the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Decisionmaking strategies implemented in SolFinder 1.0 to identify ecoefficient aircraft trajectories: application study in AirTraf 3.0
Feijia Yin
Volker Grewe
Hiroshi Yamashita
Sigrun Matthes
Simone Dietmüller
Sabine Baumann
Manuel Soler
Abolfazl Simorgh
Maximilian Mendiguchia Meuser
Florian Linke
Benjamin Lührs
The optimization of aircraft trajectories involves balancing operating costs and climate impact, which are often conflicting objectives. To achieve compromised optimal solutions, higherlevel information such as preferences of decisionmakers must be taken into account. This paper introduces the SolFinder 1.0 module, a decisionmaking tool designed to identify ecoefficient aircraft trajectories, which allow for the reduction of the flight's climate impact with limited cost penalties compared to costoptimal solutions. SolFinder 1.0 offers flexible decisionmaking options that allow users to select tradeoffs between different objective functions, including fuel use, flight time, NO_{x} emissions, contrail distance, and climate impact. The module is included in the AirTraf 3.0 submodel, which optimizes trajectories under atmospheric conditions simulated by the ECHAM/MESSy Atmospheric Chemistry model. This paper focuses on the ability of the module to identify ecoefficient trajectories while solving a biobjective optimization problem that minimizes climate impact and operating costs. SolFinder 1.0 enables users to explore trajectory properties at varying locations of the Pareto fronts without prior knowledge of the problem results and to identify solutions that limit the cost of reducing the climate impact of a single flight.
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Aviation is estimated to contribute 3 %–5 % of total anthropogenic global warming (Lee et al., 2021). Aircraft emissions affect the radiative budget of the Earth through several effects, which are usually distinguished between carbon dioxide (CO_{2}) and nonCO_{2} effects (Lee et al., 2010). NonCO_{2} effects account for about $\mathrm{2}/\mathrm{3}$ of the aviation net effective radiative forcing (Lee et al., 2021) and include, among others, the radiative forcing from contrail cirrus (Schumann, 2005; Kärcher, 2018) and the perturbations in the atmospheric concentrations of ozone (O_{3}) and methane (CH_{4}) caused by nitrogen oxide (NO_{x}) emissions (Stevenson et al., 2004; Köhler et al., 2008). The temperature perturbation resulting from CO_{2} emissions is only dependent on the amount of emitted CO_{2}, due to the long atmospheric lifetime of CO_{2}. To reduce CO_{2} emissions, several solutions are currently under development, exploring, e.g. the use of new propulsion technologies or alternative aviation fuels (Staples et al., 2018; Yin and Rao, 2020). Contrarily, nonCO_{2} effects occur over short timescales, which typically range from hours (e.g. contrails) to months or years (e.g. NO_{x}induced changes on O_{3} and CH_{4}). As a consequence, the temperature perturbation caused by an aircraft unit emission is highly dependent on the time and location of the emission (Köhler et al., 2013; Frömming et al., 2021). Many studies investigated the possibility of using this time and space dependency to reduce the climate effect of a flight, for example, by optimizing its trajectory to minimize the induced temperature increase (Stevenson and Derwent, 2009; Sridhar et al., 2011; Grewe et al., 2017; Matthes et al., 2021). Towards the implementation of this type of operational strategy, some recent studies highlighted the main challenges and opportunities related to the avoidance of climatesensitive regions by aircraft trajectory optimization (Simorgh et al., 2022; Molloy et al., 2022). Most importantly, the current level of scientific understanding of the nonCO_{2} effects of aviation is lower than the one of CO_{2} effects, as demonstrated by the uncertainty ranges of the radiative forcing estimates reported by Lee et al. (2021). Moreover, the identification of climatesensitive regions (e.g. icesupersaturated regions, supporting persistent contrails) relies on the availability of accurate weather forecast. Depending on the stability of the forecast, trajectories can be optimized via tactical adjustments during the flight or, preferably, in advance (e.g. 1 d before departure) to limit the associated penalties in operating costs, e.g. minimizing fuel use and workload of flight crew and air traffic controllers (Molloy et al., 2022).
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 (Yamashita et al., 2021). This implies that, in most cases, it is not possible to identify a solution that simultaneously minimizes both objectives. Therefore, optimizing an aircraft trajectory with respect to its economic cost and climate impact, a set of Paretooptimal solutions can be identified. To select a single trajectory among this set of optimal solutions, a wide range of decisionmaking strategies can be employed (Kou et al., 2012; Pasman et al., 2022). In this paper, we present a new decisionmaking tool, the SolFinder 1.0 module, which was developed with the aim of identifying aircraft trajectories leading to a significant reduction of the flight climate impact, while limiting the increase in economic costs: we define these options as ecoefficient aircraft trajectories. This tool satisfies the following requirements:

it is applicable to any set of Paretooptimal 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 biobjective optimization of operating cost and climate impact, it is capable of identifying ecoefficient solutions.
To satisfy these requirements, the following options have been selected and are available in the first version of SolFinder: (1) a strategy relying on the VIKOR method (abbreviation from its Serbian name: Vlse Kriterijumska Optimizacija Kompromisno Resenje, presented by Opricovic and Tzeng, 2004) to identify ecoefficient solutions; (2) a strategy selecting the Paretooptimal solution closest to a target percentage change in one of the objectives, such as the economic costs; and (3) a decisionmaking method which combines the previous two options, applying the VIKOR method while limiting the change in one of the objectives. The SolFinder module has been coupled to the ECHAM/MESSy Atmospheric Chemistry (EMAC; Jöckel et al., 2010) submodel AirTraf (Yamashita et al., 2020), as part of the AirTraf extension for the efficient resolution of multiobjective optimization problems. This modelling chain enables users to select Paretooptimal solutions matching specific preferences of decisionmakers, e.g. ecoefficient aircraft trajectories, and to explore their dependency on the atmospheric natural variability.
