its policies. The simulation-optimization approach
obtained better results than the implementation of dispatching rules.
Future work can incorporate more constraints to the analyzed problem, such as the personnel and
instrumental availability. Also, the blocks of time in operating rooms can be added. In terms of the
surgical procedures scheduled, emergency and urgency surgeries can be included. Finally, other
simulation-optimization approaches should be implemented not only for validation of the solution but
for the construction of the schedule.
References
Abdelrasol, Z., Harraz, N., & Eltawil, A. (2014). Operating Room Scheduling Problems: A Survey and
a Proposed Solution Framework. In H. K. Kim, S.-I. Ao, & M. A. Amouzegar (Eds.), Transactions
on Engineering Technologies (pp. 315–329). Springer Netherlands.
Addis, B., Carello, G., Grosso, A., & Tànfani, E. (2016). Operating room scheduling and rescheduling:
a rolling horizon approach. Flexible Services and Manufacturing Journal, 28(1–2), 206–232.
Alonso, J. M., Clifton, J., & Diaz-Fuentes, D. (2013). The Impact of New Public Management on
Efficiency: An Analysis of Madrid’s Hospitals. Health Policy, (12), 12.
Aringhieri, R., & Duma, D. (2015). The Optimization of a Surgical Clinical Pathway. In Simulation and
Modeling Methodologies, Technologies and Applications, Advances in Intelligent Systems and
Computing 256 (Vol. 256, pp. 313–331).
Aringhieri, R., Landa, P., Soriano, P., Tànfani, E., & Testi, A. (2015). A two level metaheuristic for the
operating room scheduling and assignment problem. Computers & Operations Research, 54, 21–34.
Baesler, F., Gatica, J., & Correa, R. (2015). Simulation Optimisation for Operating Room Scheduling.
International Journal of Simulation Modelling, 14(2), 215–226.
Bayer, S. (2014). Simulation modelling and resource allocation in complex services. BMJ Quality &
Safety, 23(5), 353–355.
Beaulieu, I., Gendreau, M., & Soriano, P. (2012). Advanced Decision Making Methods Applied to Health
Care, 173.
Brailsford, S., & Vissers, J. (2011). OR in healthcare: A European perspective. European Journal of
Operational Research, 212(2), 223–234.
Brucker, P. (2007). Scheduling Algorithms (5th ed.). Berlin, Heidelberg: Springer Berlin Heidelberg.
Cardoen, B., Demeulemeester, E., & Beliën, J. (2010). Operating room planning and scheduling: A
literature review. European Journal of Operational Research, 201(3), 921–932.
Cartes Rubilar, I., & Medina Duran, R. (2016). A GRASP algorithm for the elective surgeries scheduling
problem in a Chilean public hospital. IEEE Latin America Transactions, 14(5), 2333–2338.
Ceballos-Acevedo, T. M., Velásquez-Restrepo, P. A., & Jaén-Posada, J. S. (2014). Length of the
Hospitalization . Methodologies for Intervention Duração da estancia hospitalar . Metodologias para
sua intervenção, 13(27), 274–295.
Chen, Y., Hao, J.-K., & Glover, F. (2016). An evolutionary path relinking approach for the quadratic
multiple knapsack problem. Knowledge-Based Systems, 92, 23–34.
Denton, B., Viapiano, J., & Vogl, A. (2007). Optimization of surgery sequencing and scheduling
decisions under uncertainty. Health Care Management Science, 10(1), 13–24. Retrieved from
Díaz-López, L. P., Fuquen-Fraile, L., Barrera, D., González-Neira, E. M., García-Herreros, L. G., Suárez,
D. R., … Suárez, D. R. (2015). Control de la variabilidad en la programación de pacientes electivos
en salas de cirugía. Gerencia Y Políticas de Salud, 14(28), 78–87.
Dobrzykowski, D., Saboori Deilami, V., Hong, P., & Kim, S.-C. (2014). A structured analysis of
operations and supply chain management research in healthcare (1982–2011). International Journal
D. M. Díaz-López et al. / International Journal of Industrial Engineering Computations 9 (201 8)
13
of Production Economics, 147(2014), 514–530.
