multiobjective metaheuristics and parallel computing for

TESIS DOCTORAL

Multiobjective Metaheuristics and Parallel Computing

for Optimizing WDM Optical Networks

Metaheurísticas Multiobjetivo y Computación Paralelapara Optimizar Redes Ópticas WDM

Autor: Álvaro Rubio Largo

Departamento: Tecnología de los Computadores y de las ComunicacionesEscuela Politécnica – Universidad de Extremadura (Cáceres, España)

Conformidad del Director:

Fdo: Dr. Miguel Ángel Vega Rodríguez

Año de Lectura: 2013

PhD Thesis Work:

Multiobjective Metaheuristics andParallel Computing for OptimizingWDM Optical NetworksMetaheurísticas Multiobjetivo y ComputaciónParalela para Optimizar Redes Ópticas WDM

TECNOLOGÍAS INFORMÁTICAS Y COMUNICACIONES (TINC)

Department of Computer and Communications TechnologiesEscuela Politécnica – University of Extremadura, Cáceres (Spain)

Candidate

Álvaro Rubio-Largo

D.N.I. 28963485-E

Thesis Advisor

Dr. Miguel A. Vega-Rodríguez

A thesis submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Computer Science

March 2013

Multiobjective Metaheuristics and Parallel Computing for Optimizing WDM Optical Net-

works

Ph.D. thesis.Department of Computer and Communications TechnologiesEscuela Politécnica – University of Extremadura, Cáceres (Spain)

© 2013 Álvaro Rubio-Largo. All rights reserved

.

Website: http://arco.unex.es/arl

Author’s email: [email protected]

http://arco.unex.es/arl

mailto:[email protected]

This thesis is dedicated to my love, Estefanía,and my parents, who have always stood by me.

Summary

Nowadays, the number of users that use the Internet has risen exponentially. However, our currentdata networks are not able to support this exponential growth due to their bandwidth is not enough.In this way, due to the huge bandwidth of optical fiber (50Tbps), the use of these data networksis a suitable option for dealing with this drawback.

In this way, the most promising technology for exploiting the huge bandwidth of these datanetworks is based on Wavelength Division Multiplexing (WDM). This technology multiplies theavailable capacity of an optical fiber link by adding new channels, each channel on a new wavelengthof light (Gbps). The aim of WDM is to ensure fluent communications between several devices,avoiding bottlenecks. In WDM networks, a problem comes up when it is necessary to interconnecta set of connection requests. This problem is known in the literature as the Routing and WavelengthAssignment problem (RWA). On the one hand, the choice of the physical route of a connectionis based on some cost criterion such as hop length. On the other hand, the wavelength is chosenbased on the wavelength usage factor in the entire network. The optical connection establishedend-to-end from a source node to a destination node is known as lightpath.

So, the RWA problem may be stated as a Multiobjective Optimization Problem (MOOP),searching the best solutions that simultaneously minimize the number of hops and the number ofwavelength conversions.

Unfortunately, the majority of current devices or applications are constrained by their process-ing speed (a few Mbps), which is translated into a waste of bandwidth. This drawback is efficientlysolved by grooming several low-speed connection demands (Mbps) onto high-speed wavelengthchannels (Gbps). This problem is known as the Traffic Grooming problem.

Like the RWA problem, the Traffic Grooming problem may be stated as a MultiobjectiveOptimization Problem that simultaneously optimizes the total throughput of the network, thenumber of transceivers/receivers used, and the average propagation delay.

Depending on the traffic pattern, these Telecommunication problems may be considered staticor dynamic. On the one hand, if the set of demands is known in advance, the problem is the designof a virtual topology in order to accommodate as much of these demands as possible. On the otherhand, whether the traffic model is dynamic, the virtual topology is reconfigured on demand eitherarrival or departure. In this work, we focus on solving the static version of both problems due tothe fact that the Wide Area Networks (WANs) are oriented to pre-contracted services.

On the whole, with the aim of optimizing the performance of the optical networks, in thisPhD dissertation we propose the use of multiobjective optimization, evolutionary algorithms, andparallelism for tackling these real-world problems (RWA and Traffic Grooming). The main goalof this research is not only to obtain results of higher quality than other methods published inthe literature by other authors, but also obtaining such results in a reasonable time by usingparallelism.

As we will see, in this Thesis, many different multiobjective metaheuristics have been im-plemented, analyzed, and compared when solving these two networking problems. Furthermore,important results and conclusions have been obtained from the parallel computing perspective.

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Resumen

En las últimas décadas, el número de usuarios que utilizan Internet ha crecido de manera exponen-cial. Sin embargo, el ancho de banda de las redes de datos actuales no es suficiente para satisfacereste enorme crecimiento, surgiendo así la necesidad de utilizar fibra óptica, debido a su enormeancho de banda (50Tbps).

La tecnología con más futuro para explotar el ancho de banda de estas redes ópticas está basadaen la multiplexación por división de longitud de onda (Wavelength Division Multiplexing, WDM).El objetivo principal de la tecnología WDM es introducir concurrencia en las transmisiones dedatos, dividiendo cada enlace óptico en diferentes canales o longitudes de onda (Gbps).

El uso de la tecnología WDM da lugar a un problema de Enrutamiento y Asignación de Longitudde Onda (Routing and Wavelength Assignment, RWA), cuando es necesario establecer un conjuntode demandas. La ruta elegida para establecer cada demanda es seleccionada en base a criterios decoste (p.e. número de saltos), y la longitud de onda en base al factor de utilización de longitudes deonda de toda la red. De este modo, definiremos el concepto lightpath como: la conexión establecidapunto-a-punto entre dos nodos a través de una longitud de onda específica.

Por todo ello, el problema RWA ha sido considerado como un problema de Optimización Mul-tiobjetivo (MultiObjective Optimization Problem, MOOP), el cuál trata de encontrar solucionesque simultáneamente minimicen el número de saltos y el número de conmutaciones de longitudesde onda.

Desgraciadamente, la mayor parte de aplicaciones y dispositivos están limitados por la velocidadde procesamiento (unos pocos Mbps). Dado que el ancho de banda de un canal es de Gbps, seproduce un desperdicio de ancho de banda al establecer estas demandas de baja velocidad. Conel fin de resolver este problema, se pueden multiplexar varias demandas de baja velocidad sobreun mismo lightpath, dando lugar a un problema de optimización conocido como problema TrafficGrooming.

Al igual que el problema RWA, el problema Traffic Grooming ha sido considerado como un prob-lema de Optimización Multiobjetivo, el cuál trata de optimizar de manera simultánea el throughputtotal de la red, el número de transmisores/receptores utilizados y el retardo de propagación medio.

Según el patrón de tráfico, ambos problemas pueden ser considerados como estáticos o dinámi-cos. Si las demandas son conocidas a priori, el problema se soluciona diseñando una topologíavirtual para establecer el mayor número de demandas de tráfico posibles (modelo estático). Sinembargo, en el caso dinámico, la topología virtual necesita ser reconfigurada cada vez que unademanda de tráfico entra o sale de la cola de peticiones. En esta investigación, nos centraremosen resolver ambos problemas de telecomunicaciones siguiendo un modelo estático debido a que lagran mayoría de Redes de Área Extensa están orientadas a servicios precontratados.

En definitiva, con el fin de optimizar el rendimiento de las redes de fibra óptica, en este trabajoproponemos utilizar optimización multiobjetivo, algoritmos evolutivos y paralelismo, para abordardos problemas reales de Telecomunicaciones (RWA y Traffic Grooming). El objetivo principal deesta investigación es, no sólo obtener resultados de mayor calidad que los publicados por otrosautores, sino que además, utilizando paralelismo, ser capaces de obtener dichos resultados en untiempo reducido.

Como veremos, en esta Tesis, muchas metaheurísticas multiobjetivo distintas se han implemen-tado, analizado y comparado para resolver estos dos problemas de red. Además, también se hanobtenido resultados y conclusiones importantes desde el punto de vista de la computación paralela.

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Acknowledgments

There are a number of people without whom this PhD thesis might not have been written, andto whom I am greatly indebted.

I would like to start thanking to my love, Estefanía, for her unconditional support and immensepatience in those moments of hard work. We have lived thousands of special moments together,but I know there will be millions of new unforgotten moments in the near future. I promise I willdo my best to make you smile every single day because you are the one for me, I love you morethan ever!

I cannot find words to express my gratitude to my parents, brother, and sister. I am speciallygrateful to my parents, who taught me the importance of working hard; so, without your supportthroughout my whole life, this thesis would have never been possible.

It is with immense gratitude that I acknowledge the support and help of my supervisor Dr.Miguel A. Vega-Rodríguez. You have been more than just a supervisor throughout these years, afriend. Despite of having a lot of work to do, you always find time for solving any doubt; thus, thisthesis would have remained a dream had it not been for your dedication and support.

To David, a co-worker and a friend. We have started and finished our PhD thesis at the sametime, sharing a lot of journeys, conferences, and of course, great moments at the lab these years.We have realized together not only the hardness of the research career, but also how gratifying isto complete a project like this.

To all my co-workers at the research group: Chema, Jose M., Josi, Sergio, and Victor; thanksfor hours and hours of great time at work. Furthermore, thanks to Alberto and all my partners atthe University of Extremadura.

To my friends: Lidia, Sergio, Bea, and Nuria; thanks for your support and help, I hope we canrepeat the amazing summer trip to Costa da Caparica this summer!

Finally, I would like to thanks the economical support received throughout these years to:

• Gobierno de Extremadura (Consejería de Economía, Comercio e Innovación) and the Euro-pean Social Fund (ESF) for the predoctoral research grant PRE09010 (2009 – 2013).

• Spanish Ministry of Science and Innovation and ERDF (the European Regional DevelopmentFund) for the following national research projects: TIN2008-06491-C04-04 (the M* project,2009 – 2011) and TIN2012-30685 (BIO project, 2013 – 2015).

• Gobierno de Extremadura for the research grant GR10025 provided to the Computer Archi-tecture and Logic Design Group (ARCO, TIC015).

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Contents

Summary iii

Resumen v

Acknowledgments vii

List of Tables xiii

List of Figures xix

1 Introduction 11.1 Objectives and Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Optical Networks: Fundamentals and Background 72.1 Wavelength Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Routing and Wavelength Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Wavelength Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Traffic Grooming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Multiplexing Techniques in Traffic Grooming . . . . . . . . . . . . . . . . . 192.3.2 Virtual Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Proposed Methodology 273.1 Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Classification of Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Evolutionary Algorithms: Basic concepts . . . . . . . . . . . . . . . . . . . . 31

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xii

3.2 Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.1 Formal Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Multiobjective Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . 393.2.3 Quality Assessment in Multiobjective Optimization . . . . . . . . . . . . . . . 41

3.3 Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Shared memory Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.2 Distributed Memory Architecture . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.3 Hybrid Shared-Distributed Memory Architecture . . . . . . . . . . . . . . . 543.3.4 Parallel Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3.5 Performance Assessment in Parallel Computing . . . . . . . . . . . . . . . . 57

3.4 Statistical Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Routing and Wavelength Assignment problem 614.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 Example of the RWA problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Representation of Individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4 Data Sets: Optical networks and sets of demands . . . . . . . . . . . . . . . . . . . 66

5 Traffic Grooming problem 715.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Example of the Traffic Grooming problem . . . . . . . . . . . . . . . . . . . . . . . 755.3 Representation of the Individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4 Data Sets: Optical networks and sets of low-speed traffic demands . . . . . . . . . 79

6 Multiobjective Evolutionary Algorithms 836.1 Differential Evolution with Pareto Tournaments . . . . . . . . . . . . . . . . . . . . 846.2 Multiobjective Variable Neighbourhood Search . . . . . . . . . . . . . . . . . . . . 866.3 Multiobjective Artificial Bee Colony . . . . . . . . . . . . . . . . . . . . . . . . . . 876.4 Multiobjective Gravitational Search Algorithm . . . . . . . . . . . . . . . . . . . . 896.5 Multiobjective Firefly Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.6 Fast Non-dominated Sorting Genetic Algorithm . . . . . . . . . . . . . . . . . . . . 926.7 Strength Pareto Evolutionary Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 946.8 Parallel Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.8.1 Parallel DEPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.8.2 Parallel MO-ABC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7 Solving the RWA problem 1057.1 Parameter Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.2 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2.1 European Optical Network (COSTS239) . . . . . . . . . . . . . . . . . . . . 1087.2.2 National Science Foundation (NSF) Network . . . . . . . . . . . . . . . . . 1157.2.3 Nippon Telegraph and Telephone (NTT) Network . . . . . . . . . . . . . . . 1217.2.4 Conclusions of the Comparative Study . . . . . . . . . . . . . . . . . . . . . 127

7.3 Comparison with other works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3.1 Comparison with Typical Heuristics . . . . . . . . . . . . . . . . . . . . . . 1287.3.2 Comparison with MOACOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.4 Performance of the Parallel Approach . . . . . . . . . . . . . . . . . . . . . . . . . 134

xiii

8 Solving the Traffic Grooming problem 1418.1 Parameter Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428.2 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.2.1 6-node Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468.2.2 European Optical Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 1568.2.3 National Science Foundation Network . . . . . . . . . . . . . . . . . . . . . 1668.2.4 Nippon Telegraph and Telephone Network . . . . . . . . . . . . . . . . . . . 1768.2.5 Conclusions of the Comparative Study . . . . . . . . . . . . . . . . . . . . . 186

8.3 Comparison with other works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1878.4 Performance of the Parallel Approach . . . . . . . . . . . . . . . . . . . . . . . . . 192

9 MOEA/D-NBI for 3-objective optimization problems 2019.1 MOEA/D with the NBI-style Tchebycheff approach for 3-objective Optimization

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2029.1.1 Testing the MOEA/D-NBI with benchmark functions . . . . . . . . . . . . 206

9.2 Indirect Encoding and Construction Heuristics for the Traffic Grooming problem . 2109.3 MOEA/D-NBI for the Traffic Grooming problem . . . . . . . . . . . . . . . . . . . 214

9.3.1 Previous Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2149.3.2 Comparison with NSGA-II∗∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 2159.3.3 Comparison with the proposed MOEAs . . . . . . . . . . . . . . . . . . . . 2199.3.4 Comparison with other works . . . . . . . . . . . . . . . . . . . . . . . . . . 228

9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

10 Conclusions and future work 23310.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

11 Scientific Achievements 23711.1 Publications related to this PhD Thesis . . . . . . . . . . . . . . . . . . . . . . . . 23711.2 Other Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24011.3 Other scientific achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

11.3.1 International Stays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24111.3.2 Reviewer of International Journals . . . . . . . . . . . . . . . . . . . . . . . . 24111.3.3 Participation in International Conferences . . . . . . . . . . . . . . . . . . . . 24111.3.4 Participation in Research Projects . . . . . . . . . . . . . . . . . . . . . . . 24211.3.5 Collaboration with other Institutions . . . . . . . . . . . . . . . . . . . . . . 243

Bibliography 245

List of Tables

2.1 Summary of the main approaches published in the literature for the static RWAproblem. Note that, the notation used is: Wavelength Conversion (WC), Multiob-jective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART1/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Summary of the main approaches published in the literature for the static RWAproblem. Note that, the notation used is: Wavelength Conversion (WC), Multiob-jective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART2/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Summary of the main approaches published in the literature for the Traffic Groomingproblem. Note that, the notation used is: Wavelength Conversion (WC), Multiob-jective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART1/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Summary of the main approaches published in the literature for the Traffic Groomingproblem. Note that, the notation used is: Wavelength Conversion (WC), Multiob-jective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART2/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Compilation options for the most common C/C++ compilers . . . . . . . . . . . . 503.2 Advantages and Disadvantages of MPI and OpenMP . . . . . . . . . . . . . . . . . 55

4.1 COST239: Specifications of data sets for the RWA problem. . . . . . . . . . . . . . 674.2 NSF: Specifications of data sets for the RWA problem. . . . . . . . . . . . . . . . . 684.3 NTT: Specifications of data sets for the RWA problem. . . . . . . . . . . . . . . . . 69

5.1 6-node: Specifications of data sets for the Traffic Grooming problem. . . . . . . . . 795.2 COST239: Specifications of data sets for the Traffic Grooming problem. . . . . . . 805.3 NSF: Specifications of data sets for the Traffic Grooming problem. . . . . . . . . . . 815.4 NTT: Specifications of data sets for the Traffic Grooming problem. . . . . . . . . . 82

6.1 The Schemes for the DE algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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xvi List of Tables

7.1 Comparison among the MOEAs by using the Hypervolume (HV) indicator. Thenotation used is HVIQR, where HV is the median hypervolume and IQR is theinterquartile range in 30 independent runs. . . . . . . . . . . . . . . . . . . . . . . 108

7.2 COST239 network. Comparison among the MOEAs by using the Hypervolume (HV)indicator. The notation used is HVIQR, where HV is the median hypervolume andIQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . . . 109

7.3 Statistical Analysis among the MOEAs in the COST239 network. The table indi-cates in which data sets two algorithms have no statistically significant differences. 113

7.4 COST239 network. Comparison among the MOEAs by using the Set Coverage (SC)indicator, A B. Note that, SC represents the mean coverage of an algorithm Aover an algorithm B in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . 114

7.5 NSF network. Comparison among the MOEAs by using the Hypervolume (HV)indicator. The notation used is HVIQR, where HV is the median hypervolume andIQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . . . 115

7.6 Statistical Analysis among the MOEAs in the NSF network. The table indicates inwhich data sets two algorithms have no statistically significant differences. . . . . . 119

7.7 NSF network. Comparison among the MOEAs by using the Set Coverage (SC)indicator, A B. Note that, SC represents the mean coverage of an algorithm Aover an algorithm B in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . 120

7.8 NTT network. Comparison among the MOEAs by using the Hypervolume (HV)indicator. The notation used is HVIQR, where HV is the median hypervolume andIQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . . . . 121

7.9 Statistical Analysis among the MOEAs in the NTT network. The table indicates inwhich data sets two algorithms have no statistically significant differences. . . . . . 125

7.10 NTT network. Comparison among the MOEAs by using the Set Coverage (SC)indicator, A B. Note that, SC represents the mean coverage of an algorithm Aover an algorithm B in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . 126

7.11 Ranking of the MOEAs when tackling the RWA problem. . . . . . . . . . . . . . . 1277.12 Summary of the Typical Heuristics published in the literature for the RWA problem. 1287.13 Comparison between the best Typical Heuristics and the proposed MOEAs (DEPT,

MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the HV indicator. . . . . . . 1297.14 Comparison between the best Typical Heuristics and the proposed MOEAs (DEPT,

MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the SC indicator. . . . . . . 1307.15 Summary of the MOACOs published in the literature for the RWA problem. . . . . . 1317.16 Comparison between the best MOACO and the proposed MOEAs (DEPT, MO-VNS,

MO-ABC, MO-GSA, and MO-FA) by using the HV indicator. . . . . . . . . . . . . 1327.17 Comparison between the best MOACO and the proposed MOEAs (DEPT, MO-VNS,

MO-ABC, MO-GSA, and MO-FA) by using the SC indicator. . . . . . . . . . . . . 1337.18 Mean runtime and standard deviation for the sequential version of the DEPT algo-

rithm in 30 independent runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.19 Mean runtime, speedup, and efficiency for the OpenMP and MPI versions of the

pDEPT algorithm with 2 cores in 30 independent runs. Note that we report thestandard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.20 Mean runtime, speedup, and efficiency for the OpenMP and MPI versions of thepDEPT algorithm with 4 cores in 30 independent runs. Note that we report thestandard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

List of Tables xvii

7.21 Mean runtime, speedup, and efficiency for the OpenMP and MPI versions of thepDEPT algorithm with 8 cores in 30 independent runs. Note that we report thestandard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.22 Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions ofthe pDEPT algorithm with 16 cores in 30 independent runs. Note that we reportthe standard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.23 Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions ofthe pDEPT algorithm with 32 cores in 30 independent runs. Note that we reportthe standard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.24 Summary of the mean speedup and efficiency for the OpenMP+MPI and MPI ver-sions of the pDEPT algorithm in all the data sets. . . . . . . . . . . . . . . . . . . 140

8.1 Comparison among the MOEAs by using the Hypervolume (HV) indicator. . . . . 1448.2 Amount of Traffic (in OC-1 units) and Runtime (in seconds) per Optical network . 1458.3 6-node network (TM1). Comparison among the MOEAs by using the Hypervolume

(HV) indicator. The notation used is HVIQR, where HV is the median hypervolumeand IQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . 147

8.4 6-node network (TM2). Comparison among the MOEAs by using the Hypervolume(HV) indicator. The notation used is HVIQR, where HV is the median hypervolumeand IQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . 148

8.5 6-node network (TM3). Comparison among the MOEAs by using the Hypervolume(HV) indicator. The notation used is HVIQR, where HV is the median hypervolumeand IQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . 149

8.6 6-node network. Statistical Analysis among the MOEAs in the 6-node network. Thetable indicates in which data sets two algorithms have no statistically significantdifferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.7 6-node network. Comparison among the MOEAs by using the Set Coverage (SC)indicator, A B. Note that, SC represents the mean coverage of an algorithm Aover an algorithm B in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . 155

8.8 COST239 network (TM1). Comparison among the MOEAs by using the Hyper-volume (HV) indicator. The notation used is HVIQR, where HV is the medianhypervolume and IQR is the interquartile range in 30 independent runs. . . . . . . 157



8.11 Statistical Analysis among the MOEAs in the COST239 network. The table indi-cates in which data sets two algorithms have no statistically significant differences. 164

8.12 COST239 network. Comparison among the MOEAs by using the Set Coverage (SC)indicator, A B. Note that, SC represents the mean coverage of an algorithm Aover an algorithm B in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . 165

8.13 NSF network (TM1). Comparison among the MOEAs by using the Hypervolume(HV) indicator. The notation used is HVIQR, where HV is the median hypervolumeand IQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . 167

xviii List of Tables



8.16 Statistical Analysis among the MOEAs in the NSF network. The table indicates inwhich data sets two algorithms have no statistically significant differences. . . . . . 174

8.17 NSF network. Comparison among the MOEAs by using the Set Coverage (SC)indicator, A B. Note that, SC represents the mean coverage of an algorithm Aover an algorithm B in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . 175

8.18 NTT network (TM1). Comparison among the MOEAs by using the Hypervolume(HV) indicator. The notation used is HVIQR, where HV is the median hypervolumeand IQR is the interquartile range in 30 independent runs. . . . . . . . . . . . . . 177



8.21 Statistical Analysis among the MOEAs in the NTT network. The table indicates inwhich data sets two algorithms have no statistically significant differences. . . . . . 184

8.22 NTT network. Comparison among the MOEAs by using the Set Coverage (SC)indicator, A B. Note that, SC represents the mean coverage of an algorithm Aover an algorithm B in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . 185

8.23 Ranking of the MOEAs in the 6-node and COST239 optical network topologieswhen tackling the TG problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.24 Ranking of the MOEAs in the NSF and NTT optical network topologies whentackling the TG problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.25 Mean runtime and standard deviation for the sequential version of the MO-ABCalgorithm in 30 independent runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.26 Mean runtime, speedup, and efficiency for the OpenMP and MPI versions of thepMOABC algorithm with 2 cores in 30 independent runs. Note that we report thestandard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . 193



8.29 Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions ofthe pMOABC algorithm with 16 cores in 30 independent runs. Note that we reportthe standard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . 196

8.30 Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions ofthe pMOABC algorithm with 32 cores in 30 independent runs. Note that we reportthe standard deviation of the runtime. . . . . . . . . . . . . . . . . . . . . . . . . . 197

List of Tables xix

8.31 Summary of the mean speedup and efficiency for the OpenMP+MPI and MPI ver-sions of the pMO-ABC algorithm in all the data sets. . . . . . . . . . . . . . . . . 198

9.1 Test instances and parameters of MOEA/D-NBI for generating the reference points 2079.2 IGD-metric values of the non-dominated solutions found by MOEA/D, MOEA/D-

NBI, NSGA-II∗, and NSGA-II on UF8, UF9, DTLZ1, and DTLZ2 using differentscales. The notation used for pointing the statistically non-significant differences isthe following: (†) NSGA-II∗ and NSGA-II. . . . . . . . . . . . . . . . . . . . . . . . 207

9.3 Amount of Traffic (in OC-1 units) and Runtime (in seconds) per Optical network . 2149.4 Configuration of MOEA/D-NBI and NSGA-II∗∗ . . . . . . . . . . . . . . . . . . . . 2159.5 Comparison between the MOEA/D-NBI and the NSGA-II∗∗ by using the Hyper-

volume (HV) indicator. The notation used is HVIQR, where HV is the medianhypervolume and IQR is the interquartile range in 30 independent runs. . . . . . . 216

9.6 Set Coverage (AB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2199.7 6-node network. Comparison between the MOEA/D-NBI and the five proposed

algorithms in terms of HV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.8 COST239 network. Comparison between the MOEA/D-NBI and the five proposed

algorithms in terms of HV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.9 NSF network. Comparison between the MOEA/D-NBI and the five proposed algo-

rithms in terms of HV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229.10 NTT network. Comparison between the MOEA/D-NBI and the five proposed algo-

rithms in terms of HV. The notation used for pointing the statistically non-significantdifferences is the following: (†) MOEA/D-NBI and MO-GSA. . . . . . . . . . . . . 223

9.11 Summary of the Set Coverage indicator among the diverse MOEAs. We present theaverage, worst, and best coverage relation. . . . . . . . . . . . . . . . . . . . . . . . 226

List of Figures

2.1 A fiber link divided into multiple wavelengths of light (λ) . . . . . . . . . . . . . . 82.2 Illustrative view of the lightpath concept . . . . . . . . . . . . . . . . . . . . . . . . 82.3 A WXCs or wavelength router with nodal degree three and four wavelengths . . . 92.4 Wavelength continuity constraint example . . . . . . . . . . . . . . . . . . . . . . . 102.5 Classification of the different mechanisms of wavelength conversion . . . . . . . . . 102.6 Illustration of different routing approaches . . . . . . . . . . . . . . . . . . . . . . . 122.7 Physical topology (a) with four lightpaths: L1(1-3), L2(1-5), L3(2-5), and L4(5-2);

which forms the Virtual Topology (b). . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 Illustrative example of Multi-hop and Single-hop traffic grooming when it is required

to established the connection c (N1,N2). . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Classification of Metaheuristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Concepts of individual and population in Evolutionary Algorithms. . . . . . . . . . 323.3 Concepts of parent, child, selection, recombination, and mutation in Evolutionary

Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Common selection methods in EAs. . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Common crossover methods in EAs. . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Concept of bitwise mutation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.7 Concepts of Decision Space (Ω) and Objective Space (R2) . . . . . . . . . . . . . . 353.8 Ideal (z∗), utopian (z∗∗), and nadir (znadir) objective vectors. . . . . . . . . . . . . 363.9 Example of a minimization bi-objective problem with four solutions (z1, z2, z3, and

z4) in the objective space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.10 Pair-wise comparison to obtain a set of Non-dominated solutions . . . . . . . . . . 383.11 Pareto optimal solutions for four combinations of two types of objectives and Global

and Local Pareto-optimal concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.12 Number of Solutions and Number of Optimal Solutions quality indicators. . . . . . 423.13 Hypervolume, quality indicator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.14 Epsilon Indicator, Generational Distance, Inverse Generational Distance, and Spread

quality indicators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xxi

xxii List of Figures

3.15 Set Coverage, quality indicator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.16 Concepts of Single and Parallel Computing . . . . . . . . . . . . . . . . . . . . . . 463.17 Shared Memory architecture with Uniform and Non-Uniform Memory Access. . . . 473.18 Example of the Threads Model in a Shared-Memory system. . . . . . . . . . . . . . 483.19 Fork-join execution model in OpenMP. . . . . . . . . . . . . . . . . . . . . . . . . . 493.20 Distributed Memory Architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.21 Example of the Message Passing Model in a Distributed-Memory system. . . . . . 523.22 Master-slave scheme in MPI with four processes. . . . . . . . . . . . . . . . . . . . 523.23 Hybrid Shared-Distributed Memory architecture SMP or SMP/GPU units. . . . . 543.24 Master-slave scheme in a Hybrid OpenMP/MPI model with four processes and four

threads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.25 Possible situations of speedup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.26 Statistical analysis scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1 Illustrative example of the RWA problem. . . . . . . . . . . . . . . . . . . . . . . . 644.2 Representation of the Individuals for the Traffic Grooming problem . . . . . . . . . 65

5.1 Illustrative example of the Traffic Grooming problem. . . . . . . . . . . . . . . . . 765.2 Representation of the Individuals for the Traffic Grooming problem. . . . . . . . . 77

6.1 Sequential operation scheme of the DEPT algorithm. . . . . . . . . . . . . . . . . . 966.2 Parallel operation scheme of the pDEPT algorithm by using OpenMP. . . . . . . . 976.3 Parallel operation scheme of the pDEPT algorithm by using Message Passing Inter-

face (MPI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.4 Hybrid operation scheme of the pDEPT algorithm by using Message Passing Inter-

face (MPI) and OpenMP jointly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.5 Sequential operation scheme of the MO-ABC algorithm. . . . . . . . . . . . . . . . 1006.6 Legend for the operation scheme of the MO-ABC algorithm . . . . . . . . . . . . . 1006.7 Parallel operation scheme of the pMOABC algorithm by using OpenMP. . . . . . . . 1016.8 Parallel operation scheme of the pMOABC algorithm by using Message Passing

Interface (MPI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.9 Hybrid operation scheme of the pMOABC algorithm by using Message Passing In-

terface (MPI) and OpenMP jointly. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.1 The optical network topology: European Optical Network (COST239). . . . . . . . 1097.2 Comparison among NSGA-II, SPEA2 and each proposed MOEA (DEPT, MO-VNS,

MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume ob-tained in 30 independent runs with the COST239 network. . . . . . . . . . . . . . 110

7.3 Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume obtained in 30 independent runswith the COST239 network (1/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.4 Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume obtained in 30 independent runswith the COST239 network (2/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.5 The optical network topology: National Science Foundation (NSF). . . . . . . . . . 1157.6 Comparison among NSGA-II, SPEA2 and each proposed MOEA (DEPT, MO-VNS,

MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume ob-tained in 30 independent runs with the NSF network. . . . . . . . . . . . . . . . . 116

List of Figures xxiii

7.7 Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume obtained in 30 independent runswith the NSF network (1/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.8 Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume obtained in 30 independent runswith the NSF network (2/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.9 The optical network topology: Nippon Telegraph and Telephone (NTT). . . . . . . . 1217.10 Comparison among NSGA-II, SPEA2 and each proposed MOEA (DEPT, MO-VNS,

MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume ob-tained in 30 independent runs with the NTT network. . . . . . . . . . . . . . . . . 122

7.11 Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume obtained in 30 independent runswith the NTT network (1/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.12 Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume obtained in 30 independent runswith the NTT network (2/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.13 Illustrative comparison between the proposed MOEAs and the best Typical Heuris-tics by using the HV indicator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.14 Sets of non-dominated solutions obtained by the best Typical Heuristics and theworst proposed MOEA (MO-GSA). . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.15 Illustrative comparison between the proposed MOEAs and the best MOACO byusing the HV indicator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.16 Sets of non-dominated solutions obtained by the best MOACO and the worst pro-posed MOEA (MO-GSA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.17 Communication and computation time for the OpenMP and MPI versions of thepDEPT algorithm with 2 cores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.18 Communication and computation time for the OpenMP and MPI versions of thepDEPT algorithm with 4 and 8 cores. . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.19 Communication and computation time for the OpenMP+MPI and MPI versions ofthe pDEPT algorithm with 16 cores. . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.20 Communication and computation time for the OpenMP+MPI and MPI versions ofthe pDEPT algorithm with 32 cores. . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.21 Summary of the communication and computation time for the OpenMP+MPI andMPI versions of the pDEPT algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 139

