Table Of ContentModeling and Optimization of Air Traffic
Modeling and
Optimization of
Air Traffic
Daniel Delahaye
Stéphane Puechmorel
Series Editor
Narendra Jussien
First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . xi
PART 1. OPTIMIZATION AND ARTIFICIAL
EVOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 1. Optimization: State of the Art . . . . . . 3
1.1. Methodological principles in optimization . . . . 3
1.1.1. Introduction . . . . . . . . . . . . . . . . . . . . 3
1.1.2. Modeling . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3. Complexity . . . . . . . . . . . . . . . . . . . . . 12
1.1.4. Computation time . . . . . . . . . . . . . . . . . 13
1.1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . 13
1.2. Optimization algorithms . . . . . . . . . . . . . . . 14
1.2.1. Introduction . . . . . . . . . . . . . . . . . . . . 14
1.2.2. Linear programming . . . . . . . . . . . . . . . 15
1.2.3. Nonlinear programming (NLP). . . . . . . . . 16
1.2.4. Local methods subject to constraints . . . . . 19
1.2.5. Deterministic global methods. . . . . . . . . . 21
1.2.6. Stochastic global methods . . . . . . . . . . . . 25
1.2.7. Genetic algorithms . . . . . . . . . . . . . . . . 33
1.2.8. Conclusion . . . . . . . . . . . . . . . . . . . . . 34
vi ModelingandOptimizationofAirTraffic
Chapter 2. Genetic Algorithms
and Improvements . . . . . . . . . . . . . . . . . . . . . . 37
2.1. General points . . . . . . . . . . . . . . . . . . . . . 37
2.1.1. Introduction . . . . . . . . . . . . . . . . . . . . 37
2.1.2. Principle of genetic algorithms . . . . . . . . . 39
2.1.3. Coding principles . . . . . . . . . . . . . . . . . 42
2.1.4. Random generation of the initial
population . . . . . . . . . . . . . . . . . . . . . 42
2.1.5. Crossover operators . . . . . . . . . . . . . . . . 43
2.1.6. Mutation operators . . . . . . . . . . . . . . . . 45
2.1.7. Selection principles . . . . . . . . . . . . . . . . 47
2.2. Classic improvements . . . . . . . . . . . . . . . . . 48
2.2.1. Scaling . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.2. Sharing . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.3. Crowding . . . . . . . . . . . . . . . . . . . . . . 52
2.2.4. Memetic algorithms . . . . . . . . . . . . . . . 53
2.2.5. Multi-objective genetic algorithms . . . . . . 53
2.3. Our contributions . . . . . . . . . . . . . . . . . . . 57
2.3.1. Adaptive clustered sharing . . . . . . . . . . . 58
2.3.2. Association of genetic algorithms with
simulated annealing . . . . . . . . . . . . . . . 60
2.3.3. Parallel genetic algorithms . . . . . . . . . . . 64
2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 66
Chapter 3. A New Concept for Genetic
Algorithms Based on Order Statistics . . . . . . . . 67
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 67
3.2. Order statistics . . . . . . . . . . . . . . . . . . . . . 68
3.3. Estimating the probability that the global
optimum belongs to a given domain . . . . . . . . 71
3.4. Genetic algorithms and order statistics . . . . . . 71
3.4.1. Introduction . . . . . . . . . . . . . . . . . . . . 71
3.4.2. Coding . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.3. Recombination operators . . . . . . . . . . . . 73
3.4.4. Evaluation of fitness . . . . . . . . . . . . . . . 75
TableofContents vii
3.5. Application to test functions . . . . . . . . . . . . . 75
3.5.1. Results for the Griewank function . . . . . . . 77
3.5.2. Results for the Rosenbrook function . . . . . 78
3.5.3. Results for the Lennard-Jones function . . . 79
3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 81
PART 2. APPLICATIONS TO AIR TRAFFIC
CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Chapter 4. Air Traffic Control . . . . . . . . . . . . . . 85
