# Pandemic Risk Management: Resources Contingency Planning and
Allocation111First version: April 8, 2020.

Xiaowei Chen School of Finance, Nankai University. Email<EMAIL_ADDRESS>Wing Fung Chong Department of Mathematics and Department of Statistics,
University of Illinois at Urbana-Champaign. Email<EMAIL_ADDRESS>Runhuan Feng Department of Mathematics, University of Illinois at Urbana-
Champaign. Email<EMAIL_ADDRESS> Linfeng Zhang Department of
Mathematics, University of Illinois at Urbana-Champaign. Email:
<EMAIL_ADDRESS>
###### Abstract

Repeated history of pandemics, such as SARS, H1N1, Ebola, Zika, and COVID-19,
has shown that pandemic risk is inevitable. Extraordinary shortages of medical
resources have been observed in many parts of the world. Some attributing
factors include the lack of sufficient stockpiles and the lack of coordinated
efforts to deploy existing resources to the location of greatest needs.

The paper investigates contingency planning and resources allocation from a
risk management perspective, as opposed to the prevailing supply chain
perspective. The key idea is that the competition of limited critical
resources is not only present in different geographical locations but also at
different stages of a pandemic. This paper draws on an analogy between risk
aggregation and capital allocation in finance and pandemic resources planning
and allocation for healthcare systems. The main contribution is to introduce
new strategies for optimal stockpiling and allocation balancing spatio-
temporal competitions of medical supply and demand.

## Introduction

### Lessons from Recent Pandemics

An epidemic is an outbreak of a disease that spreads rapidly to a cohort of
individuals in a wide area. According to the definition of the World Health
Organization (WHO), “a pandemic is the worldwide spread of a new disease.”
Because human has little immunity to the new disease, a pandemic can emerge
quickly around the world. One of the most disastrous pandemics in the recent
history is the 1918 flu pandemic which infected around 500 million and
resulted in the deaths of estimated 50 million people worldwide, more than
those died from the World War I. Most recently, the novel coronavirus disease
of 2019 (COVID-19) is an infectious disease caused by a new virus that only
emerged in late 2019 and has since spread out to nearly every country in the
world. According to Cutler and Summers, the COVID-19 could cause financial
losses that sum up to $16 trillion, or roughly $90\%$ of the annual GDP in the
United States [8]. However, its far-reaching impact and consequent economic
fallout due to the loss of productivity is yet to be realized; as of November
28, 2020, the number of infected cases has accumulated to over 61 million
worldwide with the death toll reaching over 1.4 million (see, for example,
[44]).

The repeated history of pandemics in recent decades, such as SARS, swine flu,
Ebola, and the most recent COVID-19, has taught us that pandemic risk is
inevitable. Recent research studies (see, for instance, [29] and [34]) have
shown that the frequency of pandemics has increased over the past century due
to increased social connectivity, long-distance travel, urbanization, changes
in land use, trade and consumption of wild animals, and greater exploitation
of the natural environment. Figure 1 visualizes both frequency and severity of
well-recognized pandemics and public health emergencies of international
concern declared by the WHO since 1900s. The vertical axis represents the
number of documented infections on a logarithmic scale. Both the size and the
color scale of the circles indicate the number of deaths resulting from the
pandemics and public health emergencies. The alarming pattern of increased
frequency clearly points to the critical importance of pandemic risk
management.

Figure 1: Frequency and severity of pandemics and public health emergencies
since 1900s

Governments around the world have been taking blames for their failure to
promptly implement appropriate policies to contain the pandemic. Many
countries experienced severe shortages of resources. [38] studied the critical
role of scarce resources, such as ventilators and personal protective
equipment (PPE), in shaping the direction of COVID-19. Such an unprecedented
challenge exposes the inadequacy in contingency planning and resources
allocation strategies of public health systems. The lack of planning drives
policymakers to make impromptu decisions on resources acquisitions and
allocations that may have exacerbated the extraordinary shortage.

The United States boasts one of the best healthcare systems with a large
network of healthcare professionals, best-equipped medical facilities and
hospitals, and most advanced medical technology. Yet, the country is under-
prepared for the COVID-19 pandemic. There were severe shortages of diagnostic
and preventative medical supplies both for healthcare providers and the
general public in many states in the early stage of the pandemic, which made
it difficult for public authorities to contain the pandemic. According to a
recent report by President Obama’s former advisors on science and technology
[24], there are several contributing factors to the lack of medical resources:
(1) National reserves of critical medical supplies were not replenished
sufficiently prior to COVID-19. Strategic National Stockpile (SNS) was
established by the government in 2003 as the national repository of
pharmaceutical and vaccination stockpiles. The SNS relies on the appropriation
of funding from the Congress. Much of the mask stockpile was depleted during
the 2009 H1N1 pandemic and the Congress has not acted quickly to provide the
funding to replenish the stockpile to an appropriate level projected by many
studies. (2) In order to minimize inventory cost and improve efficiency, many
manufacturers and supply chain management of medical supplies have shifted to
just-in-time (JIT) inventory system prior to the pandemic. Goods are only
received just in time for production and distribution. The JIT system relies
the ability of manufacturers to accurately predict the demand. The initial
public policies such as lock-downs caused major disruptions to supply chains
around the world and there were not sufficient inventories to absorb surge
demand. (3) There was a lack of sufficient coordination among federal and
state governments to deploy existing resources to the most devastated areas in
the country. Healthcare professionals are put at high risk to treat patients
without sufficient personal protective equipment. It was difficult to uncover
and contain the spread of disease without adequate testing. Several states in
the United States acted on their own to secure supplies from foreign
manufactures and engaged in a price bidding war for limited supplies [11].
Existing resources are not necessarily distributed on a basis of health need
[46]. Many hardest hit states have to ration care, while other states have low
utilization of their resources.

While there are policy related issues that require policymakers’ actions,
academics can contribute to the understanding of pandemic evolution and the
resulting dynamics of demand and supply. There is a clear need for the
development of scientific foundation for adaptive strategies for balancing
demand and supply and rationing limited resources. Contingency planning and
resources allocation in a centralized form have been advocated as two coherent
strategies to mitigate catastrophic economic consequences from a pandemic; see
for example [28]. [38] argued that the government should have tracked the use
of resources and the projection of needs in all subsidiaries, and should have
coordinated allocation of resources to reduce shortage across subsidiaries and
over time in the course of a pandemic. A comparable example of centralized
planning is the Federal Emergency Management Agency, which administers many
pre-disaster risk mitigation programs, such as national flood insurance,
mitigation grants, and post-disaster response plans including search and
rescue team, medical assistance teams, monetary reliefs, etc; see [47] and
[43] for information.

Centralized planning and resources allocation have long been practiced as risk
management strategies in the financial industry. For example, banks and
insurers are heavily regulated by governments to ensure their capabilities to
absorb severe financial losses and endure adverse economic scenarios [41]. The
central hypothesis of this paper is that many risk aggregation and capital
allocation techniques drawn from financial and insurance literature can be
extended and applied to pandemic resources planning and allocation.

Figure 2: New York, Florida, and California experience different phases of
COVID-19, based on data as of September 12, 2020 from [44]

### Case Study

Figure 2 depicts the daily confirmed COVID-19 cases in the State of New York
(NY), Florida (FL), and California (CA) in 2020. One could observe that, among
these three states, NY was first hit the hardest by the pandemic in April,
while both FL and CA experienced a peak of cases around July and August;
moreover, although both NY and FL showed comparatively fewer cases in May and
June, CA was savaged by the COVID-19 around that time.

Let us take a thought experiment for the moment. Imagine that these three
states established resource pooling alliance prior to the pandemic. They could
have complemented each other by delivering one state’s surplus resources to
aid another in deficit. For example, in April, the alliance could have
coordinated the efforts to send initial stockpile and increase emergency
production to support the NY; in May and June, remaining resources, together
with additional production, should have been redirected from both NY and FL to
CA; by July and August, when both FL and CA were hit the hardest, unused
resources in the NY could be made available to both FL and CA. Such a
coalition is not unimaginable even in a decentralized political system like
the United States. In April 2020, six northwestern states including New York,
Connecticut, New Jersey, Rhode Island, Pennsylvania, and Delaware, formed a
government procurement coalition for critical medical equipment in an effort
to avoid a bidding war [25].

The purpose of this paper is to propose an overarching framework for different
regions to optimize stockpiling and resources allocation at different pandemic
stages in order to best utilize limited resources. While we use inter-state
resources pooling as an illustrative example, applications can also include
international collaboration on the production, procurement, distribution and
pooling of critical medical resources such as masks, ventilators,
pharmaceuticals, vaccines, etc.

### Pandemic Risk Management Framework and Contribution

A vast amount of recent literature on COVID-19 focus on the prediction of
transmission dynamics (e.g. [15, 26]), infected cases (e.g. [17]), economic
impact (e.g. [1, 21]), and the effect of non-pharmaceutical intervention and
other public policies (e.g. [5]). However, to the best of our knowledge,
academic research on quantitative framework for contingency planning and
resources allocation in response to pandemic risk are rare. While the banking
and insurance industries have long had a rich tradition of developing
technologies for robust risk management, the focus has been largely on
financial and insurable risks. This paper aims to take advantage of the vast
medical literature on epidemic modeling and apply classic concepts from risk
management and insurance literature such as reserving and capital allocation
to pandemic risk management.

Epidemic dynamicspredictive model Need assessmentResource
demandpredictionCentralized stockpiling& distribution strategyResources
allocationstrategyBefore pandemicDuring pandemicPillar IPillar IIPillar III
Figure 3: Three-pillar pandemic risk management framework

A strategic pandemic planning requires scientific assessment, rather than on-
the-fly ad-hoc decisions and patchworks for damage control. In accordance with
current practices of national pandemic preparedness and control strategies
around the world, we summarize and propose a three-pillar framework for
quantitative pandemic risk management:

  * Pillar I: Regional and Aggregate Resources Supply and Demand Forecast. Any pre-pandemic preparation plan should consist of supply and demand assessment and forecast. The supply side should include inventory assessments of critical resources and supplies, the maximum capacity of services, the capability of emergency acquisition and production. The demand side requires an understanding of the dynamics of a potential pandemic across regions and across borders. Historical data and predictive models can be used to project the evolution of a pandemic and the resulting surge demand around a healthcare system.

  * Pillar II: Centralized Stockpiling and Distribution. A central authority coordinates the efforts to develop a national preparedness strategy and to set up reserves of critical resources including preventative, diagnostic and therapeutic resources. A response plan is also necessary to understand how the central authority can deliver resources to different region quickly to meet surge demands and to balance competing interests and priorities.

  * Pillar III: Central-Regional Resources Allocation. A pandemic response plan is critical for a central authority to contain and control the spread of a pandemic in regions under its jurisdiction. As demand may exceed any best-effort pre-pandemic projection, the authority needs to devise optimal strategies that best utilize limited existing resources and minimize the economic cost of supply-demand imbalance. A coordination strategy needs to be in place to ensure smooth communications with regional authorities. The allocation strategy should be based scientifically sound methods taking into account spatio-temporal differences across regions to ensure fairness and impartiality.