7.22 Summary of the mean speedup and efficiency obtained by the parallel versions ofthe pDEPT in all the data sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.1 Illustrative comparison among the five proposed MOEAs and the well-known NSGA-II and SPEA2, by using the median HV in 30 independent runs . . . . . . . . . . . 144

8.2 The optical network topology: 6-node Network (6-node). . . . . . . . . . . . . . . . 1468.3 6-node network (TM1). Comparison among NSGA-II, SPEA2 and each proposed

MOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.3 for W =1,2,3,4. . . . . . . . . . . . . . 150

8.4 6-node network (TM2). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.4 for W =2,4,6,8. . . . . . . . . . . . . . . 151

xxiv List of Figures

8.5 6-node network (TM3). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.5 for W =3,6,9,12. . . . . . . . . . . . . . 152

8.6 6-node network. Illustrative summary of the performance of each proposed MOEAby using the HV indicator. Note that, each point represents the mean of the mediansof HV reported in Table 8.3 (TM1), Table 8.4 (TM2), and Table 8.5 (TM3) for thedifferent values of W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.7 The optical network topology: European Optical Network (COST239). . . . . . . . 1568.8 COST239 network (TM1). Comparison among NSGA-II, SPEA2 and each proposed


8.9 COST239 network (TM2). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.9 for W =2,4,6,8. . . . . . . . . . . . . . . 161

8.10 COST239 network (TM3). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.10 for W =3,6,9,12. . . . . . . . . . . . . 162

8.11 COST239 network. Illustrative summary of the performance of each proposedMOEA by using the HV indicator. Note that, each point represents the meanof the medians of HV reported in Table 8.8 (TM1), Table 8.9 (TM2), and Table8.10 (TM3) for the different values of W . . . . . . . . . . . . . . . . . . . . . . . . 163

8.12 The optical network topology: National Science Foundation (NSF). . . . . . . . . . 1668.13 NSF network (TM1). Comparison among NSGA-II, SPEA2 and each proposed


8.14 NSF network (TM2). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.14 for W =2,4,6,8. . . . . . . . . . . . . . . 171

8.15 NSF network (TM3). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.15 for W =3,6,9,12. . . . . . . . . . . . . 172

8.16 NSF network. Illustrative summary of the performance of each proposed MOEA byusing the HV indicator. Note that, each point represents the mean of the mediansof HV reported in Table 8.13 (TM1), Table 8.14 (TM2), and Table 8.15 (TM3) forthe different values of W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.17 The optical network topology: Nippon Telegraph and Telephone (NTT). . . . . . . 1768.18 NTT network (TM1). Comparison among NSGA-II, SPEA2 and each proposed


8.19 NTT network (TM2). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.19 for W =2,4,6,8. . . . . . . . . . . . . . . 181

8.20 NTT network (TM3). Comparison among NSGA-II, SPEA2 and each proposedMOEA by using the HV indicator. Note that, each point represents the mean ofthe medians of HV reported in Table 8.20 for W =3,6,9,12. . . . . . . . . . . . . 182

List of Figures xxv

8.21 NTT network. Illustrative summary of the performance of each proposed MOEA byusing the HV indicator. Note that, each point represents the mean of the mediansof HV reported in Table 8.18 (TM1), Table 8.19 (TM2), and Table 8.20 (TM3) forthe different values of W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.22 Best set of non-dominated solutions obtained by the best (∗) and the worst ()approach at each scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.27 Communication and computation time for the OpenMP and MPI versions of thepMO-ABC algorithm with 2 cores. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194



8.30 Communication and computation time for the OpenMP+MPI and MPI versions ofthe pMO-ABC algorithm with 16 cores. . . . . . . . . . . . . . . . . . . . . . . . . 197

8.31 Communication and computation time for the OpenMP+MPI and MPI versions ofthe pMO-ABC algorithm with 32 cores. . . . . . . . . . . . . . . . . . . . . . . . . 198

8.32 Summary of the communication and computation time for the OpenMP+MPI andMPI versions of the pMO-ABC algorithm. . . . . . . . . . . . . . . . . . . . . . . . 199

8.33 Summary of the mean speedup and efficiency obtained by the parallel versions ofthe MO-ABC in the 6-node. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

8.34 Summary of the mean speedup and efficiency obtained by the parallel versions ofthe MO-ABC in the NSF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.1 A set of uniformly distributed solutions in the normalized objective space may notbe uniformly distributed in the original (real) objective space. . . . . . . . . . . . . 203

9.2 Distribution of ri on the plane Π (CHIM) which contains the points F 1, F 2, and F 3 2049.3 Illustrative example of the DistributeReferencePoints procedure. . . . . . . . . . . 2069.4 Evolution of the mean of IGD-metric value versus the number of generations on UF8.2089.5 Evolution of the mean of IGD-metric value versus the number of generations on UF10.2099.6 Evolution of the mean of IGD-metric value versus the number of generations on

DTLZ1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2099.7 Evolution of the mean of IGD-metric value versus the number of generations on

DTLZ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2109.8 Comparison between MOEA/D-NBI and NSGA-II∗∗ by using the HV indicator.

Note that, each point represents the mean of the medians of HV reported in Table9.5 for W =3,4 (TM1), W =4,6 (TM2), and W =6,9 (TM3) – (PART 1/2). . 217

9.9 Comparison between MOEA/D-NBI and NSGA-II∗∗ by using the HV indicator.Note that, each point represents the mean of the medians of HV reported in Table9.5 for W =3,4 (TM1), W =4,6 (TM2), and W =6,9 (TM3) – (PART 2/2). . 218

9.10 Comparison among DEPT, MO-VNS, MO-ABC, MO-GSA, MO-FA, and MOEA/D-NBI by using the HV indicator. Note that, each point represents the mean of themedians of HV reported in Table 9.7 (6-node), Table 9.8 (COST239), Table 9.9(NSF), and Table 9.10 (NTT) for W =3,4 (TM1), W =4,6 (TM2), and W =6,9(TM3) – (PART 1/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

xxvi List of Figures

9.11 Comparison among DEPT, MO-VNS, MO-ABC, MO-GSA, MO-FA, and MOEA/D-NBI by using the HV indicator. Note that, each point represents the mean of themedians of HV reported in Table 9.7 (6-node), Table 9.8 (COST239), Table 9.9(NSF), and Table 9.10 (NTT) for W =3,4 (TM1), W =4,6 (TM2), and W =6,9(TM3) – (PART 2/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

9.12 Best set of non-dominated solutions obtained by the MOEA/D-NBI (∗) and the bestproposed MOEA () at each scenario. . . . . . . . . . . . . . . . . . . . . . . . . . 227

9.13 Comparison between MOEA/D-NBI and other approaches published in the litera-ture by using the 6-node network and W =3. . . . . . . . . . . . . . . . . . . . . . . 229

9.14 Comparison between MOEA/D-NBI and other approaches published in the litera-ture by using the 6-node network and W =4. . . . . . . . . . . . . . . . . . . . . . . 230

1Introduction

In the last decades, the designing and developing of efficient algorithms for solving complex prob-lems have been mainly studied in Computer Science research. Thus, the main goal is to providenew methods to solve these complex problems minimizing the required computational efforts and,consequently, improve the previous methods. In this way, the development of new methods not onlyallow us to handle new complex problems, but also tackling diverse tasks which were impossibleto deal with in the past due to their extremely computationally and/or financially expensive.

More and more engineering problems which require time-consuming computer simulation orexpensive physical experiments appear every day. Fortunately, the technology and the computa-tional resources evolve and improve constantly as well. Therefore, recent years have witnessedsignificant progress in the development of metaheuristics and exact heuristics for solving thesecomplex engineering problems and make the most of the new available computational resources.

On the one hand, the exact methods are really good approaches for certain problems due tothey are able to provide optimal solutions. However, the required computational efforts and theruntime of the method become prohibitive when the complexity of the problem raises; in addition,there exist many engineering problems that are not easily handle with exact heuristics due to itsnature. On the other hand, the metaheuristics are generally quick methods that lead to near-optimal solutions in a reasonable time; therefore, the use of metaheuristics provide a good balancebetween quality in the solutions found and the time spent in obtaining them.

In the past years, three technological developments have influenced the status of our currenttelecommunication. In the 60s, the research were focused on digital signal transmission insteadof analog transmission, which made it possible to transmit not only voice but also any kind ofinformation, emerging the Internet as a result. In the decade 1980-1990, the wireless telephonyattracted to much attention for consumers due to miniaturization of mobile telephone componentsas well as new compression techniques for digital speech. Finally, in the last decades, the opticaltransmission components are replacing the old electrical equipment due to its huge bandwidth.Therefore, it is not strange to find tens of complex optimization problems (generally NP-hardproblems) in this field. The vast majority of them aims to optimize, not only the design oftelecommunication systems, but also their operation. At first, the use of exact heuristics fordealing with Telecommunication problems was widely applied. However, due to their capabilitiesfor obtaining high quality solutions in a reasonable time, a quickly move forward the developmentof metaheuristics appeared.

1

2 1. Introduction

This is the scope of this PhD Thesis, the development of metaheuristics methods for solvingreal-world optimization problems from the Telecommunication industry. Concretely, we focuson optimizing optical network problems due to this kind of networks has attracted much moreattention from the Telecommunication industry in the last decade. The two problems tackledare: Routing and Wavelength Assignment problem and the Traffic Grooming problem. Due tothe nature of these telecommunication problems, we have developed diverse metaheuristics byusing multiobjective optimization and parallel computing. On the one hand, we use multiobjectiveoptimization because the two problems consist of two or more conflicting objective functions whichneed to be optimize simultaneously. On the other hand, depending on the size of the networktopology or the amount of traffic given, the runtime may be prohibitive; thus, we use parallelcomputing for obtaining, not only high quality results, but also in a reasonable time for theindustry.

1.1 Objectives and Steps

The objectives of this thesis are: propose innovative multiobjective metaheuristics methods toreal-world optimization problems from the Telecommunication field, study the behaviour of theseproposals in a wide variety of test problems, compare them with other approaches published in theliterature, and use parallel computing for reducing the runtime.

Since the aim of any scientific inquiry is to obtain knowledge, in this thesis, we have followed asequence of steps to reach this goal. A brief description of each step is listed below:

1. Study the state-of-the-art. As a result, we focus on optimizing optical networks due toits importance in the last decades.

2. Select the specific optical network problems. We select the Routing and WavelengthAssignment problem and the Traffic Grooming problem because they have been widely tack-led in the literature with single-objective optimization techniques, but there are not mucheffort in solving them by using multiobjective optimization.

3. Study the characteristics of each problem. We study the nature of each problemand how they have been tackled by other authors with the aim of extracting the essentialinformation for developing effective and efficient multiobjective metaheuristics.

4. Development of the multiobjective metaheuristics and their possible paralleliza-tion. After an exhaustive study of the literature, we select different techniques and al-gorithms, and we develop them for dealing with the selected problems. Some of these tech-niques/algorithms have been parallelized in order to study the improvement of their runtime.

5. Experimental phase. We carefully select a set of test instances for making a rigorous com-parison among the multiobjective approaches and other methods published in the literature.

6. Analyze the results. We study the behaviour of the approaches on the different scenarios.

7. Conclusions. We describe all the conclusions obtained from our inquiry.

In the process of providing new knowledge, we also take into account another components, suchas: statistical reliability, replication, and external review. In the first place, since we are dealingwith stochastic algorithms, we ensure a certain level of confidence in all of our experiments by

1.2 Contributions 3

analyzing statistically the results. Secondly, we provide all the necessary information about ourexperiments in order to allow the replication of the results obtained. Finally, we have submittedthe results showed in this thesis to different scientific journals or conferences with a peer reviewprocess, which ensure a certain quality in the research work presented.

1.2 Contributions

In this section we present and briefly describe the main contributions of this PhD thesis work. Themain contributions are listed as follows:

• Innovative definitions for linking the two real-world Telecommunication problems tackled inthis thesis, to the evolutionary algorithms world. In other words, a new bridge between eachoriginal problem context and the problem-solving space where evolution takes place.

• Development of five new multiobjective metaheuristics for solving these optical network prob-lems. We briefly describe each one as:

– Differential Evolution with Pareto Tournaments (DEPT). A multiobjective variant ofthe standard Differential Evolution.

– Multiobjective Variable Neighbourhood Search (MO-VNS). A new trajectory-based al-gorithm based on the Variable Neighbourhood Search for dealing with optimizationproblems with multiple objective functions.

– Multiobjective Artificial Bee Colony Algorithm (MO-ABC). A multiobjective approachbased on the behaviour of honey bees.

– Multiobjective Gravitational Search Algorithm (MO-GSA). A multiobjective methodbased on the law of motion and mass interactions.

– Multiobjective Firefly Algorithm (MO-FA). A swarm intelligence algorithm that use thebioluminescent aptitudes of fireflies to attract other ones and optimize a multiobjectiveproblem.

• Design of two parallel algorithms (pDEPT and pMOABC) for exploiting different multi-core systems in the same interconnecting data network. These models are able to speedthe runtime of the algorithms, obtaining identical high-quality solutions than the sequentialmodel in a reasonable time. It is the first time that these parallelization techniques areapplied to these real-world optical problems.

• A comprehensive comparison among diverse multiobjective metaheuristics for solving eachproblem. This study includes comparisons with the proposals of other authors, comparisonswith well-known standard multiobjective algorithms (such as NSGA-II or SPEA2), as wellas comparisons among all our multiobjective proposals. At each comparison, we use a widespectrum set of instances for measuring the performance of the metaheuristics in very differentscenarios.

• Since the complexity of the Traffic Grooming problem increases exponentially with the num-ber of nodes of the tackled topology, no approach in the literature has ever dealt with theoptical network Nippon Telegraph and Telephone (55 nodes and 144 optical links). In thisthesis, we use our approaches for solving the Traffic Grooming problem in this large opticalnetwork.

4 1. Introduction

• An indirect encoding model using a construction heuristics for solving the Traffic Groomingproblem. In this method, we use a two-phase metaheuristics for constructing a solution tothe problem.

• A formal definition of a new multiobjective framework based on decomposition for solv-ing real-world 3-objective optimization problems with objectives in very different scales. Acomparison between this framework based on decomposition and the standard Fast Non-dominated Sorting Genetic Algorithm (based on Pareto dominance) is presented with theaim of proving the effectiveness of the new framework in a real-world 3-objective optimiza-tion problem such as the Traffic Grooming problem.

In addition, we have spread the scientific knowledge obtained from this research to diversescientific international journals and several international and national conferences in the field. Wehave also developed a web-page for each problem with the aim of providing to future researchers,the description of these telecommunication problems and all the data sets used in our research.

1.3 Thesis Organization

This thesis consists of different chapters. We start describing basic concepts in the telecommuni-cation field, multiobjective optimization, and parallel computing. In the second place, we describethe proposed metaheuristics and the results obtained in the experimental phase. Then, we discussthe conclusions obtained in this research, as well as future lines of work. Finally, we report anappendix which contains the scientific merits obtained from this inquiry. A detailed description ofeach chapter is presented as follows:

• Chapter 1 Introduction. In this introductory chapter, we present a brief overview of theresearch work tackled in this thesis, the objectives and steps, and the contributions of thisinquiry. Furthermore, we include a brief description of each chapter in order to provide aglobal view of the document.

• Chapter 2 Optical Networks: Fundamentals and Background. In the second chapter,we present a description of the main concepts of Wavelength Division Multiplexing technology,as well as introducing the two optimization problems tackled in this thesis: Routing andWavelength Assignment and Traffic Grooming. For both problems, we describe the mainfundamentals and present a literature survey.

• Chapter 3 Proposed Methodology. In this chapter, we present diverse concepts re-garding metaheuristics, multiobjective optimization, and parallel computing. Furthermore,we formally describe different indicators for measuring the quality of multiobjective evolu-tionary algorithms, as well as the most common performance metrics for evaluating parallelapproaches.

• Chapter 4 Routing and Wavelength Assignment problem. A formal description of theRouting and Wavelength Assignment (RWA) problem is reported in this chapter, includingall the assumptions, variables, parameters, and constraints involved. Furthermore, we presentan illustrative example to help to understand the problem formulation. Since in this thesis weuse Multiobjective Evolutionary Algorithms (MOEAs), we present the representation of theindividuals used, which determines how the problem is structured in the algorithms. Finally,

1.3 Thesis Organization 5

we present the optical networks and data sets used in the experimental results for the RWAproblem.

• Chapter 5 Traffic Grooming problem. In this chapter, we present a formal description ofthe Traffic Grooming problem, including a general problem statement, the related parameters,variables, constraints, objective functions, and an illustrative example of the problem. Likein Chapter 4 (RWA problem), we present the individuals encoding used in the MultiobjectiveEvolutionary Algorithms (MOEAs). Finally, we present the optical networks and data setsused in the experimental results for the Traffic Grooming problem.

• Chapter 6 Multiobjective Evolutionary Algorithms. In this chapter we detail the fiveproposed MOEAs for solving the RWA problem and the Traffic Grooming problem. Thesealgorithms are multiobjective variants of the following evolutionary algorithms: DifferentialEvolution, Variable Neighbourhood Search, Artificial Bee Colony, Gravitational Search Algo-rithm, and Firefly Algorithm. In this chapter, we also describe two well-known MOEAs: FastNon-dominated Sorting Genetic Algorithm (NSGA-II) and the Strength Pareto EvolutionaryAlgorithm 2 (SPEA2). Finally, we present the parallelization of two of the five aforementionedMOEAs. For each parallel MOEA, we present three parallel operation schemes that exploitsystems with different memory architecture: shared, distributed, or shared-distributed.

• Chapter 7 Solving the RWA problem. The main aim of this chapter is to use multiob-jective evolutionary computation to solve the Routing and Wavelength Assignment problem.In the first place, we use a preliminary set of twelve data sets in order to tune the parametersof each MOEA, comparing their results in order to check the goodness of each approach.Then, we present a comparative study among our five algorithms and the two well-knownMOEAs when tackling three optical networks with different number of nodes as well as di-verse amounts of traffic. In addition, we present a comparison among the MOEAs and diversetechniques published in the literature for solving the RWA problem. Finally, since the RWAproblem is an NP-hard problem, we have used a parallel version of the Differential Evolutionwith Pareto Tournaments (pDEPT) in order to obtain high quality results in a reasonableamount of time for the industry.

• Chapter 8 Solving the Traffic Grooming problem. In this chapter we apply diversemultiobjective evolutionary algorithms (MOEAs) for solving the Traffic Grooming problem.We start tuning the parameters of each MOEA with two optical networks and two smalltraffic matrices, a total of 13 scenarios. Then, we study the performance of each MOEAwhen solving four optical networks with different sizes and different loads of traffic. In thisstudy, we compare each MOEA with two standard multiobjective approaches (NSGA-II andSPEA2) with the aim of proving their goodness solving this problem. In addition, we checkthe accuracy of the proposals by making a comparison with several techniques publishedin the literature that have dealt with the Traffic Grooming problem. Finally, since TrafficGrooming is an NP-hard problem that has proven to be computationally intractable whenthe given topology contains a large number of nodes, we apply a parallel multiobjectiveevolutionary algorithm based on the Artificial Bee Colony algorithm (pMOABC), that notonly solves the Traffic Grooming problem in a reasonable time, but also in an efficient way.

• Chapter 9 MOEA/D-NBI for 3-objective optimization problems. In MultiobjectiveOptimization the objective functions may have different scales, leading to a neglecting ofone or more objective functions as a result. In the last decades, diverse multiobjective

6 1. Introduction

evolutionary algorithms (MOEAs) have proposed the normalization of the objective spacewith the aim of solving this drawback. However, in the case of more than two objectives, aset of uniformly distributed solutions in the normalized objective space may not be uniformlydistributed in the original objective space. Therefore, in this chapter we propose an innovativeframework based on decomposition (MOEA/D-NBI) for solving this drawback in 3-objectiveoptimization problems. Since the Traffic Grooming is a 3-objective optimization problemand their three objectives are commonly in very different scales, we prove the effectivenessof the new proposal solving this telecommunication problem. Furthermore, we propose theuse of indirect encoding using a construction heuristics within the MOEA/D-NBI.

• Chapter 10 Conclusions and future work. In this chapter we summarize the main con-clusions obtained from the set of experiments and work presented in this thesis. Furthermore,since the number of plausible lines of work is large, we briefly describe the main ones.

• Chapter 11 Scientific Achievements. In this chapter we present the papers that havebeen published as a result of the research work developed throughout this PhD Thesis. Fur-thermore, we include other scientific achievements, such as International Stays, reviewer ofInternational Journals, participation in International Conferences, participation in ResearchProjects, and collaborations with other institutions.

2Optical Networks: Fundamentals and Background

The advances in semiconductor products was essential for the growth of networking that led toimprovements in the long-distance communication infrastructures in the 20th century [1]. Nowa-days, the major network of networks is the Internet. This network provides many data services,from the traditional e-mail or the access to the World Wide Web, to new services such as Voiceover Internet Protocol (VoIP) and Internet Protocol Television (IPTV). In addition, the Internethas enabled and accelerated new forms of human interactions through instant messaging, forums,and social networking, as well as online shopping.

Since the Internet traffic volume has grown considerably over the last decades, data traffic nowexceeds voice traffic. Various methods have evolved to provide high levels of Quality of Service(QoS), such as the use of light in communications. The research into optical fibers and theirapplications, revolutionized the communications industry in the 1990s and 2000s.

In optical networks, the telecommunication transmission lines employ light signals to carry dataover guided channels, known as optical fibers. The transmission signals over this guided medium,unlike air, with a very low attenuation and bit-error rates makes optical fibers a useful choice forthe medium of communication for next-generation high-speed networks.

Initially, a migration from electronic to optical transmission technology was achieved by onlyreplacing copper wires by fibers. It leads to a high reliability in data transmissions, an improvementof the signal-to-noise ratio, and a reduction bit-error rates in telecommunications. The traditionalTime Division Multiplexing (TDM), was employed in the first generation in order to allow multipleusers to share the bandwidth of a fiber link - the bandwidth sharing is in the time domain. Thesynchronous optical networks (SONET) are the most popular networks in this category. They arebased on a ring-architecture and employ circuit-switched connections in order to carry voice anddata. Other multiplexing techniques appear to make the most of these kinds of networks, such asWavelength Division Multiplexing (WDM).

2.1 Wavelength Division Multiplexing

The available bandwidth on each optical fiber link is approximately 50Tbps. Despite this hugebandwidth, the maximum speed for an end user is constrained by the processing speed of theirdevices (a few Gbps). Hence, the key to making the most of optical networks is to introduce

7

8 2. Optical Networks: Fundamentals and Background

Figure 2.1. A fiber link divided into multiple wavelengths of light (λ)

concurrency in transmission of data. This is the goal of Wavelength Division Multiplexing (WDM)technology: to use several wavelengths of light (λ) for multiplexing multiple signals. In this way,by using WDM technology, several wavelengths can be multiplexed to or demultiplexed from thesame optical fiber link. In Figure 2.1, we present a wavelength division multiplexing of an opticalfiber link.

The mechanism of this technology divides the bandwidth space into smaller portions. There-fore, the multiplexing is said to occur in the space domain [1], where diverse connections betweendifferent source and destination nodes, can share the available bandwidth on a link using differ-ent wavelength channels. Depending on the space, we can differentiate into three categories ofWavelength Division Multiplexing networks:

• WDM. It supports from 2 to 4 wavelengths per fiber with 1310-1550nm or nanometer ofspacing between the wavelengths. The original system was dual channels.

• Coarse WDM (CWDM). It supports from 4 to 8 wavelengths per fiber with 10- 20nm ofspacing between the wavelengths. Designed for Regional and Metropolitan area networks.

• Dense WDM (DWDM). It supports 8 or more wavelengths per fiber with 1-2nm of spacing.The newer systems are able to support hundreds of wavelengths.

These days, there exist several relevant telecommunication companies in USA, Europe, andJapan which are testing and using several prototypes of WDM optical networks. Therefore, every-thing points to the future of the Internet will be based on WDM.

In [2] and [3], the authors define wavelength routing to the use of wavelength to route data, theyalso refer to networks that employ this technique as wavelength-routed networks. A traffic requestestablished end-to-end from one node to another in an optical network over a specific wavelengthof light (λ) is commonly known as lightpath. In Figure 2.2, we present an illustrative view of thelightpath concept. We may observe in Figure 2.2, a lightpath for connecting the nodes N2 and N4

over the first wavelength (λ1).

Figure 2.2. Illustrative view of the lightpath concept

2.1 Wavelength Division Multiplexing 9

Figure 2.3. A WXCs or wavelength router with nodal degree three and four wavelengths

In an optical transmission system we can find three basic components: transmitter, transmissionmedium, and receiver [4]. The binary data information is converted into a sequence of light pulseswhich are transmitted through the medium, the optical fiber link. Once the light pulses arrive todestination, they are converted back to an electrical signal by using an optical detector. Therefore, aWDM system uses a multiplexer at the transmitter to join the signals together, and a demultiplexerat the receiver to split them apart. In general networks, there are more than one input and outputline at each node; therefore, we need a Wavelength Cross-Connects (WXCs) to route, add, orterminate wavelength connections through the network. In this way, they are able to connect(switch) an individual wavelength from one link to another, avoiding bottlenecks [5]. In Figure 2.3,we present a WXCs that allows wavelength conversion. Note that, it consists of three incomingand outgoing fiber links with a capacity of four wavelengths on each link.

In wavelength-routed WDM network, a lightpath is determined by the location of the signaltransmitter, the wavelength on which is transmitted, and the state of the network devices. Animportant constraint in WDM networks is the wavelength continuity constraint, that is, the samewavelength must be assigned to a connection on every link [6].

For example, let suppose a network topology with two wavelengths per link (see Figure 2.4(a)),where there are two lightpaths established: N1-N3(λ1) and N3-N4(λ2). On the one hand, if wesuppose that the nodes are equipped with WXCs (Figure 2.4(b)), we can establish a lightpathfrom the node N2 to the node N4 switching the wavelength from λ2 to λ1 at the intermediate nodeN3. On the other hand, if the nodes do not support wavelength conversion, it is not possible toestablish this lightpath. However, as is shown in Figure 2.4(c), a feasible solution for allocatingthe lightpath N2-N4 is to increase the number of wavelengths per link to three, rising the cost ofthe network as a result.

The wavelength continuity constraint restricts a connection to occupy the same wavelength onevery link of a chosen physical path from a source node to a destination node. Assuming thisconstrain could reject a data request even though the required capacity is available on all the linksof the path but not on the same wavelength. As we may observe in Figure 2.4(c), the reason forrejecting a request is due to the intermediate nodes are not equipped with WXCs that provideswitching from one wavelength to another on two consecutive links.

However, this WXCs are also expensive devices; thus, may be not all nodes in the network need


(a): Initial state of the network (b): Not assuming the wavelength continuity constraint

(c): Assuming the wavelength continuity constraint

Figure 2.4. Wavelength continuity constraint example

(a): No wavelength conversion (b): Fixed-wavelength conversion

(c): Limited-wavelength conversion (d): Full-wavelength conversion

Figure 2.5. Classification of the different mechanisms of wavelength conversion

to be equipped with these components for switching. This concept is known as sparse-wavelengthconversion networks.

Depending on the range of wavelength conversion, we can classified the wavelength conversionmechanisms as follows:

• Fixed-wavelength conversion. It allows the signal to be converted from a specific inputwavelength to a fixed output wavelength. The choice of output wavelength is fixed for aninput wavelength, hence the name.

• Full-wavelength conversion. If the signal on a wavelength can be converted into any otherwavelength.

• Limited-wavelength conversion. If the signal can be converted from one wavelength toa set of, but not all, wavelengths.

2.2 Routing and Wavelength Assignment 11

In Figure 2.5 we present these types of wavelength conversion with the aim of clarifying them.As we may observe in Figure 2.5, each optical fiber consists of five incoming wavelength of lights(λ1,λ2,. . . ,λ5) and five outgoing wavelengths.

After a brief description of the main concepts of Wavelength Division Multiplexing technol-ogy, in the following sections we focus on two different optimization problems for improving theperformance of this kind of networks: Routing and Wavelength Assignment problem and TrafficGrooming problem.

2.2 Routing and Wavelength Assignment

In WDM optical networks, one of the main challenges is to provide a suitable route and a wavelengthamong the many possible choices for establishing a single connection between a source node and adestination node. A good routing and wavelength assignment are critically important to improvethe performance of WDM networks.

Therefore, a problem comes up when is required to interconnect a set of connection requests.This problem is known in the literature as Routing and Wavelength Assignment problem (RWA)[7].

Typically, the connection requests may be of two types:

• Static. The entire set of connections is known in advance, and the problem is then to setup lightpaths for these connections minimizing network resources such as the number ofwavelengths or the number of fibers in the network. Alternatively, one may attempt to setup as many of these connections as possible for a given fixed number of wavelengths.

• Dynamic. In this case, a lightpath is set up for each connection request as it arrives, and itis released after some finite amount of time. There exist also the incremental-traffic case, inwhich the connections arrive sequentially, a lightpath is established for each connection, andthe lightpath remains in the network indefinitely. The objective in these models of trafficis to establish lightpaths and assign wavelengths in manner that minimized the amount ofconnection blocking, or that maximizes the number of connections that are set up in theoptical network at any time.

The Routing and Wavelength Assignment in these optical networks has been proven to be NP-complete by Chlamtac and Ganz [8] and thus is a challenging optimization problem of practicalrelevance. A number of research works tackled this problem using a wide range of optimizationtechniques. In order to make the problem more tractable, the vast majority of authors divide theproblem into two subproblems: routing and wavelength assignment; solving each subproblem sepa-rately. However, in more recent literature, the RWA problem has been tackled as a multiobjectiveoptimization problem.

In the following subsections, we describe the traditionally routing and wavelengths assignmentmethods applied to solve each subproblem. Then, an overview of the RWA state of the art isshown, in which we review the different heuristics and metaheuristics proposed for solving thisoptimization problem in the literature.

2.2.1 Routing

As we mentioned in the previous section, the RWA problem may be divided into two subproblems:routing and wavelengths assignment. In this section we briefly describe diverse approaches forrouting connection requests.


(a): Fixed routing. (b): Fixed-Alternate routing. (c): Adaptive routing.

Figure 2.6. Illustration of different routing approaches

There exist three approaches for routing handle the routing subproblem:

• Fixed routing. Using always the same path for a given pair of nodes is the most straightfor-ward approach to routing a connection. For example, we can use a shortest-path approachfor computing the minimum cost route from the source node to the destination node, Di-jkstra or Bellman-Ford algorithms are examples of methods commonly used for calculatingthe shortest-path between a given pair of nodes [9]. In Figure 2.6(a), the fixed route fromN1 to N3 is N1 − N2 − N3. As we may observe, this approach is very simple; however,the main disadvantage is that, if the resources (wavelengths) along the fixed route are busy,the connection will not be established. This situation can occur frecuently with more thanone connection demand; therefore, the use of a fixed routing approach may result in a largenumber of wavelengths per fiber link.

