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Reducing post-surgery recovery bed occupancy through an analytical
prediction model
Belinda Spratt
and Erhan Kozan
School of Mathematical Sciences, Queensland University of Technology (QUT)
2 George St, Brisbane, QLD 4000, Australia
Belinda Spratt b.spratt@qut.edu.au orcid.org/0000-0002-2522-2967
Erhan Kozan e.kozan@qut.edu.au orcid.org/0000-0002-3208-702X
Abstract
Operations Research approaches to surgical scheduling are becoming increasingly popular in
both theory and practice. Often these models neglect stochasticity in order to reduce the
computational complexity of the problem. In this paper, historical data is used to examine the
occupancy of post-surgery recovery spaces as a function of the initial surgical case sequence.
We show that the number of patients in the recovery space is well modelled by a Poisson
binomial random variable. A mixed integer nonlinear programming model for the surgical case
sequencing problem is presented that reduces the maximum expected occupancy in post-
surgery recovery spaces. Given the complexity of the problem, Simulated Annealing is used to
produce good solutions in short amounts of computational time. Computational experiments
are performed to compare the methodology here to a full year of historical data. The solution
techniques presented are able to reduce maximum expected recovery occupancy by 18% on
average. This reduction alleviates a large amount of stress on staff in the post-surgery recovery
spaces and improves the quality of care provided to patients.
Keywords
OR in health services; stochastic; surgical case sequencing; operating room scheduling;
Acknowledgements
This research was funded by the Australian Research Council (ARC) Linkage Grant LP
140100394. Computational resources and services used in this work were provided by the HPC
and Research Support Group, Queensland University of Technology, Brisbane, Australia.
Conflict of Interest
The authors declare that there are no conflicts of interest.
1. Introduction
In recent years, a great deal of focus has been placed on improving the performance of the
operating theatre (OT) through use of Operations Research approaches. Given an increasing
number of surgical requests, and a fixed number of resources, it is necessary to increase surgical
throughput in order to keep up with demand. For a variety of reasons the OT is often scheduled
as a stand-alone entity, with no regard for the impact on the rest of the hospital.
Surgical schedules can have a large impact on downstream wards (e.g. patient recovery
spaces). Not only is it necessary to ensure that patient demand on downstream wards does not
exceed capacity, but it is also necessary to ensure that there are sufficient staff available at
times of peak demand. As such, we present an analytical model for predicting recovery bed
occupancy based on the assumption of lognormally distributed surgery and recovery durations.
We then show that this model can be implemented in conjunction with standard Operations
Research techniques. This is done to improve the surgical case sequences (SCSs) produced by
a large Australian public hospital.
The remainder of the paper is organised as follows. In Section 2 we provide a review of the
relevant literature. In Section 3 the motivating case study is discussed and a description of the
problem is provided. Section 4 contains the details of the derivation of a Poisson Binomial
model of recovery bed occupancy. This leads to the Mixed Integer Nonlinear Programming
(MINLP) formulation in Section 5. Due to the complexity of the problem, we implement
Simulated Annealing (SA) to produce good feasible solutions (cf. Section 6). Results are
presented and discussed in Section 7. Concluding remarks are made in Section 8.
2. Literature Review
OT planning and scheduling is often classified into three main levels: strategic, tactical, and
operational. These problems are addressed on long, medium, and short term planning horizons
respectively (Cardoen, Demeulemeester, & Beliën, 2010). There is often a hierarchical approach
to OT planning and scheduling. Strategic decisions influence tactical decisions which then, in
turn, influence operational decisions. Furthermore, the surgical department has the highest
impact on hospital workload (Vanberkel et al., 2011). For a thorough review of OT planning
and scheduling, see Cardoen, et al. (2010). A review of the use of Operations Research in other
hospital wards is provided by van de Vrugt, Schneider, Zonderland, Stanford, and Boucherie
(2018).
The master surgical scheduling problem (MSSP) is the tactical problem of allocating
surgeons or surgical specialties to OT time blocks. Motivated by the effect of OT department
schedules on the entirety of the hospital, a number of authors have considered the impact of
the MSS on downstream wards. Vanberkel, et al. (2011) provide a number of distributions
including an exact form of ward occupancy based on an initial MSS. More recently, Fügener,
Hans, Kolisch, Kortbeek, and Vanberkel (2014) solve the MSSP under downstream unit
capacity constraints. In their model, Fügener, et al. (2014) produce probability distributions
for the number of patients in the ICU and in the ward on a given day based on a cyclical MSS
and aim to minimise downstream costs associated with post-surgical patient flow.
Whilst understanding the patient occupancy distributions caused by the choice of MSS is
useful in medium term staff planning, it is important to understand how downstream
occupancy changes throughout the day. To capture the daily variability in downstream ward
occupancy caused by surgical throughput, operational level OT planning and scheduling
decisions must be considered. As such, we focus on the operational level of OT planning and
scheduling to predict recovery bed occupancy.
Chow, Puterman, Neda, Wenhai, and Derek (2011) implement a surgical scheduling model
based on mixed integer programming (MIP) and Monte Carlo simulation to reduce peak bed
occupancies. The MIP model is used to assign both surgeons and patient types to OT time
blocks, whilst the Monte Carlo simulation is utilised to approximate recovery occupancy.
Min and Yih (2010) provide a closed form model for the expected number of patients in the
surgical intensive care unit (SICU) based on the assumption of normally distributed surgical
durations and uniformly distributed patient length of stay (LOS) in the SICU. The authors
use this model to minimise patient related costs whilst assigning patients to OT time blocks.
The closed form model provided by Min and Yih (2010) is inappropriate whenever these
distribution assumptions are violated. In reality, surgical durations are closer to a lognormal
distribution Strum, May, and Vargas (1998). Recovery durations are also well-modelled by the
lognormal distribution (Spratt, Kozan, & Sinnott, in press). As such, a new model is required
to predict post-surgical recovery occupancy.
Recently, a great deal of focus has been placed on ensuring patient throughput does not
result in resource shortages in downstream wards. Latorre-Núñez et al. (2016) simultaneously
schedule the OT and post anaesthesia recovery spaces when solving the surgical case
sequencing problem (SCSP). When producing their schedules, the authors consider only
average durations (both surgical and recovery). The authors indicate that using deterministic
surgery and recovery durations is a limitation and that the inclusion of stochasticity should be
considered in future.
The contribution of this paper is two-fold. Firstly, we present an analytical model for the
distribution of post-surgery recovery bed occupancy based on lognormally distributed surgical
and recovery durations (cf. Section 4). Secondly, we present a simple MINLP formulation of
the SCSP where the objective is to minimise the maximum expected recovery bed occupancy
level (cf. Section 5). In doing so, we show that the analytical model presented in Section 4 can
be implemented to produce surgical sequences that are less erratic, thus reducing the stress on
post-surgery recovery staff and providing more information to hospital planning and rostering
staff members.
3. Case Study and Problem Description
The work presented in this paper is based on a case study of an Australian public hospital.
The hospital has a large surgical department consisting of 21 operating rooms (ORs)
1
. There
are around twenty bed spaces available in the surgical care unit (SCU) and post-anaesthesia
care unit (PACU).
The hospital utilises a modified block scheduling policy when generating a four week,
rotating MSS. This means that they may reallocate OR time in the case that a surgical team
is unavailable or does not require a particular OR block. In this case, other specialties are able
to request this time based on the patient demand. The MSS is not modified often, and has
remained largely unchanged since the hospital opened.
Nurse and anaesthetist scheduling is performed more regularly and is based on the current
MSS. When scheduling nurses and anaesthetists, the hospital must consider their preference
or experience regarding the different surgical specialties. Adjustments to the MSS are made
without consideration of the availability of nurses and anaesthetists as there are usually
sufficient staff available.
At the case study hospital, surgeons work in conjunction with case managers and other
administrative staff members to plan and schedule the OT department. Each week, surgeons
produce lists of patients that they would like to see during their allocated surgical time. The
hospital holds an elective bookings meeting every Thursday to discuss potential cases in the
1
Note: The OT is the set of ORs.
upcoming week. In some cases, surgeons do not provide a list for a particular OR block as they
predict non-elective arrivals will occur in the meantime.
On the day of surgery, elective patients are required to arrive on time for their surgery. If
the patient is an inpatient (expected to stay for more than a single day) they are allocated to
a ward for their stay. The patient is then sent either to their ward or to prepare in the SCU
depending on the time until surgery and whether any other procedures are required beforehand.
Most patients recover from surgery in the PACU, with recovery times depending on the
type of surgery. Any surgical transfers from the intensive care unit (ICU) are returned directly
to the ICU following surgery. Once a patient has recovered sufficiently in the PACU, day
patients continue recovery in the SCU whereas inpatients are sent back to the ward or to the
ICU. The PACU forms a bottleneck in the system due to its limited capacity. A surgery cannot
proceed if there is overcrowding in the PACU. Similarly, a patient may be held in the OR
until a space becomes available in the PACU. This, in turn, results in a delay in subsequent
surgeries.
In the past, OT planning and scheduling at this hospital has been performed without
consideration of downstream resources. Here, we propose that patients are sequenced so as to
reduce the strain on the PACU and create a more efficient surgical department by relieving
the bottleneck in the system.
4. Predicting Recovery Bed Occupancy
The first step in sequencing patients to reduce PACU strain is to understand how the
sequence of patients affects the occupancy of the PACU. Spratt, et al. (in press) show that
both surgery and recovery durations at the case study hospital are well-modelled by the
lognormal distribution. Figure 1 shows a histogram of the historical surgery and recovery
durations of elective patients at the case study hospital. A fitted lognormal pdf is shown in
grey.
Whilst there appear to be some deviations from the fitted lognormal distribution, a better
fit is obtained when patients are further categorised by their specialty and ASA (American
Society of Anesthesiologists) physical status classification code. The ASA code identifies the
health of a patient prior to their surgery and ranges from one (healthy) to six (brain-dead).
In order to predict the occupancy of recovery beds in the PACU, we assume that patient
surgery and recovery durations are lognormally distributed, with parameters dependent on
surgical specialty and ASA code. The remainder of this section is organised as follows. First,
we derive the probability an individual patient is in recovery at any given time. We then
discuss the distribution, expectation, and variance of the total number of patients in recovery
at any given time. A Normal approximation is used to identify 95% predictive intervals. We
then compare our prediction to historical data.
Figure 1: Historical surgery and recovery durations compared to lognormal distributions.
4.1.
Predicting an Individual’s Recovery
Here, patient ’ s surgical duration and recovery duration are assumed to be lognormally
distributed, and are denoted
and
respectively. As the sum of lognormal distributions
can be approximated by a lognormal distribution,
is the random variable representing the
total duration of patient ’s surgery and recovery.
~Lognormal
,
()=1
2+1
2erfln()−
√2
, ≥0
~Lognormal̃
,̃
≈
+
~Lognormal̂
,̂
()=1
2+1
2erfln()−̂
√2̂
, ≥0
The parameter values for the random variable
are determined through moment
matching.
̂
=ln
+
̂
=ln
+
=
+
+
̃
+
=
−1
+
+
−1
̃
+
A binary variable,
()
can be used to indicate whether patient
is in recovery at time
.
It is assumed that the start time of patient
’s surgery is known and takes the value
.
()=1, patient is in recovery at time
0, otherwise
Pr
()=1=Pr
+
≤≤
+
=Pr−
≤
−Pr−
<
=PrS
≤t−Z
−Pr
<−
=F
t−Z
−F
t−Z
=1
2erfln−
−
√2
−1
2erfln−
−̂
√2̂
,
≤≤
+exp̂
−
̂
̂
−
4.2.
Predicting the Number of Patients in Recovery
Let
()
be the number of patients in recovery at time
. Here, in order to simplify remaining
calculations, it is assumed that patient surgery durations and recovery times are independent.
Based on this assumption,
()
is distributed according to a Poisson binomial distribution. It
is possible to determine the expectation and variance of the number of patients in recovery at
time
.
[()]=Pr
()=1
∈
=1
2erfln−
−
√2
−1
2erfln−
−̂
√2̂
∈
[()]= Pr
()=11−Pr
()=1
∈
=1
2erfln−
−
√2
−1
2erfln−
−̂
√2̂
∈
×1−1
2erfln−
−
√2
+1
2erfln−
−̂
√2̂
The cumulative distribution function (CDF) of
()
is determined according to Hong
(2013).
()
()=1
+1{1−exp[−(+1)]}
1−exp(−)
=
,
where ω = 2π
+1 ,
and
=1−
+
exp( )
=
.
The above CDF must be evaluated at every value of
,
as such it is too computationally
expensive to be implemented in real-world situations. Here, in order to estimate 95% prediction
intervals,
()
are approximated with a normal distribution,
()
.
()≈()~(E[()],Var[()])
Historical data indicates that this approximation is suitable when examining the 95%
prediction intervals.
Pr()≤()≤() = 0.95,
where ()=E[()]−1.96Var[()],
and ()=E[()]+1.96Var[()].
4.3.
Model Validation
We validate the model presented above by comparing predictions to historical PACU
occupancy levels through the duration of 2016. In particular, we are interested in the mean,
variance, and estimated 95% prediction intervals. Figure 2 shows the actual recovery
occupancy, the expected recovery occupancy, and the estimated 95% prediction interval for a
week in February 2016.
Figure 2 indicates that there is a good agreement between the model and historical data.
The peaks and troughs in historical PACU occupancy align well with the peaks and troughs
in expected occupancy. The historical recovery occupancy is within the shaded 95% prediction
interval for most of the week.
When validating the model against historical data, the historical and expected occupancies
are calculated every 0.1 hours for the duration of 2016. On average, there was a difference of
0.07 in the historical and expected occupancy levels. As such, the model presented here slightly
underestimates occupancy levels. The model overestimates occupancy 49.5% of the time and
underestimates 49.6% of the time. Approximately 0.9% of the time, the model and historical
data were in exact agreement. Historical data showed actual occupancy exceeded the upper
bound on the approximate 95% prediction interval 3.3% of the time. Actual occupancy was
below the lower bound on the approximate 95% prediction interval only 0.2% of the time.
Figure 2: Comparison of model to historical data.
In future it may be worth considering modelling durations with truncated distributions
rather than standard lognormal distributions. It may also be possible to refine the prediction
model by incorporating real-time schedule realisations to assist in the recovery occupancy
predictions. This is not included in the model as we focus instead on start-of-day schedules,
prior to any schedule realisations.
5.
Surgical case sequencing model
In this section, a multi-objective MINLP formulation for the surgical case sequencing problem
is presented. The maximum expected number of patients in recovery at any one time is
minimised. As such, this model can be used to level the occupancy of the PACU.
5.1.
Scalar Parameters
: the number of surgeons scheduled for the day.
̅
: the number of patients scheduled for the day.
: the number of ORs.
: the number of hours that each OR is open for during a standard working day.
∗
: the number of hours in a day.
5.2.
Index Sets
: the set of surgeons that practice at the hospital.
={1,…,
}
: the set of patients that are scheduled for the day.
={1,…,̅}
:
the set of ORs.
=1,…,
: the set of patients treated by surgeon
,
∀∈
.
: the set of patients treated in OR
,
∀∈
.
5.3.
Indices
:
index for surgeon in set
.
:
index for patient in set
.
: alternative index for patient in set
.
:
index for OR in set
.
5.4.
Vector Parameters
: the setup time surgeon
requires when starting in a new OR,
∀∈
.
: 1 if patient
requires post-surgery recovery in the recovery space, 0 otherwise,
∀∈
.
: the expected duration of patient
’s surgery,
∀∈
.
: the mean of the natural logarithm of patient
’s surgical duration,
∀∈
.
: the variance of the natural logarithm of patient
’s surgical duration,
∀∈
.
̃
: the mean of the natural logarithm of patient
’s recovery duration,
∀∈
.
̃
: the variance of the natural logarithm of patient
’s recovery duration,
∀∈
.
̂
: the mean of the natural logarithm of the total duration of patient
’s surgery and recovery,
∀∈
.
̂
: the variance of the natural logarithm of the total duration of patient
’s surgery and
recovery,
∀∈
.
:
the start of shift time of surgeon
, ∀∈.
+
: the setup time (in hours) before patient
’s surgery,
∀∈
.
−
: the clean-up time (in hours) after patient
’s surgery,
∀∈
.
5.5.
Decision Variables
: 1 if patient
’s surgery starts during patient
’s surgery, 0 otherwise,
∀,∈
.
: 1 if patient
’s surgery starts after patient
’s sugery starts, 0 otherwise,
∀,∈
.
:
the expected start time of patient
’s surgery,
∀∈
.
∗
:
the
expected end time of patient
’s surgery,
∀∈
.
Ω
: 1 if surgeon
will require overtime,
0
otherwise,
∀∈
.
: the expected overtime associated with surgeon
,
∀∈
.
5.6.
Objective Function
The main objective is to minimise the maximum expected number of patients in recovery at
any one time throughout the surgical department opening hours. In doing so, the demand on
PACU staff is levelled. This not only assists in staff planning but may result in increased staff
satisfaction and better care for patients in recovery. An example occupancy profile is provided
in Figure 3.
Figure 3: An example occupancy profile over a single day before and after optimisation.
Figure 3 includes both the original (historical) expected occupancy profile, and the
improved expected occupancy profile. Whilst the historical occupancy profile exhibits peaks
and troughs in demands, by reducing the maximum expected occupancy (MEO) throughout
OT opening hours, the improved occupancy profile is steadier and predicts level demand
throughout opening hours.
The objective function is formulated based on the work presented in Section 4.
Minimise
max
≤≤
∗
1
2erfln−
−
√2
−1
2erfln−
−̂
√2̂
,
≤≤
+exp̂
−
̂
̂
−
0, otherwise
∈
This objective is equivalent to
Minimise
max
≤
≤
∗
2
∈
erf
ln
max
−
,
0
−
√
2
−
erf
ln
max
−
,
0
−
̂
√
2
̂
,
(1)
when
+exp
−
̂
−
>
∗
. This condition holds for the parameters of interest in the case
study.
5.7.
Constraints
Surgeon
’s surgeries must be expected to start after the start of their shift.
≥
,
∀
∈
,
∈
(2)
Constraint (2) is used to ensure that surgeries are completed before the end of surgeon
’s
shift.
+
≤
∗
+
,
∀
∈
,
∈
(3)
In some instances, overtime may be required (as seen in constraint (2)). For each surgeon, the
amount of overtime available is limited by the expected duration of their patient’s surgeries,
with provision for setup and cleanup times. In most cases, surgeons will not be allocated
overtime.
≤
Ω
+
+
+
−
∈
−
+
∗
,
∀
∈
(4)
Constraints (5) and (6) determine whether patient
ends on or after the time patient
starts.
−
∗
>
−
∗
,
∀
,
∈
,
≠
(5)
∗
−
≥
∗
−
1
,
∀
,
∈
,
≠
(6)
Constraints (7) and (8) determine whether the treatment of patients overlaps. Although this
could be done with a single constraint, that constraint would be nonlinear.
≥
+
−
1
,
∀
,
∈
,
≠
(7)
2
≤
+
,
∀
,
∈
,
≠
(8)
Each surgeon must not be assigned to treat more than one patient at a time.
=
0
,
∀
∈
,
,
∈
,
≠
(9)
If two patients’ surgeries overlap, then they must not be treated in the same OR.