In Sect. 2, we describe the modelling chain, and we present the decisionmaking strategies included in SolFinder 1.0 (Sect. 2.3). In Sect. 3, we illustrate an example application of the selected decisionmaking strategies, using the Paretooptimal solutions that are identified by AirTraf when a European air traffic sample of 100 nighttime flights is optimized with respect to economic cost and climate impact. In Sect. 4, we compare our results to those obtained in previous studies, and we discuss uncertainties affecting our results. Our key messages are summarized in Sect. 5.
We conduct our simulations using the ECHAM/MESSy Atmospheric Chemistry (EMAC) model (Jöckel et al., 2010). This is a numerical climate model system that includes submodels describing tropospheric and middle atmosphere processes and their interaction with oceans, land, and human influences (Jöckel et al., 2010). This system relies on the second version of the Modular Earth Submodel System (MESSy2) to connect multiinstitutional computer codes, while the core atmospheric model is the fifthgeneration European Centre Hamburg general circulation model (ECHAM5, Roeckner et al., 2006). Figure 1 illustrates the relation between the EMAC model and the three submodels that have a major relevance in our experiments: CONTRAIL (Frömming et al., 2014), ACCF (Yin et al., 2023), and AirTraf (Yamashita et al., 2020).
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 (Burkhardt et al., 2008). The ACCF submodel employs the algorithmic climate change functions (aCCFs) in order to deliver the estimated spatially and temporally resolved climate effect of aviation emissions to AirTraf; lastly, the AirTraf submodel identifies the optimal aircraft trajectories with respect to the routing strategy selected by the user. The optimization process is performed in two steps: (1) a genetic algorithm – Adaptive Range MultiObjective Genetic Algorithms (ARMOGA; Sasaki and Obayashi, 2005) – is employed to solve a single or multiobjective optimization problem; (2) if a multiobjective optimization is solved, it is possible that more than one optimal solution is found, thus a decisionmaking module (SolFinder) intervenes to select a single recommended trajectory, based on the preferences of the user.
2.1 Base model configuration
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 ERAInterim reanalysis data (Dee et al., 2011; Jöckel et al., 2016).
2.2 AirTraf submodel
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 (Yamashita et al., 2016). Once this information is collected, the genetic algorithm (ARMOGA; Sasaki and Obayashi, 2005) intervenes to identify an optimal trajectory. The number of design variables of the optimization problem is fixed to 11, since the model describes each aircraft trajectory as a Bspline curve defined by three control points on the horizontal domain (three pairs of coordinates) and five on the vertical crosssection (as illustrated in Yamashita et al., 2016, Fig. 6). The domains of the horizontal control points are centred on the great circle connecting the airports of departure and arrival, while the vertical control points are bounded by the flight levels at 29 000 ft (FL290) and at 41 000 ft (FL410), corresponding to altitudes of about 8.8–12.5 km. To calculate the flight properties along a candidate trajectory, the path is divided into flight segments by n_{wp}=101 waypoints. In particular, to calculate the fuel used at each flight segment, AirTraf uses the aircraft performance model of Eurocontrol's Base of Aircraft Data (BADA Revision 3.9; Eurocontrol, 2011) and the Deutsches Zentrum für Luft und Raumfahrt (DLR) fuel ﬂow method (Yamashita et al., 2016).
The version AirTraf 2.0 presented by Yamashita et al. (2020) allowed the user to solve singleobjective optimization problems, minimizing one of several available objective functions, including fuel use, flight time, NO_{x} emissions, contrail distance, operating costs, and climate impact. The submodel is being expanded to use the same optimization method for the efficient resolution of multiobjective optimization problems. As a result, it is possible to simultaneously optimize two or more objective functions, without combining such functions into a single objective. This is particularly convenient when we aim to identify ecoefficient aircraft trajectories under a large number of weather conditions. In fact, a conversion factor would be necessary to combine climate impact and operating costs into a single objective functions. However, this factor (e.g. as defined by Simorgh et al., 2023, Eq. 7) is affected by a large variability over different flights and days; thus, it is difficult to find a general value of the factor which can be applicable to optimize large air traffic samples over longterm simulations. Due to this AirTraf expansion for the resolution of multiobjective optimization problems, a decisionmaking module became necessary to ensure that a single optimal solution is identified, and its properties are evaluated and stored by the model. This task can be performed by the SolFinder 1.0 module presented in this paper. Therefore, SolFinder has been coupled to AirTraf to select one optimal solution from the full Pareto set, reducing the computational effort of the model.
In the development of the decisionmaking strategies implemented in SolFinder 1.0, the underlying goal has been to find ecoefficient 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. (1):
where TIME_{i} and FUEL_{i} represent the flight time and fuel used at the ith flight segment, respectively, while c_{t} = USD 0.75 per second and c_{f}=USD 0.51 per kilogram are the unit time and unit fuel costs (Burris, 2015; Yamashita et al., 2020). The climate impact of each aircraft trajectory is measured in terms of average temperature response (ATR) over 20 years (ATR20), as provided by the ACCF submodel (van Manen and Grewe, 2019; Yin et al., 2023). The total climate impact ATR20_{tot} of each aircraft trajectory is determined by summing the contribution from the main climate effects:
where each addend represents, from left to right, the ATR20 at the ith flight segment from (1) carbon dioxide (CO_{2}), (2) water vapour (H_{2}O), (3) ozone (O_{3}) from emission of NO_{x}, (4) methane (CH_{4}) from emission of NO_{x}, and (5) contrails. The term ${\text{ATR20}}_{{\text{NO}}_{\text{x}}{\text{CH}}_{\mathrm{4}},i}$ includes the changes in primary mode ozone induced by the reduced CH_{4} atmospheric concentration, while it neglects the feedback from stratospheric water vapour (Yin et al., 2023). The present study uses ATR20 as climate metric, assuming a businessasusual future emission scenario (FATR20). Alternative climate metrics can also be used, e.g. considering a time horizon of 100 years (ATR100). A detailed description of the climate metric conversion is presented in Dietmüller et al. (2023).