• Fixed-Alternate routing. In this approach, each node in the network is required to main-tain a routing table that contains an ordered list of a number of fixed paths to each destinationnode. For example, these routes may include the shortest-path route, the second-shortest-path route, the third-shortest-path route, and so on. One of the most famous algorithms forcomputing the k shortest paths is the Yen’s algorithm [10]. Figure 2.6(b) illustrates the twoshortest paths for the pair of nodes (N1,N3), which are: N1−N2−N3 and N1−N4−N5−N2.In this way, when a connection demand arrives, the source node (N1) attempts to set up theconnection on each of the paths from the routing table in sequence, until a route with a validwavelength assignment is found. The main advantage of this method is that, depending onthe number of the alterative routes, the total cost of the network may be reduced; however,a disadvantage is that the complexity increases with the size of the network.

• Adaptive routing. Given a pair of nodes, the route from the source node to the destina-tion node is selected dynamically, depending on the network state. We can determine thenetwork state by the set of connections that are currently in established. For example, anadaptive routing technique may be based on the shortest-path routing taking into accountthe wavelengths used at each optical link. In this way, each unused optical fiber link in thenetwork has a cost equal to the number of available wavelengths (W ) and a cost equals to 0if there is no available channel. Thus, when a connection arrives, we select the shortest-costpath between the source node and the destination node; in the event of tie, it is usuallychosen randomly. In Figure 2.6(c), we present an example of the aforementioned techniquewhere the number of wavelengths per link is equal to one (W = 1). As we may observe,a connection from node N1 to N2 is required and it is set up using the shortest-cost path


N1−N4−N5−N6−N3 due to the links N2−N3 and N5−N3 do not present any availablewavelength (the cost is 0). This technique based on adaptive routing requires extensive sup-port from the control and management protocols to continuously update the routing tables atthe nodes; however, an advantage of adaptive routing is that it results in better performancethan fixed and fixed-alternate routing.

2.2.2 Wavelength Assignment

In this section we present diverse techniques to face the wavelength assignment subproblem. Inthe literature, this subproblem has been tackled as a static problem as well as a dynamic problem.

On the one hand, given a set of lightpaths and their routes, assign a wavelength to eachlightpath such that no two lightpaths share the same wavelength on a given fiber link (assumingthe wavelength continuity constraint). The most common approach to solving this subproblem isto formulate it as a graph-coloring problem [11].

On the other hand, the most common dynamic wavelength assignment techniques are: Random,First-Fit, Least-Used, and Most-Used. Now, we briefly describe how each technique works:

• Random (R). In this approach, we first search the space of wavelengths in order to determinethe set of all wavelengths that are available on the required route. Among the availablewavelengths, we choose one randomly (usually with uniform probability).

• First-Fit (FF). The first step in this method is to number all wavelengths. When it is needto assign a wavelength, we always consider to use the lower-numbered wavelength before ahigher-numbered wavelength. Then, the first available wavelength is chosen. Note that thisapproach requires no global information; therefore, compared to the Random wavelengthassignment, the computational efforts are lower because there is no need to search the entirewavelength space for each route. Typically, this technique is preferred in practice because ofits small computational overhead and low complexity. Similar to Random, First-Fit does notintroduce any communication overhead because no global knowledge is required.

• Least-Used (LU). In order to balance the load among all the wavelengths, in this methodwe select the least used wavelength in the network. According to [12], this technique isnot preferred in practice because some additional storage, computation cost are required.Furthermore, it introduces additional communication overhead for computing the least-usedwavelength.

• Most-Used (MU). It is the opposite of LU; thus, it attempts to select the most-usedwavelength in the network. The communication overhead, storage, and computation cost areall similar to those in LU.

There exist other wavelength assignment published in the literature. Some examples of thesetechniques for assigning wavelengths are: Min-Product, Least-Loaded, MAX-SUM, or Relative Ca-pacity Loss. In [13], the authors propose an approach to pack wavelengths into fibers with theaim of minimizing the number of fibers in the network, they refer to it as Min-Product. TheLeast-Loaded [14] algorithm selects the wavelength that has the largest residual capacity on themost-loaded link along route. With the aim of maximizing the remaining path capacities after thelightpath establishment, the approaches MAX-SUM [15] and Relative Capacity Loss [16] considerall possible paths (lightpaths with their preselected routes) in the network.


2.2.3 State of the Art

In the 1990s, the Routing and Wavelength Assignment problem emerged when optical networkswith full-optical switching at nodes became available. In [8], Chlamtac and Ganz proposed a novelarchitectural approach that meets the high bandwidth requirements by introducing a communi-cation architecture based on lightpaths. They proved that the problem of optimally establishinglightpaths is NP-complete.

In [17], Chan and Yum analyzed an adaptive routing rule in a WDM lightwave network. Theyfound that without any wavelength converters, the wavelength conflict possesses an inherent block-ing to alternate route traffic and that the use of wavelength converter to resolve wavelength conflictsdoes not give any significant reduction of blocking probability.

A genetic algorithm to optimize the allocation plan on the eleven central nodes of the EuropeanOptical Network was presented in [18]. They produced several route and wavelength allocationplans for the network, proving that genetic algorithms are more effective than other robust opti-mization heuristic in solving this problem.

Ramaswami and Sivarajan [7] solved the static-RWA problem by using mixed-integer linearprogramming. In order to tackle the optimization problem, they suggested dividing it into twosubproblems: routing , and wavelength assignment - solving each one separately.

In [19], the authors employed a iterative approach which combines simulated annealing to searchfor a good virtual topology and flow deviation for optimal routing of packet traffic on the virtualtopology.

Banerjee and Mukherjee [20] proposed a practical approach to solve routing and wavelengthassignment of lightpaths in large networks. They suggested that a large RWA problem may bepartitioned into several smaller subproblems, each of which may be solved independently and effi-ciently using well-known approximation techniques. In this way, they employed a multicommodityflow formulation combined with randomized rounding to calculate the routes for lightpaths, andgraph-coloring techniques for wavelength assignment.

In [21], Banerjee and Chen developed a unified method for the routing and wavelength assign-ment subproblems. They also introduced an analytical model for predicting the average numberof lightpaths requests a given physical network can support.

An alternate routing method with limited trunk reservation in which connections with morehops are prepared more alternate routes is presented in [22]. The performance improvement wasinvestigated by introducing a wavelength assignment policy and a dynamic routing method.

Sinclair [23] used a genetic algorithm (GA)/heuristic hybrid approach for minimum cost routingand wavelength allocation in multi-wavelength all-optical networks. A comparison between thehybrid approach and three heuristics was performed in this work.

In [24], Hyytiä and Virtamo considered that once routes are fixed, the wavelength assignmentis essentially a graph coloring problem. In this way, several heuristic methods for coloring a givengraph were studied. Furthermore, they presented an iterative algorithm (Tabu Search) for findinga reasonably good routing and wavelength assignment.

Ant Colony Optimisation (ACO) was applied in [25] to the RWA problem in multi-wavelengthall-optical virtual-wavelength-path routed transport network. Commonly, ants are attracted bythe pheromone trail of ants from their own colony, in [25], the artificial ants are also repelled bythe pheromone of other colonies. They presented several results of the ACO algorithm on small-and medium-sized networks.

Two dynamic routing algorithms based on path and neighbourhood link congestion in all-opticalnetworks were proposed in [26] for solving the routing subproblem in the RWA problem with a


dynamic traffic pattern. They compared an alternate routing method and their dynamic routingapproaches by using the first-fit heuristics for wavelength assignment.

Li and Shima [27] defined a new coloring problem motivated by the wavelength routing andassignment problem in WDM networks. They showed that this new coloring problem is as hardas the standard vertex coloring. Furthermore, they proposed some heuristics for this coloringproblem.

A comprehensive review of the RWA problem can be found in [12]. They examined the RWAproblem and reviewed various routing approaches and wavelength assignment techniques proposedin the literature. They also proposed a new wavelength assignment scheme, called DistributedRelative Capacity Loss (DRCL).

Ramamurthy and Mukherjee [28] proposed an approximate analytical model that incorporatesalternate routing and sparse wavelength conversion. They demonstrate that alternate routing canimprove the blocking performance of optical networks without wavelength conversion facility in allnodes.

In [29], the authors explored an alternative solution technique in the well-known maximum edgedisjoint paths (EDP) problem adapted to the RWA problem. They investigated the performanceof a simple greedy maximum edge disjoint paths algorithm applied to the RWA problem.

Banerjee and Sharan [30] solved the RWA problem by an evolutionary algorithm. A hybridapproach based on the k-shortest path for every source-destination pair is used and the wavelengthassignment to lightpaths was performed using a special graph-coloring technique.

A Tabu Search algorithm, which considers routing and wavelength assignment jointly is pre-sented in [31]. They compared the performance of their approach the Integer Linear Programming(ILP) method for minimizing wavelength usage.

In [32], the authors considered that the problem of routing and wavelength assignment in all-optical networks may be solved by a combined approach involving the computation of alternativeroutes for the lightpaths, followed by the solution of a partition coloring problem in a conflictgraph. In this way, they presented a tabu search heuristic for the partition coloring problem.

Yoon et al. [33] proposed an approach using path conflict graphs and an algorithm for findingall Edge Disjoint Paths (EDPs).

In [34], Skorin-Kapov addressed the problem of routing and wavelength assignment (RWA)of static lightpath requests in wavelength routed optical networks by minimizing the number ofwavelengths used. The application of classical bin packing algorithms was suggested.

An approach inspired by Particle Swarm Optimization (PSO) is proposed for solving staticRWA in [35] by Hassan and Phillips. The objective of the PSO is to minimize the total number ofwavelengths required, taking into account the average path length of the chosen routes.

De et al. [36] proposed two new polynomial time heuristics for wavelength assignment calledCPWA1 and CPWA2, based on clique partitioning concepts for static traffic demand with theobjective of minimizing the number of wavelengths. For routing, the authors used the Dijkstra’sshortest-path algorithm.

In [37], the authors presented a Memetic Algorithm (MA) for minimizing the number of wave-lengths used in the static RWA problem. The MA uses a recombination operator and a scheme fordistributing the computation.

In [38], two static and three dynamic routing algorithms were proposed and compared to someof the existing algorithms on the basis of blocking probability. On the one hand, the two proposedstatic routing and wavelength assignment (RWA) algorithms reduce the blocking probability tomaximize the utilization of network. On the other hand, for the dynamic algorithms, a modelwith no weights assignments was presented and three algorithms were proposed and analysed with


weight assignment resulting in reduction of blocking probability. Finally, these algorithms wereanalysed and compared with the well-known wavelength assignment schemes: first-fit, random,most used and least used.

In order to optimize the blocking probability, in [39], the authors proposed a Variable Neigh-bourhood Search algorithm. They compared the results with those obtained by the optimizationsoftware package CPLEX, a tool for solving integer programming problems.

In [40], Markovic et al. present a Bee Colony Optimization (BCO) approach for solving thelightpath scheduling problem, which implies solving the RWA problem. For solving the RWAproblem, the authors use Fixed-alternate routing and First-fit wavelength assignment heuristics.

As we can see, all heuristics and metaheuristics mentioned above, considered the RWA problemas a single-objective optimization problem. However, the RWA problem has been also tackled asa multiobjective optimization problem. In this way, using multiobjective optimization, we look fora solution for which each objective has been optimized to the extent that if we try to optimize itany further, then the other objective will suffer as a result.

In [41] and [42], several multiobjective varieties of Ant Colony Optimization algorithms aresuggested for optimizing the number of hops and the number of wavelengths conversions simulta-neously. Furthermore, in [42] the authors present a comparison among many varieties of MOACOsand typical heuristics in the Telecommunication field.

Leesuthipornchai et al. [43] consider a multi-objective network design problem with staticrouting wavelength assignment (RWA) in WDM networks. The design objective is to maximize thenumber of accepted communication requests (source-destination pairs) and to minimize the numberof wavelength channel required subject to a limited number of wavelength channels available oneach network link and at least 80% of all commodities must be accepted. They solved it by usinga Genetic Algorithm for Routing and Minimum Degree First Wavelength Assignment (GA-MDF).

In [44] and [45], Rubio-Largo et al. focused on solving the RWA problem with a static trafficpattern. To solve it, they proposed a multiobjective evolutionary algorithm based on the VariableNeighborhood Search algorithm (MO-VNS)[44], and on the Differential Evolution (DEPT) [45].

A hybrid evolutionary computation approach is applied in [46] for maximizing the number oftraffic demands to be served and minimizing the number of wavelength channels to be assigned.

In [47], Rubio-Largo et al. presented a comparative study on swarm intelligence for solving theRouting and Wavelength Assignment (RWA) problem. They evaluated three multiobjective meta-heuristics based on the behavior of honey bees (Multiobjective Artificial Bee Colony, MO-ABC)[48] [49], on the law of gravity and mass interactions (Multiobjective Gravitational Search Algo-rithm, MO-GSA) [50], and also on the flash pattern of fireflies (Multiobjective Firefly Algorithm,MO-FA) [47]. In order to study the goodness of these algorithms, they studied their capabilitiesto solve the RWA problem over three real-world optical networks, comparing their results withthose obtained by a well-known multiobjective approach (Fast Non-Dominated Sorting GeneticAlgorithm, NSGA-II).

A parallel version of the Differential Evolution with Pareto Tournaments (DEPT) [45] forsolving the RWA problem was presented in [51] and [52]. On the one hand, in [51] a multi-coreversion of the DEPT is reported and, on the other hand, in [52] a Hybrid OpenMP+MPI versionof this multiobjective evolutionary algorithm was presented. The aim of both parallel approachesis to accelerate the runtime, obtaining results of identical quality.

Finally, in Table 2.1 and Table 2.2 we summarize all the above mentioned related work in orderto provide a global view of the state of the art.

2.2R

outingand

Wavelength

Assignm

ent17

Table 2.1. Summary of the main approaches published in the literature for the static RWA problem. Note that, the notation used is:Wavelength Conversion (WC), Multiobjective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART 1/2)

Traffic pattern Methods forYear Author Static Dynamic WC MO H MH Approach Routing λ Assignment

1992 Chlamtac and Ganz [8] • • Yes/No • Lightpath Establishment Heuristics - Bin Packing1994 Chan and Yum [17] • Yes • Least Congested Path Routing Fixed-Alternate LU1995 Tan and Sinclair [18] • Yes • Genetic Algorithm (GA) - GA1995 Ramaswami and Sivarajan [7] • • Yes/No • shortest-path RWA Fixed Graph coloring1996 Mukherjee et al. [19] • - • Simulated-Annealing (SA) Dijsktra -1996 Banerjee and Mukherjee [20] • • Yes • Randomized-Rounding algorithm Fixed-Alternate Graph coloring1996 Banerjee and Chen [21] • Yes • Coloring adaptive path-graph

(CAP)Adaptive Graph coloring

1997 Masayuki et al. [22] • • Yes/No • Alternate routing heuristics Fixed-Alternate FF1998 Sinclair [23] • Yes • Genetic Algorithm (GA)/heuristic

hybrid approach- -

1998 Hyytiä and Virtamo [24] • No • Tabu Search + Graph Coloring Tabu Search Graph coloring1999 Varela and Sinclair [25] • Yes • Ant Colony Optimization (ACO) - -1999 Li and Somani [26] • No • Fix-path Least Congestion + FF Fixed FF2000 Li and Shima [27] • No • Shortest-path + Partition Coloring

problem (PCP)Fixed PCP

2000 Zhang et al. [12] • Yes • Distributed Relative Capacity Loss Adaptive DRCL2002 Ramamurthy and Mukherjee [28] • No • Fixed-Alternate Routing procedure Fixed-Alternate Reservation2002 Manohar et al. [29] • No • Greedy Edge-Disjoint path Algo-

rithm- -

2004 Banerjee and Sharan [30] • No • Genetic Algorithm (GA) GA FF2005 Wang et al. [31] • Yes • Tabu Search (TS) Fixed-Alternate FF2006 Insfrán et al. [41] • Yes • • Multiobjective varieties of ACOs - -

182.

Opt

ical

Net

wor

ks:

Fun

dam

enta

lsan

dB

ackg

roun

d

Table 2.2. Summary of the main approaches published in the literature for the static RWA problem. Note that, the notation used is:Wavelength Conversion (WC), Multiobjective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART 2/2)

Traffic pattern Methods forYear Author Static Dynamic WC MO H MH Approach Routing λ Assignment

2006 De Noronha and Ribeiro [32] • No • Tabu Search (TS) + Partition Col-oring (PC)

Fixed-Alternate PC

2006 Yoon et al. [33] • No • Edge-disjoint paths + Partition Col-oring (PC)

Fixed PC

2007 Arteta et al. [42] • Yes • • Multiobjective varieties of ACOs +Typical Heuristics

- -

2007 Skorin-Kapov [34] • No • Shortest-path + Bin packing Fixed Bin Packing2008 Hassan and Phillips [35] • No • Particle Swarm Optimization (PSO) - -2008 De et al. [36] • No • Clique Partitioning Fixed Clique Partitioning2008 Fischer et al. [37] • No • Distributed Memetic Algorithm

(DMA)- -

2009 Leesuthipornchai et al. [43] • No • • Multi-Objective Genetic Algorithm GA Minimum DegreeFirst

2010 Randhawa and Sohal [38] • • No • SRWA3 and SRWA4 Fixed/Fixed-Alternate

R/FF/LU/MU

2010 Rubio-Largo et al. [44] • Yes • • Multiobjective Variable Neighbour-hood Search

- -

2010 Leesutthipornchai et al. [46] • No • • Hybrid Evolutionary Algorithm GA LU2010 Rubio-Largo et al. [45] • Yes • • Differential Evolution with Pareto

TournamentsFixed-Alternate -

2011 Rubio-Largo et al. [50] • Yes • • Multiobjective Gravitational SearchAlgorithm

Fixed-Alternate -

2011 Tintor et al. [39] • No • Variable Neighbourhood Search Fixed-Alternate -2012 Rubio-Largo et al. [49] [48] • Yes • • Multiobjective Artificial Bee Colony - -2012 Rubio-Largo et al. [47] • Yes • • Multiobjective Firefly Algorithm Fixed-Alternate -2012 Markovic et al. [40] • No • Bee Colony Optimization Fixed-Alternate FF2012 Rubio-Largo et al. [51] [52] • Yes • • Parallel Differential Evolution with

Pareto TournamentsFixed-Alternate -

2.3 Traffic Grooming 19

2.3 Traffic Grooming

Nowadays, commercially available optical fibers can support over a hundred wavelength channels[53], each of which can have a transmission speed in the Gbps range (e.g. OC-48, OC-192, andOC-768).

Unfortunately, our current optical networks may still be required to support traffic connec-tions at rates that are lower than the full wavelength capacity. In order to use the networkresources efficiently [54], each WDM node will need to employ access stations in order to groomlow-speed connections onto a high-speed wavelength channel. These access stations can multi-plex/demultiplex and switch low-speed connections using various multiplexing techniques, suchas: Time-Division Multiplexing (TDM), Space-Division Multiplexing (SDM), Frequency-DivisionMultiplexing (FDM), or Packet-Division Multiplexing (PDM) [53].

Traffic grooming in WDM networks can be classified into two types, depending on whether thetraffic is static or dynamic. In the case of static traffic, for each connection request, the source,destination, and data rate are known in advance, so the problem is to design a virtual topologyreserving bandwidth on the lightpaths for each connection demand. On the other hand, if thetraffic model is dynamic, the virtual topology is reconfigured on demand either on the arrival oron the departure of traffic [55].

Depending on the number of lightpaths traversed by a traffic request, we can distinguish twovarieties of Traffic Grooming: single-hop and multi-hop.

Thus, given an optical network topology, a fixed number of transceivers per node, a fixednumber of wavelengths per optical fiber, a capacity of each wavelength, and a set of low-speedtraffic demands, the Traffic Grooming problem can be divided up into three subproblems:

• The lightpath routing subproblem. This consists of the establishment of lightpaths over thephysical topology; as a result, we obtain a virtual topology where each edge between twonodes represents an established lightpath.

• The wavelength assignment subproblem. This consists of assigning an available wavelengthto each established lightpath.

• The traffic routing subproblem. This consists of establishing each low-speed traffic demandthrough a lightpath or several lightpaths concatenated, depending on whether we assumesingle-hop or multi-hop traffic grooming.

Hence, the Traffic Grooming problem may be considered as a generalized version of the RWAproblem where the main goals are: the maximization of the number of demands established success-fully (total throughput), the minimization of the number of lightpaths created in order to minimizethe number of transceivers used at each node, and the minimization of the amount of time it takesfor a demand to travel from the source node to the destination node over each established lightpath.

In the following subsections, we describe some concepts of the Traffic Grooming problem. Then,a literature survey of the Traffic Grooming problem is shown, in which we review the differenttechniques proposed for solving this optimization problem.

2.3.1 Multiplexing Techniques in Traffic Grooming

Different multiplexing techniques may be used for traffic grooming in different domains of opticalWDM networks, such as:


(a): Physical Topology (b): Virtual Topology

Figure 2.7. Physical topology (a) with four lightpaths: L1(1-3), L2(1-5), L3(2-5), and L4(5-2); whichforms the Virtual Topology (b).

• Space-Division Multiplexing (SDM). The physical space is partitioned to increase trans-port bandwidth. For example, bundling a set of fibers into a single cable, or using severalcables within a network link.

• Frequency-Division Multiplexing (FDM). The available frequency spectrum is dividedinto a set of independent channels. For example, the use of FDM within an optical networkis known as wavelength-division multiplexing.

• Time-Division Multiplexing (TDM). The bandwidth’s time domain is partitioned intorepeated time-slots of fixed length. In this way, if we use TDM, multiple signals can share agiven wavelength if they are non-overlapping in time.

• Packet-Division Multiplexing (PDM). The bandwidth of a WDM channel is sharedbetween multiple traffic streams or virtual circuit service in an IP/MPLS WDM networkarchitecture.

Although most research on traffic-grooming problems in the literature concentrate on efficientlygrooming low-speed circuits onto high-capacity WDM channels using a TDM approach, the genericgrooming idea may be applied to any optical network domain using the various multiplexing tech-niques mentioned above.

2.3.2 Virtual Topology

A virtual topology is a network which consists of a set of lightpaths established between a subsetof node pairs in the network. Thus, in a virtual topology, a node corresponds to a routing node inthe network and an edge corresponds to a lightpath.

In Figure 2.7(a), we present a physical topology (six nodes and eight bidirectional optical links)whith four lightpaths established: Lightpath 1 (L1) between the nodes N1 and N3, Lightpath 2 (L2)between the nodes N1 and N5, Lightpath 3 (L3) between the nodes N2 and N5, and Lightpath4 (L4) between nodes N5 and N2). As we can see, the established lightpahts form the virtualtopology shown in Figure 2.7(b), where the nodes are N1, N2, N3, and N5; and the edges are thelightpaths L1, L2, L3, and L4.

Depending on the number of lightpaths traversed by a traffic request, we can distinguish twotypes of traffic grooming:


• Single-hop. If the requests are restricted to using no more than a single lightpath.

• Multi-hop. If the requests can use several concatenated lightpaths for being established.

Traditionally, when we need to establish a traffic request, we determine how to route it underthe current virtual topology state, applying one of the following possible operations:

• Operation 1. Route the connection onto an existing lightpath directly connecting the sourceand the destination nodes (single-hop).

• Operation 2. Route the request through multiple existing lightpaths (multi-hop).

• Operation 3. Set up a new lightpath directly between the source and the destination androute the traffic request onto this lightpath (single-hop).

• Operation 4. Set up one or more lightpaths that do not directly connect the source andthe destination nodes, and route the demand onto these lightpaths and/or some existinglightpaths (multi-hop).

In this way, each operation must satisfy certain pre-requisites in order to be applied. In somesituations, all the operations are applicable, while in other situations, only some of them can beused. If none of them can be applied, the traffic must be blocked.

(a): Multi-hop

(b): Single-hop

Figure 2.8. Illustrative example of Multi-hop and Single-hop traffic grooming when it is required toestablished the connection c (N1,N2).


For example, if we need to establish the connection c=(N1,N2) in the virtual topology shownin Figure 2.7(b), supposing it is a multi-hop virtual topology, then, c is established through thevirtual path L2-L4 (Operation 2). However, if we suppose a single-hop virtual topology, it is notpossible to establish the connection c, unless we establish a new lightpath L5 between nodes 1 and2 (Operation 3). An illustration of this example is presented in Figure 2.8.

2.3.3 State of the Art

In this section, we include some of the background of the traffic grooming problem, a brief reviewof the evolution of the problem, and how it has been tackled by other authors.

Konda and Chow [56] suggested that the Traffic Grooming problem is equivalent to a certaintraffic maximization problem. Therefore, they gave an intuitive interpretation of this equivalenceand use this interpretation to derive a greedy algorithm for transceiver minimization. Furthermore,they presented computational results comparing the heuristic solutions with the optimal solutionsfor several small example networks.

Some preliminary results on on-line schemes to provision connections with different bandwidthgranularities (dynamic traffic grooming) in WDM mesh networks were reported in [57] by Zhu andMukherjee. They employed the GMPLS distributed control plane.

In [58], the authors addressed dynamic traffic grooming in interconnected unidirectional rings.In particular, they developed two genetic algorithms, one for circle grooming and the other forcombining multiple topologies.

Two cases (best-fit and full-fit) for handling reconfigurable Synchronous Optical Networks(SONET) rings over WDM network were proposed in [59]. For each one, the authors gave aninteger linear programming model, and an algorithm based on the Tabu Search.

Xu et al. [60] proposed a genetic algorithm (GA) to solve the traffic grooming problems inunidirectional SONET/WDM rings with arbitrary asymmetric traffic requirements. The objectivewas to optimally assign calls to wavelengths in order that the total number of SONET add/dropmultiplexers (ADMs) is minimized while as few wavelengths as possible are used.

In [61], the authors tackled the problem as follows: given a set of connections (with theircorresponding route) and the grooming factor, to find an optimal wavelength assignment andgrooming such that the number of wavelengths required in the network is minimized.

An agent-based traffic grooming and management mechanism for IP over Wavelength DivisionMultiplexing (WDM) optical networks was proposed in [62]. The agent-based mechanism effectivelymanages the traffic aggregation across the optical core between the IP client networks.

Zhu and Mukherjee [63] investigated the traffic-grooming problem in a WDM-based opticalmesh topology network. Their objective was to improve the network throughput. To this end,they studied the node architecture for a WDM mesh network with traffic-grooming capability.Furthermore, they presented a mathematical formulation of the traffic grooming problem, basedon integer linear programming, and two fast heuristics for solving this telecommunication problem:Maximizing Single-hop Traffic (MST) and Maximizing Resource Utilization (MRU). Both heuristicshave been considered by other authors in order to test the accuracy of new heuristics applied tothis problem.

In [64], the authors suggested that it is not necessary to have traffic-grooming capability at everynetwork node. They called a network which has only a few grooming nodes to be a sparse-groomingnetwork. They investigated the problem of designing such a sparse-grooming WDM mesh network.Furthermore, they proposed a mathematical formulation of the problem and several design schemes.Xin and Qiao [65] developed a performance analysis model for the multi-hop traffic grooming in


mesh WDM optical networks, using load sharing for traffic allocation in limited multi-hop paths.In [66], Zhu et al. proposed an auxiliary graph model approach for traffic grooming in WDM

mesh networks. Based on the auxiliary graph, they developed an integrated grooming algorithm(IGABAG) which jointly solves the traffic-grooming subproblems for one traffic demand, and also agrooming procedure (INGPROC) which can accommodate both static and dynamic traffic groom-ing using the IGABAG algorithm. They also presented several grooming policies and traffic selec-tion methods. The grooming policies proposed in [66] are the following: Minimizing the Number ofTraffic Hops (MinTH), Minimizing the Number of Lightpaths (MinLP), and Minimizing the Numberof Wavelength-Links (MinWL). On the other hand, the proposed traffic-request-selection schemesfor static traffic grooming are: Least Cost First (LCF), Maximum Utilization First (MUF), andMaximum Amount First (MAF).

Hu and Lleida [67] considered traffic grooming in combination with traffic routing and wave-length assignment. Their objective was to minimize the total number of transponders required inthe network. They proposed a decomposition method that divides the problem into two smallerproblems: the traffic grooming and routing problem and the wavelength assignment problem, solv-ing them separately.

A performance analysis of grooming dynamic client traffic in WDM optical networks with amesh topology is reported in [68]. The authors developed two link blocking models: an exact modelbased on the stochastic knapsack problem and an approximation model based on an approximatecontinuous time Markov chain (CTMC).

Lee et al. [69] propose a traffic grooming algorithm that employs a table for shortest EdgeDisjoint Paths (EDPs) with clever selection of demands with the aim of maximizing the throughputof the optical network. However, their approach is quite similar to the MRU heuristic, in that itconstructs the lightpaths according to maximum resource utilization.

The authors in [70], studied how to design logical topologies (using minimum network resource)for dynamic traffic grooming, to meet the given traffic blocking probability requirements. Theyformulated this problem into an integer linear programming problem, concluding that it is highlyeffective for small to medium sized networks.

Wang and Byrav [71], with the aim of solving the dynamic traffic grooming problem, intro-duced a largely simplified link bundled auxiliary graph (LBAG) model and proposed the Simpli-fied Auxiliary Graph With Link Bundling (SAG-LB) method to find paths and assign wavelengthsfor new lightpath. Furthermore, they proposed a Constrained Integrated Grooming Algorithm(CIGA) based on the LBAG model. In [72], the authors proposed the maximize-lightpath-sharingmulti-hop (MLS-MH) grooming algorithm to support dynamic traffic grooming in sparse groomingnetworks. They also presented an analytical model to evaluate the blocking performance of theMLS-MH algorithm.

Yoon et al. [73] investigated the traffic grooming in WDM optical networks and proposed animproved heuristic algorithm with respect to network throughput called Shortest path First Trafficgrooming (SFT). This algorithm consider the shortest path on original graph and uses as manyshortest paths as possible for the demands. However, their approach is quite similar to the MSTheuristic, in that it constructs the lightpaths according to maximum single-hop traffic.

A study of optical networks with sparse traffic grooming and wavelength conversion resourceswas reported in [74]. The authors proposed two novel heuristics that minimize the cost of thetraffic grooming and wavelength conversion equipment used in optical network without hinderingthe network blocking performance. In [75], a simple genetic algorithm that minimizes the number ofrequired Add-Drop Multiplexers based on the shortest path and a possible alternate shortest pathwas applied for solving the traffic grooming problem in synchronous optical networks (SONET).


Bensong et al. [76] presented a hierarchical traffic grooming framework for WDM networks ofgeneral topology, with the objective of minimizing the total number of electronic ports. At thefirst level of hierarchy, they used a modified version of the K-Center algorithm to decompose thenetwork into clusters and designate one node in each cluster as the hub for grooming traffic. At thesecond level, the hubs form another cluster for grooming inter-cluster traffic. Finally, the routingand wavelength assignment is performed directly on the underlying physical topology by using theLongest First Alternate Path (LFAP) algorithm.

In [55], De et al. investigated the traffic grooming problem with the objective of maximizing thenetwork throughput for wavelength-routed mesh networks and map this problem to the clique par-titioning problem. They proposed an algorithm to handle general multi-hop static traffic groomingbased on the clique partitioning concept.

