Mitigating the COVID‐19 pandemic through data‐driven resource sharing

COVID‐19 outbreaks in local communities can result in a drastic surge in demand for scarce resources such as mechanical ventilators. To deal with such demand surges, many hospitals (1) purchased large quantities of mechanical ventilators, and (2) canceled/postponed elective procedures to preserve care capacity for COVID‐19 patients. These measures resulted in a substantial financial burden to the hospitals and poor outcomes for non‐COVID‐19 patients. Given that COVID‐19 transmits at different rates across various regions, there is an opportunity to share portable healthcare resources to mitigate capacity shortages triggered by local outbreaks with fewer total resources. This paper develops a novel data‐driven adaptive robust simulation‐based optimization (DARSO) methodology for optimal allocation and relocation of mechanical ventilators over different states and regions. Our main methodological contributions lie in a new policy‐guided approach and an efficient algorithmic framework that mitigates critical limitations of current robust and stochastic models and make resource‐sharing decisions implementable in real‐time. In collaboration with epidemiologists and infectious disease doctors, we give proof of concept for the DARSO methodology through a case study of sharing ventilators among regions in Ohio and Michigan. The results suggest that our optimal policy could satisfy ventilator demand during the first pandemic's peak in Ohio and Michigan with 14% (limited sharing) to 63% (full sharing) fewer ventilators compared to a no sharing strategy (status quo), thereby allowing hospitals to preserve more elective procedures. Furthermore, we demonstrate that sharing unused ventilators (rather than purchasing new machines) can result in 5% (limited sharing) to 44% (full sharing) lower expenditure, compared to no sharing, considering the transshipment and new ventilator costs.

infection in different local regions has led to multiple peaks in COVID-19 cases in almost all countries, with each peak posing a demand surge for limited resources, in particular mechanical ventilators.
To deal with such spikes in demand, many hospitals in the US and other countries sought to purchase new ventilators while several firms launched or increased ventilator manufacturing.In Michigan, automobile companies (Ford and General Motors) revamped their production lines to produce numerous ventilators as quickly as they could (Ford, 2020;General Motors, 2020).Many hospitals also canceled or postponed elective surgeries for patients without COVID-19 to preserve the life-saving ventilators for COVID-19 patients, which resulted in a remarkable financial loss for these organizations and poor outcomes for those without COVID-19 (American College of Surgeons, 2020).While these measures substantially helped US hospitals meet the growing demand for mechanical ventilators, many hospitals in resource-limited countries have forced to ration care due to the prevailing global ventilator shortage (Isaac, 2021;The New York Times, 2021).
Because the infection spreads at varying rates in different areas (e.g., states), there are opportunities for healthcare organizations operating close to or beyond capacity to acquire unused transportable resources from other organizations with extra capacity.Mechanical ventilators are an example of resources that are in short supply, transportable, and critical to providing care for many patients with severe COVID-19, other respiratory illnesses, and following some types of surgery.For instance, cardiac, pulmonary, abdominal, neurosurgery and orthopedic operations are among the types of surgeries that require mechanical ventilation during and sometimes after the surgery.
Ventilator sharing can be an effective strategy to mitigate the COVID-19 pandemic with fewer total ventilators.This will be significantly beneficial for both resource-limited countries, which are unable to afford the purchase of large numbers of expensive ventilators, and resource-rich countries, which will be stressed by the emergence of new vaccine-resistant variants that can lead to many breakthrough infections.For example, hospitals in Michigan had to import ventilators from other hospitals outside of the state after the emergence of the Omicron variant in December 2021 (NPR News, 2021).Prior to that, several Ohio hospitals had to borrow ventilators to cope with the demand (Cleveland News, 2020).Indeed, the state of Ohio mandated all entities to report their ventilator inventory to the Department of Health every week.As explained by the Ohio Governor, Mike DeWine, "This will allow for the identification and re-distribution of machines from healthcare providers who are no longer performing elective procedures."(Ohio Governor's Office, 2020).
In this paper, motivated by the aforementioned observations, we develop a Data-driven Adaptive Robust Simulation-based Optimization (DARSO) approach for optimal sharing of ventilators, essential to providing care for patients with COVID-19 and severe illness.This approach leverages the Clinical and Economic Analysis of COVID-19 interventions (CEACOV) model -a microsimulation model of COVID-19 spread and intervention -to obtain a daily forecast for ventilator demand.CEACOV allows us to consider the impact of population characteristics, transmission dynamics, and interventions (e.g., social distancing and mask-wearing policies) simultaneously, on disease spread and ventilator demand.When specifying resource sharing plan, the DARSO approach takes uncertainty in the disease spread rate into account, and as a result, ventilator demand in various states and different regions within a state as well as the operational burden of transferring ventilators.Our DARSO methodology provides a systematic way of sharing ventilators (and potentially other portable resources) by proactively considering the transmission dynamics and the pandemic trajectory in each region and optimizing the ventilator transshipment.To evaluate its efficacy, we apply it for allocating and relocating of ventilators during the first peak of the COVID-19 pandemic among various regions in Ohio and Michigan, two adjacent states in the US.We provide various managerial insights into how hospitals could handle surge demands for the ventilator while preserving their elective procedures to the greatest extent possible.

Related literature
Our work is relevant to the following research streams: (i) prediction of disease transmission, (ii) emergency resource allocation, and (iii) data-driven optimization.We discuss each of them below.Prediction of disease transmission.Forecasting models are used to predict new infections, hospitalization, mortality, and consumption of key resources (e.g., ventilator and ICU capacity) under a specified set of interventions (e.g., social distancing, mask mandates, prohibitions on gatherings, and lockdowns).Most studies focus on predicting the disease transmission at a population level by using both machine learning and epidemiological modeling.In this regard, several surveys on these models have been conducted (e.g., Biswas et al., 2019;Gupta et al., 2020;Parohan et al., 2020;Wynants et al., 2020).Wynants et al. (2020) surveyed 169 different COVID-19 forecasting models.They highlighted many significant shortcomings of these models, including dichotomisation of predictor variables, overfitting risk, inappropriate inclusion and exclusion of patients from the study population, and lack of accounting or controlling for censoring or competing risk factors.Unlike the existing forecasting models, our CEACOV simulation model provides a daily forecast of COVID-19 spread by considering the effects of individual characteristics, transmission dynamics, and local interventions, simultaneously.
Emergency scarce resource allocation.The allocation of life-saving scarce medical resources must be carefully managed in the advent of a pandemic.Queiroz et al. (2020) provided a comprehensive review of various approaches for optimizing the allocation of critical resources during the COVID-19 pandemic.In this regard, the recent studies include Ahn et al. (2021), Bertsimas et al. (2020), Lacasa et al. (2020), Mehrotra et al. (2020), and Parker et al. (2020), among others.
The most relevant works to our study are Bertsimas et al. (2020) and Mehrotra et al. (2020).Under the assumption of an accurate demand forecast, Bertsimas et al. (2020) formulated a deterministic optimization model for sharing ventilators among different states of US.This model allocates the ventilator federal stockpile and specifies the ventilator transshipment among states to reduce the shortage cost.Assuming the same setting, but under stochastic demand assumption, Mehrotra et al. (2020) proposed a two-stage stochastic program model for sharing and allocating ventilators.They showed how sharing ventilators among different states, even when each state is averse to the risk of not having enough machines for its own patients, can reduce shortages.
These works and ours differ in the following critical ways.Bertsimas et al. (2020) did not account for uncertainty in resource need (known demand), which might not be the case for a pandemic.While considering resource sharing only at the state-level, their model did not explicitly account for the release time of resources after they are assigned to patients.Similarly, Mehrotra et al. (2020) considered resource sharing only at the state level.They used stochastic scenarios for handling demand uncertainty, which requires knowing the full probability distribution for demand.However, in a complex pandemic such as COVID-19, local outbreaks and enforcement of non-medical interventions (such as mask-wearing and physical distancing) can dynamically alter disease spread and subsequently demand for medical resources over time.Moreover, the optimal decision in stochastic models depends on the specific scenarios, which may not readily apply to the real-world situations.Our study develops a new DARSO methodology, which considers uncertainty in demand for scarce resources and provides a framework to make implementable resource-sharing decisions in real-time as uncertainty unfolds.Our method also informs optimal resource sharing at both regional and state levels and considers practical aspects of resource sharing such as the distance between regions, the lead time for moving resources, and the random release time of resources.
Data-driven optimization.Data-driven optimization is centered around deploying pertinent data and analysis of data to model uncertainty and then making decisions accordingly.In general, this area can be divided into three main streams: (i) data-driven (distributionally) robust optimization approaches, (ii) stochastic programming with rolling horizon approaches, and (iii) integrated machine learning and optimization approaches.
The seminal works of Delage (2009) and Delage and Ye (2010) formulated data-driven decision making problems as a distributionally robust optimization (DRO) under moment uncertainty.Saif and Delage (2021) studied a data-driven DRO version of the classical capacitated facility location problem with a distributional ambiguity.The stochastic programming integrated with rolling horizon approaches were developed by Keyvanshokooh, Kazemian, et al. (2022) and Maggioni et al. (2014).These approaches can help practitioners use the latest information and adjust their decisions by dynamically utilizing the realization of uncertain parameters.The last stream of data-driven optimization methods is the ones that integrated optimization approaches with machine learning techniques.Elmachtoub and Grigas (2021) proposed a predict-then-optimize approach, which leverages the optimization problem structures for designing better machine learning models.In the context of healthcare, Bertsimas et al. (2021) showed how machine learning methods can be used for making patient flow predictions and then inform decision making by optimization models.
Both stochastic and robust optimization approaches are common techniques to formulate uncertain models in many applications, including healthcare (e.g., Denton et al., 2010;Keyvanshokooh, Kazemian, et al., 2022;Meng et al., 2015;Mehrotra et al., 2020;Neyshabouri & Berg, 2017).However, there are two central issues with these approaches (Lu & Shen, 2020).First, besides the substantial computational burden, the policy obtained by stochastic models is valid only for a restricted number of scenarios.This implies that they are not well suited to progressive adjustments and rapid adaptation to the realized uncertainty in real-time implementation.Moreover, in pandemics such as COVID-19, limited data and many uncertainty sources make it challenging to create probability distributions of uncertain parameters from which we generate a set of representative scenarios.Second, robust approaches mostly consider static uncertainty sets, thereby suffering from neglecting the dynamic and probabilistic nature of uncertain parameters over time.They may also obtain over-conservative solutions, leading to sub-optimal performance in real-world implementation.
To resolve these shortcomings, the seminal study of Ben-Tal et al. (2004) introduced an adaptive robust optimization approach, where some of the decisions can be adjusted after a subset of uncertain data is realized.Bertsimas et al. (2012), Bertsimas et al. (2019), andLorca et al. (2016) employed this approach for different problems and settings.In this paper, we propose a more powerful approach by formulating a policy-guided optimization model that would allow decision-makers to (i) integrate a complex simulation approach for estimating an uncertainty set with an optimization model, and (ii) dynamically update decisions at each period based on all the uncertain parameters and outcomes realized up to the current decision period.Accordingly, our DARSO methodology can mitigate each of the issues discussed above through a new policy-guided approach.