Chapter 5. Contributions to Airspace
Sectorization . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 91
5.2. Modeling in 2D . . . . . . . . . . . . . . . . . . . . . 93
5.2.1. Model based on a transportation network . . 93
5.2.2. Associated complexity . . . . . . . . . . . . . . 98
5.3. Continuous modeling . . . . . . . . . . . . . . . . . 99
5.3.1. Principle. . . . . . . . . . . . . . . . . . . . . . . 99
5.3.2. Chromosome coding . . . . . . . . . . . . . . . 101
5.3.3. Initial population generation principle . . . . 101
5.3.4. Crossover operator . . . . . . . . . . . . . . . . 101
5.3.5. Mutation operator. . . . . . . . . . . . . . . . . 103
5.3.6. Calculation and normalization
of the fitness function . . . . . . . . . . . . . . 104
5.3.7. Results. . . . . . . . . . . . . . . . . . . . . . . . 106
5.3.8. Conclusion . . . . . . . . . . . . . . . . . . . . . 110
5.4. Discrete modeling . . . . . . . . . . . . . . . . . . . 111
5.4.1. Principle. . . . . . . . . . . . . . . . . . . . . . . 111
5.4.2. Coding . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.3. Recombination operators . . . . . . . . . . . . 115
5.4.4. Results. . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . 119
5.5. Extension 3D . . . . . . . . . . . . . . . . . . . . . . 119
5.5.1. Introduction . . . . . . . . . . . . . . . . . . . . 119
5.5.2. Mathematical modeling . . . . . . . . . . . . . 122
viii ModelingandOptimizationofAirTraffic
5.5.3. Application of artificial evolution to the
problem . . . . . . . . . . . . . . . . . . . . . . . 127
5.5.4. Results. . . . . . . . . . . . . . . . . . . . . . . . 132
5.5.5. Conclusion . . . . . . . . . . . . . . . . . . . . . 135
5.6. Accounting for the dynamic aspect . . . . . . . . . 136
5.6.1. Formalization of objectives and associated
mathematical model . . . . . . . . . . . . . . . 136
5.6.2. Optimization using a genetic algorithm:
continuous approach . . . . . . . . . . . . . . . 140
5.6.3. Optimization using a genetic algorithm:
discrete approach . . . . . . . . . . . . . . . . . 144
Chapter 6. Contribution to Traffic Assignment . . 151
6.1. Summary of traffic assignment methods based
on transportation network theory . . . . . . . . . 152
6.1.1. Transportation networks . . . . . . . . . . . . 153
6.1.2. Static assignment . . . . . . . . . . . . . . . . . 155
6.1.3. Dynamic assignment . . . . . . . . . . . . . . . 163
6.2. Other approaches to traffic assignment . . . . . . 167
6.2.1. Temporal extension of the network . . . . . . 167
6.2.2. Optimal control . . . . . . . . . . . . . . . . . . 168
6.2.3. Dynamic programming approaches (ground
holding problem) . . . . . . . . . . . . . . . . . 169
6.2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . 171
6.3. Using artificial evolution in all-or-nothing traffic
assignment . . . . . . . . . . . . . . . . . . . . . . . 173
6.3.1. Mathematical formalization of objectives . . 173
6.3.2. Coding and operators of the genetic
algorithm . . . . . . . . . . . . . . . . . . . . . . 176
6.3.3. Introduction of an inter-chromosome
distance for sharing. . . . . . . . . . . . . . . . 179
6.3.4. Example of results . . . . . . . . . . . . . . . . 182
6.3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . 185
6.4. Allocation of routes and slots using artificial
evolution . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.4.1. System architecture . . . . . . . . . . . . . . . 187
TableofContents ix
6.4.2. The fitness function . . . . . . . . . . . . . . . . 192
6.4.3. Simple genetic algorithm . . . . . . . . . . . . 194
6.5. Modification of the algorithm –
adaptive modifications . . . . . . . . . . . . . . . . 198