It should be pointed out that, while the first pillar is not the focus of this
paper, it plays a critical role in ensuring the adequacy and effectiveness of
planning and responses in second and third pillars. The proposed framework
applies regardless of predictive models for projecting reported cases. The
main technical contribution of this paper lies in second and third pillars for
which we propose centralized resources stockpiling, distribution, and
allocation strategies.

Capital risk management | Pandemic risk management
---|---
Business line and aggregate risk | Regional and aggregate resources demand
Risk-based capital | Centralized stockpiling
Business line capital allocation | Centralized distribution and allocation
Trade-off between surplus/deficiency and cost of capital | Balance of supply/demand and economic cost
Table 1: Comparison between capital and pandemic risk managements

The paper draws inspirations from two sources in insurance and risk management
literature. (1) Insurance applications of epidemic models. Early applications
of epidemic compartment models appeared in [27], in which real option pricing
is used for operational risk management, and [14], which analyzed epidemic
insurance coverage. The study of epidemic insurance was extensively developed
in stochastic setting in [31], [30], and [32], and more recently to cyber risk
assessment by [23]. All of these compartmental models can be used in Pillar I.
(2) Capital allocation. The subject of capital management is well-studied in
the insurance literature. The applications of reserving and capital allocation
form the basis of the proposed Pillars II and III. Table 1 reveals how our
proposed framework shadows the classical capital management. While spatial
balancing of allocation is well-known in banking and insurance (see, for
instance, [9], and [6]), this paper develops a novel spatio-temporal balancing
of resources distribution and allocation, which, to the best of our knowledge,
was not previously studied in either financial or management literature.

The rest of the paper is organized as follows. Each of the next three section
provides detailed discussion of one of the three pillars in the proposed
pandemic risk management framework as well as economic interpretations of
resulting optimal strategies. Numerical examples are embedded in the
discussion for better readership. We conclude in the last section with
discussions of potential applications and future work.

## Pillar I: Regional and Aggregate Resources Demand Forecast

In the pre-pandemic time, a central authority should first model the pandemic
transmission dynamics in each region. Regardless of the choice of
epidemiological models, the central authority should calibrate the model in
each region by its preparedness and other contingency measurements. Indeed,
epidemic forecast models have been used for healthcare policy making and
public communications; see, for example, [33] and [45]. In this paper, in line
with [7] and [22], the population in each region is divided into seven
mutually exclusive compartments, namely, the susceptible ($S$), the exposed
($E$), the mild infected ($I_{1}$), the infected with hospitalization
($I_{2}$), the infected with intensive care ($I_{3}$), the recovered ($R$),
and the deceased ($D$). The dynamics, among these seven compartments, are
governed by a set of ordinary differential equations, and the model is, in
short, called the SEIRD.

This SEIRD model is characterized by a set of ordinary differential equations
that describe population flows among all aforementioned compartments:

$\displaystyle\;\mathrm{d}S(t)$ $\displaystyle=$
$\displaystyle-(\beta_{1}I_{1}(t)+\beta_{2}I_{2}(t)+\beta_{3}I_{3}(t))S(t)\;\mathrm{d}t,$
$\displaystyle\;\mathrm{d}E(t)$ $\displaystyle=$
$\displaystyle[(\beta_{1}I_{1}(t)+\beta_{2}I_{2}(t)+\beta_{3}I_{3}(t))S(t)-\gamma
E(t)]\;\mathrm{d}t,$ $\displaystyle\;\mathrm{d}I_{1}(t)$ $\displaystyle=$
$\displaystyle[\gamma E(t)-(\delta_{1}+p_{1})I_{1}(t)]\;\mathrm{d}t,$
$\displaystyle\;\mathrm{d}I_{2}(t)$ $\displaystyle=$
$\displaystyle[p_{1}I_{1}(t)-(\delta_{2}+p_{2})]I_{2}(t)]\;\mathrm{d}t,$
$\displaystyle\;\mathrm{d}I_{3}(t)$ $\displaystyle=$
$\displaystyle[p_{2}I_{2}(t)-(\delta_{3}+\mu)I_{3}(t)]\;\mathrm{d}t,$
$\displaystyle\;\mathrm{d}R(t)$ $\displaystyle=$
$\displaystyle[\delta_{1}I_{1}(t)+\delta_{2}I_{2}(t)+\delta_{3}I_{3}(t)]\;\mathrm{d}t,$
$\displaystyle\;\mathrm{d}D(t)$ $\displaystyle=$ $\displaystyle\mu
I_{3}(t)\;\mathrm{d}t.$

All parameters in the set of equations bear clinical meanings; $\beta_{i}$,
$i=1,2,3$, is the transmission rate of the infected class $I_{i}$; $1/\gamma$
is the average latency period; $1/\delta_{i}$, $i=1,2,3$, is the average
duration of infection in the class $I_{i}$ before recovery to the class $R$;
$p_{i}$, $i=1,2,3$, represents the rate at which conditions worsen and
individuals require healthcare at the next level of severity; $\mu$ is the
rate for most severe cases in the class $I_{3}$ to the deceased class $D$.
Suppose that the total number of individuals in the entire popluation is $N.$
Each of the ordinary differential equation represents a decomposition of
instantaneous change in the population of a compartment. For example, the
first equation shows that the instantaneous rate of reduction in the number of
susceptible, $-\;\mathrm{d}S(t)$ matches the sum of the rates of infection due
to contacts with the infected of all classes,
$\beta_{1}I_{1}(t)S(t)+\beta_{2}I_{2}(t)S(t)+\beta_{3}I_{3}(t)S(t)$. The
products are due to the law of mass action in biology. For example, the rate
of secondary infection by the mildly infected, $(\beta_{1}N)I_{1}(t)(S(t)/N)$
can be interpreted as the number of adequate contact each infected makes to
transmit the disease $\beta_{1}N$ multiplied by the number of infected
$I_{1}(t)$, multiplied by the percentage that each contact is made with a
susceptible, $S(t)/N.$ All other equations can be explained in similar ways.
The estimation of these model parameters are well-studied in the literature
for the COVID-19 as well as other pandemics, such as [48], [49], and [50].
Based on these parameters, the basic reproductive ratio $R_{0}$ of a pandemic
is given by:

$R_{0}=\frac{N}{p_{1}+\delta_{1}}\left(\beta_{1}+\frac{p_{1}}{p_{2}+\delta_{2}}\left(\beta_{2}+\beta_{3}\frac{p_{2}}{\mu+\delta_{3}}\right)\right).$

The basic reproductive ratio $R_{0}$ can be estimated by empirical data and is
often used to calibrate other parameters. In what follows, we shall use
discretized version of the compartmental model. For example, we use the
notation $I_{1,j}=I_{1}(j\Delta t)$ to indicate the number of mild cases
projected on the $j$-th period each with the length $\Delta t.$ We sometimes
omit the information on $\Delta t$ as the time unit may vary depending on the
reporting period.

Based on predictive models such as the above-mentioned regional SEIRD models,
a central authority could predict, prior to a pandemic or at the onset of a
pandemic, changes in demand over the course of the pandemic. Resources require
different stockpiling and allocation strategies, depending on their shelf
lives. In this paper, we consider two types of medical resources, namely
durable and single-use. Durable resources refer to those that can perform
their required functions for a lengthy period of time without significant
expenditures of maintenance or repair. Single-use resources are those that are
designed to be used once and then disposed of. Mechanical ventilators and PPE
are used as representative examples of durable and single-use resources
respectively in this paper.

| $\alpha\in\left[0,1\right]$ | Data source
---|---|---
China | $<20\%$ | [50]
Italy | $[87\%,90\%]$ | [19]
Seattle | $75\%$ | [4]
Washington | $71.4\%$ | [2]
Table 2: Percentage of severe ICU infected cases requiring ventilators

Durable Resource: Ventilator

Based on the findings in medical literature (references within Table 2), there
are estimates of the percentage $\alpha$ of the infected with intensive care
that require the use of mechanical ventilators. These regional differences can
be addressed in separate regional compartment models. We can use these
estimates to project the ventilator demand by
$X_{j}^{\text{VEN}\left(i\right)}=\alpha I_{3,j}^{\left(i\right)}$, where $i$
indicates the $i$-th region in the alliance and $j$ indicates the $j$-th day
of the pandemic. The model can also be extended to include time-varying
percentage of severe patients requiring ventilators. The calculations in the
rest of the paper would carry through. Figure 4(a) shows the projected
ventilator demands in New York, Florida and California based on the SEIRD
model proposed by [7], which is calibrated to publicly available reported
cases as of September 12, 2020, and demand assessment parameters in Appendix
B.

Single-Use Resource: Personal Protective Equipment

The assessment of need for PPE sets varies by the patients’ class and severity
of medical conditions in care, and the function of medical professionals.
Table 3 offers an example of such need assessment by [12]. Given these
estimates, we can project the regional PPE set demand by
$X_{j}^{\text{PPE}\left(i\right)}=\theta^{E}(S^{\left(i\right)}_{j-1}-S^{\left(i\right)}_{j})+\theta^{I_{2}}I^{\left(i\right)}_{2,j}+\theta^{I_{3}}I^{\left(i\right)}_{3,j}$,
where $\theta^{E}$ is the number of PPE sets per exposed case,
$\theta^{I_{2}}$ is the number of PPE sets per day per hospitalized patient,
and $\theta^{I_{3}}$ is the number of PPE sets per day per intensive care
patient. Note that $S_{j-1}-S_{j}$ represents the daily exposed cases whereas
$I_{2,j}$ and $I_{3,j}$ keeps track of existing infected cases that require
medical attention. Figure 4(b) shows how ventilator and PPE demands are
projected to evolve over time for New York, Florida, and California, based on
the model by [7] and the PPE need assessment in Appendix B.

| Suspected |  Infected Hospitalized cases |  Infected ICU cases
---|---|---|---
| Number of sets per case | Number of sets per day per patient
Healthcare staff | $\theta^{E}$ | $\theta^{I_{2}}$ | $\theta^{I_{3}}$
Nursing | 1-2 | 6 | 6-12
Medical | 1 | 2-3 | 3-6
Cleaning | 1 | 3 | 3
Assistant nursing and other services | 0-2 | 3 | 3
Total | 3–6 | 14–15 | 15–24
Table 3: Minimum amount of PPE sets for different scenarios

(a) Ventilator

(b) Personal protective equipment

Figure 4: Ventilator and personal protective equipment regional demand
prediction in New York, Florida, and California

In the first pillar, the central authority is expected to work with regional
authorities and healthcare professionals to predict the dynamics of regional
demands. All regional data are then compiled and aggregated to form the basis
of forecast for the system-wide resource demand. Suppose that there are a
total of $n$ regions in a healthcare system or medical resource alliance. For
example, the aggregate ventilator demand can be determined by
$X_{j}^{\text{VEN}}=\sum_{i=1}^{n}X_{j}^{\text{VEN}\left(i\right)}$, while the
aggregate PPE set demand may be given by
$X_{j}^{\text{PPE}}=\sum_{i=1}^{n}X_{j}^{\text{PPE}\left(i\right)}$. Figure 5
shows how the aggregate ventilator and PPE demand prediction for the COVID-19
pandemic could have been made in the hypothetical example of a three-state
resources pooling alliance. Observe that projections for ventilators and PPE
sets show very similar patterns as both were driven by the same SEIRD model.
The peaks in demand for ventilator are delayed compared with those for PPE
sets in Figure 5 due to the fact that it may take a few days before newly
diagnosed patients to develop symptoms that require ventilator intervention.
The projection of regional and aggregate demands offers health authorities a
clear understanding of the temporal competition of critical resources.

