Han, He, and Oh

Data-Driven Inpatient Bed Assignment Using the $P$ Model

Data-Driven Inpatient Bed Assignment: The $P$ Model Approach to Boarding and
Overflowing

Shasha Han Department of Analytics and Operations, NUS Business School,
National University of Singapore, Singapore 117592
<EMAIL_ADDRESS>Shuangchi He Department of Industrial Systems Engineering
and Management, National University of Singapore, Singapore 117576
<EMAIL_ADDRESS>Hong Choon Oh Health Services Research, Eastern Health
Alliance, Singapore 529541
<EMAIL_ADDRESS>
_Problem definition_ : Emergency department (ED) boarding refers to the
practice of holding patients in the ED after they have been admitted to
hospital wards, usually resulting from insufficient inpatient resources.
Boarded patients may compete with new patients for medical resources in the
ED, compromising the quality of emergency care. A common expedient for
mitigating boarding is patient overflowing, i.e., sending patients to beds in
other specialties or accommodation classes, which may compromise the quality
of inpatient care and bring on operational challenges. We study inpatient bed
assignment to shorten boarding times without excessive patient overflowing.

_Academic/practical relevance_ : As a cross-departmental issue, boarding is
caused by periodic mismatches between the demand for inpatient care and the
supply of inpatient resources. Besides reducing overall boarding times,
mitigating the time-of-day effect is also essential for patients’ safety. With
numerous bed assignment constraints, it is challenging to strike a balance
between boarding and overflowing.

_Methodology_ : We use a queue with multiple customer classes and multiple
server pools to model hospital wards. Exploiting patient flow data from a
hospital, we propose a computationally tractable approach to formulating the
bed assignment problem, where the joint probability of all waiting patients
meeting their respective delay targets is maximized.

_Results_ : By dynamically adjusting the overflow rate, the proposed approach
is capable not only of reducing patients’ waiting times, but also of
mitigating the time-of-day effect on boarding times. In numerical experiments,
our approach greatly outperforms both early discharge policies and threshold-
based overflowing policies, which are commonly used in practice.

_Managerial implications_ : We provide a practicable approach to solving the
bed assignment problem. This data-driven approach captures critical features
of patient flow management, while the resulting optimization problem is
practically solvable. The proposed approach is a useful tool for the control
of queueing systems with time-sensitive service requirements.

emergency department boarding, healthcare operations, queueing network,
routing, integer programming, $P$ model

##  1 Introduction

Emergency department (ED) crowding has been a worldwide crisis of healthcare
delivery, compromising the quality of and access to emergency care in both
developing and developed countries (Pines et al. 2011). As a major cause of ED
crowding, _boarding_ refers to the practice of holding patients in the ED
after they have been admitted to hospital wards, usually resulting from
insufficient inpatient resources. Because boarded patients may compete with
new patients for medical resources in the ED, boarding could bring on
treatment delays and higher complication rates. Boarded patients also suffer
from prolonged waiting, since they may not receive the same level of care as
in inpatient wards. Recent studies have identified an association between
prolonged boarding times and increased mortality in both EDs and inpatient
wards (Singer et al. 2011, Sun et al. 2013). As a quality indicator of
emergency care, boarding times are closely monitored by government agencies in
some countries. In the United States, the Centers for Medicare and Medicaid
Services include boarding times in the measures of their pay-for-performance
program; on their Hospital Compare website, the statistics of boarding times
from over 4,000 hospitals are accessible to the public. In Singapore, public
hospitals must report boarding times to the Ministry of Health regularly;
their daily median boarding times are published on the Ministry’s website and
updated every week. The Ministry may investigate those cases in which
patients’ waiting times for beds are excessively long.

Figure 1: Mean waiting times of ED patients at an anonymous public hospital.

Figure 2: Relative frequencies of inpatient discharge hours at the hospital.

As a cross-departmental issue, boarding is caused by _periodic mismatches_
between the demand for inpatient care and the supply of resources such as
inpatient beds and hospital staff. In hospital wards, the majority of patients
are commonly discharged in the afternoon, which may lead to a _time-of-day
effect_ on boarding times: When the inpatient bed occupancy rate is high,
patients who request beds at late night or in the early morning may experience
long delays until appropriate beds are available. This phenomenon is evident
in Figure 2, where the mean waiting times of ED patients at an anonymous
public hospital in Singapore are plotted across different bed request hours.
These results are estimated using the hospital’s patient flow records of
April–September 2015. Within this period, patients who requested beds at night
were expected to wait more than eight hours, while those who requested beds in
the early afternoon spent only about four hours. Prolonged boarding times
could be more detrimental to ED patients who request beds at late night,
because they may be in a serious condition which could be better managed in
inpatient wards. Therefore, aside from reducing overall waiting times,
mitigating the time-of-day effect on boarding times is also essential for
patients’ safety. We illustrate the relative frequencies of inpatient
discharge hours at the hospital in Figure 2. As one may expect, most patients
were discharged in the afternoon, and not until then, would appropriate beds
be available for a number of requests made on the previous night. Shi et al.
(2016) reported a similar patient discharge pattern (see Figure 2 in their
paper) and the resulting time-of-day effect on boarding times at another
hospital in Singapore.

To mitigate ED boarding, some hospitals launch initiatives encouraging more
patients to be discharged from wards before noon (Wertheimer et al. 2014, Shi
et al. 2016). Although positive outcomes are reported on patient throughput
(Wertheimer et al. 2015), there is no sufficient evidence in the literature
that discharging patients earlier would be a practical solution to prolonged
boarding times. To assess the influence of earlier discharge hours, Shi et al.
(2016) developed a high-fidelity simulation model using a set of patient flow
data from a public hospital in Singapore. They found out that in the hospital,
boarding times may be equalized over the day if a portion of patients are
discharged before 9 am. However, such a discharge policy would be difficult to
implement, because physicians and nurses are busy with ward rounds in the
morning. Discharging patients at earlier hours will inevitably affect existing
daily ward routines. As pointed out by Green (2008), physicians are usually
reluctant to adopt new operational practices owing to clinical concerns, and
in general, hospital administrators are unwilling to exert pressure on
physicians to change their behavior. In other words, resistance from medical
staff may also be a hurdle in implementing a new patient discharge policy.

A common expedient for mitigating ED boarding is _patient overflowing_ , which
refers to assigning patients to “incorrect” beds when the most appropriate
beds are not available. For example, a cardiac patient could be sent to a bed
in the general medical ward upon his admission, if there is no bed available
in the cardiac ward. As a means of resource pooling, patient overflowing can
always reduce excessive delays. However, it may compromise the quality of
inpatient care, bringing on operational challenges such as additional costs
and coordination between wards (Teow et al. 2011). Hence, a balance must be
struck between boarding and overflowing.

We study inpatient bed assignment in this paper to shorten ED boarding times
and mitigate the time-of-day effect. From a modeling perspective, hospital
wards can be viewed as a queueing system, with patients being the customers
and beds being the servers. Since both patients and beds are categorized by
specialty, gender, and accommodation class, the assignment of inpatient beds
turns out to be a routing problem for a queue with multiple customer classes
and multiple server pools. We adopt the terminology used by Shi et al. (2016),
referring to a bed that completely matches a patient’s specialty, gender, and
requested accommodation class as a _primary bed_ for the patient, and
referring to such a patient as a _primary patient_ for the bed. If a patient
is allowed to be sent to a bed that does not match all the requirements, they
are said to be a _non-primary patient_ and a _non-primary bed_ , respectively.
With requirements for boarding times, it may not be possible to find every
patient a primary bed. Sending a fraction of patients to non-primary beds
would be necessary for meeting delay targets, especially when the bed
occupancy rate is high. Since a non-primary bed for a boarded patient could be
a primary bed for others, overflowing may prevent wards from accommodating
more primary patients in the future. Without a carefully designed bed
assignment policy, overflowing could be spread over multiple wards, resulting
in more operational issues. Shi et al. (2016) reported that at a public
hospital in Singapore, nearly 30% of patients from the ED were sent to non-
primary beds in 2008–2010.

To address this problem in a practical setting, we propose a dynamic bed
assignment approach that exploits patient flow data from the hospital. When a
bed request is received and there is no primary bed available, the assignment
algorithm will determine whether to send the patient to a non-primary bed
immediately, or to let the patient wait for a primary bed. When an inpatient
bed becomes available and there are boarded patients, the algorithm will
determine whether to assign a patient to this bed and who should be sent to
this bed if there are multiple patients waiting. In other words, the
assignment algorithm will determine _whether to overflow_ and _where to
overflow_ for patients and beds.

We impose mandatory targets for patient boarding times on the bed assignment
problem, in order to meet requirements for the quality of care. Our intention
is to maximize the percentage of patients whose boarding times are within
these targets, while the overflow rate is maintained below a certain level. We
formulate an optimization problem to obtain assignment decisions by maximizing
the joint probability of all boarded patients meeting the delay targets. This
formulation allows us to shorten boarding times for bed requests at night,
thus mitigating the time-of-day effect. Taking the joint probability of target
attainment as the objective function, such an optimization problem was first
studied by Charnes and Cooper (1963), who termed the formulation the _$P$
model_. This formulation, however, has not been widely used in practice, in
part because evaluating the joint probability requires integration in high
dimensions, which is generally computationally difficult. In a recent study,
He et al. (2018) pointed out that the $P$ model could be tractable if the
objective joint probability can be written into a product form. Considering a
patient scheduling problem in EDs, they proposed a hybrid robust-stochastic
framework under which the $P$ model formulation can be modified into a
computationally amiable form. They also demonstrated that their approach may
outperform an asymptotically optimal scheduling approach proposed by Huang et
al. (2015).