2.3 SolFinder module
In this section, we describe the decisionmaking strategies implemented in SolFinder 1.0. Our aim is to solve a multiobjective optimization problem by minimizing a set of N objective functions f_{n}, with $n=\mathrm{1},\mathrm{2},\mathrm{\dots},N$. If two or more objective functions are conflicting, a set of Paretooptimal solutions, P, is identified. The values f_{n,j} of the objective functions are assigned to each Paretooptimal solution p_{j}∈P , with $j=\mathrm{1},\mathrm{2},\mathrm{\dots},J$. Subsequently, a decisionmaking strategy intervenes to select one solution p_{rec}∈P which is the recommended solution according to the decisionmaker criteria. We include the following strategies towards the resolution of our problem: (1) option selecting a solution based on its weighted distance from an ideal (usually, not feasible) solution (VIKOR method, Appendix A) for the identification of ecoefficient trajectories (Sect. 2.3.1); (2) option leading to a target percentage change in one of the objective functions, with respect to its minimum value (Sect. 2.3.2); (3) option combining the previous two strategies, limiting the change in one of the objective functions while applying the VIKOR method (Sect. 2.3.3); and (4) selection of one of the extremes of the Paretooptimal set (Sect. 2.3.4).
A large variety of multicriteria decisionmaking methods is currently available to select one solution among a set of optimal options (Wang and Rangaiah, 2017; Sałabun et al., 2020). In a preliminary phase of our research, we considered different options among the most popular techniques, including GRA (grey relational analysis; Wang and Rangaiah, 2017), TOPSIS (Technique for Order Preference by Similarity to Ideal Solution; Chen and Hwang, 1992), and VIKOR (abbreviation from its Serbian name: Vlse Kriterijumska Optimizacija Kompromisno Resenje, Appendix A). These options have been implemented in a Python library,^{1} in order to apply them on test cases and compare their effectiveness in identifying ecoefficient aircraft trajectories. We identified the VIKOR method (Opricovic, 1998; Opricovic and Tzeng, 2004) as a suitable candidate to translate our definition of ecoefficient solutions into a decisionmaking algorithm. This is due to its peculiarity of recommending more than one solution if certain criteria are not met by a single solution (Appendix A). This allows for flexible identification of a region of the Pareto front (e.g. the section of the Pareto front leading to a small change in the economic costs), within which the user is able to choose their preferred solution (e.g. the solution with the largest climate impact reduction). Moreover, no a priori knowledge on the optimization problem expected results is needed. To apply the VIKOR method, only the following information is provided as input, translating the user preferences into mathematical terms:

the relative importance of N objective functions, represented by the weights w_{n}≥0, such that ${\sum}_{n}{w}_{n}=\mathrm{1}$, with $n=\mathrm{1},\mathrm{2},\mathrm{\dots},N$.

the relative importance of group utility (preference towards achieving the greatest benefit) and individual regret (preference towards avoiding large penalties), represented by the parameter γ, with $\mathrm{0}<\mathit{\gamma}<\mathrm{1}$. If γ>0.5, the majority rule principle is applied; contrarily, γ<0.5 implies the application of the veto principle; lastly, γ≈0.5 represents a voting by consensus strategy (Opricovic and Tzeng, 2004).
2.3.1 Strategy using the VIKOR method to identify ecoefficient trajectories
The SolFinder 1.0 module identifies ecoefficient trajectories using a decisionmaking option based on the VIKOR method. This method, introduced by Opricovic (1998) and Opricovic and Tzeng (2004), makes use of the overall distance (group utility) and the maximum distance (individual regret) of a Paretooptimal solution from the minimal values of the optimization objectives to rank the Paretooptimal solutions. For more details on the version of the VIKOR method implemented in SolFinder, we refer to Appendix A. Figure 2 illustrates how this decisionmaking strategy is applied using as an example the biobjective optimization of a flight with respect to its SOC and ATR20. Firstly, the decisionmaking module collects all the Paretooptimal solutions (represented by the grey crosses in Fig. 2a). Subsequently, the VIKOR method is applied according to the prescribed values of the parameter γ and the relative weights w_{n} (Fig. 2b). This leads to a recommended subset of optimal solutions (represented by the blue circles in Fig. 2c). If the VIKOR method identifies more than one recommended solution (i.e. the solutions ${p}_{v}\phantom{\rule{0.25em}{0ex}}(v=\mathrm{1},\mathrm{2},\mathrm{\dots},M)$ are equally recommended), the model selects the one with the minimum value of the objective function assigned to the lowest weight w_{n}. In Fig. 2d, the objective with the lowest relative weight is ATR20 (${w}_{\text{ATR20}}=\mathrm{0.3},{w}_{\text{SOC}}=\mathrm{0.7}$); thus, the model selects the point among the recommended solutions (indicated by the blue circles in Fig. 2d) with the lowest ATR20 (red triangle in Fig. 2d). This last step is thus formulated to translate in mathematical terms our definition of ecoefficient aircraft trajectories, i.e. a compromise solution between costoptimal and climateoptimal solutions, such that the largest possible climate impact reduction is achieved, while keeping the operating costs nearly unchanged with respect to the costoptimal solution. Using the VIKOR method, a subset of Paretooptimal solutions is identified, according to the relative importance of the two optimization objectives. Therefore, if the highest weight is assigned to the objective function representing operating costs, the VIKOR method equally recommends a subset of Paretooptimal solutions close to – or, possibly, including – the costoptimal extreme point of the Pareto front. Among this subset of equally recommended solutions, we choose the point leading to the largest climate impact reduction, i.e. the minimum value of the objective function assigned to the lowest weight. Therefore, the objective with the highest weight plays a dominant role in the selection of the subset of equally recommended solutions (VIKOR method), while the objective with the lowest weight becomes dominant in the selection of a single solution among this subset.
The resulting strategy can be configured to follow the steps listed here:

A biobjective optimization problem is solved to simultaneously minimize the total climate impact (ATR20_{tot} as defined in Eq. 2) and operating costs (SOC, Eq. 1). This step results in the identification of J Paretooptimal solutions (Fig. 2a).