The study in [77] was aimed at traffic-grooming problem in a WDM mesh network. Theystudied the problem of static single-hop and multi-hop traffic grooming with the objective ofmaximizing the network throughput for wavelength routed mesh networks. They also proposedan algorithm TG1, based on the concept of single hop and multi hop grooming. Balma et al.[78] addressed a multi-hop traffic grooming problem in an SDH/WDM network. They proposed asimple and efficient branching strategy. This branching strategy consists on selecting the variablesfor branching according to the corresponding channel sizes by defining a sequence of mixed integersubproblems, each corresponding to a channel type.

All the aforementioned techniques tackled the Traffic grooming problem as a single-objectiveoptimization problem. However, this problem has been also solved by using multiobjective opti-mization.

In [79], the main purpose of the author is to design the virtual topology that optimizes perfor-mance and cost. Thus, their objective functions include a maximization of throughput (as foundin [63], [73], and [69]), a minimization of the network cost in terms of the number of transceivers(as found in [56] and [67]), and a minimization of the average propagation delay of the lightpaths.They propose a well-known Multiobjective Evolutionary Algorithm (MOEA), the Strength ParetoEvolutionary Algorithm (SPEA), to solve these objective functions all together at the same time.

De et al. [80] proposed a multiobjective approach based on the well-knwon Non-DominatedSorting Genetic Algorithm (NSGA). The problem of minimization of cost of a SONET/WDM unidi-rectional ring was modeled in [81] as a multiobjective optimization problem which simultaneouslyminimizes the number of add-drop multiplexers, the number of wavelengths, and the groomingratio. The authors proposed the use of the popular NSGA-II.

In [82], the design objectives were to maximize the number of accepted communication requests(source-destination pairs) as well as to minimize the number of wavelength channels. To solve themulti-objective network design problem, they applied the NSGA-II.

In recent literature [83], Rubio-Largo et al. tackled the traffic grooming problem by usinginnovative MOEAs. They proposed the Differential Evolution with Pareto Tournaments (DEPT)algorithm and the Multiobjective Variable Neighborhood Search algorithm (MO-VNS). They car-ried out several comparisons with well-known MOEAs (NSGA-II and SPEA2), as well as comparingthe DEPT algorithm with other approaches published in the literature. Furthermore, Rubio-Largoet al. present a parallel multiobjective version of a well-known evolutionary algorithm based onswarm intelligence, the Artificial Bee Colony algorithm [84], for solving the Traffic grooming prob-lem in WDM optical networks.

Finally, in order to provide a global view of the state of the art of the Traffic grooming problem,in Table 2.3 and Table 2.4, we summarize the related work in the literature.

2.3T

raffic

Groom

ing25

Table 2.3. Summary of the main approaches published in the literature for the Traffic Grooming problem. Note that, the notation used is:Wavelength Conversion (WC), Multiobjective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART 1/2)

Traffic pattern TopologyYear Author Static Dynamic WC Multi-hop Sparse (#nodes) MO H M Approach Routing λ Assignment Grooming

2001 Konda and Chow [56] • No • Mesh (16) • Greedy Heuristics for minimizingtransceivers

Fixed - -

2001 Zhu and Mukherjee [57] • No • Mesh (14) • Two-Layered Route Compu-tation grooming algorithm(TLRC)

- - TLRC

2001 Xu and Zeng [58] • No Ring • Genetic Algorithm (GA) forgrooming

- - GA

2002 Zhang and Ramamurthy [59] • • No Ring • Tabu Search Grooming approach - - Best-fit /Full-fit

2002 Xu et al. [60] • Yes Ring • Genetic Algorithm (GA) forgrooming

- - GA

2002 Li et al. [61] • No Mesh (14) • Algorithm based on binarysearch

- Graph coloring -

2002 Xin et al. [62] • No Mesh (15) • Agent Based Mechanism (AGB) Fixed FF AGB2002 Zhu and Mukherjee [63] • No • Mesh (15) • - Adaptive FF MST/MRU2002 Zhu et al. [64] • No • Mesh (24) • Nodal-Degree / Bypass-Traffic /

Random-Selection- - Random

2003 Xin and Qiao [65] • No • Mesh (15) • Online Multi-hop grooming pro-cedure (MH)

Fixed FF MH

2003 Zhu et al. [66] • • Yes/No • • Mesh (19) • INtegrated Groming Procedure(INGPROC)

Fixed MinTH/MinLP/ MinWL

LCF/MUF/MAF

2004 Hu and Lleida [67] • Yes Mesh (-) • ILP Relaxation approach (RA)for grooming and wavelength as-signment

- ILP-WA ILP-GR

2004 Xin et al. [68] • No Mesh (15) • Reducing Load Approximationwith Continous time Markovchain (CTMC)

Fixed FF CTMC

2005 Lee et al. [69] • No • Mesh (20) • Shortest Edge Disjoint Paths al-gorithm (EDP)

Fixed(EDP)

FF MRU

2005 Prathombutr et al. [79] • Yes/No • Mesh (14) • • Strength Pareto Evolutionary Al-gorithm (SPEA)

Fixed-Alternate

Graph coloring MST

262.

Opt

ical

Net

wor

ks:

Fun

dam

enta

lsan

dB

ackg

roun

d

Table 2.4. Summary of the main approaches published in the literature for the Traffic Grooming problem. Note that, the notation used is:Wavelength Conversion (WC), Multiobjective (MO), Heuristics (H), Metaheuristics (M), and Not-Available (-). (PART 2/2)

Traffic pattern TopologyYear Author Static Dynamic WC Multi-hop Sparse (#nodes) MO H M Approach Routing λ Assignment Grooming

2005 Xin et al. [70] • Yes/No Mesh (14) • Approach to design a logicaltopology

Fixed FF Random

2005 Wang and Byrav [71] • Yes/No Mesh (22) • Link Bundled Auxiliary Graph(LBAG)

SAG-LB SAG-LB CIGA

2005 Wang et al. [72] • No • • Mesh (14) • Maximize Lightpath-SharingMulti-Hop (MLS-MH)

Fixed Random MLS-MH

2005 Yoon et al. [73] • No • Mesh (14) • Shortest path First TrafficGrooming (SFT)

Fixed - MST

2006 Awwad et al. [74] • Yes • • Mesh (16) • Genetic algorithm (GA) withthe Most Contiguous Heuristics(MCH)

Fixed-Alternate

MCH GA

2007 Roy and Naskar [75] • No Ring • Genetic algorithm (GA) - - -2008 De et al. [80] • No • Mesh (14) • • Non-dominated Sorting Genetic

Algorithm (NSGA)Fixed-Alternate

FF MST

2008 Bensong et al. [76] • No • Mesh (47) • Hierarchical Approach (LFAP +K-center)

LFAP LFAP K-center

2009 Biswas et al. [81] • No Ring • • Fast Non-dominated Sorting Ge-netic Algorithm (NSGA-II)

- - -

2009 Leesutthipornchai et al. [82] • No Mesh (20) • • Fast Non-dominated Sorting Ge-netic Algorithm (NSGA-II)

GA MDF LHG

2010 De et al. [55] • No • Mesh (14) • Traffic Grooming based onClique Partitioning (CP)

Fixed CP CP

2010 Bhattacharya et al. [77] • No • Mesh (14) • Dynamic Path Selection strategy Adaptive LU MST2011 Balma et al. [78] • Yes • Ring/Mesh (14) • Branch and Bound strategy - - -2012 Rubio-Largo et al. [83] • No • Mesh(55) • • Multiobjective Differential Evo-

lution with Pareto TournamentsFixed-Alternate

FF -

Multiobjective Variable Neigh-bourhood Search

2012 Rubio-Largo et al. [84] • No • Mesh(14) • • Parallel Multiobjective ArtificialBee Colony Algorithm

Fixed-Alternate

FF -

3Proposed Methodology

In the literature, specialized heuristics were typically developed to solve complex combinatorialoptimization problems. It was need a new approach for each new optimization problem that ap-peared, and the lessons learned from one problem did not always be applied to a different class ofproblems. Fortunately, new general solution strategies came up, they were named metaheuristics.Among the most popular ones are: tabu search, evolutionary algorithms, or simulated annealing.The main advantages of the metaheuristics are both effectiveness and applicability; without forget-ting that they usually require less work in being adapted to a particular problem than developinga specialized heuristic [85].

In the literature, many optimization methods have failed to be either effective or efficient intackling complex optimization problems. However, metaheuristics were designed to solve real-world problems. The main advantages of metaheuristics are effectiveness and general applicability;therefore, these methods have come to be recognized as one the most practical approaches forsolving complex real-world problems.

In the last decades, research on real-world optimization problems with multiple objectives hasattracted much more attention [86] [87], referring to them as Multiobjective Optimization problems(MOOP) [88]. In a MOOP, no single solution can optimize all the objective functions at the sametime; thus, we require a set of solutions. In this set, each solution is denominated as Pareto-optimal solution if any improvement in one objective must lead to deterioration to at least oneother objective function. Since the number of Pareto-optimal solutions for a specific MOOP maybe very large, the use of multiobjective search methods is a good choice for attempting to find awell representative set of Pareto-optimal solutions.

Over the years, diverse multiobjective evolutionary algorithms (MOEAs) have been proposed.They work with a population of solutions and produce a set of Pareto optimal solutions in a singleexecution as a result. We may classify them into three groups: Pareto dominance, Indicator-based,and decomposition.

In the first place, among the most popular methods based on Pareto dominance are the ParetoArchived Evolution Strategy (PAES [89]), the Fast Non-Dominated Sorting Genetic Algorithm(NSGA-II [90]), or the Strength Pareto Evolutionary Algorithm 2 (SPEA2 [91]).

In the second place, those algorithms which make use of an indicator function to evaluate aset of solutions are the Indicator-based algorithms. The most popular indicator-based approachesare: Indicator-based Evolutionary Algorithm (IBEA) [92] and S-metric Selection Evolutionary

27

28 3. Proposed Methodology

Multiobjective Optimisation Algorithm (SMS-EMOA) [93].Finally, we have the Multiobjective Genetic Local Search (MOGLS [94]) or the Multiobjective

Evolutionary Algorithm based on Decomposition (MOEA/D [95]). These decomposition-basedapproaches associate each individual solution with a particular scalar optimization problem withthe aim of allocating computational resources to different parts of the Pareto front.

From a computational perspective, metaheuristics are just algorithms from which we can extractfunctional or data parallelism [96]. Therefore, parallel metaheuristics offer the possibility to addressproblems more efficiently in terms of computing efficiency or solution quality.

In this chapter, we start with a brief introduction to metaheuristics, including basic conceptsof evolutionary algorithms. In the second place, we formally present the main concepts used inMultiobjective Optimization as well as describing the most important metrics for assessing theperformance of multiobjective methods. Then, we explain diverse concepts regarding parallelcomputing (architecture models and performance metrics) and we outline the importance of usingparallel metaheuristics. Finally, we explain the necessity of ensuring statistical reliability whentreating with metaheuristics.

3.1 Metaheuristics

According to [85], metaheuristics, in the original definition, are solutions methods that orchestratean interaction between local improvement procedures and higher level strategies to create a processcapable of escaping from local optima and performing a robust search of a solution space.

The metaheuristics are stochastic search and optimization approaches that require a functionassigning fitness values to possible (or partial) solutions, and an encode mechanism between theproblem and algorithm domains. The optimal solution is not guaranteed to be found by theseapproaches, but they may find it eventually. In general, they provide a good approximation to acomplex optimization problem in which traditional methods find difficulties.

Among the most popular ones are: Simulated Annealing (SA) [97], Tabu search [98], andEvolutionary Computation (EC).

Simulated Annealing [97] is an algorithm explicitly modeled on an annealing analogy, where,for example, a piece of metal is heated and then gradually cooled until a certain temperature isreached. In this metaheuristics, SA chooses randomly movements to try finding the optimum. Inthe case of the move improves the current optimal solution it is always replaced; otherwise, it isonly replaced if a probability is satisfied (p < 1). This probability exponentially decreases eitherby time or with the amount by which the current optimum is worsened. Therefore, the analogyfor SA is that if the probability decreases slowly enough the global optimum is found.

Tabu search [98] is a metaheuristics developed to avoid getting stuck on a local optimal solution.It maintains a record of both visited optimal solutions and the paths which reached them. Thisinformation restricts the choice of solutions to evaluate next. Tabu search is often integrated withother optimization methods.

Evolutionary Computation may include techniques of genetic algorithms (GAs), evolutionstrategies, and evolutionary programming, which are known as Evolutionary Algorithms (EA)[99]. These techniques are based on natural evolution and the Darwinian concept of Survival ofthe Fittest. Basically, a general EA consists of a population of encoded solutions (individuals)manipulated by a set of operators and evaluated by some fitness function. The value of fitnessdetermines if the individual survive into the next generation or not.

In the first place, we present a classification of metaheuristics. Then, since in this thesis wefocus on Evolutionary Computation for solving real-world optimization problems, we describe the

3.1 Metaheuristics 29

basic structural terms and concepts regarding evolutionary algorithms.

3.1.1 Classification of Metaheuristics

Depending on the characteristics of these metaheuristics, they can be classified in different ways.The most common classification is by the number of solutions used at the same time; therefore, wehave trajectory-based and population-based. In the following, we enumerate and describe the mostimportant metaheuristics in both groups, some of the definition have been taken from [100].

On the one hand, the trajectory-based algorithms starts from an initial state (the initial solution)and describes a trajectory in the state space. A brief description of the main trajectory-basedevolutionary algorithms is presented:

• Simulated Annealing (SA) is commonly said to be the oldest among the metaheuristicsand surely one of the first algorithms that had an explicit strategy to avoid local minima.The origins of the algorithm are in statistical mechanics (Metropolis algorithm) and it wasfirst presented as a search algorithm for combinatorial optimization problems in [97] and[101]. The fundamental idea is to allow moves resulting in solutions of worse quality thanthe current solution (uphill moves) in order to escape from local minima. The probability ofdoing such a move is decreased during the search.

• Tabu Search (TS) is among the most cited and used metaheuristics for combinatorialoptimization problems. TS basic ideas were first introduced in [98], based on earlier ideasformulated in [102]. A description of the method and its concepts can be found in [103]. TSexplicitly uses the history of the search, both to escape from local minima and to implementan explorative strategy.

• Greedy Randomized Adaptive Search Procedure (GRASP) is a simple metaheuris-tics proposed in [104], which combines constructive heuristics and local search. Its structureis an iterative procedure, composed of two phases: solution construction and solution im-provement, where the best found solution is returned upon termination of the search process.

• Variable Neighbourhood Search (VNS) is a metaheuristics proposed in [105], whichexplicitly applies a strategy based on dynamically changing neighborhood structures. Thealgorithm is very general and many degrees of freedom exist for designing variants andparticular instantiations.

• Iterated Local Search (ILS) is a simple but powerful metaheuristic algorithm [106] thatapplies local search to an initial solution until it finds a local optimum; then it perturbs thesolution and it restarts local search. The importance of the perturbation is obvious: toosmall a perturbation might not enable the system to escape from the basin of attraction ofthe local optimum just found. On the other side, too strong a perturbation would make thealgorithm similar to a random restart local search.

On the other hand, the population-based methods are devoted to deal with a set of solutions(population) rather than with a single solution in every iteration of the algorithm. As they dealwith a population of solutions, population-based algorithms enhance the exploration of the searchspace. The most studied population-based algorithms are enumerated and described as follows:

• Scatter Search (SS) is a search strategy proposed in [107] that generates a set of solutionsfrom a chosen set of reference solutions corresponding to feasible solutions to the problem


under consideration. This is done by making combinations of subsets of the current set ofreference solutions. The resulting solutions are called trial solutions. These trial solutionsmay be infeasible solutions and are therefore usually modified by means of a repair procedurethat transforms them into feasible solutions. An improvement mechanism is then applied inorder to try to improve the set of trial solutions (usually this improvement procedure is alocal search). These improved solutions form the set of dispersed solutions. The new set ofreference solutions that will be used in the next iteration is selected from the current set ofreference solutions and the newly created set of dispersed solutions.

• Population-based Incremental Learning (PBIL) is a metaheuristics proposed in [108].It creates a real valued probability vector which when used to sample the search spacegenerates high quality solutions with high probability. Initially, the values of the probabilityvector are initialized to 0.5 (for each binary variable there is equal probability to be set to 0 or1). The goal of shifting the values of this probability vector in order to generate high qualitysolutions is accomplished as follows: a number of solution vectors are generated according tothe probability vector. Then the probability vector is shifted toward the generated solutionvector with highest quality. The distance that the probability vector is shifted depends onthe learning rate parameter. Then, a mutation operator is applied to the probability vector.After that, the cycle is repeated.

• Differential Evolution (DE) is a very simple algorithm for global optimization proposedin [109]. It optimizes a problem by maintaining a population of candidate solutions and gen-erating new candidate solutions by combining existing ones according to its simple equationof vector-crossover and mutation.

• Swarm Intelligence (SI) refers to the collective behaviour of decentralized, self-organizedsystems, natural or artificial. The inspiration often comes from nature, especially biologicalsystems. The agents follow very simple rules, and although there is no centralized controlstructure dictating how individual agents should behave, local, and to a certain degree ran-dom, interactions between such agents lead to the emergence of intelligent global behavior,unknown to the individual agents. Among the most common swarm intelligence algorithms,we can find:

– Particle Swarm Optimization (PSO) is a metaheuristics proposed in [110], which isinspired in the representation of the movement of organisms in a bird flock or fish school.PSO optimizes a problem by having a population of candidate solutions (particles) andmoving them around in the search-space according to a simple mathematical formulae,taking into account position and velocity of each particle. The movement of each particleis then influenced by its local best known position and is also guided toward the bestknown positions in the search-space, which are updated as better positions are foundby other particles.

– Ant Colony Optimization (ACO) is a metaheuristics proposed in [111]. It is inspired inthe foraging behavior of ants. This behavior enables ants to find shortest paths betweenfood sources and their nest. While walking from food sources to the nest and vice versa,ants deposit a substance called pheromone on the ground. When they decide about adirection to go, they choose with higher probability paths that are marked by strongerpheromone concentrations.

– Artificial Bee Colony (ABC) is a metaheuristics proposed in [112]. It is based on theintelligence behaviour of honey bees. A feasible solution in this algorithm is considered


Figure 3.1. Classification of Metaheuristics.

a food source and the population is called colony. The colony consists of three typesof artificial bees: the employed bee, the onlooker bee, and the scout bee. In ABC, theartificial bees fly around in a the search space and the employed and onlooker beesare in charge of choosing food sources depending on the experience of themselves andtheir nest mates, and adjust their positions. The scouts fly and choose the food sourcesrandomly without using experience.

– Gravitational Search Algorithm (GSA) is metaheuristics proposed in [113]. It is basedon the law of gravity and mass interactions. In this algorithm, the individuals areconsidered searcher agents, and the entire population is a collection of masses whichinteract with each other based on the laws of motions. Using the gravitational force,every mass in the system can see the situation of other masses. The gravitational forceis therefore a way of transferring information between different masses.

– Firefly Algorithm (FA) is a metaheuristics proposed in [114]. In this algorithm theindividuals are fireflies which use their bioluminescent aptitudes to attract other fireflieswith the aim of optimizing a mono-objective problem. In FA, the attractiveness isproportional to their brightness, and for any two fireflies, the less brighter one will beattracted by (and thus move to) the brighter one; however, the brightness can decreaseas their distance increases.

Finally, in Figure 3.1, we summarize the classification presented. On the one hand, we findthe trajectory based algorithms, and on the other hand the population-based algorithms, whichincludes the more and more important subgroup of algorithms based on swarm intelligence.

3.1.2 Evolutionary Algorithms: Basic concepts

In an Evolutionary Algorithm, we denote individual to an encoded solution to a given optimizationproblem. It is commonly represented as a string corresponding to a biological genotype. Theindividual may be composed of one or more chromosomes. A chromosome is a structure whichconsists of separate genes that take on certain values (alleles) from some genetic alphabet. We candefine the concept of locus that identifies the position of a gene within the chromosome. Therefore,


Figure 3.2. Concepts of individual and population in Evolutionary Algorithms.

Figure 3.3. Concepts of parent, child, selection, recombination, and mutation in Evolutionary Algorithms.

each individual decodes into a set of parameters used as input to the function under consideration.Each individual is associated with a numerical value (fitness) that measures how well satisfiesthat objective function. A set of several individuals is known as population. Figure 3.2 shows anillustrative example of the aforementioned concepts.

In EA, the individuals that conform the population are called parent. Therefore, at eachgeneration, two parent are selected and recombined for generating new individuals (children) which,after a process of mutation, will be the members of the next generation. A scheme of the processcarried out at each generation is reported in Figure 3.3

The selection method is key in EAs. The main reason is because the use of a good selectionmethod in these algorithms leads to a good balance between exploration and exploitation, whichenhances the process of searching of the optimal solution as a result. There exist different selectionmethods, among the most important are:

• Roulette-wheel selection. In this selection method, it assigns to each individual of thepopulation a portion of the wheel proportional to the ratio of its fitness and the averagevalue of fitness of the population. Therefore, the EA gives individual with higher fitness a


(a): Roulette-wheel

(b): Binary Tournament

Figure 3.4. Common selection methods in EAs.

higher probability of contributing one or more children in the succeeding generation. Thereexists a rank-based roulette wheel, where the probability of a chromosome to be selected isbased on its fitness rank relative to the entire population. In Rank-based selection schemes,first sort individuals in the population according to their fitness and then computes selectionprobabilities according to their ranks rather than fitness values.

• Tournament selection. It randomly selects q individuals from the current population.Then, select the best to survives to the next generation. Probably, the most common tour-nament selection method is the Binary tournament, where q = 2.

In Figure 3.4, we present a visual representation of the aforementioned selection methods usedin EAs.

In EAs, the recombination or crossover method is used to vary the programming of one or morechromosomes from one generation to the next. In this method, we take information from the givenparents in order to generate the new solutions. Among the most common crossover proceduresare:

• Point-Crossover. Given two parents, we establish one or more crossover point on both


(a): Single-point crossover

(b): Uniform Crossover

Figure 3.5. Common crossover methods in EAs.

Figure 3.6. Concept of bitwise mutation.

chromosomes. All data beyond that points in either chromosome are swapped between thetwo parent chromosomes, generating two children as a result. The most common point-crossover is the single-point-crossover, where we establish only a single point. See Figure 3.5(a).

• Uniform Crossover. Given two parents, it uses a fixed mixing ratio between them. Then,bits are uniformly copied from the first or from the second parent for generating two childrenas a result. Normally, the ratio is fixed to 0.5, see Figure 3.5 (b).

3.2 Multiobjective Optimization 35

By mutation, individuals are randomly altered with the aim of preserving the genetic diversityfrom one generation to the next. In the mutation process, the allele of the chromosome are alteredwith a low probability (mutation probability or mutation rate); otherwise, the search will turn intoa random search. Among the most common mutation procedure is bitwise mutation, where, in abinary encoding, a ’1’ is changed to a ’0’, or vice versa (see Figure 3.6).

3.2 Multiobjective Optimization

In this section, we start defining, in a formal way, a Multiobjective Optimization Problem as wellas some basic concepts in the multiobjective domain. Then, we describe diverse metrics used inthe literature for measuring the performance of the algorithm. Finally, we present a classificationof the principal multiobjective algorithms presented in the literature.

3.2.1 Formal Definitions

A Multiobjective Optimization Problem (MOOP) [88] has a number of objective functions whichare to be optimized. As in single-objective optimization, the problem usually has a number ofconstraints which must be satisfied by any feasible solution.

In the following, we state a generic minimization MOOP in a formal way:

minimize F (x) = (f1(x), f2(x), . . . , fM (x))subject to x ∈ Ω

gi(x) ≥ 0, ∀i ∈ 1, . . . , Jhi(x) = 0, ∀i ∈ 1, . . . , K

where a solution x is a vector of n decision variables: x = x1, . . . , xn. The upper and lowerbound of each component of x, constitute a decision variable space, or simply decision space (Ω).

Furthermore, we identify J inequality and K equality constraints associated to the MOOP, thatmust be satisfied. In this way, those solutions which do not satisfy all of the J+K constraints iscalled infeasible solution. Thus, a feasible solution is a solution that satisfies all of these constraints.Note that, the solutions that belong to Ω must be feasible solutions.

As we can see, we have M objective functions: (f1(x), f2(x), . . . , fM (x)); which need to beminimized (we are supposing a minimization MOOP). In this way, these M objective functionsconstitute a multi-dimensional space: F : Ω → RM ; which is commonly known as objective space(RM ).

Figure 3.7. Concepts of Decision Space (Ω) and Objective Space (R2)


Figure 3.8. Ideal (z∗), utopian (z∗∗), and nadir (znadir) objective vectors.

For each solution x = x1, . . . , xn in the decision space, there exists a point z = z1, . . . , zMin the objective space. In Figure 3.7, we illustrate the mapping of a solution with three decisionvariables (n = 3) in the objective space (R2).

There exist special solutions which are often used in multiobjective optimization algorithms.In the following, we briefly describe them:

• Ideal Objective Vector (z∗). For each of the M objective functions, there exist onedifferent optimal solution. An objective vector constructed with these individual optimalobjective values constitutes the ideal objective vector. Therefore, if the minimum solutionfor the objective function fi is the decision vector x∗

i with function value f∗i , we may state

the ideal vector as z∗ = (f∗1 , . . . , f∗

M ). Theoretically, the ideal objective vector correspondsto a non-existent solution, due to the minimum solution for each objective function need tobe the same solution, and it is only possible if the objectives are not conflicting to each other.

• Utopian Objective Vector (z∗∗). Basically, we can define the utopian objective vectorz∗∗ as that vector which has each objective function value marginally smaller than the idealobjective vector: z∗∗

i = z∗i − ǫi with ǫi > 0 for all i ∈ 1, . . . , M. Like the ideal objective

vector, it is a non-existent solution.

• Nadir Objective Vector (znadir). Unlike the ideal objective vector which represents thelower bound of each objective function in the entire feasible objective space, the nadir ob-jective vector (znadir) represents the upper bound of each objective function in the entirePareto-optimal set, and not in the entire search space. It should not be confused with avector of objectives found by using the worst feasible function values in the entire searchspace.

An illustrative example of ideal objective vector (z∗), utopian objective vector (z∗∗), and nadirobjective vector (znadir) is presented in Figure 3.8.

In multiobjective optimization is necessary to compare solutions in order to decide which one isbetter. Thus, two solutions are compared on the basis of whether one dominates the other solutionor not.

In this way, a solution x1 is said to dominate the other solution x2, if and only if the followingconditions are satisfied:

1. The solution x1 is no worse than x2 in all objective functions, or fi(x2) ≤ fi(x1) for alli ∈ 1, . . . , M.


2. The solution x1 is strictly better than x2 in at least one objective function, or fi(x1) < fi(x2)for at least one index i ∈ 1, . . . , M.

As was mentioned before, if one of the conditions are violated, the solution x1 does not dominatethe solution x2. We can use the following mathematical notation to denote that x1 dominates x2:

x1 ≺ x2 (3.1)

To illustrate the concept of dominance, in Figure 3.9 we consider a minimization bi-objectiveproblem with four solutions (z1, z2, z3, and z4) in the objective space.

In Figure 3.9, for each solution zi, where i ∈ 1, 2, 3, 4, we present in dark grey, the area ofthe objective space dominated by zi, and in light grey the area of possible solutions that dominateit. As we can see, the solution z1 (Figure 3.9(a)) does not dominate any solution; however isdominated by the solution z2. In Figure 3.9(b), we may observe how the solution z2 dominates thesolutions z1 and z3, whereas it is not dominated by any other solution. Like the solution z1, in thesolution z3 does not dominate any solution; however it is dominated by the solutions z2 and z4, aswe can see in Figure 3.9(c). Finally, the solution z4 dominates the solution z3, but it is dominatedby no solutions, as it is shown in Figure 3.9(d).

Basically, there are three possibilities that can be outcome when we compare two solutions (x1

and x2) in multiobjective optimization:

(a): Solution z1 (b): Solution z2

(c): Solution z3 (d): Solution z4

Figure 3.9. Example of a minimization bi-objective problem with four solutions (z1, z2, z3, and z4) inthe objective space.


1. The solution x1 dominates solution x2. For example, in Figure 3.9(b), the solutions z2 andz1.

2. The solution x1 is dominated by the solution x2. For example, the solution z3 and z2 inFigure 3.9(c).

3. The solutions x1 and x2 do not dominate each other. For example, in Figure 3.9(b), thesolutions z2 and z4.

It is also important to clarify the binary relations properties of the dominance operator (≺),which are:

• Not Reflexive. The dominance relation is not reflexive, since any solution does not dominateitself.

• Not Symmetric. The dominance relation is not symmetric, because x1 x2 does not implyx2 x1; therefore, the dominance operator is asymmetric.

• Not Antisymmetric. Since the relation is not symmetric, it cannot be antisymmetric aswell.

• Transitive. The dominance relation is transitive because if x1 x2 and x2 x3, thenx1 x3.

If we focus again in Figure 3.9(b), we can see that the value of the first objective function in thesolution z2 is lower than in the solution z4; however, the value of the second objective in z4 is lowerthan in z2. Therefore, we cannot say that z2 dominates z4, nor that z4 dominates z2. Actually,the goodness of both solutions is exactly the same in multiobjective optimization, they are callednon-dominated solutions.

In this way, given a set of solutions, we can perform all possible pair-wise comparisons in orderto find which solutions dominates which and which solutions are non-dominated with respect tothe other. Finally, we expect to have a set of non-dominated solutions. We refer to this set asnon-dominated set. In Figure 3.10 we present a set of solutions and the corresponding set ofnon-dominated solutions that result from a pair-wise comparison.

A set of non-dominated solutions may be defined as: Among a set of solutions P , the non-dominated set of solutions P ′ are those that are not dominated by any member of the set P .

Figure 3.10. Pair-wise comparison to obtain a set of Non-dominated solutions


(a): Minimization-Minimization (b): Minimization-Maximization (c): Maximization-Minimization

(d): Maximization-Maximization (e): Global and Local Pareto-optimalsets

Figure 3.11. Pareto optimal solutions for four combinations of two types of objectives and Global andLocal Pareto-optimal concepts.

As in single-objective optimization, where we can find global and local optimal solutions, inmultiobjective optimization we can find a global and a local Pareto-optimal set of non-dominatedsolutions. Therefore, we can say that a non-dominated set (P ) is a Pareto-optimal set when theset P is the entire search space. In Figure 3.11, we present an illustrative view of different scenar-ios in a bi-objective optimization problem, where the first and second objective functions areto be for: minimization-minimization, minimization-maximization, maximization-minimization,maximization-maximization; in order to mark the Pareto-optimal set in each case.

If for every member x in a set P there exists no solution y dominating any member of the setP ; then, solutions belonging to the set P constitute a locally Pareto-optimal set. In Figure 3.11(e)we illustrate the concept of global and local Pareto-optimal sets.

3.2.2 Multiobjective Evolutionary Algorithms

As we mentioned in previous sections, in Evolutionary Algorithms a set of individuals is known aspopulation. The population-based nature of this metaheuristics is very promising for approximat-ing a set of non-dominated solutions for a MOOP in a single run.