Although both the present paper and the study by He et al. (2018) concern
control problems in queueing networks, the respective $P$ model formulations
differ greatly. First, the hybrid formulation by He et al. (2018) imposes an
additional constraint on the $P$ model to convert the objective function into
a product form, i.e., the feasible set of service times is required to be
hyperrectangular. The optimal solution to their hybrid formulation is an
approximate solution to the original problem. In the present paper, we obtain
exact optimal solutions to the bed assignment problem, without using
approximation techniques. This is because in the bed assignment problem,
patients’ waiting times are relatively short compared with their bed occupancy
times. Such a queueing system operates in the _Halfin–Whitt regime_ (see,
e.g., Halfin and Whitt 1981, Borst et al. 2004), where the number of boarded
patients is much smaller than the number of beds. Therefore, a patient’s
boarding time must be another patient’s residual bed occupancy time. Since
patients’ discharge times within a day are assumed to be mutually independent,
the joint probability of all boarded patients meeting the delay targets has a
product form under any given assignment plan, which results in a tractable $P$
model formulation. Second, the patient scheduling problem studied by He et al.
(2018) involves optimal sequencing, which is computationally demanding even if
the objective joint probability has a product form. Only for small- to
moderate-scale queueing networks, may the sequencing problem be practically
solvable. In contrast, the $P$ model can be used to solve routing problems for
much larger queueing systems. In Section 6, we use the $P$ model formulation
to dynamically determine bed assignments for a hospital where patients are
categorized into 50 types and the 571 inpatient beds are categorized into 34
pools. To the best of our knowledge, no other approaches in the literature are
capable of solving routing problems in such a large queueing system with so
many customer classes and server pools, when practical features such as
general, yet non-identically distributed service times and time-sensitive
service requirements are involved.

The main contribution of this paper is twofold. First, we provide a
_practicable_ data-driven approach to solving the bed assignment problem in
hospitals. Unlike conventional approaches in the literature relying on highly
stylized models, the $P$ model approach is able to capture critical features
of patient flow management, such as a time-varying patient arrival process,
general but non-identical bed occupancy times, and many patient and bed types,
while the resulting optimization problem is still practically solvable. Using
a set of patient flow data from a hospital, our simulation study shows that
the proposed approach can greatly reduce patients’ boarding times and mitigate
the time-of-day effect. Second, through this bed assignment problem, we
demonstrate that the $P$ model could be a useful tool for the control of
queueing systems with time-sensitive service requirements. Solving such
problems is usually difficult under the conventional probabilistic framework
of queueing theory. We expect to see more applications of the $P$ model in
large-scale stochastic systems such as call centers and computer clusters.

The remainder of this paper is organized as follows. By reviewing related
literature, we position our study in Section 2. We introduce the multi-class,
multi-pool queueing model for the bed assignment problem in Section 3. In
Section 4, we present a tractable approach to solving this problem, using the
$P$ model formulation. We introduce a high-fidelity simulation model in
Section 5, which is primarily based on the simulation model by Shi et al.
(2016). Using the simulation model, we conduct a data-based numerical study in
Section 6 to assess the $P$ model approach. The paper is concluded in Section
7. We leave the proof of a proposition and details about the simulation model
in the appendix.

##  2 Related Literature

The assignment of inpatient beds is a routing problem for a queueing system
with many servers, where most customers should be sent to matched servers. In
the literature, routing problems for many-server queues are primarily motived
by call center applications. Optimal call routing policies have been studied
by Bassamboo et al. (2006), Dai and Tezcan (2008), Bassamboo and Zeevi (2009),
Gurvich and Whitt (2009a, b), Gurvich and Whitt (2010), Armony and Mandelbaum
(2011), and Stolyar and Tezcan (2011) under various operational regimes and
cost structures. Most of these studies rely on asymptotic analysis and
Markovian assumptions such as independent, exponentially distributed service
times. The bed assignment problem is generally more difficult, because some
assumptions in call routing are no longer applicable in the hospital setting.
For example, Shi et al. (2016) pointed out that the discharge pattern in
hospital wards may affect boarding times significantly. As a consequence, one
should not assume identically distributed bed occupancy times (see Dai and Shi
2017b, Chan et al. 2017 for more discussion). With constraints on specialty,
gender, and accommodation class, there may be tens of patient types and bed
pools, while some pools may have only several beds. Even though there are
hundreds of beds in a hospital, the routing policies that are asymptotically
optimal for large queues may not work well for all patient types and bed
pools. Therefore, we cannot adopt routing policies for call centers to solve
the bed assignment problem. One may refer to Armony et al. (2015) for an
overview of the general patient transfer process to hospital wards.

As a major concern of inpatient bed assignment, how to balance excessive
delays against overflowing has attracted considerable attention from
operations researchers. Using a simulation model calibrated with patient flow
data, Harrison et al. (2005) analyzed bed occupancy rates in a hospital in
Australia, pointing out that both the seasonal variations of patient arrival
rates and the variable patient discharge rates over days contributed a lot to
frequent overflowing in the hospital. Thompson et al. (2009) studied periodic
reallocation of patients across multiple hospital wards to reduce waiting
times for beds. They modeled this problem as a finite-horizon Markov decision
process (MDP), using random sampling to simplify computation. It was reported
that this patient reallocation approach helped a hospital in Connecticut to
cut the average waiting time for beds by half and to increase the hospital’s
revenue by 1%. Transferring patients between wards during their inpatient
stays, however, may add to staff’s workload and interfere with the wards’
daily routines. It may not be a viable option for all hospitals. Samiedaluie
et al. (2017) studied a patient admission problem within a neurology ward.
They also used an MDP to determine whether an incoming patient should be
admitted to the ward and which patient should be sent to each available bed.
They assumed that patients would be transferred to other hospitals if not
admitted to the ward. Patient overflowing across wards is not an option in
their paper.

Using a high-fidelity simulation model, Shi et al. (2016) examined the bed
assignment policy employed by a public hospital in Singapore, where
overflowing was triggered when a patient’s waiting time exceeded a
predetermined threshold. They also analyzed the factors that may cause
prolonged delays and proposed early discharge policies to mitigate the time-
of-day effect on boarding times. By analyzing a simplified queueing model with
two customer classes and two server pools, Kilinc et al. (2016) found some
structural properties of optimal bed assignment policies. They proved that
under such a policy, a patient will be sent to a non-primary bed only when his
waiting time exceeds a threshold that depends on the system’s current state.
Based on their findings, the authors proposed heuristic policies that are
implementable in hospitals.

An independent, concurrent study by Dai and Shi (2017a) is the most relevant
work to our paper. They adopted a two-time-scale model proposed by Shi et al.
(2016) for bed occupancy times, and formulated the bed assignment problem as a
discrete-time, infinite-horizon MDP. Their formulation allows for practical
features such as time-varying patient arrival and discharge patterns, thus
having a high-dimensional state space. To cope with the curse of
dimensionality, they resorted to the technique of approximate dynamic
programming (ADP). Although our study also follows the two-time-scale model to
formulate bed occupancy times, the approach proposed in this paper is
methodologically distinct, not relying on the MDP framework. By solving the
$P$ model problem repeatedly, we provide a computationally efficient approach
that is readily implementable for hospitals with various constraints on their
inpatient management.

We would point out that our dynamic $P$ model approach does not satisfy
Bellman’s principle of optimality. This approach is _myopic_ in nature,
although future bed request rates could be incorporated in the formulation
(through the overflow budget given by (3)). We would like to explore such an
approach because patient arrival patterns in hospitals are highly variable. In
an uncertain environment, a computationally tractable myopic policy may
perform well, because its simple structure allows us to incorporate some
realistic features that are difficult to be included in the MDP framework (see
He et al. 2018 for more discussion). In Section 4, we translate the $P$ model
formulation into a tractable integer linear program, the solution to which can
be quickly obtained using standard solvers.

##  3 A Multi-Class, Multi-Pool Queueing Model

We use a multi-class, multi-pool queueing system to model hospital wards,
which are managed by a centralized bed assignment system to meet requirements
for boarding times without sending excessive patients to non-primary beds.
Based on the state of boarded patients and the availability of beds, the bed
assignment system will make sequential recommendations for incoming patients
and available beds.

In our partner hospital, there is a department in charge of inpatient bed
management, called the Bed Management Unit (BMU). The general bed assignment
process in the hospital is as follows. When a bed request is received by the
BMU, a staff member will first search the wards for a primary bed that matches
the patient’s specialty, gender, and requested accommodation class. If a
primary bed is available, the BMU will assign the patient to this bed
immediately and initiate the patient transfer process. Otherwise, the patient
may have to wait until a primary bed is available. If a patient has been
waiting for a long time or no primary bed is expected to be available within
hours, the BMU may assign the patient to an available non-primary bed.

Figure 3: The multi-class, multi-pool queueing model for hospital wards. Solid
lines represent primary patient-bed pairs and dashed lines represent non-
primary pairs.

Both patients and beds are categorized by specialty, gender, and accommodation
class. In other words, this queueing system has multiple customer types and
multiple server pools. The gender constraint requires that each inpatient room
must accommodate patients of the same gender, i.e., while a single room is
gender-neutral, a room with multiple beds can only accept patients of the same
gender as the current patients. As a result, each bed is labeled as _male
only_ , _female only_ , or _gender-neutral_. With this requirement, some beds
may have more than one type of primary patients, e.g., a single room in the
surgical ward (which has a gender-neutral bed) may accept either a male or
female surgical patient. Therefore, the number of patient types could be
different from the number of bed pools in this queueing model. We allow a bed
pool to have more than one primary patient type, and a patient type to have
more than one primary bed pool. This assumption makes our queueing model
slightly different from the models in Kilinc et al. (2016) and Dai and Shi
(2017a), where each bed pool is assumed to have one and only one primary
patient type.

We assume that bed requests are sent to the BMU in a stochastic manner. The
bed request process may have a time-varying rate. The BMU must process all
requests. In other words, the queueing system has infinite buffers holding all
waiting customers. We illustrate this multi-class, multi-pool queueing model
in Figure 3. We assume that each patient is required to be transferred to an
inpatient bed within a given delay target. To meet this requirement, patient
overflowing is generally inevitable. However, the BMU must also ensure that
most patients are sent to primary beds so that the overall quality of care
will not be compromised. An optimization model should thus be formulated in
order for the hospital to strike a balance between reducing excessive delays
and matching patients with primary beds.

##  4 The $\boldsymbol{P}$ Model Approach

We will design a dynamic bed assignment algorithm driven by patient flow data.
Our intention is to maximize the percentage of patients whose boarding times
do not exceed their respective delay targets, while maintaining the overflow
rate at a reasonable level. We assume that a decision iteration is triggered
at the following event times: (i) when a bed becomes available, (ii) when a
bed request is received and there is at least one available bed for the
patient, (iii) when a patient’s boarding time is about to exceed the delay
target. Each time, the bed assignment algorithm will recommend the next
patient to be assigned to an available bed.

Let $t$ be a moment when the bed assignment algorithm is triggered. We use
$\mathcal{I}$ to denote the set of patients whose bed requests are pending at
time $t$ and $\mathcal{J}$ the set of beds that are either available at time
$t$ or to be available soon (see more discussion later). For the sake of
convenience, the dependence on $t$ is suppressed in notation. We assume that
both $\mathcal{I}$ and $\mathcal{J}$ are nonempty. For $i\in\mathcal{I}$, we
use $\mathcal{J}_{i}\subset\mathcal{J}$ to denote the set of primary and non-
primary beds for patient $i$. Without loss of generality, we assume that
$\mathcal{J}=\bigcup_{i\in\mathcal{I}}\mathcal{J}_{i}$.

For $j\in\mathcal{J}$, let $\tilde{d}_{j}$ be the time when bed $j$ becomes
available. If bed $j$ is available at time $t$, we simply take
$\tilde{d}_{j}=t$. If $\tilde{d}_{j}>t$, i.e., bed $j$ will be available
later, $\tilde{d}_{j}$ is a random variable whose distribution can be
estimated from data, e.g., using the discharge time distribution illustrated
in Figure 2. We assume that $\\{\tilde{d}_{j}:j\in\mathcal{J}\\}$ is a set of
independent random variables. (This assumption can be verified by a standard
$\chi^{2}$ test of independence. It is generally valid for hospitals having
many inpatient wards.) Let $a_{i}$ be patient $i$’s bed request time, with
$a_{i}\leq t$ for all $i\in\mathcal{I}$. Patient $i$ is allowed to be assigned
to bed $j$ only if $j\in\mathcal{J}_{i}$, in which case the boarding time of
patient $i$ will be $\tilde{d}_{j}-a_{i}$. We use $\tau_{i}$ to denote the
delay target of patient $i$, assuming $\tau_{i}>0$ for all $i\in\mathcal{I}$.
We would find a bed assignment plan for the boarded patients to maximize the
joint probability of all of them meeting their respective delay targets.

We use a binary variable $z_{ij}\in\\{0,1\\}$ to indicate the assignment
decision of patient $i$ to bed $j$, i.e., $z_{ij}=1$ if he is assigned to the
bed and $z_{ij}=0$ otherwise. An assignment plan
$\boldsymbol{z}=(z_{ij})_{i\in\mathcal{I},j\in\mathcal{J}}$ is said to be
_admissible_ if every waiting patient is assigned to exactly one bed that is
primary or non-primary for him, i.e.,

$\sum_{j\in\mathcal{J}_{i}}z_{ij}=1\quad\mbox{for }i\in\mathcal{I},$

and each bed is allowed to accommodate at most one patient, i.e.,

$\sum_{i\in\mathcal{I}}z_{ij}\leq 1\quad\mbox{for }j\in\mathcal{J}.$

We use $\mathcal{A}$ to denote the set of all admissible assignment plans. We
assume that $\mathcal{J}$ includes sufficient beds so that $\mathcal{A}$ is
nonempty. Typically, we may take $\mathcal{J}$ to be the set of beds that are
either available now or to be available on the present day. If $\mathcal{A}$
is empty with such a choice of $\mathcal{J}$, we may include more beds that
will be available later (e.g., all beds to be available the next day) to make
$\mathcal{A}$ nonempty. This assumption is reasonable because in general, a
patient’s discharge is scheduled at least one day in advance. Under an
arbitrary assignment plan $\boldsymbol{z}\in\mathcal{A}$, the joint
probability of all waiting patients meeting their delay targets is given by

$\mathbb{P}\big{(}\tilde{d}_{j}-a_{i}\leq\tau_{i}:z_{ij}=1,\ i\in\mathcal{I},\
j\in\mathcal{J}_{i}\big{)}=\prod_{i\in\mathcal{I}}\prod_{j\in\mathcal{J}_{i}}\mathbb{P}\big{(}\tilde{d}_{j}-a_{i}\leq\tau_{i}\big{)}^{z_{ij}},$

where the product form on the right side follows from the independence
assumption of $\\{\tilde{d}_{j}:j\in\mathcal{J}\\}$.

We need to send most patients to primary beds. To this end, a cost is incurred
when a patient is assigned to a non-primary bed. Let $u_{ij}\geq 0$ be the
cost of assigning patient $i\in\mathcal{I}$ to bed $j\in\mathcal{J}_{i}$. We
assume that $u_{ij}=0$ if bed $j$ is a primary bed for patient $i$, and that
$u_{ij}>0$ if bed $j$ is a non-primary bed. To refrain from sending too many
patients to non-primary beds, we add an extra constraint, requiring the total
overflow cost not to exceed an _overflow budget_ $B>0$. Hence, we may solve
the following $P$ model problem to find an optimal assignment plan:

$\begin{array}[]{l@{\quad}l@{}l@{\quad}l@{}}\text{max}&\lx@intercol\displaystyle{\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}_{i}}z_{ij}\ln\mathbb{P}\big{(}\tilde{d}_{j}-a_{i}\leq\tau_{i}\big{)}}\hfil\lx@intercol\\\
\text{s.t.}&\displaystyle{\sum_{j\in\mathcal{J}_{i}}z_{ij}=1},&&i\in\mathcal{I}\\\
&\displaystyle{\sum_{i\in\mathcal{I}}z_{ij}\leq 1},&&j\in\mathcal{J}\\\
&\displaystyle{\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}_{i}}z_{ij}u_{ij}\leq
B},&&\\\ &z_{ij}\in\\{0,1\\},&&i\in\mathcal{I},\ j\in\mathcal{J}.\end{array}$
(1)

In this formulation, patient overflowing is controlled by the overflow budget
$B$. We may take $u_{ij}=1$ if bed $j$ is a non-primary bed for patient $i$.
In this case, the overflow budget turns out to be the maximum number of non-
primary assignments allowed by the current iteration. By adjusting the
overflow budget, the overflow rate of patients can be maintained at a
reasonable level.

A greater overflow budget allows more patients to be assigned to non-primary
beds, thus yielding shorter boarding times. However, a greater overflow budget
may also result in worse quality of care, so $B$ should be kept as small as
possible. In order to get a feasible solution to (1), the minimum overflow
budget required could be obtained by solving the following integer program:

$\begin{array}[]{ll@{\quad}l@{}l@{\quad}l@{}}B^{\star}=&\text{min}&\displaystyle{\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}_{i}}x_{ij}u_{ij}}\\\
&\text{s.t.}&\displaystyle{\sum_{j\in\mathcal{J}_{i}}x_{ij}=1},&&i\in\mathcal{I}\\\
&&\displaystyle{\sum_{i\in\mathcal{I}}x_{ij}\leq 1},&&j\in\mathcal{J}\\\
&&x_{ij}\in\\{0,1\\},&&i\in\mathcal{I},\ j\in\mathcal{J},\end{array}$ (2)

which is an _assignment problem_. This problem is efficiently solvable because
the linear programming (LP) relaxation of that has integral solutions. We may
thus solve the LP relaxation instead (see, e.g., Balinski 1986). More
specifically, in the multi-class, multi-pool queueing model, all compatible
customer-server pairs form a bipartite graph (see Figure 3). Since the node-
edge incidence matrix of a bipartite must be totally unimodular, routing
constraints in the queueing model specify an integral polyhedron in the LP
relaxation (see, e.g., Corollary III.1.2.9 in Nemhauser and Wolsey 1988).
These constraints also appear in the integer linear program (1), the feasible
region of which is nonempty if $B\geq B^{\star}$. Although the feasible region
of (1) may not be an integral polyhedron with the extra constraint on the
overflow cost, this formulation may still have a “nice” structure, allowing us
to quickly solve (1) using existing integer programming solvers. A
computational comparison is made between the $P$ model formulation (1) and the
LP relaxation in Section 6.3. We find out that it is generally efficient to
obtain an optimal solution to the $P$ model formulation.

When a bed becomes available, the bed assignment algorithm determines the next
patient for this bed by solving (1). The following proposition states that
there exists an optimal assignment plan under which this bed will take a
patient whose bed request time is the earliest in his type. For
$i,k\in\mathcal{I}$, we say patients $i$ and $k$ are of _the same type_ if
$\mathcal{J}_{i}=\mathcal{J}_{k}$, $\tau_{i}=\tau_{k}$, and $u_{ij}=u_{kj}$
for all $j\in\mathcal{J}_{i}$. The proof of Proposition 4.1 can be found in
Section 8 in the online appendix.

###### Proposition 4.1

Assume that $B\geq B^{\star}$ in (1), where $B^{\star}$ is the minimum
overflow budget given by (2). For an arbitrary bed $j\in\mathcal{J}$ that is
available to take the next patient, there exists an optimal solution
$\boldsymbol{z}^{\star}$ to (1) that satisfies the following condition: If
patient $k$ is assigned to bed $j$ under $\boldsymbol{z}^{\star}$, then his
bed request time must be the earliest in his type, i.e., $a_{k}\leq a_{\ell}$
for any patient $\ell$ of the same type as patient $k$.

Assume that bed $j$ is available and patient $k$ is assigned to bed $j$
according to an optimal solution to (1). If patient $k$ does not have the
earliest bed request time in his type, we may find the earliest one and send
that patient to bed $j$ instead. As shown in the proof of Proposition 4.1,
switching beds between these two patients will not affect the optimality of
the assignment plan. In this way, we can obtain an optimal assignment plan
that satisfies the condition in Proposition 4.1 based on an arbitrary optimal
solution to (1).

We determine bed assignments for incoming patients by solving (1) repeatedly.
In this process, a decision iteration is triggered when a bed becomes
available, when a bed request arrives with at least one bed being available,
or when a patient’s boarding time is about to exceed the delay target. In
general, patients are not sent to beds on a first-come, first-served (FCFS)
basis, because different patients may have different urgency levels and delay
requirements. However, if we follow an optimal assignment plan that satisfies
the condition in Proposition 4.1 in each iteration, patients belonging to each
type will be sent to beds in the FCFS manner. This feature of the $P$ model
formulation not only guarantees fairness in serving patients of the same type,
but also prevents excessively long waiting times patients might experience
when their delay targets cannot be met.

Even if there are available beds when a decision iteration is triggered, it is
still possible that by an optimal solution to (1), all patients should be
assigned to beds to be available later. This may happen when all patients’
boarding times are far below their delay targets, so it is not necessary to
send anyone to a non-primary bed immediately. Such a phenomenon implies that
our dynamic bed assignment policy is _not_ non-idling. As pointed out by
Kilinc et al. (2016) and Dai and Shi (2017a), it is necessary to idle some
beds from time to time so that the overflow rate can be maintained at a
reasonable level. The practice of strategic idling is also helpful for
improving operational performance in other service systems, e.g., Baron et al.
(2017) studied idling policies in a health examination network.

A decision iteration is triggered when a bed request is received and there is
at least one bed available for the patient. If a primary bed is available, one
may expect the incoming patient to be sent to this bed immediately. However,
the $P$ model approach may not always make such a decision. An optimal
solution to (1) occasionally specifies a non-primary patient for the available
bed, even if the incoming primary patient will be waiting. This may happen
when the non-primary patient’s boarding time is about to exceed the delay
target, in which case it is necessary to assign the patient to a non-primary
bed immediately. Although such a decision is important in preventing a
patient’s boarding time from exceeding the delay target, it will not happen
under the ADP-based policy by Dai and Shi (2017a), who assumed that if a
primary bed is available upon a bed request, the incoming patient will always
be assigned to the primary bed immediately. Their assumption is reasonable
because no delay targets are imposed on their formulation.

To ensure that an optimal solution to (1) exists, we should take $B\geq
B^{\star}$ in each decision iteration. The selection of the overflow budget
will affect the solution to (1), thus having significant influence on the
performance of the bed assignment algorithm. We adjust this parameter
dynamically to mitigate the time-of-day effect. More specifically, if a
decision iteration is triggered at time $t$, we determine the overflow budget
for this iteration by the following formula:

$B=\max\Big{\\{}B^{\star},\
\alpha\cdot|\mathcal{I}|+\beta\int_{t}^{t+\Delta}\lambda(u)\,\mathrm{d}u\Big{\\}},$
(3)

where $B^{\star}$ is the minimum overflow budget given by (2), $\lambda(u)$ is
the patient arrival rate at time $u$, and $\alpha$, $\beta$, and $\Delta$ are
three positive numbers to be determined heuristically. Clearly, the overflow
budget given by (3) satisfies $B\geq B^{\star}$. It takes such a form because
of the following considerations. First, in order to mitigate the time-of-day
effect, we should impose a greater overflow budget when the system is more
congested. The term $\alpha\cdot|\mathcal{I}|$ in (3) allows us to adjust $B$
based on the number of waiting patients. Second, if bed requests are expected
to surge in the next few hours, sending more patients to non-primary beds in
advance would be helpful in preventing congestion. Hence, we also adjust the
overflow budget using an estimate of the number of bed requests in the next
$\Delta$ hours, which is the integral term in (3). When all $u_{ij}$’s are
integer-valued, so is $B^{\star}$. The problem (1) can be solved in a more
efficient way by integer programming solvers if $B$ is also an integer. In
this case, we would modify (3) into

$B=\max\Big{\\{}B^{\star},\
\Big{\lfloor}\alpha\cdot|\mathcal{I}|+\beta\int_{t}^{t+\Delta}\lambda(u)\,\mathrm{d}u\Big{\rfloor}\Big{\\}}$
(4)

for the convenience of computation.

##  5 Simulation Model

In this section, we introduce the simulation model for inpatient operations in
our partner hospital. This simulation model is primarily based on the
stochastic network model proposed by Shi et al. (2016), but including several
features absent from their model, such as gender and accommodation class
constraints. This simulation platform is populated with parameters estimated
from patient flow records, allowing us to faithfully reproduce patient flow
dynamics in this hospital. We replicate important performance measures from
the data set to validate this model, including ED patients’ mean waiting
times, service levels, and overflow rates. This simulation model will be used
for assessing the proposed bed assignment approach in Section 6.

###  5.1 Categorization of Patients and Beds

The hospital provided us with their patient flow records of April–September
2015. On average, the hospital wards admitted about 134 patients per day from
four different sources: 81.4% of inpatient admissions were requested from the
ED, 8.4% were same-day-admissions (SDA) requested for patients having day
surgery, 5.2% were elective (EL) admissions scheduled in advance, and 5.0%
were requested from specialist outpatient clinics (SOC). The hourly bed
request rates from the four sources are plotted in Figure 5, where those from
ED and SOC patients are time-varying throughout the day and almost all bed
requests from EL and SDA patients were sent to the BMU around 7 pm. According
to the hospital’s weekly schedule, there are no bed requests from EL patients
on Friday and no bed requests from SDA patients on Saturday. In the simulation
model, we assume that bed requests from ED and SOC patients follow two
periodic, non-homogeneous Poisson processes with the period being one day,
while bed requests from EL and SDA patients are received at 7 pm, except on
Friday and Saturday, respectively.

Figure 4: Hourly bed request rates from four sources.

Figure 5: Bed request proportions of six specialties.

All inpatient beds in the hospital are broadly categorized into six
specialties, i.e., cardiology, gastroenterology, geriatrics, general medicine,
respiratory medicine, and surgery. The proportions of bed requests from these
specialties are illustrated in Figure 5, where the fractions from different
sources are also specified. We differentiate ED patients who were admitted
before and after 12 pm, because these two groups of patients have different
length-of-stay distributions. This phenomenon was discussed by Shi et al.
(2016), who speculated that ED patients admitted in the morning could undergo
some diagnostic tests in the afternoon, whereas those admitted later may have
to wait until the next day. A fraction of patients would thus spend one more
day in the hospital if admitted after 12 pm. We report the mean lengths of
stay and the corresponding standard deviations of the six specialties from
different patient sources in Table 3 in Section 9. The mean lengths of stay of
ED patients admitted in the morning are shorter in all specialties.

In this hospital, inpatient rooms are categorized into four accommodation
classes, A1, B1, B2, and C, having one, four, six, and eight beds,
respectively. The percentages of bed requests in the six specialties for these
four classes are reported in Table 4 in Section 9. The beds of classes B2 and
C are subsidized, and thus demand for these two classes is much higher than
the other two classes. As we mentioned earlier in Section 3, each room must
accommodate patients of the same gender. While A1 beds are gender-neutral, all
other beds can only accommodate patients of a particular gender. The gender
distributions in the six specialties are reported in Table 5 in Section 9. We
would emphasize that patient overflowing refers to sending patients not only
to wards in other specialties, but also to beds belonging to the requested
specialties but in other accommodation classes.

According to the patient flow records, the number of beds in several wards
varied across the period of interest owing to renovation or maintenance,
opening of new wards, etc. For the convenience of implementation, we assume
that the number of beds is fixed in each ward in simulation. To match the
performance statistics of the data, we have slightly adjusted the number of
beds in a few wards. There are totally 571 inpatient beds in the simulation
model, divided into 34 pools by specialty, gender, and accommodation class.
Please refer to Table 6 in Section 9 for the indexes and capacities of these
bed pools. All patients are categorized into 50 types accordingly. The primary
and non-primary bed pools for male and female patients are specified in Tables
7 and 8 in Section 9, respectively, according to the hospital’s guidelines. In
these two tables, both _preferred_ and _secondary_ pools refer to non-primary
bed pools for a particular type of patients. When a patient has to be sent to
a non-primary bed, the BMU will try their best to find the patient a preferred
bed because the conditions are more similar to a primary bed. Only if a
preferred bed is not available, would the BMU assign the patient to a
secondary bed. The overflow rate refers to the percentage of patients sent to
either preferred or secondary beds.

###  5.2 Bed Occupancy Times and Pre- and Post-Allocation Delays

Shi et al. (2016) pointed out that a patient’s bed occupancy time generally
follows a two-time-scale model, i.e.,

$T=L-A+D,$ (5)

where $L$ is the length of stay or number of nights the patient stays at the
ward, $A$ is the time on the admission day when the patient is admitted to the
bed, and $D$ is the time on the discharge day when the bed is vacated.
Therefore, if measured in days, $L$ must be an integer, $0\leq A<1$, and
$0\leq D<1$. Although it is reasonable to assume that both lengths of stay and
discharge times are independent and identically distributed (i.i.d.) for each
type of patients, bed occupancy times are not i.i.d. because they depend on
bed admission times. In the simulation model, we assume that the lengths of
stay of patients of each type follow the empirical distribution obtained from
the data, and that the discharge times of all patients follow the distribution
specified in Figure 2.

Aside from the availability of beds, the bed assignment and patient transfer
process may also be affected by secondary resources such as BMU staff and
nurses in the ED and wards. To faithfully reproduce patient flow dynamics, we
add pre- and post-allocation delays to patients’ waiting times, following the
simulation model by Shi et al. (2016). From the instant when a bed request is
received till the point when the patient is assigned to an available bed,
several activities can be performed by the BMU staff, such as searching
through the bed management system, batching the bed requests for processing
later, liaising with ward staff and patients whenever operational needs arise.
A patient’s pre-allocation delay is the duration of this process. Once the
patient is assigned to an available bed, the BMU will initiate the transfer
process that entails handing over the patient’s records to ward staff and
moving the patient from the ED to the ward. A patient’s post-allocation delay
is the duration of the transfer process. Please refer to Section 3.4 in Shi et
al. (2016) for more details. We illustrate the mean pre- and post-allocation
delays from the records of ED and SOC patients in Figure 14 in Section 9,
along with the corresponding standard deviations. In the simulation model, we
assume that pre- and post-allocation delays follow log-normal distributions
with time-varying means and standard deviations specified in Figure 14, as Shi
et al. (2016) did in their paper.

###  5.3 The Hospital’s Bed Assignment Policy

To replicate performance statistics from the patient flow data, we must
faithfully represent the hospital’s bed assignment practice in the simulation
model. Our partner hospital is relatively conservative toward patient
overflowing, always trying their best to minimize overflow rates. The
following guidelines are recommended for their inpatient bed management.

When a bed request is received and a primary bed is found available, the BMU
should assign the patient to the primary bed immediately. If there is no
primary bed, the BMU may either assign the patient to a primary bed to be
available later, or put the bed request on hold until a primary bed is
available or overflowing is triggered. We adopt the method proposed by Shi et
al. (2016) to simulate the latter cases: If a bed request is raised at time
$t$ of a day and no primary bed is available, then with probability $p(t)$,
the patient will be assigned to a primary bed to be available later, and with
probability $1-p(t)$, the bed request will be put on hold. In simulation, we
take

$p(t)=\begin{cases}0&\mbox{for $0\leq t<15$,}\\\ 0.7&\mbox{for $15\leq
t<21$,}\\\ 0&\mbox{for $21\leq t<24$}\end{cases}$

for all days, where $t$ is the time of the day measured in hours. This formula
can be roughly verified by the data and interpreted according to the BMU’s
practice. From 3 pm to 9 pm, some patients are assigned to beds to be
available later, because most beds vacated on the present day have been taken
by waiting patients by 3 pm. The BMU would thus reserve some beds for incoming
patients to control the overflow rate.

When a bed becomes available and a patient has been assigned to this bed, the
BMU will initiate the patient transfer process immediately. If no patient has
been assigned to this bed and there are primary patients on hold, the BMU will
generally follow the FCFS rule, assigning the primary patient who has the
earliest bed request time to this bed. If no patient has been assigned to the
bed and there is no primary patient waiting, the BMU will leave the bed vacant
until a primary patient’s bed request is received or patient overflowing is
triggered.

The BMU begins to conduct patient overflowing at 5 pm each day. Bed requests
received before 3 pm and still on hold will be processed in the FCFS manner.
If a non-primary bed is available, a patient will be sent there immediately.
If the BMU cannot find a bed for a patient, the bed request will be on hold
again. Since primary beds are unlikely to be available later, patient
overflowing will be triggered repeatedly at the beginning of each hour after 5
pm, until all patients who requested beds before 3 pm are assigned to beds.
While bed requests received after 3 pm are eligible for overflowing on the
next day, most of them will be assigned to primary beds before the next day’s
overflowing is triggered.

###  5.4 Model Validation

(a) Mean waiting times

(b) 8-hour service level

(c) 12-hour service level

(d) 16-hour service level

Figure 6: Performance comparison between statistics from the data and
simulation results.

To validate the simulation model, we compare some performance estimates
obtained by simulation with corresponding statistics from the patient flow
records. Using a software platform coded in Python, we simulate the hospital’s
inpatient operations for 1000 days, excluding the first 100 days as the burn-
in period. The mean waiting times and several service levels of ED patients
across the day are plotted in Figure 6. Here, a patient’s waiting time refers
to the duration from the bed request time to the bed admission time, including
both pre- and post-allocation delays, and the $n$-hour service level is the
percentage of patients whose waiting times exceed $n$ hours. This figure
clearly shows that our simulation model can generally capture the hospital’s
performance in terms of these metrics. The mean waiting times from the ED to
various bed pools are reported in Table 9 in Section 9, where the simulation
results generally agree with the results from the data for large pools (e.g.,
the pools of B2 and C beds in medicine and surgery).

We compare overflow rates in different specialties in Figure 8. The overflow
rates produced by simulation are generally close to those from the data,
except in geriatrics and respiratory medicine. These two specialties are
relatively small, having 24 and 43 beds, respectively. We would admit that our
simulation model are unable to represent all bed assignment decisions in
practice. For example, the BMU staff may occasionally send patients in a
serious condition to non-primary beds before 5 pm, to ensure that the patients
can receive inpatient care as soon as possible. Such special cases are not
captured in our simulation model. As a result, we may slightly underestimate
the overflow rate by simulation. The overflow rate of all ED patients from the
data is 12.25%, while that produced by simulation is 10.88%.

Figure 7: Overflow rates in different specialties.

Figure 8: Early discharge time distributions.

##  6 Numerical Assessment of the $\boldsymbol{P}$ Model Approach

We conduct a numerical study in this section to assess the proposed bed
assignment approach, using the simulation model introduced in Section 5 as the
platform. We consider three performance measures, including the mean waiting
times and 12-hour service levels of ED patients and the overflow rates of all
patients, for comparison between the proposed approach and several bed
assignment approaches commonly used in practice. Although we consider the
first two performance measures only for ED patients, we do not assume that
these patients have priority over patients from other sources in the bed
assignment process. In our simulation, all bed requests are scheduled by the
same assignment policy.

We repeat the simulation process described in Section 5.4 with different bed
assignment approaches. In particular, the $P$ model approach is implemented by
calling Gurobi Optimizer 6.5.2. In this approach, a decision iteration is
triggered when a bed is available, when a bed request is received and there is
at least one available bed for the patient, or when a patient’s boarding time
is about to exceed the delay target. The overflow cost is taken to be
$u_{ij}=1$ if bed $j$ is a non-primary bed for patient $i$, so the overflow
budget turns out to be the maximum number of non-primary assignments allowed
in an iteration. We first solve (2) for the minimum overflow budget, next
follow (4) to obtain the overflow budget with some $\alpha$, $\beta$, and
$\Delta$, and then solve (1) for an optimal assignment plan. This optimal
assignment plan is used to determine the next patient for the available bed.
If there is a patient whose boarding time has already exceeded the delay
target, we will use the assignment plan obtained by solving (2) to determine
the next patient for the available bed. We include pre- and post-allocation
delays in simulation for all bed assignment approaches, in order for the
comparison to be fair. Therefore, a patient’s waiting time is the sum of the
boarding time and the pre- and post-allocation delays. Since we would reduce
the 12-hour service level, delay targets for all patients are set to be 10
hours in the $P$ model approach. We use $P(\alpha,\beta,\Delta)$ to denote the
$P$ model policy with parameters $\alpha$, $\beta$, and $\Delta$ in (4). Then
for $\Delta\geq 0$, $P(0,0,\Delta)$ refers to the $P$ model policy using
minimum overflow budgets in all iterations.

###  6.1 Comparison with the Hospital’s Current Practice and Early Discharge
Policies

(a) Mean waiting times

(b) 12-hour service levels

Figure 9: Performance comparison between early discharge policies and the
$\boldsymbol{P}$ model approach.

In the first numerical example, we compare the $P$ model approach with the
hospital’s current practice as well as two early discharge policies. More
specifically, the following bed assignment policies are considered.

The BMU’s policy

: This policy is designed according to the hospital’s current guidelines (see
Section 5.3), denoted by BMU in figures and tables.

The BMU’s policy with the first early discharge distribution

: To mitigate the time-of-day effect, Shi et al. (2016) proposed a
hypothetical policy under which 26% of patients are discharged before noon and
the first discharge peak time appears between 8 am and 9 am (see Figure 2b in
their paper). They demonstrated that if the distributions of pre- and post-
allocation delays are time-invariant, this early discharge policy is capable
of equalizing waiting times for ED patients in their simulation model. We
consider a similar discharge time distribution, allowing 26% of patients to be
discharged before noon and the first discharge peak time to appear before 9 am
(see Figure 8). We use Early-1 to denote the BMU’s bed assignment policy with
this early discharge time distribution.

The BMU’s policy with the second early discharge distribution

: We follow the BMU’s bed assignment policy, but discharging patients even
more aggressively in the morning. We assume that 55% of patients are
discharged before noon and 26% of patients are discharged before 9 am (see
Figure 8). This policy is denoted by Early-2 in figures and tables.

We plot the mean waiting times and 12-hour service levels for the above bed
assignment policies in Figure 9. The waiting times of ED patients are
shortened if a significant portion of patients are discharged in the morning.
A more aggressive early discharge policy can further reduce patients’ waiting
times. However, it turns out that under the BMU’s current practice,
discharging patients several hours earlier does not suffice to synchronize the
patient discharge process with the bed request process. Even if more than one
half of patients are discharged before noon, the reduction in the mean waiting
time is merely around one hour for patients who request beds at night. Many of
them may still wait more than eight hours for beds. As we discussed in Section
1, it is generally difficult to discharge a significant portion of patients
before noon, because physicians and nurses are busy with ward rounds in the
morning. For many hospitals, an aggressive early discharge policy is not a
viable option.

By maximizing the joint probability of all patients meeting their delay
targets in each iteration, the $P$ model approach can reduce waiting times
significantly. Even if the overflow budget is kept minimum in all iterations
(i.e., $\alpha=\beta=0$), the mean waiting times can still be reduced up to
6.4 hours at night. As a result, the time-of-day effect on ED boarding is
greatly mitigated. If a larger overflow budget is allowed, we may further
equalize the waiting times throughout the day. We also test the $P$ model
approach with $\alpha=0.15$ and $\beta=0$. In this case, we may send up to 15%
of waiting patients to non-primary beds when $B^{\star}$ is small. This
adjustment allows us to further reduce waiting times in the early morning at
the expense of an increased overflow rate. By setting $\alpha=0.15$,
$\beta=0.02$, and $\Delta=4$ hours, we may further improve the delay
performance of the $P$ model approach. In this case, we also adjust the
overflow budget according to the bed request rate in the next four hours.

Table 1: Overflow rates under various bed assignment policies. | BMU | Early-1 | Early-2 | $P(0,0,0)$ | $P(.15,0,0)$ | $P(.15,.02,4)$ | TB-1 | TB-2
---|---|---|---|---|---|---|---|---
Overflow rate | 10.89% | 10.76% | 10.55% | 13.12% | 14.11% | 15.52% | 19.50% | 15.96%

Figure 10: Overflow rates in different specialties under various bed
assignment policies.

Figure 11: The mean waiting times of the $\boldsymbol{P}$ model formulation
with a matched overflow rate.

We report the total overflow rates under these bed assignment policies in
Table 1 and illustrate the overflow rates in each specialties in Figure 11.
The total overflow rate is slightly decreased under the early discharge
policies, since more primary beds would be available before patient
overflowing is triggered. As we discussed in Section 5.3, the hospital’s
current bed assignment guidelines are generally conservative toward patient
overflowing. In this table, the overflow rate under the $P$ model approach is
noticeably larger than that under the BMU’s policy, even though the overflow
budget takes the minimum values in all iterations. This implies that the BMU’s
policy would be too conservative if we take 10 hours as the delay target
(excluding pre- and post-allocation delays). Since more patients are sent to
non-primary beds by the $P$ model approach, one may raise the question whether
it is fair to compare its performance with that of the BMU’s policy. To
address this concern, we increase the delay target to $\tau_{i}=17$ hours for
all patients and take $\alpha=\beta=0$ in (4). With these parameters, the $P$
model approach yields an overflow rate of 10.67%, comparable to the rate of
10.89% by the BMU’s policy. We plot the mean waiting times under this policy
in Figure 11. Although the $P$ model approach cannot mitigate the time-of-day
effect on boarding times with such a large delay target, the improvement in
waiting times is apparent compared with the performance of the BMU’s policy.
For the convenience of illustration, we re-plot the mean waiting times under
the two early discharge policies in Figure 11, where the $P$ model approach
outperforms both of them with a comparable overflow rate. One may further
improve the performance of the $P$ model approach by adjusting the parameters,
while maintaining the overflow rate at a similar level. We would omit such an
experiment since it is not our focus.

Figure 12: Hourly overflow rates under the three $\boldsymbol{P}$ model
policies.

We plot hourly overflow rates under the three $P$ model policies in Figure 12.
The hourly overflow rate under the $P$ model approach with $\alpha=\beta=0$
increases with time throughout the day. This is because in our implementation,
we assume that the set $\mathcal{J}$ includes the beds that are either
available at the present time or to be available on the present day. At the
beginning of a day, all beds to be available on the present day are included
in $\mathcal{J}$. A bed request received in the early morning is thus more
likely to be assigned to a primary bed that will be available later. As a
result, the overflow rate in the early morning would be relatively low
compared with later hours on the same day, while waiting times in the early
morning would be relatively long. We may mitigate this issue using larger
$\alpha$ and $\beta$, which allows us to send more patients to non-primary
beds in the morning, thus equalizing boarding times across the day.

###  6.2 Comparison with Threshold-Based Overflowing Policies

In many hospitals, patient overflowing is triggered on an individual basis
when a patient’s boarding time exceeds a threshold. We refer to such a policy
as a _threshold-based overflowing policy_. Simple threshold-based policies are
widely used in practice (Shi et al. 2016, Kilinc et al. 2016). In the second
numerical example, we compare the performance of the $P$ model approach with
the following two threshold-based policies.

The first threshold-based policy

: Since patients requesting beds at late night or in the early morning are
likely to experience excessive delays, we should send them to non-primary beds
sooner than others. In the first policy, the threshold time for patients
requesting beds from 10 pm to 3 am is taken to be two hours, while that for
other patients is ten hours. Patients cannot be sent to non-primary beds until
overflowing is triggered. When a bed becomes available, the eligible primary
or non-primary patient who has the earliest bed request time will be assigned
to this bed. We use TB-1 to denote this policy.

The second threshold-based policy

: In the second policy, the threshold time for patients requesting beds from 7
pm to 12 am is taken to be two hours, while that for other patients is ten
hours. The second policy is denoted by TB-2.

(a) Mean waiting times

(b) 12-hour service levels

Figure 13: Performance comparison between a threshold-based overflowing policy
and the $\boldsymbol{P}$ model approach.

We plot the mean waiting times and 12-hour service levels for these two
threshold-based policies in Figure 13. The performance of the first policy is
comparable to that of the $P$ model policies. However, the overflow rate
reaches 19.50% under the first threshold-based policy (see Table 1), much
larger than the overflow rates under the $P$ model policies. We may adjust
threshold times to reduce the overflow rate. Under the second threshold-based
policy, the overflow rate is 15.96%, comparable to that under the $P$ model
approach with $\alpha=0.15$, $\beta=0.02$, and $\Delta=4$ hours. However, the
waiting time performance under the second policy is not as good as that under
the $P$ model policies. Therefore, by exploiting information about beds to be
available later, the $P$ model approach outperforms threshold-based
overflowing policies that are widely used in practice.

###  6.3 Computational Performance

As we discussed in Section 4, the $P$ model approach is computationally
tractable because the feasible region of the integer linear program (1) is
sometimes an integral polyhedron. Even if it is not integral, the $P$ model
formulation may still have a “nice” structure, allowing us to quickly solve
(1) by a relaxation-based method (such as the cutting plane method) which is
embedded in standard integer programming solvers. Although it is beyond our
scope to analyze the computational complexity of (1), we would demonstrate the
computational performance of the $P$ model approach through numerical
experiments.

We repeat the simulation process with the $P$ model policies and record
computation times for solving the $P$ model formulation in all decision
iterations. For the sake of comparison, we also solve the LP relaxation of (1)
in each iteration using the same solver and record the computation time. We
report the mean and maximum computation times for solving the $P$ model and
the LP relaxation in Table 2, along with the percentage of iterations in which
the LP relaxation has integral solutions. In all three cases, the mean
computation time for solving the $P$ model is just slightly greater than that
for solving the LP relaxation, and the maximum computation time for solving
the $P$ model is not significantly greater than that for solving the LP
relaxation. If the LP relaxation has an integral solution, it must also be an
optimal solution to the $P$ model. For these three $P$ model policies, the LP
relaxation has integral solutions in around one half of iterations. In
general, a larger overflow budget allows the LP relaxation to have integral
solutions in more iterations.

Table 2: Computation times (in seconds) for solving the $\boldsymbol{P}$ model and the LP relaxation. |  $P$ model |  | LP relaxation
---|---|---|---
| Mean | Max |  | Mean | Max | Integral solutions
$P(0,0,0)$ | 0.034 | 1.500 |  | 0.033 | 0.953 | 47.13%
$P(0.15,0,0)$ | 0.030 | 2.250 |  | 0.029 | 1.422 | 52.00%
$P(0.15,0.02,4)$ | 0.024 | 0.266 |  | 0.023 | 0.250 | 59.22%

##  7 Concluding Remarks

We proposed a data-driven bed assignment approach to balancing boarding and
overflowing in hospital wards. The objective is maximizing the percentage of
patients whose boarding times are within mandatory targets, without sending
excessive patients to non-primary beds. Allowing for critical features of
patient flow management, this $P$ model approach is computationally tractable
and easy to implement using existing integer programming solvers. We expect to
see more applications of the $P$ model in the management of large-scale
stochastic systems with time-sensitive service requirements.

A data-driven bed assignment platform based on the $P$ model approach is
currently in development by our partner hospital. We may study several more
practical issues about inpatient operations using the $P$ model formulation
and incorporate more features into the hospital’s platform. For example, how
to improve fairness in bed assignment across multiple wards (Mandelbaum et al.
2012)? How to schedule elective patients to reduce boarding times in the ED
(Helm and Van Oyen 2014)? How to determine ward partitioning to reduce
overflow rates (Best et al. 2015)? We will provide practicable solutions using
the $P$ model approach in the future.

The authors would like to thank Melvyn Sim for the constructive discussion
that initiated this paper.

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Appendix

##  8 Proof of Proposition 4.1

###### Proof 8.1

Proof of Proposition 4.1. Let $t$ be the present time. Then,
$\tilde{d}_{j}=t$. Since $B\geq B^{\star}$, the feasible region of (1) is
nonempty and an optimal solution exits. Suppose that there is an optimal
solution $\hat{\boldsymbol{z}}$ to (1) under which patient $k$ is assigned to
bed $j$ but does not have the earliest bed request time among the patients of
the same type. In other words, with $\hat{z}_{kj}=1$, there exists
$\ell\in\mathcal{I}$ such that $a_{\ell}<a_{k}$ where

$a_{\ell}=\min\\{a_{i}:i\in\mathcal{I},\ \mathcal{J}_{i}=\mathcal{J}_{k},\
\tau_{i}=\tau_{k},\ u_{im}=u_{km}\mbox{ for }m\in\mathcal{J}_{k}\\}.$

Let $h\in\mathcal{J}$ be the bed to which patient $\ell$ is assigned under
$\hat{\boldsymbol{z}}$, i.e., $\hat{z}_{\ell h}=1$. Then, $\tilde{d}_{h}\geq
t$ since bed $h$ may or may not be available at time $t$. We modify
$\hat{\boldsymbol{z}}$ by assigning patient $k$ to bed $h$ and patient $\ell$
to bed $j$. Let $\boldsymbol{z}^{\star}$ be the resulting assignment plan.
Then,

$z^{\star}_{im}=\begin{cases}1&\mbox{if $(i,m)=(k,h)$ or $(i,m)=(\ell,j)$,}\\\
\hat{z}_{im}&\mbox{otherwise.}\end{cases}$

Clearly, $\boldsymbol{z}^{\star}$ satisfies all constraints in (1). Consider
the objective function in (1), given by

$V(\boldsymbol{z})=\sum_{i\in\mathcal{I}}\sum_{m\in\mathcal{J}_{i}}z_{im}\ln\mathbb{P}\big{(}\tilde{d}_{m}-a_{i}\leq\tau_{i}\big{)}.$

Note that $\tilde{d}_{j}=t\leq\tilde{d}_{h}$. If $t-a_{\ell}>\tau_{\ell}$, we
have
$\mathbb{P}\big{(}\tilde{d}_{j}-a_{\ell}\leq\tau_{\ell}\big{)}=\mathbb{P}\big{(}\tilde{d}_{h}-a_{\ell}\leq\tau_{\ell}\big{)}=0$.
In this case, $V(\boldsymbol{z}^{\star})=V(\hat{\boldsymbol{z}})=-\infty$. If
$t-a_{\ell}\leq\tau_{\ell}$ and $V(\hat{\boldsymbol{z}})>-\infty$, we have

$\displaystyle V(\boldsymbol{z}^{\star})-V(\hat{\boldsymbol{z}})$
$\displaystyle=\ln\mathbb{P}\big{(}\tilde{d}_{j}-a_{\ell}\leq\tau_{\ell}\big{)}+\ln\mathbb{P}\big{(}\tilde{d}_{h}-a_{k}\leq\tau_{k}\big{)}-\ln\mathbb{P}\big{(}\tilde{d}_{j}-a_{k}\leq\tau_{k}\big{)}-\ln\mathbb{P}\big{(}\tilde{d}_{h}-a_{\ell}\leq\tau_{\ell}\big{)}$
$\displaystyle=\ln\mathbb{P}\big{(}\tilde{d}_{h}-a_{k}\leq\tau_{k}\big{)}-\ln\mathbb{P}\big{(}\tilde{d}_{h}-a_{\ell}\leq\tau_{\ell}\big{)}$
$\displaystyle\geq 0.$

Since $\hat{\boldsymbol{z}}$ is an optimal solution to (1), we must have
$V(\boldsymbol{z}^{\star})=V(\hat{\boldsymbol{z}})$ and
$\boldsymbol{z}^{\star}$ is also optimal. This implies that by switching the
beds for patients $k$ and $\ell$, we can obtain an optimal assignment plan
$\boldsymbol{z}^{\star}$ that satisfies the condition specified in the
proposition.

##  9 More on the Simulation Model

More details about the simulation model is provided in this appendix.

In Table 3, we report the mean lengths of stay and the corresponding standard
deviations of the six specialties from different patient sources. From this
table, we can tell that in all six specialties, patients admitted in the
morning have shorter mean lengths of stay than those admitted after 12 pm.

We report the distributions of bed requests for the four accommodation classes
in the six specialties in Table 4. Beds of classes B2 and C have much higher
demand than the other two classes because they are subsidized by the
government.

The gender distributions of patients in the six specialties are reported in
Table 5. There are more male patients than female patients in all specialties
except for geriatrics.

We list the indexes of the 34 bed pools in Table 6, along with the numbers of
beds belonging to these pools. In the surgical wards of this hospital, some B2
beds are reserved for EL and SDA patients. These beds are denoted by “Surg.
(EL)” in Table 6. The hospital also reserves some C beds as two common
(secondary) pools for male and female patients in all specialties requesting
B2 and C beds. We create a new “mixed” specialty in Table 6 for these beds.

We list the primary and non-primary bed pools for male and female patients in
Tables 7 and 8, respectively, according to the hospital’s guidelines. The BMU
will try to send patients to preferred beds when primary beds are not
available. Only when a preferred bed is not available, may the BMU send the
patient to a secondary bed.

We plot the means and standard deviations of pre- and post-allocation delays
across a day obtained from the patient flow data in Figure 14. In the
simulation model, we assume that these delays follow log-normal distributions
with time-varying means and standard deviations specified in this figure.

In Table 9, we compare the mean waiting times from the ED to various bed pools
obtained by simulation with those from the patient flow records. The
simulation results generally agree with the results from the data for large
bed pools (e.g., the pools of B2 and C beds in medicine and surgery). However,
it is difficult for our simulation model to replicate mean waiting times for
small bed pools. This is because the sample sizes of data about such pools are
also small.

Table 3: Mean lengths of stay (in days) in the six specialties from various sources, with corresponding standard deviations in the parentheses. A superscript $\boldsymbol{*}$ denotes a sample of less than 5 admissions. | ED(am) | ED(pm) | SDA | EL | SOC
---|---|---|---|---|---
Cardio. | 2.56 (3.23) | 3.10 (3.83) | 1.23 (1.16) | 2.21 (1.62) | 3.68 (4.79)
Gastro. | 4.55 (4.37) | 4.70 (4.37) | NA | NA | NA
Geria. | 8.40 (9.29) | 8.90 (10.02) | NA | 19.30 (11.93)* | 11.86 (15.28)
Med. | 5.74 (6.87) | 6.16 (7.37) | 2.60 (2.60)* | 2.50 (2.28) | 5.74 (5.84)
Respir. | 3.87 (4.04) | 4.49 (4.68) | 1.33 (1.53)* | 1.33 (0.77) | 4.30 (3.02)
Surg. | 4.45 (5.44) | 5.03 (5.90) | 2.11 (2.42) | 2.71 (3.09) | 6.12 (7.07)
Table 4: The percentages of requested accommodation classes in the six specialties. | Cardio. | Gastro. | Geria. | Med. | Respir. | Surg.
---|---|---|---|---|---|---
A1 |  06% |  07% |  04% |  05% |  04% |  09%
B1 |  08% |  09% |  03% |  07% |  08% |  08%
B2 | 36% | 39% | 32% | 32% | 36% | 39%
C | 50% | 45% | 61% | 56% | 52% | 44%
Table 5: Gender distributions in the six specialties. | Cardio. | Gastro. | Geria. | Med. | Respir. | Surg.
---|---|---|---|---|---|---
Male | 64% | 55% | 34% | 51% | 63% | 60%
Female | 36% | 45% | 66% | 49% | 37% | 40%

(a) Pre-allocation delays

(b) Post-allocation delays

Figure 14: The means and standard deviations of pre- and post-allocation delays. Table 6: The indexes of 34 bed pools in the hospital, with corresponding bed numbers in the parentheses. | Cardio. | Gastro. | Geria. | Respir. | Med. | Surg. | Surg. (EL) | Mixed
---|---|---|---|---|---|---|---|---
B1 Male | 1 (4) | NA | NA | NA | 9 (12) | 12 (18) | NA | NA
B2 Male | 2 (24) | 3 (6) | 5 (4) | 7 (10) | 10 (26) | 13 (36) | 15 (14) | NA
C Male | NA | 4 (8) | 6 (4) | 8 (12) | 11 (47) | 14 (42) | NA | 16 (19)
B1 Female | 17 (4) | NA | NA | NA | 25 (8) | 28 (16) | NA | NA
B2 Female | 18 (12) | 19 (6) | 21 (8) | 23 (12) | 26 (24) | 29 (20) | 31 (20) | NA
C Female | NA | 20 (8) | 22 (8) | 24 (9) | 27 (38) | 30 (36) | NA | 32 (16)
A1 | NA | NA | NA | NA | 33 (20) | 34 (20) | NA | NA
Table 7: Twenty-five types of male patients and their respective primary, preferred, and secondary bed pools. | Primary |  | Preferred |  | Secondary
---|---|---|---|---|---
Cardio. A1 | 33 |  | NA |  | 1, 9, 34
Cardio. B1 | 1 |  | NA |  | 9, 33, 34
Cardio. B2 | 2 |  | NA |  | 3, 4, 7, 8, 10, 11, 13, 16
Cardio. C | 2 |  | NA |  | 3, 4, 7, 8, 10, 11, 14, 16
Gastro. A1 | 33 |  | 9 |  | 1, 12
Gastro. B1 | 9 |  | 33 |  | 12
Gastro. B2 | 3, 7, 10 |  | 4, 8, 11 |  | 13, 16
Gastro. C | 4, 8, 11 |  | 3, 7, 10 |  | 14, 16
Geria. A1 | 33 |  | NA |  | 9, 34, 12
Geria. B1 | 9 |  | NA |  | 1, 12, 33
Geria. B2 | 5 |  | 6 |  | 2, 3, 4 ,7, 8, 10, 11, 16
Geria. C | 6 |  | 5 |  | 2, 3, 4, 7, 8, 10, 11, 16
Respir. A1 | 33 |  | 9 |  | 34
Respir. B1 | 9 |  | 33 |  | 1, 12
Respir. B2 | 3, 7, 10 |  | 4, 8, 11 |  | 2, 16
Respir. C | 4, 8, 11 |  | 3, 7, 10 |  | 14, 16
Med. A1 | 33 |  | 9, 12 |  | 34
Med. B1 | 9 |  | 33 |  | 1, 12
Med. B2 | 3, 7, 10 |  | 4, 8, 11 |  | 13, 14, 16
Med. C | 4, 8, 11 |  | 3, 7, 10 |  | 13, 14, 16
Surg. A1 | 34 |  | 12 |  | 9
Surg. B1 | 12 |  | 34 |  | 9
Surg. B2 | 13 |  | 14, 15 |  | 10, 11, 16
Surg. C | 14 |  | 13, 15 |  | 10, 11, 16
Surg. B2 (EL) | 15 |  | 14 |  | 10, 11, 16
Table 8: Twenty-five types of female patients and their respective primary, preferred, and secondary bed pools. | Primary |  | Preferred |  | Secondary
---|---|---|---|---|---
Cardio. A1 | 33 |  | NA |  | 17, 25, 34
Cardio. B1 | 17 |  | NA |  | 25, 33, 34
Cardio. B2 | 18 |  | NA |  | 19, 20, 23, 24, 26, 27, 29, 32
Cardio. C | 18 |  | NA |  | 19, 20, 23, 24, 26, 27, 30, 32
Gastro. A1 | 33 |  | 25 |  | 17, 28
Gastro. B1 | 25 |  | 33 |  | 28
Gastro. B2 | 19, 23, 26 |  | 20, 24, 27 |  | 29, 32
Gastro. C | 20, 24, 27 |  | 19, 23, 26 |  | 30, 32
Geria. A1 | 33 |  | NA |  | 25, 34, 28
Geria. B1 | 25 |  | NA |  | 17, 28, 33
Geria. B2 | 21 |  | 22 |  | 18, 19, 20, 23, 24, 26, 27, 32
Geria. C | 22 |  | 21 |  | 18, 19, 20, 23, 24, 26, 27, 32
Respir. A1 | 33 |  | 25 |  | 34
Respir. B1 | 25 |  | 33 |  | 17, 28
Respir. B2 | 19, 23, 26 |  | 20, 24, 27 |  | 18, 32
Respir. C | 20, 24, 27 |  | 19, 23, 26 |  | 30, 32
Med. A1 | 33 |  | 25, 28 |  | 34
Med. B1 | 25 |  | 33 |  | 17, 28
Med. B2 | 19, 23, 26 |  | 20, 24, 27 |  | 29, 30, 32
Med. C | 20, 24, 27 |  | 19, 23, 26 |  | 29, 30, 32
Surg. A1 | 34 |  | 28 |  | 25
Surg. B1 | 28 |  | 34 |  | 25
Surg. B2 | 29 |  | 30, 31 |  | 26, 27, 32
Surg. C | 30 |  | 29, 31 |  | 26, 27, 32
Surg. B2 (EL) | 31 |  | 30 |  | 26, 27, 32
Table 9: Estimates of ED patients’ waiting times of various types, with 95% confidence intervals. | Male |  | Female
---|---|---|---
| Simulation | Data |  | Simulation | Data
Cardio. A1 | 3.78 $\pm$ 0.54 | 5.54 $\pm$ 1.91 |  | 4.59 $\pm$ 0.74 | 6.08 $\pm$ 2.69
Cardio. B1 | 3.48 $\pm$ 0.51 | 5.26 $\pm$ 2.00 |  | 2.96 $\pm$ 0.69 | 4.65 $\pm$ 2.29
Cardio. B2 | 6.03 $\pm$ 0.21 | 5.63 $\pm$ 0.74 |  | 6.29 $\pm$ 0.29 | 6.43 $\pm$ 0.94
Cardio. C | 6.13 $\pm$ 0.19 | 6.51 $\pm$ 0.68 |  | 5.92 $\pm$ 0.26 | 6.89 $\pm$ 0.97
Gastro. A1 | 2.65 $\pm$ 0.44 | 3.71 $\pm$ 2.66 |  | 3.90 $\pm$ 0.49 | 5.84 $\pm$ 2.26
Gastro. B1 | 3.42 $\pm$ 0.42 | 2.76 $\pm$ 2.31 |  | 5.96 $\pm$ 0.47 | 4.60 $\pm$ 2.31
Gastro. B2 | 7.91 $\pm$ 0.19 | 7.78 $\pm$ 1.05 |  | 6.51 $\pm$ 0.21 | 7.09 $\pm$ 1.08
Gastro. C | 5.72 $\pm$ 0.18 | 6.52 $\pm$ 0.91 |  | 6.79 $\pm$ 0.19 | 6.71 $\pm$ 1.14
Geria. A1 | 4.23 $\pm$ 1.07 | 7.36 $\pm$ 3.05 |  | 4.05 $\pm$ 0.79 | 3.95 $\pm$ 3.56
Geria. B1 | 3.33 $\pm$ 1.21 | 4.03 $\pm$ 3.94 |  | 5.15 $\pm$ 0.81 | 6.73 $\pm$ 3.12
Geria. B2 | 6.68 $\pm$ 0.38 | 7.42 $\pm$ 1.47 |  | 7.51 $\pm$ 0.27 | 9.56 $\pm$ 1.00
Geria. C | 8.59 $\pm$ 0.27 | 8.12 $\pm$ 1.08 |  | 9.38 $\pm$ 0.19 | 8.72 $\pm$ 0.75
Respir. A1 | 4.11 $\pm$ 0.68 | 6.13 $\pm$ 2.79 |  | 3.97 $\pm$ 0.92 | 5.34 $\pm$ 2.86
Respir. B1 | 4.22 $\pm$ 0.49 | 4.70 $\pm$ 1.83 |  | 5.54 $\pm$ 0.63 | 6.02 $\pm$ 1.88
Respir. B2 | 8.80 $\pm$ 0.23 | 8.48 $\pm$ 0.81 |  | 4.44 $\pm$ 0.31 | 6.99 $\pm$ 0.95
Respir. C | 8.82 $\pm$ 0.19 | 7.17 $\pm$ 0.60 |  | 6.48 $\pm$ 0.25 | 6.68 $\pm$ 0.89
Med. A1 | 4.09 $\pm$ 0.41 | 5.02 $\pm$ 1.18 |  | 4.21 $\pm$ 0.43 | 5.85 $\pm$ 1.14
Med. B1 | 4.05 $\pm$ 0.33 | 5.73 $\pm$ 1.01 |  | 5.30 $\pm$ 0.33 | 4.90 $\pm$ 0.87
Med. B2 | 7.49 $\pm$ 0.15 | 7.59 $\pm$ 0.46 |  | 7.64 $\pm$ 0.15 | 7.97 $\pm$ 0.43
Med. C | 5.76 $\pm$ 0.12 | 6.54 $\pm$ 0.33 |  | 8.42 $\pm$ 0.12 | 8.15 $\pm$ 0.35
Surg. A1 | 4.84 $\pm$ 0.30 | 5.33 $\pm$ 0.74 |  | 5.10 $\pm$ 0.37 | 4.46 $\pm$ 1.06
Surg. B1 | 2.99 $\pm$ 0.33 | 5.26 $\pm$ 0.90 |  | 2.90 $\pm$ 0.40 | 5.43 $\pm$ 0.91
Surg. B2 | 6.10 $\pm$ 0.15 | 6.58 $\pm$ 0.41 |  | 8.72 $\pm$ 0.18 | 8.23 $\pm$ 0.45
Surg. C | 8.55 $\pm$ 0.13 | 7.75 $\pm$ 0.36 |  | 6.23 $\pm$ 0.16 | 6.19 $\pm$ 0.46