The VIKOR method is applied, following the steps described in Appendix A (Fig. 2b).

A set of equally recommended solutions are selected in the sections in the Pareto front closest to the costoptimal solution by assigning a relatively high weight to the operating costs, i.e. w_{SOC}>0.5 (with γ=0.5). Depending on the shape of the Pareto front, this set of solutions extends towards the best ideal solution, i_{best}, allowing for higher climate impact reductions with respect to the costoptimal solution while avoiding cost penalties that are not compensated by a climate impact reduction (Fig. 2c).

Among this set of equally recommended solutions, the solution leading to the largest climate impact reduction with respect to the costoptimal solution is selected, since w_{ATR}<w_{SOC}. We define this point as the ecoefficient solution among the set of Paretooptimal options (Fig. 2d).
Sensitivity of the VIKOR parameterization
To understand the effectiveness of the VIKOR method with various configurations of γ and w, we take as example a set of Paretooptimal solutions, resulting from the biobjective optimization of an aircraft trajectory with respect to the SOC and ATR20 of the flight. Within this set, a subset of solutions is recommended by the VIKOR method with different values of γ and w=[w_{SOC}, w_{ATR20}]. The results of this analysis are shown in Fig. 3. This figure illustrates the impact of varying the weights w by comparing different rows. It is possible to see how by increasing the value of w_{SOC} from 0.2 (Fig. 3a, b, c) to 0.8 (Fig. 3j, k, l), the set of solutions recommended by VIKOR moves closer to the costoptimal extreme of the Pareto front (0 % change in SOC and ATR20). It is less intuitive to understand the impact of changing the parameter γ, which represents the relative importance of group utility. As explained in Sect. 2.3 and according to the formulas included in Appendix A, with γ=0.5 the same relative importance is assigned to avoiding large penalties in one of the objectives and to achieving the greatest overall benefit. In the results presented in Sect. 3.2, we always set the default value γ=0.5. A value of γ<0.5 leads to the application of the veto principle; that is, if one of the objectives is heavily penalized by selecting a certain Paretooptimal solution, then it will have a low likelihood to be recommended. Therefore, setting γ=0.25 (as in Fig. 3a, d, g, j) leads to the exclusion of elements located in the external sections of the Pareto front, because of their distance to the opposite extreme of the Pareto front. On the other hand, when the veto principle is not applied (γ≥0.5), the set of recommended solutions (blue circles) can include the solution minimizing the objective with the highest relative weight. For example, when w_{SOC}=0.8 and γ≥0.5, the solution with minimum SOC is included in the set of Paretooptimal solutions (Fig. 3k, l).
This observation is the base for the definition of ecoefficient solution given at the end of Sect. 2.3.1, i.e. the solution selected with a high relative weight of SOC (w_{SOC} larger than 0.5) and γ=0.5. Following this definition, the selected solution shown in Fig. 3k (represented by a red triangle) is an ecoefficient solution. In fact, this solution leads to the maximum ATR20_{tot} reduction among the subset of the solutions recommended by VIKOR (blue circles) while being equally recommended as the costoptimal solution by the VIKOR method. In other words, the selected solution in Fig. 3k follows the definition of the ecoefficient solutions given in the Introduction, i.e. it allows for a significant reduction in the flight climate impact while avoiding significant increases in operating costs. However, we note that determining the values of γ and w remains an arbitrary choice, which reflects the specific preferences of the decisionmaker. Further elements to consider when setting these parameters are discussed later in Sect. 3.
2.3.2 Target percentage change in one of the objective functions
In some scenarios, the decisionmaker 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 decisionmaking strategy, we propose to select the solution p_{rec} leading to a percentage change in one of the objective functions, f_{p}. On the other hand, to obtain the expected outcome, this option requires certain information on the optimization results to be known before solving the problem, such as the typical shape of the Pareto fronts and reference values at their extreme points. The threshold for the allowable change is specified as a target value x_{t}. This is translated in the following process:
Therefore, the selected Paretooptimal solution p_{rec}∈P is the solution corresponding to the relative change x_{rec}, such that $\mathrm{\Delta}{x}_{\text{rec}}={min}_{j}\mathrm{\Delta}{x}_{j}$.
Figure 4 illustrates which solution is selected within a Paretooptimal set resulting from the resolution of a triobjective optimization problem, simultaneously minimizing flight time, fuel use, and ATR20_{tot}. In this example, ${x}_{\mathrm{t}}=+\mathrm{0.5}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$ is set as the target percentage change in flight time.
2.3.3 Hybrid option: the VIKOR method with target percentage change in one of the objectives
To combine the advantages of the two decisionmaking 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 decisionmaker provides (1) the configuration of the VIKOR method, setting the parameters γ and w, and (2) a target percentage increase x_{t} in the objective f_{p}. Subsequently, the following decisionmaking process is followed:

Apply the VIKOR method and select the recommended solution minimizing the objective function having the lowest weight w_{n}, as described in Appendix A.

Calculate x_{p}, representing the relative change in f_{p} of the recommended solution with respect to the minimum value of f_{p}.

If x_{p}>x_{t}, replace the recommended solution identified in step 1 with the Paretooptimal solution leading to a target percentage change x_{t} in the objective function f_{p} (in other words, apply the strategy described in Sect. 2.3.2).
This strategy addresses the fact that, with the VIKOR method, no limit in the percentage increase of the objective is set; thus, a fraction of the solutions can be affected by changes much larger than the average, as is shown in a later section of this paper (Sect. 3.2, Fig. 7). With this hybrid option, the VIKOR method can be employed to identify ecoefficient trajectories, while introducing a constraint on the operating costs, to prevent increases in the operating costs of some flights that the decisionmaker does not accept.
2.3.4 Selecting one extreme of the Paretooptimal set
Lastly, an additional option is considered to select a solution minimizing one of the objective functions f_{n}, which we indicate as f_{min}. This simple decisionmaking process selects the optimal solution corresponding to the result of a singleobjective optimization minimizing f_{min}. Nevertheless, it can be useful to implement this method to identify which values are used as reference during the resolution of the multiobjective optimization problem and to verify the performance of the model.