The last years are witnessed of a exponential growing interest in using these metaheuristics for


solving MOOP, they are commonly known as Multiobjective Evolutionary algorithms (MOEAs).According to [115], by January 2011, more than 5600 publications1 have been published on

evolutionary multiobjective optimization. Among these papers, 66.8% have been published in thelast eight years, 38.4% are journal papers and 42.2% are conference papers.

In the last years, diverse MOEAs have been proposed. We may classify them into three groups:Pareto dominance, Indicator-based, and Decomposition-based.

Pareto Dominance MOEAs

These Pareto dominace based algorithms treat a MOOP as a whole and use the Pareto dominancerelationship among the solutions for driving them towards the Pareto front; however, their set ofPareto-optimal solutions are not uniformly distributed along the Pareto front.

Among the most popular methods based on Pareto dominance are:

• Pareto Archived Evolution Strategy (PAES) [89]. It is a multiobjective optimizerwhich uses a simple (1+1) local search evolution strategy. Nonetheless, it is capable of findingdiverse solutions in the Pareto optimal set because it maintains an archive of nondominatedsolutions which it exploits to estimate accurately the quality of new candidate solutions.There exist three versions of PAES, (1 + 1), (1 + λ), and (µ + λ).

• Fast Non-Dominated Sorting Genetic Algorithm (NSGA-II) [90]. This MOEA triesto obtain a new population (offspring population Q) from an original one (parent populationP ) by applying classical genetic operators, such as selection, crossover, and mutation. Then,both populations, offspring and parent, are mixed into a new population R. This new popula-tion is sorted into categories (ranks) according to their relationship of dominance. After that,the best individuals are selected to create a new parent population for the next generation.In the case of having to choose among individuals with the same rank, the crowding distanceof the individuals belonging to the same rank is calculated, in order to decide which are thebest individuals.

• Strength Pareto Evolutionary Algorithm 2 (SPEA2) [91]. In SPEA2, a fitness valuethat is the sum of its strength raw fitness plus a density estimation is assigned to eachindividual. Then, the best individuals (non-dominated ones) of both (population and archive)are copied into a new population, truncating it with the aim of not exceeding the size of thepopulation. Moreover, SPEA2 also applies selection, crossover, and mutation operators, withthe aim of generating the next population.

Indicator-based MOEAs

The Indicator-based MOEAs make use of an indicator function to evaluate a set of solutions. Thehypervolume measure [116] has been often used as indicator function, the most popular indicator-based approaches are:

• Indicator-based Evolutionary Algorithm (IBEA) [92]. This MOEA uses an arbitraryindicator to compare a pair of candidate solutions. Furthermore, any additional diversitypreservation mechanism such as fitness sharing, is no longer required. In comparison to otherapproaches, the IBEA only compares pairs of individuals instead of entire approximation sets.

1The statistical data is based on the paper repository in the EMOO web site,http://delta.cs.cinvestav.mx/ ccoello/EMOO/, which is maintained by Professor Coello Coello.


• S-metric Selection Evolutionary Multiobjective Optimisation Algorithm (SMS-EMOA) [93]. The idea in this MOEA, is to aim explicitly for the maximisation of thedominated hypervolume within the optimisation process. A steady-state EMOA featuresa selection operator based on the hypervolume measure combined with the concept of non-dominated sorting. Furthermore, the population of the algorithm evolves to a well-distributedset of solutions, focussing on interesting regions of the Pareto front. However, the maindisadvantage of this MOEA is the high computational cost in calculating the hypervolumewhen the number of objective functions is more than two.

Decomposition-based MOEAs

In this algorithms, the MOOP is divided into diverse single-objective problems. Therefore, theyassociate each individual solution with a particular scalar optimization problem with the aim ofallocating computational resources to different parts of the Pareto front.

Among the most important MOEAs based on decomposition are:

• Multiobjective Genetic Local Search (MOGLS) [94]. In this MOEA, a weighted scalar-ization function is used at each generation during selection, where the weights are generatedrandomly. The mating population in the MOGLS consists of a few individuals selected fromthe current population in terms of the current scalarization function, which are used for gen-erating new solutions (offspring solutions) by using genetic operators. Then, a local searchprocedure is applied with the aim of improving the quality of the new solution. Furthermore,the MOGLS use not only the current population, but also an external population.

• Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D) [95].It is based on conventional aggregation approaches in which an MOOP is decomposed intoa number of scalar objective optimization subproblems. The objective of each subproblem,is a weighted aggregation of the individual objectives. Neighborhood relations among thesesubproblems are defined based on the distances between their aggregation weight vectors;thus, a subproblem i is a neighbor of a subproblem j if and only if the weight vector ofsubproblem i is close to that of subproblem j. Each subproblem is optimized in the MOEA/Dby using information mainly from its neighboring subproblems. The main advantage ofMOEA/D is that it commonly obtains a well representative set of Pareto-optimal solutionsuniformly distributed along the Pareto front.

3.2.3 Quality Assessment in Multiobjective Optimization

In order to evaluate the quality of a set of non-dominated solutions or an approximation to Pareto-optimal points, it is normally considered in the objective space towards the global Pareto-optimalfront or simply Pareto front.

It is reasonable to think that a fair comparison between two approximations should be per-formed by using the Pareto dominance concept. However, since the Pareto dominance is not a totalorder, not all points are comparable. For example, a point located in the middle of a bi-dimensionalobjective space is able to dominate 1/4 of it, is dominated by 1/4, and is no-comparable to 1/4+1/4(see Figure 3.9). Therefore, this point is only comparable with 1/2 of the total objective space. Ifwe consider M dimensions, then this fraction decreases exponentially, therefore, the use of just thePareto dominance concept is insufficient for measuring the quality of an approximation set.

In general, the quality indicator reflect a cardinality value related to the number of solutionpoints, a real value for proximity to the Pareto front or other set of points, or real values for


measuring some approximation set characteristics (diversity of points, uniformity of points, etc.).A good quality measure or indicator should respect the following aspect:

• Convergence. The approximation set should minimize the distance to the Pareto front.

• Diversity. The approximate non-dominated solutions should be spread along the whole Paretofront.

• Cardinality. The number of non-dominated solutions should be appropriate.

The order established by the quality indicator should be conform with the Pareto dominancerelation as far as possible, we denominate these indicators as Pareto Compliant. Thus, we cansay that a quality indicator is Pareto Compliant if, given two approximation sets A and B whereA B and B A, the quality indicator reports that A is better than B.

We can differentiate between two groups of quality indicators, those in which is required toknow the optimal Pareto front, and those in which it is not needed. In the following, we brieflydescribe the most common used quality indicators in the literature.

• Number of Solutions. It indicates the number of solutions in the approximation set.Despite of how easy this indicator is calculated, it does not show the quality of the front. Inthe example showed in Figure 3.12(a), the approximation set contains five non-dominatedsolutions.

• Number of Optimal Solutions. It indicates the number of optimal solutions in the approx-imation set. In this indicator is necessary to know the optimal Pareto front. Furthermore, itdoes not provide any information about the distribution of the solutions. In Figure 3.12(b),we can see that two out of the five solutions are optimal solutions.

• Hypervolume (HV) [117]. It measures the volume (in the objective function space) coveredby members of a non-dominated set of solutions. In a multiobjective optimization problemwith M minimization objective functions, we calculate the size of the region of the objectivespace (hypervolume) of a non-dominated set of solutions A = a1, ..., ak bounded by areference point r = (r1, ..., rM ). The corresponding hypercube for each member ai of set A iscalculated as follows: h(ai) = [ai

1, r1]× ...× [aiM , rM ]. In this way, as it is shown in equation

(a): #Solutions = 5 (b): #Optimal Solutions = 2

Figure 3.12. Number of Solutions and Number of Optimal Solutions quality indicators.


(a): 2-dimensional HV (b): 3-dimensional HV

Figure 3.13. Hypervolume, quality indicator.

3.2, the hypervolume of A is computed by the union of these |A| hypercubes, with repeatedlycovered hypercubes being counted once, where L refers to Lebesgue measure.

HV (A, r) = L

|A|⋃

i=1

h(ai) | ai ∈ A

(3.2)

In Figure 3.13 we present an illustration of the 2-dimensional and 3-dimensional hypervolumeof a set of non-dominated solutions.

• Epsilon Indicator (Iε) [118]. Given two approximate sets, A and B, this ε-indicatormeasures the smallest amount, ε, that must be used to translate the set, A, so that everypoint in B is covered, see Figure 3.14(a). In a more formal way, it is defined as:

Iε(A, B) = minε ∈ R | ∀b ∈ B ∃a ∈ A : a ≺ b (3.3)

So, when Iε(A, B) < 1, all solutions in B are dominated by a solution in A. If Iε(A, B) = 1and Iε(B, A) = 1, then A and B represent the same Pareto front approximation. If Iε(A, B) >1 and Iε(B, A) > 1, then they both contain solutions not dominated by the other set.

• Generational Distance (GD) [119], [120]. It computes how far, on average, the approx-imation set (A) is from the optimal front (P ∗) using the Euclidean distance (measured inobjective space) between each vector and the nearest member of the optimal front (see Figure3.14(b)), in a more formal way, it is defined as:

GD(A, P ∗) =

∑

a∈A dist(a, P ∗)|A|

(3.4)

where dist(a, P ∗) is the minimum Euclidean distance between a and the points in P ∗. Themain disadvantage of this indicator is that only considers the distance to the optimal Paretofront, but not the spread of the solutions.

• Inverse Generational Distance (IGD) [118]. It computes how far, on average, theoptimal front (P ∗) is from the approximation set (A) using the Euclidean distance (measured


(a): Epsilon Indicator (b): Generational Distance (c): Inverse Generational Distance

(d): Spread

Figure 3.14. Epsilon Indicator, Generational Distance, Inverse Generational Distance, and Spread qualityindicators.

in objective space) between each vector and the nearest member of the optimal front (seeFigure 3.14(b)), in a more formal way, it is defined as:

IGD(A, P ∗) =

∑

v∈P ∗ dist(v, A)|P ∗|

(3.5)

where dist(v, A) is the minimum Euclidean distance between v and the points in A. Themain advantage of this indicator is that it measures both convergence and diversity.

• Spread [90]. It is a diversity indicator which measures the distribution of the solutions inan approximation set, see Figure 3.14(d). Given an approximation set A = a1, ..., ak, thespread indicator may be formulated as:

Spread(A) =(dL + dU +

∑k−1i=1

∣

∣di − d∣

∣)

dL + dU + (k − 1)d(3.6)

where di is the Euclidean distance between two consecutive solutions, d is the average of thesedistances, and dL, dU are the Euclidean distances to the extreme solutions of the optimalPareto front. In this metric, a value equals to zero indicates a uniform distribution of thesolutions in the set.

3.3 Parallel Computing 45

(a): SC(A,B) (b): SC(B,A)

Figure 3.15. Set Coverage, quality indicator.

• Set Coverage (SC) [121]. If we suppose two sets of non-dominated solutions A = a1, ..., akand B = b1, ..., bj, this indicator measures the fraction of non-dominated solutions in B;which are covered by the non-dominated solutions in A as:

SC(A, B) =|b ∈ B; ∃ a ∈ A : a b|

|B|(3.7)

If SC(A, B)=1, all points in B are dominated by or equal to points in A, whereas SC(A, B)=0means that none of the points in B are covered by the set of A. As the dominance operatoris not symmetric, it is necessary to calculate both SC(A, B) and SC(B, A), since SC(B, A)is not necessarily equal to 1 - SC(A, B).

An example of this indicator is shown in Figure 3.15. In Figure 3.15, the black points arethe solutions in A, and the grey points are the solutions in B. As we may observe, the frontA dominates 30% of the front B, whereas the front B dominates 60% of the front A.

3.3 Parallel Computing

Traditionally, software has been written for serial computation, that is to say, to be run on asingle computer having a single Central Processing Unit (CPU). In serial computing, a problemis commonly broken into a discrete series of instructions which are executed one after another;therefore, only one instruction may execute at any moment in time [122].

On the other hand, we can define Parallel Computing as the use of multiple compute resourcesto solve a computational problem, that is to say, to be run using multiple CPUs. Unlike serialcomputing, in parallel computing the problem is divided into discrete parts that can be solvedconcurrently, where each part is further divided into a discrete series of instructions which areexecuted one after another. Therefore, the instructions of each part execute simultaneously ondifferent CPUs.

The compute resources in parallel computing might be:

• A single computer with multiple processors.

• Several computers interconnected by the same data network.


(a): Single Computing

(b): Parallel Computing

Figure 3.16. Concepts of Single and Parallel Computing

• A combination of both: Several computers with multiple processors interconnected by thesame data network.

Commonly, a computational problem should be able to be broken into discrete pieces of workthan can be solved simultaneously, execute multiple program instructions at any moment in time,and be solved in less time with multiple compute resources than with a single computer resource.

The use of Parallel computing is critical in many fields. Historically, it has been used to modeldifficult problems in many areas of science and engineering, such as: atmosphere and environment,physics, bioscience, molecular sciences, geology and seismology, mechanical engineering, electricalengineering, computer science, or telecommunications.

Nowadays, more and more commercial applications provide a greater force in the developmentof faster computers. Therefore, the use of parallel computing is a great option to face the largeamounts of data in sophisticated ways. Among the most common industrial and commercialapplication are: databases, oil exploration, web search engines, medical imaging and diagnosis,pharmaceutical design, financial and economic modeling, or networked video and multimedia tech-nologies.

To sum up, in Figure 3.16, we present an illustrative example of single and parallel computing.As is pointed in [122], there exist many reasons for using parallel computing in both research

and development of applications, among the most important reasons are:

• Save time and/or money. In theory, throwing more resources at a task will shorten itstime to completion, with potential cost savings. Parallel computers can be built from cheap,commodity components.


• Solve larger problems. Many problems are so large and/or complex that it is impracticalor impossible to solve them on a single computer, especially given limited computer memory.

• Provide concurrency. A single compute resource can only do one thing at a time. Multiplecomputing resources can be doing many things simultaneously.

• Use of non-local resources. Using compute resources on a wide area network, or even theInternet when local compute resources are scarce.

• Limits to serial computing. Both physical and practical reasons pose significant con-straints to simply building ever faster serial computers: transmission speeds, limits to minia-turization, or economic limitations. Nowadays, current computer architectures are increas-ingly relying upon hardware level parallelism to improve performance by using multipleexecution units, pipelined instructions, or multi-core systems.

Depending on the access to memory, we can clearly differentiate three types of architectures:shared-memory, distributed-memory, and hybrid distributed-shared memory.

In this section, we start explaining each of the aforementioned parallel computer memory archi-tectures, enumerating the most important parallel application program interfaces used for makingthe most of each one. Then, we present a brief description of different parallel metaheuristicsmodels in the literature, and the power of using both approaches jointly for solving real-worldoptimization problems.

3.3.1 Shared memory Architecture

In this architecture, all the processors in the computer have direct (usually bus based) accessto common physical memory. From a programming point of view, several parallel tasks have thesame picture of memory and thus can directly address and access the same logical memory locationwithout taking into account where the physical memory is located.

Shared memory parallel computers vary widely, but it is common that all processors access allmemory as global address space. In this way, the processor operate independently but share thesame memory resources; so, changes in any part of memory effected by one processor are visibleto all the other processors as a result.

Depending on the way to access memory, the shared memory computers may be classified in:

(a): Uniform Memory Access (UMA) (b): Non-Uniform Memory Access (NUMA)

Figure 3.17. Shared Memory architecture with Uniform and Non-Uniform Memory Access.


Figure 3.18. Example of the Threads Model in a Shared-Memory system.

• Uniform Memory Access (UMA). Those computers with an uniform memory access arecommonly known as Symmetric Multi-Processors (SMP). In SMP, the hardware character-istics of all the processors is exactly the same, so, equal access times to memory. In Figure3.17(a), we present a graphical representation.

• Non-Uniform Memory Access (NUMA). They are commonly made by physically link-ing two or more SMPs. In this type of access, one SMP is able to access to the memory spaceof another SMP through a bus/interconnection network. Obviously, the access times to mem-ory are not the same due to the memory access across the bus/interconnection network isslower. Figure 3.17(b) illustrates a common non-uniform memory access model.

On the one hand, the use of machines with a shared memory space provides (global addressspace of memory) provides a user-friendly programming perspective to memory. Furthermore,between tasks, data sharing is both fast and uniform due to the proximity of memory to CPUs.

On the other hand, there exist a lack of scalability between the memory and CPU, that is tosay, adding more CPUs leads to an increment of traffic on shared-memory CPU paths and trafficassociated with cache/memory management. In this shared-memory systems, the user should beresponsible for synchronization among threads for ensuring a safe access of global memory. Finally,the design becomes increasingly expensive as we add new processors to the system.

The most common parallel programming model used in shared-memory computers is the threadsmodel. In the threads model of parallel programming, single process can have multiple and con-current execution paths. In order to understand easily the threads model, in Figure 3.18 we showa simple program that includes four subroutines that are executed by four threads. In this model,threads has local data, but also, share the entire resources. The communication among threads iscarried out through global memory (updating address location), so a synchronization constructionis required to ensure that more than one thread is not modifying the same global address at anytime.

From a programming point of view, a thread implementation is commonly a library of subrou-tines that are called from within parallel source code, but there exist other implementations whichjust include a set of compiler directives embedded in either serial or parallel source code. However,in both cases, the final user is in charge of determining the parallelism. Among the most importantimplementations, we can find:


• POSIX Threads. It is a library based implementation specified by the IEEE POSIX 1003.1cstandard (1995). It only supports C language and it is commonly referred as Pthreads. Sincethe parallelism in this implementation is very explicit, the programmer should pay significantattention to detail.

• OpenMP. It is a compiler directive based implementation which is available in C/C++ andFortran languages. Unlike the Pthreads, it is very easy and simple to use. Further details areprovided in the next subsection.

In order to exploit shared-systems and because it is easy to use, in this thesis we use OpenMPimplementation for exploiting the capabilities of SMP systems. In the following subsection weintroduce several aspects of OpenMP.

OpenMP

Open Multi-Processing (OpenMP) is an application programming interface that was create by themost important hardware and software vendors. The directives, library routines, and environmentvariables defined in OpenMP allow users to create and manage parallel programs while permittingportability across different shared memory architectures. The proposed set of directives in OpenMPextend the C, C++, and Fortran base languages with single program multiple data (SPMD) con-structs, work-sharing constructs, and synchronization constructs; and they provide support for thesharing and privatization of data. Furthermore, the library routines and environment variablesprovide the functionality to control the runtime environment. Thus, those compilers that supportthe OpenMP API often include a command line option to the compiler that activates and allowsinterpretation of all OpenMP directives.

Using this API, the user explicitly specifies the actions to be taken by the compiler in orderto execute the program in parallel. In this way, OpenMP does not check dependencies, conflicts,deadlocks, race conditions, or other problems that results from non-conforming programs. Inconclusion, the user is completely responsible for producing a conforming parallel program.

Figure 3.19. Fork-join execution model in OpenMP.

OpenMP uses the fork-join model of parallel execution (see Figure 3.19). In this way, anOpenMP program begins as a single thread of execution, called the initial thread. The initialthread executes sequentially, as if enclosed in an implicit inactive parallel region surrounding thewhole program.


On the one hand, when any thread encounters a parallel construct, the thread creates a team ofitself and zero or more additional threads and becomes the master of the new team. All membersof the new team execute the code inside the parallel construct. On the other hand, when any teamencounters a work-sharing construct, the work inside the construct is divided among the membersof the team and executed co-operatively instead of being executed by every thread. There is anoptional barrier at the end of work-sharing constructs. Execution of code by every thread in theteam resumes after the end of the work-sharing construct.

In OpenMP there exists synchronization constructs and library routines to co-ordinate threadsand data in parallel and work-sharing constructs. In addition, library routines and environmentvariables are available to control or query the runtime environment of OpenMP programs. Unfor-tunately, this parallel API makes no guarantee that input or output to the same file is synchronouswhen executed in parallel; in this case, the programmer is responsible for synchronizing input andoutput routines using the provided synchronization constructs or library routines.

There exists several compilers that support OpenMP directives, such as: GNU, INTEL, HP,MS, Cray. . . . In Table 3.1, we present the compilation options for the most common C/C++compilers: GNU and INTEL.

Table 3.1. Compilation options for the most common C/C++ compilers

GNU gcc (4.3.2) Compile with the -fopenmp optionIntel icc (10.1) Compile with the -openmp option

A full OpenMP 3.1 API Specification for C/C++/Fortran languages is able to download in[123]. In the following, we present a simple C/C++ program example with the aim of checkingwhether the compiler supports OpenMP directives:

Algorithm 1 HelloWorld_OpenMP.c1: #include <omp.h>2: #include <stdio.h>3: int main()4: #pragma omp parallel5: printf("Hello World! I am Thread#%d", omp_get_thread_num() );6:

For compiling the program HelloWorld_OpenMP.c with GCC or ICC, we can compile like this:

gcc -fopenmp HelloWorld_OpenMP.c -o HelloWorld_OpenMP

icc -openmp HelloWorld_OpenMP.c -o HelloWorld_OpenMP

Finally, if we run the HelloWorld_OpenMP program in a SMP system with 4 cores, the outputshould be:

Hello World! I am Thread#2Hello World! I am Thread#3Hello World! I am Thread#0Hello World! I am Thread#1


3.3.2 Distributed Memory Architecture

From a hardware perspective, it refers to a network based memory access for physical memorythat is not common. In this way, the different tasks only access local machine memory and mustuse messages to communicate with other machines where other tasks are running (the memory ofevery node/machine is private).

Like shared memory systems, distributed computers vary widely but share a common character-istic: they require a communication network (for example Ethernet) to connect different computers.Figure 3.20 shows an example of a distributed system with four computers interconnected througha data network.

Figure 3.20. Distributed Memory Architecture.

The processors in a distributed memory systems have their own local memory, and thus, thememory addresses in one processor do not map to another processor, so the concept of globaladdress space is not accepted in these computers. Due to their own local memory, each processorwork independently, and the changes made in local memory have no effect on the memory of theother processors.

In case of a processor need data from a memory located in other machine in the network, it isresponsibility of the programmer to explicitly define how and when data is communicated.

On the one hand, in this architecture, the memory is perfectly scalable with the number ofprocessors, so the higher the number of processors, larger amount of memory.

On the other hand, the programmer is completely responsible of the details associated with datacommunication between processors. Furthermore, the mapping of global data structures becomesdifficult in this memory organization.

The most common parallel programming model used in distributed-memory systems is themessage passing model. In this model, there exists a set of tasks that use their own local memory,where multiple tasks may reside on the same physical machine or across diverse machines. In orderto communicate, the tasks exchange data by sending and receiving messages. This data transferusually requires cooperative operation to be performed by each process. In Figure 3.21 we illustratea communication between a machine A and a machine B by messaging passing.

From a programming point of view, the message passing implementations usually comprise alibrary of subroutines. The most important implementation of this programming model appearedin 1994: the Message Passing Interface (MPI).

Since in this thesis we also deal with distributed memory systems with MPI, a further expla-nation of this implementation is reported in the next subsection.


Figure 3.21. Example of the Message Passing Model in a Distributed-Memory system.

Message Passing Interface (MPI)

The Message Passing Interface (MPI) is a specification for the developers and users of messagepassing libraries. The main goal of the MPI is to provide a widely used standard for writing messagepassing programs, attempting to be: practical, portable, efficient, and flexible. Like OpenMP, theinterface specification was defined for C/C++ and Fortran languages.

Since MPI is supported on virtually all high performance computing platforms, it is the onlymessage passing library which can be considered a standard. In fact, there is no need to modify amessage passing application when you port it to a different platform that supports MPI.

In MPI, different processes communicate one each other by using different routines, such assend or recv, for sending or receiving data respectively. In Figure 3.22 we present a typical master-slave example. As we may observe, the master role is usually assigned to the process with rank 0,and the rest of processes are designated as slaves. In this example, the master process sends to theslaves some work to do and remains waiting until the slaves finished their assigned tasks. Oncefinished the work, each slave process sends its results to the master process which is in charged ofprocessing them.

Every program that makes use of this application program interface presents the following basicscheme:

1. Step 1. Include the library mpi.h, which includes more than a hundred routines.

2. Step 2. Initialize the MPI environment. Before using any MPI routine, it is necessary to

Figure 3.22. Master-slave scheme in MPI with four processes.


initialize the MPI environment.

3. Step 3. Obtain the number of processes in the MPI environment.

4. Step 4. Who am I?. It is required that each process in the MPI environment knows theirunique identification or rank.

5. Step 5. Closure of the MPI environment.

Although the MPI programming interface has been standardized, implementations will differ,as will the way MPI programs are compiled and run on different platforms. Among the mostimportant implementations are: MPICH, Open MPI, LAM/MPI, or IBM MPI.

For a detailed explanation about the specifications of MPI, please refer to [124]. Like wedo in the OpenMP subsection, we present a simple C/C++ program example with the aim ofunderstanding how MPI works. In this example, we present the basic MPI scheme. In the body ofthe program, the processes print a simple message of Hello World and report their rank and thenumber of processes in the MPI environment.

Algorithm 2 HelloWorld_MPI.c1: #include <mpi.h> // Step 12: #include <stdio.h>3: int main(int argc, char** argv)4: int rank, size;5: MPI_Init(&argc, &argv); // Step 26: MPI_Comm_size(MPI_COMM_WORLD, &size); // Step 37: MPI_Comm_rank(MPI_COMM_WORLD, &rank); // Step 48: printf("Hello World! I am Process#%d of %d \n", rank+1, size);9: MPI_Comm_Finalize(); // Step 5

10: return 0;11:

For compiling the program HelloWorld_MPI.c (Algorithm 2), we use the MPI implementationMPICH. In this way, we can compile like this:

mpicc HelloWorld_MPI.c -o HelloWorld_MPI

If we run the HelloWorld_MPI program in a system with four interconnected machines, wehave to use the following command:

mpiexec -n 4 HelloWorld_MPI

and the output should be:

Hello world! I am Process#2 of 4Hello world! I am Process#3 of 4Hello world! I am Process#1 of 4Hello world! I am Process#4 of 4


3.3.3 Hybrid Shared-Distributed Memory Architecture

In the last years, the vast majority of high performance computers in the world employ a hybridshared-distributed memory architecture. In this way, they use the advantages of the shared anddistributed models.

In this architecture (see Figure 3.23), the shared memory component can be a cache coherentSMP machine and/or graphics processing units (GPUs). On the other hand, the distributedmemory component is the networking of diverse SMP/GPU machines. Each SMP/GPU unit onlyhas their own local memory and, therefore, in case of requiring a data from another SMP/GPUunit, it has to use communications.

In Figure 3.23 we present a basic scheme of a hybrid model. Concretely, in Figure 3.23(a), thesystem consists of four SMP units interconnected through a data network. In the second image,Figure 3.23(b), we present a model in which each unit consists of two CPUs and two GPUs.

In this architecture the programming model is also hybrid. Therefore, it combines more thanone of the previously described programming models (threads model and message passing model).

The most common programming model in this architecture is the combination of the messagepassing model MPI with the threads model OpenMP. In this way, the threads perform computa-tionally intensive kernels using local, on-node data, whereas communications between processeson different nodes is carried out over the data network by using MPI. On the whole, this hybridmodel is perfectly suitable for being used in enviroment of clustered multi/many-core machines.

(a): SMP units

(b): SMP/GPU units

Figure 3.23. Hybrid Shared-Distributed Memory architecture SMP or SMP/GPU units.


Table 3.2. Advantages and Disadvantages of MPI and OpenMP

(a) MPI

+ Portable to distributed andshared memory machines

+ Scales beyond one node+ No data placement problem

- Difficult to develop and debug- High latency, low bandwidth- Explicit communication- Large granularity- Difficult load balancing

(b) OpenMP

+ Easy to implement parallelism+ Low latency, high bandwidth+ Implicit Communication+ Coarse and fine granularity+ Dynamic load balancing

- Only on shared memory machines- Scale within one node- Possible data placement problem- No specific thread order

However, as we saw in the previous sections, both OpenMP and MPI have advantages anddisadvantages, in Table 3.2 we summarize the main advantages and disavantages of both.

The immediate question is then why using a hybrid programming model?. Because using thismodel, we can avoid extra communication overhead with MPI within node, as well as increasing theload balancing by using OpenMP, leading to a maximization of the global efficiency, performance,and scalability.

In Figure 3.24 we present a typical master-slave example in a hybrid shared-distributed memorysystem using MPI and OpenMP jointly. As we may observe, the master process (rank 0) sendsto the slaves some work to do and remains waiting until the slaves finished their assigned tasks.In this hybrid models, the slaves processes use OpenMP directives to divide their assigned loadof work among four threads. Once finished the work, each slave process sends its results to themaster process which is in charged of processing them.

In the following, we present a simple example of a C/C++ program with the aim of understand-ing how to use OpenMP and MPI together. In this example, each process executes the OpenMPdirective in order to generate a number of threads, then each thread report its parent process andits thread rank.

Figure 3.24. Master-slave scheme in a Hybrid OpenMP/MPI model with four processes and four threads.


Algorithm 3 HelloWorld_Hybrid.c1: #include <mpi.h>2: #include <omp.h>3: #include <stdio.h>4: int main(int argc, char** argv)5: int rank, size, mpisupport;6: MPI_Init_thread(&argc, &argv, MPI_THREAD_MULTIPLE,7: &mpisupport);8: MPI_Comm_size(MPI_COMM_WORLD, &size);9: MPI_Comm_rank(MPI_COMM_WORLD, &rank);

10: #pragma omp parallel11: printf("Hello World! I am Process#%d of %d (Thread#%d)\n",12: rank+1, size, omp_get_thread_num() );13: MPI_Comm_Finalize();14: return 0;15:

For compiling the program HelloWorld_Hybrid.c (Algorithm 3), we use the MPI implementa-tion MPICH with GCC. In this way, we can compile like this:

mpicc HelloWorld_Hybrid.c -o HelloWorld_Hybrid -fopenmp

If we run the HelloWorld_Hybrid program in a system with two interconnected SMP machineswith four cores, we have to use the following command:

mpiexec -n 2 HelloWorld_Hybrid

and the output should be:

Hello world! I am Process#1 of 2 (Thread#0)Hello world! I am Process#1 of 2 (Thread#1)Hello world! I am Process#1 of 2 (Thread#2)Hello world! I am Process#1 of 2 (Thread#3)Hello world! I am Process#2 of 2 (Thread#0)Hello world! I am Process#2 of 2 (Thread#1)Hello world! I am Process#2 of 2 (Thread#2)Hello world! I am Process#2 of 2 (Thread#3)

3.3.4 Parallel Metaheuristics

In the literature, the parallel metaheuristics has been widely explored, we can find a number ofsurveys, taxonomies, and syntheses: [125], [126], [127], [103], [128], [129], [96].

From a computational perspective, metaheuristics are just algorithms from which we can extractfunctional or data parallelism [96]. Unfortunately, data and functional parallelism are in shortsupply for many metaheuristics. For example, in a local search procedure we find strong datadependencies between successive iterations, or in the passage from one generation to another instandard evolutionary algorithms is essentially a sequential process.


