# A Simulation–Based Optimization approach for analyzing the ambulance
diversion phenomenon in an Emergency Department network

Christian Piermarini  Dipartimento di Ingegneria Informatica, Automatica e
Gestionale “A. Ruberti”
SAPIENZA, Università di Roma
via Ariosto, 25 – 00185 Roma, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>Massimo Roma

###### Abstract

Most of the studies dealing with the increasing and well–known problem of
Emergency Department (ED) overcrowding usually mainly focus on modeling the
patient flow within a single ED, without considering the possibilities offered
by the cooperation among EDs. Instead, it is important to analyze the
overcrowding phenomenon considering an ED network rather than a single ED. In
this paper, the Simulation–Based Optimization approach is adopted for studying
an ED network under different conditions, by means of Discrete Event
Simulation models. In particular we consider the so called Ambulance Diversion
problem, analyzing different diversion policies. Such models are carried out
from real data collected from six big EDs in the Lazio region of Italy. The
aim is to optimize the performances of the entire network, in order to provide
the best service to the patients without sustaining too high costs. The
obtained experimental results show which are the best diversion policies both
in terms of patient waiting time and costs for the service providers.

###### keywords:

Simulation–based Optimization , Emergency Department , Discrete Event
Simulation , Ambulance Diversion.

##  1 Introduction

The Simulation–Based Optimization (SBO) approach is a methodology oriented to
the optimization of specific Key Performance Indicators (KPIs) and applicable
to a wide range of phenomena and different situations (see e.g., [1, 2]). It
is adopted to perform deep analyses on management and operative problems
arising in systems which are represented by means of simulation models. The
great potentialities of this approach are nowadays well known. Of course, the
SBO approach considers Black–Box Optimization problems, since the objective
function and the constraints of the optimization problem in hand are not
available in analytic form [3]. As a consequence, algorithms belonging to the
class of Derivative–Free Optimization (DFO) methods (see, e.g., [4, 5]) must
be applied (see also [6]). Since both the objective function and the
constraints are evaluated from the output of the simulation model, and since
the simulation runs could be very time expensive, it is that clear the
optimization algorithm adopted should be as effective as possible. Moreover,
the interface between the optimization algorithm and the simulation runs
should be build so that the communication is handy and not time consuming.

In the framework of emergency medical services, and in particular in tackling
ED overcrowding problem, the SBO approach has be successfully applied (see,
e.g., the paper [7] where a systematic review of SBO methods applied in
hospital EDs is reported and the papers [8, 9, 10, 11]). In particular, here
we focus on a phenomenon strongly related to the ED overcrowding, namely the
_Ambulance Diversion_ (AD) (see, e.g. [12, 13] and the references reported
therein). AD is a procedure adopted by ED managers during an emergency
situation, when severe patients cannot be accepted and treated anymore.
Therefore, incoming ambulances can be diverted to neighboring EDs. More in
details, when an ED goes on _AD status_ , two plans can be carried out:
_boarding_ , i.e patient joins a queue into the ED hall or inside an ambulance
in the parking lot, waiting for his turn; _redirection_ , i.e. patient is
transferred towards another available nearby ED which is not on AD status. The
adoption of the AD represents a great inefficiency of the service provided by
an ED. Moreover, this practice is also source of heated discussion: on one
hand, the scientific community of physicians does not approve the AD, since
every patient should receive the adequate treatment and care whenever it is
needed. On the other hand, the ED board and executives aim at obtaining an
effective and efficient management of the first–aids, thus possible
investments for improving service capacity should always be sustained in the
face of an actual request of a specific level of service. As is often the
case, the answer lies somewhere in between: even if the AD should not be
considered as a standard response to the ED overcrowding, it should not be
prohibited when an emergency situation occurs, in particular when the waiting
minutes for the interhospital transportation, compared to the hours spent
waiting in the ED halls, can make the difference between life or death of a
patient.