Main contributions and focus
Below, we summarize the main contributions of this paper.
(a) Problem definition and setting.This paper formulates a new mathematical optimization model to inform data-driven optimal allocation and relocation of scarce resources (in particular, mechanical ventilators).To the best of our knowledge, our paper extends the literature by modeling the following practical features that were not considered in earlier studies (e.g., Bertsimas et al., 2020;Mehrotra et al., 2020;Parker et al., 2020).
First, we incorporate the release times for resources after their usage.This time is stochastic and has a significant role in optimal resource allocation decisions.Our policy-guided model enables us to monitor the discharge of used ventilators periodically and leverages the real-time data on the availability of ventilators.Second, we take into account distance-based lead times for ventilator transshipment.Our optimization model considers the distance between regions and minimizes the total transshipment distance when making ventilator sharing decisions.Third, the objective function of our model can be adjusted to suite the immediate need of the healthcare system.Our first priority is to minimize ventilator shortages as they can be life-threatening.The second priority is to minimize calling on new ventilators as purchasing new ventilators is costly.Minimizing the number of ventilator transshipments is the third priority as more transshipments pose more operational challenges.In contrast to prior studies, time-dependent weights are also considered for our second priority (minimizing new ventilators), ensuring that new ventilators are called on as late as possible to (i) increase the likelihood that used ventilators will be released, and (ii) have enough time for obtaining new ventilators.
Fourth, we consider a two-level allocation and relocation optimization model where ventilator sharing is possible not only at the state level but also among different regions within the same state.This framework allows modeling various sharing levels, such as scenarios in which hospital-owned ventilators can versus cannot be shared, or when additional ventilator supplies (from state or federal stockpiles) must remain within the state boundaries or can be shared across multiple states.We exploit this flexibility to define and evaluate multiple practical sharing strategies in our case study.Finally, unlike previous studies, integration of an optimization model with an advanced microsimulation model enables us to consider unique location-specific disease spread, population characteristics, and non-medical interventions (social distancing and mask-wearing).This is key to tailoring the decisions to specific locations and making the method readily adaptable to other states, regions, and time frames (e.g., with the emergence of new variants).
(b) Methodology.We have proposed a new DARSO methodology in this paper.Methodologically, this study proposes a new data-driven decision-making framework that enables the integration of an optimization model with a complex simulation model.This methodology resolves two main shortcomings of stochastic and robust optimization approaches, namely the requirement of having full knowledge of probability distributions of uncertain parameters and relying on static uncertainty sets, respectively.The DARSO methodology consists of two essential components: (i) a simulation model of COVID-19 (CEACOV) and (ii) a policy-guided optimization model, described below.
Our CEACOV model provides estimates for the number of ventilators needed on each day based on the natural history of COVID-19 progression, each region's unique characteristics, and local pandemic mitigation policies, which can subsequently alter the pandemic's growth rate as captured by the effective reproduction number (R e ).This simulation model resolves a critical limitation of current robust optimization models with static uncertainty sets through creating and updating a dynamic uncertainty set based on the realized data, capturing the joint temporal and spatial correlations of ventilator demand at various regions across the states.
Leveraging the CEACOV model, our policy-guided optimization model incorporates the impact of prior decisions and uncertainty realization on each day via a real-time implementation procedure.Therefore, our methodology enables policymakers to harness the latest information revealed up to the current time step to make the next decision.Using our policy-guided model, the ventilator allocation and relocation decisions can be made in real-time, which makes the approach directly implementable in practice.Our policy-guided model is an adaptive robust optimization model; thus, we develop a set of efficient approximations by combining the worst-case constraint generation and the duality-based techniques, to derive a tractable mixed-integer linear program.This model can then be solved efficiently by commercial solvers such as CPLEX in a reasonable amount of time.
(c) Case study and managerial insights.In close collaboration with epidemiologists and infectious disease doctors, we conducted a case study of ventilator sharing in Ohio and Michigan during the first peak of COVID-19 pandemic to assess the benefits of our DARSO methodology.Our case study provides valuable insights into the impact of various ventilator sharing strategies using region-specific and pandemic data from the states of Ohio and Michigan.First, our results highlight the benefits of data-driven resource sharing as an effective strategy to mitigate the excessive demand for healthcare resources (e.g., ventilators) caused by local surges in cases and hospitalizations.Those benefits include satisfying the demand with markedly fewer total ventilators as well as substantial cost savings.Second, our analyses shed light on when resource sharing is most effective.In short, strategies that allow ventilator sharing over long distances (e.g., across states) yield the highest benefits as distant regions are more likely to experience their peaks over distinct time periods, thereby providing more opportunities for resource sharing.Third, emergence of new virus variants with higher transmissibility may stress the healthcare system again, even in resource-rich countries, and warrants additional resources and updated resource sharing plans.Our results support resource sharing as a powerful strategy for combating the pandemic.