Metaheuristics are limited in data or functional parallelism, but they usually offer other oppor-tunities for parallel computing [96]. A classification of different parallelization strategies appearsin [96], where the authors differentiate three models:

• Type 1. Those strategies in which we can find the parallelism within an iteration of themetaheuristics approach. In this type, the limited functional or data parallelism of a moveevaluation is exploited or moves are evaluated in parallel. In [96], the authors refer to theseparallel strategies as low-level parallelism, which are rather straightforward and aim solely tospeedup computations, without any attempt at achieving better exploration or higher qualitysolutions.

• Type 2. In these approaches the parallelism is focused on partitioning the set of decisionvariables and reducing the search space as result. The main disadvantage of these parallelstrategies is that is necessary to be repeated to allow the exploration of the complete searchspace. Type 2 strategies are generally implemented in some sort of master-slave framework,where the master process partitions the decision variables and send different parts to theslaves processes. Concurrently and independently, the slaves explore their assigned partitions.Finally, once the master receives the results, it must perform a more complex operation ofcombining the partial solutions to form a complete solution to the problem.

• Type 3. The parallelism is obtained from multiple concurrent explorations of the solutionspace. In this type, each concurrent thread or process may or may not execute the sameheuristic method. They may start from the same or different initial solutions and maycommunicate during the search or only at the end to identify the best overall solutions(single objective) or the set of non-dominated solutions (multiobjective).

On the whole, parallel metaheuristics offer the possibility to address problems more efficientlyin terms of computing efficiency or solution quality. Since the tackled problems in this thesisare NP-complete and we solve large instances, the use of parallelism is clearly justified. Moreconcretely, we present two parallel multiobjective evolutionary algorithms in which the goal is tospeedup the runtime without any attempt at achieving better exploration (Type 1).

3.3.5 Performance Assessment in Parallel Computing

Since measuring the performance of any parallel evolutionary algorithm is important, some perfor-mance measures, such as speedup, have been borrowed from traditional algorithms [130].

As is defined in [131], Amdahl’s Law states that the performance improvement to be gainedfrom using some faster mode of execution is limited by the fraction of the time the faster modecan be used.

Therefore, Amdahl’s Law defines the speedup that can be gained by using a particular feature.In a more formal way, let Tm be the runtime for an algorithm using m processors and T1 theruntime of the sequential version, the speedup is computed according to the following equation:

Speedup =T1

Tm(3.8)

In this way, the speedup metrics reports us how much faster an algorithm will run as opposedto the sequential version. According to [130], since evolutionary algorithms are non-deterministic,we have to adapt the speedup definition. Thus, if we denote the mean runtime on a uni-processoras T1 and the mean runtime on m processors as Tm, the adapted speedup metrics should be:


(a): Speedup < m (b): Speedup = m (c): Speedup > m

Figure 3.25. Possible situations of speedup.

Speedup =T1

Tm

(3.9)

Depending on the obtained value of speedup, we can distinguish three cases:

1. Speedup < m. It is the most common case and may occur for diverse reasons, for example,if the algorithm is not completely parallelized or parallelizable.

2. Speedup = m. It indicates linear speedup (ideal case).

3. Speedup > m. It is not theoretically possible, but it rarely happens. According to [130],in parallel evolutionary algorithms, super-linear speedup is possible, both in theory andpractice, for homogeneous as well as heterogeneous computational resources. Anyway, inthis case, maybe something is wrong in the experimentation.

In Figure 3.25 we present an example for each of the aforementioned cases. In Figure 3.25(a),we present the common case, in which the speedup < m. In Figure 3.25(a) we can see a plotthat shows the linear case, in which the speedup is always exactly the number of processors used.Finally in Figure 3.25(c), we present the rare case of super-linear speedup.

Other related metrics widely used in parallel computing is the efficiency. The efficiency iscomputed dividing the obtained speedup with m processors by the number of processors used (m),therefore:

Efficiency =Speedup

m(3.10)

In this thesis we use the notation Sm and Em to denote the speedup and efficiency obtainedon m processors; respectively. Furthermore, we report the efficiency as a percentage.

3.4 Statistical Reliability

In this thesis, with the aim of making a comparison among the algorithms with a certain level ofconfidence, and because we are dealing with stochastic algorithms, we have performed a statisticalanalysis of the results obtained.

3.4 Statistical Reliability 59

Test for Residual Normality

(Kolmogorov-Smirnov and Saphiro-Wilk)

Non-Parametric Analysis

(Kruskal-Wallis)

Test for Homoscedasticity

(Levene)

Variance Analysis

(ANOVA)

No

No

Yes

Yes

Figure 3.26. Statistical analysis scheme.

In Figure 3.26, we present a scheme of the statistical analysis that we have applied in this thesisfor the adjustment of the algorithms, for comparing the proposals, and for validating the parallelproposals [132].

As we can see in Figure 3.26, in the first place, a test for calculating residual normality is applied,in our case, Kolmogorov-Smirnov [133]. The main objective of this test is to check whether thevalues of the results follow a gaussian distribution or not. For non-gaussian distributions, weperform a non-parametric analysis, such as Kruskal-Wallis [134]. However, if the values follow agaussian distribution, a test to check the homogeneity of the variances (Levene test [135]) is alsocarried out.

Finally, if this is positive, we apply an ANOVA analysis [136]; otherwise, we perform theKruskal-Wallis analysis.

The confidence level considered in this work is always 95% in the statistical tests (a significancelevel of 5% or p-value under 0.05). This means that the differences are unlikely to have occurredby chance with a probability of 95%.

4Routing and Wavelength Assignment problem

Nowadays, the number of users of the Internet has raised exponentially. Unfortunately, our currentdata networks are not able to support this exponential growth because their bandwidth is notsufficient. The usage of optical data networks is a suitable option for dealing with this drawback.

Since the vast majority of devices require only a few Gbps of bandwidth, there exists a wastageof bandwidth in each optical link (an optical fiber is around 50 Tbps). To optimize the usage ofbandwidth in optical networks, a new technology has been introduced. This technique is knownas wavelength division multiplexing (WDM). This technology multiplies the available capacity ofan optical fiber link by adding new channels, with each channel on a new wavelength of light. Theaim of WDM is to ensure fluent communications between several devices, avoiding bottlenecks

However, a problem occurs when it is necessary to establish a set of traffic demands. Thisproblem is known in the literature as routing and wavelength assignment (RWA) problem. TheRWA problem consists of two subproblems: routing and wavelength assignment.

On the one hand, we calculate a physical route for the given connection. Afterward, we assignto each physical link of this route an available wavelength. This connection carried end-to-endfrom a source node to a destination node over a wavelength on each intermediate optical fiber linkis known as lightpath.

The RWA problem may be classified in two groups, depending on the traffic pattern adopted.On the one hand, we could refer to a static problem when the demands are given in advance(static RWA problem). On the other hand, we consider a dynamic problem when the demandsare given in real time (dynamic RWA problem). Since wide area networks (WANs) are oriented toprecontracted services [5], in this thesis we focus on tackling the static RWA problem.

As we mentioned in Chapter 2, the RWA problem has been proven to be an NP-completeoptimization problem [8] and thus, a challenging optimization problem of practical relevance. Aswe saw, many research works deal with this problem by dividing it into two subproblems: routingand wavelength assignment, solving them separately. However, in more recent literature, the RWAproblem has been tackled as a multiobjective optimization problem.

This chapter is devoted to present a formal definition of the multiobjective RWA problem, in-cluding all assumptions, variables, parameters, and constraints involved. Furthermore, we showan illustrative example to facilitate the understanding of this telecommunication problem. Sincein this thesis we use Multiobjective Evolutionary Algorithms (MOEAs), we present the representa-tion of the individuals used, which determines how the problem is structured in these algorithms.

61

62 4. Routing and Wavelength Assignment problem

Finally, we present the optical networks and data sets used in the experimental results for theRWA problem.

4.1 Problem Formulation

In this thesis, an optical network is modeled as a directed graph G = (N, E), where N is the setof nodes and E is the set of links between nodes. In the first place, we enumerate the assumptionstaken into account for solving the RWA problem:

− In all nodes of N , the number of outgoing physical links is identical to the number of in-coming links. Furthermore, all nodes are equipped with facilities for supporting wavelengthconversion; therefore, we assume that a wavelength may be switched from one wavelength toanother one in all nodes of the network.

− All nodes are equipped with infinite number of transceivers, that is to say, it is possible to es-tablish as much lightpaths as required without exceeding the number of available wavelengthsper optical fiber links.

− All physical links in E will be divided into the same number of channels or wavelengths oflight (W ). Furthermore, we assume a fix capacity at each link equals to OC-48 (2.5 Gbps).

− We suppose that, for all links (m, n) ∈ E, where m, n ∈ N ; the propagation delay (dmn) isequal to 1, dmn = 1.

− The set of data connections is known in advance; thus, we suppose a static traffic pattern.Furthermore, all data connections have the same bandwidth requirement, OC-48 (2.5 Gbps).Furthermore, as in [41] and [42], for any connection request there exists at least one solution;therefore, the event of blocked requests is not possible.

In the following, we describe the main parameters and variables involved in the RWA problem:

• N : the set of nodes. Note that, |N | indicates the number of optical nodes.

• E: the set of links.

• Λ: traffic matrix

Λ = [ Λsd; s, d ∈ N ]|N |×|N |,

where the rows represent source nodes (s) and columns represent destination nodes (d). Thus,Λsd is the number of connection requests between the node pair (s, d) with OC-48 bandwidthrequirement.

• (m, n) ∈ E. Optical link from node m to node n.

• Pmn, ∀m, n ∈ N : number of fibers interconnecting nodes m and n. If Pmn = Pnm = 0, thereare no physical links between m and n, otherwise Pmn=Pnm=1.

• W . Number of channels or different wavelengths per link (m, n) ∈ E.

4.2 Example of the RWA problem 63

• φm, ∀m ∈ N : indicates the number of wavelength conversions at node m. It is increasedevery time that a wavelength λa is switched to λb, where a, b ∈ 1 . . . W.

• P ij,wmn , ∀w ∈ 1 . . . W, ∀i, j, m, n ∈ N : number of lightpaths between node i to node j routed

through fiber link (m, n) on wavelength w.

Using the above definitions, the RWA problem may be stated as a Multiobjective OptimizationProblem (MOOP) [88], searching the best solution that simultaneously minimizes the followingtwo objective functions:

1. Number of hops (f1): It is the number of routers traversed by a packet between its sourceand destination.

Minimize

|N |∑

i=1

|N |∑

j=1

|N |∑

m=1

|N |∑

n=1

(dmn ×W∑

w=1

P ij,wmn )

(4.1)

2. Number of wavelength conversions (f2): A wavelength conversion occurs when the inputwavelength in a WDM router must be converted into another wavelength of light.

Minimize

|N |∑

m=1

φm

(4.2)

Furthermore, we have to fulfill the wavelength conflict constraint: Two different unicast trans-missions must be allocated with different wavelengths when they are transmitted through the sameoptical link (i, j).

4.2 Example of the RWA problem

An example helps to understand the formulation problem and the objective functions of the static-RWA problem.

Statement: Given the optical network topology of Figure 4.1 (|N |=5), suppose the followingtraffic matrix (Λ) and number of available wavelengths per link (W ):

Λ =

0 0 1 0 10 0 0 0 10 0 0 1 00 0 0 0 00 0 0 0 0

W = 2

As we can see in the solution of Figure 4.1, the demands (1,3), (2,5), and (3,4) do not presentany wavelength switching; however, the demand (1,5) presents one wavelength conversion at node3. Therefore, the solution has seven hops (f1=7), and one wavelength conversion (f2=1).

In Figure 4.1, we present all necessary calculations to obtain the value of the two objectivefunctions, number of hops (f1) and number of wavelength conversions (f2). The solution presented(Figure 4.1) for this specific topology could not be the best one; this example only tries to help tounderstand the problem formulation and the objective functions.


Figure 4.1. Illustrative example of the RWA problem.

4.3 Representation of Individuals

The representation of the individual is critical for determining how the problem is structured; thus,in Figure 4.2 we present the individual encoding used in this thesis for solving the RWA problem.

Before running the Evolutionary Algorithm, we obtain the k shortest paths for each demand(s,d) in the traffic matrix (Λ) by using the Yen’s algorithm [10]. Let p1 . . . pk be the set of k shortestpaths computed between the nodes s and d. The total number of feasible lightpaths (Ls,d) betweenthe nodes s and d is:

∑ki=1 W ni , where ni is the number of hops corresponding to pi.

For each pi, where i ∈ 1 . . . k, we calculate the position of its first lightpath or index (p0i ), as

follows:

4.3 Representation of Individuals 65

Figure 4.2. Representation of the Individuals for the Traffic Grooming problem

p0i =

0 if i = 1;p0

i−1 + W ni−1 otherwise.

In order to generate a random individual, for each demand (s,d) in Λ, we select randomly afeasible lightpath in the range [0, Ls,d) (line 5 in Algorithm 4) and we store it in the chromosome(see Figure 4.2). Let ls,d be a random number between 0 and Ls,d − 1, we can calculate easily thelightpath by using the CalculateLightpath procedure (line 6). In the first place, we set Ps,d = pj

and P 0s,d = p0

j if p0j ≤ lsd < p0

j+1, where j ∈ 1 . . . k − 1; otherwise, Ps,d = pk and P 0s,d = p0

k.Then, we calculate the wavelengths at each link of the physical path by converting (ls,d − P 0

s,d)from Base-10 to Base-W :

(ls,d − P 0s,d)(10) → (wsd)(W )

Finally, we update the resources of the network as well as the number of conversions per node(φm, ∀m ∈ N). To do this, we use the UpdateResources procedure (line 7 in Algorithm 4).

For example, in Figure 4.2, we suppose two wavelengths per link (W =2) and the following setof shortest paths between nodes 3 and 4: p1 = 3− 5− 4 (n1 = 2) and p2 = 3− 1− 2− 4 (n2 = 3).Therefore, the indexes of p1 and p2 are: p0

1 = 0 and p02 = p0

1 +W n1 = 0+22 = 4. The total numberof feasible lightpaths between the nodes 3 and 4 is L3,4 = 22 + 23 = 12.


Algorithm 4 Random Individual for the RWA problem1: x ← ∅ // new individual2: for s = 0 to N -1 do3: for d = 0 to N -1 do4: if s 6= d then5: xsd ← ramdom (0, Lsd)6: CalculateLightpath(xsd)7: UpdateResources(xsd)8: end if9: end for

10: end for11: x.f1 ←

∑|N |i=1

∑|N |j=1

∑|N |m=1

∑|N |n=1 (dmn ×

∑Ww=1 P ij,w

mn )

12: x.f2 ←∑|N |

m=1 φm

As we may observe in Figure 4.2, in the chromosome of the individual we have selected randomlythe following feasible lightpath: l3,4 = 3. Since, l3,4 is greater than p0

1 but is not greater than p02;

then, P3,4 = p1 and P 03,4 = p0

1. The wavelengths are calculated by converting (l3,4 − P 03,4) from

Base-10 to Base-2 (W = 2): (3− 0) = 3(10) = 11(2). Therefore, due to the obtained value is 11(2),the used wavelengths between the nodes in the physical route 3-5-4 are: λ1 and λ1, respectively.Since the wavelengths are in the range [1 . . . W ], we have to label the wavelengths as λ2 and λ2,respectively; so, the final lightpath is 3− λ2 − 5− λ2 − 4 instead of 3− λ1 − 5− λ1 − 4.

Now, lets suppose that l3,4 is 10 instead of 3 (l3,4 = 10). In this case, l3,4 is greater than p01

and also greater than p02; thus, P3,4 = p2 and P 0

3,4 = p02. Like we did before, the wavelengths are

calculated by converting (l3,4 − P 03,4) from Base-10 to Base-2 (W =2): (10 − 4) = 6(10) = 110(2).

Therefore, the final lightpath should be 3− λ2 − 1− λ2 − 2− λ1 − 4.

4.4 Data Sets: Optical networks and sets of demands

In this thesis, we have used diverse optical network topologies as well as different amount of trafficin order to compare the goodness and effectiveness of proposed multiobjective approaches whensolving the RWA problem.

In this way, we have used three real-world optical networks. The first one is the well-documentedPan European network (COST239, Europe) [137], which consists of 11 nodes and 52 links of fiber.The second one is the National Science Foundation network (NSF, U.S.A.) [138], which consistsof 14 nodes and 42 links. Finally, the last optical network corresponds with the Nippon Telegraphand Telephone (NTT, Japan) [139], which consists of 55 nodes and 144 links.

Furthermore, the traffic matrices or sets of demands used were generated by using a gravitydemand model [140]. As we can see in Equation 4.3, the gravity-based model assumes that thenumber of demands exchanged between two nodes is proportional to the product of the degrees ofthe two nodes, and inversely proportional to the Euclidean distance between them.

demands(s, d) =[

nodal degrees∗nodal degreed

distances−d∗ constant

]

(4.3)

where the constant is simply a uniform scaling factor for adjusting the traffic to the desiredvolume level. In this way, we have generated 12 data sets for each optical topology, a total of 36scenarios with very different characteristics.

4.4 Data Sets: Optical networks and sets of demands 67

In Table 4.1, Table 4.2, and Table 4.3, we present the specifications of each optical networktopology, as well as the corresponding constant value used for generating the different sets ofdemands. Those data sets in which there is no constant value is due to the fact that they wereobtained from [141]. For further information about them, please refer to [141]. In the use of thereference points, the ideal point always is (0,0) for all the scenarios. Therefore, the reference pointindicated in the tables always is the nadir point.

Note that, the notation #Demands used in the aforementioned tables refers to the total numberof demands in a traffic matrix, therefore:

#Demands =|N |∑

s=1

|N |∑

d=1

Λsd (4.4)

The three optical network topologies, with their 36 corresponding data sets can be downloadedfrom [142].

Table 4.1. COST239: Specifications of data sets for the RWA problem.

#Demands W Runtime Reference constant(seconds) Point

COST239#01 10 2 6 (46, 9) 0.042COST239#02 20 2 65 (103, 25) 0.044COST239#03 30 2 70 (145, 44) 0.047COST239#04 20 3 6 (74, 14) 0.044COST239#05 30 3 65 (109, 25) 0.047COST239#06 40 3 110 (147, 37) 0.051COST239#07 104 10 20 (480, 30) 0.09COST239#08 202 16 40 (950, 90) 0.15COST239#09 303 24 70 (1350, 110) 0.19COST239#10 406 32 120 (1800, 150) 0.26COST239#11 505 40 175 (2200, 160) 0.31COST239#12 609 48 250 (2650, 200) 0.38

Number of Nodes 11Number of Links 52


Table 4.2. NSF: Specifications of data sets for the RWA problem.


NSF#01 10 6 1 (70, 10) -NSF#02 20 6 3 (150, 20) -NSF#03 30 6 6 (230, 50) -NSF#04 20 8 3 (150, 20) -NSF#05 30 8 6 (240, 30) -NSF#06 40 8 12 (300, 70) -NSF#07 104 16 20 (324, 72) 0.22NSF#08 202 22 40 (749, 226) 0.34NSF#09 303 36 70 (1062, 305) 0.45NSF#10 406 50 120 (1383, 397) 0.51NSF#11 505 60 175 (1824, 530) 0.63NSF#12 609 72 250 (2139, 620) 0.76


4.4 Data Sets: Optical networks and sets of demands 69

Table 4.3. NTT: Specifications of data sets for the RWA problem.


NTT#01 10 10 6 (220, 20) -NTT#02 20 10 65 (530, 70) -NTT#03 40 10 110 (790, 190) -NTT#04 10 8 6 (230, 20) -NTT#05 20 8 65 (520, 110) -NTT#06 30 8 70 (560, 80) -NTT#07 104 12 20 (212, 12) 0.15NTT#08 202 18 40 (445, 44) 0.17NTT#09 303 26 70 (747, 107) 0.22NTT#10 406 34 120 (1131, 210) 0.25NTT#11 505 42 175 (1405, 250) 0.3NTT#12 609 50 250 (2077, 504) 0.37


5Traffic Grooming problem

As we saw in previous chapters, the most encouraging technique for exploiting the bandwidthof optical networks is Wavelength Division Multiplexing (WDM), which multiplies the availablecapacity of a fiber through using over a hundred of parallel channels [53], each channel on awavelength of light (λ). These channels support traffic demands in Gbps range (e.g. OC-48,OC-192, and OC-768).

Unfortunately, the majority of current devices or applications are constrained by their process-ing speed (a few Mbps), which is translated into a waste of bandwidth. This drawback is efficientlysolved by grooming several low-speed connection demands (Mbps) onto high-speed wavelengthchannels (Gbps). The optical connection established end-to-end from a source node to a destina-tion node is known as lightpath. This problem is known as Traffic Grooming problem.

Like the RWA problem, this problem may be considered static or dynamic, depending on thetraffic pattern,. On the one hand, if the set of demands and data rates are known in advance; theproblem is the design of a virtual topology in order to accommodate as much of these demandsas possible. On the other hand, whether the traffic model is dynamic, the virtual topology isreconfigured on demand either arrival or departure [55]. In this work, we focus on solving thestatic version of this problem.

Furthermore, depending on the number of lightpaths traversed by a traffic request, we candistinguished: single-hop traffic grooming, if the demands are constrained to use no more thana single lightpath; and multi-hop traffic grooming, if requests can traverse several concatenatedlightpaths.

Since Traffic Grooming is an NP-hard problem that has proven to be computationally in-tractable when the given topology contains a large number of nodes [63], in this thesis we proposethe use of Multiobjective Evolutionary Computation for solving this telecommunication problemefficiently.

In this chapter, we present a formal description of the Traffic Grooming problem, includinga general problem statement, the related parameters, variables, constraints, objective functions,and an illustrative example of the problem. Like in the RWA problem, we present the individualsencoding used in the Multiobjective Evolutionary Algorithms (MOEAs). Finally, we present theoptical networks and data sets used in the experimental results for the Traffic Grooming problem.

71

72 5. Traffic Grooming problem

5.1 Problem Formulation

In this work, an optical network topology has been designed as a directed graph G=(N ,E), whereN is the set of nodes and E is the set of physical links connecting nodes.

The following assumptions have been taken into account in this thesis in order to model theproblem:

− There exist an equal number of fibers joining two nodes in both directions. Furthermore, wesuppose that all links have the same weight 1, which corresponds to the fiber hop distance.

− All optical network nodes are equipped with Di×Di wavelength-routing switches, where Di

is the number of incoming fiber links to node i. Note that, for any node i, the number ofincoming fiber links is equal to the number of outgoing fiber links.

− We assume wavelength continuity constraint [6]. Therefore, the nodes do not support wave-length conversion, so a lightpath must use the same wavelength in all physical links of itspath, and all lightpaths using the same fiber link must use distinct wavelengths.

− Traffic demands are known in advance (static traffic demands). Since the granularity oflow-speed demands is x ∈1, 3, 12, and 48 (where the bandwidth of an OC-x channel isx × 51.84Mb/s), these traffic demands cannot be split into several lower speed connectionsand routed separately.

− We suppose multi-hop grooming facility, a low-speed connection can traverse different light-paths.

In the first place, we describe the main parameters used in the problem:

• N : the set of nodes. Note that, |N | indicates the number of optical nodes. The architectureof a node is presented in [63].

• W : number of wavelengths (λ) that can be multiplexed on a single fiber. We assume thatall fiber-links support an equal number of wavelengths.

• Ti, ∀i ∈ N : number of transmitters at node i; Ti ≥ 1.

• Ri, ∀i ∈ N : number of receivers at node i; Ri ≥ 1 .

• K: number of shortest paths (in terms of number of hops).

• Pmn, ∀m, n ∈ N : number of fibers interconnecting nodes m and n. If Pmn = Pnm = 0, thereare no physical links between m and n, otherwise Pmn=Pnm=1.

• C: capacity of each channel, i.e. C = 48 for OC-48.

• Λ: traffic demand matrix

Λx = [ Λxsd; s, d ∈ N ]|N |×|N |,

where Λxsd is the number of OC-x connection requests between node pair (s, d); x ∈1, 3, 12,

and 48.

5.1 Problem Formulation 73

• dmn, ∀m, n ∈ N : propagation delay on fiber link from node m to node n. We assumedmn = 1 .

Then, we present the variables,

• V wij , ∀w ∈ 1 . . . W, ∀i, j ∈ N : number of lightpaths from node i to node j on wavelength

w (Virtual Topology). If V wij >1, the lightpaths between node i and j on wavelength w may

take different paths. Note that, Vij is the total number of lightpaths established from nodei to node j.

• P ij,wmn , ∀w ∈ 1 . . . W, ∀i, j, m, n ∈ N : number of lightpaths between node i to node j routed

through fiber link (m, n) on wavelength w (Physical Topology route).

• ξsd,tij,x , x ∈ 1, 3, 12, and 48, ∀s, d, i, j ∈ N, t ∈ 1 . . . Λx

sd: represents the tth OC-x low-speedtraffic request from source node (s) to destination node (d) employing lightpath (i,j) as anintermediate virtual link.

• Sxsd, x ∈ 1, 3, 12, and 48, ∀s, d ∈ N : number of OC-x streams requested from node s to

node d that are successfully routed. If the tth OC-x low-speed traffic request from node s tonode d has been successfully routed, then Sx,t

sd =1; otherwise Sx,tsd =0 .

For this problem, there exist different constraints. We can classify them into three groups:

• On virtual topology connection variables:

1.∑|N |

j=1 Vij ≤ Ti, ∀i ∈ N : ensures that the number of lightpaths between node pair (i,j)is less than or equal to the number of transmitters at node i.

2.∑|N |

i=1 Vij ≤ Rj , ∀j ∈ N : ensures that the number of lightpaths between node pair (i,j)is less than or equal to the number of receivers at node j.

3.∑W

w=1 V wij = Vij , ∀i, j ∈ N : means that lightpaths between nodes (i,j) are composed of

lightpaths on distinct wavelengths between nodes i and j.

• On physical route variables (employing the wavelength constraint):

1.∑|N |

m=1 P ij,wmk =

∑|N |n=1 P ij,w

kn if k 6= i, j, ∀i, j, k ∈ N, ∀w ∈ 1 . . . W: the number ofincoming lightpaths is equal to the number of outgoing lightpaths for an intermediatenode k of the lightpath (i,j) on wavelength w.

2.∑|N |

m=1 P ij,wmi = 0, ∀i, j ∈ N, ∀w ∈ 1 . . . W: there are no incoming streams in origin

node i of the lightpath (i,j) on wavelength w.

3.∑|N |

n=1 P ij,wjn = 0, ∀i, j ∈ N, ∀w ∈ 1 . . . W: there are no outgoing streams in termina-

tion node j of the lightpath (i,j) on wavelength w.

4.∑|N |

n=1 P ij,win = V w

ij , ∀i, j ∈ N, ∀w ∈ 1 . . . W: ensures that, for the origin node i oflightpath (i,j) on wavelength w, the number of outgoing lightpaths is equal to the totalnumber of lightpaths between node pair (i,j) on wavelength w.

5.∑|N |

m=1 P ij,wmj = V w

ij , ∀i, j ∈ N, ∀w ∈ 1 . . . W: ensures that, for the termination node jof lightpath (i,j) on wavelength w, the number of incoming lightpaths is equal to thetotal number of lightpaths between node pair (i,j) on wavelength w.


6.∑|N |

i,j=1 P ij,wmn ≤ 1, ∀m, n ∈ N, ∀w ∈ 1 . . . W: the wavelength w is only used on a fiber

link (m,n) in at most one lightpath.

• On virtual-topology traffic variables: The following equations ensure the correct routing oflow-speed traffic requests on the virtual topology, taking into account that the aggregatetraffic through lightpaths cannot exceed the overall wavelength capacity.

1.∑|N |

i=1 ξsd,tid,x = Sx,t

sd , ∀s, d ∈ N, x ∈ 1, 3, 12, and 48, t ∈ 1 . . . Λxsd

2.∑|N |

j=1 ξsd,tsj,x = Sx,t

sd , ∀s, d ∈ N, x ∈ 1, 3, 12, and 48, t ∈ 1 . . . Λxsd

3.∑|N |

i=1 ξsd,tik,x =

∑|N |j=1 ξsd,t

kj,x if k 6= s, d; ∀s, d, k ∈ N,x ∈ 1, 3, 12, and 48, t ∈ 1 . . . Λx

sd

4.∑|N |

i=1 ξsd,tis,x = 0, ∀s, d ∈ N, x ∈ 1, 3, 12, and 48, t ∈ 1 . . . Λx

sd

5.∑|N |

j=1 ξsd,tdj,x = 0, ∀s, d ∈ N, x ∈ 1, 3, 12, and 48, t ∈ 1 . . . Λx

sd

6.∑|N |

s,d=1(x× ξsd,tij,x ) ≤ V w

ij × C, ∀i, j ∈ N, x ∈ 1, 3, 12, and 48,t ∈ 1 . . . Λx

sd, ∀w ∈ 1 . . . W

In this way, given an optical network topology, a fixed number of transmitters and receivers ateach node, a fixed number of available wavelengths per fiber, the capacity of each wavelength anda set of connection requests with different bandwidth granularity, the Traffic Grooming problemmay be stated as a Multiobjective Optimization Problem (MOOP) [88], in which our objectivesare maximizing the performance of the traffic throughput or the successful routed traffic, minimiz-ing the cost of the network in term of the number of transceivers or number of lightpaths, andminimizing the average propagation delay or average hop count of the lightpaths, simultaneously.In a more formal way, they are defined as:

• Traffic Throughput (f1): Maximize the total successfully routed low-speed traffic demandson virtual topology.

Maximize(

∑|N |s=1

∑|N |d=1

∑Λxsd

t=1 (x× Sx,tsd ))

x ∈ 1, 3, 12, and 48

(5.1)

• Number of Transceivers (transmitters/receivers, Ti/Ri) or Ligthpaths (f2): Minimize thetotal number of transceivers used or the total number of lightpaths established.

Minimize

|N |∑

i=1

|N |∑

j=1

W∑

w=1

V wij

(5.2)

5.2 Example of the Traffic Grooming problem 75

• Average Propagation Delay (APD, f3). Minimize the average hop count of lightpaths estab-lished, because we assume dmn = 1 in all physical fiber links (m, n).

Minimize

(

∑|N |i=1

∑|N |j=1

∑|N |m=1

∑|N |n=1 (dmn ×

∑Ww=1 P ij,w

mn )∑|N |

i=1

∑|N |j=1

∑Ww=1 V w

ij

)

(5.3)

5.2 Example of the Traffic Grooming problem

In Figure 5.1 we present an illustrative example of the traffic grooming problem, using a smallsix-node network (|N |=6). Each link of the network has one wavelength channel (W=1), and eachchannel presents a capacity of OC-48 (C=48, approximately 2.5 Gbps). Each node of the opticalnetwork contains a tunable transmitter and a tunable receiver (Ti=1, Ri=1). The set of low-speedconnection requests consists of: 3 connections between N5 and N4 with bandwidth requirementOC-12, 5 connections between N2 and N5 with OC-3, 4 connections between N4 and N3 withOC-3, 4 connections between N3 and N6 with OC-1, 4 connections between N5 and N3 with OC-1,and 3 connections between N4 and N6 with OC-1; therefore, the corresponding OC-12, OC-3, andOC-1 traffic matrices are:

Λ12 =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 3 0 00 0 0 0 0 0

Λ3 =

0 0 0 0 0 00 0 0 0 5 00 0 0 0 0 00 0 4 0 0 00 0 0 0 0 00 0 0 0 0 0

Λ1 =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 40 0 0 0 0 30 0 4 0 0 00 0 0 0 0 0

So, the total amount of traffic is 74 OC-1 units. Note that, in this example, we assumewavelength continuity constraint and multi-hop facility.