We now present an example study, in which different settings of the decisionmaking strategies are compared. This application exemplifies how the decisionmaking strategies can be employed and what to consider to determine the settings that best translate the decisionmaker preference. In this example, we focus on the suitable settings to identify ecoefficient aircraft trajectories. Nevertheless, SolFinder can also be used to comply with alternative decisionmaking preferences, by changing the settings of AirTraf and SolFinder.
3.1 Simulations setup
As previously stated, we intend to identify ecoefficient aircraft trajectories, i.e. trajectories reducing the climate impact with limited changes in the operating costs. Therefore, we solve a biobjective optimization problem, aiming to simultaneously minimize SOC and ATR20, as defined in Sect. 2.2. We conduct 1month simulations, from 1 January 2018 to 31 January 2018. On each simulation day, 100 nighttime flights departing at 00:00 UTC are optimized. Figure 5 shows the locations of the airports of origin and destination, which were selected considering the available seat kilometres (ASK) for the European Civil Aviation Conference (ECAC) area in 2018 (Meuser et al., 2022). The same criterion was used to select the A320214 (CFM565B4) as the aircraft/engine type to be simulated. The climate impact of each aircraft trajectory is estimated using the aCCF 1.0A (Matthes et al., 2023). Table 1 summarizes the model configuration.
To compare the effects of using different decisionmaking strategies, we perform two sets of experiments, whose characteristics are summarized in Table 2 and explained below:

In the first set of experiments, the VIKOR method is employed as we described in Sect. 2.3.1, fixing γ=0.5 and varying the relative weight w_{SOC} between 0.2 and 0.9 (Table 2). Therefore, we obtain a total of six selected tradeoff solutions. The density of w_{SOC} values increases for higher values of w_{SOC}, as these values lead to the selections of solutions in the section of the Pareto front of major interest when searching for ecoefficient trajectories, i.e. the section closer to the costoptimal solution. Additionally, we select the two solutions located at the Paretofront extremes, i.e. the costoptimal and climateoptimal solutions, with the routine mentioned in Sect. 2.3.4. No preliminary knowledge about the expected problem results is needed to conduct these simulations. As a result of this first set of experiments, for each optimized flight we obtain information on eight solutions that determine the shape and extension of the Pareto fronts relative to individual flights, and the relation between penalties and benefit aggregated over the whole air traffic sample.

In the second set of experiments, we explore the effects of selecting a solution leading to a target change in SOC (Sect. 2.3.2). The target values are chosen using the results form the first set of experiments, which obtained average SOC increases up to about +3.0 % (see Table 3 and Fig. 6). In light of these results, we vary x_{t} from 0.5 % to 3.0 % (Table 2), selecting four solutions per flight. Moreover, we set w_{SOC}=0.7 to exemplify the effects of using the option of using VIKOR while setting a target SOC change (Sect. 2.3.3) of +1.0 %.
3.2 Results
As explained in Sect. 2.3, one of the main advantages of the VIKOR method is that no knowledge of the expected results is needed before solving our problem. For this reason, the first set of experiments applies the VIKOR method with a range of values of the parameter w_{SOC}. This enables us to perform a preliminary examination of the characteristics of the relation between benefit (in our application, ATR20_{tot} reduction) and penalty (increase in SOC). Relative to our problem, this relation is shown in Fig. 6, illustrating the change in SOC that is required to achieve a certain reduction in ATR20. The grey curves in Fig. 6a are obtained by aggregating the 100 Paretooptimal solutions selected using the VIKOR method on each simulation day and varying the relative weight w_{SOC} of the objective function SOC. The coordinates of the points connected by the black line (representing the average relation over the 31 simulated days) can be found in Table 3. Setting a low value of w_{SOC} (e.g. w_{SOC}=0.2), we obtain a reduction in ATR20_{tot} of about 13 %, which is almost as large as the maximum potential reduction achieved by selecting a climateoptimal routing strategy. Moving along the curve towards the costoptimal strategy, the magnitude of the ATR20_{tot} reduction decreases. This occurs with a simultaneous decrease of the cost per kelvin reduction in climate impact, measured by the climatecost coefficient k [USD per kelvin] (Table 3). This coefficient is defined as (Matthes et al., 2017)
where ATR20_{costopt} and SOC_{costopt} are relative to the costoptimal solution, while ATR20 and SOC are relative to the considered decisionmaking strategy. From this first set of experiments, we understand that (1) the average ΔATR20_{tot} ranges from −3.5 to −14.4 %, with an increase in SOC of 0.1 %–3.0 %, respectively; (2) about half of the maximum feasible climate change reduction can be achieved with only +0.5 % in SOC.
As previously mentioned, the red and green points highlighted in Fig. 6a result from summing all the solutions selected for the 100 flights and averaging over the month of simulation. However, each optimized trajectory is characterized by a different change in ATR20_{tot} and SOC with respect to its corresponding costoptimal solution. We note here that this costoptimal solution (shown in red in Fig. 6a), which we take as reference to calculate the relative changes in ATR20_{tot} and SOC, is specific to each route and each simulation day; thus, it varies between different flights. The full variability of the relationship between relative changes in ATR20_{tot} and SOC is illustrated in Fig. 6b. In this panel, one can see that specific solutions can show large deviations from the average values, as some solutions reach an absolute ΔATR20_{tot} larger than 60 % at penalties lower than 2.5 % in terms of ΔSOC. Moreover, Fig. 6c–f show that the subsets of Paretooptimal solutions obtained setting different w_{SOC} behave as expected: moving from lower to higher values of w_{SOC} leads to subsets that are confined below lower increases in SOC, while they still stretch towards high ATR20_{tot} reductions (e.g. see the points selected with w_{SOC}=0.8 in Fig. 6f). The distributions of ΔATR20_{tot} and ΔSOC values across the optimized flights are represented in Fig. 7, which compares the results obtained by varying the parameter w_{SOC} applying the VIKOR method and the climateoptimal scenario. Figure 7b shows the percentage of flights characterized by a certain ΔSOC. One can see that, employing VIKOR, the mode of each curve is close to ΔSOC ∼ 0 %, while the mean and maximum ΔSOC values increase when the weight w_{SOC} decreases. Looking at the ΔATR20_{tot} distributions in Fig. 7a, we can observe larger differences also in the modal values for varying w_{SOC}.