(a) Ventilator

(b) Personal protective equipment

Figure 5: Ventilator and personal protective equipment aggregate demand
prediction in New York, Florida, and California

It should be pointed out that predictive models in Pillar I, such as the SEIRD
model introduced in this section, are used for multiple purposes as shown in
Figure 3. First, they need to be developed prior to a pandemic using
historical data and to form the basis of demand forecast for contingency
planning in Pillar II. Then, as a pandemic starts to emerge, the predictive
models also need to be re-calibrated and updated with latest medical knowledge
and reported cases. New forecasts would then be fed into models to determine
optimal allocation strategies in Pillar III. As medical knowledge of the viral
disease evolves and predictive models improve over time, Pillars I and III may
be revisited from period to period. When a distribution schedule of resources
requires an update, we can go back to Pillar II. Therefore, the three-pillar
framework may be utilized in circles such as Pillars I, II, III, I, III, I,
III, I, II, III, etc.

## Pillar II: Centralized Stockpiling and Distribution

As the pandemic unfolds, many hospitals and healthcare facilities may run out
of pharmaceuticals and other essential resources before emergency production
can pick up and additional supplies become available. To meet the surge demand
at the onset of a pandemic, many countries maintain national repositories of
antibiotics, vaccines, chemical antidotes, antitoxins, and other critical
medical supplies. A centralized stockpiling strategy is intended to provide a
stopgap measure to meet the surge in resources demand at the early stage of
the pandemic. There has been well-established literature on stockpiling
strategies for influenza pandemics; see, for example, [20] and [42].

One should keep in mind that a practical stockpiling strategy is often an act
of balance between adequate supply and economic cost. On one hand, under-
stocking is a common issue as resources and their storage can post heavy cost,
and the actual demand during the pandemic outbreak could deviate from the
projection; for example, [10] claimed that as many as $20$ states in the
United States are expected to encounter shortage in ICU beds when the COVID-19
cases peak. On the other hand, excessive stockpiling for long term could lead
to unnecessary waste, especially for disposable and perishable resources; for
instance, [13] reported that, in March 2020 during the COVID-19 pandemic, the
Strategic National Stockpile in the United States stocked $13$ million N95
masks, of which as many as $5$ million may have expired, partly contributing
to the nation-wide shortage of masks.

In the second pillar of our proposed framework, based on the estimated
aggregate resources demand, the central authority could then develop
stockpiling and distribution strategies in regular time before a pandemic.
Notice that durable resources such as ventilators can be reused throughout the
pandemic, while single-use resources such as PPE sets must be disposed of
after one-time usage. Hence, we have to treat them separately for optimal
centralized stockpiling and distribution strategies.

### Durable Resource: Ventilator

It is typical that a central authority has to determine an optimal initial
stockpile size $K_{0}$ of resources to maintain in some centralized location.
In addition, to meet surge demand, the authority may need to reach contractual
agreements with suppliers for emergency orders, which may be limited by the
maximum production rate of $a$ units per day during a pandemic. Since
ventilators are durable, the stock of ventilators does not decrease over time
due to usage. We assume that they can be deployed to different regions at
negligible cost. Therefore, the total number of available ventilators in the
entire alliance is given by $K_{j}=K_{0}+aj$, on the $j$-th day since the
onset of the pandemic. Hence, the only decision variable of the central
authority in the case of ventilators is the initial stockpile size $K_{0}$.

(a) Extreme understocking

(b) Extreme oversupply

Figure 6: Two extreme scenarios of initial stockpile size $K_{0}$ for
ventilators

To better explain the need for an optimal initial stockpile size $K_{0}$,
consider two extreme cases in Figures 6(a) and 6(b) for the three-state
alliance. On one end of the extreme, the central authority may decide not to
hold any initial stockpile but simply rely on the maximum emergency production
limit during the pandemic; Figure 6(a) shows a clear extreme shortage at the
first peak time of aggregate ventilator demand. On the other end, suppose that
the central authority decides to hold an extraordinary amount of initial
stockpile for ventilators to meet the highest peak of aggregate ventilator
demand; Figure 6(b) illustrates a clear extreme oversupply of ventilators
during most of the time of pandemic; also, in this case, the economic cost of
severe storage can be huge. Therefore, the central authority has to take a
delicate balance on an initial stockpile size $K_{0}$ that takes into account
the economic cost of shortage, oversupply, as well as storage costs.

Consider the following optimization model for an initial stockpiling size.

$\min_{K_{0}\geq
0}\sum_{j=1}^{m}\omega_{j}\left(\frac{\theta^{+}_{j}}{2}\left(X^{\text{VEN}}_{j}-\left(K_{0}+aj\right)\right)_{+}^{2}+\frac{\theta^{-}_{j}}{2}\left(X^{\text{VEN}}_{j}-\left(K_{0}+aj\right)\right)_{-}^{2}+c_{j}\left(K_{0}+aj\right)\right)+c_{0}K_{0},$
(1)

where $m$ is the number of days of the pandemic, $\omega_{j}$ is a weight for
significance of precision for the costs on the $j$-th day of the pandemic,
$\theta_{j}^{+}$ is an economic cost per squared unit of shortage,
$\theta_{j}^{-}$ is an opportunity cost per squared unit of oversupply,
$c_{j}$ is the aggregate cost of possession per unit of ventilators per day,
$c_{0}$ is the initial stockpile cost, which may include both the acquisition
cost and expected cost of possession (storage, maintenance, inventory
logistics, opportunity cost). The quadratic form can be interpreted as
follows. While one copy of the quantity
$X^{\text{VEN}}_{j}-\left(K_{0}+aj\right)$ represents the amount of resource
imbalance (shortage or surplus), the other copy
$(\theta^{\pm}_{j}/2)[X^{\text{VEN}}_{j}-\left(K_{0}+aj\right)]_{\pm}$ can be
viewed as the (linear) variable cost of the imbalance. In other words, the
larger the imbalance, the higher the price to pay. The quadratic form is the
product of cost per unit and the unit of imbalance, which yields the overall
economic cost of imbalance. The weight $w_{j}$ can be used for different
purposes. For example, it may be reasonable to assume that the precision of
meeting demands in near future is more important than that in the far-future
given the uncertainty with prediction. Another case may be to make the weight
proportional to the daily demand $X_{j}$ as the demand-supply imbalance can
have a greater impact on population dense areas than otherwise.

$Y_{[1]}$$Y_{[2]}$$\dots$$Y_{[J-2]}$$Y_{[J-1]}$$Y_{[J]}$$Y_{[J+1]}$$\dots$$Y_{[m-1]}$$Y_{[m]}$$K_{0}$

Figure 7: Optimal initial stockpile $K_{0}$ relative to projected shortages
without initial stockpile

To understand the analytical solution to this problem, we need to look at the
projected shortage without any initial stockpile,
$Y_{j}:=X_{j}^{\text{VEN}}-aj$, for $j=1,\cdots,m,$ which is the accumulated
demand less the accumulated supply apart from the initial stockpile. Note that
we consider the accumulated supply because the resources are durable and can
be reused. When $Y_{j}>0,$ there is a drain on the initial stockpile as
current demand exceeds the accumulated supply. Otherwise, the stockpile
increases as supply exceeds demand. Because the economic costs of shortage and
surplus are weighed differently, the value of this objective function depends
on the number of days with decreasing stockpile $(Y_{j}>0)$ and those with
increasing stockpile $(Y_{j}<0)$. The analytical solution to this problem
requires sorting projected shortages in an ascending order. Let us denote the
sorted sequence by $\\{Y_{[j]},j=1,\cdots,m\\},$ where $Y_{[j]}$ represents
the $j$-th smallest projected shortage. In the objective function (1), the
cost coefficient $\theta^{\pm}_{j}$ applies according to whether or not
stockpile exceeds demand. If $K_{0}$ is placed below $Y_{[j]}$, there is a
shortage in the healthcare system and hence the cost coefficient $\theta^{+}$
is applied. Otherwise, there is a surplus in the system and the cost efficient
$\theta^{-}$ is applied. The optimality is achieved when $K_{0}$ is kept at a
delicate position. See Figure 7. The nature of the sum of squared shortages in
(1) determines that the optimal initial stockpile $K^{\ast}_{0}$ should be
squeezed between $Y_{[J-1]}$ and $Y_{J}$ in such a way that

$Y_{[J-1]}\leq\frac{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}\left(Y_{[j]}-\frac{c_{[j]}}{\theta_{[j]}^{-}}\right)+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}\left(Y_{[j]}-\frac{c_{[j]}}{\theta_{[j]}^{+}}\right)-c_{0}}{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}}\leq
Y_{[J]}.$

Once $J$ is identified, the optimal stockpile $K_{0}^{*}$ is given by

$K_{0}^{*}=\max\left\\{\frac{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}\left(Y_{[j]}-\frac{c_{[j]}}{\theta_{[j]}^{-}}\right)+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}\left(Y_{[j]}-\frac{c_{[j]}}{\theta_{[j]}^{+}}\right)-c_{0}}{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}},\;\;0\right\\}.$

The proof of this result can be found in Appendix A.1. This result shows that
the optimal initial stockpile $K_{0}$ is the weighted average of all projected
shortages discounted by the cost of possession relative to the economic cost
of shortage, $Y_{[j]}-c_{[j]}/\theta^{\pm}_{[j]}$. The adjustment term
$c_{[j]}/\theta^{\pm}_{[j]}$ indicates that, the higher cost of possession
relative to economic cost of imbalance, the fewer ventilators should be
acquired. It is logical that, if the cost of possession for durable resource
is too high, the central authority in a poor country may have little financial
means to pay for stockpiling and be left with no choice but to deal with the
demand-supply imbalance. In contrast, if the economic cost of imbalance is too
high due to lost productivity or even the society’s resentment on government’s
failure to meet demand, then the central authority would ignore the cost of
possession and do everything possible to reduce the shortage.

(a) $\theta_{j}^{+}=\theta_{j}^{-}$ (Shortage costs the same as surplus)

(b) $\theta_{j}^{+}=20\theta_{j}^{-}$ (Shortage costs more)

Figure 8: Optimal initial stockpile size $K_{0}$ for ventilators according to
different weights of economic cost

Figure 8 depicts optimal initial stockpile size in the case study. The model
parameters are provided in Appendix B. Observe that optimal initial stockpiles
are chosen to reduce shortage in the early stages and oversupply in the late
stages of the pandemic, compared with those strategies shown in Figure 6. When
the resource shortage costs the same or less than the resource surplus, Figure
8(a) shows that the strategy requires less initial stockpile due to the
excessive amount of supply after the pandemic dies down. In contrast, if the
shortage weighs more than the surplus, the strategy is to reduce shortage in
early stages at the expense of increasing oversupply in late stages; see
Figure 8(b).