Duma, D., & Aringhieri, R. (2015). An online optimization approach for the Real Time Management of
operating rooms. Operations Research for Health Care, 7, 40–51.
Elkhuizen, S. G., Das, S. F., Bakker, P. J. M., & Hontelez, J. a M. (2007). Using computer simulation to
reduce access time for outpatient departments. Quality and Safety in Health Care, 16(5), 382–386.
Estupiñán, A. M., Torres, M. J., Caro, M. P., González-neira, E. M., Barrera, D., Pérez, N., … Suárez,
D. R. (2016). Reglas de despacho en la programación de procedimientos quirúrgicos electivos :
impacto en los indicadores de ocupación y oportunidad. Ciencias de La Salud, 14(2), 211–222.
Retrieved from http://revistas.urosario.edu.co/index.php/revsalud/article/view/4948/3387
Figueira, G., & Almada-Lobo, B. (2014). Hybrid simulation-optimization methods: A taxonomy and
discussion. Simulation Modelling Practice and Theory, 46, 118–134.
Framinan, J. M., & Perez-Gonzalez, P. (2015). On heuristic solutions for the stochastic flowshop
scheduling problem. European Journal of Operational Research, 246(2), 413–420.
Ghazalbash, S., Sepehri, M. M., Shadpour, P., & Atighehchian, A. (2012). Operating Room Scheduling
in Teaching Hospitals. Advances in Operations Research, 2012, 1–16.
Hasvold, P. E., & Scholl, J. (2011). Flexibility in interaction: sociotechnical design of an operating room
scheduler. International Journal of Medical Informatics, 80(9), 631–45.
Holm, L. B., Lurås, H., & Dahl, F. a. (2013). Improving hospital bed utilisation through simulation and
optimisation. With application to a 40% increase in patient volume in a Norwegian general hospital.
International Journal of Medical Informatics, 82(2), 80–89.
Hulshof, P. J. H., Kortbeek, N., Boucherie, R. J., Hans, E. W., & Bakker, P. J. M. (2012). Taxonomic
classification of planning decisions in health care: a structured review of the state of the art in OR/MS.
Health Systems, 1(2), 129–175.
Juan, A. A., Faulin, J., Grasman, S. E., Rabe, M., & Figueira, G. (2015). A review of simheuristics:
Extending metaheuristics to deal with stochastic combinatorial optimization problems. Operations
Research Perspectives, 2, 62–72.
Juan, A. a., Grasman, S. E., Caceres-Cruz, J., & Bektaş, T. (2014). A simheuristic algorithm for the
Single-Period Stochastic Inventory-Routing Problem with stock-outs. Simulation Modelling Practice
and Theory, 46, 40–52.
Landa, P., Aringhieri, R., Soriano, P., Tànfani, E., & Testi, A. (2016). A hybrid optimization algorithm
for surgeries scheduling. Operations Research for Health Care, 8, 103–114.
Li, S., Jia, Y., & Wang, J. (2012). A discrete-event simulation approach with multiple-comparison
procedure for stochastic resource-constrained project scheduling. The International Journal of
Advanced Manufacturing Technology, 63(1–4), 65–76.
Litvak, E. (2010). Managing Patient Flow in Hospitals: Strategies and Solutions. (E. Litvak, Ed.) (2nd
ed.). Joint Commission Resources. Retrieved from
http://books.google.com.co/books/about/Managing_Patient_Flow_in_Hospitals.html?id=snQPQgA
ACAAJ&pgis=1
Liu, Y., Chu, C., & Wang, K. (2011). A new heuristic algorithm for the operating room scheduling
problem. Computers & Industrial Engineering, 61(3), 865–871.
M’Hallah, R., & Al-Roomi, A. H. (2014). The planning and scheduling of operating rooms: A simulation
approach. Computers & Industrial Engineering, 78, 235–248.
Molina-Pariente, J. M., Hans, E. W., & Framinan, J. M. (2016). A stochastic approach for solving the
operating room scheduling problem. Flexible Services and Manufacturing Journal, 1–28.
