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A Model of Supply-Chain Decisions for Resource Sharing with an Application to Ventilator Allocation to Combat COVID-19
Sanjay Mehrotra, Hamed Rahimian, Masoud Barah, Fengqiao Luo, Karolina Schantz
TL;DR
The paper addresses how to allocate scarce ventilators when states experience uncertain demand peaks at different times and must preserve inventory for non-COVID-19 patients. It develops a stochastic multi-period allocation and sharing model with risk-averse reallocation, then evaluates it using U.S. COVID-19 demand scenarios. The results indicate that stockpile adequacy depends strongly on the fraction reserved for non-COVID-19 care, with substantial shortfalls under severe reservation and demand conditions.
Problem
Different state demand peaks and competing non-COVID-19 needs create a problem of coordinating scarce ventilator allocation and reallocation.
Method
The paper develops a stochastic multi-period model for central allocation, state sharing, future production, and risk-averse reallocation of ventilators.
Results
When 75% or more of available ventilators must serve non-COVID-19 patients, worst-day shortfalls in Cases III and IV reach 1,500-2,700 ventilators.
Takeaways & Limitations
FEMA’s 20,000-ventilator stockpile is sufficient under lower non-COVID-19 reservation levels but becomes inadequate under more severe reservation or demand scenarios.
Takeaways & Limitations
The computational study relies on ventilator-need forecasts that are difficult to validate against actual hospital and state operational data.
Abstract
from arXiv · showhide
This paper presents a stochastic optimization model for allocating and sharing a critical resource in the case of a pandemic. The demand for different entities peaks at different times, and an initial inventory from a central agency is to be allocated. The entities (states) may share the critical resource with a different state under a risk-averse condition. The model is applied to study the allocation of ventilator inventory in the COVID-19 pandemic by the Federal Emergency Management Agency of the US Department of Homeland Security (FEMA) to different states in the US. Findings suggest that if less than 60% of the ventilator inventory is available for non-COVID-19 patients, FEMA's stockpile of 20,000 ventilators (as of 03/23/2020) would be nearly adequate to meet the projected needs. However, when more than 75% of the available ventilator inventory must be reserved for non-COVID-19 patients, various degrees of shortfall are expected. In an extreme case, where the demand is assumed to be concentrated in the top-most quartile of the forecast confidence interval, the total shortfall over the planning horizon (till 05/31/20) is about 28,500 ventilator days, with a peak shortfall of 2,700 ventilators on 04/12/20. The results also suggest that in the worse-than-average to severe demand scenario cases, NY requires between 7,600-9,200 additional ventilators for COVID-19 patients during its peak demand. However, between 400 to 2,000 of these ventilators can be given to a different state after the peak demand in NY has subsided.
1 Introduction
The paper addresses the need to allocate and reallocate scarce ventilators as states face different demand peaks and possible shortages. It proposes a model that accounts for non-COVID-19 demand, future production, state sharing, and risk-averse reallocation.
- Mechanical ventilation is critical for severe COVID-19 cases involving respiratory failure and insufficient oxygenation.Approximately 5% of patients were estimated to require ICU treatment with mechanical ventilation.
- States may reach ventilator demand peaks at different times, creating a need for coordinated federal allocation and reallocation.Projected demand indicates that the timing of capacity gaps varies by state.
- The paper presents a model for allocating and reallocating national-stockpile ventilators while estimating state shortfalls under alternative demand scenarios.
- The framework reserves part of existing state inventory for non-COVID-19 patients and incorporates future production and risk-averse sharing.
2 Literature Review
Prior work spans healthcare-resource allocation, infectious-disease modeling, and ventilator stockpiling. This paper builds on those strands by formulating a stochastic, multi-period ventilator allocation problem with possible reallocation under uncertainty.
- Related research has used operations research and optimization to allocate healthcare resources across disasters, disease prevention, and developing-country settings.
- COVID-19 and pandemic studies have addressed quarantine effectiveness, ventilator allocation frameworks, demand estimation, and stockpile optimization.
- Existing ventilator studies include estimates of additional national needs and models comparing local with central storage under pandemic scenarios.One cited high-severity estimate projected 35,000 to 60,500 additional ventilators.
- The paper assumes a finite planning horizon with reallocation decisions made at discrete times and formulates the problem as a stochastic program.
3 A Model for Ventilator Allocation
The model is a multi-period stochastic program in which a central agency allocates ventilators across regions while states may reserve, share, or return inventory. Its objective is to minimize expected shortages subject to inventory, safety-stock, and scenario constraints.
