*[enumerate,1]label=(),

Data-driven Fair Resource Allocation For Novel Emerging Epidemics: A COVID-19
Convalescent Plasma Case Study

Maryam Akbari-Moghaddam1,*, Na Li1,2,3, Douglas G. Down1, Donald M. Arnold3,4,
Jeannie Callum5,6, Philippe Bégin7,8, Nancy M. Heddle 3,4.

1 Department of Computing and Software, McMaster University, Hamilton, Ontario
L8S 4L7, Canada

2 Community Health Sciences, University of Calgary, Calgary, Alberta T2N 1N4,
Canada

3 McMaster Centre for Transfusion Research, Department of Medicine, McMaster
University, Hamilton, Ontario L8N 3Z5, Canada

4 Department of Medicine, McMaster University, Hamilton, Ontario L8N 3Z5,
Canada

5 Department of Pathology and Molecular Medicine, Kingston Health Sciences
Centre, Kingston, Ontario K7L 2V7, Canada

6 Department of Pathology and Molecular Medicine, Queen’s University,
Kingston, Ontario K7L 3N6, Canada

7 Section of Allergy, Immunology and Rheumatology, Department of Pediatrics,
CHU Sainte-Justine, Montreal, Quebec H3T 1C5, Canada

8 Department of Medicine, CHUM, Université de Montréal, Montreal H3T 1J4,
Quebec, Canada

* Corresponding author

Email<EMAIL_ADDRESS>
###### Abstract

Epidemics are a serious public health threat, and the resources for mitigating
their effects are typically limited. Decision-makers face challenges in
forecasting the supply and demand for these resources as prior information
about the disease is often not available, the behaviour of the disease can
periodically change (either naturally or as a result of public health
policies) and can differ by geographical region. Randomized controlled trials
(RCTs) using scarce resources such as blood products as a randomized
intervention are affected by epidemics. In this work, we discuss a model that
is suitable for short-term real-time supply and demand forecasting during
emerging outbreaks. We consider a case study of demand forecasting and
allocating scarce quantities of COVID-19 Convalescent Plasma (CCP) in an
international multi-site RCT involving multiple hospital hubs across Canada
(excluding Québec). We propose a data-driven mixed-integer programming (MIP)
resource allocation model that assigns available resources to maximize a
notion of fairness among the resource-demanding entities. Numerical results
from applying our MIP model to the case study suggest that our approach can
help balance the supply and demand of limited products such as CCP and
minimize the unmet demand ratios of the demand entities. We analyze the
sensitivity of our model to different allocation settings and show that our
model assigns equitable allocations across the entities.

_K_ eywords Resource Allocation; Epidemics; COVID-19 Convalescent Plasma;
Data-driven Optimization; Demand Forecasting

##  1 Introduction

Epidemics ††footnotetext: This paper has been submitted to the INFOR:
Information Systems and Operational Research journal. have impacted the world
many times, and will occur again in the future. Emergency responses to these
epidemics can either be pre-event or post-event, where here event refers to
the initiation of an infectious disease (Dasaklis ., 2012). Forecasting the
potential dangers and planning the necessary steps to deal with an epidemic
are considered as pre-event tasks. Post-event responses occur after the
disease has started spreading and is still in progress. The corresponding
actions at these points are associated with treatment and allocating the
corresponding available resources. Our focus in this paper is on post-event
response situations.

Resource allocation decisions during emerging epidemics are challenging due to
several key reasons. First, limited knowledge and historical data about
disease demographics make it difficult to predict the demand for particular
resources. Second, because epidemics are usually unexpected and can spread
rapidly, there are often limited health care resources (vaccines, blood
products, medical equipment, etc.) compared to the total number of entities
requesting them. Determining how to fairly allocate the limited resources
becomes a challenge. Third, the demand can vary significantly between
geographically dispersed entities. Finally, the decisions must be made in a
timely manner. These issues motivate the investigation of supply and demand
forecasting models for scenarios where there are small amounts of available
data and the data can exhibit fundamental changes in behaviour. It is of
interest to incorporate these models into algorithms that yield fair
allocations.

Clinical trials can have different durations depending on the underlying
research, with many lasting as short as several months (Piantadosi, 2017).
Some of these clinical trials may require rapid roll-out to different
participating sites right after the planning phase, which might not allow for
prior data collection by the sites. Randomized Controlled Trials (RCTs) can
face additional limitations in data collection within the study duration as
different experimental interventions can be associated with specific time
marks. During uncertain situations like emerging epidemics, RCTs that use
scarce resources such as blood products as a randomized intervention can be
affected when there is a need for a timely and fair resource allocation
(Stawicki ., 2020). Convalescent plasma is a blood product which has been used
as a potential treatment for a number of diseases such as Ebola (Kraft .,
2015; Van Griensven ., 2016), influenza (Hung ., 2011; Zhou ., 2007), and
COVID-19 (Chen ., 2020). COVID-19 Convalescent Plasma (CCP) contains
antibodies, or special proteins, generated by the body’s immune system in
response to the novel coronavirus (Chen ., 2020). It has been considered as an
experimental treatment for hospitalized COVID-19 patients in a number of RCTs
worldwide (Bégin ., 2021; L. Li ., 2020; Simonovich ., 2021; Gharbharan .,
2020). Despite the importance of blood supply and demand management for scarce
products in RCTs, there is a lack of published research that provides insights
on the problem. This can be related to several challenges encountered when
making decisions on CCP allocation in an RCT setting. When such a scarce blood
product serves as a randomized intervention in an RCT, poor linkage across
blood donor recruitment, inventory management and the randomization process
can raise biases (N. Li ., 2022).

In this work, we tackle the real-time allocation of scarce resources during
epidemics to entities located in widespread geographical locations and with
different resource requirements. Furthermore, we consider an RCT setting which
requires rapid roll-out and has limited historical data due to its short study
duration. We are interested in demand forecasting models that can predict
short-term demand in real-time. The demand forecasts directly impact the
resource allocation decisions as inaccurate forecasts may lead to inefficient
and unfair use of limited (and valuable) resources. We note that there are
different notions of fairness (balance) in terms of resource allocation in the
literature (Kumar Kleinberg, 2000). In our case, we define fairness as
minimization of the entities’ unmet demand ratios, but one could also
generalize fairness to other notions. For instance, Karsu Erkan (2020) propose
an approach where imbalance is defined as the deviation from a reference
distribution determined by the decision-maker. They show that in resource
allocation problems, it is possible to maintain a mixed-integer programming
(MIP) structure even after generalizing the notion of fairness. In general,
there are wide-ranging views of fairness and how fairness metrics can be
incorporated into optimization problems (see Karsu Morton, 2014; Smith .,
2013; Mestre ., 2012; Heitmann Brüggemann, 2014; Karsu Morton, 2015). We
evaluate our proposed model on a case study of demand forecasting and CCP
distribution in a multi-site RCT, where  1. there were limited historical
supply and demand data,  2. the supply was limited and restricted by
manufacturing policies and donor recruitment, 3. the demand arising from the
demand entities was heterogeneous, and  4. specific clinical requirements were
needed for administrating CCP transfusion. More details of the RCT, including
how the approach presented in this paper is embedded and the resulting
observations from a clinical perspective, can be found in (N. Li ., 2022).

We make the following contributions: First, we discuss real-time short-term
forecasting of supply and demand of scarce resources in epidemics with limited
historical data. We propose the use of a forecasting model that does not
require indeterminate epidemiological parameters (such as location and time-
specific parameters) and thus does not require periodically updating the
parameters. Secondly, we address challenges that may arise in an online
setting due to extrapolation and sparse data. Next, we propose a data-driven
MIP model for real-time multi-location allocation of scarce resources
regularly and fairly to entities, which have heterogeneous demand. This
approach maximizes a notion of fairness among the resource-demanding entities.
Finally, numerical results of applying our model in a CCP case study show that
our approach yields fair allocations that are both close to the scenario where
supply and demand are known (rather than forecast) and are preferable to what
was used in practice.

The rest of the paper is organized as follows. Section 2 presents the existing
literature on demand forecasting and resource allocation approaches during
infectious disease outbreaks and our motivation for this work. We describe our
data-driven resource allocation problem in Section 3.1. Section 3.2 discusses
in depth the supply and demand forecasting methods that we use and we define
our proposed MIP resource allocation model in Section 3.3. The CCP case study,
the numerical results of applying our MIP model to the case study, and
sensitivity analysis of the model are discussed in Section 4. We conclude this
work and discuss how it may inform responses to future pandemics in Section 5.

