The optimization of aircraft trajectories involves balancing operating costs and climate impact, which are often conflicting objectives. To achieve compromised optimal solutions, higher-level information such as preferences of decision-makers must be taken into account. This paper introduces the SolFinder 1.0 module, a decision-making tool designed to identify eco-efficient aircraft trajectories, which allow for the reduction of the flight's climate impact with limited cost penalties compared to cost-optimal solutions. SolFinder 1.0 offers flexible decision-making options that allow users to select trade-offs between different objective functions, including fuel use, flight time,

Aviation is estimated to contribute 3 %–5 % of total anthropogenic global warming

Currently, air traffic optimization focuses on minimizing economic penalties, e.g. identifying aircraft trajectories that lead to minimal operating cost. Minimizing the operating cost and the climate impact of a single flight are expected to be conflicting objectives of aircraft trajectory optimization

it is applicable to any set of Pareto-optimal solutions resulting from the optimization of a single aircraft trajectory;

it is suitable for identifying compromise solutions between any number of objective functions;

in particular, when applied for the bi-objective optimization of operating cost and climate impact, it is capable of identifying eco-efficient solutions.

In Sect.

We conduct our simulations using the ECHAM/MESSy Atmospheric Chemistry (EMAC) model

Overview of the relations between EMAC and the three submodels CONTRAIL, ACCF, and AirTraf. The present study focuses on the description of the decision-making module SolFinder 1.0, which is highlighted in red in the diagram.

The EMAC model provides the atmospheric conditions at a specific time and location (e.g. wind, temperature, potential vorticity, relative humidity) to determine the fuel consumption, emission indexes, and climate effects of aircraft emissions. The CONTRAIL submodel computes the potential contrail coverage, i.e. the fraction of the model grid box where persistent contrails can exist

For the present study, we applied EMAC (ECHAM5 version 5.3.02, MESSy version 2.55.0) in the T42L31ECMWF resolution. This resolution has a spherical truncation of T42 (corresponding to a quadratic Gaussian grid of approximately 2.8 by 2.8° in latitude and longitude) and includes 31 vertical hybrid pressure levels up to 10 hPa (i.e. to an altitude of approximately 30 km). We describe the model output obtained by simulating the atmospheric conditions occurring from 1 January 2018 to 31 January 2018, employing a temporal resolution of 12 min. To obtain weather conditions aligned with those observed in January 2018, the simulations are conducted nudging by Newtonian relaxation the prognostic variable divergence, vorticity, temperature and the (logarithm of the) surface pressure down to the surface towards the respective ECMWF ERA-Interim reanalysis data

The air traffic simulator AirTraf is responsible for the optimization of the aircraft trajectories, according to the routing strategy prescribed by the user. The submodel requires as input information (1) the atmospheric conditions at the time and location of the flight, provided by the EMAC model, and (2) the air traffic sample, including the location of the airports of departure and arrival, the departure time of each flight, and characteristics of the aircraft and engine type to be simulated

The version AirTraf 2.0 presented by

In the development of the decision-making strategies implemented in SolFinder 1.0, the underlying goal has been to find eco-efficient aircraft trajectories, compromising between the optimization of climate impact and operating costs. Hence, we include here the definitions of these objective functions within AirTraf. The economic costs of the flights are represented by the simple operating costs (SOC), defined as in Eq. (

In this section, we describe the decision-making strategies implemented in SolFinder 1.0. Our aim is to solve a multi-objective optimization problem by minimizing a set of

A large variety of multi-criteria decision-making methods is currently available to select one solution among a set of optimal options

See Supplement and

the relative importance of

the relative importance of group utility (preference towards achieving the greatest benefit) and individual regret (preference towards avoiding large penalties), represented by the parameter

The SolFinder 1.0 module identifies eco-efficient trajectories using a decision-making option based on the VIKOR method. This method, introduced by

Illustration of the steps performed by the eco-efficient decision-making strategy relying on VIKOR. The aircraft trajectories are optimized to minimize SOC and ATR20, resulting in a set of Pareto-optimal solutions (grey crosses). We set

The resulting strategy can be configured to follow the steps listed here:

A bi-objective optimization problem is solved to simultaneously minimize the total climate impact (ATR20

The VIKOR method is applied, following the steps described in Appendix

A set of equally recommended solutions are selected in the sections in the Pareto front closest to the cost-optimal solution by assigning a relatively high weight to the operating costs, i.e.

Among this set of equally recommended solutions, the solution leading to the largest climate impact reduction with respect to the cost-optimal solution is selected, since

Variability of the selected solution (red triangle) using the eco-efficient decision-making method. The grey crosses represent the Pareto-optimal solutions, while the blue circles indicate the subset of solutions recommended by the VIKOR method. The axes show percentage changes in the objective functions, relative to the solution minimizing SOC. In this example, the Pareto front consists of 308 solutions.

To understand the effectiveness of the VIKOR method with various configurations of

This observation is the base for the definition of eco-efficient solution given at the end of Sect.