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. 3.1. The result of this second set of experiments are shown in Table 4. To illustrate the difference of employing different decisionmaking methods, it is useful to compare Fig. 7 with Fig. 8. The most evident difference emerges comparing Fig. 7b with Fig. 8b. As intended, when we set a target increase in SOC (e.g. +1.0 %), most of the selected flights are affected by an increase in SOC equal or similar to the target; thus, the curves are centred on this ΔSOC value. However, we can also deduce that some Pareto fronts do not extend to the target ΔSOC, since values of ΔSOC which are lower than the target are assigned to a fraction of the flights. This results in average ΔSOC values that are lower than the targets, as we can see in Table 4. For example, a total increase of 0.8 % in SOC is obtained, when the target is set to 1.0 %. Moreover, Fig. 8b shows what distribution of ΔSOC the user can expect employing the hybrid method described in Sect. 2.3.3. The histogram relative to the hybrid option (yellow, with hatching) corresponds to the one obtained with VIKOR (with w_{SOC}=0.7), with an additional peak at the determined target SOC increase (1.0 %), replacing the larger SOC increases observed in Fig. 7b. The results illustrated in Figs. 7b and 8b confirm that the strategies available in SolFinder 1.0 are correctly implemented and lead to the expected selection of Paretooptimal solutions. Moreover, the possibility of relying on the VIKOR method to identify ecoefficient trajectories is confirmed by Fig. 9. These curves represent the distributions of the climatecost coefficient k [USD per kelvin], obtained using VIKOR (blue curves) or the target SOC change (red curves). It is possible to observe that lower values of k are obtained using VIKOR, in particular when a relatively high weight is assigned to SOC.
Employing different decisionmaking strategies, we obtain trajectories which are characterized by different properties. How these properties vary is shown in Fig. 10, which compares the mean flight altitudes and flown distances obtained when different decisionmaking strategies are selected. Firstly, we can notice that costoptimal flights are characterized by the highest mean flight altitudes and the shortest trajectories among the solutions considered, due to the presence of fuel consumption in the optimization objective. On the opposite situation, the lowest altitude and the longest distances are obtained for climateoptimized flights. This confirms previous studies, which commented that (1) aircraft emissions have lower climate impact at lower altitudes, e.g. due to shorter residence time of emitted species (Castino et al., 2021; Matthes et al., 2021; Frömming et al., 2012), and (2) lateral deviations may be necessary to avoid climatesensitive regions (Matthes et al., 2020). Moreover, the variability of the flight properties across the set of simulated flights is larger for climateoptimized flights than for costoptimal ones. For example, comparing the different variability in flight altitudes, we see the impact of the lowest aerodynamic drag, which allows for minimal fuel use, is systematically found at higher altitudes. Contrarily, the altitude leading to the minimal ATR20 is highly variable, due to the high temporal and spatial variability of the atmospheric conditions determining the net flight ATR20. Tradeoff solutions between these two extreme scenarios show intermediate properties, with median values and interval bars which gradually evolve, moving from one extreme of the curve in Fig. 6 to the other. We can also see that employing the VIKOR method rather than fixing a target increase in costs can lead to different tendencies in the average characteristics of the selected trajectories. For example, comparing the VIKOR method with w=0.4 (causing ΔSOC $\sim +\mathrm{1.2}$ %, see Table 3) and setting a target +1.0 % change in SOC, we can see that the latter strategy leads to a less frequent selection of lower flight altitudes, while flying longer trajectories, than the former case. Therefore, the user is recommended to identify their preferred decisionmaking strategy by also considering these secondary aspects and not exclusively the resulting distributions in objective function values.
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_{2}, H_{2}O, contrails, NO_{x} via perturbation of ozone, and NO_{x} via methane depletion. The relative importance of these effects under different decisionmaking strategies is illustrated in Fig. 11. Firstly, we can notice that the ATR20 from CO_{2} emissions increases moving from costoptimal to climateoptimal and compromise solutions. This increase is expected, as the climate impact of CO_{2} is independent of the background atmospheric conditions at time and location of emission; thus it is simply proportional to the amount of fuel used, which is minimized by costoptimal flights. On the other hand, the increase in the ATR20 from CO_{2} is largely compensated by the reduction in the ATR20 from contrails and, secondarily, from the impact of NO_{x} on ozone. Moreover, Fig. 11 shows that reduction in climate impact of contrails becomes increasingly dominant over the reduction of the other effects, when the relative weight w_{SOC} of SOC increases. This aspect should be considered when selecting the preferred settings of the decisionmaking strategy. For example, if the decisionmaker is interested in reducing both the climate impact of contrails and NO_{x}ozone via trajectory optimization, this goal can be achieved, allowing larger changes in SOC than those needed to only reduce the ATR20 from contrails effects. The dominant contributions of contrails and, secondarily, of the NO_{x} climate impact to the ATR20 reduction is confirmed by the results obtained by optimizing aircraft trajectories with different tools (Lührs et al., 2021; Simorgh et al., 2022). However, large uncertainties affect the estimates of nonCO_{2} effects of aviation (see Sects. 1 and 4), which can heavily affect the relative importance of the different effects of aviation emissions. Therefore, it is relevant to update the modelling chain presented here when enhancements in the scientific understanding and in our ability to model nonCO_{2} effects are available, e.g., via updated aCCF versions.
