The research, led by William Kirschenman, who completed the work as a Ph.D. student at NC State and is now an assistant professor at the Naval Postgraduate School, appears in two peer-reviewed journals: Naval Research Logistics and Omega. For Kirschenman, the problem is not an abstract puzzle. As a former company commander, he experienced firsthand how the order and placement of equipment can determine whether a mission succeeds. Wargaming amphibious landings, he says, revealed just how difficult it becomes to keep a unit organized in the sequence that best supports the mission when the environment is chaotic or contested. That practical frustration became the mathematical motivation for a new generation of packing models.

At the technical core of the first study is a two-dimensional orthogonal packing problem, a variant of the classic bin packing problem in which rectangular items must be placed without overlap inside a confined rectangular container. Traditional formulations of this problem typically minimize the number of bins used or the amount of unused space. The NC State team instead embedded a prioritization matrix into the bin packing framework, allowing items to be clustered with one another or pulled toward specific access points based on assigned priority weights. In other words, the model does not merely ask whether an item fits; it asks whether the item sits where it should relative to the doors, hatches or exits through which it will be retrieved, and whether it sits near the other items it belongs with.

This blending of ideas is what the authors describe as combining bin packing principles with facility layout concepts. Facility layout design, a longstanding field within industrial engineering, concerns itself with the spatial arrangement of departments and workstations to optimize flows and adjacencies. By importing that spatial-priority logic into packing, the researchers created a single mixed-integer linear programming model, or MILP, capable of balancing space efficiency against proximity to access points and adjacency among functionally related items. The result is a formulation that extends the utility of packing optimization to applications demanding far more nuanced layout preferences than simple volume maximization.

There was, however, an immediate computational obstacle. Every additional constraint, every priority level, every adjacency requirement multiplies the number of possible configurations the optimizer must consider. The researchers found that the potential combinations quickly outstripped available computing power when the full model was handed to a commercial solver as one monolithic problem. Their answer was a sliding-window matheuristic, a hybrid technique that breaks the large optimization problem into a sequence of smaller, overlapping subproblems. The algorithm solves each window of the packing space in turn, carrying forward the consequences of earlier decisions, and thereby obtains a good overall solution in a fraction of the time a direct solve would require.

The computational experiments were striking. When the sliding-window matheuristic was tested against a direct MILP solve using a commercially available optimization solver, the matheuristic consistently outperformed the monolithic approach in both runtime and solution quality. It also proved superior among the adapted heuristic and metaheuristic alternatives the team considered. For practitioners, that means high-quality, priority-aware packing plans can be produced on ordinary computing hardware rather than waiting hours or days for an exact solver to grind through an intractable search space, or settling for a crude rule-of-thumb arrangement.

The second study pushed the framework into an even more demanding arena: loading large maritime vessels. On a ship, geometry is only half the battle. Too much weight placed too high or too far to one side can cause the vessel to list, tilting dangerously, so load stability becomes a hard constraint that the packing plan must satisfy. As co-author Brandon McConnell, a research associate professor in NC State’s Edward P. Fitts Department of Industrial and Systems Engineering, explains, adding the stability constraint means working around a fixed center of gravity, and the correctness of the balance cannot be confirmed until the entire area has been packed, which usually forces adjustments. The team studied three solution approaches for this globally constrained problem: a monolithic MILP model, a sliding-window matheuristic, and a sliding-window matheuristic augmented with in-stride load balancing penalties applied during the solve.

Testing revealed a clear winner. The most efficient path was to run the existing sliding-window model first and then adjust selected lower-priority items at the end as needed to achieve stability, a post-processing strategy that selectively relaxes and re-optimizes item positions with minimal disruption to the prioritized layout. This simpler pipeline of matheuristic followed by repair emerged as the recommended practical configuration, offering the strongest overall combination of balance success, solution quality and runtime. The in-stride variant, which enforces balance during the initial solve, remains a narrower alternative for cases where achieving balance in the first stage is paramount. Notably, the matheuristic pipelines generated high-quality, load-balanced solutions for single-vessel scenarios within a few minutes on average, fast enough to let planners rapidly evaluate multiple loading configurations during time-critical deployment planning.