##  2 Motivation and Related Work

There has been a variety of work in various fields that tackle the problem of
resource allocation during infectious disease outbreaks. In what follows, we
divide the body of work into six categories and discuss the motivation behind
our proposed methodology.
Compartmental Models and Resource Allocation

Epidemiological compartmental models, consisting of a set of nonlinear
ordinary differential equations, can help model the dynamics of different
epidemiological variables during a pandemic (Brauer, 2008). These models can
give insight into disease-related information such as spread rate, the
duration of an epidemic, and the total number of infected and recovered
patients. Decision-makers can employ compartmental models to derive demand for
medical resources to guide resource allocation decisions. Focusing on the
COVID-19 outbreak (Velavan Meyer, 2020), there have been many applications and
tools developed by different organizations worldwide to forecast infections,
hospitalizations, and deaths using compartmental models (Tomar Gupta, 2020;
Gong ., 2020; Chowdhury ., 2021). For instance, CHIME (Weissman ., 2020) is a
tool based on a Susceptible-Infectious-Recovered (SIR) model that can be used
for forecasting the number of daily hospitalized COVID-19 patients in the
short-term (e.g., up to $30$ days).

Demand Forecasting Models and Resource Allocation

Another approach that researchers have studied for forecasting healthcare
resources is using time series models or machine learning methods. The
references for this approach are extensive, thus we only discuss a few studies
as examples. Ferstad . (2020) introduce a time series model to forecast the
availability and utilization of intensive and acute care beds. Nikolopoulos .
(2021) use epidemiological and deep learning models to forecast the excess
demand for products and services considering auxiliary data and simulating
governmental decisions, while N. Li . (2021) combine ideas from statistical
time series modelling and machine learning to develop a hybrid demand
forecasting model for red blood cell components using clinical predictors.

Segmented Regression Models and Resource Allocation

Piecewise Linear Regression (PLR), also known as segmented linear regression,
forecasting models are a special case of a larger set of models known as
spline functions (Suits ., 1978). Modelling the regression function in
"pieces" can be helpful when dealing with sparse data because we can still use
linear regression models for data that does not fit a single line. To be more
specific, PLR is a simple model that makes understanding the data easier by
solving several linear regressions. Points at which the behaviour changes are
called breakpoints, which act as boundaries between each piece. There have
been a few studies on finding the number of breakpoints and their locations.
Rosen Pardalos (1986) propose a method for finding the minimum number of
equally spaced breakpoints within a given error tolerance, a sequential method
is proposed in Strikholm (2006) for finding the number of breakpoints, and
Yang . (2016) propose a discontinuous piecewise linear approximation and how
to determine the optimal breakpoint locations.

Piecewise linear models have been used in different applications when
modelling the structural shifts in data and forecasting based on the most
recent behaviour in data is desired. We will discuss a PLR model, namely MARS,
in more detail in Section 3.2.1, where we discuss the supply and demand
forecasting model used in this work. The MARS model has been used in
healthcare for medical diagnosis using classification approaches (see Butte .,
2010; Chou ., 2004; Yao ., 2013; Senthilkumar Paulraj, 2013; Lasheras ., 2020;
Serrano ., 2020). In the area of time series forecasting, López-Lozano .
(2019) evaluate the generalizability of MARS for identifying thresholds for
antibiotic consumption and Katris (2021) studies MARS and other time series
approaches for predicting the evolution of reported COVID-19 cases to track
the outbreak in Greece.

Disaster Management and Resource Allocation

In terms of resource allocation, the literature on emergency logistics
involves disaster operations performed before or after the impact of a
disaster where short-term post-disaster planning begins immediately. This
includes activities among which relief distribution is more related to our
scope of the study. Relief distribution consists of providing medical
supplies, shelters, manpower, and other associated resources, and thus,
resource allocation is a significant component (Caunhye ., 2012). A number of
resource allocation models for relief distribution that consider resource
assignment without determining flow quantities among arcs focus on allocating
equipment to amend maritime disasters, such as cleaning up oil spills. Unlike
other disasters, oil spills create a single demand point, and studies in this
area only consider direct equipment assignment to the spill areas (see
Psaraftis ., 1986; Srinivasa Wilhelm, 1997; Wilhelm Srinivasa, 1997).

Optimization Models and Resource Allocation

Many studies have focused on developing allocation models for medical
resources during infectious disease outbreaks. A dynamic linear programming
model based on an epidemic diffusion model is introduced in Liu Liang (2013)
to allocate medical resources. Preciado . (2013) analyze a networked version
of a susceptible-infected-susceptible (SIS) epidemic model when different
susceptibility levels are present. They propose a convex optimization approach
for distributing vaccination resources in a cost-optimal manner and test their
approach in a real social network. Yarmand . (2014) consider two-phase vaccine
allocation to different geographical locations. They capture each region’s
epidemic dynamics for different vaccination phases by a two-stage stochastic
linear program (2-SLP) model and show that their model helps to reduce vaccine
production and administration costs. Yin Büyüktahtakın (2021) define two new
equity metrics for fair resource allocation and develop a multi-stage
stochastic programming vaccine allocation model and apply it to an Ebola virus
disease case study. Furthermore, two resource allocation problems during
outbreaks are discussed in Preciado . (2014), where they use geometric
programming to solve the problems. Following the work in Preciado . (2014),
Han . (2015) propose a data-driven robust optimization framework based on
conic geometric programming. Their model is used to determine an optimal
allocation of medical resources such as vaccines and antidotes and can help
control an SIS viral spreading process in a directed contact network with
unknown contact rates.

Other Epidemiological Frameworks and Resource Allocation

A number of works have investigated resource allocation frameworks using
outbreak case studies. The problem of scheduling limited available resources
between multiple infected areas is discussed in NP. Rachaniotis . (2012), and
their proposed deterministic scheduling model is studied in a case study of
mass vaccination against A(H1N1)v influenza. A real-time synchronous heuristic
algorithm is proposed in N. Rachaniotis . (2017) and is tested on the same
case study as in NP. Rachaniotis . (2012). Sun . (2014) focus on allocating
patients and resources between hospitals located in a healthcare network and
propose a multi-objective optimization model. They discuss the application of
their model in an influenza outbreak case study. Finally, a large integer
programming problem framework for optimally allocating a resource donation is
introduced in Anparasan Lejeune (2019), and results of applying the framework
to a 2010 cholera outbreak case study are reported. Closely related to our
work is Du . (2021) where they study a multi-period location-specific resource
allocation problem for cholera outbreak intervention. They consider a rolling
time horizon and periodically determine an optimal intervention resource
allocation strategy with their data-driven optimization approach. Also similar
in spirit to our approach is Bekker . (2023), who propose a model for making
daily short-term predictions of the number of occupied ICU and clinical beds
in the Netherlands due to COVID-19.

Motivation Behind Proposed Methodology

With the aforementioned approaches in mind, we now discuss the motivation
behind the prediction methodology that we consider for real-time short-term
demand and supply forecasting for experimental products in clinical trials. We
observed the following limitations with the application of the methodologies
above to our problem setting:

  1. 1.

When an epidemic is first emerging, the epidemic state is only partially
observable, and the parameters that the epidemiological models require are
often indeterminate, as the disease information can only be obtained over a
period of time and after sufficient cases are reported and required data is
collected. Moreover, different geographical locations can show various
characteristics in terms of the disease spread pattern. The estimates that
compartmental models make are sensitive to the model’s structure (Silal .,
2016). Even in the presence of reasonable parameter estimates and simple model
structures, real-time resource allocation requires the parameters to be
periodically updated based on the disease spread rate and the number of people
involved, whether susceptible or infected. Therefore, CHIME-like models may
lead to poor approximation of the actual demand when used in a real-time
setting where obtaining the most recent updated parameters may not always be
possible.

  2. 2.

The aforementioned demand forecasting models work best when large datasets are
available and the models can capture the trend and seasonality, which is not
possible during emerging epidemics. Furthermore, real-time demand forecasting
can be challenging with any forecasting model if the demand is affected by
external factors, such as population characteristics, geographical locations,
operational procedures, guidelines, and governmental policies. As far as we
are aware, none of the mentioned approaches are consistent with the challenges
we introduced in Section 1. It is quite difficult to directly apply these
approaches to a real-time setting where short-term supply and demand
forecasting is desired, there is a limited supply of resources, and the
available data is sparse and shows fundamental changes in behaviour during
different periods. This motivates us to avoid models that are reliant on a
large number of parameters (as for CHIME-like models), as they require
continual updates and they may not always yield accurate forecasts for
resource supply and demand. Nonetheless, we seek a model that makes reasonable
predictions even when facing such challenges. We will revisit these challenges
in the case study in Section 4.

  3. 3.

More sophisticated demand forecasting models, such as ARIMA, can also be used
for time series analysis, especially demand forecasting with univariate data.
However, satisfaction of basic requirements for such models is necessary which
might not be possible in our setting. For instance, ARIMA and ARMA work best
in the presence of stationary data and the rule of thumb for these models
suggests having at least $50$ but preferably more than $100$ observations (Box
Tiao, 1975). We cannot meet that requirement, and as we start forecasting with
almost no historical data, any forecasting model might fail to exhibit its
best performance. We found a simple forecasting model such as MARS to be a
good choice considering our non-stationary and limited data.

  4. 4.

Our study is different from the aforementioned disaster management literature
in the sense that although there is the common objective of resource
allocation of limited resources, the time scale is different. Epidemics tend
to be much longer than disasters such as earthquakes, and there is more
uncertainty in the quantity of the effective resources due to the unknown
nature of the disease. Furthermore, the evolution of epidemics might involve
unexpected changes in supply and demand patterns, requiring suitable
forecasting models to be incorporated into the resource allocation model.

This work focuses on  1. how the methodologies above are combined to solve a
real-time short-term supply and demand forecasting and resource allocation
problem, and  2. how the combined methodology is applied to a clinical problem
setting with various human interventions and limited historical data. We
demonstrate that using MARS to determine inputs (supply and demand forecasts)
to a resource allocation problem is an effective combination in an emerging
epidemic setting. The prediction model in Bekker . (2023) consists of a linear
programming model inspired by smoothing splines for predicting the arrivals
and methods stemming from queueing theory to convert arrivals into occupancy.
The motivation for choosing their model is similar to ours in the sense that
it works with little historical data, which is a consequence of an emerging
epidemic setting. To the best of our knowledge, no other study has tackled the
problem of real-time multi-location allocation of scarce resources with
limited historical data and without relying on epidemiological models.

##  3 Data-driven Resource Allocation Model

###  3.1 Problem Description

Our work considers the patients of a COVID-19 case study (details in Section
4) where the distribution of supply and demand is similar in terms of the
population; however, there is a limited and highly-varied pattern for both in
the collected data. The supply for this study comes from the CCP of the
patients who have recovered from COVID-19, and donated blood. The demand comes
from the patients who are currently infected with COVID-19 and require CCP,
but will be considered as supply once they recover and donate CCP.

Hub-and-spoke structures can help effective access to blood products in a
public health network and clinical trials (Fernandez-Sojo ., 2022; Roberts .,
2018). A hub-and-spoke structure was established for our case study where CCP
units were distributed from regional blood centers to hospital hubs for
subsequent CCP allocation to patients. Therefore, our model also considers a
hub-and-spoke structure where we have a centralized supplier (hub) that
interacts with $H$ customers (spokes) and is responsible for satisfying their
demand for $R$ types of resources. Figure 1 shows a flowchart of our data-
driven resource allocation process.

Figure 1: Data-driven Resource Allocation Process

We are interested in a setting where limited resources must be allocated on a
regular basis (e.g. every week) to the entities requesting them, the demand
for the resources can be heterogeneous and arises from geographically
dispersed locations. We forecast both supply and demand for the following
reasons:  1. Common source: The same population defines both supply and
demand, but at different points in time.  2. Dependence between supply and
demand: Changes in the behaviour of the disease naturally lead to correlated
changes in supply and demand.  3. Uncertainty in supply and demand: A classic
resource allocation problem considers known resource quantities. However, we
do not know the number of resources that are available to be allocated at
every point in time and we face uncertainty in both supply and demand due to
the unknown nature of the disease. The uncertainty in inventory collection
also indicates the impact of CCP collection in our study, resulting in a
resource collection and allocation model.

We choose to work with the cumulative supply or demand as a cumulative sum of
data samples over time is helpful in situations where one needs to smooth
heterogeneous and sparse data and still make quantitative predictions for
future supply and demand without altering the original data. Ellaway (1978)
investigates the application of the cumulative sum technique in a
neurophysiology study. The cumulative sum is shown to be a powerful technique
for finding periods of change in data, as well as reducing real-time decision-
making uncertainty. In our case, we forecast weekly supply and demand based on
historical cumulative supply and demand data for a particular resource prior
to resource allocations. In particular, we would like to fit a forecasting
model weekly to our data where for each week, only the most recent week’s
observation is added to the dataset, and we cannot modify the predictions the
model previously made (as real-time allocations are made based on the
forecasts). We then perform error analysis of the forecasts, and allocate the
available resources to the customers in a fair manner by solving an
appropriate optimization problem. We assume that both the supplier and
customers can hold inventories of the resources and can use them to satisfy
future demand. Once the supplier allocates resources to customers, the
decision is final, and the resources cannot be reassigned. Thus, we need to
update the inventories at the end of each week, and consider them when solving
the optimization model in the next week.

We discuss a PLR supply and demand forecasting model and our resource
allocation optimization model in more detail in Section 3.2 and Section 3.3,
respectively.

###  3.2 Supply and Demand Forecasting

We would like to forecast the supply and demand for a particular week $t+1$
where we only have data available up to week $t$. Consider a dataset
consisting of $t$ observations where $X$ is an array of elements $x_{i}$
($i=1,2,{\dots},t$) representing the input variables (in our case,
$X={1,2,3,{\dots},t}$). Consider $y$ to be a vector of observed (supply or
demand) data samples $y_{i}$ ($i=1,{\dots},t$). We use Multivariate Adaptive
Regression Splines (MARS), a PLR model, to forecast the supply and demand.

####  3.2.1 Multivariate Adaptive Regression Splines (MARS)

Multivariate Adaptive Regression Spline (MARS) is a nonparametric regression
approach that was introduced by Friedman (1991). The MARS model consists of a
collection of simple linear models that can capture patterns and trends
related to interactions and nonlinearities. MARS uses a series of piecewise
linear pieces (splines) of different gradients. These pieces, also known as
basis functions (BFs), are connected at positions called knots which allow
thresholds, bends, and other departures from linear functions. A MARS model is
specified as follows:

${\hat{y}}^{\prime}=\beta_{0}+\sum_{p=1}^{P}\beta_{p}\lambda_{p}(x),$

where $P$ is the number of BFs and each $\lambda_{p}(x)$ is a BF, which can be
a spline function or the product of two or more spline functions. The
parameters $\beta_{0}$ and $\beta_{p}$, $p=1,\ldots,P$ are estimated using the
least-squares method. The basis functions are described as:

$BF(x)=\\{\max(0,x-c_{p}),\max(0,c_{p}-x)\\},$

where $c_{p}$ is the knot of the spline (threshold value).

A forward stage and a backward stage are considered in the MARS algorithm. In
the forward stage, BF functions and their potential knots are chosen, which
may result in a complicated and over parameterized model. In the backward
stage, to prevent overfitting, the algorithm considers deleting the BFs in
increasing order of the amount that they reduce the training error (Friedman,
1991).

####  3.2.2 Error Analysis and Model Enhancement

Based on the slope of the last piece, the forecast supply and demand for week
$t+1$ under MARS is simply:

${\hat{y}_{t+1}}^{\prime}={\hat{y}_{t}}^{\prime}+\frac{{\hat{y}_{t}}^{\prime}-{\hat{y}_{t-1}}^{\prime}}{x_{t}-x_{t-1}}.$

We can improve the forecast value for week $t+1$ (${\hat{y}_{t+1}}^{\prime}$)
by calculating its forecast error (Jarrett Khumuwala, 1987; Lu Chen, 2020). We
first fit an autoregressive (AR) model of order $l$ using Conditional Maximum
Likelihood to the residual error
(${\epsilon_{i}=y_{i}-{\hat{y}_{i}}^{\prime}}$) data up to week $t$ and use it
to forecast the error for week $t+1$:

$\hat{\varepsilon}_{t+1}=a_{0}+a_{1}\varepsilon_{t}+a_{2}\varepsilon_{t-1}+{\cdots}+a_{l}\varepsilon_{t-l+1},$
(1)

where ${a_{0},a_{1},{\dots},a_{l}}$ are the coefficients obtained from the AR
model and ${\hat{\epsilon}_{t+1}}$ is the forecast error for week $t+1$. We
finally calculate
${\hat{y}_{t+1}={\hat{y}_{t+1}}^{\prime}+\hat{\varepsilon}_{t+1}}$, i.e., the
improved supply or demand forecast value for week $t+1$, and use it as our
final forecast value for that week.

####  3.2.3 Challenges

We now discuss general challenges that may be faced when MARS is employed in
an online manner:

  * •

Predictions made using MARS may degrade if there is a sudden transitory change
in data. For instance, we found that holidays may affect the amount of data
collected in a particular week but the data follows its previous pattern after
the holidays have passed. We discuss this issue in greater detail in the case
study discussed in Section 4.

  * •

MARS is fitting piecewise linear models, and thus the slope of the segment
that ends with the most recent observation has a significant effect on the
forecast for the next week (it may lead to a large
underestimation/overestimation).

Although working with the cumulative sums for forecasting future supply and
demand can help address the issue of data sparsity, a particular challenge
arises when cumulative sums are employed in an online setting. Consider a
situation where the demand before week $t$ has a steep slope resulting in a
(relatively) high forecast for week $t$. However, upon observing the demand
for week $t$, one finds out that the actual demand was considerably lower.
Thus, the slope that affects week $t+1$’s forecast demand will be less steep
than what it was when only data up to week $t-1$ was available. This may
result in the forecast cumulative demand for week $t+1$ being lower than the
previously forecast cumulative demand for week $t$, which is not possible. For
instance, Figure 2a and Figure 2b demonstrate the cumulative forecast demand
obtained by MARS for week $21$ and week $22$, respectively. The cumulative
forecast demand for week $21$ and week $22$ are $22$ and $19$, respectively,
under MARS, which cannot happen in practice. This is one consequence of
considering an online setting where modifying the predictions is not possible
as real-time decisions are made. The sparser the data, the more this issue can
affect the model’s predictions. A possible solution for dealing with such
situations is to assume that the cumulative forecast value for week $t+1$ is
equal to the cumulative forecast value for week $t$. We will observe this
issue in our case study discussed in Section 4 and deal with it in the
suggested manner.

(a) cumulative forecast demand for week $21$

(b) cumulative forecast demand for week $22$

Figure 2: Forecasting a negative non-cumulative value

###  3.3 Resource Allocation Optimization

We allow for demand for resource $r$ to be satisfied by resource $r^{\prime}$.
This can be represented with a matrix $C$ of size $R\times R$, where an
element ($r$,$r^{\prime}$) of $C$ is $1$ if demand for resource $r$ can be
satisfied by resource $r^{\prime}$ and is $0$ otherwise. Furthermore, the
supply used in our MIP model for resource $r$ for week $t$ is constrained as
follows:

$c_{r,t}=\left\\{\begin{array}[]{@{}ll@{}}\hat{s}_{r,t},&\text{if}\
\hat{s}_{r,t}\leq s_{r,t}\\\ s_{r,t},&\text{otherwise},\end{array}\right.$ (2)

where $s_{r,t}$ and $\hat{s}_{r,t}$ are the actual and forecast supply for
resource $r$ on week $t$, respectively. The second case in (2) indicates that
we cannot allocate resources beyond the actual available supply. We will
formulate our resource allocation as a Mixed Integer Program (MIP). To do so,
we require the following notation:

Indices

$t$ |  index of time periods, $t=1,{\dots},T$
---|---
$h$ |  index of customers, $h=1,{\dots},H$
$r,r^{\prime}$ |  index of resource, $r,r^{\prime}=1,{\dots},R$

Data

$c_{r,t}$ |  the amount of resource $r$ available at the supplier for assignment at time $t$ (see (2) above)
---|---
$i_{r,h,t-1}$ |  the inventory of resource $r$ stored at customer $h$ at time $t-1$
$\hat{d}_{r,h,t}$ |  the estimated demand for resource $r$ by customer $h$ at time $t$

Decision variables

$v_{r,r^{\prime},h,t}$ |  the number of units of resource $r^{\prime}$ assigned to customer $h$ to satisfy demand for resource $r$ at time $t$. We only consider the set of $v_{r,r^{\prime},h,t}$ that correspond to $C(r,r^{\prime})=1$.
---|---

We formulate our resource allocation problem as follows:

Objective function

$\min{\mathop{\max}\limits_{h:\sum_{r=1}^{R}\hat{d}_{r,h,t}>0}}\frac{\sum_{r=1}^{R}(\hat{d}_{r,h,t}-\sum_{r^{\prime}=1}^{R}v_{r,r^{\prime},h,t}-i_{r,h,t-1})}{\sum_{r=1}^{R}\hat{d}_{r,h,t}}.$
(3)

Constraints

$\sum_{h=1}^{H}\sum_{r=1}^{R}v_{r,r^{\prime},h,t}\leq
c_{r^{\prime},t}{\rm\;,\;}\forall{\rm\emph{r}^{\prime}},$ (4)
$v_{r,r^{\prime},h,t}\geq 0{\rm\;and\;integer\;valued},$ (5)
$\sum_{r^{\prime}=1}^{R}v_{r,r^{\prime},h,t}\leq\hat{d}_{r,h,t}{\rm\;\;\;,\;}\forall{\rm\emph{r}\;,\;}\forall{\rm\emph{h}}.$
(6)

Constraint (4) prevents the over-allocation of available resources. Constraint
(5) ensures the integrality and non-negativity of the resource allocations and
constraint (6) keeps the allocation to each customer below the corresponding
estimated demand. If all of the estimated demand at time $t$ can be met, any
excess supply is held in inventory at the hub.

The objective function (3) captures our notion of fairness: minimizing the
largest ratio of unmet demand over all customers. It could be modified to
capture other notions of fairness. For example, one can penalize larger
deviations between the demand and what is allocated by using a quadratic
objective function, which would create a Mixed Integer Quadratic Program
(MIQP). If the problem size becomes too large, one could also look at Linear
Programming (LP) relaxations of the problem.

##  4 The CONCOR-1 trial: A case study for a proposed application of the
resource allocation model

The Randomized, Open-Label Trial of CONvalescent Plasma for Hospitalized
Adults With Acute COVID-19 Respiratory Illness (CONCOR-1) was an RCT involving
72 academic and community sites across Canada, the USA, and Brazil (Bégin .,
2021). The randomization in this RCT was performed at a ratio of 2:1
allocation to receive CCP or standard of care for a planned study population
of $1200$ patients, stratified by age ($<60$ and $\geq 60$ years). The first
CCP unit for the trial was collected on April $24$, $2020$, and the first
patient was randomized on May $14$, $2020$. The trial ceased on January $29$,
$2021$ with a total of $940$ randomized patients. The objective of the trial
was to assess whether transfusing CCP reduces deaths or the proportion of
patients requiring intubation at day 30 compared to standard of care for
hospitalized COVID-19 infected adult patients (Bégin ., 2021). Canadian Blood
Services (CBS) is responsible for collecting the CCP units and is the national
blood supplier across all provinces in Canada except Québec. The supply and
demand management of the CCP products for patients enrolled in the trial was
conducted by CBS and the CONCOR-1 team at the McMaster Centre for Transfusion
Research. In what follows, as we are evaluating our approach, we will take the
viewpoint that the trial is in progress.

Canada’s blood supply chain network is currently centralized and comprises two
levels: regional CBS distribution sites and hospital blood banks. Nine CBS
blood distribution sites are currently located across Canada, and each centre
attempts to meet the CCP demand from the hospital blood banks in its network.
We use all available data from the trial which comes from $18$ hospital hubs,
$30$ hospital sites, and $8$ CBS distribution sites.

CCP is stored frozen and ideally must be transfused as soon as possible after
being thawed, but at most within five days (Bloch ., 2021). A patient who is
randomized to the CCP arm in the trial requires a single dose of approximately
$500$ or two doses of $250$ (from a single or two different donors) and the
CCP unit is transfused to the patient within the first 24 hours after
randomization (Bégin ., 2021).

It is challenging for CBS to make decisions on CCP allocation, because  1. the
CCP supply is limited and restricted by manufacturing policies and may not
meet the total CCP demand,  2. the trial involves hospitals from
geographically dispersed locations and multiple blood distribution centres,
3. the demand for CCP exhibits heterogeneity between different geographic
regions,  4. there is limited knowledge or historical data about the disease
demographics making it difficult to forecast the supply and demand of CCP
products,  5. there are specific clinical requirements for administrating CCP
transfusions, such as specific product dose, ABO blood group compatibility,
and medical condition requirements, and  6. the decisions must be made in
real-time and once the CCP units are shipped to a hospital hub, redistribution
to other hospital hubs is undesirable. Hence, the decision for every unit
matters. The key observations of our work suggest that our proposed data-
driven MIP model can help balance the supply and demand of CCP products and
lead to a fair allocation of limited CCP products among hospital hubs.

The underlying network of our case study is shown in Figure 3. The
corresponding definitions and notations are listed as follows:

  * •

$l$: represents a cluster and is defined as a geographic region with only one
CBS distribution site but one or more CBS donor collection sites, hospital
hubs and individual hospital sites.

  * •

$g$: represents an individual CBS donor collection site in cluster $l$,
$g=1,{\dots},G$.

  * •

$b$: represents the CBS distribution site in cluster $l$.

  * •

$h$: represents an individual hospital hub under CBS distribution site $b$ in
cluster $l$, $h=1,{\dots},H$.

  * •

$p$: represents an individual hospital site for hospital hub $h$ in cluster
$l$, $p=1,{\dots},P$.

Figure 3: CCP allocation network

The structure in Figure 3 is a hub-and-spoke structure (a CBS distribution
site is a hub and its underlying hospital hubs are the spokes) and is
consistent with our resource allocation MIP model proposed in Section 3.3. CBS
distribution sites can centrally decide to reallocate the blood products if
there is excess supply in a particular CBS region. Thus, we only consider a
single CBS distribution site in our optimization problem. Based on the
agreements between CBS and Héma-Québec, the blood supplier in Québec, less
common blood groups (blood groups AB and B) can be shared (Bégin ., 2021).

The CBS distribution site provides CCP to hospital hubs that in turn allocate
units to other hospital sites in the area. We only focus on the allocations
from the CBS distribution site to the hospital hubs.

As of May $2021$, the distribution of ABO blood groups in Canada is O:$46\%$,
A:$42\%$, B:$9\%$, and AB:$3\%$ (Canadian Blood Services, 2021). In general,
plasma for AB and O blood groups are universal donor and recipient,
respectively. In practice, transfusing blood-specific units is prioritized. In
the CONCOR-1 trial, due to the limited resources for B and AB plasma, patients
with O and B blood groups can receive A and AB plasma, respectively, when an
exact match is not available.

The dataset that we work with includes the CBS distribution site’s available
CCP units after shipment at an aggregate level for each blood group on
specific dates starting from September $1$, $2020$, up to January $25$,
$2021$. It also contains data for the received CCP units from the CBS
distribution site for each hospital hub, whether randomized to a patient or
stored in inventory, from May $11$, $2020$, up to January $25$, $2021$.
Furthermore, CCP-related information such as product dose, ABO group, and
whether a unit was broken or leaking after being thawed for transfusion at a
hospital hub are recorded. Table 1 provides a summary of the dataset’s
attributes, their description, and their format.

Table 1: Dataset description Dataset |  Attribute |  Description |  Format
---|---|---|---
CBS supply data |  date |  CBS distribution site’s aggregate available units after shipment on a date |  Date
A |  Total number of blood group A CCP ($250$) units |  Integer
AB |  Total number of blood group AB CCP ($250$) units |  Integer
B |  Total number of blood group B CCP ($250$) units |  Integer
O |  Total number of blood group O CCP ($250$) units |  Integer
Total |  Total number of CCP ($250$) units |  Integer
Hospital Hub data |  hospitalhub_ID |  Unidentifiable unique ID of a hospital hub |  String
receiveddate |  Hospital hubs received CCP units from the CBS distribution site on a date |  Date
DNL |  The CCP unit’s de-identified ID |  String
productABOgroup |  The ABO blood group of the CCP unit |  String
productdose |  The CCP unit dose with $1$ indicating $250$ units and $2$ indicating $500$ units |  Integer
matched |  Boolean variable with 1 indicating the CCP unit was matched with a randomized patient (indicating a demand), $0$ if the unit was stored in the hospital hub’s inventory |  Boolean
thawed |  Boolean variable with $1$ indicating that the unit broke when thawing, $0$ otherwise |  Boolean

We first create a cumulative weekly dataset of the number of new $500$ units
assigned to randomized patients at each hospital hub (considered as their CCP
demand) and the total received units from CBS for each resource. We
cumulatively sum hospital hubs’ weekly received CCP units and CBS weekly
inventory difference to calculate CBS cumulative new weekly supply. To
maintain and promote efficient CCP allocation, prior estimation of the
hospital hub’s demand and CBS supply is necessary. One could either translate
the CHIME model outputs to CCP supply or demand, or build a CHIME-like model
that incorporates our variables. However, we found it more effective to work
with the supply and demand directly and use PLR forecasting models. One reason
is that the proportion of patients consenting to the trial also affects the
total CCP demand. The consent rate can be affected by many factors such as a
patient’s religion and other competing treatment strategies. So, the patient
population may not be the same as the population considered in the CHIME or
CHIME-like models. Furthermore, given the limited and highly-varied pattern of
CCP supply and demand in our dataset, PLR models appear to be an effective
tool for forecasting CCP units.

###  4.1 Model Assumptions

We have created an environment for weekly allocation of CCP units from the CBS
distribution site to the hospital hubs participating in the CONCOR-1 trial.
Our dataset contains CBS supply data from a later date than when the first
patient was randomized to receive a CCP unit. So, we consider the week that
CCP supply was first reported as the starting week in our model (week of
August $31$, $2020$). The last week in the dataset is the week of January
$25$, $2021$, so the duration $T$ of our model is $22$ weeks. We combined
hospital hubs that are very close in distance, under the same distribution
network and with few COVID-19 patients. Small hospitals in distant areas with
a very low number of hospitalized COVID-19 patients were removed from the
study dataset.

Table 2 shows the actual supply and demand ($500$ units) reported for each
resource and in total in this period. There are a total of $15$ A, $14$ O,
$9.5$ B, and $14$ AB $500$ CCP units received from Héma-Québec recorded in the
dataset. The numbers in Table 2 are only associated with CBS supply (excluding
any units received from Héma-Québec). We observe in the data that $4.5$ A, $3$
O, $2$ B, and $0.5$ AB $500$ CCP units are unused at the hospital hubs because
they were leaking or broken after being thawed for transfusion. These units
are not included in our model as the wastage due to the thawed CCP units is
negligible.

CCP Units ($500$) | A | O | B | AB | Total
---|---|---|---|---|---
Supply | $157$ | $84.5$ | $29$ | $26$ | $296.5$
Demand | $131.5$ | $94.5$ | $34.5$ | $33.5$ | $294$
Table 2: CBS CCP supply and CCP demand (August $31$, $2020$ \- January $25$,
$2021$)

We assume that we have no information on the actual CCP supply and demand when
deciding on how to allocate the CCP units. Thus, we perform weekly forecasts
considering only the available data up to that week. The open source _scikit-
learn-contrib_ library _py-earth_ (Rudy, 2016) is used for applying the MARS
model, where the maximum degree of interaction terms generated by the forward
stage (the $max\\_degree$ parameter) is set to $2$ to better deal with the
nonlinearities in the data, and the remaining parameters follow the default
values. We are not performing any validations because our dataset is sparse
and the performance of MARS cannot be significantly enhanced as using the
cumulative sums has already smoothed the dataset. In general, a $k$-fold cross
validation can be used with MARS to obtain a less biased estimate. We start
making predictions four weeks from the start of available data to decrease the
likelihood of severe underestimation or overestimation. Finally, a lag $l=1$
is used in the autoregressive model of residual errors in (1). This choice is
due to a week’s supply and demand tending to be closer to the amounts for the
most recent week, as well as trying to avoid sudden changes that we might face
when considering a larger lag.

We deal with the challenges that might occur in our forecasting process, as
discussed in Section 3.2.3; if a negative non-cumulative forecast value on
week $t+1$ arises, we instead set the cumulative forecast value equal to the
cumulative forecast value for week $t$. Furthermore, we account for expected
sudden transitory changes in data, such as weeks containing holidays. We
observe that the hospital hubs’ requested CCP units are relatively lower on
weeks containing the Christmas, Boxing Day, and New Year’s Day statutory
holidays. This issue is due to fewer working days or reduced staffing for
these weeks. Thus, it is important to account for these weeks as they can be
falsely detected as breakpoints. We have chosen the value of $0.7$ as a
reasonable adjustment factor and multiply the slope of forecast line going
through the weeks containing Christmas, Boxing Day, and New Year’s Day
holidays by this factor. One might need different adjustment factors depending
on the extent the supply and demand are affected on a particular holiday.

From this point on, wherever we refer to MARS, we are considering the model’s
forecasts after accounting for the mentioned challenges and performing error
analysis. Because CCP is stored in $250$ units, we allocate $250$ units from
CBS to the hospital hubs in our model. The forecast values are decimal values;
therefore, we round the supply and demand forecast values to the nearest
integer so that all values correspond to $250$ units. All the results reported
in the figures and tables correspond to $500$ units.

We are interested in making a fair allocation of different ABO blood group CCP
units among the hospital hubs while minimizing their unmet CCP demand
proportions. We consider four resources ($R=4$) of A, O, B, and AB and examine
two compatibility matrices $C$ for assigning the CCP units in our MIP problem
, as described in Section 3.3:  1. the identity compatibility matrix, and  2.
the ABO compatibility matrix used in the CONCOR-1 trial which allows the
transfusion of A and AB plasma to patients with O and B blood groups,
respectively, when the same blood group is not available (CONCOR-1
compatibility matrix). For our primary study, we use the forecast supply and
demand for each of the resources in our model for each week. Furthermore, to
evaluate different allocation scenarios, we also forecast the aggregate supply
and demand for each week and consider the distribution of ABO blood groups in
Canada as the probabilities for calculating the forecast CCP for each resource
(Zietz ., 2020). For this sensitivity analysis, we use the following
probabilities based on the distribution of Canadian blood groups on each run
of our model for generating the supply and demand for each blood group:
$w_{c_{A}}=w_{h_{A}}=0.42$, $w_{c_{O}}=w_{h_{O}}=0.46$,
$w_{c_{B}}=w_{h_{B}}=0.09$, and $w_{c_{AB}}=w_{h_{AB}}=0.03$, where
$w_{c_{r}}$ and $w_{h_{r}}$ are the supply and demand probabilities for blood
group $r$, respectively. The goal of the sensitivity analysis is to determine
the performance of the model in situations where it is only possible to
forecast the aggregate amount of resources in terms of supply and demand. This
analysis also helps gain insights as to the generalizability of the results of
our primary study.

We assume that both CBS and hospital hubs can store the excess units at the
end of each week to use them in later weeks. We solve our MIP model based on
forecast supply and demand for each resource, and the actual inventories held
at CBS and hospital hubs on the week under study. We assume that the hospital
hubs can use a compatible CCP unit according to the CONCOR-1 compatibility
matrix (if available) for a patient when the same blood group CCP is not
available. We examine both identity and CONCOR-1 compatibility matrices for
solving our MIP model; however, at the inventory level of the hospital hubs,
only the CONCOR-1 compatibility matrix is used. We shall see later that this
is a good combination for the situations where forecast supply and demand are
used, and the forecasting errors are not too large. The identity compatibility
matrix at the MIP level prevents issues in terms of greedy allocation of
scarce resources, and the CONCOR-1 compatibility matrix at the inventory level
of the hospital hubs allows for the efficient compensation of forecasting
errors.

Because we know the actual demand for a particular week only after that week
has passed and because the supply is limited, we might not fully meet all
demands. In these cases, the unmet demand is carried over to the next week. We
note that the optimization model ensures that the CCP units of a resource
allocated to a hospital hub are never more than its forecast demand. Excess
units stored in hospital hubs’ inventories are due to the possible difference
between the actual and forecast values.

A total of $m$ runs are used to calculate the mean final unmet demand
($\overline{u}_{T}$) and the mean ratio of final unmet demand to total demand
($\overline{z}_{T}$) (for ease of reading, we will subsequently refer to these
quantities as unmet demand and unmet demand ratio, respectively) for each
hospital hub:

$\overline{u}_{T}=\frac{\sum_{r=1}^{R}u_{T_{r}}}{m}$

where the duration of our model is $T=22$ weeks, and $u_{T_{r}}$ is the
hospital hub’s final unmet demand for resource $r$ and,

$\overline{z}_{T}=\frac{\sum_{r=1}^{R}u_{T_{r}}}{m\sum_{t=1}^{T}\sum_{r=1}^{R}d_{t_{r}}}$

where $d_{t_{r}}$ is the hospital hub’s actual newly-added demand on week $t$
for resource $r$.

###  4.2 Results

In Figure 4, we show the (real-time) cumulative weekly supply and demand
forecasts over time, respectively, in terms of total, A, O, B, and AB CCP
units after using MARS for the dataset considering only the available data up
to that week. The real-time demand forecasting for each week in Figure 4 uses
the cumulative aggregate demand data over all hospital hubs up to that week.
We observe that MARS performs well even with our limited dataset. Table 3
shows Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE)
of our supply and demand forecasts under MARS for total, A, O, B, and AB CCP
units; lower values are better. We observe that MARS fits the data well after
performing error analysis and accounting for the mentioned challenges
discussed in Section 3.2.2 and Section 3.2.3. We note that we are making
forecasts from week $4$, which explains the severe supply overestimation for
week $4$ in Figure 4e where the data is sparse. In such situations, the
cumulative value remains the same until a subsequent cumulative forecast is
higher.

(a) Total units

(b) A units

(c) O units

(d) B units

(e) AB units

Figure 4: Model performance in forecasting CCP supply and demand Hub | RMSE | MAPE
---|---|---
Total | A | O | B | AB | Total (%) | A (%) | O (%) | B (%) | AB (%)
Hospital Hub 1 | 1.06 | 0.43 | 0.84 | 0.43 | 0.41 | 16.07 | 17.05 | 19.73 | 20.45 | 22.73
Hospital Hub 2 | 1.74 | 1.21 | 0.91 | 0.84 | 0.58 | 12.87 | 18.79 | 15.50 | 17.82 | 17.86
Hospital Hub 3 | 2.97 | 1.88 | 1.56 | 0.81 | 0.87 | 11.32 | 19.41 | 12.59 | 19.89 | 18.67
Hospital Hub 4 | 2.33 | 1.59 | 0.95 | 0.61 | 0.49 | 21.70 | 25.45 | 53.45 | 16.75 | 16.67
Hospital Hub 5 | 1.76 | 1.10 | 0.80 | 1.02 | 0.46 | 7.31 | 15.26 | 11.52 | 22.35 | 17.36
Hospital Hub 6 | 3.00 | 1.38 | 1.73 | 1.06 | 0.66 | 15.12 | 13.86 | 14.60 | 22.66 | 14.94
Hospital Hub 7 | 2.64 | 2.01 | 1.04 | - | - | 27.91 | 25.58 | 27.12 | - | -
All Hospital Hubs | 7.01 | 4.27 | 1.99 | 1.57 | 1.57 | 3.67 | 6.45 | 3.78 | 18.36 | 14.46
CBS Distribution Site | 10.68 | 5.25 | 5.41 | 1.78 | 3.35 | 6.12 | 6.01 | 9.90 | 7.59 | 18.00
Table 3: RMSE and MAPE of supply and demand forecasts under MARS

All the experiments are run on Spyder Python IDE on a computer with an
Intel(R) Core(TM) i7-4510U processor running at $2.60$ GHz using $94$ MB of
RAM, running Windows 10 version 21H2. The weekly supply and demand forecasting
took on average approximately $0.04$ and $0.28$ seconds, respectively. Our MIP
model has $43$ variables and $81$ constraints, and solving the weekly MIP
model took on average approximately $0.02$ seconds. Table 4 reports our CCP
allocation model’s performance in terms of unmet demand and unmet demand ratio
for our primary study, i.e., allocating the resources based on the forecast
supply and demand for each blood group. In Table 5, we compare the results to
both when no forecasting is required, i.e., the actual values of supply and
demand for each resource are known for each week and the actual allocations in
the CONCOR-1 trial. In both tables, two different compatibility matrices
(identity and CONCOR-1) are chosen for the MIP allocation model. Finally, in
Table 6, we analyze the sensitivity of our model to different allocation
settings by forecasting the aggregate supply and demand and using the
distribution of Canadian blood groups to calculate the supply and demand for
each resource. A total of $m=300$ runs are considered for calculating unmet
demand and unmet demand ratio and the standard error (SE) of their
corresponding $95$% confidence intervals is reported.

|  |  | Identity | CONCOR-1
---|---|---|---|---
Hub | Total Demand | Total Forecast Demand | $\overline{u}_{T}$ |  $\overline{z}_{T}$ (%) | $\overline{u}_{T}$ |  $\overline{z}_{T}$ (%)
Hospital Hub 1 | 12.5 | 12 | 1.00 | 8.00 | 1.00 | 8.00
Hospital Hub 2 | 29 | 30.5 | 0.50 | 1.72 | 1.00 | 3.45
Hospital Hub 3 | 74 | 79 | 4.50 | 6.08 | 10.00 | 13.51
Hospital Hub 4 | 26 | 30.5 | 1.00 | 3.85 | 1.50 | 5.77
Hospital Hub 5 | 30 | 30 | 1.50 | 5.00 | 1.00 | 3.33
Hospital Hub 6 | 105 | 110.5 | 8.50 | 8.10 | 10.00 | 9.52
Hospital Hub 7 | 17.5 | 20.5 | 0 | 0 | 0 | 0
Table 4: Resource allocation results under forecast supply and demand for each resource |  | MIP Allocations |  |
---|---|---|---|---
Hub | Total Demand | Identity | CONCOR-1 | Allocations in CONCOR-1
|  | $\overline{u}_{T}$ |  $\overline{z}_{T}$ (%) | $\overline{u}_{T}$ |  $\overline{z}_{T}$ (%) | $\overline{u}_{T}$ |  $\overline{z}_{T}$ (%)
Hospital Hub 1 | 12.5 | 1.50 | 12.00 | 1.00 | 8.00 | 3.00 | 24.00
Hospital Hub 2 | 29 | 1.00 | 3.45 | 0.50 | 1.72 | 1.50 | 5.17
Hospital Hub 3 | 74 | 7.00 | 9.46 | 4.00 | 5.41 | 20.00 | 27.02
Hospital Hub 4 | 26 | 2.00 | 7.69 | 0.50 | 1.92 | 4.50 | 17.31
Hospital Hub 5 | 30 | 1.50 | 5.00 | 0.50 | 1.67 | 5.00 | 16.67
Hospital Hub 6 | 105 | 10.00 | 9.52 | 7.00 | 6.67 | 27.00 | 25.71
Hospital Hub 7 | 17.5 | 0 | 0 | 0 | 0 | 1.00 | 5.71
Table 5: MIP model allocations under actual supply and demand versus actual allocations in CONCOR-1  |  |  | Identity | CONCOR-1
---|---|---|---|---
Hub | Total Demand | Total Forecast Demand | $\overline{u}_{T}\pm SE$ |  $\overline{z}_{T}\pm SE$ (%) | $\overline{u}_{T}\pm SE$ |  $\overline{z}_{T}\pm SE$ (%)
Hospital Hub 1 | 12.5 | 10.5 | $1.00\pm 0.03$ | $8.04\pm 0.26$ | $0.93\pm 0.03$ | $7.47\pm 0.27$
Hospital Hub 2 | 29 | 30 | $1.62\pm 0.10$ | $5.57\pm 0.36$ | $1.65\pm 0.10$ | $5.70\pm 0.34$
Hospital Hub 3 | 74 | 77 | $3.66\pm 0.15$ | $4.94\pm 0.20$ | $5.97\pm 0.20$ | $8.07\pm 0.27$
Hospital Hub 4 | 26 | 28.5 | $1.29\pm 0.08$ | $4.97\pm 0.31$ | $1.52\pm 0.08$ | $5.85\pm 0.30$
Hospital Hub 5 | 30 | 29.5 | $1.26\pm 0.06$ | $4.20\pm 0.19$ | $1.21\pm 0.05$ | $4.03\pm 0.16$
Hospital Hub 6 | 105 | 110 | $8.57\pm 0.17$ | $8.16\pm 0.16$ | $7.63\pm 0.19$ | $7.27\pm 0.18$
Hospital Hub 7 | 17.5 | 20.5 | $0.02\pm 0.01$ | $0.14\pm 0.07$ | $0.42\pm 0.03$ | $2.38\pm 0.15$
Table 6: Resource allocation results under aggregate forecast supply and
demand (Sensitivity Analysis)

The key observations obtained from using our data-driven resource allocation
model in the CONCOR-1 case study are as follows:

  1. 1.

The role of hospital hubs’ proportion of CCP demand — _A hospital hub’s
proportion of CCP demand for a resource can lead to its demand not being fully
met if the available supply of the resource on a particular week is limited
compared to its total demand for the same week. Our approach can help minimize
the unmet CCP demand ratios and lead to balanced and fair CCP allocation
decisions._

We note that there is a shortage of $10$ O, $5.5$ B, and $7.5$ AB CCP units
due to the limitation in supply of these blood groups, as reported in Table 2.
Thus, the CCP demand proportion of a hospital hub for a particular resource
and the variance in the total demand between the hospital hubs in each week
can affect the unmet demand and unmet demand ratio. For instance, if the
available supply on week $t$ can meet all the demand on week $t$, a hospital
hub’s proportion of CCP demand for a resource is not an issue. On the other
hand, if the available supply on week $t$ is limited compared to the total
demand for the same week, a hospital hub’s high demand proportion for a
limited resource CCP can lead to its demand not being fully met. This issue is
unavoidable in our setting where we make predictions and allocations on a
weekly basis and the optimization model finds a solution based only on the
current situation.

We observe in Table 4 that Hospital Hub $1$, Hospital Hub $3$, and Hospital
Hub $6$ have larger unmet demand ratios under both compatibility matrices.
Hospital Hub $1$’s unmet demand ratio is high because of its unmet CCP demand
proportions. However, its unmet demand is not high and is due to the demand
that was not met on the last week in our model (and may have been satisfied if
we continued for additional weeks). In fact, the supply and demand for the
last week can highly affect the final unmet demands of all hospital hubs.
Hospital Hub $3$ and Hospital Hub $6$ require a higher proportion of B and AB
CCP units compared to the other hospital hubs on weeks when the supply of
these units is limited. A hospital hub’s unmet demand on a particular week is
moved to the next week and will have an effect on its future allocations. We
observe that the rest of the hospital hubs have similar unmet demand and unmet
demand ratio values. This suggests that our proposed data-driven MIP model
leads to a reasonable and fair balance of limited CCP products between the
hospital hubs under the MARS forecasting model.

  1. 2.

The role of the compatibility matrix

    1. 2.1.

_When the actual supply and demand is unbalanced, not known before allocation,
and hence error due to forecasting is present, using the identity
compatibility matrix in the MIP level is preferred as it prevents the
allocation of limited resources to demand for a more abundant resource._

We notice in Table 4 that the total unmet demand for the identity
compatibility matrix is lower than for the CONCOR-1 compatibility matrix.
While the CONCOR-1 compatibility matrix satisfies as much demand as possible
in the current week, its use in this real-time setting can cause issues for
future demand, in particular by allocating resources with lower long term
supply to demands that can (eventually) be satisfied by more abundant
resources. We found that under the CONCOR-1 compatibility matrix, the shortage
of O units was compensated by excess A units, and excess B demands were met by
AB units. Because the supply of A units was the highest during the trial, and
the supply of AB units was the lowest, using AB units to meet demand from
other blood groups led to shortages of AB units in future weeks. Furthermore,
because the supply and demand forecasts are used instead of the actual values,
we might overestimate the demand for a rare resource, such as the AB blood
group CCP that is also compatible with another resource, so its allocation
increases the final unmet demand of some hospital hubs. However, under the
identity compatibility matrix, greedy allocations at the MIP level are
prevented. If the forecasting error is not too large, the combination of the
identity compatibility matrix at the MIP level and the CONCOR-1 compatibility
matrix at the inventory level of hospital hubs is preferred under forecast
supply and demand as the identity compatibility matrix at the MIP level
prevents greedy allocations that cause issues for future demand, but the
forecasting error results in the allocation of some compatible resources that
combined with the CONCOR-1 compatibility matrix at the hospital hubs’
inventory level can help meet more demand. Therefore, for our primary study,
we recommend this combination of compatibility matrices that prevents future
misallocation and absorbs the forecasting errors in an effective manner.

    1. 2.2.

_The CONCOR-1 compatibility matrix results in lower unmet demand when the
actual supply and demand for each week is known. The total unmet demand under
this compatibility matrix is close to the lowest achievable value, the actual
shortage in supply._

While using the identity compatibility matrix in Table 4 where supply and
demand forecasts are used exhibits better results, the opposite is true when
solving the MIP model based on the assumption of knowing the actual supply and
demand for each week. This can be observed by comparing the total unmet demand
in Table 5 under "MIP Allocations" for both compatibility matrices ($23$
versus $13.5$). When the actual supply and demand are used, over-allocation of
limited resources will not happen and the final unmet demand is close to the
actual supply shortage. We note again that demand for $5.5$ B and $7.5$ AB CCP
units cannot be met due to supply shortage which is almost exactly the total
unmet demand result of our MIP model under the CONCOR-1 compatibility matrix
($13.5$). This observation also reinforces the notion in the previous
observation that forecasting errors (combined with unbalanced supply and
demand) are what drive the recommendation of the identity compatibility matrix
under forecast supply and demand. The identity compatibility matrix does not
allow cross-transfusion, so $10$ O units cannot be satisfied by the excess A
units due to supply shortage. The total unmet demand under the identity
compatibility matrix ($23$) further validates the performance of our model in
efficiently allocating the available resources as it is equal to the actual
shortage in supply, as shown in Table 2.

    1. 2.3.

_The identity compatibility matrix results in lower unmet demand under
forecast supply and demand compared to when it is used under actual supply and
demand, as reasonable forecast error can better help in meeting the demand._

Comparing the unmet demand ratio values in Table 4 and Table 5 under the
CONCOR-1 compatibility matrix for each hospital hub, we observe that the unmet
demand ratio for all hospital hubs is improved (or is the same) in Table 5,
significantly so for Hospital Hub $3$ and Hospital Hub $6$. Furthermore, the
unmet demand ratio values in Table 5 are closer to each other under the
CONCOR-1 compatibility matrix than those reported in Table 4, where the
results are affected by the forecasting errors. However, we notice that all
unmet demand ratio values in Table 4 under the identity compatibility matrix
are lower than (or the same as) those in Table 5. The reason is that using the
identity compatibility matrix at the MIP level under actual supply and demand
is not a good choice considering that the MIP model never assigns more CCP
units than the actual demand. This choice leads to the CONCOR-1 compatibility
matrix at the inventory level of the hospital hubs not helping at all in
meeting the demand. However, under forecast supply and demand, although the
identity compatibility matrix prevents the allocation of a resource to a
demand for another resource, the forecasts induce some compatible assignments.
Hence, when the identity compatibility matrix at the MIP level is combined
with the CONCOR-1 compatibility matrix at the inventory level of the hospital
hubs, and the supply and demand forecasts do not have large errors, there is
more chance to meet the demand, as what is observed for Hospital Hub $3$ and
Hospital Hub $6$.

    1. 2.4.

_The MIP model’s results after using the aggregate forecast supply and demand
(sensitivity analysis) are close to the primary study when the identity
compatibility matrix is used, and improved under the CONCOR-1 compatibility
matrix. However, the identity compatibility matrix is a better choice than the
CONCOR-1 compatibility matrix when using the aggregate forecast supply and
demand and in the presence of forecast errors, consistent with what is
observed in the primary study._

We notice that although the results under the identity compatibility matrix
remain almost unchanged, the total unmet demand and unmet demand ratio values
under the CONCOR-1 compatibility matrix in Table 6 are lower than those in
Table 4. This appears to arise due to the fact that including randomness for
generating the supply and demand for each blood group based on the aggregate
supply and demand may slightly lower the forecasting error. Furthermore,
performing sufficient runs helps to smooth the effect of scenarios where
resources with limited supply, such as the AB blood group CCP, are allocated
in a greedy manner and cause issues for future demand. We note that the
results under the identity compatibility matrix are better than those under
the CONCOR-1 compatibility matrix in Table 6 as the identity compatibility
matrix prevents any cross-allocation of units and lowers the effect of
forecast errors, consistent with what we observed in Table 4. Comparing the
unmet demand and unmet demand ratio values under the CONCOR-1 compatibility
matrix in Table 5 and Table 6 under "MIP Allocations", it is important to note
that while having exact knowledge of supply and demand leads to a fairer
allocation, the degradation in performance is not unreasonable if we use the
forecast values instead, further supporting the efficacy of our approach.
However, as previously discussed, using the identity compatibility matrix in
the MIP model under actual supply and demand leads to more unmet demand (as
compared to using the CONCOR-1 compatibility matrix) due to not allocating
excess units of compatible resources, which prevents the CONCOR-1
compatibility matrix at the inventory level of the hospital hubs to help meet
the demand. Similar to our primary study, if the forecast errors are not too
large, using the identity compatibility matrix at the MIP level under forecast
supply and demand results in making some compatible assignments that can
better help meet the demand when combined with the CONCOR-1 compatibility
matrix at the hospital hubs’ inventory level.

For this case study and based on all the above observations, we conclude that
our primary study when the identity compatibility matrix is used at the MIP
level is the most promising choice for our real-time setting. The reason is
that as long as the forecast errors for the individual blood groups are not
too large, using the combination of the identity compatibility matrix at the
MIP level and the CONCOR-1 compatibility matrix at the inventory level helps
in both preventing greedy allocations and compensating for forecasting errors.
Furthermore, no randomness due to supply and demand probabilities is included
in this choice for calculating the supply and demand for each blood group (as
what is assumed in our sensitivity analysis), and the assignments of the model
are not affected by any ABO blood distributions.

  1. 3.

The role of the MIP model — _The results obtained from our MIP model both
under actual and forecast supply are preferable to what was used in practice._

We compare the unmet demand and unmet demand ratio for both compatibility
matrices using the MIP model allocations under actual supply and demand, and
the actual allocations in the CONCOR-1 trial. We observe that the results
under "MIP Allocations" are more fair than those under "Allocations in
CONCOR-1" (the unmet demand ratio values are closer to each other).
Furthermore, the total unmet demand under "MIP Allocations" is notably lower
($23$ and $13.5$ versus $62$) which further supports the use of our
optimization model, as it accounts for the hospital hubs’ unmet demand and
inventories.

##  5 Conclusion and Future Work

Decision-makers often face challenges in terms of  1. allocating limited
resources, such as vaccines, blood products, and medical equipment, and  2.
forecasting the supply and demand for these resources during epidemics as
there is limited knowledge and historical data about disease demographics.
This work has considered the problem of real-time short-term supply and demand
forecasting and fair allocation of limited resources during epidemics by using
PLR forecasting models and introducing a data-driven MIP resource allocation
model. We have studied the application of our proposed MIP model in a CCP
clinical trial case study with the objective of minimizing each hospital hub’s
unmet ratio of CCP demand. We showed that as long as a hospital hub does not
have a high demand proportion for a limited blood group CCP on a particular
week (in which case a fair allocation is not possible), our MIP model leads to
a balanced and fair final unmet ratio of CCP demand between the hospital hubs.
We also showed that allocating a compatible resource to satisfy the demand for
a resource helps in situations where the actual supply and demand is known,
but might be problematic when we are forecasting the supply and demand and the
actual supply of the compatible resource is limited. It would be interesting
to investigate the range of the forecasting errors within which a particular
compatibility matrix is preferred.

Examining PLR forecasting models on larger datasets and comparing their
performance with more advanced machine learning and time-series forecasting
models could be investigated in future work. We have addressed several
challenges that arise when dealing with sparse data in a real-time setting. We
are interested in evaluating other forecasting models that can adapt to these
challenges while yielding reasonable forecasts. It would be worthwhile to
investigate our MIP model’s performance when other supply and demand
forecasting models are used.

Finally, multiple objective functions and different notions of fairness can be
considered in a single resource allocation problem. Using more than one
objective function and focusing on other notions of fairness such as
minimizing the aggregate unmet demand over all hospital hubs or minimizing the
transportation costs of shipping CCP units with respect to the location of the
hospital hubs are examples of problems of interest. It would also be
interesting to see our MIP model’s performance when applied to other
allocation settings with limited supply. Our MIP model only considers a
centralized supplier, thus we leave the model’s evaluation with the presence
of multiple suppliers to future investigation.

## Acknowledgments

The authors would like to thank Julie Carruthers, Erin Jamula, and Melanie St
John at the McMaster Centre for Transfusion Research for their administrative
support.

## Data Availability Statement

Due to the ethically, legally, and commercially sensitive nature of the
research, participants of this study did not agree for their data to be shared
publicly, so supporting data is not available.

## Funding

The authors acknowledge the support and funding provided by a Mitacs Research
Training Award (Award IT22358), the McMaster Centre for Transfusion Research,
and the Natural Sciences and Engineering Research Council of Canada (NSERC)
Discovery Grant Program (RGPIN-2016-04518 and RGPIN-2022-02999).

## Disclosure statement

No potential competing interest was reported by the authors.

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