### Single-Use Resource: Personal Protective Equipment

Similar to the case of durable resources, the central authority needs to set
up an initial stockpile size $K_{0}$ of single-use resources such as PPE and
make contractual agreements with emergency suppliers, which can provide
additional supply at the production rate $a$ units per day. Since PPE is of
single-use, during the pandemic, the central authority has to stockpile PPE
sets not only for the present but also to potentially deploy them for later
time in order to meet surge demand. Therefore, the central authority needs to
plan for both initial stockpile size $K_{0}$ and the amount of distribution
$k_{j}$ to all regions on day $j$. The dynamics of the centralized storage
$\\{K_{j},j=0,1,\dots,m\\}$ is determined by the recursive relation
$K_{j}=K_{j-1}+a-k_{j}+\left(k_{j}-X_{j}\right)_{+}$ for $j=1,\dots,m$. The
relation can be interpreted as follows. The current stockpile $K_{j}$ is based
off the previous period’s stockpile $K_{j-1}$, increased by the net surplus of
new supply $a$ less the arranged distribution up to the total demand,
$\max\\{k_{j},X_{j}\\}.$ In other words, if we arrange to distribute $k_{j}$
units but can only consume $X_{j}<k_{j},$ then the unused amount should count
towards the centralized storage for future use.

(a) Aggressive distribution

(b) Conservative distribution

Figure 9: Two extreme scenarios of distribution schedule
$k_{1},k_{2},\dots,k_{m}$ for personal protective equipment Figure 10: Optimal
distribution schedule $k_{1},k_{2},\dots,k_{m}$ and initial stockpile size
$K_{0}$ of personal protective equipment for the three-state resource pooling
alliance

Consider two extreme cases in Figures 9(a) and 9(b) in the case study. In an
extreme case, assume that the central authority decides to distribute as much
as possible to meet the demand until the centralized storage is exhausted.
This aggressive early distribution strategy is depicted in Figure 9(a). After
the storage depletion, the system relies only on the new supply, which clearly
is not sufficient to meet demands and can cause severe shortage at the time of
second peak. In the other extreme, the central authority may choose to hold
off dispersing any equipment at all till the point that the storage is
believed to be sufficient to cover all future demands. Such a conservative
distribution strategy is illustrated in Figure 9(b). The challenge with this
strategy is that the central authority would have to deal with the
repercussion of not providing any assistance in the early stage of the
pandemic. Therefore, it is sensible that the central authority develop a
distribution schedule that takes a temporal balance of varying needs from all
regions. Here we introduce the optimization problem for both an initial
stockpile and the distribution schedule of single-use resources.

$\displaystyle\min_{K_{0}\geq 0,k_{1},\dots,k_{m}}$
$\displaystyle\;\sum_{j=1}^{m}\omega_{j}\left(\frac{\theta^{+}_{j}}{2}\left(X^{\text{PPE}}_{j}-k_{j}\right)_{+}^{2}+\frac{\theta^{-}_{j}}{2}\left(X^{\text{PPE}}_{j}-k_{j}\right)_{-}^{2}+c_{j}K_{j}\right)+c_{0}K_{0}$
such that
$\displaystyle\;K_{j}=K_{j-1}+a-k_{j}+\left(k_{j}-X^{\text{PPE}}_{j}\right)_{+}\geq
0\text{ and }k_{j}\geq 0,\quad\text{for }j=1,2,\dots,m,$

where $c_{j}$ is the centralized cost of possession per unit of PPE per day.
It should be pointed out that the centralized storage should be kept non-
negative for practical purposes and the distribution amount should also be
kept non-negative. Negative distribution could mean confiscation of regional
resources for system wide re-distribution, which is not considered in this
paper.

Figure 10 depicts the case of optimal distribution schedule for the three-
state alliance. Observe that the optimal supply distribution schedule stays
below the trajectory of demand. The slight shortage results from the
consideration of the cost of possession. Should the cost of possession be
zero, the optimal supply would be to match the demand exactly at all times.
The initial stockpile can be set artificially high so that any desired amount
can be carried over from period to period and last long enough to support all
future demands. In the presence of possession cost, Figure 10 also reveals a
distribution strategy that in essence ignores the demands at the start of a
pandemic and after the pandemic dies down and instead focuses on meeting
demands at the first peak of the pandemic. Keep in mind that the weight of
significance $\omega_{j}$ in this example is set to be proportional to the
size of demand. Therefore, the strategy prioritizes meeting the demand in the
first peak over other periods due to its high demand. If we were to choose the
same weight $\omega_{j}$ for all periods, the shortage would be more balanced
among all periods.

This optimization problem is cumbersome to be solved analytically. However, it
is straightforward to show that any oversupply distribution scheme
$k_{j}>X^{\text{PPE}}_{j}$ must be sub-optimal, and hence the problem can be
simplified as follows.

$\displaystyle\min_{K_{0}\geq 0,k_{1},\dots,k_{m}}$
$\displaystyle\;\sum_{j=1}^{m}\omega_{j}\left(\frac{\theta^{+}_{j}}{2}\left(X^{\text{PPE}}_{j}-k_{j}\right)^{2}+c_{j}K_{j}\right)+c_{0}K_{0}$
such that $\displaystyle\;K_{j}=K_{j-1}+a-k_{j}\geq 0\text{ and }0\leq
k_{j}\leq X^{\text{PPE}}_{j},\quad\text{for }j=1,2,\dots,m,$

Because of the convexity, it can be solved numerically using Disciplined
Convex Programming (DCP), which requires minimal computational time [18]. The
solution shown in Figure 10 is obtained with the help of an R package CVXR
developed based on the DCP method [16].

## Pillar III: Centralized Resources Allocation

In the time of severe resources shortage, a coordinated effort becomes
necessary to obtain additional supplies and to ration limited existing
resources. Existing resources are not necessarily distributed on a basis of
health need or justice [46]. Many hardest hit states have to ration care,
while other states have low utilization of their resources. As alluded to
earlier, not all regions experienced surge in demands at the same time (see
Figure 2), it has been long argued that United States federal government
should have tracked the current use and the projection of needs in all states
and coordinated allocation of resources to reduce shortage across regions and
over time, during the COVID-19 pandemic [38].

There are two common types of resources allocation problems in the course of a
pandemic, both of which can be formulated and cast in the Pillar III of our
proposed framework.

  1. 1.

Macro level resources pooling. A central authority acts in the best interest
of a union of many regions to increase supply as well as to coordinate the
distribution of existing and additional resources among different regional
healthcare providers.

  2. 2.

Micro level rationing. Facing an imbalance of demands and supplies in medical
equipment and resources, hospitals often have to make difficult but necessary
decisions to ration limited existing resources as well as new supplies.

While in both cases the aim of allocation is to deliver limited resources to
where they are needed the most, the macro level pooling large addresses
spatio-temporal differences and the micro level rationing focuses on
healthcare effectiveness and justice. The setting of standards, protocols and
policies can have profound impact on the functioning of a healthcare system at
the time of crisis. Therefore, the best practice of resources allocation
should be based on scientific assessment and evaluation rather than on-the-fly
ad-hoc decisions and a patchwork of damage-control rules.

  1. 1.

Resources allocation should be based on a holistic approach to address
concerns of all stakeholders.

There are often conflicting interests and priorities for using limited
resources. For example, when medical supplies are scarce, many countries,
states and cities are competing for resources. While each state acts in its
best interest to acquire medical devises and protective equipment, a federal
government may see the urgent need to seize control of the cargo to boost a
centralized stockpile. A holistic approach aims to strike a balance among
different objectives for various stakeholders.

  2. 2.

Scientific methods for resources allocation should be developed under a set of
optimization objectives, meet certain ethical and humanitarian criteria, take
into account logistic and budgetary constraints.

When a pandemic breaks out, it often spreads from one cluster to another in
geographic areas due to its transmission dynamics and affects different
sectors of a healthcare system in a chain reaction. Medical needs can vary
greatly by demographics and other socio-economics factors. While there is no
universal “one-size-fits-all” solution for allocation problems, there are a
set of quantifiable and justifiable criteria. While it is difficult address
all of these criteria in a single model, we believe that they can be
formulated similarly as in this section.

     * •

Minimization of shortage and oversupply

Decision makers need to take into account spatio-temporal differences in
demand and supply over the course of a pandemic. It is imperative for
authorities to allocate more resources to epicenters of a pandemic than other
regions under less imminent threat. For example, New York City was the first
in the State of New York to witness the COVID-19 pandemic, when other counties
had little to no reported cases [3]. The State governor issued an executive
order to take ventilators and other protective gears from underutilized
private hospitals and companies. As infected cases stabilize or even decline
in pandemic-savaged regions, a central authority may need to shift its
attention to other areas of potential outbreaks and allocate resources in
anticipation of new waves. This was also evident when many states in the
United States in early stages of outbreak during the COVID-19 took preemptive
measures to procure medical supplies from countries like China and South Korea
which have developed production capacities after the local epidemics are under
control. Therefore, it is sensible to develop an allocation strategy that
minimizes shortage and oversupply across different regions and over the life-
cycle of a pandemic.

     * •

Promoting and rewarding instrumental value

Critical preventive gears and medical care should be provided first to
healthcare workers in the front line, employees in essential businesses and
critical infrastructures. Not only because they are at high risk due to their
exposure to infectious disease, but also the society bears heavy economic cost
when these workers fall ill and are unable to return to work. The lack of
sufficient front-line workforce may cause severe disruptions to public
services, which can have rippling effect on the rest of the economy. Priority
access to medical care can be a critical incentive for retention.

     * •

Prioritizing the worst off

The ultimate goal of a healthcare system is to save lives. Access to critical
medicare treatment should be reserved for patients facing life-threatening
conditions, when there is an insufficient supply of equipment such as
ventilators.

     * •

Maximization of benefit from treatment

Maximization of the benefit requires prognosis on how patients are likely to
survive with treatment. Some recent study of COVID-19 patients in the United
States finds that most patients do not survive after being placed on
mechanical ventilators [37]. To maximize the benefit, access to ventilator
treatment should be prioritized for younger patients who can benefit the most
and have the higher chance of survival. For example, many hospitals in Italy
lowered the age cutoff from 80 to 75 in order to ration limited ventilators
[39]. Such a strategy often leads to ethical dilemma when in conflict with
prioritizing the worst off.

The third pillar of the proposed pandemic risk management framework is to
allocate limited resources for different regions, building on the proposed
optimal centralized stockpiling and distribution strategies. Figures 11(a) and
12(a) put the regional resources demand and optimal aggregate supply together
for the ease of exposition.

