Modular Multi-Objective Genetic Algorithm for Large Scale Bi-level

Problems Stefano Costanzo, Desirée Rigonat,

L. Castelli, A. Turco

Problem – Air Traffic Congestion

Problem: air traffic in Europe is too congested, leading to flight delays and unhappy passengers

Goal: redistribution of the air traffic according to peak and off-peak hours to reduce congestion and maintain profits

Proposed solution: the system was modeled as a bi-level problem and subsequently optimized with a ad-hoc multi-objective genetic algorithm

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Peak-Load Pricing

Peak-load pricing (PLP) is a pricing mechanism aimed at achieving efficient capacity management and commonly used in the transport sector.

The idea is to charge a higher toll where and when a peak in demand is expected and a lower toll for off-peak areas and/or times.

We have applied the PLP policy to the European Air Traffic Management (ATM) system to achieve our goal.

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PLP in Air Traffic Management

Key actors: European Air Navigation Service Providers (ANSP) - entities which finance their operations by charging Airspace Users (AUs) to EC Regulations.

What do they charge?

Air Navigation Service (ANS) charges are composed of en-route and terminal charges and depend on several factors (distance, national unit rate, aircraft weight etc.)

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Pricing example – how does it work?

For an aircraft weighing 80 metric

tonnes the price per kilometre

(July 2013) is €1.00 in Italy and

€0.53 in Croatia.

The longer flight route from Milan to Brindisi using the Croatian air space is €177.19 cheaper !

From PRB Annual monitoring Report 2012, Volume 1, European overview and PRB recommendations, Section 3.2

But, such situations may lead to undesired consequences:

• Lower charges, as in the above example, might encourage airspace users to fly on longer routes with a detrimental effect on flight efficiency

• They might also create a competition between ANSPs based on Unit Rates to attract traffic

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Tackling the problem

• Assumptions:

– Peaks in demand are periodic in time and location (and therefore predictable)

– Demand has some degree of elasticity concerning time and location in which a service is provided

• Action:

– Time intervals and locations for which a peak in demand is expected are assigned higher charges than other times and locations (off-peak)

• Objective:

– Reduce the amount of shift on the network, where shift is intended as the difference between the requested (by Airspace Users) and assigned departure or arrival time

• Expected outcomes:

– A portion of the peak demand will be redirected to different (cheaper) travel/service consumption options

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Centralized Peak Load Pricing

We introduced a new entity – the Central Planner which sets peak and off-peak rates for the whole network:

• These rates should guarantee the following:

– Total schedule shift and airspace capacity violations are minimised

– ANSPs are able to recover their costs

– AUs are able to perform flights without imbalances between the

amount of traffic and available airspace capacity

• Each AU can choose the cheapest route for each flight

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Assumptions (I)

• Fixed demand A fixed number of flights can be performed between any two airports in the network.

• Infrastructure capacity constraints are known

• Finite set of possible 4D trajectories for each Origin/Destination/Aircraft combination: AUs can select a route from a set of pre-determined routes

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Assumptions (II)

• AUs are rational decision makers – we assume that they will choose the cheapest option

• Revenue neutrality ANSP revenues should be balanced with their expenses

• Heterogeneous demand Flights using different aircraft types will have different costs and consequently a different sensitivity to imposed location/time rates.

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Bi-level formulation of the CPLP

The problem was modelled as a bi-level (CP vs. AU) linear programming problem and formulated as a Stackelberg game with the CP acting as leader and the AUs acting as followers.

Leader (CP)

• Manages the network

• Can set arc costs (rates)

• Can predict the reaction of the follower to a pricing strategy

Follower (AU)

• User(s) of the network

• Can set arc flows

• Reacts to leader’s strategy

Route choices

Rates choices

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Bilevel formulation for MILP solver

To identify the Stackelberg equilibrium some reformulations have been necessary.

• The bi-level problem was transformed into a single level problem. This reformulation required a complete enumeration of all possible routing choices for all users.

• A further reformulation was then carried out to linearize all terms that are expressed as the product of two variables.

The result is a linear programming problem that can be solved using any MILP solver.

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Dimensionality Testing

The model was tested on a regional scale using real air traffic data: all aircrafts departing between 6:00 AM and 10:00 AM on 12 Sep 2014 that cross, depart from or arrive in the French air space (2,376 flights in total). Each aircraft has to choose a route from a subset of 4 routes in average. The maximum allowed shift per route is limited to -30/+30 minutes, measured in 5 min intervals, thus 13 shift possibilities.

This bi-level CPLP model has ~250.000 variables

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Experimental Design

• Chosen day: Friday, 12 SEP 2014: 4th busiest day in 2014, approx. 30,000 flights

• Departure/arrival times: last filed flight plan in all EU countries

• Set of flight routes between each O/D pair: from the two preceding weeks

• Airspace configuration: dynamic sectorization in place on 12 SEP 2014 with nominal declared capacities

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Genetic Algorithm

Genetic Algorithms are popular stochastic optimization methods inspired by the evolutionist theory on the origin of species and natural selection.

GAs are particularly suitable for solving multi-objective problems and finding reasonably good trade-off solutions (i.e. Pareto solutions).

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Genetic Algorithm for Leader Level

A multi-objective GA called MOGASI is used to solve the bi-level CPLP, simultaneously considering two objectives :

– Minimization of the global shift for all flights

– Minimization of the maximum revenue neutrality violation

To drive the search through the solution space towards feasible regions we set additional constraints:

– average capacity violation must be lower than 20%

– revenue neutrality violation must not exceed 10% for each ANSP.

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Custom Exact Solver for Follower Level

• The AU objective function should minimize the strategic flight operation costs.

• For this purpose we developed an exact solver that would create the minimum flight cost assignment.

• The computational time of this solver with fixed peak and off-peak charges is about 3 seconds. This enables the GA Leader to explore thousands of charge sets in a very short time.

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Experiments

• The model with the original Unit Rates is used both as the baseline and starting point.

• The MOGASI termination criterion parameter was set to 20,000 evaluations, which took nearly 15 hours.

• MOGASI was able to identify many feasible optimal solutions.

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Results

0%

2%

4%

6%

8%

10%

24700 24800 24900 25000 25100 25200

Max

imu

m D

elt

a R

eve

nu

e o

n A

NSP

s

Total shifts

Baseline

Baseline is already optimized to respect capacities,

but using the normal unit rates. Different from the actual traffic.

Thousands of different possible solutions

Results

Results

-€ 10,00

-€ 8,00

-€ 6,00

-€ 4,00

-€ 2,00

€ -

€ 2,00

€ 4,00

€ 6,00

€ 8,00

€ 10,00

€ 12,00

EB ED EF EH ES LE LI LJ LU LZ

Off-peak Grey

Peak Grey

Off-peak Green

Peak Green

Off-peak Blue

Peak Blue

Only peak and/or off-peak rates > |4 EUR| than SEP 14 unit rate

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Conclusions

• The work presented here shows that the modification of en-route charges indeed redistributes the traffic.

• We analyzed one full and very busy day of air traffic over the entire Europe

• Our custom GA found a good range of possible Pareto solutions, but it has also created improvements in other variables.

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Future Work

• Detailed analysis of the distribution of costs across different AUs to check if any equity issues might arise

• Expand the MOGASI with specialized structures to improve its convergence rate and manageable variables

• Application of PLP in a decentralized manner, thus having ANSPs set the charges instead of the CP

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Thank you for your attention

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