PROBLEM STATEMENT
Many countries around the world (including the United States), have experienced multiple peaks of COVID-19 infections.These peaks are often the result of community outbreaks or emergence of new and more transmissible variants of the virus.These peaks have placed the healthcare system under unprecedented strain due to the rapid growth in demand for scarce healthcare resources, specifically ICU beds and mechanical ventilators.Hospitals attempted to alleviate the ventilator shortage by taking the following two key actions.(i) Many hospitals increased their ventilator inventory by obtaining extra ventilators from state and federal stockpiles.(ii) Elective operations for non-COVID-19 patients were either delayed or postponed at many hospitals in order to save life-saving ventilators for COVID-19 patients who required mechanical ventilation (American College of Surgeons, 2020).To alleviate the ICU bed shortage, many hospitals converted in-patient units to ICU by including mechanical ventilators in those units and altering their patient care routines and Infection Prevention and Control (IPC) practices (Xiong et al., 2021;Fadaak et al., 2021;Strubinger, 2021).
While these measures were mostly effective in preventing catastrophic shortages of needed care in the United States, care had to be rationed in many resource-constrained nations owing to the worldwide ventilator shortage (Isaac, 2021;The New York Times, 2021).Even in the US, purchasing new ventilators came at a substantial cost for many hospitals (Kaiser Health News, 2020).Moreover, canceling elective surgeries and other procedures undermined care for patients without COVID-19 and resulted in a substantial financial loss for these healthcare organizations (Tonna et al., 2020).
In this context, we develop a novel decision support tool to inform optimal ventilator sharing plans, which can help mitigate some of the aforementioned problems.Our tool is built on the premise that different states, and even different regions within the same state, often experience the peak of infections at varying time points.This creates a unique opportunity for continuously moving unused portable resources, such as ventilators, from regions that will not require them in the near future to others that will.Such a resource sharing strategy can help avert shortages while also lowering costs.
The primary challenge, however, is that future ventilator demand is unknown, and a myopic strategy may not work effectively due to the operational constraints associated with transporting ventilators across long distances.Therefore, an ideal method should take the uncertain demand into account and design proactive ventilator sharing schemes that are robust to variability in future demand.Consequently, our proposed method integrates a powerful Monte Carlo state-transition microsimulation model of COVID-19 transmission and intervention with a novel data-driven adaptive robust optimization model to determine the optimal ventilator sharing plans that consider transmission dynamics and future ventilator demand patterns tailored to a specific location.Figure 1 provides a high-level overview of our simulation-based optimization methodology.
We combine several neighboring counties in each state to form a "region."Thus, each state is divided into several regions.We then consider ventilator sharing among regions of the same state as well as between states.Our simulation model (CEACOV) is populated with region-specific population characteristics, COVID-19 transmission dynamics, and interventions (local COVID-19 mitigation strategies).The model is then run to estimate daily transmissions along with the number of individuals with a severe or critical COVID-19 illness who require hospital stay with supplemental oxygen or mechanical ventilation over the next few weeks in each region.These estimates (in particular, demand for ventilators) are then passed to the robust optimization model.The optimization model considers the average ventilator demand in each region over the next few weeks (informed by CEACOV) along with an estimate of the variability in demand to optimize ventilator sharing schemes.The optimization model dynamically updates the ventilator sharing plan in each time period (day) based on new information revealed, including the number of ventilators released in each region (because patients using those have recovered or died) and updated projected ventilator demand from CEACOV.
Specifically, our DARSO methodology determines the optimal ventilator allocation (assignment of new ventilators to states and regions) and transshipment (relocation of existing ventilators between regions) to minimize shortages with as few ventilators as possible while also minimizing ventilator transshipments for better operational feasibility.The primary objective of the optimization model is to minimize ventilator shortages to ensure that every COVID-19 patient who needs a ventilator will receive one.The secondary objective is to minimize the additional ventilators needed to care for patients with COVID-19 by sharing (moving) ventilators across different regions, rather than seeking new supplies of ventilators.The tertiary objective is to minimize ventilator transshipments to reduce the operational challenges of moving ventilators over long distances.
The benefits of our DARSO method for optimal ventilator sharing during the pandemic (or other periods of high demand) are threefold.First, our method ensures that all ventilator demand of patients with COVID-19 is met with the minimum number of new ventilators despite the inherent uncertainty in demand.This is extremely important, especially in resource-constrained countries in which purchasing large quantities of new ventilators may not be feasible.Thus, sharing the available resources is likely the best way to avoid or minimize rationing care.Second, the healthcare system in both resource-rich and resource-limited countries benefits financially by reducing the number of new (additional) ventilators that must be acquired.Given the high cost of purchasing new mechanical ventilators compared to transporting them, ventilator sharing offers a significant cost savings.Third, meeting the ventilator needs of COVID-19 patients with fewest possible ventilators allows more ventilators to be saved for patients without COVID-19.This is important because many surgeries require some period of ventilation during or after the operation.By preserving more ventilators for non-COVID-19 purposes, hospitals may provide more elective procedures, resulting in improved patient outcomes and increased revenue for the institutions.
In Section 4, we demonstrate all of these benefits using a case study of ventilator sharing in Ohio and Michigan during the first peak of COVID-19 pandemic in the two states.We chose these two states for our case study primarily because we had more complete data about the population characteristics, transmission rates, and local COVID-19 mitigation strategies in various counties of Ohio and Michigan.We note that our DARSO method is even more valuable in resource-constrained countries where hospitals started with a more limited initial inventory of ventilators and the countries' healthcare budgets are tighter.Moreover, while our case study focuses on sharing ventilators in Ohio and Michigan during the first peak of the COVID-19 pandemic, our method can be readily adapted to other areas, other periods of demand surge, and other portable resources such as healthcare personnel, personal protective equipment (PPE), and point-of-care testing units.

DATA-DRIVEN ADAPTIVE ROBUST SIMULATION-BASED OPTIMIZATION METHODOLOGY
Our methodology synergizes a simulation model of COVID-19 spread and intervention with an adaptive robust optimization approach.Sections 3.1 and 3.2 describe these two main components of our DARSO methodology, respectively.Section 3.3 describes our proposed data-driven framework for the real-time implementation of the policy-guided model in practice.

Simulation model of COVID-19
CEACOV is a dynamic agent-based and state-transition microsimulation model of COVID-19 transmission and intervention.CEACOV simulates a defined group of individuals (e.g., persons living in a region) for a set number of days (called the simulation horizon) with a one-day time unit.CEACOV comprises several modules, including natural history, transmission, and intervention, that together determine individual trajectories and pandemic growth.Unlike an SIR model, which divides a population into three compartments (susceptible, infectious, or recovered) and tracks them as a cohort, CEACOV is a microsimulation model that tracks individual people over the simulation horizon (Vandepitte et al., 2021).Hence, CEACOV allows for incorporating individual characteristics into natural history disease progression, resource usage, and transmission dynamics.For example, while an SIR model generally applies the same transition rate to everyone in a compartment, CEACOV incorporates age-dependent disease progression, hospitalization, and ventilator need for infected individuals.This is especially important when modeling a disease like COVID-19, which is known to be more serious in older adults (Haridy, 2020;Mizumoto et al., 2020).
Individuals who are susceptible to SARS-CoV-2 infection can contract the virus from infected members of the cohort.Once infected, a person goes through a series of infection states until he or she recovers or dies.The disease states in CEACOV include pre-infectious latency, asymptomatic infection, mild/moderate disease, severe disease, critical disease, and recuperation.Those individuals who are in the severe disease state experience dyspnea or hypoxemia and require hospital care with standard supplemental oxygen.Individuals in the critical disease state need high-flow supplemental oxygen or mechanical ventilation.
After the pre-infectious latency state, individuals transition into one of the four "paths" (natural history disease trajectories) that determine the eventual severity of their disease.Those include the asymptomatic, mild/moderate, severe, and critical disease paths.Age-specific transition probabilities for COVID-19 natural history are used to determine the disease state and the length of time spent in each state.
Patients are considered recovered when they no longer experience severe or critical symptoms and pose no risk of transmitting to others.We assume recovered individuals are immune from repeat infection for the duration of simulation (60 days in our case study).Figure 2 illustrates the health state transition diagram for CEACOV.The CEACOV model has been validated in previous publications (Baggett et al., 2020;Losina et al., 2020;Neilan et al., 2020).
Infected individuals in the pre-infectious latency state and those recovered do not transmit the virus.Patients in other infection states (Figure 2) can transmit COVID-19 to susceptible individuals.Transmission rates vary across different infection states.The natural history transmission rates are adjusted by a transmission multiplier to capture the effect of social distancing, mask-wearing, and other interventions that can alter contact rate or infectivity in the regions being modeled.
The basic reproduction number (R 0 ) is defined as the daily rate at which an infected individual contacts susceptible individuals and infects them in a fully susceptible cohort, multiplied by the duration of infectivity.This idea can be modeled as follows: where C is the number of daily contacts between an infected individual and susceptible individuals in a fully susceptible cohort, b is the per contact probability of transmission between an infectious and a susceptible individual, and L is the mean duration of infectivity.R 0 estimates the expected number of secondary cases produced by an infected individual when the cohort is fully susceptible.
The CEACOV model calculates the effective transmission rate on each day of simulation as follows: Effective Transmission Rate = Nominal Transmission Rate * Transmission Multiplier.
The nominal transmission rate is defined as R 0 in a fully susceptible cohort divided by the average duration of infectivity (L).Equivalently, it can be expressed as a function of the average daily contact rate of an infected person with susceptible individuals in a fully susceptible cohort (C) times the probability of transmission per contact between an infectious person and a susceptible individual (b).This nominal transmission rate captures the ratio (rather than the magnitude) of daily infectivity across different infection states (e.g., asymptomatic to mild/moderate, or mild/moderate to severe).The transmission multiplier is a time-dependent adjusting factor that accounts for population density and interventions that can alter the number of contacts (e.g., social distancing, shelter in place, and curfew orders) and infectivity per contact (e.g., mask-wearing policies) in the region being modeled, which ultimately change the effective reproduction number (R e ).Thus, the effective transmission rate on day t, r(t), is calculated as follows in CEACOV: where R 0 ∕L is the nominal transmission rate and m(t) is the transmission multiplier on simulation day t.r(t) determines the effective transmission rate of infected individuals on each day of simulation and is equivalent to the effective reproduction number (R e ) divided by the duration of infectivity (L).
As more people in the cohort get infected over time, the number of susceptible people declines.Thus, not all of the daily contacts of infected individuals will be with susceptible people.CEACOV calculates the force of infection to each susceptible individual on each day of simulation as follows: where f (t) is the force of infection to each susceptible person on day t, I s (t) is the number of infected individuals in infection state s on day t, S is the set of infection states, r(t) is the effective transmission rate on day t, and M is the cohort size.This leads to an expected daily number of infections equal to the number of susceptible individuals multiplied by the force of infection, f (t), on that day.We summarize the most important natural history and transmission parameters in CEACOV in Appendix A.