(a) Regional ventilator demands

(b) Optimal ventilator allocations

Figure 11: Optimal ventilator allocations in New York, Florida, and California

(a) Regional PPE demands

(b) Optimal PPE allocations

Figure 12: Optimal personal protective equipment allocations in New York,
Florida, and California

### Durable Resource: Ventilator

Throughout this section we consider the allocation of existing resources in a
healthcare system with $n$ regions during a pandemic that lasts for $m$ days.
We always use the superscript $(i)$ to indicate quantities for the $i$-th
region. Bear in mind that there could still be aggregate shortage of supply
for ventilators and PPE sets to all regions in the alliance. The central
authority would have to take a holistic view of competing interests of
participating regions. On each day in the pandemic, when the aggregate demand
exceeds the aggregate supply, the central authority should choose to allocate
resources taking into account spatial differences in demand and supply. This
motivates the optimization model for ventilator allocations:

$\displaystyle\underset{j=1,2,\dots,m}{\min_{K_{j}^{(i)}\geq
0;i=1,2,\dots,n;}}\;\;$
$\displaystyle\sum_{j=1}^{m}\sum_{i=1}^{n}\omega_{j}^{(i)}\left(\frac{\theta_{j}^{+(i)}}{2}\left(X_{j}^{\text{VEN}(i)}-K_{j}^{(i)}\right)_{+}^{2}+\frac{\theta_{j}^{-(i)}}{2}\left(X_{j}^{\text{VEN}(i)}-K_{j}^{(i)}\right)_{-}^{2}\right)$
such that $\displaystyle\sum_{i=1}^{n}K_{j}^{(i)}=K_{j},\quad\text{for
}j=1,2,\dots,m,$

where $\omega_{j}^{\left(i\right)}$ is a weight to the $j$-th day of the
pandemic in the $i$-th allied region, $\theta_{j}^{+(i)}$ is an economic cost
per squared unit of shortage, $\theta_{j}^{-(i)}$ is an opportunity cost per
squared unit of oversupply. The quadratic terms
$\theta_{j}^{\pm(i)}/2\left(X_{j}^{(i)}-K_{j}^{(i)}\right)^{2}$ represents the
economic cost due to the demand-supply imbalance. Note that
$\theta_{j}^{\pm(i)}/2$ measures the rate of increase in cost per unit, and
hence $\theta_{j}^{\pm(i)}/2\left(X_{j}^{(i)}-K_{j}^{(i)}\right)$ represents
the linear variable cost per unit. The variable cost in principle reflects the
law of demand that the price increases with the quantity demanded. Therefore,
the total cost is the product of variable cost per unit
$\theta_{j}^{\pm(i)}/2\left(X_{j}^{(i)}-K_{j}^{(i)}\right)$ and the total unit
of imbalance $\left(X_{j}^{(i)}-K_{j}^{(i)}\right)$. The economic cost is used
to account for both potential loss of lives due to the lack of resources and
the opportunity cost of idle medical sources due to oversupply. The structure
of economic cost is used not only for its mathematical tractability, but also
to penalize large imbalance of demand and supply. The weight
$\omega_{j}^{(i)}$ can be used to measure the relative importance of resource
allocation for region $i$ at time $t_{j}$ to other regions and time points.
There are some examples of its applications under various criteria for
resource pooling or rationing. For example, in a national contingency
planning, where $X^{(i)}$ is used as predicted demand from each region, a
metropolitan area with a large population may carry more weight than a rural
area with a small region for political reasons. When hospitals have to ration
limited resources, they may implement the strategy to maximize the benefit
from treatment. In such a setup, $X^{(i)}$ represents the demand from a
particular cohort. A decision maker may give higher weight to age cohorts with
more remaining life years than age cohorts with less remaining years. It is
also a common strategy to give priorities for access to medical resources to
healthcare workers. In both cases, the set of weights $\omega^{(i)}_{j}$
reflects the management’s priorities and preferences over time.

The constraint $\sum^{n}_{i=1}K^{(i)}_{j}=K_{j}$ indicates that resources
allocated to different regions must add up to the total amount of supply
available to the central authority. The evolution of supply
$\\{K_{j},j=1,2,\dots,m\\}$ is based on the centralized stockpiling strategy
discussed in previous sections. The evolution of demand
$\\{X^{(i)}_{j},i=1,\dots,n,j=1,2,\cdots,m\\}$ can be based on forecasts from
epidemiological models fitted to most recent local data.

### Single-Use Resources: Personal Protective Equipment

The allocation of single-use resources is similar to that of durable
resources. The key difference lies in the amount of stockpile to be released
each period. For durable resources, the central authority distributes the
accumulated stock $K_{j}$ at any given time $j$. Because single-use resources
cannot be reused, the central authority can only distribute incremental amount
according to some distribution schedule. With this difference in mind, we
formulate the allocation of single-use resources by an optimization problem.

$\displaystyle\underset{j=1,2,\dots,m}{\min_{k_{j}^{(i)}\geq
0;i=1,2,\dots,n;}}\;\;$
$\displaystyle\sum_{j=1}^{m}\sum_{i=1}^{n}\omega_{j}^{(i)}\left(\frac{\theta_{j}^{+(i)}}{2}\left(X_{j}^{\text{PPE}(i)}-k_{j}^{(i)}\right)_{+}^{2}+\frac{\theta_{j}^{-(i)}}{2}\left(X_{j}^{\text{PPE}(i)}-k_{j}^{(i)}\right)_{-}^{2}\right)$
such that $\displaystyle\sum_{i=1}^{n}k_{j}^{(i)}=k_{j},\quad\text{for
}j=1,2,\dots,m.$

Note that the distribution amount $k_{j}$ could be determined prior to a
pandemic by some contingency planning or during a pandemic by an adjusted
distribution schedule. The “Single-Use Resources: Personal Protective
Equipment” section offers an example of how such a distribution schedule can
be determined to take into temporal competition of single-use resources.

### Holistic Allocation Algorithms

Because both allocation problems take the same form, their analytical
solutions can be derived in the same way. Because the allocation is done from
period to period in the solution, we shall suppress the subscript $j$ for
brevity. To simplify notation in the solution, we will use $X^{(i)}$ without
the indicator of resource type for the demand in region $i$, and use $K^{(i)}$
for the quantity of allocated resources in region $i$.

Here we discuss the analytical solutions to the optimization problems
presented above, from which we can glean economic insights. The proofs of
these solutions are relegated to Appendix A.2. The central authority has to
first determine whether or not there is a system wide surplus or shortage. The
allocation strategy differs under these scenarios.

#### System wide surplus

If there is an overall surplus in the healthcare system at time $j$, i.e.,
$K>\sum\limits_{r=1}^{n}X^{(r)}=X$, then only the economic cost for oversupply
$\theta^{-(i)}$ applies and the optimal allocation of existing supply to the
$i$-th region is given by

$\displaystyle
K^{(i)}=\left(1-\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\right)X^{(i)}+\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\left(K-\sum\limits_{r\neq
i}X^{(r)}\right),\forall i=1,2,\cdots,n.$ (2)

Observe that the allocation formula (2) has an explicit economic
interpretation, which shows that the optimal supply for region $i$ results
from a balance of two competing optimal solutions.

  * •

Self-concerned optimal supply: $X^{(i)}$

If region $i$ can ask for as much as it needs, then this amount shows the
ideal supply in the best interest of the region alone. The demand and supply
for all other regions are ignored in its consideration.

  * •

Altruistic optimal supply: $K-\sum_{r=1;r\neq i}^{n}X^{(r)}$

If the region $i$ places interests of all other regions above its own, then
the medical supply goes to other regions and region $i$ ends up with the
leftover amount.

The central authority has the responsibility to mediate among regions
competing for resources. The formula (2) indicates that the optimality for
region $i$ in consideration of the entire system is the weighted average of
two extremes, namely the self-concerned optimal and the altruistic optimal
supplies. It should be pointed that the average of two optimal supplies is
determined by the harmonic weighting
$\frac{1}{\omega^{(i)}\theta^{-(i)}}\Big{/}\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}$
as opposed to arithmetic weight
$\omega^{(i)}\theta^{-(i)}\Big{/}\sum\limits_{r=1}^{n}\omega^{(r)}\theta^{-(r)}$.
It is known in [6] that in multi-objective Pareto optimality the harmonic
weighting is always used for balancing competing interests of participants in
a group whereas the arithmetic weighting serves the purpose of balancing
competing objectives of the same participant. The pandemic resource allocation
problem is in essence a model of Pareto optimality with regards to competing
interests of members in a group.

An alternative interpretation of formula (2) can be obtained from the
equivalent formula

$K^{(i)}=X^{(i)}+B^{(i)}\left(K-\sum_{r=1}^{n}X^{(r)}\right),\qquad
B^{(i)}=\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}.$
(3)

It follows from (3) that the allocated resource is always presented as an
adjustment to the actual demand. When there is a surplus in the system supply
after optimal supplies have been fully distributed to all regions, then
additional resource can be made available for region $i$ and each region
obtains a portion determined by harmonic weighting. Observe that
$\sum^{n}_{i=1}K^{(i)}=K$ as expected since $\sum^{n}_{i=1}B^{(i)}=1.$

#### System wide shortage

If there is an overall shortage in the healthcare system at time $j$, i.e.,
$K\leq\sum_{r=1}^{n}X^{(r)}=X$, it turns out that the optimal allocation
strategy is to deliver the resources where they are needed the most. We can
summarize the algorithm in three steps:

  * Step 1: Demand ranking.  The first order of business is to sort regional demands $\\{X^{(i)},i=1,\dots,n\\}$ in a descending order. We use the subscript $[i]$ to indicate the $i$-th largest order statistic, i.e., $X^{[1]}\geq\cdots\geq X^{[n]}\geq 0$. The ranking of regional demands determines the order in which the regions are considered for resource allocation in the next step.

  * Step 2: Frugality test.  The algorithm first tests cases that perform allocation rules in a similar way to (3). For any fixed $I=1,\cdots,n$, consider the holistic allocation rule that provides for $I$ regions with largest demand by

$\displaystyle\tilde{K}^{[i]}=X^{[i]}+\tilde{B}^{[i]}\left(K-\sum_{r=1}^{I}X^{[r]}\right),\qquad\tilde{B}^{[i]}=\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum_{r=1}^{I}\frac{1}{\omega^{[r]}\theta^{+[r]}}}.$
(4)

To find the optimal number $I$ of regions to provide support to, the algorithm
ensures that the allocation rule should be frugal to meet the following
criteria:

    * (i)

The total supply $K$ is only almost enough to meet the demands for all $I$
regions;

$K\leq\sum_{r=1}^{I}X^{[r]}.$

    * (ii)

When the allocation rule (4) is forcefully applied to all regions, the $I$
regions with highest demands should receive non-negative allocation and the
rest of the group receive negative allocation.

$\tilde{K}^{[1]},\cdots,\tilde{K}^{[I]}\geq
0>\tilde{K}^{[I+1]},\cdots,\tilde{K}^{[n]}.$

There is a unique value of $I$ that passes the frugality test. As the aim of
the strategy is to cover as many regions of highest demand as possible, the
search algorithm stops after the total demand of $I$ regions exceeds the
available supply. The algorithm would reach a rule that can be rewarding for
those $I$ regions but discourages allocations to the rest.