In this paper, we illustrated the decisionmaking strategies implemented in the SolFinder 1.0 module and how they can be used to identify ecoefficient 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, Grewe et al. (2017) optimized a set of transatlantic flights during eight representative weather patterns in winter and summer, simulating the air traffic with the system for traffic assignment and analysis at a macroscopic level (SAAM) and using the climate change functions (CCFs). That study found that a 10 % reduction in climate impact can be achieved with a 1.0 % increase in operating costs. In our study, we considered a different region of the airspace (European flights), and we included a larger number of weather patterns (every day in the month of January 2018) but limited to the winter season and to nighttime; nevertheless, we found the results in Grewe et al. (2017) to be aligned with those we presented in Sect. 3, since a $\mathrm{\Delta}\text{SOC}\sim +\mathrm{1.2}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$ corresponds to a $\mathrm{\Delta}\text{ATR20}\sim \mathrm{10.3}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$ (Table 3). The maximum feasible climate impact reduction is higher (about 20 %) in Grewe et al. (2017), but this is achieved with an approximately double increase in costs. The discrepancy in the section of the Pareto front closer to the climateoptimal solution may be due to the different temporal and spatial domains. The mitigation potential of optimized aircraft trajectories is expected to be affected by seasonal and daily variability (e.g. Reutter et al., 2020; Castino et al., 2021); thus, the methodology presented in this paper can be extended to cover multiple years to investigate this aspect. Under specific weather conditions, cost–benefit relations were found to show higher ecoefficiency than were found aggregating the model output over all the routes and all the simulated days. For example, Lührs et al. (2021) analysed a day characterized by a weather situation with strong contrail formation and found that a 0.75 % increase in fuel used halved the climate impact. This study suggests the possibility of identifying weather situations that allow for a higher ecoefficiency than others. The modelling chain presented in this paper enables us to optimize aircraft trajectories under a large number of different atmospheric conditions, within a feasible computational time. Therefore, ongoing research is using the newly developed version of AirTraf, coupled with the SolFinder module, to analyse under which weather conditions ecoefficient aircraft trajectories are most likely to be identified. To this end, additional decisionmaking strategies are being investigated to exploit the ability of VIKOR to identify lower values of the climatecost coefficient k than a strategy applying a target increase in SOC to all flights (Fig. 9). Two candidate additional strategies are illustrated in Fig. 12. With these options, only a fraction of the flights identified by VIKOR are climateoptimized, due to an additional condition. This is obtained by setting a threshold value of the coefficient k, in order to only optimize the topranked half of the flights (Fig. 12a) or until a certain budget is spent (e.g. +0.5 % of SOC in Fig. 12b). A disadvantage of these decisionmaking strategies is that their setting configuration relies on information on the whole system of optimized flights, which is not available when we consider the Pareto front resulting from a singleflight optimization. Therefore, preliminary simulations have to be run to derive some parameter values, e.g. the specific threshold value of the climatecost coefficient k [USD per kelvin]. Because of the large variability of k, these decisionmaking strategies are more reliant on the results obtained from previous simulations than the strategies included in SolFinder 1.0.
The present work estimates the climate effect of aviation resulting from the emission of CO_{2}, H_{2}O, NO_{x}, and from the formation of contrail cirrus. Estimating the radiative forcing caused by nonCO_{2} effects is a complex process, leading to results that are affected by large uncertainties due to, e.g. incomplete scientific understanding and modelling capabilities (Lee et al., 2021). As described in Sect. 3.1, we use the aCCFs version 1.0A to estimate the climate impact of aviation (Matthes et al., 2023), which are calibrated towards the results of a climate response model (AirClim; Dahlmann et al., 2016) to align the relative importance of individual aCCFs. This is an update of the consistent set of aCCF 1.0 (Yin et al., 2023), calculating the climate impact of CO_{2}, H_{2}O, NO_{x}ozone, NO_{x}methane, and contrails in terms of ATR20, assuming pulse emissions (PATR20). This prototype set of functions is the focus of ongoing research to address their sources of uncertainties. Moreover, we employed factors to (1) convert the aCCFs values to a different climate metric, FATR20, which assumes a businessasusual future emission scenario, and (2) include the efficacy of each climate impact effect (Dietmüller et al., 2023). These assumptions introduce additional sources of uncertainties (Dietmüller et al., 2023). Moreover, the majority of icesupersaturated regions (supporting persistent contrail formation) have characteristic dimensions that are smaller than the horizontal resolution of our model. To take this factor into account, we employ a parameterization developed by Burkhardt et al. (2008) to estimate the fraction of model grid box which is supporting persistent contrails. The potential of reducing the flights climate impact by contrail avoidance could be reduced by factors not included in this study. For example, here we use weather conditions simulated by an atmospheric model (EMAC) rather than weather forecast. In real life applications, this mitigation strategy relies on the accuracy and stability of weather forecast, as discussed in Sect. 1. Lastly, we note that we use a simplified representation of the operating cost to limit the computational time required for their evaluation within each optimization step. To this end, we assume a linear relationship between cost of time and flight time (see Eq. 1 and Yamashita et al., 2020). Therefore, we neglect additional costs caused by delays. Further research will explore the impact of optimizing not only the location of the flight trajectory (see Sect. 2.2), but also the airspeed, improving fuel efficiency compared to this study, which assumed a constant Mach number for all solutions. Other studies found that speed changes can be important for reducing fuel flow, NO_{x} emissions, and, ultimately, NO_{x} climate effects (Simorgh et al., 2023). This improvement of AirTraf is expected to impact the SolFinder results by reducing the penalties in terms of operating costs and CO_{2} emissions for a certain gain in terms of reduction of ATR20. On the other hand, including the airspace structure and capacity, which are neglected in this study, can reduce the estimated climate impact mitigation potential of this operational strategy.