Optimization model
In this section, we first formulate a two-stage adaptive robust optimization model for the problem described in Section 2. Table 1 presents the notation for sets, parameters, and decisions in our formulation.We then explain how a dynamic uncertainty set for the ventilator demand can be designed using the CEACOV model.Next, we formulate a policy-guided adaptive robust optimization model in which this dynamic uncertainty set is employed.Using the structural properties of this model, we convert it into a tractable mixed-integer linear program.The first step for building a robust optimization model is to construct an uncertainty set.In this study, the uncertain parameter is the ventilator demand.The uncertainty set of ventilator demand, denoted by  as a real-valued random vector is defined as follows:

𝜏 r
Release time of a ventilator after it is assigned to a patient.

dj,t
The uncertain ventilator demand of region j ∈  in period t ∈  .We use dt to denote the vector of uncertain demands in period t for all regions, and d[t] = ( dt 0 , dt 0 +1 , ..., dt ) to present the uncertain demand from the beginning period until the end of period t.

𝜈 j,j ′
The distance for transshipment between regions j ∈  and j ′ ∈  .

𝜆 j
Lead time for transporting ventilators to region j ∈  from its state depot ( j ≥ 1, ∀j ∈  ).

𝛾 t
The maximum number ventilator transshipments allowed in period t ∈  .

S j,t 0
The number of available ventilators in region j ∈  at the beginning of planning horizon.

𝛼
The penalty weight for the ventilators' shortage in the objective function.

𝛽
The penalty weight for the total transshipments in the objective function.
Decisions made prior to uncertainty realization

N t
The number of additional ventilators obtained from federal stockpile in period t ∈  .

K i,t
The number of ventilators allocated to state i ∈  from federal stockpile at the beginning of period t ∈  .

Z i,j,t
The number of new ventilators, which are sent from state depot i ∈  to region j ∈  (i) at the beginning of period t ∈  .

I i,t
The number of available ventilators in state depot i ∈  at the beginning of period t ∈  .

Y j,j ′ ,t
The number of ventilators transshipped from region j ∈  to region j ′ ∈  at the beginning of period t ∈  .
Decisions made after uncertainty realization The number of ventilators used in region j ∈  during period t ∈  .

Δ j,t
The amount of ventilator shortage in region j ∈  in period t ∈  .

S e j,t
The number of available ventilators in region j ∈  at the end of period t ∈  .

S b j,t
The auxiliary variable that presents the number of available ventilators in region j ∈  at the beginning of period t ∈  .
where in region j ∈  in period t ∈  , djt is the uncertain ventilator demand, djt is the nominal value for ventilator demand, Note that here Γ is between 0 and 1 and controls the conservativeness of our optimization model.For Γ = 0, the uncertainty set only contains the nominal demand values and for Γ = 1 the uncertainty set is the largest one that contains all possible demand values.
In our model formulation, the decisions have a two-stage nature.Ventilator allocation decisions (N, K, Z), the available ventilators at state depots (I), and transshipment decisions (Y) take into account all possible future demands represented in the uncertainty set.However, the ventilator usage, ventilator shortage, and ventilator inventory level on each day, presented by X, , and S e , are all dependent on the realized ventilator demand from the beginning of the planning horizon until that day (period), and ventilator inventory level at the beginning of each day (S b ) is dependent on the ventilator demand from the beginning of the planning horizon until the day before that day.To highlight the dependency of these decisions on the forecasted demand in period t, they are shown as where (Z, Y, d) is the optimal value of the following problem: The objective function (2) has two parts, reflecting the two-stage nature of decisions.The first part minimizes the total new ventilators and total transshipments.The second part minimizes the worst-case number of cumulative shortages over the planning horizon, which is mathematically shown in relation (10).It is worth noting that the optimal value of (Z, Y, d) is a function of the first stage decision variables Z and Y and uncertain demand d.In (2), the known penalty weight parameters  and  are set such that the shortage minimization (the third term in (2)) has the highest importance, while the secondary goal is to minimize additional ventilators that the model calls on (the first term in (2)).Note that our main purpose is to employ as few ventilators as possible to preserve as many ventilators as possible for non-COVID-19 patients.Further, it is desirable for states to have as much time as possible before they need to purchase new ventilators.Therefore, the coefficient (| | + 1 − t) is added for ventilator supplies in each time t.Lastly, we minimize the number of transshipments (the second term in (2)), which is the tertiary goal of our model with a lower priority (weight) than others.Constraints (3) imply that the number of new allocated ventilators to various states should be equal to their available number in each time period.Constraints (4) and ( 5) illustrate the balance constraints for each state's inventory.Constraints (6) prevent sending new ventilators of a given depot state to regions in other states.Constraints (7) guarantee that only new ventilators from each region can be moved to out-of-state regions.Note that constraints ( 7) are used to model one of the strategies investigated in the case study in Section 4. Constraints (8) limit the number of ventilator transshipments per day to the defined limit (1000 per day) to ensure operational feasibility.Constraints (9) define variable types in outer optimization model.
The objective function (10) minimizes the worst-case number of cumulative shortages over the planning horizon for the inner optimization model.The initial available inventory in each region at the start and end of planning periods is derived via constraints (11)-( 13).The number of used ventilators and ventilator shortages in each region are calculated through constraints ( 14).Variable types are defined by constraints (15).These variables are considered in the real space in order for the optimization model to become computationally tractable, which is common in resource sharing literature (see e.g., Mehrotra et al., 2020).