  * Step 3: Holistic allocation. Once the algorithm settles on the value of $I$, all existing resources are divided among the $I$ states according to the holistic allocation principle. In other words, the allocation of supply to the $i$-th region is given by

$\displaystyle K^{[i]}$ $\displaystyle=\tilde{K}^{[i]},\qquad\forall
i=1,2,\cdots,I;$ $\displaystyle K^{[I+1]}$ $\displaystyle=\cdots=K^{[n]}=0.$

The general idea of the holistic allocation is illustrated in Figure 13. The
burden of shortage $Y:=\sum^{I}_{r=1}X^{[r]}-K$ is carried by all $I$ states
in proportion to their respective harmonic weight $\tilde{B}^{[i]}$.
Therefore, each region receives its demand less its “fair” portion of system-
wide shortage, i.e. $X^{[i]}-\tilde{B}^{[i]}Y.$

System wide total supply $K$Shortage$\tilde{B}^{[1]}$$\cdots$$\tilde{B}^{[I]}$
Regional demand $X^{[1]}$$\dots$$X^{[I]}$ Allocation
$\dots$$K^{[1]}$$\cdots$$K^{[I]}$$K$

Figure 13: Holistic allocation of resources in face of system wide shortage

The solution to the three-state alliance for pooling ventilators and PPE sets
are shown in Figures 11 and 12. Figure 11(b) depicts the case of optimal
allocation strategies for ventilators, which confirms the intuition in the
“Case Study” Section. In April, the central authority could have optimally
reallocated all available aggregate ventilators in the alliance to NY; This is
owing to the fact that NY has the highest demand of all three states. By May
and June, ventilators in NY could have gradually reallocated to both FL and
CA; in July and August, with the reallocated resources from NY, both FL and CA
should have experienced no shortage of ventilators at all. Figure 12(b)
illustrates optimal allocation strategies for PPE sets in the case study. In
April, the central authority could distribute stockpiled PPE sets and have all
sent to NY for emergency response; As the pandemic dies down for NY and picks
up for CA and FL, the resources are more evenly spread in June. By August, all
PPE sets should be released to CA and FL both with high demands.

## Conclusions and Limitations

The COVID-19 pandemic has placed extraordinary demands and constraints on
public healthcare systems, exposing many problems such as the lack of adequate
planning and coordination among others. This paper investigates what could
have been done better to reduce the imbalance of medical resources demand and
supply. Inspired by classical theory of risk aggregation and capital
allocation, this paper proposes a three-pillar resources planning and
allocation framework — demand forecast, centralized stockpiling and
distribution, and centralized resources allocation. This paper further
develops a novel spatio-temporal balancing of resources and can potentially
used by public policymakers as quantitative basis for making informed
decisions on planning, funding and rationing of critical resources.

It should be pointed out that the paper focuses on managerial insights that
can be drawn from the optimization framework and its analytical solutions.
There are admittedly a number of limitations in the hypothetical example of a
three-state resources pooling arrangement. The three states are chosen for the
most drastic effects of planning and allocation of existing resources for the
purpose of illustration. It is beyond the scope of this paper to consider the
political reality that may prevent such arrangements. In theory, the
methodology can be applied to the actual voluntary coalition formed by six
northwestern states in the US, although the coalition was formed largely to
avoid price competition in government procurement. Another limitation of this
example is the egalitarian approach to shortages in different regions, which
ignores ethical issues that may arise from freely moving resources from one
region to another. While it may be economically optimal to deliver all
resources in the system to the place where they are needed the most, it may be
politically challenging to leave other places with less severe shortage
without support. A potential remedy would be to introduce additional
constraints in the optimization problems that require some minimal support for
each region.

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##  Appendix A Analytical Solutions and Proofs

###  A.1 Stockpiling of durable resources

The optimization problem for the stockpiling of durable resources is as
follows:

$\displaystyle\min_{K_{0}\geq
0}\sum_{j=1}^{m}\omega_{j}\left(\frac{\theta_{j}^{+}}{2}(X_{j}-(K_{0}+aj))_{+}^{2}+\frac{\theta_{j}^{-}}{2}(X_{j}-(K_{0}+aj))_{-}^{2}+c_{j}(K_{0}+aj)\right)+c_{0}K_{0}.$

######  Theorem A.1.

Let $Y_{j}=X_{j}-aj,\forall j=1,2,\cdots,m$. Let
$S=\sum_{j=1}^{m}\omega_{j}c_{j}+c_{0}.$ Let $Y_{[1]}\leq
Y_{[2]}\leq\cdots\leq Y_{[m]}$ be the increasingly ordered sequence of
$Y_{1},Y_{2},\cdots,Y_{m}$. Let $J=1,2,\cdots,m$ such that

$\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J]})\leq
S\leq\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J-1]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J-1]}),$

where we define that when $J=1$,

$\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J]})=\sum_{j=1}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[1]}),$
$\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J-1]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J-1]})=\infty.$

Let

$K_{0}^{\prime}=\frac{\sum_{j=1}^{J-1}\omega_{(j)}{\theta_{(j)}^{-}}Y_{(j)}+\sum_{j=J}^{m}\omega_{(j)}{\theta_{(j)}^{+}}Y_{(j)}-S}{\sum_{j=1}^{J-1}\omega_{(j)}{\theta_{(j)}^{-}}+\sum_{j=J}^{m}\omega_{(j)}{\theta_{(j)}^{+}}}.$

If $K_{0}^{\prime}<0$, then the optimal initial stockpile, which minimizes the
objective function above, $K_{0}^{*}=0$, and if $K_{0}^{\prime}\geq 0$, then
$K_{0}^{*}=K_{0}^{\prime}$.

###### Proof.

Let $Y_{j}=X_{j}-aj,\forall j=1,2,\cdots,m$. Let $Y_{[1]}\leq
Y_{[2]}\leq\cdots\leq Y_{[m]}$ be the increasingly ordered sequence of
$Y_{1},Y_{2},\cdots,Y_{m}$. In that case, $Y_{j}$ represents the daily
shortage when the initial stockpile is completely missing. Then the objective
function becomes

$F(K_{0}):=\sum_{j=1}^{m}\omega_{[j]}\left(\frac{\theta_{[j]}^{+}}{2}(Y_{[j]}-K_{0})_{+}^{2}+\frac{\theta_{[j]}^{-}}{2}(K_{0}-Y_{[j]})_{+}^{2}\right)+\left(\sum_{j=1}^{m}\omega_{j}c_{j}+c_{0}\right)K_{0}+\sum_{j=1}^{m}\omega_{j}c_{j}aj.\\\
$

Note that $F(K_{0})$ is a convex function in $K_{0}$ for any
$K_{0}\in\mathbb{R}$. Let

$\displaystyle G(K_{0})$
$\displaystyle=\sum_{j=1}^{m}\omega_{[j]}\left(\frac{\theta_{[j]}^{+}}{2}(Y_{[j]}-K_{0})_{+}^{2}+\frac{\theta_{[j]}^{-}}{2}(K_{0}-Y_{[j]})_{+}^{2}\right),$
$\displaystyle S$ $\displaystyle=\sum_{j=1}^{m}\omega_{j}c_{j}+c_{0},$

where $G(\cdot)$ is convex in $K_{0}$.

The first order derivatives of $G$ at $Y_{[1]}$ and $Y_{[m]}$ are as follows:

$G^{\prime}(Y_{[1]})=\sum_{j=1}^{m}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[1]}-Y_{[j]})\leq
0.$
$G^{\prime}(Y_{[m]})=\sum_{j=1}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[m]})(-1)=\sum_{j=1}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[m]}-Y_{[j]})\geq
0.$

Then $\exists$ $\tilde{K_{0}}\in\left[Y_{[1]},Y_{[m]}\right]$ such that
$G^{\prime}(\tilde{K_{0}})=0$. Since
$S=\sum_{j=1}^{m}\omega_{j}c_{j}+c_{0}\geq 0$, $\exists$
$K_{0}^{\prime}\in(-\infty,Y_{[m]}]$ such that $F^{\prime}(K_{0}^{\prime})=0$,
i.e., $\exists$ $J=1,\cdots,m$ such that
$K_{0}^{\prime}\in\left[Y_{[J-1]},Y_{[J]}\right]$, where we define
$Y_{(0)}=-\infty$, and $F^{\prime}(K_{0}^{\prime})=0$.

Here, we are relaxing the constraint that the initial stockpile has to be non-
negative, but we will add it back in the end. The key observation is that
given all the $(Y_{[j]})_{j\in\\{0,\dots,m\\}}$, we can always find a
$K_{0}^{\prime}$ between two adjacent $Y_{[j]}$’s, $Y_{[J-1]}$ and $Y_{[J]}$,
that minimize the objective function. That is,

$F^{\prime}(K_{0}^{\prime})=\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(K_{0}^{\prime}-Y_{[J]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-K_{0}^{\prime})(-1)+S=0.$

Then

$K_{0}^{\prime}=\frac{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}Y_{[j]}+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}Y_{[j]}-S}{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}}.$

Therefore, the condition that $K_{0}^{\prime}\in[Y_{[J-1]},Y_{[J]}]$ is given
by

$Y_{[J-1]}\leq\frac{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}Y_{[j]}+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}Y_{[j]}-S}{\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}}\leq
Y_{[J]}.$

Or equivalently, $\exists$ $J=1,2,\cdots,m$, such that

$\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J]})\leq
S\leq\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J-1]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J-1]}),$

wherein if $J=1$, we define

$\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J]})=\sum_{j=1}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[1]}),$
$\sum_{j=1}^{J-1}\omega_{[j]}{\theta_{[j]}^{-}}(Y_{[j]}-Y_{[J-1]})+\sum_{j=J}^{m}\omega_{[j]}{\theta_{[j]}^{+}}(Y_{[j]}-Y_{[J-1]})=\infty.$

Finally, given the non-negativity of stockpile, if $K_{0}^{\prime}<0$, then
the optimal initial stockpile $K_{0}^{*}=0$, and if $K_{0}^{\prime}\geq 0$,
then $K_{0}^{*}=K_{0}^{\prime}$. ∎

###  A.2 Centralized resources allocation

The optimization problem for allocating resources amount regions is as
follows:

$\displaystyle\underset{j=1,2,\dots,m}{\min_{K_{j}^{(i)}\geq
0;i=1,2,\dots,n;}}\;\;$
$\displaystyle\sum_{j=1}^{m}\sum_{i=1}^{n}\omega_{j}^{(i)}\left(\frac{\theta_{j}^{+(i)}}{2}\left(X_{j}^{(i)}-K_{j}^{(i)}\right)_{+}^{2}+\frac{\theta_{j}^{-(i)}}{2}\left(X_{j}^{(i)}-K_{j}^{(i)}\right)_{-}^{2}\right)$
such that $\displaystyle\sum_{i=1}^{n}K_{j}^{(i)}=K_{j},\quad\text{for
}j=1,2,\dots,m,$

######  Theorem A.2.

The optimization above is done from period to period, and thus to simplify
notation, the time indicator $j$ can be dropped at each time point. let
$X^{[1]}\geq\dots\geq X^{[n]}>0$ be the decreasingly ordered sequence of
$X^{(1)},X^{(2)},\dots,X^{(n)}$.

