Pandemic risk management: resources contingency planning and allocation
PPandemic Risk Management: Resources ContingencyPlanning and Allocation ∗ Xiaowei Chen † , Wing Fung Chong ‡ , Runhuan Feng § , Linfeng Zhang ¶ December 8, 2020
Abstract
Repeated history of pandemics, such as SARS, H1N1, Ebola, Zika, and COVID-19, has shown thatpandemic risk is inevitable. Extraordinary shortages of medical resources have been observed in manyparts of the world. Some attributing factors include the lack of sufficient stockpiles and the lack ofcoordinated efforts to deploy existing resources to the location of greatest needs.The paper investigates contingency planning and resources allocation from a risk management per-spective, as opposed to the prevailing supply chain perspective. The key idea is that the competitionof limited critical resources is not only present in different geographical locations but also at differentstages of a pandemic. This paper draws on an analogy between risk aggregation and capital allocation infinance and pandemic resources planning and allocation for healthcare systems. The main contribution isto introduce new strategies for optimal stockpiling and allocation balancing spatio-temporal competitionsof 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 spreadof a new disease.” Because human has little immunity to the new disease, a pandemic can emerge quicklyaround the world. One of the most disastrous pandemics in the recent history is the 1918 flu pandemic whichinfected around 500 million and resulted in the deaths of estimated 50 million people worldwide, more thanthose died from the World War I. Most recently, the novel coronavirus disease of 2019 (COVID-19) is aninfectious disease caused by a new virus that only emerged in late 2019 and has since spread out to nearlyevery country in the world. According to Cutler and Summers, the COVID-19 could cause financial lossesthat sum up to $
16 trillion, or roughly 90% of the annual GDP in the United States (Cutler and Summers,2020). However, its far-reaching impact and consequent economic fallout due to the loss of productivity is ∗ First version: April 8, 2020. † School of Finance, Nankai University. Email: [email protected]. ‡ Department of Mathematics and Department of Statistics, University of Illinois at Urbana-Champaign. Email: [email protected]. § Department of Mathematics, University of Illinois at Urbana-Champaign. Email: [email protected]. ¶ Department of Mathematics, University of Illinois at Urbana-Champaign. Email: [email protected]. a r X i v : . [ q -f i n . R M ] D ec et to be realized; as of November 28, 2020, the number of infected cases has accumulated to over 61 millionworldwide with the death toll reaching over 1.4 million (see, for example, The Atlantic (2020)).The repeated history of pandemics in recent decades, such as SARS, swine flu, Ebola, and the most recentCOVID-19, has taught us that pandemic risk is inevitable . Recent research studies (see, for instance, K. E.Jones et al. (2008) and Morse (1995)) have shown that the frequency of pandemics has increased over the pastcentury due to increased social connectivity, long-distance travel, urbanization, changes in land use, trade andconsumption of wild animals, and greater exploitation of the natural environment. Figure 1 visualizes bothfrequency and severity of well-recognized pandemics and public health emergencies of international concerndeclared by the WHO since 1900s. The vertical axis represents the number of documented infections on alogarithmic scale. Both the size and the color scale of the circles indicate the number of deaths resultingfrom the pandemics and public health emergencies. The alarming pattern of increased frequency clearlypoints to the critical importance of pandemic risk management.Figure 1: Frequency and severity of pandemics and public health emergencies since 1900sGovernments around the world have been taking blames for their failure to promptly implement appro-priate policies to contain the pandemic. Many countries experienced severe shortages of resources. Ranney,Griffeth, and Jha (2020) studied the critical role of scarce resources, such as ventilators and personal protec-tive equipment (PPE), in shaping the direction of COVID-19. Such an unprecedented challenge exposes theinadequacy in contingency planning and resources allocation strategies of public health systems. The lackof planning drives policymakers to make impromptu decisions on resources acquisitions and allocations thatmay have exacerbated the extraordinary shortage.The United States boasts one of the best healthcare systems with a large network of healthcare profession-als, best-equipped medical facilities and hospitals, and most advanced medical technology. Yet, the countryis under-prepared for the COVID-19 pandemic. There were severe shortages of diagnostic and preventativemedical supplies both for healthcare providers and the general public in many states in the early stage ofthe pandemic, which made it difficult for public authorities to contain the pandemic. According to a recentreport by President Obama’s former advisors on science and technology (Holdren et al., 2020), there areseveral contributing factors to the lack of medical resources: (1) National reserves of critical medical supplieswere not replenished sufficiently prior to COVID-19.
Strategic National Stockpile (SNS) was established by2he government in 2003 as the national repository of pharmaceutical and vaccination stockpiles. The SNSrelies on the appropriation of funding from the Congress. Much of the mask stockpile was depleted duringthe 2009 H1N1 pandemic and the Congress has not acted quickly to provide the funding to replenish thestockpile to an appropriate level projected by many studies. (2)
In order to minimize inventory cost andimprove efficiency, many manufacturers and supply chain management of medical supplies have shifted tojust-in-time (JIT) inventory system prior to the pandemic.
Goods are only received just in time for produc-tion 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 worldand there were not sufficient inventories to absorb surge demand. (3)
There was a lack of sufficient coordi-nation among federal and state governments to deploy existing resources to the most devastated areas in thecountry.
Healthcare professionals are put at high risk to treat patients without sufficient personal protectiveequipment. It was difficult to uncover and contain the spread of disease without adequate testing. Severalstates in the United States acted on their own to secure supplies from foreign manufactures and engaged ina price bidding war for limited supplies (Estes, 2020). Existing resources are not necessarily distributed ona basis of health need (Tobin-Tyler, 2020). Many hardest hit states have to ration care, while other stateshave low utilization of their resources.While there are policy related issues that require policymakers’ actions, academics can contribute tothe understanding of pandemic evolution and the resulting dynamics of demand and supply. There is aclear need for the development of scientific foundation for adaptive strategies for balancing demand andsupply and rationing limited resources. Contingency planning and resources allocation in a centralizedform have been advocated as two coherent strategies to mitigate catastrophic economic consequences froma pandemic; see for example Jamison et al. (2017). Ranney, Griffeth, and Jha (2020) argued that thegovernment should have tracked the use of resources and the projection of needs in all subsidiaries, andshould have coordinated allocation of resources to reduce shortage across subsidiaries and over time in thecourse of a pandemic. A comparable example of centralized planning is the Federal Emergency ManagementAgency, 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 assistanceteams, monetary reliefs, etc; see Vanajakumari, Kumar, and Gupta (2016) and Stauffer and Kumar (2020)for information.Centralized planning and resources allocation have long been practiced as risk management strategies inthe financial industry. For example, banks and insurers are heavily regulated by governments to ensure theircapabilities to absorb severe financial losses and endure adverse economic scenarios (Segal, 2011). The centralhypothesis of this paper is that many risk aggregation and capital allocation techniques drawn from financialand insurance literature can be extended and applied to pandemic resources planning and allocation.
Case Study
Figure 2 depicts the daily confirmed COVID-19 cases in the State of New York (NY), Florida (FL), andCalifornia (CA) in 2020. One could observe that, among these three states, NY was first hit the hardestby 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 savagedby the COVID-19 around that time.Let us take a thought experiment for the moment. Imagine that these three states established resourcepooling alliance prior to the pandemic. They could have complemented each other by delivering one state’s3 pr Jul050001000015000 Apr Jul Apr Jul D a il y c o n f i r m e d c a s e s New York Florida California
Figure 2: New York, Florida, and California experience different phases of COVID-19, based on data as ofSeptember 12, 2020 from The Atlantic (2020)surplus resources to aid another in deficit. For example, in April, the alliance could have coordinated theefforts 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 FLto CA; by July and August, when both FL and CA were hit the hardest, unused resources in the NY couldbe made available to both FL and CA. Such a coalition is not unimaginable even in a decentralized politicalsystem like the United States. In April 2020, six northwestern states including New York, Connecticut, NewJersey, Rhode Island, Pennsylvania, and Delaware, formed a government procurement coalition for criticalmedical equipment in an effort to avoid a bidding war (Holveck, Racioppi, and Shanes, 2020).The purpose of this paper is to propose an overarching framework for different regions to optimize stock-piling and resources allocation at different pandemic stages in order to best utilize limited resources. Whilewe use inter-state resources pooling as an illustrative example, applications can also include internationalcollaboration on the production, procurement, distribution and pooling of critical medical resources such asmasks, 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.Fern´andez-Villaverde and C. I. Jones (2020) and Horta¸csu, Liu, and Schwieg (2020)), infected cases (e.g.Giordano et al. (2020)), economic impact (e.g. Acemoglu et al. (2020) and Gregory, Menzio, and Wiczer(2020)), and the effect of non-pharmaceutical intervention and other public policies (e.g. Charpentier et al.(2020)). However, to the best of our knowledge, academic research on quantitative framework for contingencyplanning and resources allocation in response to pandemic risk are rare. While the banking and insuranceindustries have long had a rich tradition of developing technologies for robust risk management, the focushas been largely on financial and insurable risks. This paper aims to take advantage of the vast medicalliterature on epidemic modeling and apply classic concepts from risk management and insurance literaturesuch as reserving and capital allocation to pandemic risk management.A strategic pandemic planning requires scientific assessment, rather than on-the-fly ad-hoc decisions andpatchworks for damage control. In accordance with current practices of national pandemic preparedness andcontrol strategies around the world, we summarize and propose a three-pillar framework for quantitative4 pidemic dynamicspredictive modelNeed assessment Resource demandprediction Centralized stockpiling& distribution strategyResources allocationstrategyBefore pandemicDuring pandemic
Pillar I Pillar IIPillar III
Figure 3: Three-pillar pandemic risk management frameworkpandemic risk management:
Pillar I: Regional and Aggregate Resources Supply and Demand Forecast.