In this work we formulate the AD problem as an Integer Optimization problem
with bound constraints on the decision variables and with simulation
constraints. In particular, as case study, we consider a network of six big
EDs located in the Lazio region of Italy and we construct a Discrete Event
Simulation (DES) model of this network by using ARENA simulation software
[14], The availability of the collected real data enables to build a reliable
model. Then an interface written in Python has been created for executing
simulation runs via an external code. By using this code we also obtained the
communication between ARENA simulation software and an external optimization
algorithm and, as far as the authors are aware, this represents a novelty with
respect to the standard use of Visual Basic for Application (VBA) included
within ARENA software. This allowed us to use the efficient DFO algorithm for
black–box optimization problems with integer variables recently proposed in
[15]. The obtained results show which is the best AD policy to be adopted in
terms of operative performance.

The paper is organized as follows: in Section 2 the case study is described
along with the most commonly used AD policies. Section 3 reports a description
of the DES model we built. In Section 4 we describe the formulation of the SBO
problem for the case study considered and in Section 5 its implementation. The
results of the experimentation we performed is included in Section 6. Finally,
in Section 7 some final remarks are reported.

##  2 The case study and the different diversion policies

The purpose of this paper concerns the ED organization and the management,
focusing on the well–known problem of the overcrowding. However, this issue is
not faced by means of the analysis of a single ED, but considering a network
constituted by EDs connected to each other via the transportation on ambulance
of severe patients. Thus, the focus is not to manage in detail a single ED in
order to optimize some specific KPIs, but to treat an emergency situation
widespread in a specific geographic area in order to ensure the safety and the
rescue of life–threatening people.

In particular, the focus is on the _Ambulance Diversion_ phenomenon. AD is a
procedure put in place by the hospital ED managers when an emergency situation
occurs, and severe patients cannot be accepted and treated anymore, usually
indicated as ED on AD status. Two possibilities are usually considered, with
possible different policies adopted, when an ED goes on an AD status:

  * 1.

the _boarding_ , namely the patients are anyhow accepted but, due to
unavailability of ED resources, they wait (in a possible long queue) for
visits and treatments. They are usually placed into and ED hall or inside the
ambulance in the parking lot during the waiting time;

  * 2.

the _redirection_ , namely the patients are transferred to another available
nearby ED.

In particular, as case study we consider a network composed by six EDs located
in the Lazio region of Italy. For these EDs, data for the whole year 2012 have
been collected, only considering yellow and red tagged patients111We refer to
the standard four–colors tag scale used in the ED triage, since AD only
concerns severely damaged or life–threatening patients. For each single ED and
for each patient the following timestamps are available: the starting time of
the triage, the starting time of the first medical visit and the discharge
time. Moreover, the times needed for the transportation of patients between
each ED of the network is known.

Another important concept related to the AD phenomenon is the definition of
_diversion policies_ , which refers to the set of organizational choices that
defines unequivocally which are the conditions that establish when an ED must
be considered on AD status, if an ED has to execute the patient’s boarding or
the redirection, and which are the destinations of the redirected patients. Of
course, the diversion policies strongly affect the dynamics of the whole ED
network. In this paper, we consider the following four most commonly adopted
policies:

  1. P1:

Ambulance stoppage, which does not allow any redirection of patients inside
the network, authorizing the boarding instead.

  2. P2:

Redirect towards the nearest ED (complete diversion) when all the ED resources
are occupied and patients are redirected to the nearest ED, if the latter is
not on diversion.

  3. P3:

Redirect towards the nearest ED, with priority to the red–tagged patients
(partial diversion), which is identical to the previous one, but it involves
the diversion for only yellow–tagged patients. More specifically, when the
number of ED occupied resources reaches a certain threshold value, the
yellow–tagged patients are redirected to the nearest ED, if the latter is not
on partial or complete diversion.

  4. P4:

Redirection to the least occupied ED of the network, regardless of the
distance between the “sending” and the “receiving” ED.

##  3 The Discrete Event Simulation model

In order to build the DES model to be used for studying the AD policies for
the ED network considered, we first introduce the concept of sanitary
resource. We define sanitary resource an integer valued parameter which
corresponds to the set of all human (like physicians, nurses) and physical
(like ED rooms, beds, stretchers) resources required to treat a single
patient. The introduction of this notion enables to model each single ED as
single queuing system, in which the sanitary resource represents the number of
servers as represented in Figure 3.1

Figure 3.1: ED network and single ED structure

As we will describe afterward, the value of the sanitary resource associated
to each ED will be determined by means of a model calibration phase, so that
this simplified representation of each single ED allows us to obtain a good
accuracy in evaluating the KPIs of interest computed from the simulation
output. Note these sanitary resources actually will result as control
variables during the optimization phase.