If $K>\sum\limits_{r=1}^{n}X^{(r)}=X$, then

$\displaystyle
K^{(i)}=\left(1-\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\right)X^{(i)}+\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\left(K-\sum\limits_{r\neq
i}X^{(r)}\right),\forall i=1,2,\dots,n.$

If $K\leq\sum_{r=1}^{n}X^{(r)}=X$, we can find an $I=1,2,\dots,n$ such that
$K\leq\sum_{r=1}^{I}X^{[r]}$,

$\displaystyle X^{[i]}$
$\displaystyle\geq\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=1}^{I}X^{[r]}-K\right),\forall
i=1,\dots,I,$ $\displaystyle X^{[i]}$
$\displaystyle<\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=1}^{I}X^{[r]}-K\right),\forall
i=I+1,I+2,\dots,n.$

Then $K^{[I+1]}=\dots=K^{[n]}=0$.

$\displaystyle
K^{[i]}=\left(1-\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\right)X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum\limits_{r=1,r\neq
i}^{I}X^{[r]}\right),\forall i=1,\dots,I.$

###### Proof.

At each time point $j$, we want to solve the following optimization problem,

$\displaystyle\min_{K^{(i)};i=1,2,\cdots,n}$
$\displaystyle\sum_{i=1}^{n}\omega^{(i)}\left(\frac{\theta^{+(i)}}{2}(X^{(i)}-K^{(i)})^{2}_{+}+\frac{\theta^{-(i)}}{2}(K^{(i)}-X^{(i)})^{2}_{+}\right)$
such that $\displaystyle\sum_{i=1}^{n}K^{(i)}=K,K^{(i)}\geq 0,\forall
i=1,2,\cdots,n.$

  1. Case 1:

Coexisting surpluses and shortages

First, let us consider the case in which some regions are having surpluses,
whereas other regions are experiencing shortages at the same time. We shall
show below that this case is impossible regardless of the non-negative
constraints. That is, suppose that $K^{(1)},K^{(2)},\dots,K^{(n)}$ lie locally
in a feasible set, such that $I$ of them satisfy $K^{(i)}>X^{(i)}$, where
$I=1,2,\dots,n-1$. The remaining $n-I$ of them satisfy $K^{(i)}\leq X^{(i)}$.
Without loss of generality, assume the first $I$ of $K^{(i)}$ are in the
former group. Then the local problem becomes

$\displaystyle\min_{K^{(i)};i=1,2,\cdots,n}$
$\displaystyle\sum_{i=I+1}^{n}\omega^{(i)}\frac{\theta^{+(i)}}{2}(X^{(i)}-K^{(i)})^{2}+\sum_{i=1}^{I}\omega^{(i)}\frac{\theta^{-(i)}}{2}(X^{(i)}-K^{(i)})^{2}$
such that $\displaystyle\sum_{i=1}^{n}K^{(i)}=K.$

The solution to this problem is given by

$\displaystyle
K^{(i)}=\left(1-\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{(r)}\theta^{-(r)}}+\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{(r)}\theta^{+(r)}}}\right)X^{(i)}+\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{(r)}\theta^{-(r)}}+\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{(r)}\theta^{+(r)}}}\left(K-\sum\limits_{r\neq
i}X^{(r)}\right),$

for $i=1,2,\dots,I$, and

$\displaystyle
K^{(i)}=\left(1-\frac{\frac{1}{\omega^{(i)}\theta^{+(i)}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{(r)}\theta^{-(r)}}+\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{(r)}\theta^{+(r)}}}\right)X^{(i)}+\frac{\frac{1}{\omega^{(i)}\theta^{+(i)}}}{\sum\limits_{r=1}^{I}\frac{1}{\omega^{(r)}\theta^{-(r)}}+\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{(r)}\theta^{+(r)}}}\left(K-\sum\limits_{r\neq
i}X^{(r)}\right),$

for $i=I+1,\dots,n$.

However, $\forall i=1,2\dots,I$, $K^{(i)}>X^{(i)}$, and $\forall
i=I+1,\dots,n$, $K^{(i)}\leq X^{(i)}$, which implies
${K>\sum_{r=1}^{n}X^{(r)}}$ and $K\leq\sum_{r=1}^{n}X^{(r)}$, and thus we have
a contradiction. This result shows that it is impossible for some regions to
have surpluses while other regions are experiencing shortages. Therefore, it
suffices to only consider the scenarios that there is a system wide surplus
and that there is a system wide shortage.

  2. Case 2:

System wide surplus

If there is a system wide surplus, i.e., all regions have surpluses, then the
problem becomes

$\displaystyle\min_{K^{(i)};i=1,2,\cdots,n}$
$\displaystyle\sum_{i=1}^{n}\omega^{(i)}\frac{\theta^{-(i)}}{2}(K^{(i)}-X^{(i)})^{2}$
such that $\displaystyle\sum_{i=1}^{n}K^{(i)}=K,$ $\displaystyle
K^{(i)}>X^{(i)},\forall i=1,\dots,n.$

The solution is given by

$\displaystyle
K^{(i)}=\left(1-\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\right)X^{(i)}+\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\left(K-\sum\limits_{r\neq
i}X^{(r)}\right).$

For this result to hold, we only need the condition that
$K>\sum_{r=1}^{n}X^{(r)}=X$. Due to uniqueness, under this condition, this
$K^{(i)}$ is optimal.

  3. Case 3:

System wide shortage

It remains to solve the case that there is a system wide shortage, i.e., all
regions have shortages. In that case, the problem becomes

$\displaystyle\min_{K^{(i)};i=1,2,\cdots,n}$
$\displaystyle\sum_{i=1}^{n}\omega^{(i)}\frac{\theta^{+(i)}}{2}(X^{(i)}-K^{(i)})^{2}$
such that $\displaystyle\sum_{i=1}^{n}K^{(i)}=K;$ $\displaystyle 0\leq
K^{(i)}\leq X^{(i)},\forall i=1,\cdots,n.$

Apart from the given condition that $\sum_{i=1}^{n}K^{(i)}=K$, there are
additional inequality constraints in this optimization problem. Hence, we make
use of Karush-Kuhn–Tucker (KKT) conditions as follows.

$\displaystyle\omega^{(i)}\theta^{+(i)}(K^{(i)}-X^{(i)})-\lambda_{1}^{(i)}+\lambda_{2}^{(i)}+\mu=0,$
$\displaystyle 0\leq K^{(i)}\leq X^{(i)},$ $\displaystyle\lambda_{1}^{(i)}\geq
0,\quad\lambda_{1}^{(i)}K^{(i)}=0,$ $\displaystyle\lambda_{2}^{(i)}\geq
0,\quad\lambda_{2}^{(i)}(K^{(i)}-X^{(i)})=0,$

for all $i=1,\cdots,n$. Regarding the values of $\lambda_{1}^{(i)}$ and
$\lambda_{2}^{(i)}$, we can consider the following four mutually exclusive
cases,

    1. (i)

$\lambda_{1}^{(i)}>0$ for some $i=1,\cdots,n$, and $\lambda_{2}^{(i)}>0$ for
some $i=1,\cdots,n$;

    2. (ii)

$\lambda_{1}^{(i)}=0$ for all $i=1,\cdots,n$, and $\lambda_{2}^{(i)}>0$ for
some $i=1,\cdots,n$;

    3. (iii)

$\lambda_{1}^{(i)}>0$ for some $i=1,\cdots,n$, and $\lambda_{2}^{(i)}=0$ for
all $i=1,\cdots,n$;

    4. (iv)

$\lambda_{1}^{(i)}=0$ for all $i=1,\cdots,n$, and $\lambda_{2}^{(i)}=0$ for
all $i=1,\cdots,n$.

Each of them will be discussed as follows.

    1. Case 3.1:

$\lambda_{1}^{(i)}>0$ for some $i=1,\cdots,n$, and $\lambda_{2}^{(i)}>0$ for
some $i=1,\cdots,n$.

We will show that this case will lead to a contradiction, and thus is
impossible. Because the ordering of $\lambda_{1}^{(i)}$ and
$\lambda_{2}^{(i)}$ does not affect the conditions, for simplicity, we
rearrange them in such a way that there is an $I=1,2,\dots,n-1$, and an
$\tilde{I}=1,2,\dots,n-1$, for which

$\displaystyle\lambda_{1}^{[i]}>0$ $\displaystyle,\quad\forall i=1,\dots,I,$
$\displaystyle\lambda_{1}^{[i]}=0$ $\displaystyle,\quad\forall i=I+1,\dots,n,$
$\displaystyle\lambda_{2}^{[i]}=0$ $\displaystyle,\quad\forall
i=1,\dots,n-\tilde{I},$ $\displaystyle\lambda_{2}^{[i]}>0$
$\displaystyle,\quad\forall i=n-\tilde{I}+1,\dots,n,$

where $[i]$ are indices after the rearrangement.

We also have the condition that $I\leq n-\tilde{I}$, because for each $i$,
$\lambda_{1}^{[i]}>0$ implies $K^{[i]}=0$, which further implies
$\lambda_{2}^{[i]}=0$. Therefore, the number of $\lambda_{2}^{[i]}$ that are
equal to 0, i.e., $n-\tilde{I}$, is at least the number of $\lambda_{1}^{[i]}$
that are greater than 0, i.e., $I$. Then by the complementary slackness
conditions in the KKT conditions above, $K^{[i]}=0,\forall i=1,2,\cdots,I$
and, $K^{[i]}=X^{[i]},\forall i=n-\tilde{I}+1,\cdots,n$.

Then, the KKT conditions are simplified.

$\displaystyle-\omega^{[i]}\theta^{+[i]}X^{[i]}-\lambda_{1}^{[i]}+\mu=0$
$\displaystyle,\quad\forall i=1,\dots,I$
$\displaystyle\omega^{[i]}\theta^{+[i]}(K^{[i]}-X^{[i]})+\mu=0$
$\displaystyle,\quad\forall i=I+1,\dots,n-\tilde{I}$
$\displaystyle\lambda_{2}^{[i]}+\mu=0$ $\displaystyle,\quad\forall
i=n-\tilde{I}+1,\dots,n$ $\displaystyle 0\leq K^{[i]}\leq X^{[i]}$
$\displaystyle,\quad\forall i=I+1,\dots,n-\tilde{I}$
$\displaystyle\sum_{r=I+1}^{n-\tilde{I}}K^{[r]}=K-\sum_{r=n-\tilde{I}+1}^{n}X^{[r]}$
,

from which we can observe the following contradiction that,

$\displaystyle\mu=-\lambda_{2}^{[n]}<0$
$\displaystyle\mu=\omega^{[1]}\theta^{+[1]}X^{[1]}+\lambda_{1}^{[1]}>0.$

Therefore, we can conclude that if $\tilde{I}=1,\cdots,n-1$, then $I=0$ or
$n$, and its contrapositive is also true, which states that if
$I=1,\cdots,n-1$, then $\tilde{I}=0$ or $n$. These two statements correspond
the second and third cases respectively, and they are considered as follows.

    2. Case 3.2:

$\lambda_{1}^{(i)}=0$ for all $i=1,\cdots,n$, and $\lambda_{2}^{(i)}>0$ for
some $i=1,\cdots,n$.

As in the previous case, We use the rearranged $\lambda_{1}^{[i]}$ and
$\lambda_{2}^{[i]}$, so now $\lambda_{2}^{[i]}=0$, for all
$i=1,\dots,n-\tilde{I}$, and $\lambda_{2}^{[i]}>0$, for all
$i=n-\tilde{I}+1,\dots,n$, for some $\tilde{I}=1,2,\dots,n-1$. This implies
$K^{[i]}=X^{[i]}$, for all $i=n-\tilde{I}+1,\dots,n$.