Any pre-pandemic prepa-ration plan should consist of supply and demand assessment and forecast. The supply side shouldinclude inventory assessments of critical resources and supplies, the maximum capacity of services, thecapability of emergency acquisition and production. The demand side requires an understanding ofthe dynamics of a potential pandemic across regions and across borders. Historical data and predictivemodels can be used to project the evolution of a pandemic and the resulting surge demand around ahealthcare system.
Pillar II: Centralized Stockpiling and Distribution . A central authority coordinates the efforts to developa 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 centralauthority can deliver resources to different region quickly to meet surge demands and to balancecompeting interests and priorities.
Pillar III: Central-Regional Resources Allocation.
A pandemic response plan is critical for a centralauthority to contain and control the spread of a pandemic in regions under its jurisdiction. As demandmay exceed any best-effort pre-pandemic projection, the authority needs to devise optimal strategiesthat best utilize limited existing resources and minimize the economic cost of supply-demand imbal-ance. A coordination strategy needs to be in place to ensure smooth communications with regional5uthorities. The allocation strategy should be based scientifically sound methods taking into accountspatio-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 criticalrole in ensuring the adequacy and effectiveness of planning and responses in second and third pillars. Theproposed framework applies regardless of predictive models for projecting reported cases.
The main technicalcontribution 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 managementBusiness line and aggregate risk Regional and aggregate resources demandRisk-based capital Centralized stockpilingBusiness line capital allocation Centralized distribution and allocationTrade-off between surplus/deficiency and cost of capital Balance of supply/demand and economic costTable 1: Comparison between capital and pandemic risk managementsThe paper draws inspirations from two sources in insurance and risk management literature. (1) Insuranceapplications of epidemic models. Early applications of epidemic compartment models appeared in Hua andCox (2009), in which real option pricing is used for operational risk management, and Feng and Garrido(2011), which analyzed epidemic insurance coverage. The study of epidemic insurance was extensivelydeveloped in stochastic setting in Lef`evre, Picard, and Simon (2017), Lef`evre and Picard (2018), and Lef`evreand Simon (2020), and more recently to cyber risk assessment by Hillairet and Lopez (2020). All of thesecompartmental models can be used in Pillar I. (2) Capital allocation. The subject of capital managementis well-studied in the insurance literature. The applications of reserving and capital allocation form thebasis of the proposed Pillars II and III. Table 1 reveals how our proposed framework shadows the classicalcapital management. While spatial balancing of allocation is well-known in banking and insurance (see, forinstance, Dhaene et al. (2012), and Chong, Feng, and Jin (2020)), this paper develops a novel spatio-temporalbalancing of resources distribution and allocation, which, to the best of our knowledge, was not previouslystudied in either financial or management literature.The rest of the paper is organized as follows. Each of the next three section provides detailed discussionof one of the three pillars in the proposed pandemic risk management framework as well as economic in-terpretations of resulting optimal strategies. Numerical examples are embedded in the discussion for betterreadership. 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 ineach region. Regardless of the choice of epidemiological models, the central authority should calibrate themodel in each region by its preparedness and other contingency measurements. Indeed, epidemic forecastmodels have been used for healthcare policy making and public communications; see, for example, Leunget al. (2020) and Tian et al. (2020). In this paper, in line with
Covid Act Now (2020) and Hill et al. (2020),the population in each region is divided into seven mutually exclusive compartments, namely, the susceptible( S ), the exposed ( E ), the mild infected ( I ), the infected with hospitalization ( I ), the infected with intensive6are ( I ), the recovered ( R ), and the deceased ( D ). The dynamics, among these seven compartments, aregoverned 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 populationflows among all aforementioned compartments:d S ( t ) = − ( β I ( t ) + β I ( t ) + β I ( t )) S ( t ) d t, d E ( t ) = [( β I ( t ) + β I ( t ) + β I ( t )) S ( t ) − γE ( t )] d t, d I ( t ) = [ γE ( t ) − ( δ + p ) I ( t )] d t, d I ( t ) = [ p I ( t ) − ( δ + p )] I ( t )] d t, d I ( t ) = [ p I ( t ) − ( δ + µ ) I ( t )] d t, d R ( t ) = [ δ I ( t ) + δ I ( t ) + δ I ( t )] d t, d D ( t ) = µI ( t ) d t. All parameters in the set of equations bear clinical meanings; β i , i = 1 , ,
3, is the transmission rate of theinfected class I i ; 1 /γ is the average latency period; 1 /δ i , i = 1 , ,
3, is the average duration of infection inthe class I i before recovery to the class R ; p i , i = 1 , ,
3, represents the rate at which conditions worsen andindividuals require healthcare at the next level of severity; µ is the rate for most severe cases in the class I to the deceased class D . Suppose that the total number of individuals in the entire popluation is N. Eachof the ordinary differential equation represents a decomposition of instantaneous change in the populationof a compartment. For example, the first equation shows that the instantaneous rate of reduction in thenumber of susceptible, − d S ( t ) matches the sum of the rates of infection due to contacts with the infectedof all classes, β I ( t ) S ( t ) + β I ( t ) S ( t ) + β I ( t ) S ( t ). The products are due to the law of mass action inbiology. For example, the rate of secondary infection by the mildly infected, ( β N ) I ( t )( S ( t ) /N ) can beinterpreted as the number of adequate contact each infected makes to transmit the disease β N multipliedby the number of infected I ( 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 parametersare well-studied in the literature for the COVID-19 as well as other pandemics, such as Wu and McGoogan(2020), P. Yang et al. (2020), and X. Yang et al. (2020). Based on these parameters, the basic reproductiveratio R of a pandemic is given by: R = Np + δ (cid:18) β + p p + δ (cid:18) β + β p µ + δ (cid:19)(cid:19) . The basic reproductive ratio R can be estimated by empirical data and is often used to calibrate otherparameters. In what follows, we shall use discretized version of the compartmental model. For example,we use the notation I ,j = I ( j ∆ t ) to indicate the number of mild cases projected on the j -th period eachwith the length ∆ t. We sometimes omit the information on ∆ t as the time unit may vary depending on thereporting period.Based on predictive models such as the above-mentioned regional SEIRD models, a central authoritycould predict, prior to a pandemic or at the onset of a pandemic, changes in demand over the course of thepandemic. 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 resourcesrefer to those that can perform their required functions for a lengthy period of time without significant7xpenditures of maintenance or repair. Single-use resources are those that are designed to be used onceand then disposed of. Mechanical ventilators and PPE are used as representative examples of durable andsingle-use resources respectively in this paper. α ∈ [0 ,
1] Data sourceChina <
20% X. Yang et al. (2020)Italy [87% , .
4% Arentz et al. (2020)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 per-centage α of the infected with intensive care that require the use of mechanical ventilators. These regionaldifferences can be addressed in separate regional compartment models. We can use these estimates to projectthe ventilator demand by X VEN( i ) j = αI ( i )3 ,j , where i indicates the i -th region in the alliance and j indicatesthe j -th day of the pandemic. The model can also be extended to include time-varying percentage of severepatients 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 modelproposed by Covid Act Now (2020), which is calibrated to publicly available reported cases as of September12, 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 conditionsin care, and the function of medical professionals. Table 3 offers an example of such need assessment byEuropean Centre for Disease Prevention and Control (2020). Given these estimates, we can project theregional PPE set demand by X PPE( i ) j = θ E ( S ( i ) j − − S ( i ) j ) + θ I I ( i )2 ,j + θ I I ( i )3 ,j , where θ E is the number of PPEsets per exposed case, θ I is the number of PPE sets per day per hospitalized patient, and θ I is the numberof PPE sets per day per intensive care patient. Note that S j − − S j represents the daily exposed caseswhereas I ,j and I ,j keeps track of existing infected cases that require medical attention. Figure 4(b) showshow ventilator and PPE demands are projected to evolve over time for New York, Florida, and California,based on the model by Covid Act Now (2020) and the PPE need assessment in Appendix B.In the first pillar, the central authority is expected to work with regional authorities and healthcareprofessionals to predict the dynamics of regional demands. All regional data are then compiled and aggre-gated to form the basis of forecast for the system-wide resource demand. Suppose that there are a totalof n regions in a healthcare system or medical resource alliance. For example, the aggregate ventilator de-mand can be determined by X VEN j = (cid:80) ni =1 X VEN( i ) j , while the aggregate PPE set demand may be given by X PPE j = (cid:80) ni =1 X PPE( i ) j . Figure 5 shows how the aggregate ventilator and PPE demand prediction for theCOVID-19 pandemic could have been made in the hypothetical example of a three-state resources poolingalliance. Observe that projections for ventilators and PPE sets show very similar patterns as both weredriven by the same SEIRD model. The peaks in demand for ventilator are delayed compared with those8uspected InfectedHospitalized cases InfectedICU casesNumber of sets per case Number of sets per day per patientHealthcare staff θ E θ I θ I Nursing 1-2 6 6-12Medical 1 2-3 3-6Cleaning 1 3 3Assistant nursingand other services 0-2 3 3Total 3–6 14–15 15–24Table 3: Minimum amount of PPE sets for different scenarios
Jul 2020 Jan 2021 Jul 202105k10k New York Florida California V e n t il a t o r d e m a n d (a) Ventilator Jul 2020 Jan 2021 Jul 202101M2M3M New York Florida California PP E d e m a n d (b) Personal protective equipment Figure 4: Ventilator and personal protective equipment regional demand prediction in New York, Florida,and Californiafor PPE sets in Figure 5 due to the fact that it may take a few days before newly diagnosed patients todevelop symptoms that require ventilator intervention. The projection of regional and aggregate demandsoffers health authorities a clear understanding of the temporal competition of critical resources.