In order to develop the DES model based on the structure now described, it is
required to know:

  * 1.

the patient arrival stochastic process to each ED of the network;

  * 2.

the probability distribution of the Length of Stay (LOS) at each ED of the
network.

To this aim, we consider 8-hours time slots of each day (from 00:00 to 08:00,
from 08:00 to 16:00 and from 16:00 to 24:00). The choice of 8-hours time slots
is motivated by the fact that the ED staff is scheduled on 8-hours shifts,
hence an analysis based on these time slots results quite realistic.

We evaluate patient interarrival time and LOS probability distributions for
each time slot and for each yellow and red–tagged patient. The data were
collected into Excel files where each line is associated to a patient and
reports: the ED name, the triage tag assigned, date and time of the starting
of the triage, date and time of the starting of the visit and date and time of
discharge from the ED. Of course, from these data we can easily obtain the
time difference between the starting of the triage and the starting of the
visit. Since the starting of the visit is the real tacking charge of the
patient by the ED, this time difference actually represents the patient
waiting time before the ED tacking charge and we will use this as the main KPI
to calibrate and validate our DES model.

As regards, the stochastic process of patient arrival at each ED, according to
the standard assumption in the literature (see, e.g.,[16, 17, 18, 19, 20, 21,
22]), we model patient arrivals by a Nonhomogeneous Poisson Process (NHPP) and
following a usual procedure we approximate the arrival rate by a piecewise
constant function. On the basis of the data, we determine the patient arrival
rate for each ED of the network, for each time slot and for each color tag. In
Table 3.1, we report the number of patient arrivals during the whole year at
each ED of the network, for red and yellow–tagged patients, in each time slot.

| Color tag | 00:00-08:00 | 08:00-16:00 | 16:00-24:00 | Total
---|---|---|---|---|---
ED1 | Yellow | 875 | 2688 | 2279 | 5842
Red | 94 | 299 | 187 | 580
ED2 | Yellow | 390 | 1592 | 1299 | 3281
Red | 56 | 153 | 107 | 316
ED3 | Yellow | 599 | 2331 | 1839 | 4769
Red | 84 | 102 | 92 | 278
ED4 | Yellow | 439 | 2118 | 1503 | 4060
Red | 70 | 295 | 202 | 567
ED5 | Yellow | 399 | 2007 | 1597 | 4003
Red | 55 | 299 | 188 | 542
ED6 | Yellow | 324 | 1138 | 832 | 2294
Red | 37 | 58 | 72 | 167
Table 3.1: Number of patient arrivals at each ED, for each color tag and for
each time slot

Note that the red–tagged patient arrivals in each time slot (and in particular
during the night time slot) are few. This is another motivation for
considering 8-hours time slots, since smaller time slots would imply an even
smaller number of patient arrivals, making meaningless every statistical
analysis.

Regarding the LOS probability distributions, the time interval between the
take charge and the relative discharge has been evaluated from available data
for each ED of the network, for each patient and for each color tag. Then, a
standard statistical analysis has been performed to obtain the best fitting.

The model has been implemented by using ARENA 16.1 simulation software [14].
On the basis of the assumptions previously described, each ED can be
represented as reported in Figure 3.2.

Figure 3.2: ARENA model of a single ED

The model entities are the patients which flows through the ED; the first two
modules on the left create the flow of red and yellow–tagged patients inside
the model. An attribute (_delay_code_) is assigned to each entity, depending
on the color tag and the time slot in which the patients entered the ED. Then,
the entities join a queue, with priority to the red tagged–patients, and then
they are “taken in charge” by one free sanitary resource (if any) which is
seized to this aim. The patients stay inside the ED for a time that depends on
the LOS probability distribution of the associated _delay_code_. At the end,
the entity releases the seized resource and exit the ED. If one sanitary
resource is not available, entity is queued waiting his turn. Record modules
are used to store the waiting time value of each patient, distinguishing each
color tag and each time slot. This waiting time is computed as time difference
between the starting of the triage and the actual take charge of the patient
(the starting of the visit), and thus it represents the waiting time of each
entity before seizing a sanitary resource.