In this case, the KKT conditions become

$\displaystyle\omega^{[i]}\theta^{+[i]}(K^{[i]}-X^{[i]})+\mu=0$
$\displaystyle,\quad\forall i=1,\dots,n-\tilde{I}$
$\displaystyle\lambda_{2}^{[i]}+\mu=0$ $\displaystyle,\quad\forall
i=n-\tilde{I}+1,\dots,n$ $\displaystyle 0\leq K^{[i]}\leq X^{[i]}$
$\displaystyle,\quad\forall i=1,\cdots,n-\tilde{I}$
$\displaystyle\sum_{r=1}^{n-\tilde{I}}K^{(r)}=K-\sum_{r=n-\tilde{I}+1}^{n}X^{(r)}$
.

By solving this system, we get

$\displaystyle K^{[i]}$
$\displaystyle=\left(1-\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{n-\tilde{I}}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\right)X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{n-\tilde{I}}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum\limits_{r\neq
i}^{n}X^{[r]}\right)$
$\displaystyle=X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{n-\tilde{I}}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum_{r=1}^{n}X^{[r]}\right),\quad\forall
i=1,2,\cdots,n-\tilde{I}$ $\displaystyle\mu$
$\displaystyle=\frac{1}{\sum\limits_{r=1}^{n-\tilde{I}}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=1}^{n}X^{[r]}-K\right)$
$\displaystyle\lambda_{2}^{[i]}$
$\displaystyle=-\mu=-\frac{1}{\sum\limits_{r=1}^{n-\tilde{I}}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=1}^{n}X^{[r]}-K\right),\quad\forall
i=n-\tilde{I}+1,\cdots,n.$

By the system wide shortage assumption, $\sum\limits_{r=1}^{n}X^{[r]}-K\geq
0$, and therefore, $\lambda_{2}^{[i]}\leq 0$ for all
$i=n-\tilde{I}+1,\cdots,n$. But this contradicts the assumption that
$\lambda_{2}^{[i]}>0$ for all $i=n-\tilde{I}+1,\cdots,n$. Therefore, we can
tell that this is another impossible case.

    3. Case 3.3:

$\lambda_{1}^{(i)}>0$ for some $i=1,\cdots,n$, and $\lambda_{2}^{(i)}=0$ for
all $i=1,\cdots,n$.

Again, for this case, we use the rearranged $\lambda_{1}^{[i]}$ and
$\lambda_{2}^{[i]}$ for $i=1,\dots,n$. And we assume that there is an
$I=1,2,\dots,n-1$, such that $\lambda_{1}^{[i]}>0$ for $i=1,\dots,I$, and
$\lambda_{1}^{[i]}=0$ for $i=I+1,\dots,n$. This implies $K^{[i]}=0$ for
$i=1,\dots,I$, and we get the following conditions,

$\displaystyle-\omega^{[i]}\theta^{+[i]}X^{[i]}-\lambda_{1}^{[i]}+\mu=0$
$\displaystyle,\quad\forall i=1,\dots,I$
$\displaystyle\omega^{[i]}\theta^{+[i]}(K^{[i]}-X^{[i]})+\mu=0$
$\displaystyle,\quad\forall i=I+1,\dots,n$ $\displaystyle 0\leq K^{[i]}\leq
X^{[i]}$ $\displaystyle,\quad\forall i=I+1,\cdots,n$
$\displaystyle\sum_{i=I+1}^{n}X^{[r]}=K$ .

They together give us

$\displaystyle K^{[i]}$
$\displaystyle=\left(1-\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\right)X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum\limits_{r=I+1}^{n}X^{[r]}\right)$
$\displaystyle=X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum\limits_{r=I+1}^{n}X^{[r]}\right),\quad\forall
i=I+1,\cdots,n.$ $\displaystyle\mu$
$\displaystyle=\frac{1}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=I+1}^{n}X^{[r]}-K\right)$
$\displaystyle\lambda_{1}^{[i]}$
$\displaystyle=\frac{1}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=I+1}^{n}X^{[r]}-K\right)-\omega^{[i]}\theta^{+[i]}X^{[i]},\quad\forall
i=1,\cdots,I.$

Since $\lambda_{1}^{[i]}>0$ for $i=1,\dots,I$ and $0\leq K^{[i]}\leq X^{[i]}$
for $i=I+1,\cdots,n$, the following conditions hold.
$K\leq\sum\limits_{r=I+1}^{n}X^{[r]}$,

$\displaystyle X^{[i]}$
$\displaystyle<\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=I+1}^{n}X^{[r]}-K\right),\forall
i=1,2,\cdots,I,$ $\displaystyle X^{[i]}$
$\displaystyle\geq\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=I+1}^{n}X^{[r]}-K\right),\forall
i=I+1,\cdots,n.$

    4. Case 3.4:

$\lambda_{1}^{(i)}=0$ for all $i=1,\cdots,n$, and $\lambda_{2}^{(i)}=0$ for
all $i=1,\cdots,n$.

Now it only remains to consider the case in which $\lambda_{1}^{(i)}=0$ and
$\lambda_{2}^{(i)}=0$ for all $i=1,\dots,n$. Since there is no divergence in
the values of $\lambda_{1}^{(i)}$ and $\lambda_{2}^{(i)}$, rearrangement does
not make a difference. Nevertheless, we will use the ordered indices $[i]$
here for consistency.

In this case, conditions now become

$\displaystyle\omega^{[i]}\theta^{+[i]}(K^{[i]}-X^{[i]})+\mu=0$
$\displaystyle,\quad\forall i=1,\dots,n$ $\displaystyle 0\leq K^{[i]}\leq
X^{[i]}$ $\displaystyle,\quad\forall i=1,\dots,n$
$\displaystyle\sum_{i=1}^{n}K^{[i]}=K$ .

Solving this system gives us

$\displaystyle K^{[i]}$
$\displaystyle=\left(1-\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\right)X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum\limits_{r\neq
i}X^{[r]}\right)$
$\displaystyle=X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum_{r=1}^{n}X^{[r]}\right),\quad\forall
i=1,\cdots,n.$

Since we assume there is a system wide shortage, $K-\sum_{r=1}^{n}X^{[r]}\leq
0$, and thus we get $K^{[i]}\leq X^{[i]}$ for all $i=1,\dots,n$. Then, the
only required condition for this result to hold is $0\leq K^{[i]}$ for all
$i=1,\cdots,n$, which gives us

$X^{[i]}\geq\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum_{r=1}^{n}X^{[r]}-K\right),\quad\forall
i=1,2,\cdots,n.$

This result can actually be seen as a special case of Case 3.3, where $I=0$,
and therefore, we can combine the results from these two cases.

In summary, the result depends on whether there is a system wide surplus or
shortage. In the event of a system wide surplus, i.e., $K^{(i)}>X^{(i)}$ for
$i=1,\dots,n$, the optimal allocation in each region is given by

$\displaystyle
K^{(i)}=\left(1-\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\right)X^{(i)}+\frac{\frac{1}{\omega^{(i)}\theta^{-(i)}}}{\sum\limits_{r=1}^{n}\frac{1}{\omega^{(r)}\theta^{-(r)}}}\left(K-\sum\limits_{r\neq
i}X^{(r)}\right).$

In the event of a system wide shortage, i.e., $0\leq K^{(i)}\leq X^{(i)}$ for
$i=1,\dots,n$, it has been demonstrated that the solution can be found by
sorting $\lambda_{1}^{(i)}$ in such a way that the first $I=0,1,\dots,n-1$ of
them are greater than $0$. If $I$ is $0$, then all $\lambda_{1}^{(i)}=0$. The
$I$ here should satisfy the following conditions,
$K\leq\sum\limits_{r=I+1}^{n}X^{[r]}$,

$\displaystyle X^{[i]}$
$\displaystyle<\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=I+1}^{n}X^{[r]}-K\right),\forall
i=1,2,\cdots,I,$ $\displaystyle X^{[i]}$
$\displaystyle\geq\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(\sum\limits_{r=I+1}^{n}X^{[r]}-K\right),\forall
i=I+1,\cdots,n,$

where if $I=0$, then the first inequality can be discarded. Once $I$ is
identified, the optimal allocation in each region is given by
$K^{[1]}=\dots=K^{[I]}=0$,

$\displaystyle K^{[i]}$
$\displaystyle=\left(1-\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\right)X^{[i]}+\frac{\frac{1}{\omega^{[i]}\theta^{+[i]}}}{\sum\limits_{r=I+1}^{n}\frac{1}{\omega^{[r]}\theta^{+[r]}}}\left(K-\sum\limits_{r=I+1}^{n}X^{[r]}\right),\quad\forall
i=I+1,\cdots,n.$

Since $\lambda_{1}^{(i)}$ are sorted, the final result should have been re-
sorted accordingly. However, the conditions are the same regardless of the
ordering. Finally, we relabel $I$ and $n-I$ for the sake of simplifying the
notations in the main text, and hence $X^{(i)}$ should be sorted in descending
order instead.

∎

##  Appendix B Parameter Values in Numerical Examples

This section offers an inventory of all model parameters used in earlier
sections. The same sets of parameters are used for all calculations in the
three-pillar framework.

Parameter | Value
---|---
Percentage of intensive care patients requiring ventilators (class $I_{3}$) | 0.9
Units of PPE required per exposed patient (class $E$) | 5
Units of PPE required per hospitalized patient (class $I_{2}$) | 15
Units of PPE required per intensive care patient (class $I_{3}$) | 20
Table 4: Demand assessment parameters; values are chosen in the ranges provided in Tables 2 and 3. Parameter | Value
---|---
Participating states | New York, Florida, California
Cost of possession per unit per day ($c_{j}$) | 1
Initial stockpile cost per unit ($c_{0}$) | 25120
Daily production rate ($a$) | 10 Units
Shortage/surplus cost ($\theta_{j}^{+}$/$\theta_{j}^{-}$) in Figure 8(a) | 1000/1000
Shortage/surplus cost ($\theta_{j}^{+}$/$\theta_{j}^{-}$) in Figure 8(b) | 1000/20
Time varying weight ($\omega_{j}$) | Proportional to daily demand $X_{j}$
Table 5: Ventilator planning parameters [36, 40, 35] Parameter | Value
---|---
Participating states | New York, Florida, California
Cost of possession per 1000 units per day ($c_{j}$) | 0.01
Initial stockpile cost per 1000 units ($c_{0}$) | 0.5
Daily production rate ($a$) | 50000 units
Shortage cost ($\theta_{j}^{+}$) | 1
Time varying weight ($\omega_{j}$) | Proportional to daily demand $X_{j}$
Table 6: Personal protective equipment planning parameters Parameter | Value
---|---
Participating states | New York, Florida, California
Shortage/surplus cost ($\theta_{j}^{(i)+}$/$\theta_{j}^{(i)-}$) | 1
Weight for resources allocation in region $i$ at time $j$ ($\omega_{j}^{(i)}$) | Proportional to $\sum_{t=j}^{m}X_{t}^{(i)}$
Table 7: Ventilator/Personal protective equipment allocation parameters