Jul 2020 Jan 2021 Jul 202105k10k V e n t il a t o r d e m a n d aggregate demand in three states (a) Ventilator Jul 2020 Jan 2021 Jul 202101M2M3M PP E d e m a n d aggregate demand in three states (b) Personal protective equipment Figure 5: Ventilator and personal protective equipment aggregate demand prediction in New York, Florida,and California 9t should be pointed out that predictive models in Pillar I, such as the SEIRD model introduced in thissection, are used for multiple purposes as shown in Figure 3. First, they need to be developed prior to apandemic using historical data and to form the basis of demand forecast for contingency planning in PillarII. Then, as a pandemic starts to emerge, the predictive models also need to be re-calibrated and updatedwith latest medical knowledge and reported cases. New forecasts would then be fed into models to determineoptimal allocation strategies in Pillar III. As medical knowledge of the viral disease evolves and predictivemodels improve over time, Pillars I and III may be revisited from period to period. When a distributionschedule of resources requires an update, we can go back to Pillar II. Therefore, the three-pillar frameworkmay 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 otheressential resources before emergency production can pick up and additional supplies become available. Tomeet the surge demand at the onset of a pandemic, many countries maintain national repositories of antibi-otics, vaccines, chemical antidotes, antitoxins, and other critical medical supplies. A centralized stockpilingstrategy is intended to provide a stopgap measure to meet the surge in resources demand at the early stage ofthe pandemic. There has been well-established literature on stockpiling strategies for influenza pandemics;see, for example, Greer and Schanzer (2013) and Siddiqui and Edmunds (2008).One should keep in mind that a practical stockpiling strategy is often an act of balance between adequatesupply and economic cost . On one hand, under-stocking is a common issue as resources and their storage canpost heavy cost, and the actual demand during the pandemic outbreak could deviate from the projection;for example, Ellison (2020) claimed that as many as 20 states in the United States are expected to encountershortage in ICU beds when the COVID-19 cases peak. On the other hand, excessive stockpiling for longterm could lead to unnecessary waste, especially for disposable and perishable resources; for instance, Facher(2020) reported that, in March 2020 during the COVID-19 pandemic, the Strategic National Stockpile inthe United States stocked 13 million N95 masks, of which as many as 5 million may have expired, partlycontributing 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 apandemic. Notice that durable resources such as ventilators can be reused throughout the pandemic, whilesingle-use resources such as PPE sets must be disposed of after one-time usage. Hence, we have to treatthem 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 of resources tomaintain in some centralized location. In addition, to meet surge demand, the authority may need toreach contractual agreements with suppliers for emergency orders, which may be limited by the maximumproduction rate of a units per day during a pandemic. Since ventilators are durable, the stock of ventilatorsdoes not decrease over time due to usage. We assume that they can be deployed to different regions atnegligible cost. Therefore, the total number of available ventilators in the entire alliance is given by K j = K + aj , on the j -th day since the onset of the pandemic. Hence, the only decision variable of the centralauthority in the case of ventilators is the initial stockpile size K .10 ul 2020 Jan 2021 Jul 202105k10k15k demand supply N u m b e r o f v e n t il a t o r s (a) Extreme understocking Jul 2020 Jan 2021 Jul 202105k10k15k demand supply N u m b e r o f v e n t il a t o r s (b) Extreme oversupply Figure 6: Two extreme scenarios of initial stockpile size K for ventilatorsTo better explain the need for an optimal initial stockpile size K , consider two extreme cases in Figures6a and 6b for the three-state alliance. On one end of the extreme, the central authority may decide not tohold any initial stockpile but simply rely on the maximum emergency production limit during the pandemic;Figure 6a shows a clear extreme shortage at the first peak time of aggregate ventilator demand. On theother end, suppose that the central authority decides to hold an extraordinary amount of initial stockpile forventilators to meet the highest peak of aggregate ventilator demand; Figure 6b illustrates a clear extremeoversupply of ventilators during most of the time of pandemic; also, in this case, the economic cost of severestorage can be huge. Therefore, the central authority has to take a delicate balance on an initial stockpilesize K 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 ≥ m (cid:88) j =1 ω j (cid:32) θ + j (cid:0) X VEN j − ( K + aj ) (cid:1) + θ − j (cid:0) X VEN j − ( K + aj ) (cid:1) − + c j ( K + aj ) (cid:33) + c K , (1)where m is the number of days of the pandemic, ω j is a weight for significance of precision for the costs onthe j -th day of the pandemic, θ + j is an economic cost per squared unit of shortage, θ − j is an opportunitycost per squared unit of oversupply, c j is the aggregate cost of possession per unit of ventilators per day, c 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 asfollows. While one copy of the quantity X VEN j − ( K + aj ) represents the amount of resource imbalance(shortage or surplus), the other copy ( θ ± j / X VEN j − ( K + aj )] ± can be viewed as the (linear) variablecost of the imbalance. In other words, the larger the imbalance, the higher the price to pay. The quadraticform is the product of cost per unit and the unit of imbalance, which yields the overall economic cost ofimbalance. The weight w j can be used for different purposes. For example, it may be reasonable to assumethat the precision of meeting demands in near future is more important than that in the far-future given theuncertainty 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.To understand the analytical solution to this problem, we need to look at the projected shortage without11 [1] Y [2] . . . Y [ J − Y [ J − Y [ J ] Y [ J +1] . . . Y [ m − Y [ m ] K Figure 7: Optimal initial stockpile K relative to projected shortages without initial stockpileany initial stockpile, Y j := X VEN j − aj , for j = 1 , · · · , m, which is the accumulated demand less the accu-mulated supply apart from the initial stockpile. Note that we consider the accumulated supply because theresources are durable and can be reused. When Y j > , there is a drain on the initial stockpile as currentdemand 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 func-tion depends on the number of days with decreasing stockpile ( Y j >
0) and those with increasing stockpile( Y j < sorting projected shortages in an ascending order.Let us denote the sorted sequence by { Y [ j ] , j = 1 , · · · , m } , where Y [ j ] represents the j -th smallest projectedshortage. In the objective function (1), the cost coefficient θ ± j applies according to whether or not stockpileexceeds demand. If K is placed below Y [ j ] , there is a shortage in the healthcare system and hence the costcoefficient θ + is applied. Otherwise, there is a surplus in the system and the cost efficient θ − is applied.The optimality is achieved when K is kept at a delicate position. See Figure 7. The nature of the sum ofsquared shortages in (1) determines that the optimal initial stockpile K ∗ should be squeezed between Y [ J − and Y J in such a way that Y [ J − ≤ (cid:80) J − j =1 ω [ j ] θ − [ j ] (cid:18) Y [ j ] − c [ j ] θ − [ j ] (cid:19) + (cid:80) mj = J ω [ j ] θ +[ j ] (cid:18) Y [ j ] − c [ j ] θ +[ j ] (cid:19) − c (cid:80) J − j =1 ω [ j ] θ − [ j ] + (cid:80) mj = J ω [ j ] θ +[ j ] ≤ Y [ J ] . Once J is identified, the optimal stockpile K ∗ is given by K ∗ = max (cid:80) J − j =1 ω [ j ] θ − [ j ] (cid:18) Y [ j ] − c [ j ] θ − [ j ] (cid:19) + (cid:80) mj = J ω [ j ] θ +[ j ] (cid:18) Y [ j ] − c [ j ] θ +[ j ] (cid:19) − c (cid:80) J − j =1 ω [ j ] θ − [ j ] + (cid:80) mj = J ω [ j ] θ +[ j ] , . The proof of this result can be found in Appendix A.1. This result shows that the optimal initial stockpile K is the weighted average of all projected shortages discounted by the cost of possession relative to the12conomic cost of shortage, Y [ j ] − c [ j ] /θ ± [ j ] . The adjustment term c [ j ] /θ ± [ j ] indicates that, the higher cost ofpossession 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 havelittle financial means to pay for stockpiling and be left with no choice but to deal with the demand-supplyimbalance. In contrast, if the economic cost of imbalance is too high due to lost productivity or even thesociety’s resentment on government’s failure to meet demand, then the central authority would ignore thecost of possession and do everything possible to reduce the shortage. Jul 2020 Jan 2021 Jul 202105k10k15k observed demand predicted demandoptimal supply N u m b e r o f v e n t il a t o r s (a) θ + j = θ − j (Shortage costs the same as surplus) Jul 2020 Jan 2021 Jul 202105k10k15k observed demand predicted demandoptimal supply N u m b e r o f v e n t il a t o r s (b) θ + j = 20 θ − j (Shortage costs more) Figure 8: Optimal initial stockpile size K for ventilators according to different weights of economic costFigure 8 depicts optimal initial stockpile size in the case study. The model parameters are provided inAppendix B. Observe that optimal initial stockpiles are chosen to reduce shortage in the early stages andoversupply in the late stages of the pandemic, compared with those strategies shown in Figure 6. When theresource shortage costs the same or less than the resource surplus, Figure 8a shows that the strategy requiresless initial stockpile due to the excessive amount of supply after the pandemic dies down. In contrast, if theshortage weighs more than the surplus, the strategy is to reduce shortage in early stages at the expense ofincreasing oversupply in late stages; see Figure 8b. 