In this study, we described the decisionmaking 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 multiobjective optimization problems. We showed here how the selected decisionmaking strategies can be used to identify solutions matching specific preferences (e.g. ecoefficient 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 biobjective optimization problem by minimizing the climate impact of the aircraft trajectory (FATR20_{tot}) and its simple operating costs (SOC), and we compared the solutions selected by different configurations of SolFinder 1.0. Comparing the strategies using VIKOR and a target change in SOC, we found that lower values of the climatecost coefficient k [USD per kelvin] (i.e. a higher ecoefficiency) are obtained with the former option. The decisionmaking strategies included in SolFinder 1.0 are applied on sets of Paretooptimal solutions relative to a single aircraft trajectory. In the next SolFinder versions, we plan to take into account the mitigation potential variability across all flights. As a result, only the best performing fraction of the flights is optimized with respect to their climate impact, and the cost of the operational mitigation strategy is lowered. Ongoing research is using the modelling chain presented in this paper to identify those weather situations that are allowing for the largest reductions in the temperature response from aviation emissions via the optimization of aircraft trajectories.
In this appendix, we quote the steps characterizing the VIKOR method, as they were introduced and described in Opricovic (1998) and Opricovic and Tzeng (2004). Figure A1 illustrates the main working principles of the VIKOR method. The x axis represents the normalized distance from the minimum value of f_{1}, scaled by its relative weight w_{1}. As an example, this weight is set to w_{1}=0.2; thus, the x axis ranges from 0 to 0.2. Similarly, the y axis represents the normalized distance from the minimum value of f_{2}, weighted using w_{2}=0.8. The axes intersect at the reference point i_{best}, defined as the best ideal (i.e. usually not feasible) solution. Opposite to i_{best}, it is possible to identify i_{worst}, which assumes the worst values of f_{n} (f_{n,worst}) found among the set of Paretooptimal solutions. For example, when aiming at minimizing f_{n}, i_{worst} corresponds to the maximum f_{n} among the Pareto set. The grey points in Fig. A1 indicate the location of the Paretooptimal solutions, relative to i_{best}. In fact, the VIKOR method ranks the Paretooptimal solutions using i_{best} as a reference point. Hence, the first step consists of identifying such a reference point by determining the best values of each of the objectives, f_{n,best}. Since we aim to minimize the objective functions, we use the following definitions of f_{n,best} and f_{n,worst}:
In Fig. A1 we highlight the Paretooptimal solution p_{j}. In the second step of the VIKOR method, two quantities are calculated for each point in the Pareto set: S(p_{j}), which measures the group utility of the solution p_{j}, and R(p_{j}), which represents its individual regret. In other words, S(p_{j}) measures the overall distance of p_{j} from i_{best}, taking into account all the optimization objectives. On the other hand, R(p_{j}) measures the largest distance of p_{j} from i_{best} considering each objective individually. These quantities are defined by Eqs. (4) and (5), respectively:
where w_{n} is the relative weight of each objective f_{n}.
The geometric representation of S(p_{j}) and R(p_{j}) is illustrated in Fig. A1. It is possible to deduce from Eqs. (4) and (5) and from Fig. A1 that lower values of S(p_{j}) and R(p_{j}) are preferable. These measures of the distance of p_{j} from i_{best} are combined in the value Q(p_{j}), which is used as main ranking parameter by the VIKOR method. The value of Q(p_{j}) is calculated using Eq. (6):
The next step consists of creating three ranking lists of the Paretooptimal solutions, sorting them by S, R, and Q. We define p_{i} as the Paretooptimal solution at the ith position in the list sorted by Q. Consequently, the first compromise solution to be recommended is p_{1}, which minimizes Q: $Q\left({p}_{\mathrm{1}}\right)={min}_{j}Q\left({p}_{j}\right)$. The following conditions are then evaluated:

acceptable advantage. $Q\left({p}_{\mathrm{2}}\right)Q\left({p}_{\mathrm{1}}\right)\ge \frac{\mathrm{1}}{J\mathrm{1}}$
If this condition is not verified, a set of Paretooptimal solutions ${p}_{v}\phantom{\rule{0.25em}{0ex}}(v=\mathrm{1},\mathrm{2},\mathrm{\dots},M)$ is recommended, where M is the maximum value for which $Q\left({p}_{M}\right)Q\left({p}_{\mathrm{1}}\right)\le \frac{\mathrm{1}}{J\mathrm{1}}$ is true.

acceptable stability. p_{1} is the best ranked solution not only by Q, but also by S and R.
If this condition is not satisfied, both p_{1} and p_{2} are recommended.
Therefore, the application of the VIKOR method results in the identification of either one optimal solution, p_{1}, or a subset of optimal solutions, ${p}_{v}\phantom{\rule{0.25em}{0ex}}(v=\mathrm{1},\mathrm{2},\mathrm{\dots},M)$, which are recommended to the decisionmaker.
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 (http://www.messyinterface.org, last access: 2 May 2024). The code presented here has been based on MESSy version 2.55.0 and will be available after the official release of AirTraf 3.0, a submodel of MESSy. An open access version of SolFinder (see Supplement) is available from the 4TU.ResearchData repository (https://doi.org/10.4121/3744b922ffa64f92ad57dc93f94e969b, Castino, 2023) under the GNU Lesser General Public License v3.0, as are the scripts to produce the plots presented in this paper. The simulation output analysed in this paper is archived in the 4TU.ResearchData repository (https://doi.org/10.4121/57b969746ec74e869d952b7bf3c18d8f, Castino et al., 2023).
The supplement related to this article is available online at: https://doi.org/10.5194/gmd1740312024supplement.
FC, FY, VG, and HY developed the concepts presented in this paper. FC and HY implemented the decisionmaking 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 coauthors 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 Geoscientific Model Development. The peerreview process was guided by an independent editor, and the authors also have no other competing interests to declare.
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 highperformance 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.
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 Abstract
 Introduction
 Methods
 Application of decisionmaking method to analyse trajectories' variability along the Pareto front
 Discussion
 Conclusions
 Appendix A: VIKOR method
 Appendix B: Air traffic sample
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement
 Abstract
 Introduction
 Methods
 Application of decisionmaking method to analyse trajectories' variability along the Pareto front
 Discussion
 Conclusions
 Appendix A: VIKOR method
 Appendix B: Air traffic sample
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement