A genetic algorithm is used to solve the Centralised Peak-Load Pricing model on the European Air Traffic Management system. The Stackelberg equilibrium is obtained by means of an optimisation problem formulated as a bilevel linear programming model where the Central Planner sets one peak and one off-peak en-route charge and the Airspace Users choose the route among the available alternatives.

1. Modular Multi-Objective Genetic
Algorithm for Large Scale Bi-level Problems
University of Trieste
Stefano Costanzo
Supervisor: L. Castelli
Co-Supervisor: A. Turco
>Abstract
GeneticAlgorithmsaresearchheuristicsthatmimictheprocessofnaturalselection.Theybelongtothelargerclassofevolutionaryalgorithms.Thiskindofoptimizationalgorithmsarewell-suitedforcomplexrealapplication
problems that do not have an exact solution. Furthermore, GAs can be integrated with the Pareto Dominance Criteria to solve multi-objective problems.
The proposed Multi-Objective Genetic Algorithm for Structured Inputs, MOGASI, incorporates modules and operators of both general purpose and custom GA. It is designed for an efficient solving of generic problems and
those with a specific data structure. It has been tested and its efficiency confirmed on different multi-objective optimization problems against other genetic algorithms, well-known in literature. The module-based structure
of this algorithm has been represented in form of a BPMN workflow in the framework of a regional European project, which opens up new possibilites for the definition of engineering and scientific workflows.
The MOGASI algorithm has been successfully used for the solution of a real-world problem, i.e. the application of a peak-load pricing policy in the management of the European air traffic with the goal of reducing congestion,
minimizing delays and maintaining air navigation service providers’ revenues within a specified threshold. Results show that significant improvements in traffic distribution can be achieved through this simple charges mod-
ulation scheme.
1. Genetic Approach
The concept of modularity has been intensively
studied in evolutionary computation as a way of
improvingtheinnovativenessandscalabilityofthe
evolutionary search. Many authors suggest that
evolutionary algorithms are suitable for being as-
sembled progressively to enhance the optimization process. In our vision, modularity is the ability to freely combine
and reuse algorithm components. If we consider the classic genetic algorithm (GA) phases as assemblable modules, in
theorysuchmodulescouldbechosenfromawidevarietyofspecializedGAsdiscussedinliterature.MOGASI(Multi-Ob-
jective Genetic Algorithm for Structured Inputs) was construed having precisely this idea in mind.
The proposed decomposition has two aspects. The first one consists in splitting the GA phases in communicating sep-
arate modules, so that each block would remain independent from the specific implementation of the other modules
relying only on the data produced and shared by those modules. The second aspect consists in the identification of tar-
get data structures based on decision variable characteristics, which are grouped in macro-categories and handled by
specificstrategies.Apre-processingphasehasbeenimplementedinMOGASItoreducetheglobalproblemcomplexity.
Pre-processing methods include a number of simplification, elimination and shrinking methods discussed in literature,
aimed in particular at simplifying constraints and narrowing the bounds imposed on the decision variables. Further-
more, since during the optimization standard genetic operators may break the structure imposed by the encoding of
structured decision variables, a generic repair mechanism (FIXER) has been introduced for correcting such structure
violations and increasing the possibility that only feasible solutions are generate.
MOGASI has been designed for high-performance on both general
problems and those with specific structures. Its behavior was as-
sessed based on test taken from the famous Deb library. The goal
was to identify a better Pareto front with respect to other compara-
ble genetic algorithm approaches, in particular MOGA-II (Multi-Ob-
jective Genetic Algorithm) and NSGA-II (Non-Dominated Sorting Al-
gorithm). The Inverted Generational Distance (IGD) was selected as
the performance metric due to its ability to measure the distance
of the found non-dominated front from a reference sample of the
real Pareto front and to collect information regarding the accura-
cy and the uniformity of the computed front (lower IGD values are
preferred). The charts on the left clearly show the MOGASI superior
performance on 20,000 design evaluations in two multi-objective
minimization problems.
2. BPMN Decomposition
Engineering and scientific workflows are widely used, but
there is currently no accepted standard for their definition.
In an effort to provide a standard approximation to work-
flowrepresentationinengineeringapplications,Cominetal.
propose the use of a well-defined standard from the area of
business processes, BPMN 2.0, as a formal representation
of optimization workflows, which can be used for both pro-
cess representation and model execution. BPMN workflows
can be easily used for GA decomposition.
The workflow on the right has been developed and imple-
mented in collaboration with ESTECO SpA in form of a pro-
totype in the framework of the regional European project titled “Collaborative Methodology for the Definition of Engi-
neering Design Processes”. A Genetic Algorithm (GA) is decomposed into its main sub-procedures: offspring solution
generation, evaluation, selection and stopping condition. It would be possible to represent these phases with call activi-
tiesandtoprovideaBPMNmodelalsofortheirinnerstructures,butthiswillbeaddressedinfutureworks.Theaimofthis
process is to show that it is possible to define the buildingblocks of any optimization algorithm, which opens the door to
the possibility of customizing and hybridizing optimization algorithms depending on the specific application problem.
3. Bi-level Central Scheme
Peak-load pricing (PLP), a two-tariff charging
scheme commonly used in public transport and
utilities,hasbeentestedontheEuropeanAirTraffic
Management (ATM) system as a means for reduc-
ing airspace congestion. In particular, a centralised
approach to PLP with a Central Planner (CP) set-
ting out peak and off-peak tariffs on entire Europe-
annetworkisproposed.Inthefirstphasecongested
airspace sectors and their peak and off-peak hours
are identified. In a second phase the CP assesses
and imposes tariffs to reduce the overall shift on the network, where shift is intended as difference between the actual
and the scheduled flight departure time. Such differentiated charges should guarantee that Air Navigation Service Pro-
viders (ANSPs) are able to recover their operational costs (the so called Revenue Neutrality condition) while inducing
the Airspace Users (AUs) to route their traffic in a configuration that the network is able to sustain. The interaction be-
tweenCPandAUsismodelledasaStackelberggame,wheretheactionsoftheleaderaffectthechoicesofthefollower.
InthiscontexttheCPistheleader,settingthepeakandoff-peaktariffs,andtheAUsarethefollowers,makingarouting
choice based on those tariffs. This scheme has been formulated as a bilevel linear programming problem.
The leader level is tackled by MOGASI and has the peak and off-peak tariffs as variables for each ANSP and two ob-
jectives: minimization of the total shift time for all flights and minimization of the maximum difference in the revenue
neutrality of every ANSP. The follower level is tackled by a specially developed exact solver, the objective of which is to
perform each flight at minimum total cost. The optimization problem at leader-level has been constrained by a maxi-
mum allowed violation of the average capacity of all sectors and of the revenue neutrality of 20%.
Theformulationoftheproblemasbi-levelmulti-objectiveoptimisationwaseffective,butthepresenceofalargenumber
ofsolutionsrequiredapost-processinganalysis.TheParallelCoordinateschartonthefollowingfiguredepictsmultivar-
iate data sets and solutions in predefined ranges represented by vertical lines, whereas each solution is represented by
a coloured polyline. Four representative Pareto solutions have been chosen.
Variables shown on the chart:
• TGS-TotalsumofGlobalShiftoverallflights
• ACV - Average Capacity Violation
• RNV - Maximum Revenue Neutrality Violation
• nCB - Number of Capacity Breaches (number
of flights violating the capacities)
• nCV - Number of Capacity Violations (number
of (sector/airport, hour) pairs that violate their
nominal capacity levels)
The results of the optimization driven by MOGASI are compared with the baseline solution presented as the red line. It
exhibitsthehighestTGSwhiletherevenueneutralityisperfectlymatchedforeachANSP.Themostdesirablesolutionof
all is represented by the green line as it significantly improves TGS and keeps the revenue neutrality violation at around
the 10% with respect to the baseline solution.
The presented solutions show that the modulation of en-route charges indeed redistributes the traffic under each point
of view. A range of interesting alternative solutions has been obtained with the applied heuristics.
The presented PhD research work has been financed by ESTECO S.p.A.