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 of single-use resources such as PPE and make contractual agreements with emergency suppliers, which canprovide additional supply at the production rate a units per day. Since PPE is of single-use, during thepandemic, the central authority has to stockpile PPE sets not only for the present but also to potentiallydeploy them for later time in order to meet surge demand. Therefore, the central authority needs toplan for both initial stockpile size K and the amount of distribution k j to all regions on day j . Thedynamics of the centralized storage { K j , j = 0 , , . . . , m } is determined by the recursive relation K j = K j − + a − k j + ( k j − X j ) + for j = 1 , . . . , m . The relation can be interpreted as follows. The currentstockpile K j is based off the previous period’s stockpile K j − , 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 todistribute k j units but can only consume X j < k j , then the unused amount should count towards thecentralized storage for future use. 13 ul 2020 Jan 2021 Jul 20210100020003000 demand supply U n i t s o f PP E ( t h o u s a n d s ) (a) Aggressive distribution Jul 2020 Jan 2021 Jul 20210100020003000 demand supply U n i t s o f PP E ( t h o u s a n d s ) (b) Conservative distribution Figure 9: Two extreme scenarios of distribution schedule k , k , . . . , k m for personal protective equipment Jul 2020 Jan 2021 Jul 20210100020003000 observed demand predicted demandoptimal supply U n i t s o f PP E ( t h o u s a n d s ) Figure 10: Optimal distribution schedule k , k , . . . , k m and initial stockpile size K of personal protectiveequipment for the three-state resource pooling allianceConsider two extreme cases in Figures 9a and 9b in the case study. In an extreme case, assume thatthe central authority decides to distribute as much as possible to meet the demand until the centralizedstorage is exhausted. This aggressive early distribution strategy is depicted in Figure 9a. After the storagedepletion, the system relies only on the new supply, which clearly is not sufficient to meet demands and cancause severe shortage at the time of second peak. In the other extreme, the central authority may chooseto hold off dispersing any equipment at all till the point that the storage is believed to be sufficient to coverall future demands. Such a conservative distribution strategy is illustrated in Figure 9b. The challenge withthis strategy is that the central authority would have to deal with the repercussion of not providing anyassistance in the early stage of the pandemic. Therefore, it is sensible that the central authority develop adistribution schedule that takes a temporal balance of varying needs from all regions. Here we introduce theoptimization problem for both an initial stockpile and the distribution schedule of single-use resources.min K ≥ ,k ,...,k m m (cid:88) j =1 ω j (cid:32) θ + j (cid:0) X PPE j − k j (cid:1) + θ − j (cid:0) X PPE j − k j (cid:1) − + c j K j (cid:33) + c K K j = K j − + a − k j + (cid:0) k j − X PPE j (cid:1) + ≥ k j ≥ , for j = 1 , , . . . , m, where c j is the centralized cost of possession per unit of PPE per day. It should be pointed out that thecentralized storage should be kept non-negative for practical purposes and the distribution amount shouldalso be kept non-negative. Negative distribution could mean confiscation of regional resources for systemwide re-distribution, which is not considered in this paper.Figure 10 depicts the case of optimal distribution schedule for the three-state alliance. Observe thatthe optimal supply distribution schedule stays below the trajectory of demand. The slight shortage resultsfrom the consideration of the cost of possession. Should the cost of possession be zero, the optimal supplywould be to match the demand exactly at all times. The initial stockpile can be set artificially high so thatany desired amount can be carried over from period to period and last long enough to support all futuredemands. In the presence of possession cost, Figure 10 also reveals a distribution strategy that in essenceignores the demands at the start of a pandemic and after the pandemic dies down and instead focuses onmeeting demands at the first peak of the pandemic. Keep in mind that the weight of significance ω j inthis example is set to be proportional to the size of demand. Therefore, the strategy prioritizes meeting thedemand in the first peak over other periods due to its high demand. If we were to choose the same weight ω 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 toshow that any oversupply distribution scheme k j > X PPE j must be sub-optimal, and hence the problem canbe simplified as follows.min K ≥ ,k ,...,k m m (cid:88) j =1 ω j (cid:32) θ + j (cid:0) X PPE j − k j (cid:1) + c j K j (cid:33) + c K such that K j = K j − + a − k j ≥ ≤ k j ≤ X PPE j , for j = 1 , , . . . , m, Because of the convexity, it can be solved numerically using Disciplined Convex Programming (DCP), whichrequires minimal computational time (Grant, Boyd, and Ye, 2006). The solution shown in Figure 10 isobtained with the help of an R package
CVXR developed based on the DCP method (Fu, Narasimhan, andBoyd, 2020).
Pillar III: Centralized Resources Allocation
In the time of severe resources shortage, a coordinated effort becomes necessary to obtain additional suppliesand to ration limited existing resources. Existing resources are not necessarily distributed on a basis ofhealth need or justice (Tobin-Tyler, 2020). Many hardest hit states have to ration care, while other stateshave low utilization of their resources. As alluded to earlier, not all regions experienced surge in demandsat the same time (see Figure 2), it has been long argued that United States federal government should havetracked the current use and the projection of needs in all states and coordinated allocation of resources toreduce shortage across regions and over time, during the COVID-19 pandemic (Ranney, Griffeth, and Jha,2020).There are two common types of resources allocation problems in the course of a pandemic, both of whichcan be formulated and cast in the Pillar III of our proposed framework.1.
Macro level resources pooling.
A central authority acts in the best interest of a union of many regions15o increase supply as well as to coordinate the distribution of existing and additional resources amongdifferent regional healthcare providers.2.
Micro level rationing.
Facing an imbalance of demands and supplies in medical equipment and re-sources, hospitals often have to make difficult but necessary decisions to ration limited existing resourcesas 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 onhealthcare effectiveness and justice. The setting of standards, protocols and policies can have profoundimpact on the functioning of a healthcare system at the time of crisis. Therefore, the best practice ofresources allocation should be based on scientific assessment and evaluation rather than on-the-fly ad-hocdecisions and a patchwork of damage-control rules.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, whenmedical supplies are scarce, many countries, states and cities are competing for resources. Whileeach state acts in its best interest to acquire medical devises and protective equipment, a federalgovernment may see the urgent need to seize control of the cargo to boost a centralized stockpile. Aholistic approach aims to strike a balance among different objectives for various stakeholders.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 itstransmission dynamics and affects different sectors of a healthcare system in a chain reaction. Medicalneeds 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 formulatedsimilarly as in this section. • Minimization of shortage and oversupply
Decision makers need to take into account spatio-temporal differences in demand and supply overthe course of a pandemic. It is imperative for authorities to allocate more resources to epicentersof a pandemic than other regions under less imminent threat. For example, New York City was thefirst in the State of New York to witness the COVID-19 pandemic, when other counties had littleto no reported cases (Associated Press, 2020). The State governor issued an executive order totake ventilators and other protective gears from underutilized private hospitals and companies. Asinfected cases stabilize or even decline in pandemic-savaged regions, a central authority may needto shift its attention to other areas of potential outbreaks and allocate resources in anticipation ofnew waves. This was also evident when many states in the United States in early stages of outbreakduring the COVID-19 took preemptive measures to procure medical supplies from countries likeChina and South Korea which have developed production capacities after the local epidemics areunder control. Therefore, it is sensible to develop an allocation strategy that minimizes shortageand oversupply across different regions and over the life-cycle of a pandemic.16
Promoting and rewarding instrumental value
Critical preventive gears and medical care should be provided first to healthcare workers in thefront line, employees in essential businesses and critical infrastructures. Not only because they areat high risk due to their exposure to infectious disease, but also the society bears heavy economiccost when these workers fall ill and are unable to return to work. The lack of sufficient front-lineworkforce may cause severe disruptions to public services, which can have rippling effect on therest 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 treatmentshould be reserved for patients facing life-threatening conditions, when there is an insufficientsupply 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 notsurvive after being placed on mechanical ventilators (Preidt, 2020). To maximize the benefit,access to ventilator treatment should be prioritized for younger patients who can benefit the mostand have the higher chance of survival. For example, many hospitals in Italy lowered the agecutoff from 80 to 75 in order to ration limited ventilators (Rosenbaum, 2020). Such a strategyoften 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 resourcesfor different regions, building on the proposed optimal centralized stockpiling and distribution strategies.Figures 11a and 12a put the regional resources demand and optimal aggregate supply together for the easeof exposition.