Since the AD phenomenon strictly depends on patient waiting time, we use this
KPI to calibrate our model. Namely, we performed an accurate model calibration
phase in order to determine the sanitary resource value of each ED so that the
average patient waiting time computed from simulation output is a good
approximation of the real average patient waiting time determined from the
available data. This has been obtained by determining the sanitary resource
values which minimize

$\sum_{i=1}^{3}\sum_{j=1}^{2}\ \lvert W^{sim}_{i,j}-W^{real}_{i,j}\rvert,$

where $i=1,2,3$ represents the time slot, $j=1,2$ the color tag (red and
yellow ones) and $W^{sim}_{i,j}$ and $W^{real}_{i,j}$ are the average patient
waiting time obtained from simulation output and from data, respectively. The
results of the calibration procedure are reported in Table 3.2.

| 00:00–08:00 | 08:00–16:00 | 16:00–24:00
---|---|---|---
ED1 | 4 | 4 | 4
ED2 | 4 | 5 | 4
ED3 | 4 | 4 | 3
ED4 | 4 | 5 | 2
ED5 | 4 | 5 | 3
ED6 | 3 | 2 | 2
Table 3.2: Sanitary resource values obtained from the calibration procedure

By using this values of the sanitary resources, we obtain a model for each ED
reproducing its “as–is” status, without considering any AD.

Then we implemented the ED network composed by the six EDs under study,
reproducing the four AD policies (P1, P2, P3, P4) described in Section 2. The
first one (P1) does not consider any redirection, so that each ED is
considered as a single first–aid point. In the second one (P2), to implement
complete redirection to the nearest ED, some decision modules are added to
each ED model in order to represent the diversion condition and the actual
redirection of the patients when all the sanitary resources are seized and a
red/yellow–tagged patient arrives. Policy P3 considers partial diversion and
hence only redirection of yellow tagged–patients has been implemented,
depending on a threshold value of the sanitary resources. Finally, policy P4
involves an additional central decision module for redirection towards the ED
of the network with the least number of sanitary resources seized. Figure 3.3
report a part of such ARENA model, where the mentioned decision module is
highlighted.

Figure 3.3: ARENA module (part) for the AD policy P4. The red square
highlights the central decision module.

##  4 Statement of the optimization problem

In this section we report the optimization problem we formulate in order to
determine which is the AD policy to be considered the most effective to
provide prompt first–aid to red and yellow–tagged patients. The goal is to
determine the value of the sanitary resource of each ED and the AD policy to
be adopted, so that some suited KPIs are optimized.

We introduce the integer valued decision variables $n_{ij}$ to indicate the
value of the sanitary resource assigned to the $i$-th ED ($i=1,\ldots,6$)
during the $j$-th time slot ($j=1,\ldots,3$). These 18 variables represent the
control variables of the SBO problem.

As regards the KPIs of interest, we consider the patient average Non-Valued
Added (NVA) time associated to each ED, for red and yellow–tagged patients,
corresponding to waiting and transfer times. We denoted by $t^{\rm NVA/R}_{i}$
and $t^{\rm NVA/Y}_{i}$ the average NVA time of the $i$-th ED ($i=1,\ldots,6$)
for red and yellow–tagged patients, respectively. It is important to note that
such times actually are function of the variables $n_{ij}$ and they can be
only computed from the output of the simulation runs. Therefore the following
expressions

$\sum_{i=1}^{6}t^{\rm NVA/R}_{i}\qquad\hbox{and}\qquad\sum_{i=1}^{6}t^{\rm
NVA/Y}_{i}$

constitute two important KPIs to be minimized.