6.5.1. Establishing congestion levels in the
chromosome . . . . . . . . . . . . . . . . . . . . 198
6.5.2. Establishment of trends . . . . . . . . . . . . . 200
6.5.3. New coding and biased initial population . . 203
6.5.4. New crossover operator . . . . . . . . . . . . . 203
6.5.5. New mutation operator . . . . . . . . . . . . . 204
6.5.6. New results . . . . . . . . . . . . . . . . . . . . . 205
6.5.7. Dynamic bi-allocation . . . . . . . . . . . . . . 207
6.5.8. Multi-objective approach . . . . . . . . . . . . 210
6.5.9. Conclusion . . . . . . . . . . . . . . . . . . . . . 211
6.6. Sequencing flights for landing . . . . . . . . . . . 211
6.6.1. Introduction . . . . . . . . . . . . . . . . . . . . 212
6.6.2. Single runway formulation . . . . . . . . . . . 213
6.6.3. Modeling using GA . . . . . . . . . . . . . . . . 214
6.6.4. Results. . . . . . . . . . . . . . . . . . . . . . . . 217
6.7. Trajectory planning . . . . . . . . . . . . . . . . . . 220
6.7.1. Introduction . . . . . . . . . . . . . . . . . . . . 220
6.7.2. The light propagation algorithm . . . . . . . . 222
6.7.3. Approach using genetic algorithms on
B-splines . . . . . . . . . . . . . . . . . . . . . . 234
6.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 241
Chapter 7. Airspace Congestion Metrics . . . . . . . 243
7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 243
7.2. Flow-based approach . . . . . . . . . . . . . . . . . 248
7.2.1. Mathematical modeling of the control
workload . . . . . . . . . . . . . . . . . . . . . . 253
7.3. Geometrical approaches . . . . . . . . . . . . . . . 253
7.3.1. Proximity metric . . . . . . . . . . . . . . . . . 254
7.3.2. Convergence . . . . . . . . . . . . . . . . . . . . 258
7.3.3. Clusters . . . . . . . . . . . . . . . . . . . . . . . 263
7.3.4. Grassmannian indicator . . . . . . . . . . . . . 265
x ModelingandOptimizationofAirTraffic
7.4. Approach based on dynamic systems . . . . . . . 268
7.4.1. Linear dynamic systems . . . . . . . . . . . . . 268
7.4.2. Spatial extension using nonlinear dynamic
systems . . . . . . . . . . . . . . . . . . . . . . . 273
7.4.3. Spatiotemporal extension using nonlinear
dynamic systems . . . . . . . . . . . . . . . . . 281
7.4.4. Local linear models . . . . . . . . . . . . . . . . 285
7.4.5. Stochastic extension . . . . . . . . . . . . . . . 288
Conclusion and Future Perspectives . . . . . . . . . 291
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Introduction
This book presents the main research topics that we have
worked on over the last 15 years. Research is a fascinating
occupation that allows permanent enrichment of knowledge
and sustains rich intellectual activity. Moments of doubt do
arise when a problem seems impossible to solve, but this is
intrinsically linked to one of the great advantages of
research: the feeling of satisfaction when a solution is finally
found. As researchers at the Ecole Nationale de l’Aviation
Civile (French Civil Aviation University), our research
activityfocusesonthethemeofairtransportand,specifically,
on the domain of mathematical optimization. It is always
fascinating to look at real problems taken from the
operationaldomainastheircharacteristicsareoftencomplex,
representing a significant challenge with applicable results.
When observing real situations, we note that they are often
considerably different from theoretical models, and our
mathematical tools are of limited use when attempting to
tackle a complex problem as a whole. These practical
problems present a certain interest for researchers as they
require the development of new tools.
Our research focuses on issues linked to reducing
congestion in airspace. These types of problems are generally