Jul 2020 Jan 2021 Jul 202105k10k New York demand Florida demandCalifornia demand total supply N u m b e r o f v e n t il a t o r s (a) Regional ventilator demands Jul 2020 Jan 2021 Jul 202102000400060008000 New York allocation Florida allocationCalifornia allocation total supply N u m b e r o f v e n t il a t o r s (b) Optimal ventilator allocations Figure 11: Optimal ventilator 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 regionsduring a pandemic that lasts for m days. We always use the superscript ( i ) to indicate quantities for the17 ul 2020 Jan 2021 Jul 20210100020003000 New York demand Florida demandCalifornia demand total supply U n i t s o f PP E ( t h o u s a n d s ) (a) Regional PPE demands Jul 2020 Jan 2021 Jul 20210100020003000 New York allocation Florida allocationCalifornia allocation total supply U n i t s o f PP E ( t h o u s a n d s ) (b) Optimal PPE allocations Figure 12: Optimal personal protective equipment allocations in New York, Florida, and California i -th region. Bear in mind that there could still be aggregate shortage of supply for ventilators and PPE setsto all regions in the alliance. The central authority would have to take a holistic view of competing interestsof participating regions. On each day in the pandemic, when the aggregate demand exceeds the aggregatesupply, the central authority should choose to allocate resources taking into account spatial differences indemand and supply. This motivates the optimization model for ventilator allocations:min K ( i ) j ≥ i =1 , ,...,n ; j =1 , ,...,m m (cid:88) j =1 n (cid:88) i =1 ω ( i ) j (cid:32) θ +( i ) j (cid:16) X VEN( i ) j − K ( i ) j (cid:17) + θ − ( i ) j (cid:16) X VEN( i ) j − K ( i ) j (cid:17) − (cid:33) such that n (cid:88) i =1 K ( i ) j = K j , for j = 1 , , . . . , m, where ω ( i ) j is a weight to the j -th day of the pandemic in the i -th allied region, θ +( i ) j is an economic cost persquared unit of shortage, θ − ( i ) j is an opportunity cost per squared unit of oversupply. The quadratic terms θ ± ( i ) j / (cid:16) X ( i ) j − K ( i ) j (cid:17) represents the economic cost due to the demand-supply imbalance. Note that θ ± ( i ) j / θ ± ( i ) j / (cid:16) X ( i ) j − K ( i ) j (cid:17) represents the linear variablecost per unit. The variable cost in principle reflects the law of demand that the price increases with thequantity demanded. Therefore, the total cost is the product of variable cost per unit θ ± ( i ) j / (cid:16) X ( i ) j − K ( i ) j (cid:17) and the total unit of imbalance (cid:16) X ( i ) j − K ( i ) j (cid:17) . The economic cost is used to account for both potentialloss 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 largeimbalance of demand and supply. The weight ω ( i ) j can be used to measure the relative importance of resourceallocation for region i at time t j to other regions and time points. There are some examples of its applicationsunder 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 maycarry more weight than a rural area with a small region for political reasons. When hospitals have to rationlimited resources, they may implement the strategy to maximize the benefit from treatment. In such asetup, X ( i ) represents the demand from a particular cohort. A decision maker may give higher weight toage cohorts with more remaining life years than age cohorts with less remaining years. It is also a common18trategy to give priorities for access to medical resources to healthcare workers. In both cases, the set ofweights ω ( i ) j reflects the management’s priorities and preferences over time.The constraint (cid:80) ni =1 K ( i ) j = K j indicates that resources allocated to different regions must add up tothe total amount of supply available to the central authority. The evolution of supply { K j , j = 1 , , . . . , m } is based on the centralized stockpiling strategy discussed in previous sections. The evolution of demand { X ( i ) j , i = 1 , . . . , n, j = 1 , , · · · , m } can be based on forecasts from epidemiological models fitted to mostrecent 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 theamount of stockpile to be released each period. For durable resources, the central authority distributes theaccumulated stock K j at any given time j . Because single-use resources cannot be reused, the central au-thority can only distribute incremental amount according to some distribution schedule. With this differencein mind, we formulate the allocation of single-use resources by an optimization problem.min k ( i ) j ≥ i =1 , ,...,n ; j =1 , ,...,m m (cid:88) j =1 n (cid:88) i =1 ω ( i ) j (cid:32) θ +( i ) j (cid:16) X PPE( i ) j − k ( i ) j (cid:17) + θ − ( i ) j (cid:16) X PPE( i ) j − k ( i ) j (cid:17) − (cid:33) such that n (cid:88) i =1 k ( i ) j = k j , for j = 1 , , . . . , m. Note that the distribution amount k j could be determined prior to a pandemic by some contingency planningor during a pandemic by an adjusted distribution schedule. The “Single-Use Resources: Personal ProtectiveEquipment” section offers an example of how such a distribution schedule can be determined to take intotemporal competition of single-use resources. Holistic Allocation Algorithms
Because both allocation problems take the same form, their analytical solutions can be derived in the sameway. 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 forthe 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 canglean economic insights. The proofs of these solutions are relegated to Appendix A.2. The central authorityhas to first determine whether or not there is a system wide surplus or shortage. The allocation strategydiffers under these scenarios. System wide surplus
If there is an overall surplus in the healthcare system at time j , i.e., K > n (cid:80) r =1 X ( r ) = X , then only theeconomic cost for oversupply θ − ( i ) applies and the optimal allocation of existing supply to the i -th region is19iven by K ( i ) = − ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) X ( i ) + ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) K − (cid:88) r (cid:54) = i X ( r ) , ∀ i = 1 , , · · · , n. (2)Observe that the allocation formula (2) has an explicit economic interpretation, which shows that the optimalsupply 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 interestof the region alone. The demand and supply for all other regions are ignored in its consideration. • Altruistic optimal supply: K − (cid:80) nr =1; r (cid:54) = i X ( r ) If the region i places interests of all other regions above its own, then the medical supply goes to otherregions 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 oftwo extremes, namely the self-concerned optimal and the altruistic optimal supplies. It should be pointedthat the average of two optimal supplies is determined by the harmonic weighting ω ( i ) θ − ( i ) (cid:46) n (cid:80) r =1 1 ω ( r ) θ − ( r ) asopposed to arithmetic weight ω ( i ) θ − ( i ) (cid:46) n (cid:80) r =1 ω ( r ) θ − ( r ) . It is known in Chong, Feng, and Jin (2020) that inmulti-objective Pareto optimality the harmonic weighting is always used for balancing competing interests ofparticipants in a group whereas the arithmetic weighting serves the purpose of balancing competing objectivesof the same participant. The pandemic resource allocation problem is in essence a model of Pareto optimalitywith 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 ) (cid:32) K − n (cid:88) r =1 X ( r ) (cid:33) , B ( i ) = ω ( i ) θ − ( i ) (cid:80) nr =1 1 ω ( r ) θ − ( 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 byharmonic weighting. Observe that (cid:80) ni =1 K ( i ) = K as expected since (cid:80) ni =1 B ( i ) = 1 . System wide shortage
If there is an overall shortage in the healthcare system at time j , i.e. , K ≤ (cid:80) nr =1 X ( r ) = X , it turns outthat the optimal allocation strategy is to deliver the resources where they are needed the most. We cansummarize the algorithm in three steps: Step 1: Demand ranking.
The first order of business is to sort regional demands { X ( i ) , i =1 , . . . , n } in a descending order. We use the subscript [ i ] to indicate the i -th largest order statistic, i.e. , X [1] ≥ · · · ≥ X [ n ] ≥
0. The ranking of regional demands determines the order in which the regions areconsidered for resource allocation in the next step.20 tep 2: Frugality test.