- A central agency makes multi-period allocation and reallocation decisions for stochastic regional ventilator demand.Decisions occur at the beginning of each period before states use inventory to treat critical patients.
- The model includes regional inventory, central inventory, production, non-COVID-19 reservations, sharing willingness, risk aversion, demand, and relocation variables.
- The objective minimizes expected total ventilator shortage across regions and periods.
- Inventory-conservation constraints track regional and central ventilators, while safety-stock constraints limit transfers from regions with insufficient inventory.
- The formulation uses linearization for the nonlinear shortage objective and transfer constraint, with a big-M parameter.
- Demand is represented through a finite set of scenarios with associated probabilities, and the reformulated model is solved as an extensive-form mixed-binary program.
4 Ventilator Allocation Case Study: The US
The case study generates demand scenarios from forecast confidence intervals, then solves the allocation model under four cases to evaluate ventilator shortages, flows, and sharing across states.
- Demand Scenario Generation: 70 days: The planning period runs from 03/23/2020 through 05/31/2020, with random ventilator demands generated under four mitigation-related cases.Each case uses demand scenarios for all days and states, with the forecast confidence interval representing the demand-distribution support.
- Demand Scenario Generation: Case I assigns equal probability to scenarios sampled uniformly across the forecast confidence interval.The procedure samples one of two tails with probability 0.5 and then uniformly selects among 50 partitions within that tail.
- Demand Scenario Generation: Case II weights the top 25% of the confidence interval at 0.25 probability and the bottom 75% at 0.75 probability.Cases III and IV increase the top-quartile probability to 0.50 and 0.75, respectively, while assigning 0.50 and 0.25 to the bottom 75%.
- Demand Scenario Generation: 24 scenarios: The study generates 24 demand scenarios for each of the four cases and illustrates their trajectories for the US, New York, and California.Figure 1 summarizes the generated scenario cases for these geographic areas.
- Ventilator Inventory, Stockpile and Production: 20,000 ventilators: FEMA’s reserve inventory as of 03/23/2020 is used as the model’s initial central-agency stockpile.State inventory estimates come from hospital-survey-based estimates, while production estimates use information from presidential briefings.
- Model Parameters: The model uses γ for the fraction reserved for non-COVID-19 patients, τ for willingness to share initial COVID-19-use ventilators, and ρ for risk aversion to returning idle ventilators to FEMA.The study assumes common values γ_n = γ, ρ_n = ρ, and τ_n = τ across states.
- Numerical Results: The study solves model (2) for each (γ, τ) setting under Cases I–IV and reports total shortage, worst-day shortage, worst day-state shortage, and ventilator flows.Table 2 focuses on Cases III and IV under (γ, τ) = (0.75, 0), including inflow, outflow, and net flow.
- Discussion: 50%: FEMA’s 20,000-ventilator stockpile is sufficient when no more than half of state inventory is used for non-COVID-19 patients.With sharing of up to 50% of excess inventory, the threshold increases to 60%; without sharing, approximately 300 additional ventilators are needed.
5 Concluding Remarks
The paper presents a stochastic model for risk-averse procurement, allocation, and sharing of life-saving resources, applied to ventilator planning under varying demand and inventory assumptions. Results show that COVID-19 coverage depends strongly on inventory reserved for non-COVID-19 patients, while sharing and adaptable inputs support broader planning use.
- The model represents stochastic demand whose peaks differ across states and captures each state's risk aversion toward sharing excess inventory through a safety threshold.It was evaluated with realistic ventilator forecasts and availability under varied parameter settings.
- When more than 40% of existing ventilator inventory is available for COVID-19 patients, the national stockpile is sufficient to meet demand.If less than 25% is available for COVID-19 patients, the stockpile and anticipated production may be insufficient under extreme demand scenarios.
- Shortfalls increase as demand scenarios become more extreme, especially when less than 25% of existing inventory is available for COVID-19 patients.
- The model can serve as a planning framework for state and federal agencies acquiring and allocating ventilators, with inputs adjustable for refined results.A state's willingness to share idle inventory can help address overall shortfall.
- The framework may be adapted for international coordination because countries can experience different demand peaks and cycles.
- The model makes a one-time planning decision and excludes wait-and-see recourse decisions as information about stochastic demand and past decisions evolves.A time-dynamic multistage stochastic extension is identified as future work.