In some scenarios, the decision-maker wishes to limit the penalty of one of the objective functions, e.g. to avoid unsustainable increases in the operating costs. Therefore, in the second decision-making strategy, we propose to select the solution

Figure

Example of selecting the solution among the Pareto-surface matching a target increase in 0.5 % in flight time (indicated by red triangles). The blue circles indicate the Pareto-optimal solutions, which result from a tri-objective optimization problem minimizing flight time, fuel use, and ATR20

To combine the advantages of the two decision-making strategies presented, a hybrid option is considered, limiting the variability in one of the objective functions while applying the VIKOR method. When this strategy is selected, the decision-maker provides (1) the configuration of the VIKOR method, setting the parameters

Apply the VIKOR method and select the recommended solution minimizing the objective function having the lowest weight

Calculate

If

Location of the 100 flights included in the air traffic sample. Each curve represents the great circle connecting an origin/destination pair. Note that most origin/destination pairs are connected by two flights, i.e. one for each direction; thus, the number of curves is lower than 100. The list of ICAO airport codes is included in Table

Lastly, an additional option is considered to select a solution minimizing one of the objective functions

We now present an example study, in which different settings of the decision-making strategies are compared. This application exemplifies how the decision-making strategies can be employed and what to consider to determine the settings that best translate the decision-maker preference. In this example, we focus on the suitable settings to identify eco-efficient aircraft trajectories. Nevertheless, SolFinder can also be used to comply with alternative decision-making preferences, by changing the settings of AirTraf and SolFinder.

As previously stated, we intend to identify eco-efficient aircraft trajectories, i.e. trajectories reducing the climate impact with limited changes in the operating costs. Therefore, we solve a bi-objective optimization problem, aiming to simultaneously minimize SOC and ATR20, as defined in Sect.

Main settings of ECHAM5, ACCF, and AirTraf.

Overview of the two sets of experiments performed to exemplify the use of the different decision-making methods.

To compare the effects of using different decision-making strategies, we perform two sets of experiments, whose characteristics are summarized in Table

In the first set of experiments, the VIKOR method is employed as we described in Sect.

In the second set of experiments, we explore the effects of selecting a solution leading to a target change in SOC (Sect.

Properties of the trajectories selected within the first set of experiments. The table provides the total monthly percentage changes in climate impact,

Relation between the relative changes in climate impact,

As explained in Sect.

Relative frequencies [%] of different values of

As previously mentioned, the red and green points highlighted in Fig.

Properties of the trajectories selected within the second set of experiments. The table provides the total monthly percentage changes in climate impact,

The results of the first set of experiments can then be used to conduct the second set, fixing a target or a limit increase in SOC as described in Sect.

Relative frequencies [%] of different values of

Relative frequencies [%] of different values of the climate-cost coefficient

Employing different decision-making strategies, we obtain trajectories which are characterized by different properties. How these properties vary is shown in Fig.

Variability of mean flight altitudes [km]

Contribution of different climate effects of aviation to the absolute

Lastly, we analyse the contribution to the total change in ATR20 of each effect of aviation emissions that we considered in our optimization process: CO

In this paper, we illustrated the decision-making strategies implemented in the SolFinder 1.0 module and how they can be used to identify eco-efficient trajectories. The climate optimization of aircraft trajectories has been increasingly researched in the last decade, as efforts to reduce the climate impact of aviation lead to the investigation of operational mitigation strategies. For example,

Relation between the percentage changes in climate impact,

The present work estimates the climate effect of aviation resulting from the emission of CO

In this study, we described the decision-making strategies implemented in the SolFinder 1.0 module. The SolFinder 1.0 module has been coupled to the AirTraf 3.0 submodel, as part of its development to efficiently solve multi-objective optimization problems. We showed here how the selected decision-making strategies can be used to identify solutions matching specific preferences (e.g. eco-efficient aircraft trajectories). Moreover, using this modelling chain, it is possible to explore the results variability under a large number of consecutive days, due to the coupling between SolFinder and an atmospheric chemistry model (EMAC), via the EMAC submodel AirTraf. To demonstrate the usage of the tool, this paper showed results for the period of 1 winter month (1–31 January 2018). We solved a bi-objective optimization problem by minimizing the climate impact of the aircraft trajectory (F-ATR20

In this appendix, we quote the steps characterizing the VIKOR method, as they were introduced and described in

Illustration of the VIKOR method applied to a bi-objective optimization problem, minimizing

The geometric representation of

If this condition is not verified, a set of Pareto-optimal solutions

If this condition is not satisfied, both

List of origin/destination airport pairs included in the air traffic sample, as illustrated in the map in Fig.

The Modular Earth Submodel System (MESSy) is continuously developed and applied by a consortium of institutions. The usage of MESSy and access to the source code is licenced to all affiliates of institutions which are members of the MESSy Consortium. Institutions can become a member of the MESSy Consortium by signing the MESSy Memorandum of Understanding. More information can be found on the MESSy Consortium website (

The supplement related to this article is available online at:

FC, FY, VG, and HY developed the concepts presented in this paper. FC and HY implemented the decision-making module in the AirTraf submodel. FC performed the simulations and analysed the results presented in this paper. SM and SD provided the aCCF formulas and factors employed in this study. BL, FL, and MMM provided the air traffic sample employed in this study. All co-authors contributed to the discussion and the revision of the paper.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The computing resources to conduct simulations with the ECHAM/MESSy Atmospheric Chemistry (EMAC) model were provided by the TU Delft high-performance cluster (HPC12).

This research has been supported by the EU H2020 European Research Council (grant no. 891317).

This paper was edited by David Ham and reviewed by two anonymous referees.