In the first place, we construct the virtual topology. To achieve this, we try to establish alightpath for each pair of nodes which contains at least one unresolved low-speed traffic demand.We start by establishing four lightpaths: L1 (N5,N4), L2 (N2,N5), L3 (N4,N3), and L4 (N3,N6);however, we can establish no lightpaths between the pairs of nodes (N5,N3) and (N4,N6), becausethere are not enough available resources: transmitter T5, transmitter T4, receiver R3, and receiverR6 are busy.

Next, we have to route the low-speed connection requests. As we can see in the illustrativeexample, those low-speed requests of the pair of nodes (N5,N4), (N2,N5), (N4,N3), and (N3,N6)are carried out through the lightpaths L1, L2, L3, and L4 respectively. As we assume multi-hop facility in our example, on the one hand, the low-speed connections between the nodes N5

and N3 are carried out by spare capacity of the two existing lightpaths: L1 and L3 (see Figure5.1). On the other hand, the low-speed connections between the nodes 4 and 6 are successfullyestablished through the lightpaths: L3 and L4. Therefore, the value of each objective function, forthis particular example, is f1=74, f2=4, and f3=1.75.

In general, low-speed connections may traverse multiple lightpaths if no resources are availablefor establishing a lightpath between its source node and its destination node directly, as occurs inour example with the connection requests between nodes (N5,N3) and (N4,N6).

In Figure 5.1 a detailed description of the final state of the network resources is presented.


Figure 5.1. Illustrative example of the Traffic Grooming problem.

5.3 Representation of the Individuals

The representation of the individual determines how the problem is structured in the algorithms.It gives us the necessary knowledge to understand the behaviour of each algorithm.

In Figure 5.2 the representation of an individual used in this work is shown. The chromosomeof our individual is a |N | × |N | matrix, where |N | is the number of nodes in the given networktopology. This matrix contains, at each position, a vector which stores Ti transmitters, and aweighting factor (WFij). So, for each pair of nodes, we can establish a maximum of Ti lightpaths.

5.3 Representation of the Individuals 77

Figure 5.2. Representation of the Individuals for the Traffic Grooming problem.

Note that low-speed connection requests will be established onto the lightpaths stored in thechromosome.

In each position of the transmitter vector, a value greater than zero means that there exists alightpath established between node i and node j routed through the k path in the set of shortestpaths (K-shortest-paths, computed in terms of number of hops). This set of shortest paths iscalculated in advance by using Yen’s algorithm [10], at the beginning of the algorithm (we assumeN/3+1 shortest paths).

In Algorithm 5, a pseudocode for generating random individuals is shown. First of all, weinitialize the new individual X , the Virtual adjacency matrix (V ), the Streams requested matrix(S), and the list of lightpaths as empty (lines 1-3 in Algorithm 5). Since the number of transceiversper node, and available wavelengths per fiber are limited, it is very important to take into accountthe order of setting up lightpaths. In this paper, we have applied a similar procedure to the typicalMST heuristics [63] for sorting the lightpaths. The MST heuristics attempts to establish lightpathsbetween source-destination nodes (i,j) with the highest Traffic(i,j) value, where Traffic(i,j) isthe aggregate traffic between nodes i and j. However, we have a weighting factor (WF ) in thechromosome of the individual for each pair (i,j). It aims to obtain a good random-greedy balancein the evolutionary algorithms (line 11). Note that the weighting factor is in range [0.5, 1], so inthe worst case it will be halved.

Once the new individual is initialized, we continue by setting up an order for establishing thelightpaths (lines 4-19), in which the slow stream requests will be multiplexed. As we may beobserve, for each pair (i,j), we create a new element which is inserted in the list of lightpaths L.Each element contains the following information: source node (i), destination node (j), transmitterused (t), the selected path (kpath), and the value (v) of WFij*Traffic(i,j).

After sorting L in descending order of v (line 20), we start to establish the lightpaths. Forall elements in the list, we obtain, from each element e (line 22), its corresponding information:source node (i), destination node (j), transmitter used (t), and the selected path (kpath). Then, wecheck whether there exist enough available resources (line 23). In function isPossibleToEstablish,we verify that there exists an available transmitter at node i and an available receiver at nodej. Furthermore, we check whether it is possible to satisfy the wavelength continuity constraint,in other words, we try to assign the same wavelength on each physical link of the selected path


Algorithm 5 Random Individual for the Traffic Grooming problem1: x ← ∅ // new individual2: V , S ← ∅ // Virtual adjacency matrix (V ) and Streams requested (S)3: L ← ∅ // List of lightpaths4: for i = 0 to N -1 do5: for j = 0 to N -1 do6: if i 6= j then7: x.WFij ← random (0.5, 1)8: for t = 0 to Ti-1 do9: x.transmitterijt ← ramdom (0, K-shortest-paths)

10: if x.transmitterijt > 0 then11: v ← x.WFij * Traffic(i,j)12: kpath ← x.array_transmitterijt

13: e ← NewElement(i,j,t,kpath,v)14: L ← AddElement(L, e)15: end if16: end for17: end if18: end for19: end for20: L ← Sort(L) // sort in descending order of v21: for all e in L do22: i,j,t,kpath ← e23: if isPossibleToEstablish(i,j,t,kpath) then24: UpdateResources() // #Transceivers and available wavelengths25: Vij ← Vij + 126: else27: x.transmitterijt ← 028: end if29: end for30: RoutingLowSpeedConnectionDemands(V , S)31: x.f1 ←

∑|N |s=1

∑|N |d=1

∑Λxsd

t=1 (x× Sx,tsd )

32: x.f2 ←∑|N |

i=1

∑|N |j=1

∑Ww=1 V w

ij

33: x.f3 ←

∑

|N|

i=1

∑

|N|

j=1

∑

|N|

m=1

∑

|N|

n=1(dmn×

∑

W

w=1P ij,w

mn )∑

|N|

i=1

∑

|N|

j=1

∑

W

w=1V w

ij

(kpath). For the wavelength assignment we have used a simple heuristic: First-Fit (FF). In FF, allwavelengths are numbered; in this way, a lower numbered wavelength is considered before a highernumbered wavelength, so the first available wavelength is assigned.

If it is possible to establish a lightpath between nodes i and j, we update the resources ofthe network and the virtual adjacency matrix (V ) (lines 24 and 25). Otherwise, we unmark theoccurrence of a lightpath in (i,j) (line 27). A virtual topology of Nv×Nv (where Nv is the numberof lightpaths established) is obtained for routing the traffic over it.

For routing the traffic over the virtual topology (line 30), we apply Yen’s algorithm to calculatethe k-shortest paths (we assume Nv/3+1 shortest paths). In the first place, we route the low

5.4 Data Sets: Optical networks and sets of low-speed traffic demands 79

speed traffic connections over a single-hop lightpath and after that, the remaining requests areestablished over multi-hop lightpaths. Finally, we calculate the objective functions (lines 31-33),which are shown in equations (1), (2) and (3).

5.4 Data Sets: Optical networks and sets of low-speed traf-

fic demands

To solve the Traffic Grooming problem, we have carefully selected scenarios with different features:size of the network, amount of traffic, and available resources.

In the first place, we have chosen the following optical network topologies: a six node network (6-node), the European Optical network (COST239) [137], the National Science Foundation network(NSF) [138], and the Nippon Telegraph and Telephone network (NTT) [139]. So, we have a smallnetwork with 6 nodes (6-node), two medium size networks (COST239 and NSF) with 11 and 14nodes, and a large network (NTT) with 55 nodes.

In the second place, for each optical network, we have generated three traffic matrices (TM) orsets of low-speed traffic requests with small, medium, and large amount of traffic. To do this, wehave followed the methodology proposed in [63]. In this way, the number of demands for each pairof nodes is randomly generated between 0 and 2r for OC-1 demands, 0 and 2r−1 for OC-3, and

Table 5.1. 6-node: Specifications of data sets for the Traffic Grooming problem.

6-node Network

Runtime 30 secondsNodes (|N|) 6 nodesLinks (|E|) 16 linksCapacity OC-48Traffic Matrix 1 (TM1)− Amount of Traffic: 988 OC-1 units− #Transceivers (T): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12− #Wavelengths (W): 1, 2, 3, 4

Traffic Matrix 2 (TM2)− Amount of Traffic 1976 OC-1 units− #Transceivers (T): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24− #Wavelengths (W): 2, 4, 6, 8



between 0 and 2r−3 for OC-12. Thus, using different values of r, we can generate diverse trafficmatrices with different amount of traffic. In this thesis, we have used r = 4, r = 5, and r = 6, forgenerating TM1, TM2, and TM3, respectively.

Finally, we propose the use of different available resources per node and per fiber link. In thisway, for each traffic matrix, we use different number of transceivers per node (T ) and availablewavelengths per optical fiber link (W ).

As we can see, we compare the algorithms in four optical network topologies, each one withthree different sized traffic matrices, and each traffic matrix is tested in 48 scenarios with differentavailable resources; thus, a total of 576 different scenarios.

The four optical network topologies and the 576 scenarios can be downloaded from [143].

Table 5.2. COST239: Specifications of data sets for the Traffic Grooming problem.

European Optical Network (COST239)

Runtime 120 secondsNodes (|N|) 11 nodesLinks (|E|) 52 linksCapacity OC-96




5.4 Data Sets: Optical networks and sets of low-speed traffic demands 81

Table 5.3. NSF: Specifications of data sets for the Traffic Grooming problem.

National Science Foundation Network (NSF)






Table 5.4. NTT: Specifications of data sets for the Traffic Grooming problem.

Nippon Telegraph and Telephone Network (NTT)





6Multiobjective Evolutionary Algorithms

This chapter is devoted to detail the operation of the different Multiobjective Evolutionary Al-gorithms proposed for solving the two real-world optimization problems tackled in this thesis(RWA problem and Traffic Grooming problem). Furthermore, we present a detailed explanationabout two parallel MOEAs. For each parallel MOEA, we present three parallel operation schemesthat exploit different systems: shared-memory systems, distributed-memory systems, and hybridshared/distributed-memory systems.

In this work, all the MOEAs use exactly the same representation of the individuals. A detaileddescription about the chromosome encoding and the generation of a random individual is presentedin Chapter 4 (RWA problem) and in Chapter 5 (Traffic Grooming problem).

In the first place, Differential Evolution (DE) is an evolutionary algorithm created by Storn andPrice with the aim of solving single-objective optimization problems by generating new individualsby combining existing ones according to its simple equation of vector-crossover and mutation [109].The multiobjective version of the DE used in this work is the Differential Evolution with ParetoTournaments (DEPT) which incorporates the Pareto tournament concept.

The Variable Neighbourhood Search (VNS) algorithm is an evolutionary algorithm created byHansen and Mladenovic [105]. In this work, we have used a multiobjective version of the VNSalgorithm (MO-VNS) in which more than one solution is obtained per run by using an archive ofnon-dominated solutions.

The Artificial Bee Colony (ABC) algorithm is a swarm intelligence algorithm created byKaraboga [112]. This algorithm is based on the intelligence behaviour of honey bees. A feasi-ble solution in this algorithm is considered a food source and the population is called colony. Inthis way, the colony consists of three types of artificial bees: the employed bee, the onlooker bee,and the scout bee. In this work, we have adapted the standard ABC algorithm for dealing withmultiobjective optimization problems (MO-ABC).

The Gravitational Search Algorithm (GSA) is also a swarm intelligence algorithm created byRashedi et al. [113]. This algorithm is based on the law of gravity and mass interactions. In thisalgorithm, the individuals are considered searcher agents, and the entire population is a collectionof masses which interact with each other based on the laws of gravity. A multiobjective version ofthis algorithm (MO-GSA) is used in this work for solving these MOOPs.

The Firefly Algorithm (FA) is a swarm intelligence algorithm created by Yang [114]. In thisalgorithm the individuals are fireflies which use their bioluminescent aptitudes to attract other

83

84 6. Multiobjective Evolutionary Algorithms

Table 6.1. The Schemes for the DE algorithm

S Scheme Generated Individual

1 Best/1/Exponential xtrial = xbest + F · (xr1 − xr2)2 Rand/1/Exponential xtrial = xr3 + F · (xr1 − xr2)3 RandToBest/1/Exponential xtrial = xr3 + F · (xbest − xr3) + F (xr1 − xr2)4 Best/2/Exponential xtrial = xbest + F · (xr1 + xr2 − xr3 − xr4)5 Rand/2/Exponential xtrial = xr5 + F · (xr1 + xr2 − xr3 − xr4)6 Best/1/Binomial xtrial = xbest + F · (xr1 − xr2)7 Rand/1/Binomial xtrial = xr3 + F · (xr1 − xr2)8 RandToBest/1/Binomial xtrial = xr3 + F · (xbest − xr3) + F (xr1 − xr2)9 Best/2/Binomial xtrial = xbest + F · (xr1 + xr2 − xr3 − xr4)10 Rand/2/Binomial xtrial = xr5 + F · (xr1 + xr2 − xr3 − xr4)

fireflies with the aim of optimizing a mono-objective problem. In the multiobjective version (MO-FA), we use the dominance concept in order to determine if a firefly is attracted by another one.

Finally, we describe two well-known MOEAs in the Multiobjective domain. On the one hand,the Fast Non-Dominated Sorting Genetic Algorithm (NSGA-II) was created by Deb et al. [90] andit is a revised version of the NSGA [144]. On the other hand, the Strength Pareto EvolutionaryAlgorithm 2 (SPEA2) was created by Ziztler et al. [91], and it is a revised version of the SPEA[145].

As we saw in Chapter 4 and in Chapter 5, the chromosome encoding in both problems usediscrete values. Therefore, we have adapted those algorithms that were designed for solving con-tinuous problems, such as DEPT, MO-ABC, MO-GSA, or MO-FA. There exist several approachesfor combinatorial problems [146], such as: Forward/Backward Transformation Approach, RelativePosition Indexing Approach, Smallest Position Value Approach, Discrete/Binary Approach, andDiscrete Set Handling Approach.

In this thesis, the MOEAs are based on the Discrete Set Handling Approach. In this way, wekeep using continuous values at each position of the chromosome, but we truncate the value ofeach position for computing the objective functions. For further information about the DiscreteSet Handling Approach, please refer to [146].

6.1 Differential Evolution with Pareto Tournaments

Differential Evolution (DE) is a population-based algorithm for global optimization created byRainer Storn and Kenneth Price [109]. It optimizes a problem by maintaining a population ofcandidate solutions and generating new candidate solutions by combining existing ones accordingto its simple equation of vector-crossover and mutation.

In this thesis, we propose the use of a new multiobjective variant of the standard DE. Wehave modified the well-known single-objective DE, by including the Pareto Tournaments (DEPT)concept and a multiobjective fitness (MOfitness) [147], in order to solve MOOPs.

In Differential Evolution, we take into account three important parameters: the crossoverprobability (CR), amount of mutation or mutation rate (F ), and the selection scheme (S). In[109], Price and Storn propose several selection schemes (see Table 6.1). If we suppose the scheme

6.1 Differential Evolution with Pareto Tournaments 85

Rand/1/Binomial, the DEPT algorithm works as follows:

Input:

- RWA problem or Traffic Grooming problem;

- Stopping criterion;

- Population Size (Ns);

- Crossover probability (CR);

- Mutation rate (F );

- Selection scheme (S).

Output:

- Set of non-dominated solutions.

Step 1) Initialization:

Step 1.1) Initialize the set of non-dominated solutions as empty.

Step 1.2) Generate an initial population with Ns random individuals x1, x2, . . . , xNs .

Step 2) Update:

For each i ∈ 1, . . . , Ns, do

Step 2.1) Reproduction. Select randomly three indexes r1, r2, r3 from the population(Rand/1/Binomial, see Table 6.1), and generate a new solution xtrial from xi, xr1,xr2, and xr3, as:

xtrial =

xr3 + F · (xr1 − xr2) with probability CR;xi with probability 1-CR.

In case of any value in the chromosome of xtrial is out of the boundary, repair it byreseting the value in chromosome with a randomly value inside the boundary.

Step 2.2) Pareto Tournament. Replace xi by xtrial if and only if MOFitness(xi) is greaterthan MOFitness(xtrial). In case of tie, that is to say, if MOFitness(xi) is equal toMOFitness(xtrial), it means that xi and xtrial belong to the same Pareto front; so,the tie is solved by calculating their crowding distance value, replacing xi by xtrial ifand only if the crowding distance of xtrial is greater than the crowding distance of xi.Note that this crowding value is an agglomeration comparison operator that prioritizessolutions that are less crowded. For further information about the crowding distanceconcept, please refer to [90].

Step 2.3) Update the Pareto set. Update the set of non-dominated solutions with xi. Addxi to the set only if no other solution dominates it, and remove from the set all solutionsdominated by xi.

Step 3) Stopping Criterion: If the stopping criterion is satisfied, then stop and output the setof non-dominated solutions, otherwise, go to Step 2.


Note that, in the Pareto Tournament the MOFitness function return a scalar value that indi-cates the quality of the evaluated individual in regard to the remaining individuals. In this way,given an individual xi where i ∈ 1 . . . Ns, its value of MOFitness is calculated as follows:

MOFitness(xi) = |isDominated(xi)| ·Ns + |Dominates(xi)| (6.1)

where isDominated returns the number of individuals that dominate the individual xi and Dom-inates returns the number of individuals dominated by xi. For further information about thisequation, please refer to [147].

6.2 Multiobjective Variable Neighbourhood Search

The Variable Neighborhood Search (VNS) algorithm is a trajectory-based algorithm created byPierre Hansen and Nenad Mladenovic [105]. In this thesis, we apply a multiobjective version of VNS(MO-VNS) to the two telecommunication problems. Furthermore, MO-VNS has the peculiaritythat, unlike the original VNS algorithm, it can find more than one solution per run.

The basic idea in MO-VNS is the systematic change of neighborhood within a local search. Inthis way, the MO-VNS algorithm starts from a random individual (x) and performs many changesof neighborhood. However, when it is not able to improve the quality of x by exploring more distantneighborhoods, the metaheuristics generates a new random individual, restarting the process andallowing the existence of several solutions in the final Pareto front. This version of creating aPareto front is similar to the Pareto Local Search (PLS) method proposed by Paquete and Stützlein [148].

In the following, we explain the main steps in the MO-VNS algorithm.

Input:



- Neighbourhood degrees (nmax).

Output:


Step 1) Initialization: Initialize the set of non-dominated solutions as empty.

Step 2) Update:

Step 2.1) Generate a random individual x and initialize the neighbourhood degree (n); so, n = 1.

Step 2.2) Update the set of non-dominated solutions with x. Remove from the set all solutionsdominated by x and add x to the set only if no other solution dominates it.

Step 2.3) Explore the Neighbourhood of the individual x.

Step 2.3.2) Compute the Mutation rate (F ) according to the neighbourhood degree. There-fore, the mutation rate should be:

F = n · 15% (6.2)

6.3 Multiobjective Artificial Bee Colony 87

Step 2.3.3) Mutate the individual x by using the Mutation rate F in order to obtain themutated individual x′. Therefore, the new individual x′ should be computed as:

x′ = F · (x)

Step 2.3.4) Apply a Local Search procedure to the mutated individual x′ with the aim ofmaking some quality improvements. The new individual is denoted as x′′.

Step 2.3.5) Update the set of non-dominated solutions with x′′. Remove from the set allsolutions dominated by x′′ and add x′′ to the set only if no other solution dominatesit.

Step 2.3.6) Replace x by x′′ if and only if x′′ 4 x, then, set n = 1. Otherwise, increase thesize of the neighbourhood degree (n = n + 1).

Step 2.3.7) Check if n is greater than nmax, in that case, go to Step 2.1; otherwise, go toStep 2.3.


As we may observe, in the MO-VNS, the mutation depends directly on the neighborhood degree(n), because if the evolution of the solution stagnates, then the mutation is intensified. In this thesis,we have used 6 and 5 degrees of neighbourhood for the RWA problem and the Traffic Groomingproblem, respectively. In this way, the degree enhances the exploration of neighborhoods, so weincrease the mutation factor (F) as n is increased. If we suppose nmax = 5, with n=1 (initial value)we apply a mutation of F=15%, with n=2 a mutation of F=30%, and so on until n=5, where weapply the maximum value of mutation, F=75%.

Note that we use a Local Search procedure in the MO-VNS algorithm for improving the qualityof the mutated solution. Therefore, we have developed a specific Local Search procedure for eachoptical network problem.

On the one hand, for the RWA problem the Local Search procedure tries to improve severalpositions of the chromosome (Figure 4.2) by checking n random paths from the list of feasiblelightpaths.

On the other hand, for the Traffic Grooming problem, the local search consists on improvingthose positions that indicate the existence of a lightpath between nodes i and j in the chromosomeof x′ (see Figure 5.2). Therefore, to improve a lightpath i-j, we check all available shortest pathsfrom node i to node j.

6.3 Multiobjective Artificial Bee Colony

The Artificial Bee Colony (ABC) algorithm is a population-based evolutionary algorithm createdby Dervis Karaboga [112]. It is inspired by the intelligent behavior of honey bees.

In this algorithm, the population of individuals is defined as a colony in which we can differ-entiate three groups of bees: employed bees associated with specific food sources, onlooker beeswatching the dance of employed bees within the hive with the aim of choosing a food source, andscout bees searching new food sources randomly. The nectar of food sources are exploited by em-ployed bees and onlooker bees, so this continual exploitation will ultimately cause them to becomeexhausted. Then, the bee whose food source has been exhausted becomes a scout bee. In ABC,the position of a food source represents a feasible solution to the problem and the nectar amountof a food source corresponds to the quality or fitness of the associated solution.


Since we are dealing with MOOPs, we have adapted the standard ABC algorithm to themultiobjective domain (MO-ABC). An outline of MO-ABC is presented below:

Input:




- Maximum Limit value (limitmax);

- Mutation rate (F ).

Output:




Step 1.2) Fill the first half (Ns/2) of the population/colony with random individuals or employedbees, x1, x2, . . . , xNs/2.

Step 2) Update:

Step 2.1) Employed Bees Phase.

For each i ∈ 1, . . . , Ns/2, doStep 2.1.1) Select randomly an index k from the first half of the colony and generate a new

solution yi from xi and xk, according to the following equation:

yi = xi + F · (xi − xk) (6.3)

Step 2.1.2) Replace xi by yi if and only if yi ≺ xi. In case of not replacing xi by yi, increasethe stagnation limit of the solution xi.

Step 2.2) Onlooker Bees Phase.

Step 2.2.1) Since onlooker bees probabilistically choose their food sources depending on theinformation provided by the employed bees, calculate a probability vector by usingthe following equation, where i ∈ 1, . . . , Ns/2:

Pi =MOFitness2(xi)

∑Ns/2j=1 MOFitness2(xj)

(6.4)

For each j ∈ Ns/2 + 1, . . . , Ns, doStep 2.2.2) From the first half of the colony, select an index i by using the probability

vector P and an index k randomly. Then, generate a new solution yj from xi andxk, according to the following equation:

yj = xi + F · (xi − xk) (6.5)

Step 2.2.3) Set xj=yj if and only if yj 4 xi; otherwise, set xj=xi.

6.4 Multiobjective Gravitational Search Algorithm 89

Step 2.3) Scout Bees Phase.For each i ∈ 1, . . . , Ns/2, doStep 2.3.1) Check if the stagnation limit of the employed bee xi is greater than limitmax,

in that case, replace xi by a scout bee. Note that, in this work, a scout bee is aramdom bee.

Step 2.4) Update the Pareto Set.For each i ∈ 1, . . . , Ns, doStep 2.4.1) Compute MOFitness2 (xi).Step 2.4.2) Update the set of non-dominated solutions with xi. Add xi to the set only if

no other solution dominates it, and remove from the set all solutions dominated byxi.

Step 2.5) With the aim of preserving the best bees for the next generation, sort the colony byMOFitness2. Therefore, the best solutions found so far will be the employed bees in thenext generation.


As we can see, to calculate the probability of an employed bee it is necessary to compute, for eachemployed bee, its corresponding multiobjective value of fitness (MOFitness2 ). The MOFitness2is computed as follows:

MOFitness2(xi) = (2Rank(xi) +1

1 + CrowdingDistance(xi))−1 (6.6)

In this way, the MOFitness2 needs to be maximized. If we compare the worst individual withbest rank (rank 1) and lowest value of crowding distance (≈ 0); with the best individual with rank2 and highest crowding distance (∞), the value of MOFitness2 is 1/3 and 1/4, respectively. TheRank and CrowdingDistance are well-known concepts in the multiobjective field; thus, for furtherinformation about them, please refer to [90].

The main difference between MOFitness (equation 6.1) and MOFitness2 (equation 6.6) is thatin MOFitness2 we take into account not only the dominance among the solutions, but also thedistribution of the solutions along the Pareto front.

6.4 Multiobjective Gravitational Search Algorithm

The Gravitational Search Algorithm (GSA) is a population-based algorithm created by EsmatRashedi et al. [113] in 2009. This new optimization algorithm is based on the law of gravity andmass interactions. The searcher agents (individuals) are a collection of masses which interact witheach other based on the Newtonian gravity laws of motion.

In the GSA, we find the following input parameters: kbest, α, and G0. On the one hand, thekbest parameter is used to control the exploration and exploitation of the algorithm. On the otherhand, G0 is the initial value of the Gravitational constant (G) and α is a parameter for decreasingthe value of G throughout the execution of the algorithm. Note that G is crucial in the algorithmdue to it is in charge of controlling the search accuracy.

Since in the RWA problem and in the Traffic Grooming problem we optimize more than oneobjective function simultaneously, we have adapted the standard GSA to the multiobjective context(MO-GSA) as follows:


Input:




- Initial Gravitational Constant (G0);

- Alpha (α);

- Number of best agents (kbest).

Output:




Step 1.2) Generate an initial population with Ns random individuals/agents x1, x2, . . . , xNs .

Step 2) Update:

Step 2.1) For each agent xi, where i ∈ 1, . . . , Ns, compute its value of MOFitness2 (xi), seeequation 6.6.

Step 2.2) Update the value of the Gravitational constant (G) according to the following equation:

G = G0e−α trTr (6.7)

where tr is the elapsed time/generation and Tr is the total runtime or maximum numberof generations established.

Step 2.3) Sort the agents by quality (MOFitness2 ) and select the best and the worst agentsfrom x1, x2, . . . , xNs . In the MO-GSA, the best (xbest) and the worst (xworst) agentscorrespond to those agents with the highest and the lowest value of MOFitness2, respec-tively.

Step 2.4) For each agent xi, where i ∈ 1, . . . , Ns, compute its mass (m) according to its quality(MOFitness2 ). Therefore, to compute the mass of xi we use the following equations:

Q(xi) =MOFitness2(xi)−MOFitness2(xworst)

MOFitness2(xbest)−MOFitness2(xworst)(6.8)

m(xi) =Q(xi)

∑Ns

j=1 Q(xj)(6.9)

Step 2.5) For each agent xi compute the force (F ) exerted by each agent xj , where i ∈1, . . . , Ns and j ∈ 1, . . . , kbest, thus:

F (xi, xj) = G ·m(xi) ·m(xj)||xi, xj ||

· (xj − xi) (6.10)

f(xi) =kbest∑

j=1

random[0, 1] · F (xi, xj) (6.11)

where ||xi, xj || refers to the Euclidean distance between the agent xi and the agent xj .

6.5 Multiobjective Firefly Algorithm 91

Step 2.6) For each agent xi, where i ∈ 1, . . . , Ns, compute its acceleration (a) as:

a(xi) =f(xi)m(xi)

(6.12)

Step 2.7) Update each agent by using the following equations:

v(xi) = random[0, 1] · v′(xi) + a(xi) (6.13)

xi = xi + v(xi) (6.14)

where v′(xi) refers to the velocity of the agent xi in the previous generation.

Step 2.8) Update the Pareto Set. For each i ∈ 1, . . . , Ns, update the set of non-dominatedsolutions. Add xi to the set only if no other solution dominates it, and remove from theset all solutions dominated by xi.

Step 2.9). Decrease linearly kbest to 1.


According to [113], the use of kbest improves the performance of the algorithm because theagents are only affected by the kbest best agents in the population, instead of being affected by allthe agents.

In this work, the value of kbest is initialized as Ns (population size) at the beginning, and it isdecreased linearly throughout the execution of the algorithm.

6.5 Multiobjective Firefly Algorithm

A population-based evolutionary algorithm inspired by the flash pattern and characteristics offireflies is proposed by Xin-She Yang in [114], the Firefly Algorithm (FA).

This innovative approach uses the bioluminescent aptitudes of fireflies to attract other fireflieswhich are flying around.

However, in this work, we propose a multiobjective version of the FA (MO-FA) with the aimof dealing with the RWA and the Traffic Grooming problems. The MO-FA works as follows:

Input:




- Attractiveness (β0);

- Absorption coefficient (γ);

- Control parameter for exploration (α).

Output:





Step 1.2) Generate an initial population with Ns random individuals/fireflies x1, x2, . . . , xNs .

Step 2) Update:

For i, j = 1,...,Ns do

Step 2.1) If and only if xj attracts xi (xj ≺ xi), then move xi towards xj by using the followingequation:

xi = xi + β0e−γ·(||xi,xj||)2

(xj − xi) + α ·

(

random[0, 1]−12

)

(6.15)

where ||xi, xj || refers to the Euclidean distance between the firefly xi and the firefly xj .

Step 2.2) Update the Pareto set. Update the set of non-dominated solutions with xi. Addxi to the set only if no other solution dominates it, and remove from the set all solutionsdominated by xi.


The most important issues of the Firefly algorithm are: the variation of light intensity andformulation of the attractiveness. The author postulates that, in a single-objective problem, fireflieswith better value of fitness are considered more attractive, so the other ones will move towardthem. In MO-FA, we have used the well-known Dominance (≺) multiobjective concept in order tocompare a pair of fireflies.

As we may observe in Step 2.1, in case of a firefly xj attracts (dominates) other firefly xi,then, xi moves towards xj . To move a firefly, we calculate the Euclidean distance between bothfireflies, and apply the attractiveness equation (see equation 6.15). In this equation, the secondterm corresponds with the attraction, and the third term with the control parameter α (addingdispersion).

6.6 Fast Non-dominated Sorting Genetic Algorithm

The Fast Non-Dominated Sorting Genetic Algorithm (NSGA-II) is a multiobjective population-based algorithm created by Kalyanmoy Deb et al. [90]. It is a revised version of NSGA [144].Next, we present the main steps of the NSGA-II algorithm:

Input:




- Selection method;


- Mutation probability (F ).

6.6 Fast Non-dominated Sorting Genetic Algorithm 93

Output:




Step 1.2) Generate an initial population (parent population, P ) with Ns random individualsx1, x2, . . . , xNs .

Step 2) Update:

Step 2.1) Generate an offspring population (Q) with Ns new solutions y1, y2, . . . , yNs ; thus,

For i = 1,...,Ns doStep 2.1.1) Select randomly two indexes r1 and r2 from the parent population (P ) and,

according to the selection method (S), select the index of the first parent p1.Step 2.1.2) Select randomly another two indexes r3 and r4 from the parent population (P )

and, according to the selection method (S), select the index of the second parentp2.

