Airlines around the world are exploring several tactics to meet aggressive CO2 commitments set by the International Civil Aviation Organization (ICAO). This effort has been emphasized in Europe, where aviation accounts for 13.9% of the transportation industry’s carbon emissions. The largest push comes from the European Green Deal, which aims to decrease carbon emissions from transportation by 90% by 2051. The Lufthansa Group has gone even further, committing to a 50% reduction in emissions compared to 2019 by the year 2030 and to reach net-zero emissions by 2050.

One unexpected approach that airlines can use to lower carbon emissions is through optimizing their *tail assignment*, i.e., how to assign aircraft (identified by the aircraft registration painted on their tails) to legs in a way that minimizes the total operating cost, of which fuel is a major contributor. More fuel needed to operate the aircraft means higher operating costs and more carbon ejected into the atmosphere. For example, a typical long-haul flight (longer than ~4,100km or ~2,500mi) emits about a ton of CO2.

The amount of fuel needed to fly between origin and destination can vary widely — e.g., larger aircraft weigh more and therefore require more fuel, while modern and younger aircraft tend to be more fuel-efficient because they use newer technology. The mass of the fuel itself is also significant. Aircraft are less fuel-efficient early in their flights when their fuel tanks are full than later when the volume of fuel is reduced. Another important factor for the tail assignment is the number of passengers on board; as the number of bookings changes, a smaller or larger aircraft might be required. Other factors can affect fuel consumption, both negative (e.g., headwinds or the age of the engines) or positive (e.g., tailwinds, sharklets, skin).

During the past year, Google’s Operations Research team has been working with the Lufthansa Group to optimize their tail assignment to reduce carbon emissions and the cost of operating their flights. As part of this collaboration, we developed and launched a mathematical tail assignment solver that has been fully integrated to optimize the fleet schedule for SWISS International Air Lines (a Lufthansa Group subsidiary), which we estimate will result in significant reductions in carbon emissions. This solver is the first step of a multi-phase project that started at SWISS.

A Mathematical Model for Tail Assignment

We structure the task of tail assignment optimization as a network flow problem, which is essentially a directed graph characterized by a set of nodes and a set of arcs, with additional constraints related to the problem at hand. Nodes may have either a supply or a demand for a *commodity*, while arcs have a flow capacity and a cost per unit of flow. The goal is to determine flows for every arc that minimize the total flow cost of each commodity, while maintaining flow balance in the network.

We decided to use a flow network because it is the most common way of modeling this problem in literature, and the commodities, arcs, and nodes of the flow network have a simple one-to-one correspondence to tails, legs, and airports in the real-life problem. In this case, the arcs of the network correspond to each leg of the flight schedule, and each individual tail is a single instance of a commodity that “flows” along the network. Each leg and tail pair in the network has an associated assignment cost, and the model’s objective is to pick valid leg and tail pairs such that these assignment costs are minimized.

A simple example of the tail assignment problem. There are four legs in this schedule and four possible tails that one can assign to those legs. Each tail and leg pair has an associated operational cost. For example, for Leg 1, it costs $50 to assign Tail 1 to it but $100 to assign Tail 2. The optimal solution, with the minimum cost, is to assign Tail 4 to Legs 3 and 2 and Tail 1 to Legs 1 and 4.

Aside from the standard network flow constraints, the model takes into account additional airline-specific constraints so that the solution is tailored to Lufthansa Group airlines. For example, aircraft turnaround times — i.e., the amount of time an aircraft spends on the ground between two consecutive flights — are airline-specific and can vary for a number of reasons. Catering might be loaded at an airline’s hub, reducing the turnaround time needed at outstations, or a route could have a higher volume of vacation travelers who often take longer to board and disembark than business travelers. Another constraint is that each aircraft must be on the ground for a nightly check at a specified airport’s maintenance hub to receive mandated maintenance work or cleaning. Furthermore, each airline has their own maintenance schedule, which can require aircraft to undergo routine maintenance checks every few nights, in part to help maintain the aircraft’s fuel efficiency.

Preliminary Results & Next Steps

After using our solver to optimize their fleet schedule in Europe, SWISS Airlines estimates an annual savings of over 3.5 million Swiss Francs and a 6500 ton reduction in CO2 emitted. We expect these savings will multiply when the model is rolled out to the rest of the airlines in the Lufthansa Group and again when traffic returns to pre-COVID levels. Future work will include ensuring this model is usable with larger sets of data, and adding crew and passenger assignment to the optimization system to improve the flight schedules for both passengers and flight crew.

If you are interested in experimenting with your own network flow models, check out OR-Tools, our open source software suite that can be used to build optimization solutions similar to the solver presented in this post. Refer to OR-Tools related documentation for more information.

Acknowledgements

Thanks to Jon Orwant for collaborating extensively on this blog post and for establishing the partnership with Lufthansa and SWISS, along with Alejandra Estanislao. Thanks to the Operations Research Team and to the folks at SWISS, this work could not be possible without their hard work and contributions.