Dynamic uncertainty set for the ventilator demand
There are various factors that can influence disease spread and transmission during a pandemic like COVID-19.This inherent uncertainty leads to two major problems in the decision-making process.First, it is almost impossible to elicit a complete probability distribution from historical data to model the demand uncertainty.Second, the forecasted ventilator demand should capture both spatial and temporal correlations, because the ventilator demand varies in different regions and also over time.We address these two challenges in the decision-making process by proposing a dynamic uncertainty set through which we can integrate the CEACOV model with the optimization model in order to inform data-driven resource sharing decisions by updating uncertainty set  through the planning horizon.Using this uncertainty set, the policy-guided model presented later in this section captures the dynamics of uncertainty based on the realized uncertainty in each time.To our best knowledge, this is the first attempt to resolve these challenges in a data-driven fashion.
Accordingly, the CEACOV model forecasts the expected ventilator demand (nominal value in our uncertainty set) for each time period (day) over a pre-specified planning horizon.The demand forecast depends on various factors, including population characteristics, transmission rates, past interventions, and the realized demand up to the current time period.Given that new information unfolds progressively over time, we can dynamically update uncertainty set , and the param- Proposition 1.We denote the ventilator demand of region j on day t + 1 by dj,t+1 .The variance of dj,t+1 is bounded as follows.
Var( dj,t+1 ) ≤ dj,t+1 where dj,t+1 is the expected value (nominal value) of dj,t+1 , and n s j,t and n c j,t are the number of patients in region j on day t who are in severe and critical disease states, respectively.
The proof of Proposition 1 is provided in Appendix B. According to Proposition 1, we let the minimum and maximum of uncertain ventilator demand in region j on day t ( dj,t ) be as ) )+ and dj,t , respectively, and then we can easily obtain d min j,t and d max j,t in uncertainty set . Based on the result of Proposition 1, we observe that, at each time period t, ventilator demand variability depends on data realization in the previous period t − 1, through the terms n s j,t−1 and n c j,t−1 .Thus, at the end of period t−1, we estimate the minimum and maximum ventilator demand change in each region and update the uncertainty set .

3.2.2
Policy-guided adaptive robust optimization The proposed robust optimization model ( 2)-( 15) is an infinitely constrained optimization problem and thus it is computationally intractable.This is because constraints ( 12)-( 15) should be valid for any values of ventilators demand in uncertainty set .To make the problem computationally tractable, we restrict our attention to the affine policies, which consider approximation schemes with decision rules (Ben-Tal et al., 2004;Jabr, 2013;Kuhn et al., 2011;Lorca et al., 2016).
To this objective, we first denote the vector of decisions that are dependent on ventilator demand as w.A full affine policy then formulates these decisions through the relation . Here,  w j,t is the value of the decision variable for region j ∈  in period t ∈  when the demand is equal to the nominal one, and the parameter  w j,t,j ′ , for region j ∈  in period t ∈  is the sensitivity coefficient of the deviation of nominal demand from realized demand in region j ′ ∈  and period  ≤ t.In this full affine policy, the decisions w depend on the demand uncertainty in all regions and periods prior to t.
In our optimization problem, X, S b , S e , and  are dependent on ventilator demand and in the case of full affine policy, these decisions in each time period are dependent on the entire history of realized demand at each location and every time period up to the current time.So this full affine dependency requires defining a large number of variables  (sensitivity coefficients), which can quickly lead to scalability issues in real-word problems and this full affine policy is computationally burdensome to solve in practice.To address this issue, similar to Lorca et al. (2016), we consider affine policies with simplified structures by restricting them to only depend on the most recently revealed demand at the current period and location, rather than on the whole history.As a consequence, by assuming d j,t as ( dj,t − dj,t ), the following simplified affine policy ( 16)-( 19) is adopted: ) where  X j,t ,  SE j,t ,  SB j,t , and  Δ j,t are the sensitivity coefficients of the deviation of the realized demand from nominal demand in region j and period t corresponding to each decision.Also,  X j,t ,  SE j,t ,  SB j,t , and  Δ j,t are the decision variables for region j and period t when the nominal demand is equal to the realized one.
Based on these decisions rules ( 16)-( 19), in our optimization problem, the decisions for region j ∈  in period t ∈  would be dependent on d jt as djt − djt , and we can define new uncertainty set  instead of , which is the set of possible deviations of uncertain ventilator demand in comparison to the nominal data as follows: Note that based on our explanations in Section 3.2.1,deriving uncertainty set  and the nominal value of ventilator demand from our CEACOV model is straightforward.
Using the affine policy ( 16)-( 19) and the developed uncertainty set , we develop a policy-guided adaptive robust optimization model.This is a multi-period model in which the decision maker observes the ventilator demand before each period of the planning horizon, and accordingly updates the uncertainty sets and the parameters of the optimization problem.In other words, by solving this policy-guided model, the obtained solution can be readily implemented for the first period of the planning horizon.Given the decisions in the first period, after the realization of the uncertainty, the input data parameters and uncertainty set will be updated before making decisions for the next period.This process is repeated successively over the planning period and makes the decision-making process implementable.In this way, our policy-guided model ensures that the decisions for each period are made as a function of the prior decisions and latest realized information.
The policy-guided optimization model for decision making in period t * is presented below.In this model, t * is the first period of model's planning horizon that contains | | periods.In the model, the uncertainty set  is used instead of  and the affine version of the demand-dependent decisions are replaced based on the relations ( 16)-( 19).Another important issue in our implementation is the release of ventilators after their usage and a deterministic duration ( r ) is considered for it.However, available ventilators can be captured based on the reality, which is considered in our policy-guided model.More explanations about the ventilators' release time are provided in the case study section.For our policy-guided model, the parameters used for updating the model based on prior decisions and uncertainty outcomes are reported in Table 2. min ) ) ∑ Summary of notation used in the policy-guided optimization model.

Xj,t
The number of ventilators that are used before period t * and supposed to be available in period t > t * after the usage, but they became available in previous periods.

X j,t
The number of ventilators that are used before period t * and will be released in period t > t * after  r days.

Īi,t *
The number of available ventilators in state depot i in period t * .

Y j,j ′ ,t
The number of ventilators transshipped from region j ∈  to region j ′ ∈  in period t ∈  , where t 0 ≤ t < t * .

Z i,j,t
The number of new ventilators, which are sent from state depot i ∈  to region j ∈  at the beginning of period t ∈  , where t 0 ≤ t < t * .

S j,t *
The number of available ventilators in region j in period t * .

𝜅 j,t
The number of ventilators that will be available for region j in period t ∈  , where t > t * , based on decisions made before period t The detailed explanation of constraints was previously provided, where we presented the two-stage robust optimization model ( 2)-( 15).In above, the constraints (21a)-( 21p), and the 'bar' and 'tilde' symbols are used to distinguish decisions X, Z, S, I, and Y from parameters X, X, Z, S, I, and Y.Note that the values of these parameters are obtained based on the decisions and realized uncertainty in periods prior to t * , which are known in period t * .

3.2.3
Tractable reformulation of the policy-guided model The policy-guided model is an infinitely constrained optimization problem.A deterministic counterpart of this model is therefore needed.Among two main approaches to handle this, the most popular one is a duality-based approach that replaces robust constraints by their dual program with additional constraints and variables (Ben-Tal et al., 2009).
The second one is based on constraint generation, which dynamically adds violated scenarios and associated deterministic constraints (Bertsimas et al., 2016).By exploiting the unique structure of the proposed policy-guided model and the affine policy, we obtain the deterministic counterparts of robust constraints ( 21h)-( 21o) without any dualization of the uncertainty set and and use duality-based approach for reformulating the objective function (21a).
In the following, we derive a mixed-integer linear program to obtain the optimal decisions of the policy-guided model.The detailed proofs for the propositions are provided in the Appendix B. In accordance to Ang et al. (2014) and Lorca et al. (2016), we assumed that d min jt < d max jt , ∀ j ∈  , t ∈  , and thus the uncertainty set  is full-dimensional, equivalently, there exists at least an interior point in the set.We used the full-dimensionality feature of uncertainty set  in our proofs.
Proposition 2. Constraints (21h) are equivalent to the following constraints ( 22a), (22b) whenever the uncertainty set is full-dimensional: A similar result to Proposition 2 holds for the robust constraints related to the balance of available ventilators in various regions as developed in Proposition 3.