The algorithm first tests cases that perform allocation rules in a similarway to (3). For any fixed I = 1 , · · · , n , consider the holistic allocation rule that provides for I regionswith largest demand by˜ K [ i ] = X [ i ] + ˜ B [ i ] (cid:32) K − I (cid:88) r =1 X [ r ] (cid:33) , ˜ B [ i ] = ω [ i ] θ +[ i ] (cid:80) Ir =1 1 ω [ r ] θ +[ r ] . (4)To find the optimal number I of regions to provide support to, the algorithm ensures that the allocationrule 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 ≤ I (cid:88) r =1 X [ r ] . (ii) When the allocation rule (4) is forcefully applied to all regions, the I regions with highest demandsshould receive non-negative allocation and the rest of the group receive negative allocation.˜ K [1] , · · · , ˜ K [ I ] ≥ > ˜ K [ I +1] , · · · , ˜ K [ n ] . There is a unique value of I that passes the frugality test. As the aim of the strategy is to cover asmany 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 aredivided among the I states according to the holistic allocation principle. In other words, the allocationof supply to the i -th region is given by K [ i ] = ˜ K [ i ] , ∀ i = 1 , , · · · , I ; K [ I +1] = · · · = K [ n ] = 0 . The general idea of the holistic allocation is illustrated in Figure 13. The burden of shortage Y := (cid:80) Ir =1 X [ r ] − K is carried by all I states in proportion to their respective harmonic weight ˜ B [ i ] . There-fore, each region receives its demand less its “fair” portion of system-wide shortage, i.e. X [ i ] − ˜ B [ i ] Y. The solution to the three-state alliance for pooling ventilators and PPE sets are shown in Figures 11 and12. Figure 11b depicts the case of optimal allocation strategies for ventilators, which confirms the intuitionin the “Case Study” Section. In April, the central authority could have optimally reallocated all availableaggregate ventilators in the alliance to NY; This is owing to the fact that NY has the highest demand ofall 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 noshortage of ventilators at all. Figure 12b illustrates optimal allocation strategies for PPE sets in the casestudy. In April, the central authority could distribute stockpiled PPE sets and have all sent to NY foremergency response; As the pandemic dies down for NY and picks up for CA and FL, the resources are moreevenly spread in June. By August, all PPE sets should be released to CA and FL both with high demands.21 ystem widetotal supply K Shortage˜ B [1] · · · ˜ B [ I ] Regionaldemand X [1] . . . X [ I ] Allocation . . . K [1] · · · K [ I ] K Figure 13: Holistic allocation of resources in face of system wide shortage
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 paperinvestigates what could have been done better to reduce the imbalance of medical resources demand and sup-ply. Inspired by classical theory of risk aggregation and capital allocation, this paper proposes a three-pillarresources planning and allocation framework — demand forecast, centralized stockpiling and distribution,and centralized resources allocation. This paper further develops a novel spatio-temporal balancing of re-sources and can potentially used by public policymakers as quantitative basis for making informed decisionson planning, funding and rationing of critical resources.It should be pointed out that the paper focuses on managerial insights that can be drawn from theoptimization framework and its analytical solutions. There are admittedly a number of limitations in thehypothetical example of a three-state resources pooling arrangement. The three states are chosen for themost drastic effects of planning and allocation of existing resources for the purpose of illustration. It is beyondthe scope of this paper to consider the political reality that may prevent such arrangements. In theory, themethodology 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. Anotherlimitation of this example is the egalitarian approach to shortages in different regions, which ignores ethicalissues that may arise from freely moving resources from one region to another. While it may be economicallyoptimal to deliver all resources in the system to the place where they are needed the most, it may bepolitically challenging to leave other places with less severe shortage without support. A potential remedywould be to introduce additional constraints in the optimization problems that require some minimal supportfor each region. 22 eferences [1] Daron Acemoglu et al.
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A Analytical Solutions and Proofs
A.1 Stockpiling of durable resources
The optimization problem for the stockpiling of durable resources is as follows:min K ≥ m (cid:88) j =1 ω j (cid:32) θ + j X j − ( K + aj )) + θ − j X j − ( K + aj )) − + c j ( K + aj ) (cid:33) + c K . Theorem A.1.
Let Y j = X j − aj, ∀ j = 1 , , · · · , m . Let S = (cid:80) mj =1 ω j c j + c . Let Y [1] ≤ Y [2] ≤ · · · ≤ Y [ m ] bethe increasingly ordered sequence of Y , Y , · · · , Y m . Let J = 1 , , · · · , m such that J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J ] ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J ] ) ≤ S ≤ J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J − ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J − ) , where we define that when J = 1 , J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J ] ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J ] ) = m (cid:88) j =1 ω [ j ] θ +[ j ] ( Y [ j ] − Y [1] ) , J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J − ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J − ) = ∞ . Let K (cid:48) = (cid:80) J − j =1 ω ( j ) θ − ( j ) Y ( j ) + (cid:80) mj = J ω ( j ) θ +( j ) Y ( j ) − S (cid:80) J − j =1 ω ( j ) θ − ( j ) + (cid:80) mj = J ω ( j ) θ +( j ) . If K (cid:48) < , then the optimal initial stockpile, which minimizes the objective function above, K ∗ = 0 , and if K (cid:48) ≥ , then K ∗ = K (cid:48) .Proof. Let Y j = X j − aj, ∀ j = 1 , , · · · , m . Let Y [1] ≤ Y [2] ≤ · · · ≤ Y [ m ] be the increasingly ordered sequenceof Y , Y , · · · , Y m . In that case, Y j represents the daily shortage when the initial stockpile is completelymissing. Then the objective function becomes F ( K ) := m (cid:88) j =1 ω [ j ] (cid:32) θ +[ j ] Y [ j ] − K ) + θ − [ j ] K − Y [ j ] ) (cid:33) + m (cid:88) j =1 ω j c j + c K + m (cid:88) j =1 ω j c j aj. Note that F ( K ) is a convex function in K for any K ∈ R . Let G ( K ) = m (cid:88) j =1 ω [ j ] (cid:32) θ +[ j ] Y [ j ] − K ) + θ − [ j ] K − Y [ j ] ) (cid:33) , = m (cid:88) j =1 ω j c j + c , where G ( · ) is convex in K .The first order derivatives of G at Y [1] and Y [ m ] are as follows: G (cid:48) ( Y [1] ) = m (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [1] − Y [ j ] ) ≤ .G (cid:48) ( Y [ m ] ) = m (cid:88) j =1 ω [ j ] θ +[ j ] ( Y [ j ] − Y [ m ] )( −
1) = m (cid:88) j =1 ω [ j ] θ +[ j ] ( Y [ m ] − Y [ j ] ) ≥ . Then ∃ ˜ K ∈ (cid:2) Y [1] , Y [ m ] (cid:3) such that G (cid:48) ( ˜ K ) = 0. Since S = (cid:80) mj =1 ω j c j + c ≥ ∃ K (cid:48) ∈ ( −∞ , Y [ m ] ] such that F (cid:48) ( K (cid:48) ) = 0, i.e. , ∃ J = 1 , · · · , m such that K (cid:48) ∈ (cid:2) Y [ J − , Y [ J ] (cid:3) , where we define Y (0) = −∞ , and F (cid:48) ( K (cid:48) ) = 0.Here, we are relaxing the constraint that the initial stockpile has to be non-negative, but we will add itback in the end. The key observation is that given all the ( Y [ j ] ) j ∈{ ,...,m } , we can always find a K (cid:48) betweentwo adjacent Y [ j ] ’s, Y [ J − and Y [ J ] , that minimize the objective function. That is, F (cid:48) ( K (cid:48) ) = J − (cid:88) j =1 ω [ j ] θ − [ j ] ( K (cid:48) − Y [ J ] ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − K (cid:48) )( −
1) + S = 0 . Then K (cid:48) = (cid:80) J − j =1 ω [ j ] θ − [ j ] Y [ j ] + (cid:80) mj = J ω [ j ] θ +[ j ] Y [ j ] − S (cid:80) J − j =1 ω [ j ] θ − [ j ] + (cid:80) mj = J ω [ j ] θ +[ j ] . Therefore, the condition that K (cid:48) ∈ [ Y [ J − , Y [ J ] ] is given by Y [ J − ≤ (cid:80) J − j =1 ω [ j ] θ − [ j ] Y [ j ] + (cid:80) mj = J ω [ j ] θ +[ j ] Y [ j ] − S (cid:80) J − j =1 ω [ j ] θ − [ j ] + (cid:80) mj = J ω [ j ] θ +[ j ] ≤ Y [ J ] . Or equivalently, ∃ J = 1 , , · · · , m , such that J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J ] ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J ] ) ≤ S ≤ J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J − ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J − ) , wherein if J = 1, we define J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J ] ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J ] ) = m (cid:88) j =1 ω [ j ] θ +[ j ] ( Y [ j ] − Y [1] ) , J − (cid:88) j =1 ω [ j ] θ − [ j ] ( Y [ j ] − Y [ J − ) + m (cid:88) j = J ω [ j ] θ +[ j ] ( Y [ j ] − Y [ J − ) = ∞ . Finally, given the non-negativity of stockpile, if K (cid:48) <
0, then the optimal initial stockpile K ∗ = 0, andif K (cid:48) ≥
0, then K ∗ = K (cid:48) . 27 .2 Centralized resources allocation The optimization problem for allocating resources amount regions is as follows:min K ( i ) j ≥ i =1 , ,...,n ; j =1 , ,...,m m (cid:88) j =1 n (cid:88) i =1 ω ( i ) j (cid:32) θ +( i ) j (cid:16) X ( i ) j − K ( i ) j (cid:17) + θ − ( i ) j (cid:16) X ( i ) j − K ( i ) j (cid:17) − (cid:33) such that n (cid:88) i =1 K ( i ) j = K j , for j = 1 , , . . . , m, Theorem A.2.