Of course, to keep costs low, the overall number of sanitary resources should
be minimized, too. Of course, this is a conflicting objective with respect to
the minimization of the NVA times defined above. Therefore, actually, we have
a black box multiobjective problem which we tackle by reducing into a single
objective problem by means of the standard scalarization procedure (see, e.g.,
[23]). More in detail, we consider the following objective function

$w_{1}\cdot 480\sum_{i=1}^{6}\sum_{j=1}^{3}n_{ij}+w_{2}\sum_{i=1}^{6}t^{\rm
NVA/Y}_{i}+w_{3}\sum_{i=1}^{6}t^{\rm NVA/R}_{i}$ (4.1)

where $w_{1}$, $w_{2}$ and $w_{3}$ are suited weights. The last two terms
represent the overall NVA times (in minutes) for red an yellow–tagged
patients, respectively; the first term measures the overall time (in minutes)
for which the sanitary resources are scheduled in the 8 hours (480 minutes)
time slots. This latter term takes into account the number of sanitary
resources assigned to each ED and it has been expressed in terms of time (in
minutes) for homogeneity with respect the other two terms of the objective
function. This term is meant to depict the total cost to be sustained in order
to maintain a certain number of sanitary resources active in each time slot of
each ED.

As regards the constraints, we have bound constraints on the variables
$n_{ij}$ and simulation constraints which prevent the NVA times to exceed
specific threshold values for each color tag. Such values are 40 minutes for
the yellow–tagged patients and 20 minutes for the red–tegged ones.

The complete formulation of the SBO problem we consider is the following:

$\displaystyle\min$ $\displaystyle w_{1}\cdot
480\sum_{i=1}^{6}\sum_{j=1}^{3}n_{ij}+w_{2}\sum_{i=1}^{6}t^{\rm
NVA/Y}_{i}+w_{3}\sum_{i=1}^{6}t^{\rm NVA/R}_{i}$ (4.2) $\displaystyle{\rm
s.t.}$ $\displaystyle t^{\rm NVA/Y}_{i}\leq 40,\qquad i=1,\ldots,6,$
$\displaystyle t^{\rm NVA/R}_{i}\leq 20,\qquad i=1,\ldots,6,$ $\displaystyle
2\leq n_{ij}\leq 10,\quad\leavevmode\nobreak\ \leavevmode\nobreak\
i=1,\ldots,6;\leavevmode\nobreak\ j=1,\ldots,3,$ $\displaystyle
n_{ij}\quad\hbox{integer},$

which is a Black Box Integer Optimization (BBIO) problem with 18 variables
(with the corresponding bound constraints) and 12 simulation constraints.

##  5 The SBO implementation

To solve the BBIO Problem (4.2), we use an algorithm belonging to the class of
Derivative Free Optimization methods recently proposed in [15], named
DFLINT222Publicly available in the DFL Library at the URL
www.iasi.cnr.it/$\sim$liuzzi/DFL. It is an algorithm for black box inequality
and box constrained integer nonlinear programming problems, hence well suited
for solving Problem (4.2). Since the DFLINT code is available in Python, we
decided to code Problem (4.2) in Python as well. Moreover, we also created a
specific interface between the optimization algorithm and the ARENA simulation
package by using Python. Note that ARENA is a Windows desktop application and
no information is provided for executing simulation runs from an external
software or for building an interface with any external sources. Only VBA
modules can be used within ARENA package and we believe that their use for
connecting an external optimization engine with an ARENA model is really
cumbersome. As far as the authors are aware, the use of an interface coded in
Python for enabling execution of simulation runs and communication between
ARENA and external software is a novelty which allowed us to significantly
improve the overall performance of the SBO approach applied to the problem
under study. For this reason, we now report a brief description of our
implementation, with some technical details. A more insight description is
available in [24].

Starting from the values reported in Table 3.2 (assumed as the starting point
of the optimization algorithm), at each iteration of the algorithm, the values
of the control variables (the sanitary resources) are updated and written in
an Excel file so that they can be used as input parameters of the simulation.
Then simulation run is executed via the Python code. This is performed by
using four executable ARENA files expmt.exe, model.exe, linker.exe and
siman.exe which must be executed in this specific order. After creating the
two files of the model (with .exp and .mod extension), by using the
subprocess.run Python module, the aforementioned files are converted into two
other files (with .e and .m extensions). Those last two files (in this order)
are required for the linker.exe to work. As result, a .p file is produced and
the siman.exe program uses it to run a simulation. Finally, the last part of
the code, aims at opening the ARENA output file and extracting the requested
values from it.