Proposition 3. Constraints (21i), (21j) are equivalent to the following constraints (23a)-(23d) whenever the uncertainty set is full-dimensional:
Considering the real-time demand for ventilators, in constraints (21k), the demand in each period t is realized in the implementation phase.However, at the time of decision-making, we make decisions dependent on the uncertain demand, which is full-dimensional based on the construction of the uncertainty set in our approach.In Proposition 4, we derive the equivalent finite constraints related to constraints (21k).
Proposition 4. Constraints (21k) are equivalent to the following constraints ( 24a) and ( 24b) whenever the uncertainty set is full-dimensional: Note that in constraints (24a), d is the nominal value of the uncertain demand, which is equal to the expected value of ventilator demand obtained by the CEACOV simulation model.We next derive the worst-case scenarios for the infinite constraints (21l)-(21o) in Proposition 5. 21o) in the form of  j,t +  j,t d j,t , then they are equivalent to the following constraints ( 25a) and ( 25b), where the maximum and minimum demand deviation in region j at time t is defined by d max j,t and d min j,t , respectively:

Proposition 5. If we present constraints (21l)-(
The values of d max j,t and d min j,t in constraints (25a) and ( 25b) are obtained based on Proposition 1, and their values will be set based on the observations in the previous period for the first period t * and the CEACOV simulation model for other next periods.
Based on Propositions 2-5, we now derive the deterministic counterpart of the robust constraints in the policy-guided optimization model ( 21a)-(21p).To convert the objective function (21a) into a tractable form, by considering W as a real-valued decision variable, we first rewrite it as: The robust constraints (26b) can then be written as follows: where the uncertainty set  [t * ,t * +| |−1] is equal to: We next employ a duality-based approach to reformulate the resulting robust constraint.Proposition 6. Constraints ( 27) is equivalent to constraints ( 28a)-( 28d) if there exist vectors , ,  1 ,  2 , and  satisfying the following constraints, where the maximum and minimum deviations in region j and time t is defined by d max j,t and d min j,t , respectively: Therefore, we can derive the following mixed-integer linear program to obtain the optimal decisions of the policy-guided model, which is solvable by commercial solvers such as CPLEX.

Data-driven framework for real-time implementation
In this section, we describe our proposed data-driven framework for the real-time implementation of the policy-guided model in practice.Firstly, we should distinguish between the planning horizon in the policy-guided optimization model, the considered time periods in the policy-guided model, and the decision-making planning horizon, which contains the days (periods) in which we determine sharing and allocation decisions by solving the optimization model.Here, the former is denoted by  and the latter is introduced as  D .In each period t ∈  D , we solve the policy-guided model ( 29a)-(29c) over a planning horizon  starting from t where t = t * as {t * , t * + 1, ..., t * + |T| − 1} to achieve implementable policy.In the reality, for solving the policy-guided model, the ventilator sharing and allocation decisions, the release of used ventilators, and the realized ventilator demand of different regions in the previous periods are known.Moreover, we estimate the nominal value of ventilator demand d by using the CEACOV model and construct the dynamic uncertainty set  for the planning horizon based on Proposition 1.Given the decisions for time period t * obtained from solving the policy-guided model, based on the realized uncertainty, we update the input parameters and the uncertainty set for solving this model in the next period.In the next period (t + 1), we consider t * = t + 1 and repeat this process.
In our problem, the realized uncertainty on a given day includes the ventilator demand of different regions, the release of previously-assigned ventilators, and the number of patients who are hospitalized with severe COVID-19 disease in different regions.These pieces of information are not available for future planning periods at the time of decision-making and will unfold progressively over the planning horizon.Algorithm 1 explains the steps of our proposed data-driven real-time implementation framework.Note that in this algorithm, a trajectory of the realization of the uncertain parameters over the planning horizon is denoted by s.

CASE STUDY ON COVID-19 PANDEMIC
In this section, we demonstrate the application of our DARSO methodology using a case study of allocating and relocating ventilators among different regions in Ohio and Michigan, two neighboring states in the Midwest of the US.We consider a 2-month period, from mid-March to mid-May 2020, which was when the two states experienced their first peak of COVID-19 cases.Over this time period, many hospitals required to cancel or defer their elective surgeries and other procedures for patients without COVID-19 to maintain access to life-saving ventilators for their COVID-19 patients.

7:
Obtain the optimal decisions (N t , K it , Z ijt , Y jj′t ) for period t. 8: Step III: Update the input parameters for the next period.

9:
Consider the trajectory of uncertainty realization s for the ventilator demand and released ventilators up to period t.

10:
Given the decisions in period t and the trajectory of uncertainty realization s: 11: (i) Obtain X jt (s) , Δ jt (s), and S e jt (s) . 12: (ii) Calculate the objective value for period t and the uncertainty realization s.

13:
Consider the uncertainty realization s for the release of previously-assigned ventilators in period t to update parameters for the next period.14: End For 15: Return N, Y, and the objective function for uncertainty realization s.
While we provide proof of concept for our method in the states of Ohio and Michigan, our method can similarly be applied to other locations that experienced a surge in COVID-19 cases or will experience future surges, as well as potential future pandemics.Indeed, when compared to resource-rich countries, our ventilator sharing method has the potential to save more lives and significantly improve outcomes in resource-limited countries that may be unable to purchase large quantities of new ventilators in response to a demand surge.We chose to demonstrate the application of our DARSO method in Ohio and Michigan because we had more complete data and were more familiar with local pandemic prevention policies in these states.We note that both resource-rich and resource-limited countries will financially benefit from data-driven resource sharing as it provides substantial cost savings, which we will discuss later in this section.
Hence, the pandemic size and peak, and subsequently, the demand for ventilators at a given time, varies in different regions, creating a unique opportunity for ventilator sharing.
We calibrate CEACOV to each region's population characteristics and transmission dynamics to estimate ventilator demand in different regions of each state based on the number of individuals who reach to the critical disease state over time.We provide validation results for simulation predictions in Appendix C.

Ventilator sharing strategies
We model and compare four ventilator sharing strategies.
Each strategy imposes different limitations on which ventilators can be shared between the two states and which ventilators must remain in the same state.We divide all ventilators into two groups: (a) "initial ventilators" represent the initial inventory of ventilators in each region; (b) "new ventilators" represent the ventilators acquired from the state or federal stockpiles after the start of the pandemic.The four strategies are defined below.
1.No sharing strategy, where regions do not share ventilators (status quo).To model this strategy, we added the following constraint to the optimization problem.
2. Limited sharing strategy, in which initial and new ventilators can be shared among regions of the same state, but they cannot leave the state.To model this strategy, we define Λ jj ′ , ∀j, j ′ ∈  as a binary parameter, which is equal to 1 if j and j ′ are regions of the same state, and 0 otherwise.Then the following constrain is added to optimization model.
3. Expansive sharing strategy, in which initial ventilators must be kept within the same state but new ventilators can be shared among all regions across the two states (including from one state to the other).This strategy is modeled via constraint (21f) in the optimization model.4. Full sharing strategy, in which regions are willing to share all of their unused ventilators (i.e., both initial and new ventilators) with other regions across the two states.This strategy is modeled by relaxing constraint (21f).