Molina-Sánchez, L. P., & González-Neira, E. M. (2016). GRASP to minimize total weighted tardiness
in a permutation flow shop environment. International Journal of Industrial Engineering
Computations, 7(1), 161–176.
Ozcan, Y. A., Tànfani, E., & Testi, A. (2017). Improving the performance of surgery-based clinical
pathways: a simulation-optimization approach. Health Care Management Science, 20(1), 1–15.
Pinedo, M. L. (2012). Scheduling: Theory, algorithms and systems. Springer (4th ed., Vol. 4). New York:
Springer Science & Business Media.
Pitt, M., Monks, T., Crowe, S., & Vasilakis, C. (2015). Systems modelling and simulation in health
14
service design, delivery and decision making. BMJ Quality & Safety, bmjqs-2015-004430.
Resende, M. C., & Ribeiro, C. (2010). Greedy randomized adaptive search procedures: Advances,
hybridizations, and applications. In M. Gendreau & J.-Y. Potvin (Eds.), Handbook of Metaheuristics
SE - 10 (Vol. 146, pp. 283–319). Springer US.
Saadouli, H., Jerbi, B., Dammak, A., Masmoudi, L., & Bouaziz, A. (2015). A stochastic optimization
and simulation approach for scheduling operating rooms and recovery beds in an orthopedic surgery
department. Computers & Industrial Engineering, 80, 72–79.
Samudra, M., Van Riet, C., Demeulemeester, E., Cardoen, B., Vansteenkiste, N., & Rademakers, F. E.
(2016). Scheduling operating rooms: achievements, challenges and pitfalls. Journal of Scheduling,
19(5), 493–525.
Sarin, S. C., Sherali, H. D., & Liao, L. (2013). Minimizing conditional-value-at-risk for stochastic
scheduling problems. Journal of Scheduling, 17(1), 5–15.
Shylo, O. V., Prokopyev, O. a., & Schaefer, A. J. (2013). Stochastic operating room scheduling for high-
volume specialties under block booking. INFORMS Journal on Computing, 25(July 2014), 682–692.
Tako, A. A., Kotiadis, K., Vasilakis, C., Miras, A., & le Roux, C. W. (2014). Improving patient waiting
times: a simulation study of an obesity care service. BMJ Quality & Safety, 23(5), 373–381.
Testi, A., Tanfani, E., & Torre, G. (2007). A three-phase approach for operating theatre schedules. Health
Care Management Science, 10(2), 163–172.
Van De Klundert, J., Muls, P., & Schadd, M. (2008). Optimizing sterilization logistics in hospitals.
Health Care Management Science, 11(1), 23–33.
van Sambeek, J. R. C., Joustra, P. E., Das, S. F., Bakker, P. J., & Maas, M. (2011). Reducing MRI access
times by tackling the appointment-scheduling strategy. BMJ Quality & Safety, 20(12), 1075–1080.
Velasco, N., Barrera, D., & Amaya, C. A. (2012). Logística hospitalaria: Lecccciones y retos para
Colombia. In La salud en Colombia (pp. 309–343).
Velásquez-Restrepo, P. A., Rodríguez-Quintero, A. K., & Jaén-Posada, J. S. (2012). Aproximación
metodológica a la planificación y a la programación de las salas de cirugía. Revista Gerencia Y
Politicas de Salud, 12(24), 249–266. Retrieved from
http://www.scopus.com/inward/record.url?eid=2-s2.0-84880281976&partnerID=tZOtx3y1
Wang, T. K., Chan, F. T. S., & Yang, T. (2014). The Integration of Group Technology and Simulation
Optimization to Solve the Flow Shop with Highly Variable Cycle Time Process: A Surgery
Scheduling Case Study. Mathematical Problems in Engineering, 2014, 1–10.
Zhang, Z., & Xie, X. (2015). Simulation-based optimization for surgery appointment scheduling of
multiple operating rooms. IIE Transactions, 47(9), 998–1012.
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