The implications stretch well beyond the military context that inspired the work, although combat loading, the art of arranging equipment on maritime transport so that mission-critical gear can be offloaded first while units stay cohesive and the vessel stays stable, remains the flagship application. The same mathematics applies to warehouse storage, where fast-moving goods should sit nearest the packing stations, and to multi-drop delivery services, where parcels for the last stops on a route must be loaded deepest and parcels for the first stops must be within easy reach. In every case, the model’s ability to encode multiple levels of prioritization, enforce groupings of related items, and respect physical constraints like balance turns a notoriously fiddly human judgment call into a reproducible computation.

The research team also included H. Sebastian Heese, Owens Distinguished Professor of Supply Chain Management; Michael Kay, associate professor of industrial and systems engineering; and Russell King, the Dopaco Distinguished Professor of Industrial and Systems Engineering, all of NC State. Kirschenman, a two-time selectee for the General Omar Bradley Research Fellowship in Mathematics, which helped support the research, sees the work as a bridge between abstract optimization and the messy realities of logistics. As McConnell puts it, the model could help ensure that mission-critical gear is offloaded first, save a warehouse worker’s time and energy, and lead to faster goods retrieval with fewer mistakes for any delivery business. In a world where supply chains are under unprecedented strain, teaching computers to pack with both intelligence and speed may prove one of the quiet but consequential advances of applied operations research.

Subject of Research: Prioritized two-dimensional orthogonal packing optimization with load balancing for military and commercial logistics

Article Title: Cracking the packing code – new models incorporate item priority, location and weight distribution

Article References: Cracking the packing code – new models incorporate item priority, location and weight distribution. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: packing problem, optimization, mixed-integer linear programming, matheuristic, military logistics, load planning, warehouse storage, supply chain, vessel stability, operations research, bin packing, Naval Research Logistics

The study, published in the Journal of Industrial Ecology by Hessel Roodenburg, Ernst Worrell and Javanshir Fouladvand, addresses a striking gap in the scientific literature. While the humanitarian and economic costs of war have been extensively documented — armed conflicts caused more than 100 million deaths in the twentieth century, and the Kiel Institute for the World Economy estimates that a war-affected country’s economic output falls by roughly 30 percent within five years — the underlying material basis of warfare has remained empirically unquantified. Previous environmental studies documented consequences such as habitat destruction, deforestation and heavy-metal contamination, but never traced the causal flows of materials that generate these effects.

The framework borrows its structure from military science itself. The researchers organized warfare using the doctrinal warfighting functions defined by NATO and US Army doctrine, focusing on the four physically resource-intensive functions: Movement and Manoeuvre, Fires, Protection, and Sustainment. Command and Control, Intelligence and Information were excluded because they lack direct material throughput. Material inputs were then categorized using the NATO and US Army Classes of Supply taxonomy, covering fuels, fortification materials, ammunition, repair parts and medical supplies, all organized across air, ground, naval and defensive domains.

At the heart of the model lies a deceptively simple multiplicative equation: the mass of each supply class consumed equals the number of deployed platforms of each type, multiplied by a standardized monthly intensity per platform, multiplied by a logistical factor accounting for fuel self-consumption in distribution. Intensity values were derived using a quartile-based approach, with the 25th, 50th and 75th percentiles of compiled datasets representing low, medium and high operational scenarios. Where data were scarce, a multiplier approach scaled baseline consumption by factors of 0.5, 1 or 2. Platform losses were estimated at 6.5 percent of deployed platforms per month, based on observed conflict data.

The framework was tested against two contrasting case studies: the Russia-Ukraine war, a prolonged high-intensity conflict with a 950-kilometer active frontline and more than 1.1 million frontline personnel, and Operation Desert Storm of 1991, a short but high-tempo 43-day campaign dominated by air and naval operations. To enable direct comparison, the researchers constructed a normalized two-month scenario in which all parameters remained case-specific but the time horizon was standardized.