The optimization above is done from period to period, and thus to simplify notation, thetime indicator j can be dropped at each time point. let X [1] ≥ · · · ≥ X [ n ] > be the decreasingly orderedsequence of X (1) , X (2) , . . . , X ( n ) .If K > n (cid:80) r =1 X ( r ) = X , then K ( i ) = − ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) X ( i ) + ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) K − (cid:88) r (cid:54) = i X ( r ) , ∀ i = 1 , , . . . , n. If K ≤ (cid:80) nr =1 X ( r ) = X , we can find an I = 1 , , . . . , n such that K ≤ (cid:80) Ir =1 X [ r ] , X [ i ] ≥ ω [ i ] θ +[ i ] I (cid:80) r =1 1 ω [ r ] θ +[ r ] (cid:32) I (cid:88) r =1 X [ r ] − K (cid:33) , ∀ i = 1 , . . . , I,X [ i ] < ω [ i ] θ +[ i ] I (cid:80) r =1 1 ω [ r ] θ +[ r ] (cid:32) I (cid:88) r =1 X [ r ] − K (cid:33) , ∀ i = I + 1 , I + 2 , . . . , n. Then K [ I +1] = · · · = K [ n ] = 0 . K [ i ] = − ω [ i ] θ +[ i ] I (cid:80) r =1 1 ω [ r ] θ +[ r ] X [ i ] + ω [ i ] θ +[ i ] I (cid:80) r =1 1 ω [ r ] θ +[ r ] K − I (cid:88) r =1 ,r (cid:54) = i X [ r ] , ∀ i = 1 , . . . , I. Proof.
At each time point j , we want to solve the following optimization problem,min K ( i ) ; i =1 , , ··· ,n n (cid:88) i =1 ω ( i ) (cid:18) θ +( i ) X ( i ) − K ( i ) ) + θ − ( i ) K ( i ) − X ( i ) ) (cid:19) such that n (cid:88) i =1 K ( i ) = K, K ( i ) ≥ , ∀ i = 1 , , · · · , n. Case 1: Coexisting surpluses and shortages
First, let us consider the case in which some regions are having surpluses, whereas other regions areexperiencing shortages at the same time. We shall show below that this case is impossible regardless of thenon-negative constraints. That is, suppose that K (1) , K (2) , . . . , K ( n ) lie locally in a feasible set, such that I
28f them satisfy K ( i ) > X ( i ) , where I = 1 , , . . . , n −
1. The remaining n − I of them satisfy K ( i ) ≤ X ( i ) .Without loss of generality, assume the first I of K ( i ) are in the former group. Then the local problembecomes min K ( i ) ; i =1 , , ··· ,n n (cid:88) i = I +1 ω ( i ) θ +( i ) X ( i ) − K ( i ) ) + I (cid:88) i =1 ω ( i ) θ − ( i ) X ( i ) − K ( i ) ) such that n (cid:88) i =1 K ( i ) = K. The solution to this problem is given by K ( i ) = − ω ( i ) θ − ( i ) I (cid:80) r =1 1 ω ( r ) θ − ( r ) + n (cid:80) r = I +1 1 ω ( r ) θ +( r ) X ( i ) + ω ( i ) θ − ( i ) I (cid:80) r =1 1 ω ( r ) θ − ( r ) + n (cid:80) r = I +1 1 ω ( r ) θ +( r ) K − (cid:88) r (cid:54) = i X ( r ) , for i = 1 , , . . . , I , and K ( i ) = − ω ( i ) θ +( i ) I (cid:80) r =1 1 ω ( r ) θ − ( r ) + n (cid:80) r = I +1 1 ω ( r ) θ +( r ) X ( i ) + ω ( i ) θ +( i ) I (cid:80) r =1 1 ω ( r ) θ − ( r ) + n (cid:80) r = I +1 1 ω ( r ) θ +( r ) K − (cid:88) r (cid:54) = i X ( r ) , for i = I + 1 , . . . , n .However, ∀ i = 1 , . . . , I , K ( i ) > X ( i ) , and ∀ i = I + 1 , . . . , n , K ( i ) ≤ X ( i ) , which implies K > (cid:80) nr =1 X ( r ) and K ≤ (cid:80) nr =1 X ( r ) , and thus we have a contradiction. This result shows that it is impossible for someregions to have surpluses while other regions are experiencing shortages. Therefore, it suffices to only considerthe scenarios that there is a system wide surplus and that there is a system wide shortage. Case 2: System wide surplus
If there is a system wide surplus, i.e. , all regions have surpluses, then the problem becomesmin K ( i ) ; i =1 , , ··· ,n n (cid:88) i =1 ω ( i ) θ − ( i ) K ( i ) − X ( i ) ) such that n (cid:88) i =1 K ( i ) = K,K ( i ) > X ( i ) , ∀ i = 1 , . . . , n. The solution is given by K ( i ) = − ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) X ( i ) + ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) K − (cid:88) r (cid:54) = i X ( r ) . For this result to hold, we only need the condition that
K > (cid:80) nr =1 X ( r ) = X . Due to uniqueness, underthis condition, this K ( i ) is optimal. 29 ase 3: System wide shortage It remains to solve the case that there is a system wide shortage, i.e. , all regions have shortages. In thatcase, the problem becomes min K ( i ) ; i =1 , , ··· ,n n (cid:88) i =1 ω ( i ) θ +( i ) X ( i ) − K ( i ) ) such that n (cid:88) i =1 K ( i ) = K ;0 ≤ K ( i ) ≤ X ( i ) , ∀ i = 1 , · · · , n. Apart from the given condition that (cid:80) ni =1 K ( i ) = K , there are additional inequality constraints in thisoptimization problem. Hence, we make use of Karush-Kuhn–Tucker (KKT) conditions as follows. ω ( i ) θ +( i ) ( K ( i ) − X ( i ) ) − λ ( i )1 + λ ( i )2 + µ = 0 , ≤ K ( i ) ≤ X ( i ) ,λ ( i )1 ≥ , λ ( i )1 K ( i ) = 0 ,λ ( i )2 ≥ , λ ( i )2 ( K ( i ) − X ( i ) ) = 0 , for all i = 1 , · · · , n . Regarding the values of λ ( i )1 and λ ( i )2 , we can consider the following four mutuallyexclusive cases,(i) λ ( i )1 > i = 1 , · · · , n , and λ ( i )2 > i = 1 , · · · , n ;(ii) λ ( i )1 = 0 for all i = 1 , · · · , n , and λ ( i )2 > i = 1 , · · · , n ;(iii) λ ( i )1 > i = 1 , · · · , n , and λ ( i )2 = 0 for all i = 1 , · · · , n ;(iv) λ ( i )1 = 0 for all i = 1 , · · · , n , and λ ( i )2 = 0 for all i = 1 , · · · , n .Each of them will be discussed as follows. Case 3.1: λ ( i )1 > i = 1 , · · · , n , and λ ( i )2 > i = 1 , · · · , n .We will show that this case will lead to a contradiction, and thus is impossible. Because the ordering of λ ( i )1 and λ ( i )2 does not affect the conditions, for simplicity, we rearrange them in such a way that there is an I = 1 , , . . . , n −
1, and an ˜ I = 1 , , . . . , n −
1, for which λ [ i ]1 > , ∀ i = 1 , . . . , I,λ [ i ]1 = 0 , ∀ i = I + 1 , . . . , n,λ [ i ]2 = 0 , ∀ i = 1 , . . . , n − ˜ I,λ [ i ]2 > , ∀ i = n − ˜ I + 1 , . . . , n, where [ i ] are indices after the rearrangement.We also have the condition that I ≤ n − ˜ I , because for each i , λ [ i ]1 > K [ i ] = 0, which furtherimplies λ [ i ]2 = 0. Therefore, the number of λ [ i ]2 that are equal to 0, i.e. , n − ˜ I , is at least the number of λ [ i ]1 that are greater than 0, i.e. , I . Then by the complementary slackness conditions in the KKT conditionsabove, K [ i ] = 0 , ∀ i = 1 , , · · · , I and, K [ i ] = X [ i ] , ∀ i = n − ˜ I + 1 , · · · , n .30hen, the KKT conditions are simplified. − ω [ i ] θ +[ i ] X [ i ] − λ [ i ]1 + µ = 0 , ∀ i = 1 , . . . , Iω [ i ] θ +[ i ] ( K [ i ] − X [ i ] ) + µ = 0 , ∀ i = I + 1 , . . . , n − ˜ Iλ [ i ]2 + µ = 0 , ∀ i = n − ˜ I + 1 , . . . , n ≤ K [ i ] ≤ X [ i ] , ∀ i = I + 1 , . . . , n − ˜ I n − ˜ I (cid:88) r = I +1 K [ r ] = K − n (cid:88) r = n − ˜ I +1 X [ r ] , from which we can observe the following contradiction that, µ = − λ [ n ]2 < µ = ω [1] θ +[1] X [1] + λ [1]1 > . Therefore, we can conclude that if ˜ I = 1 , · · · , n −
1, then I = 0 or n , and its contrapositive is also true,which states that if I = 1 , · · · , n −
1, then ˜ I = 0 or n . These two statements correspond the second andthird cases respectively, and they are considered as follows. Case 3.2: λ ( i )1 = 0 for all i = 1 , · · · , n , and λ ( i )2 > i = 1 , · · · , n .As in the previous case, We use the rearranged λ [ i ]1 and λ [ i ]2 , so now λ [ i ]2 = 0, for all i = 1 , . . . , n − ˜ I ,and λ [ i ]2 >
0, for all i = n − ˜ I + 1 , . . . , n , for some ˜ I = 1 , , . . . , n −