In the ARENA output file, the NVA times of interest, are stored for every
single replication and, based on the Sample Average Approximation approach
[25, 26], we compute the average values of the NVA times as mean over
independent replications. The corresponding confidence interval is computed,
too.

##  6 Experimental results

In this section we report the results obtained by means of the SBO approach we
propose, applied to the ED network under study. The aim is to determine which
AD policy (among those described in Section 2) and which ED settings (in terms
of sanitary resources to be allocated) perform the best in the sense that the
objective function (4.1) is minimized. In our experimentation the weights
$w_{1}$, $w_{2}$ and $w_{3}$ in (4.1) are chosen so that the order of
magnitude of the three terms of the objective function (4.1) is the same,
namely $w_{1}=1$, $w_{2}=300$ and $w_{3}=600$. Of course different values
could be experimented, leading to different weighing of the three terms of the
objective function. This shows the flexibility of our approach, since it
enables to assign higher weights to terms which are to be considered most
relevant, for instance to the term which refers to the most critical
(red–tagged) patients.

As regards the simulation runs, we perform 30 independent replications for
each run, since we experimented that this choice guarantees a good accuracy of
the output results of the simulation. The length of each replication is 365
days and we adopt a warm–up period of 48 hours since this ensures the system
to reach the steady state.

As concerns the parameters of the DFLINT algorithm, we use default parameters
of the original code. Moreover, in order to limit the overall computational
burden, the maximum number of function evaluations (which is used as stopping
criterion of the algorithm) is set to 700.

In Table 6.1 we report the optimal value of the sanitary resources for each
considered policy, for each ED and for each time slot determined by the
optimization algorithm. The values corresponding to the starting point (see
Table 3.2) are also reported for an easy comparison.

|  | Starting |  |  |  |
---|---|---|---|---|---|---
| Time slot | point | P1 | P2 | P3 | P4
ED1 | 00:00–08:00 | 4 | 7 | 4 | 6 | 5
08:00–16:00 | 4 | 5 | 4 | 3 | 5
16:00–24:00 | 4 | 4 | 3 | 5 | 5
ED2 | 00:00–08:00 | 4 | 6 | 5 | 5 | 5
08:00–16:00 | 5 | 5 | 5 | 4 | 5
16:00–24:00 | 4 | 7 | 4 | 4 | 4
ED3 | 00:00–08:00 | 4 | 4 | 3 | 4 | 4
08:00–16:00 | 4 | 6 | 4 | 5 | 4
16:00–24:00 | 4 | 3 | 4 | 3 | 4
ED4 | 00:00–08:00 | 4 | 7 | 5 | 6 | 4
08:00–16:00 | 5 | 4 | 6 | 6 | 3
16:00–24:00 | 2 | 6 | 4 | 4 | 4
ED5 | 00:00–08:00 | 4 | 7 | 7 | 6 | 5
08:00–16:00 | 5 | 4 | 6 | 4 | 3
16:00–24:00 | 2 | 6 | 4 | 5 | 5
ED6 | 00:00–08:00 | 3 | 4 | 4 | 4 | 3
08:00–16:00 | 3 | 4 | 4 | 3 | 3
16:00–24:00 | 2 | 4 | 3 | 3 | 2
Table 6.1: Values of the sanitary resources corresponding to the starting
point and to the optimal point for each AD policy

It can be easily observed that an increase of the number of sanitary resources
(with respect to the starting point) in many cases is required to ensure the
viability of the corresponding AD policy. This is mainly due to the
satisfaction of the simulation constraints in Problem 4.2 which impose that
the NVA times must be below a specific value.

In Table 6.2 we report the corresponding optimal function values for each AD
policy.

| Objective function value | Optimal objective
---|---|---
| at the starting point | function value
P1 | 127454,63 | 51975,77
P2 | 92956,13 | 43506,65
P3 | 93758,85 | 46180,20
P4 | 47926,04 | 42037,83
Table 6.2: Optimal values of the objective function for each AD policy

The obtained improvement is clearly evidenced for all the policies considered,
and it is significant in most cases.

To give more complete detailed results, in the Appendix we report the values
of all the NVA times corresponding to the starting point (Table A.1) and to
the optimal solution (Table A.2), for each AD policy, for each ED and for each
color tag.

By observing the results in Table 6.2, the best diversion policy in terms of
the objective function (4.1) can be easily determined: it is the policy P4,
followed by the policy P2. The former one, in particular, provides good
operative performances even before the optimization, i.e. in correspondence of
the starting point. Furthermore, the worst diversion policy is P1 for which
the best improvement of the objective function is registered. Giving higher
priority to the red code while imposing the redirect towards the nearest ED
(policy P3) seems to worsen the performances of the ED network. The reasons
behind this effect can be explained by the analysis of the average NVA times
registered for this diversion policy (see Tables A.1 and A.2 in the Appendix).
In fact, the average times referred to the red–tagged patients are
significantly low when compared to the ones obtained with the other diversion
policies, but this leads to an increase of the average NVA times referred to
the yellow–tagged patients. On the overall, the result is an overbalance
towards a general worsening of the operative performances of the network,
which presumably derives from an excessive priority given to the red–tagged
patients that have a lower probability of showing up in an ED.

By observing in detail Tables A.2 in the Appendix corresponding to the
obtained optimal point, it is clear that in most cases the NVA times are
similar across the different diversion policies. This is also due to the queue
discipline adopted for boarding patients that gives highest priority to
red–tagged patients. It can be also noticed that, the average NVA times do not
exceed 5 minutes, which can be considered a really good operative result.

Finally, it is possible to understand which is the main issue associated with
the ambulance stoppage diversion policy P1. Indeed, the inability to redirect
patients toward other EDs has the direct consequence of an allocation of an
higher number of sanitary resources to maintain good operative performances.
However, similar performances, in terms of NVA times, can be obtained by
applying other diversion policies with a reduced number of sanitary resources,
thanks to the interhospital transportation of the patients. In other words,
without adopting redirection policies, the same average NVA times are obtained
but with higher costs due to an increase of the sanitary resources needed. The
key element of this consideration is that a good trade–off is needed between
the reduction of the patient average waiting time inside the ED and the
average transportation time towards another ED. This analysis can be performed
by different choices of the weights $w_{1}$, $w_{2}$ and $w_{3}$ in the
objection function of the SBO Problem (4.2) we formulated. In fact, this
enables to obtain a different balance of the three terms (referred to three
single KPIs) that compose the objective function. For instance, an higher cost
containment can be obtained by increasing $w_{1}$, or the same priority level
(as regards the AD policies) can be assigned to red and yellow–tagged patients
by selecting $w_{2}=w_{3}$.

##  7 Conclusions

In this paper we deal with the AD phenomenon by the SBO approach. In
particular, we formulate the problem as a Black–Box Integer Optimization
problem which we solve by means of an effective DFO algorithm recently
proposed in literature. In order to experiment the approach proposed, we
consider a network composed by six EDs located in the Lazio region of Italy.
The data collected for these EDs (related to one year) have been used for
constructing a DES model which we implemented by using ARENA simulation
software. A specific interface, written in Python, has been built for
connecting the ARENA model and the optimization algorithm.

The obtained results on the case study considered show that the best diversion
policy in terms of operative performance seems to be the redirection towards
the least occupied ED. Anyhow, our experimentation clearly evidences that the
ambulance diversion and the interhospital transportation of the patients are
procedures that should be always considered, especially during important
emergency situations.

As future work, we believe that it would be very interesting to further deepen
the analysis on the ambulance diversion policies through dedicated research
activities, possibly supported by higher quality information and more
efficient data management tools.

##  Appendix A Detailed results on the NVA times

In this appendix we include detailed results concerning the NVA times $t^{\rm
NVA/Y}_{i}$ and $t^{\rm NVA/R}_{i}$, $i=1,\ldots,6$ for each AD policies, for
each ED and for each color tag. The tables report the average values and the
half–width of the $95\%$ confidence interval. In particular, in Table A.1 we
report the values of $t^{\rm NVA/Y}_{i}$ and $t^{\rm NVA/R}_{i}$ corresponding
to the starting point. In Table A.2 we report the values of $t^{\rm
NVA/Y}_{i}$ and $t^{\rm NVA/R}_{i}$ corresponding to the optimal point. All
the times are expressed in minutes.