The importance of ventilator sharing
In this sub-section, we compare the cumulative number of additional required ventilators (i.e., on top of the region's initial inventory of ventilators) over 60 days under the four strategies.First, the CEACOV simulation model is populated with region-specific population and pandemic data.Each week, transmission rates are adjusted to consider the fluctuation in the effective reproduction number (R e ) as a result of the gradual implementation of the closure of gyms, restaurants, schools, and other unessential businesses, as well as stay-at-home orders, which were introduced in Michigan and Ohio in multiple stages (Centre for Mathematical Modeling of Infectious Diseases, 2021).The CEACOV model predicts the pandemic trajectory in each region and estimates the number of infected individuals in various stages of COVID-19 disease on each day.Individuals with severe COVID-19 disease (the 'Severe Disease' state in Figure 2) require hospital stay to receive standard supplemental oxygen.A subset of infected individuals who develop critical illness (the 'Critical Disease' state in Figure 2) would require a mechanical ventilator.
CEACOV estimates the number of individuals who require a ventilator on each day of the simulation horizon for each of the 16 regions of Ohio and Michigan.These data are then fed into the optimization model based on the proposed data-driven framework described in Section 3.3.The model informs the optimal ventilator sharing policy to ensure that demand is satisfied in all regions with the minimum number of new (additional) ventilators possible where the budget uncertainty is set for having the largest uncertainty set. Figure 3 shows the cumulative new ventilators that are needed for each of the four strategies averaged over 200 sample paths.The trajectory of the realization of the uncertain parameters over the planning horizon is explained in Section 3.3.The average computational time of solving the policy-guided model is about 3 min, which is solved via the commercial software GAMS 24.11 using the CPLEX 12.0 solver.As seen in Figure 3, in the absence of resource sharing (No Sharing), compared with the case of sharing ventilators, the supply of new ventilators is required earlier.Under sharing strategies (particularly, full sharing), the early shortages are addressed by transporting ventilators from regions having extra ventilators to regions facing with a deficit.For instance, the surge of infections in the Detroit region (Region 1 in Michigan, or MI1, in short) creates a notable shortage of ventilators around day 22.To address this shortage, the model moves ventilators from other nearby regions, including Grand Rapids (MI2), Lansing (MI5), Jackson (MI7), and Toledo (OH1), to Detroit.As the surge continues in the Detroit region, all strategies eventually call on an additional supply of ventilators (e.g., from state or federal stockpile) to satisfy the demand.
Furthermore, we can observe in Figure 3 that the no sharing strategy (Strategy 1) needs substantially higher numbers of new ventilators than other strategies to satisfy the demand.Under the no sharing strategy, 1904 additional ventilators are needed through the two states to avoid shortage, however, the full sharing strategy (Strategy 4) needs only 714 new ventilators to attain the same result.Moreover, the limited (Strategy 2) and expansive (Strategy 3) sharing strategies can satisfy the demand with 1633 and 1462 additional ventilators, respectively.
Figure 4    either one way or bidirectional transshipments).It can be seen that under the full sharing strategy (Figure 4, left), a number of regions in each state send their excess ventilators to other regions within the same state as well as to the other state's regions.However, under the limited sharing strategy (Figure 4, right), each region only shares its excess ventilators with other regions within the same state.It is worth noting that the lead time varies from 1 to 3 days depending on distance between regions.
Figure 5 illustrates the cumulative number of ventilator transshipments under the limited and full sharing strategies over the planning horizon (averaged over the 200 sample paths).Note that the full sharing strategy allows more flexibility for ventilator sharing within and between states, thus requiring fewer ventilators.However, as seen in Figure 5, the full sharing strategy uses this greater flexibility and requires more ventilator transshipments compared to the limited sharing strategy, thus incurring more transshipment cost.We compare the four strategies in terms of transshipment and new ventilator costs in Section 4.6.
Table 3 summarizes the mean and standard deviation (SD) for additional ventilator needs, the total number of transshipments (within and between states), and the number of transshipments between states under the four strategies.Compared to the no sharing strategy (status quo), the limited and expansive sharing strategies satisfy the demand with 14% and 23% fewer ventilators, respectively.However, the full sharing strategy is able to satisfy the demand with 63% fewer new ventilators compared to the status quo.Also, about three-quarter of all transshipments in the full sharing strategy are between the two states, demonstrating the benefit of transporting ventilators over large distances.While nearby regions are more likely to experience a surge at or around the same time, far distant regions (e.g., regions in different states) are more likely to peak at separate times, enhancing the benefits of ventilator sharing.
Note that, in the case study, we consider a 60-day horizon in our optimization model.We also investigated the impact of the optimization horizon's choice on outcomes in Appendix D and found that expanding the horizon beyond 60 days yields little benefits.

Additional operational constraints on ventilator sharing
In the optimization model, we used constraint (21g) to limit the quantity of transshipments to 1000 ventilators per day in the base case.In this sub-section, we investigate the impact of additional operational constraints on our ventilator sharing results.
Limit on the frequency of transshipments.The results presented in Table 3 assume that ventilators can be moved on every day.In this sensitivity analysis, we evaluate the sharing strategies under the conservative operational constraint that ventilators can only be moved once every week (e.g., on Mondays).Table 4 (Part A) summarizes the results.Results did not change dramatically with this additional constraint.The limited, expansive, and full sharing strategies provide an alternative approach to satisfying the demand and require 9%, 20%, and 61% fewer ventilators, respectively, compared to no sharing.
Further limit on the quantity of transshipments.Our base case results assume up to 1000 ventilators can be transshiped on each day.In this sensitivity analysis, we investigate the sharing strategies under a conservative constraint that limits the number of ventilator transshipments to 250 ventilators per day.As seen in Table 4 (Part B), with the additional constraint on transshipment capacity, the limited, expansive, and full sharing strategies call for 12%, 22%, and 62% fewer ventilators, respectively, compared to the no sharing strategy.These results are comparable to the base case results (Table 3).Further limits on both the frequency and the quantity of transshipments.In another sensitivity analysis, we evaluate the sharing strategies under both operational constraints discussed above.That is, transshipments are allowed only once a week, and when that occurs, up to 250 ventilators can be moved in total.Table 4 (Part C) presents the results with both constraints considered simultaneously.In this case, the limited, expansive, and full sharing strategies satisfy the demand with 6%, 16%, and 59% fewer ventilators, respectively, compared to the no sharing strategy.With both constraints enforced concurrently, the full sharing strategy still offers a notable improvement upon the no sharing strategy, demonstrating the robustness of our solution to stringent operational constraints.

4.5
The importance of robustness in decision-making Our goal in using a robust optimization framework is to reduce (or eliminate) ventilator shortage despite the uncertainty surrounding ventilator demand.To investigate the impact of robustness on additional ventilator need (i.e., the number of ventilators that the model calls on) and ventilator shortage, we compared the optimal policy derived from our data-driven robust model with that of deterministic and two-stage stochastic models.The two-stage stochastic model is presented in Appendix B.6.
Table 5 shows the cumulative additional ventilator need over time for the full sharing strategy with robust, deterministic, and stochastic models.Compared with the robust model, the stochastic and deterministic models call for 7% and 15% fewer ventilators, respectively (Table 5).However, this comes at the cost of experiencing a ventilator shortage.Table 5 presents ventilator allocation results informed by the optimal policy of each model along with the mean (SD) and maximum shortage.Of note, based on our 200 samples, the policy derived from the stochastic and deterministic models resulted in an average of 32 and 79 ventilator shortages, respectively, while the robust model was able to secure a ventilator for everyone who needed one.
These results further illustrate the importance of considering a robust approach, which is more conservative than the stochastic and deterministic counterparts and achieves better outcomes in worst-case scenarios.While quantifying the clinical impact of ventilator shortage is not within the scope of our study, it is reasonable to believe that the ventilator shortages that occur under the deterministic and stochastic policies could result in poor patient outcomes and likely several deaths.
Our extensive numerical experiments suggest that ventilator shortfalls are mainly dependent on two key parameters: (1) the transshipment lead times, and (2) the variability in ventilator demand (i.e., upper and lower bounds of daily ventilator   demand compared to its average).Figure 6 presents the heat map of ventilator shortage for the two-way sensitivity analysis on lead demand variability.
In this sensitivity analysis, we consider various multipliers for the demand variability (x-axis) and lead time (y-axis).In general, increased demand variability and lead time result in an increase in ventilator shortage.However, Figure 6 illustrates that the robust model is very resistant to the variability in demand and lead time as it incurs notably fewer ventilator shortages compared to the stochastic and deterministic counterparts.

Economic evaluation of ventilator sharing
One of the primary benefits of resource sharing is that it allows satisfying the demand while using fewer overall resources.This is especially crucial in resource-limited countries where obtaining new (additional) resources may be challenging.Another benefit of sharing expensive resources such as mechanical ventilators is the potential cost savings, which can benefit the healthcare system in both resource-rich and resource-limited countries.
To better understand the financial implications of ventilator sharing, we estimated the total cost of each strategy over the studied horizon (2 months) using the best available data.The average cost of a new ventilator was $32,000 (Medtronic, 2019).The cost of a new ventilator was assumed to depreciate linearly over a useful life of 9 years (Queensland Health, 2020).We assumed up to 100 ventilators can be put in a truck, and the average cost of sending a truck from one region to another is $2000 (based on online quotes).Figure 7 illustrates the total transshipment and new ventilator costs for the four strategies over the study horizon (2 months).
Figure 7 shows that the no sharing strategy costs $1.13 million in new ventilator costs as it requires over 1900 new ventilators to satisfy the demand over the study horizon.On the other hand, the full sharing strategy incurs $206,000 in transshipment cost and $423,000 in new ventilator cost (total of $629,000 during the study horizon) as it only requires 714 new ventilators to meet the demand.Furthermore, the limited and expansive sharing strategies cost a total of $1.07 and $1.01 million, respectively.In other words, compared to the no sharing strategy, the limited, expansive, and full sharing strategies satisfy ventilator demand with 5%, 11%, and 44% lower cost during the 2-month study horizon.This highlights the importance of full sharing from an economic standpoint.
We further investigated the effect of more and less effective interventions on ventilator requirement and costs in Appendix E, the importance of using dynamic uncertainty sets in Appendix F, the sensitivity of our results to the weight parameters  and  in Appendix G, and the impact of virus transmissibility on ventilator needs in Appendix H.