1. This implies K [ i ] = X [ i ] , for all i = n − ˜ I + 1 , . . . , n .In this case, the KKT conditions become ω [ i ] θ +[ i ] ( K [ i ] − X [ i ] ) + µ = 0 , ∀ i = 1 , . . . , n − ˜ Iλ [ i ]2 + µ = 0 , ∀ i = n − ˜ I + 1 , . . . , n ≤ K [ i ] ≤ X [ i ] , ∀ i = 1 , · · · , n − ˜ I n − ˜ I (cid:88) r =1 K ( r ) = K − n (cid:88) r = n − ˜ I +1 X ( r ) . By solving this system, we get K [ i ] = − ω [ i ] θ +[ i ] n − ˜ I (cid:80) r =1 1 ω [ r ] θ +[ r ] X [ i ] + ω [ i ] θ +[ i ] n − ˜ I (cid:80) r =1 1 ω [ r ] θ +[ r ] K − n (cid:88) r (cid:54) = i X [ r ] = X [ i ] + ω [ i ] θ +[ i ] n − ˜ I (cid:80) r =1 1 ω [ r ] θ +[ r ] (cid:32) K − n (cid:88) r =1 X [ r ] (cid:33) , ∀ i = 1 , , · · · , n − ˜ Iµ = 1 n − ˜ I (cid:80) r =1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r =1 X [ r ] − K (cid:33) [ i ]2 = − µ = − n − ˜ I (cid:80) r =1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r =1 X [ r ] − K (cid:33) , ∀ i = n − ˜ I + 1 , · · · , n. By the system wide shortage assumption, n (cid:80) r =1 X [ r ] − K ≥
0, and therefore, λ [ i ]2 ≤ i = n − ˜ I +1 , · · · , n . But this contradicts the assumption that λ [ i ]2 > i = n − ˜ I + 1 , · · · , n . Therefore, we cantell that this is another impossible case. Case 3.3: λ ( i )1 > i = 1 , · · · , n , and λ ( i )2 = 0 for all i = 1 , · · · , n .Again, for this case, we use the rearranged λ [ i ]1 and λ [ i ]2 for i = 1 , . . . , n . And we assume that there is an I = 1 , , . . . , n −
1, such that λ [ i ]1 > i = 1 , . . . , I , and λ [ i ]1 = 0 for i = I + 1 , . . . , n . This implies K [ i ] = 0for i = 1 , . . . , I , and we get the following conditions, − ω [ i ] θ +[ i ] X [ i ] − λ [ i ]1 + µ = 0 , ∀ i = 1 , . . . , Iω [ i ] θ +[ i ] ( K [ i ] − X [ i ] ) + µ = 0 , ∀ i = I + 1 , . . . , n ≤ K [ i ] ≤ X [ i ] , ∀ i = I + 1 , · · · , n n (cid:88) i = I +1 X [ r ] = K. They together give us K [ i ] = − ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] X [ i ] + ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) K − n (cid:88) r = I +1 X [ r ] (cid:33) = X [ i ] + ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) K − n (cid:88) r = I +1 X [ r ] (cid:33) , ∀ i = I + 1 , · · · , n.µ = 1 n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r = I +1 X [ r ] − K (cid:33) λ [ i ]1 = 1 n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r = I +1 X [ r ] − K (cid:33) − ω [ i ] θ +[ i ] X [ i ] , ∀ i = 1 , · · · , I. Since λ [ i ]1 > i = 1 , . . . , I and 0 ≤ K [ i ] ≤ X [ i ] for i = I + 1 , · · · , n , the following conditions hold. K ≤ n (cid:80) r = I +1 X [ r ] , X [ i ] < ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r = I +1 X [ r ] − K (cid:33) , ∀ i = 1 , , · · · , I,X [ i ] ≥ ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r = I +1 X [ r ] − K (cid:33) , ∀ i = I + 1 , · · · , n. ase 3.4: λ ( i )1 = 0 for all i = 1 , · · · , n , and λ ( i )2 = 0 for all i = 1 , · · · , n .Now it only remains to consider the case in which λ ( i )1 = 0 and λ ( i )2 = 0 for all i = 1 , . . . , n . Since thereis no divergence in the values of λ ( i )1 and λ ( i )2 , rearrangement does not make a difference. Nevertheless, wewill use the ordered indices [ i ] here for consistency.In this case, conditions now become ω [ i ] θ +[ i ] ( K [ i ] − X [ i ] ) + µ = 0 , ∀ i = 1 , . . . , n ≤ K [ i ] ≤ X [ i ] , ∀ i = 1 , . . . , n n (cid:88) i =1 K [ i ] = K. Solving this system gives us K [ i ] = − ω [ i ] θ +[ i ] n (cid:80) r =1 1 ω [ r ] θ +[ r ] X [ i ] + ω [ i ] θ +[ i ] n (cid:80) r =1 1 ω [ r ] θ +[ r ] K − (cid:88) r (cid:54) = i X [ r ] = X [ i ] + ω [ i ] θ +[ i ] n (cid:80) r =1 1 ω [ r ] θ +[ r ] (cid:32) K − n (cid:88) r =1 X [ r ] (cid:33) , ∀ i = 1 , · · · , n. Since we assume there is a system wide shortage, K − (cid:80) nr =1 X [ r ] ≤
0, and thus we get K [ i ] ≤ X [ i ] for all i = 1 , . . . , n . Then, the only required condition for this result to hold is 0 ≤ K [ i ] for all i = 1 , · · · , n , whichgives us X [ i ] ≥ ω [ i ] θ +[ i ] n (cid:80) r =1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r =1 X [ r ] − K (cid:33) , ∀ i = 1 , , · · · , n. This result can actually be seen as a special case of
Case I = 0, and therefore, we can combinethe results from these two cases.In summary, the result depends on whether there is a system wide surplus or shortage. In the event of asystem wide surplus, i.e. , K ( i ) > X ( i ) for i = 1 , . . . , n , the optimal allocation in each region is given by K ( i ) = − ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) X ( i ) + ω ( i ) θ − ( i ) n (cid:80) r =1 1 ω ( r ) θ − ( r ) K − (cid:88) r (cid:54) = i X ( r ) . In the event of a system wide shortage, i.e. , 0 ≤ K ( i ) ≤ X ( i ) for i = 1 , . . . , n , it has been demonstratedthat the solution can be found by sorting λ ( i )1 in such a way that the first I = 0 , , . . . , n − I is 0, then all λ ( i )1 = 0. The I here should satisfy the following conditions, K ≤ n (cid:80) r = I +1 X [ r ] , X [ i ] < ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r = I +1 X [ r ] − K (cid:33) , ∀ i = 1 , , · · · , I, [ i ] ≥ ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) n (cid:88) r = I +1 X [ r ] − K (cid:33) , ∀ i = I + 1 , · · · , n, where if I = 0, then the first inequality can be discarded. Once I is identified, the optimal allocation in eachregion is given by K [1] = · · · = K [ I ] = 0, K [ i ] = − ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] X [ i ] + ω [ i ] θ +[ i ] n (cid:80) r = I +1 1 ω [ r ] θ +[ r ] (cid:32) K − n (cid:88) r = I +1 X [ r ] (cid:33) , ∀ i = I + 1 , · · · , n. Since λ ( i )1 are sorted, the final result should have been re-sorted accordingly. However, the conditions arethe same regardless of the ordering. Finally, we relabel I and n − I for the sake of simplifying the notationsin the main text, and hence X ( i ) should be sorted in descending order instead. B Parameter Values in Numerical Examples
This section offers an inventory of all model parameters used in earlier sections. The same sets of parametersare used for all calculations in the three-pillar framework.
Parameter Value
Percentage of intensive care patients requiring ventilators (class I ) 0.9Units of PPE required per exposed patient (class E ) 5Units of PPE required per hospitalized patient (class I ) 15Units of PPE required per intensive care patient (class I ) 20Table 4: Demand assessment parameters; values are chosen in the ranges provided in Tables 2 and 3. Parameter Value
Participating states New York, Florida, CaliforniaCost of possession per unit per day ( c j ) 1Initial stockpile cost per unit ( c ) 25120Daily production rate ( a ) 10 UnitsShortage/surplus cost ( θ + j / θ − j ) in Figure 8a 1000/1000Shortage/surplus cost ( θ + j / θ − j ) in Figure 8b 1000/20Time varying weight ( ω j ) Proportional to daily demand X j Table 5: Ventilator planning parameters (Porpora, 2020; Rowland, 2020; Patel, 2020)34 arameter Value
Participating states New York, Florida, CaliforniaCost of possession per 1000 units per day ( c j ) 0.01Initial stockpile cost per 1000 units ( c ) 0.5Daily production rate ( a ) 50000 unitsShortage cost ( θ + j ) 1Time varying weight ( ω j ) Proportional to daily demand X j Table 6: Personal protective equipment planning parameters
Parameter Value
Participating states New York, Florida, CaliforniaShortage/surplus cost ( θ ( i )+ j / θ ( i ) − j ) 1Weight for resources allocation in region i at time j ( ω ( i ) j ) Proportional to (cid:80) mt = j X ( i ) tt