Vaccinating millions of people is not only a medical challenge; it is also a vast mathematical puzzle. Before a dose reaches a patient, planners must decide how many vaccines should be sent to each location, which communities should be prioritized, how demand can be met with limited supplies, and how to keep transportation costs under control. A new computational method developed by researchers at North Carolina State University could make those decisions dramatically faster, potentially helping public agencies respond more effectively during pandemics and other large-scale emergencies.
The work combines machine learning with a mathematical optimization technique known as column generation. In tests using a detailed vaccine-distribution model, the hybrid approach reduced computational runtime by 79.1 percent compared with conventional column generation, while still producing solutions within 6 percent of the mathematically optimal result. The researchers say the method could transform optimization problems that are theoretically solvable but practically too large for routine use.
“We had developed an optimization model that ensures supplies of vaccines are distributed efficiently and equitably to the places they are most needed,” says Leila Hajibabai, an associate professor in NC State’s Edward P. Fitts Department of Industrial and Systems Engineering and corresponding author of the study. “However, the computational power needed to run that optimization model at a statewide level was not practical. Our goal was to develop a methodology that allows us to use that model but requires far less computing power.”
The underlying model is a maximal covering location-allocation problem. In simplified terms, it attempts to cover as much population as possible with available vaccine supplies while accounting for distribution costs, demand, transportation, and equity. The researchers previously applied the model to vaccine shipments and demand in Pennsylvania during the COVID-19 pandemic, using data from the Centers for Disease Control and Prevention and state and local health departments. The dataset included information detailed enough to support decisions at the level of individual census blocks.
That level of detail is valuable for identifying underserved communities, but it creates an enormous computational burden. “Running that model involved approximately 1.6 billion potential binary and integer decision variables,” says Ali Hajbabaie, an associate professor of civil, construction and environmental engineering at NC State and co-author of the study. Binary variables represent yes-or-no decisions, such as whether a particular allocation is selected, while integer variables can represent quantities such as shipment volumes. Evaluating billions of possible combinations can overwhelm even powerful optimization systems, particularly when planners need to update decisions repeatedly as new information arrives.
Column generation addresses this problem by avoiding the need to consider every possible decision at once. Instead, the method begins with a smaller version of the optimization problem and gradually introduces additional variables, or “columns,” when they appear capable of improving the solution. A secondary calculation, commonly called a pricing problem, searches for promising columns based on the current solution. This approach can shrink the active problem dramatically, but the search may still be expensive when the underlying decision space is exceptionally large.
The NC State researchers added machine learning to guide that search. Their algorithm learns from optimization problems solved previously and uses patterns in those solutions to predict which decisions are likely to be useful in subsequent iterations. Rather than examining every potential variable equally, the system directs column generation toward the most promising candidates. Less promising decisions can be estimated or postponed, reducing the number of optimization calculations required.
“Machine learning can learn from previously solved instances to predict promising decisions for one step of the optimization process,” Hajibabai explains. “That allows the algorithm to focus its computational effort where it is most likely to improve the solution. It reduces the search space and the number of optimization calculations in later iterations.” The machine-learning component does not replace the optimization model; instead, it acts as an intelligent filter that helps determine where the exact mathematical search should spend its time.
In computational experiments, the machine learning-guided column generation method offered the strongest overall balance between speed and solution quality among the techniques evaluated. It did not always reproduce the exact optimum, but its results remained close enough to be operationally useful. The researchers report that solutions were within 6 percent of the optimal outcome while the computational time fell substantially compared with benchmark methods. That trade-off could be especially important in fast-moving crises, when a slightly less-than-perfect plan delivered quickly may be more valuable than an exact plan that arrives too late.
The researchers emphasize that the method is not limited to vaccines. The same combination of predictive algorithms and mathematical optimization could support disaster-relief logistics, humanitarian supply chains, transportation planning, energy-system design, and other infrastructure problems involving millions of possible decisions. In each case, machine learning could identify promising regions of the search space, while optimization methods would evaluate the most important choices rigorously. The study, led by former NC State doctoral student Kuangying Li and co-authored by Ph.D. student Hiruni Niwunhella, was published open access in Sustainability Analytics and Modeling. The research was supported by the National Science Foundation under grant 2124825.
Subject of Research: Computational optimization of vaccine distribution and large-scale resource allocation
Article Title: Machine Learning-Guided Column Generation for a Maximal Covering Location-Allocation Problem
News Publication Date: 30-Jul-2026
Web References: https://www.sciencedirect.com/science/article/pii/S2667259626000196?via%3Dihub; https://www.sciencedirect.com/science/article/pii/S1093968726000939; https://pubsonline.informs.org/doi/10.1287/trsc.2022.1134
References: Li, Kuangying et al., “Machine Learning-Guided Column Generation for a Maximal Covering Location-Allocation Problem,” Sustainability Analytics and Modeling, DOI: 10.1016/j.samod.2026.100069
Keywords: vaccine distribution, machine learning, column generation, mathematical optimization, computational modeling, logistics, resource allocation, supply chains, disaster relief, healthcare planning