=
0
,
∀
∈
,
,
∈
,
≠
(10)
Constraint (11) is used to calculate the expected finish time of each surgery.
∗
=
+
,
∀
∈
(11)
If two patients are treated by the same surgeon, or in the same OR, then they require clean-
up and setup time between surgeries.
≥
∗
+
+
+
−
−
1
−
,
∀
∈
,
,
∈
,
≠
(12)
≥
∗
+
+
+
−
−
1
−
,
∀
∈
,
,
∈
,
≠
(13)
Overtime must be non-negative.
≥
0
,
∀
∈
(14)
5.8.
Model Assumptions
A number of assumptions are made in order to simplify the model presented. The following
assumptions are made:
Surgical and recovery durations are independent.
Non-elective patients are not considered as the hospital currently has other
reservation techniques for their surgeries.
Each patient’s surgeon and OR is known in advance based on the surgical case
assignments made on a weekly basis.
There is no delay between a patient’s surgery and recovery.
Overtime may be unavoidable depending on the predetermined surgical case
assignments.
6.
Solution Approach
The SCSP is NP-hard as it is reducible to a machine scheduling problem. In addition to this,
the objective function presented in Section 5.6 is nonlinear, nonconvex, and not continuously
differentiable. Commercial solvers are inadequate.
Simulated Annealing (SA) (Kirkpatrick, Gelatt, & Vecchi, 1983) is a simple metaheuristic
based on the cooling of metals. By occasionally accepting worsening solutions, SA is able to
escape local optima in the search of global optima. Given the simplicity, ease of
implementation, and computational efficiency, SA is applied to solve the SCSP presented in
Section 5. The metaheuristic is implemented in MATLAB R2017b on a desktop computer with
an Intel
®
Core
™
i7-6700 CPU with 16GB of RAM.
6.1.
Constructive Heuristic and Local Search
At each iteration, a schedule can be constructed by defining the order of patient’s surgeries,
and determining the earliest start and latest completion of each patient’s surgery based on
this order. These surgeries are then allocated such that they start after their earliest start, and
finish before their latest completion times, unless overtime is necessary. The pseudocode is
provided in Algorithm 1.
Algorithm 1: Constructive Heuristic.
Latest Completion (LC)
←
OR closing time
Earliest Start (ES)
←
OR opening time
FOR p in reverse patient sequence
Successors (S)
←
the set of patients after patient p in the same OR or treated by the
same surgeon.
IF Successors exist
LC(p)
←
min(LC(S)-Duration(S)-Setup(S))-Cleanup(p)
END
END
FOR p in patient sequence
Predecessors (P)
←
the set of patients before patient p in the same OR or treated by the
same surgeon.
IF Predecessors exist
ES(p)
←
max(ES(P)+Duration(P)+Cleanup(P))+Setup(p)
END
Start(p)
←
ES(p) + max(0, rand
×
(LC(p)-ES(p)-Duration(p)))
ES(p)
←
Start(p)
END
When implementing SA, it is necessary to define local search heuristics to explore solution
neighbourhoods. Here, we exploit the problem structure in order to find local neighbours. At
each iteration, two patients are randomly selected and their order in the sequence is swapped.
A schedule is then randomly generated using Algorithm 1.
6.2.
Parameter Tuning
To obtain good solutions through SA, metaheuristic parameter tuning is required. A parameter
sweep was performed and suitable parameter combinations were considered (see Table 1).
Parameter tuning was performed in parallel using MATLAB
®
R2017b on the university’s
High Performance Computing (HPC) facility.
Table 1: Set of metaheuristic parameters tested in parameter sweep.
Parameter
Description Values
alpha
The proportional decrease in
temperature. 0.85, 0.9, 0.95
Between
The number of iterations between
each temperature decrease. 50, 100, 200
MaxIters
The maximum number of iterations. 1000, 1500, 2000, 2500, 3000
To simplify the parameter tuning, the starting temperature was set to one. SA was run ten
times under each parameter combination on historical data from February 2016. The average
of the sum of MEO on each day was used to compare against other parameter combinations.