DISCUSSION AND CONCLUSIONS
In this study, we developed a novel DARSO methodology for optimal sharing of scarce healthcare resources during a pandemic that causes an upsurge in demand for those resources.
To demonstrate and assess our method, we conducted a case study of sharing mechanical ventilators during the first months of the COVID-19 pandemic in Ohio and Michigan, two states in the Midwest of the United States for which we had detailed population and pandemic data.During the study period (mid-March to mid-May 2020), the two states experienced their first peak of infections, which stressed the healthcare system in both states.We investigated four strategies: no sharing (status quo), limited sharing, expansive sharing, and full sharing.Limited sharing only allows unused ventilators to be shared with other regions of the same state.In contrast, expansive and full sharing strategies allow ventilator sharing with other states, though full sharing also permits sharing of the regions' initial inventory.Our integrated simulation and robust optimization model ensures that all patients with COVID-19 who need a ventilator can receive one on time even though the demand is uncertain.Our numerical results suggest that the ventilator demand of patients with COVID-19 could be satisfied with fewer ventilators if unused ventilators were moved to the regions in need according to the optimal plan informed by our model.Specifically, the limited, expansive, and full sharing strategies called on 14%, 23%, and 63% fewer new ventilators (in addition to the initial inventory) compared to the no sharing strategy.
In light of these results, we believe the full sharing strategy is the appropriate strategy for most settings because its performance is superior to other strategies.The full sharing strategy outperforms other strategies because it allows all the unused capacity of ventilators to be shared with other states that need them.Intuitively, this makes sense because there is typically a longer time lag between when regions of different states (that are geographically separated) experience the peak of infections, implying that interstate resource sharing is more beneficial.
Our DARSO methodology can be adapted and used as a decision support tool to design resource sharing strategies in future pandemics of the same kind, or if new variants of COVID-19 cause new surges in demand for healthcare resources both in resource-rich and resource-limited countries.We demonstrated two major benefits of ventilator sharing, as an example of a crucial portable resource, using our DARSO method.First, it ensures that, despite demand uncertainty, everyone who needs a ventilator can receive one on time using the minimum number of total ventilators.Second, ventilator sharing comes with a substantial cost saving benefit.In our case study, we illustrated that the full sharing strategy resulted in a 44% reduction in total costs over the study horizon as compared to no sharing.Furthermore, our numerical results demonstrated how hospitals could have retained many of their elective procedures during the first peak of COVID-19 infections in Michigan and Ohio and still been able to deal with the surge increment in mechanical ventilator demand.This could have had a substantial impact on improving the health outcomes of patients without COVID-19 and would have Benefited the healthcare organizations financially.
While our DARSO methodology is meant to inform the optimal ventilator sharing strategies across multiple states, its practical application would necessitate coordination on logistics and execution.We believe that the implementation of optimal ventilator sharing strategies can best be handled by a centralized agency that works closely with the department of health in each participating state and local public health departments.Fortunately, ventilator sharing does not require a complex logistics system and infrastructure as hundreds of ventilators can be placed in a single truck and moved to another region.Indeed, in our case study, we assumed daily transshipments are performed by trucks rented from a commercial truck rental company.
Our study has few limitations, which the future research can address them.First, our model only considers sharing one type of portable resource, namely the ventilators.Extending the model to consider multiple resource types can provide operational benefits, including the economy of scale for transshipments when multiple resources (e.g., ventilators, PPE, and testing kits) are bundled and moved together.Second, we did not evaluate the clinical outcomes of satisfying the ventilator demand for patients with COVID-19 with fewer ventilators.While the resource sharing plan informed by our model is likely to improve health outcomes and benefit hospitals financially by allowing them to maintain many of their elective procedures, which subsequently improves outcomes for patients without COVID-19, a detailed assessment of these impacts could shed additional light on the importance of resource sharing during a pandemic.Lastly, our problem and setting can be extended to online optimization and learning methodologies (see e.g., Keyvanshokooh et al., 2021;Keyvanshokooh, Zhalechian, et al., 2022;Zhalechian et al., 2022) In conclusion, we extended the theory of adaptive robust optimization to develop a novel data-driven policy-guided model that combines simulation projections with robust optimization in an implementable framework.Our approach addresses a critical practical gap in stochastic program and robust optimization methods and obtains implementable planning decisions in practice.We gave proof of concept for our model using a case study of sharing mechanical ventilators among different Ohio and Michigan regions during the COVID-19 pandemic.We demonstrated that, depending on the approach to ventilator sharing, demand could be satisfied with 14% to 63% fewer additional ventilators and 5% to 44% lower total costs.Our empirical findings may encourage hospitals to use resource sharing as a powerful strategy to combat the pandemic.Our model can be used as a decision support tool to provide actionable insights into how best to allocate and relocate portable healthcare resources, such as mechanical ventilators, to satisfy demand surges with the minimum total resources possible.This type of approach can substantially improve the way hospitals manage the care of patients with and without COVID-19 and result in cost containment.Our data-driven adaptive robust simulation-based optimization methodology is flexible and can be readily adopted to inform optimal sharing of other portable resources such as healthcare personnel, personal protective equipment (PPE), and point-of-care testing units.

FIGURE 1
FIGURE 1 Overview of the DARSO methodology for informing data-driven ventilator sharing.

FIGURE 2
FIGURE2CEACOV simulation model health states.
and maximum possible values for ventilator demand change compared to its nominal value, which can be obtained based on the CEACOV model as explained in Section 3.2.1.In (1), d min jt ≤ 0, d max jt ≥ 0, and eters djt , d min jt , and d max jt for ∀j ∈  , ∀t ∈  .The dynamic nature of this uncertainty set is due to the fact that the uncertainty at later time periods is dependent on the realization of uncertainty at previous time periods.In addition to forecasting the expected ventilator demand for each time period, we also require an estimate of the d min jt and d max jt for ∀j ∈  , ∀t ∈  .Proposition 1 obtains a dynamic uncertainty set by estimating upper and lower bounds for ventilator demand leveraging the CEACOV model.

FIGURE 3
FIGURE 3Cumulative number of new (additional) ventilators needed to cope with the surge in demand under the four strategies for both states.

FIGURE 4
FIGURE 4 Main ventilator transshipments between Michigan and Ohio regions under the full (left) and limited (right) sharing strategies.Arrows indicate transshipments of 10 or more ventilators.

FIGURE 5
FIGURE 5Cumulative ventilator transshipments for the limited and full sharing strategies over 60 days.

FIGURE 6
FIGURE 6 Two-way sensitivity analysis on the impact of demand variability and lead time on ventilator shortage.Green: average shortage of ventilators < 40, Yellow: average shortage of ventilators 40-70, Red: average shortage of ventilators ≥ 71.

FIGURE 7
FIGURE 7The total transshipment and new ventilator costs in different scenarios over the study horizon.

TABLE 1
Summary of notation used in our model formulations.
SetsSet of periods indexed by t, t ′ ∈  , where  = {t 0 , t 0 + 1, ..., | |}.Set of regions indexed by j, j ′ ∈  .Set of states indexed by i ∈ , and i(j) is the state index corresponding to region j ∈  .(i)Set of regions in state i ∈  indexed by j ∈  (i) .Uncertaintyset for modeling uncertain ventilator demand obtained by the CEACOV model.

TABLE 3
New ventilator needs and transshipment outcomes under the four sharing strategies.

TABLE 4
New ventilator needs and transshipment outcomes under different operational constraints.

TABLE 5
Comparison of additional ventilator allocation and ventilator shortage under the robust, stochastic, and deterministic policies.