| Color tag | P1 | P2 | P3 | P4
---|---|---|---|---|---
ED1 | Yellow | $15.89\pm 0.96$ | $8.55\pm 0.45$ | $12.78\pm 0.55$ | $2.98\pm 0.09$
Red | $6.78\pm 0.44$ | $4.86\pm 0.42$ | $4.19\pm 0.36$ | $2.70\pm 0.18$
ED2 | Yellow | $25.77\pm 4.15$ | $2.05\pm 0.39$ | $6.89\pm 0.48$ | $2.65\pm 0.17$
Red | $11.18\pm 1.62$ | $1.72\pm 0.22$ | $0.74\pm 0.12$ | $2.43\pm 0.30$
ED3 | Yellow | $5.96\pm 0.97$ | $2.10\pm 0.03$ | $5.06\pm 0.87$ | $1.32\pm 0.17$
Red | $3.94\pm 0.74$ | $1.53\pm 0.33$ | $1.98\pm 0.42$ | $1.21\pm 0.28$
ED4 | Yellow | $50.28\pm 2.75$ | $34.13\pm 1.65$ | $43.43\pm 1.54$ | $5.10\pm 0.27$
Red | $24.40\pm 1.08$ | $20.19\pm 1.08$ | $15.83\pm 0.75$ | $4.46\pm 0.34$
ED5 | Yellow | $54.63\pm 2.98$ | $37.20\pm 1.57$ | $46.48\pm 1.60$ | $5.27\pm 0.32$
Red | $23.70\pm 1.43$ | $17.85\pm 1.01$ | $14.22\pm 0.81$ | $4.37\pm 0.31$
ED6 | Yellow | $12.51\pm 1.26$ | $12.98\pm 1.25$ | $9.05\pm 0.93$ | $2.33\pm 0.19$
Red | $7.10\pm 1.22$ | $7.47\pm 0.96$ | $4.66\pm 0.70$ | $2.08\pm 0.41$
Table A.1: Values of the NVA times $t^{\rm NVA/Y}_{i}$ and $t^{\rm
NVA/R}_{i}$, $i=1,\ldots,6$ corresponding to the starting point

| Color tag | P1 | P2 | P3 | P4
---|---|---|---|---|---
ED1 | Yellow | $1.29\pm 0.29$ | $2.39\pm 0.14$ | $3.23\pm 0.15$ | $1.18\pm 0.05$
Red | $0.87\pm 0.14$ | $1.97\pm 0.16$ | $0.40\pm 0.12$ | $1.16\pm 0.10$
ED2 | Yellow | $2.62\pm 0.43$ | $1.25\pm 0.17$ | $4.88\pm 0.16$ | $1.18\pm 0.11$
Red | $1.54\pm 0.33$ | $1.07\pm 0.22$ | $0.33\pm 0.05$ | $1.09\pm 0.16$
ED3 | Yellow | $0.99\pm 0.23$ | $1.22\pm 0.20$ | $1.78\pm 0.07$ | $0.84\pm 0.07$
Red | $0.65\pm 0.20$ | $1.08\pm 0.23$ | $0.11\pm 0.05$ | $0.80\pm 0.11$
ED4 | Yellow | $2.00\pm 0.25$ | $0.99\pm 0.14$ | $2.61\pm 0.24$ | $2.17\pm 0.09$
Red | $1.45\pm 0.25$ | $0.75\pm 0.10$ | $0.51\pm 0.10$ | $2.05\pm 0.12$
ED5 | Yellow | $2.20\pm 0.26$ | $0.86\pm 0.14$ | $3.79\pm 09.33$ | $1.69\pm 0.16$
Red | $1.30\pm 0.20$ | $0.60\pm 0.11$ | $0.66\pm 0.13$ | $1.56\pm 0.14$
ED6 | Yellow | $1.08\pm 0.12$ | $1.14\pm 0.18$ | $2.67\pm 0.14$ | $1.76\pm 0.13$
Red | $0.53\pm 0.18$ | $0.70\pm 0.24$ | $0.67\pm 0.27$ | $1.39\pm 0.19$
Table A.2: Values of the NVA times $t^{\rm NVA/Y}_{i}$ and $t^{\rm
NVA/R}_{i}$, $i=1,\ldots,6$ corredsponding to the optimal point

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