The results are striking. Within a two-month window, total carbon dioxide emissions reach approximately 6 megatonnes for the Russia-Ukraine war — comparable to the monthly emissions of Denmark — and 28 megatonnes for Desert Storm, equivalent to the monthly emissions of the entire Dutch industrial sector. In mass terms, the model indicates that 8 to 10 megatonnes of fuels and ammunition are mobilized over two months, comparable to the global monthly cement trade or Sweden’s annual steel output. Fuel accounts for 70 to 80 percent of the total mass flow, with ammunition dominating the remainder and fortification materials adding hundreds of kilotonnes.

The case studies reveal fundamentally different material signatures. Over the full 43-month modeled period of the Russia-Ukraine war, fuel consumption is estimated at 13.1 billion liters for Ukraine and 32.9 billion liters for Russia, generating 36 and 90 megatonnes of carbon dioxide respectively. Residual ammunition mass totals 5.85 megatonnes for Ukraine and 10.77 for Russia, with unguided missiles alone accounting for more than 6.21 megatonnes on the Russian side. The model also estimates roughly 14.5 million unexploded rounds across the war, corresponding to 39 megatonnes of TNT-equivalent on the Russian side and 17 megatonnes for Ukraine — a legacy of contamination that will persist for decades.

Desert Storm tells a different story. Despite lasting only 43 days, the Coalition consumed an estimated 6,652 megaliters of fuel, with fuel transport and logistical activities accounting for roughly three-quarters of the total. The campaign produced 2.27 million unexploded rounds and destroyed thousands of heavy platforms, generating 24 kilotonnes of non-recoverable debris for the Coalition and 12 for Iraq. The comparison between the two conflicts illustrates a key insight: high-tempo air campaigns amplify jet-fuel emissions, while sustained ground warfare accumulates fortification debris and mine-related outputs. In the Russia-Ukraine case, sandbags and concrete dominate fortification consumption, together representing about 90 percent of the total mass.

Perhaps the most consequential finding concerns logistics. In the Russia-Ukraine case, approximately 72 percent of fuel-related emissions arise not from combat platforms but from fuel transport and logistical fuel — the energy cost of moving energy itself. Platform fuel and ammunition contribute smaller shares. This decomposition demonstrates that logistics, not combat, is the primary driver of warfare’s emissions profile, a structural dominance that had previously been suggested but never quantified within a standardized framework. Capturing only the active warfare component, the study estimates that 126 megatonnes of carbon dioxide were emitted in the first three years of the Russia-Ukraine conflict — roughly half of broader estimates that include infrastructure destruction, fires and displacement.

The authors are careful to frame the work as a foundation rather than a definitive account. The system boundaries exclude upstream manufacturing, peacetime military activity and post-war reconstruction, meaning the true footprint of any war would be significantly higher over a full life cycle. Data limitations forced reliance on proxies, such as a spare-parts ratio based on a single US Army maintenance guide, and no formal uncertainty analysis was conducted, so results should be read as order-of-magnitude estimates. Nevertheless, the framework provides something previously absent: a reproducible, transparent method for converting diffuse military resource use into traceable parameters. The researchers emphasize that the goal is not to make war more environmentally friendly, but to raise awareness of its material costs — offering policymakers, humanitarian organizations and researchers evidence-based data to assess warfare’s place within broader climate and resource governance, and to inform post-conflict recovery planning.

Subject of Research: Quantifying the material and carbon footprint of modern warfare using material flow analysis

Article Title: Material footprint of warfare: a systematic framework and empirical analysis

Article References: Roodenburg, H., Worrell, E., & Fouladvand, J. (2026). Material footprint of warfare: a systematic framework and empirical analysis. Journal of Industrial Ecology. https://doi.org/10.1007/s44498-026-00130-z

Image Credits: AI Generated

DOI: 10.1007/s44498-026-00130-z

Keywords: material footprint, warfare, material flow analysis, industrial ecology, Russia-Ukraine war, Operation Desert Storm, military logistics, carbon emissions, fuels, ammunition, environmental impact, sustainability