Based on these computational experiments, the best performing metaheuristic parameters are
highlighted in bold (Table 1).
Figure 4: Parameter tuning results.
The average results obtained under the best performing parameter combination are
displayed in Figure 4. These results are compared to the historical MEO on each weekday in
February 2016. The mean MEO for each day is shown in teal.
7.
Results and Discussion
In this section, we perform computational experiments to compare the actual (historical) MEO
to the improved MEO for each day in 2016. The metaheuristic parameters were selected based
on the parameter tuning performed in Section 6.2. Computational experiments were performed
using MATLAB
®
R2017b on a desktop computer with an Intel
®
Core
TM
i7 processor @
3.40GHz with 16GB of RAM.
For each working day in 2016 (i.e. any day that elective surgeries were performed), SA was
run ten times. Figure 5 shows the historical MEO compared to improved occupancy for each
day in 2016. The teal line shows a cubic spline through the data.
Figure 5: Maximum expected occupancy.
From Figure 5 it can be seen that whilst there are a few instances where the historical
MEO was not improved, in most instances, the MEO was decreased significantly. On average,
there was an 18% reduction in MEO. In some instances this reduction was as large as 47%.
Such large reductions in MEO have the effect of levelling the workload of staff in the PACU.
This, in turn, may result in improved patient care and reduced levels of stress in staff members.
Figure 6 shows how the throughput of the OT (in number of elective patients treated)
affects the MEO of the PACU. In black, historical throughput on each day in 2016 is compared
to the MEO of the PACU on that day. The improved maximum occupancy is plotted in teal.
A cubic spline is fitted to each set of data to show the general trend. Larger relative reductions
in MEO are seen as the number of elective patients treated increases. It appears that in either
the historical or improved case, the MEO is approximately proportional to the number of
elective patients treated. The exception to this is the slight downward trend in improved MEO
seen at around 50 patients. It is unclear whether this is a true trend, or if this is an artefact
of having few observations at those values of throughput.
Figure 6: Throughput compared to MEO.
The metaheuristic approach presented in Section 6 runs in time
()
, where
is the number
of elective patients to be sequenced on a given day. Computational experiments were performed
using MATLAB R2017b on a desktop computer with an Intel
®
Core
™
i7-6700 CPU with 16GB
of RAM. Under the parameters reported in Table 1, the metaheuristic approach requires
under six seconds to run on a standard desktop computer. This ensures that the solution
methodology is accessible to hospital planning staff. Given such a short runtime, it would be
reasonable to produce several schedules for the day and allow staff members to use their expert
knowledge to select the most appropriate schedule.
8.
Concluding Remarks
Hospital processes are subject to a wide range of uncertainty. The surgical department is often
one of the main sources of uncertainty in the hospital. To ensure that resources are used
efficiently, Operations Research techniques may be implemented to improve planning and
scheduling. Given the highly random nature of the surgical department, it is necessary to
consider stochasticity in a variety of forms, but particularly in task durations.
In this paper we presented an analytical model to predict the occupancy of the post-surgery
recovery spaces. This model is based on the assumption of lognormally distributed surgery and
recovery durations. The only inputs required to predict recovery bed occupancy are the
expected start times of each patient’s surgery, and the parameters associated with the patients’
surgery and recovery duration distributions. Validation on historical data shows that this
model is effective at capturing the mean recovery bed occupancy levels.
Based on the recovery occupancy model, a MINLP formulation is provided for the SCSP.
The objective is to minimise the maximum expected recovery bed occupancy. Given the
complexity of the SCSP, and the intricacies of the objective function, it is necessary to
implement metaheuristic approaches to produce good feasible solutions. We show that, through
the solution techniques implemented here, the maximum expected recovery bed occupancy is
reduced by 18% on average, compared to historical data at the case study hospital.
The work presented in this paper will ensure more level demand on the PACU. This may
have the effect of reducing stress on PACU staff, improving the quality of care for patients,
and reducing bottlenecks associated with this limited resource.
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