Step 2.1.3) Generate a new solution yi from xp1 and xp2 by using genetic operators. Inthis work, we have used single point crossover (CR) and mutation (F ).

Step 2.2) Combine all the new solutions created in Step 2.1 and all the solutions in P togetherand form a combined population R of size 2 ·Ns; that is to say,

R = P ∪Q (6.16)

Step 2.3) Sort the population R into categories (ranks) according to their relationship of domi-nance, R = (F1, F2, . . . , Fr).

Step 2.4) Select the best solutions from R in order to form the new parent population (P ) forthe next generation. In the case of having to choose among individuals with the samerank, the crowding distance of the individuals belonging to the same rank is calculated,in order to decide which are the best individuals, that is to say, those individuals withhigher value of crowding distance are prefered.

Step 2.5) Update the Pareto Set. For each i ∈ 1, . . . , Ns in P , update the set of non-dominated solutions. Add xi to the set only if no other solution dominates it, andremove from the set all solutions dominated by xi.


Note that the original NSGA-II does not use a set of non-dominated solutions; however, in thiswork, we incorporate it in order to be fair with the rest of MOEAs.

NSGA-II tries to obtain a new population (offspring population Q) from an original one (parentpopulation P ) by applying classical genetic operators, such as selection, crossover, and mutation.Then, both populations, offspring and parent, are mixed into a new population R.

This new population is sorted into categories (ranks) according to their relationship of domi-nance. After that, the best individuals are selected to create a new parent population for the nextgeneration. In the case of having to choose among individuals with the same rank, the crowding


distance of the individuals belonging to the same rank is calculated, in order to decide which arethe best individuals. Due to its improvements, the NSGA-II is much more efficient than its prede-cessor (NSGA), and its use has become very popular in the last few years, such that it has becomea reference algorithm.

6.7 Strength Pareto Evolutionary Algorithm

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) is a multiobjective population-basedalgorithm created by Eckart Zitzler et al. in [91].

Input:




- Archive Size (Ns);


- Mutation probability (F ).

Output:

- Set of non-dominated solutions (P ).



Step 1.2) Initialize the archive (P ) as empty.

Step 1.3) Generate an initial population (P ) with Ns random individuals x1, x2, . . . , xNs .

Step 2) Update:

Step 2.1) Fitness Assignment. For each individual xi in the population P and in the archive P ,assign a strength value representing the number of solutions it dominates. To calculatethe strength of an individual xi, use the following equation:

strength (xi) = | xj | (xj ∈ P ∪ P ) and (xi ≺ xj)| (6.17)

where | · | denotes the cardinality of a set. On the basis of the strength values, the rawfitness of an individual xi is calculated as:

Raw fitness (xi) =∑

xj∈(P ∪P ), xj≺xi

strength (xj) (6.18)

Although the raw fitness assignment provides a sort of niching mechanism based on theconcept of Pareto dominance, it may fail when most individuals do not dominate eachother. Therefore, additional density information is incorporated to discriminate betweenindividuals having identical raw fitness values. The density estimation technique usedin SPEA2 is an adaptation of the k-th nearest neighbor method, where the density at

6.8 Parallel Approaches 95

any point is a (decreasing) function of the distance to the k-th nearest data point. Theauthors simply take the inverse of the distance to the k-th nearest neighbor as the densityestimate. To be more precise, for each individual xi the distances to all individuals xj

in archive (P ) and population (P ) are calculated and stored in a list. After sortingthe list in increasing order, the k-th element gives the distance sought, denoted as σk

i .According to [91], the value of k should be equal to the square root of the populationsize and the archive size; thus, k =

√

Ns + Ns. Afterwards, the density correspondingto xi is defined by

Density (xi) =(

σki + 2

)−1(6.19)

Finally, the fitness (to minimize) of each individual xi is computed as follows:

Fitness (xi) = Raw fitness (xi) + Density (xi) (6.20)

Step 2.2) Environmental Selection. Copy all non-dominated individuals in the population Pand in the archive P to P ′. If |P ′| exceeds Ns, then reduce P ′ by means of the truncationoperator; otherwise, if |P ′| is less than Ns, then fill P ′ with dominated individuals in Pand P .

Step 2.3) Mating Selection. Perform binary tournament selection with replacement on P ′ inorder to fill the mating pool.

Step 2.4) Variation. Apply crossover (CR) and mutation (F ) operators to the mating pooland set P to the resulting population.

Step 3) Stopping Criterion: If the stopping criterion is satisfied, then stop and output P ,otherwise, go to Step 2.

As we can see, in the SPEA2, a fitness value that is the sum of its strength raw fitness plus adensity estimation is assigned to each individual. Then, the best individuals (non-dominated ones)of both (population and archive) are copied into a new population, truncating it with the aim ofnot exceeding the size of the population. Moreover, SPEA2 also applies selection, crossover, andmutation operators, with the aim of generating the next population.

6.8 Parallel Approaches

Since the complexity of the optical network optimization problems raises exponentially when thenumber of nodes increase, the use of parallel computing is convenient in order to solve the problemsin a reasonable time.

In this thesis, we have parallelized the Differential Evolution with Pareto Tournaments (DEPT)and the Multiobjective Artificial Bee Colony algorithm (MO-ABC) because their operation schemeis highly paralellizable.

In the following sections we detail parallel operation scheme of each MOEA. We have designedthree parallel versions for each MOEA in order to exploit different parallel architectures. In this way,we have a shared memory version, a distributed memory version, and a hybrid shared-distributedmemory version. We have used OpenMP, MPI, and OpenM/MPI for parallelizing both MOEAs,getting the most of each architecture as a result.


Figure 6.1. Sequential operation scheme of the DEPT algorithm.

6.8.1 Parallel DEPT

As we explained in Section 6.1, the Differential Evolution (DE) is a population-based algorithmthat optimizes a problem by maintaining a population of individuals; furthermore, it creates newindividuals by combining existing ones according to its simple formulae of vector-crossover andmutation. In order to tackle MOOPs, in this thesis, we incorporate the Pareto Tournamentsconcept to the standard DE (DEPT).

In this section we describe three parallel versions of the DEPT algorithm. In this way, eachversion focuses on exploiting a specific memory architecture system. In the first place, we presenta shared-memory version which uses OpenMP directives for distributing the computational tasksamong a set of threads in the same memory space; thus, no communication is required. In thesecond place, we present a distributed-memory version of the DEPT algorithm developed withMessage Passing Interface (MPI). In this version, the tasks are divided among a set of processeswhich use messages for communicating. Finally, we present a hybrid shared/distributed-memoryversion of the DEPT algorithm for exploiting several SMP systems interconnected through thesame data network; thus, we use a hybrid operation scheme based on OpenMP and MPI.

The sequential operational scheme of the DEPT algorithm is shown in Figure 6.1. As we cansee, a single process is in charge of generating Ns new solutions at each generation (g) by usingcrossover and mutation operators.

In the DEPT algorithm, the generation of new individuals is carried out one by one in asequential way. This loop for generating new solution does not present any data dependency, that isto say, we can generate the new solutions in parallel with no risks. Therefore, the proposed parallelversion of the DEPT algorithm divides the total population (Ns individuals) among differentthreads or processes at each generation of the algorithm until the stopping criterion is reached. Ina more formal way, if we dispose an M threads/processes, then each thread/process is in charge of


Figure 6.2. Parallel operation scheme of the pDEPT algorithm by using OpenMP.

generating ⌈Ns

M ⌉ new solutions. The parallel operation scheme used in the three versions is almostthe same, but there exists slightly required changes for each memory architecture.

In the first place, we present the shared-memory parallel version of the DEPT algorithm whichuses OpenMP. As we may observe in Figure 6.2, each thread generates ⌈Ns

M ⌉ new solutions ateach generation. All threads share the memory space; so, they use their unique identificationnumber (thread_id) in order to know their portion of population. In this way, a thread with anidentification number thread_id is in charge of generating new solutions in the range [a + 1, a + b],where a is (thread_id−1)·b, b = ⌈Ns

M ⌉, M is the total number of threads, and thread_id ∈ [1 . . . M ].For example, if we suppose a multi-core system with 4 cores (M=4) and a population size of 12individuals (Ns=12), the second thread thread_id=2 generates the following individuals: x4, x5,and x6.

In the second place, we have a parallel version of the DEPT algorithm for distributed-memorysystems, see Figure 6.3. Like in the OpenMP version, the Ns individuals of the population aredistributed among the available processes. Since the processes do not share the memory space, it iscompulsory to use messages for communications. In this work, we have used the Message PassingInterface (MPI). As we can see in Figure 6.3, the master process (process_id=1) is in charge ofsending the whole population to the slave processes by using the function MP I_Bcast; therefore,each slave process receives Ns individuals; however, like in the OpenMP version, each process onlygenerates new solutions in the range [a+1, a+b], where a is (process_id−1)·b, b = ⌈Ns

M ⌉, M is the


Figure 6.3. Parallel operation scheme of the pDEPT algorithm by using Message Passing Interface (MPI).

total number of processes, and and process_id ∈ [1 . . . M ]. Once each slave process has finishedits assigned task, it only sends the set of new individuals to the master process; therefore, themaster process only receives ⌈Ns

M ⌉ individuals from each slave process. Note that, in this version,the master process is in charge of composing the new population for the next generation.

Finally, with the aim of making the most of multi-core systems interconnected through adata network, we have designed a hybrid parallel version of the DEPT algorithm which usesOpenMP directives and MPI jointly. As we may observe in Figure 6.4, it works like the MPIversion, that is to say, the master process broadcasts the whole population to the slave pro-cesses. Then, each process makes use of OpenMP directives in order to divide the assignedworkload among diverse threads. In this parallel version, each thread generates new individu-als in the range [a + 1, a + b], where a is (thread_id − 1) + ((process_id − 1) · b), b = ⌈Ns

M ⌉,M=(Number_of_threads)·(Number_of_processes), process_id ∈ [1 . . .Number_of_processes], andthread_id ∈ [1 . . .Number_of_threads]. Once the threads have finished the generation of the newsolutions, each slave process sends the new solutions to the master like in the MPI version.

For example, if we suppose a population size of 12 individuals (Ns=12) and three SMP systems(three processes) with four cores (four threads) interconnected through a data network, the second


process process_id=2 divides its assigned four individuals among its four available cores; so, eachcore/thread will generate one new solution.

Figure 6.4. Hybrid operation scheme of the pDEPT algorithm by using Message Passing Interface (MPI)and OpenMP jointly.


Figure 6.5. Sequential operation scheme of the MO-ABC algorithm.

Figure 6.6. Legend for the operation scheme of the MO-ABC algorithm

6.8.2 Parallel MO-ABC

The Artificial Bee Colony (ABC) algorithm is a population-based evolutionary algorithm inspiredby the intelligent behavior of honey bees. As we saw in Section 6.3, in the ABC algorithm, thepopulation of individuals is defined as a colony with three groups of bees: employed bees, onlookerbees, and scout bees. Since we are dealing with two MOOPs, we have adapted the standard ABCalgorithm to the Multiobjective domain (MO-ABC).

Figure 6.5 shows the sequential operational scheme of the MO-ABC algorithm. Note that thedifferent tones of grey represents the different steps within each generation, see Figure 6.6. Aswe may observe in Figure 6.5, a single process is in charge of generating the new employed bees,onlooker bees, and scout bees at each generation.

In this way, in the design of the parallel operation scheme of the MO-ABC, we can distinguishthree regions: Employed Bees Region, Onlooker Bees Region, and Scout Bees Region. These regionsare perfectly suitable for being parallelized because there are no dependencies among the taskscarried out by threads/processes in the same region.

As we may observe in Figure 6.5, the number of bees which reach the maximum number ofiteration with no improvements is different at each generation (number of scout bees). Therefore,we divide the number of bees which reach the limit, trying to optimize the available computationalresources dynamically, minimizing the communication among processes.

In the first place, we present the shared-memory parallel version of the MO-ABC algorithm.Like the shared-memory version of the DEPT algorithm, this parallel version of the MO-ABCalgorithm also uses OpenMP directives.

6.8P

arallelA

pproaches101

Figure 6.7. Parallel operation scheme of the pMOABC algorithm by using OpenMP.


As we may observe in Figure 6.7, in the employed bees phase, each thread is in charge ofmutating ⌈Ns/2

M ⌉ employed bees in order to exploit their corresponding sources of food. As we cansee, each thread mutates the employed bees in the range [a+1, a+b]; where a is (thread_id−1) ·b,b = ⌈Ns/2

M ⌉, M is the total number of threads, and thread_id ∈ [1 . . . M ].In the onlooker bees phase, the probability vector is computed only by one thread. Then, the

same parallel procedure followed for mutating the employed bees is applied for generating theonlooker bees in the range [Ns

2 + a + 1, Ns

2 + a + b].In the scout bees phase, we store in a vector the indexes of those employed bees which have

reached the limit number of generation with no improvements. Then, we divide this vector ofindexes among the threads; therefore, it is possible that whereas some threads generate the scoutbees (random bees), other threads remain inactive. Finally, only one thread sorts the colony byquality, in order to designate the best bees as employed bees for the next generation.

For example, if we suppose a multi-core system with 4 cores (M=4) and a colony size of 16 bees(Ns=16), the second thread thread_id=2 mutates the employed bees x3 and x4, and generates thefollowing onlooker bees: x11 and x12.

In the second place, we use Message Passing Interface for exploiting distributed-memory systemswith a parallel version of the MO-ABC algorithm (see Figure 6.8). The parallelization is nearlythe same as in the OpenMP version; however, since the processes do not share the same memoryspace, we have a master-slave model in the mutation of the employed, generation of onlooker bees,and scout bees, just like in the parallel model of the DEPT algorithm for distributed-memoryarchitectures.

In the employed bees phase, the master process broadcasts the first half of the colony to theslave processes; therefore, ⌈NS/2

M ⌉ bees to each slave process. Then, each process mutates its

corresponding employed bees in the range [a + 1, a + b]; where a is (process_id−1) · b, b = ⌈Ns/2M ⌉,

M is the total number of processes, and process_id ∈ [1 . . . M ]. When a slave process finishes, itsends to the master its corresponding ⌈Ns/2

M ⌉ mutated employed bees.In the onlooker bees phase, once the master process has computed the probability vector, it

broadcasts not only all the mutated bees to all the slave processes, but also the probability vector.Then, each process is in charge of generating ⌈Ns/2

M ⌉ onlooker bees by using the probability vectorand the mutated employed bees, the range is [Ns

2 + a + 1, Ns

2 + a + b]. Finally, the scout bees phaseis conducted as in the shared-memory version and the sorting of the colony by quality, is carriedout by the master process.

A parallel version of the MO-ABC algorithm which makes use of OpenMP directives and MPIis designed for exploiting systems with a hybrid distributed/shared-memory architecture. As wemay observe in Figure 6.9, it basically works like the MPI version, that is to say, the masterprocess broadcasts/receives the sets of bees to/from the slaves processes at each phase: employed,onlooker, or scout. However, in this version, the workload of each process is divided among aset of threads by using OpenMP directives. In this parallel version, each thread generates newindividuals in the range [a + 1, a + b] for the employed bees mutation and [Ns

2 + a + 1, Ns

2 + a + b]for the generation of the onlooker bees, where a is (thread_id−1)+((process_id−1)·b), b = ⌈Ns

2 ⌉,M=(Number_of_threads)·(Number_of_processes), process_id ∈ [1 . . .Number_of_processes], andthread_id ∈ [1 . . .Number_of_threads].

Since it is possible that some processes or threads remains idle in the scout bees phase, wetry to minimize the communication among processes; therefore, we divide the workload amongthreads (shared-memory) instead of among processes (distributed-memory) whenever possible, thusavoiding unnecessary messages among processes (see Figure 6.9).

6.8P

arallelA

pproaches103

Figure 6.8. Parallel operation scheme of the pMOABC algorithm by using Message Passing Interface (MPI).

104

6.M

ulti

obje

ctiv

eE

volu

tion

ary

Alg

orit

hms

Figure 6.9. Hybrid operation scheme of the pMOABC algorithm by using Message Passing Interface (MPI) and OpenMP jointly.

7Solving the RWA problem

The main aim of this chapter is to use multiobjective evolutionary computation to solve the Routingand Wavelength Assignment problem.

The Multiobjective Evolutionary Algorithms (MOEAs) that we have used are multiobjectivevariants of Differential Evolution (DE), Variable Neighborhood Search algorithm (VNS), ArtificialBee Colony Algorithm (ABC), Gravitational Search Algorithm (GSA), and Firefly Algorithm (FA);we will refer to them as DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA, respectively. For afull description of these multiobjective algorithms please refer to Chapter 6.

In order to test the accuracy of our proposals, we have compared them with the most pop-ular algorithms for multiobjective optimization: the Non-Dominated Sorting Genetic Algorithm(NSGA-II [90]) and the Strength Pareto Evolutionary Algorithm 2 (SPEA2 [91]). We also presenta comparison with several approaches published in the literature by other authors, with the aim ofdemonstrating the great performance of evolutionary computation for solving real-world problems.

In the first place, we use a preliminary set of twelve data sets in order to tune the parametersof each MOEA, comparing their results in order to check the goodness of each approach. Then,we present a comparative study among our five algorithms and the two well-known MOEAs whentackling three optical networks with different number of nodes as well as diverse amounts of traffic.For further information about the data sets, please refer to Section 4.4 (Chapter 4).

In addition, we present a comparison between the MOEAs and diverse techniques published inthe literature for solving the RWA problem. In this way, we have compared not only with typicalheuristics in the Telecommunication field, but also with different varieties of Multiobjective AntColony Optimization algorithms.

Finally, since the RWA problem is an NP-hard problem where the complexity of the problemraises exponentially when we tackle a large optical network or a traffic matrix with a large amountof traffic, in this chapter, we have used a parallel version of the Differential Evolution with ParetoTournaments (pDEPT) in order to obtain high quality results in a reasonable amount of time forthe industry.

105

106 7. Solving the RWA problem

7.1 Parameter Tuning

In this section we briefly describe the parameter tuning of the five proposed MOEAs (DEPT,MO-VNS, MO-ABC, MO-GSA, and MO-FA), as well as the parameter tuning of the well-knownapproaches NSGA-II and SPEA2. Then, we report the different values tested and the best param-eter configuration found for each approach.

In order to tune the parameters of the MOEAs we have used 2 network topologies and six setsof demands for each one. The selected network topologies are the National Science Foundation(NSF) and the Nippon Telegraph and Telephone (NTT), and those data sets proposed in [141] byother authors:

• NSF#01, NSF#02, NSF#03, NSF#04, NSF#05, and NSF#06.

• NTT#01, NTT#02, NTT#03, NTT#04, NTT#05, and NTT#06.

To decide the best value of each parameter in each MOEA, we have performed 30 independentruns and measured the Hypervolume (HV) of each run.

In this way, we compute the median HV in the 30 runs, and select the value with the highestmedian of HV. Furthermore, we have conducted the statistical analysis presented in Section 3.4with the aim of ensuring a statistical relevance.

For further information about the data sets, and their corresponding runtime and referencepoints for computing the HV indicator, please refer to Chapter 4 and see Table 4.2 and Table 4.3.All the data sets were run by using g++ (GCC) 4.4.5 on a 2.3GHz Intel PC with 1GB RAM.

In this thesis, we use the Yen’s algorithm [10] to obtain the k shortest paths between eachpair of nodes (see Section 4.3). Therefore, we have used exactly the same value of k in all theapproaches, k=10.

In the following, we present the main parameters for each MOEA, as well as the values tested.Note that, for each parameter in each approach, we have highlighted in bold the value with thehighest median of HV (and, therefore, the configuration used for that parameter).

• DEPT

– Population size (Ns): 5, 10, 25, 50, 75, 100.

– Crossover probability (CR): 10%, 25%, 50%, 75%, 95%.

– Mutation rate (F ): 10%, 25%, 50%, 75%, 95%.

– Selection scheme (S):

1. Best/1/Exponential2. Rand/1/Exponential3. RandToBest/1/Exponential4. Best/2/Exponential5. Rand/2/Exponential6. Best/1/Binomial7. Rand/1/Binomial8. RandToBest/1/Binomial9. Best/2/Binomial

10. Rand/2/Binomial

7.1 Parameter Tuning 107

• MO-VNS

– Neighbourhood degrees (nmax): 6 degrees

– Mutation rate (F ): from 15% to 90% depending on the neighbourhood.

• MO-ABC

– Population Size (Ns): 5, 10, 25, 50, 75, 100.

– Maximum Limit value (limitmax): 2, 5, 10, 15, 20, 25.

– Mutation rate (F ): 1%, 7.5%, 10%, 25%, 50%, 75%.

• MO-GSA


– Initial Gravitational Constant (G0): 101, 102, 103, 104, 105.

– Alpha (α): 2, 5, 20, 40, 60, 80, 100.

– Number of best agents (kbest): it starts with Ns best agents and decreases linearly toNs

4 throughout the execution.

• MO-FA


– Attractiveness (β0): 0.05, 0.1, 0.25, 0.5, 0.75, 1.

– Absorption coefficient (γ): 0.05, 0.1, 0.25, 0.5, 0.75, 1.

– Control parameter for exploration (α): 0.05, 0.1, 0.25, 0.5, 0.75, 1.

• NSGA-II


– Selection method: Binary tournament.

– Crossover: Single-point (SPX).


– Mutation probability (F ): 1%, 7.5%, 10%, 25%, 50%, 75%.

• SPEA2




– Mutation probability (F ): 1%, 7.5%, 10%, 25%, 50%, 75%.

After performing the parameter tuning of each algorithm, we obtain the best parameter con-figuration for each MOEA. In Table 7.1, we present the median value of HV obtained by eachMOEA with its corresponding best configuration. As we may observe, all the algorithms are wellconfigured and obtain very promising results in all scenarios. However, we highlight the particularperformance of the MOEAs based on swarm intelligence, such as MO-FA and MO-ABC.


Table 7.1. Comparison among the MOEAs by using the Hypervolume (HV) indicator. The notation usedis HVIQR, where HV is the median hypervolume and IQR is the interquartile range in 30 independentruns.

DEPT MO-VNS MO-ABC MO-GSA MO-FA NSGA-II SPEA2

NSF#01 67.14%1.87e−05 67.14%3.89e

−05 67.14%9.17e−05 67.14%9.54e

−05 67.14%9.50e−06 67.14%4.11e

−05 67.14%5.24e−05

NSF#02 72.67%6.34e−06 72.67%1.33e

−05 72.67%7.16e−05 72.67%8.22e

−05 72.67%3.70e−05 72.67%4.46e

−05 72.67%1.89e−05

NSF#03 71.74%6.12e−03 71.70%9.28e−03 71.74%6.68e

−03 70.16%4.64e−03 71.74%6.28e−03 71.68%5.75e−03 68.70%1.25e−03

NSF#04 72.67%7.43e−05 72.67%6.12e

−05 72.67%4.55e−05 72.67%7.15e

−06 72.67%1.19e−06 72.67%4.59e

−05 72.67%3.71e−05

NSF#05 72.92%9.28e−04 72.90%5.99e−03 72.92%4.33e

−03 72.92%7.33e−03 72.92%8.62e

−04 72.92%4.74e−03 72.92%1.05e

−03

NSF#06 67.01%8.23e−03 65.99%1.06e−03 68.00%5.86e−04 66.96%2.92e−03 68.00%6.97e

−03 67.01%6.53e−03 65.01%3.70e−03

NTT#01 69.55%4.74e−05 69.55%5.24e−05 69.55%7.02e−05 69.55%2.60e−06 69.55%3.61e−05 69.55%8.41e−06 69.55%8.23e−05

NTT#02 70.73%2.82e−03 70.70%4.40e−03 70.75%3.12e−03 69.73%2.31e−03 71.13%7.22e−03 70.75%4.34e−03 70.74%1.57e−03

NTT#03 64.49%2.13e−03 65.07%2.27e−03 66.98%6.24e−04 63.96%8.17e−03 67.47%1.35e−03 65.20%1.07e−03 64.12%6.81e−03

NTT#04 70.87%5.55e−05 70.87%3.23e−05 70.87%4.65e−06 70.87%2.52e−05 70.87%8.09e−05 70.87%6.16e−05 70.87%8.75e−05

NTT#05 69.42%7.45e−03 68.78%8.87e−03 69.42%4.83e−03 68.79%5.31e−03 70.58%4.21e−03 69.42%5.20e−03 69.42%9.66e−03

NTT#06 66.49%2.41e−03 63.90%8.05e−03 66.49%1.54e−03 65.21%4.28e−03 69.29%5.15e−03 66.49%7.65e−03 65.62%1.26e−03

7.2 Comparative Study

In this section we present a comparative study among the MOEAs proposed in Chapter 6 forsolving diverse scenarios of the RWA problem.

In this study, we use three different network topologies: Pan European network (COST239,Europe), National Science Foundation network (NSF, U.S.A.), and the Nippon Telegraph andTelephone network (NTT, Japan). As we present in Chapter 4, these optical networks consist ofdifferent number of nodes: 11, 14, and 55 nodes; respectively.

For each optical network topology, we use different amounts of traffic in order to provide aglobal view of the performance of each MOEA when tackling the RWA problem. A total of 12data sets have been solved by the different multiobjective approaches for each optical network;therefore, a total of 36 different scenarios are involved in this comparative study.

The number of independent runs of the MOEAs is 30 for each data set. The stop criterion isbased on the runtime and depend on the data set, see Table 4.1, Table 4.2, and Table 4.3. All thedata sets were run using g++ (GCC) 4.4.5 on a 2.3GHz Intel PC with 1GB RAM.

The multiobjective indicators used in this section are the Hypervolume (HV) and the SetCoverage (SC). On the one hand, the HV indicator reports the quality of a set of non-dominatedsolutions obtained by an algorithm; thus, we make a comparison among the approaches by quality.On the other hand, we measure the percentage of non-dominated solutions achieved by an algorithmA which are dominated by the non-dominated solutions obtained by an algorithm B. To do that,we use the SC indicator.

Since we are dealing with stochastic algorithms, we have performed a statistical analysis of theresults obtained (see Section 3.4). We consider a significance level of 5% in the statistical tests,in other words, a p-value under 0.05; therefore, the differences are unlikely to have occurred bychance with a probability of 95%.

7.2.1 European Optical Network (COSTS239)

In the first place, we start with the Pan European Optical network (COST239, Europe), whichconsists of 11 nodes and 52 optical links. In Figure 7.1, we show the physical topology of COST239.

7.2 Comparative Study 109

Figure 7.1. The optical network topology: European Optical Network (COST239).

In Table 4.1 we present for each data set, its corresponding amount of traffic, runtime, and referencepoint required for computing the HV indicator.

On the one hand, we compare the proposed MOEAs and the standard approaches NSGA-IIand SPEA2 by using the median value of HV in 30 independent runs. As we may observe in Table7.2, all approaches obtain similar values of HV in those data sets with a lower number of demands(COST239#01-COST239#06). Furthermore, we can check in Table 7.3 that there are not relevantstatistical differences among the approaches in the majority of these data sets.

However, in the data sets COST239#07 - COST239#12, we can notice that differences of HVare remarkable among the approaches. As we can observe in Table 7.2, when the number of de-mands is over 100, the MO-FA obtains higher values of HV than the other multiobjective proposals.According to the results obtained by the statistical study, the differences of HV among each pairof MOEAs is statistically significant in almost all data sets, we may observe that the differencesamong the algorithms DEPT and MO-FA are not significant in the case of COST239#07.

In order to prove the goodness of each proposed MOEA (DEPT, MO-VNS, MO-ABC, MO-GSA,and MO-FA), diverse isolated comparisons are shown in Figure 7.2. In these isolated comparisons,we illustrate the median value of HV obtained by each one, as well as the values obtained by thestandard approaches NSGA-II and SPEA2.

Table 7.2. COST239 network. Comparison among the MOEAs by using the Hypervolume (HV) indicator.The notation used is HVIQR, where HV is the median hypervolume and IQR is the interquartile rangein 30 independent runs.


COST239#01 78.26%5.38e−06 73.64%4.49e−04 78.26%5.68e−05 78.26%9.51e−05 78.26%5.59e−05 76.09%7.42e−04 75.82%7.52e−04

COST239#02 80.58%4.31e−05 80.42%4.55e−03 80.58%8.32e−06 80.58%9.10e−03 80.58%5.80e−03 80.58%9.90e−05 80.58%3.05e−05

COST239#03 77.92%7.53e−03 76.54%7.13e−03 79.31%8.99e−05 72.85%4.50e−03 79.31%4.77e−03 76.55%9.35e−03 76.79%6.22e−03

COST239#04 72.97%9.29e−06 71.52%4.20e−03 72.97%2.92e−05 71.62%8.16e−03 72.97%1.56e−05 72.97%8.44e−06 72.97%6.78e−05

COST239#05 72.48%6.33e−05 72.40%1.03e−04 72.48%1.74e−05 72.48%5.30e−03 72.48%6.96e−06 72.48%6.54e−03 71.87%6.71e−03

COST239#06 70.75%1.47e−05 68.07%4.50e−03 70.75%2.62e−05 66.61%2.82e−03 70.75%9.97e−05 70.07%9.88e−03 69.88%8.13e−03

COST239#07 69.58%5.47e−03 64.01%4.45e−03 66.72%2.72e−03 66.71%1.10e−05 69.58%3.04e−04 64.00%6.89e−04 63.94%5.66e−03

COST239#08 63.86%2.04e−03 60.50%8.36e−04 63.99%9.14e−03 66.30%2.75e−04 70.32%7.75e−04 62.63%2.08e−03 62.91%7.95e−03

COST239#09 60.24%2.87e−03 60.31%6.94e−03 60.62%9.15e−03 64.44%5.09e−03 68.82%9.91e−04 61.96%5.27e−03 60.96%9.67e−03

COST239#10 58.05%2.16e−03 59.68%5.39e−03 61.60%3.38e−03 64.27%2.57e−03 68.61%2.81e−03 51.44%9.42e−03 52.01%9.99e−03

COST239#11 56.14%9.26e−03 58.97%3.63e−03 60.26%5.46e−03 60.71%3.19e−03 68.23%3.16e−04 58.00%4.57e−03 56.64%7.13e−03

COST239#12 56.18%9.20e−04 56.39%4.11e−03 59.47%3.38e−03 59.16%5.62e−04 68.11%3.66e−03 54.55%5.44e−04 55.89%6.85e−03


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Figure 7.2. Comparison among NSGA-II, SPEA2 and each proposed MOEA (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the median value of Hypervolume obtained in 30 independentruns with the COST239 network.


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Figure 7.3. Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) byusing the median value of Hypervolume obtained in 30 independent runs with the COST239 network(1/2).


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Figure 7.4. Comparison by pairs of MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) byusing the median value of Hypervolume obtained in 30 independent runs with the COST239 network(2/2).


Table 7.3. Statistical Analysis among the MOEAs in the COST239 network. The table indicates inwhich data sets two algorithms have no statistically significant differences.

Pairs of Number of statisticallyAlgorithms non-significant data sets Statistically non-significant data sets

DEPT MO-VNS 0 out of 12 data sets -MO-ABC 5 out of 12 data sets #01 #02 #04 #05 #06MO-GSA 3 out of 12 data sets #01 #02 #05MO-FA 6 out of 12 data sets #01 #02 #04 #05 #06 #07NSGA-II 3 out of 12 data sets #02 #04 #05SPEA2 2 out of 12 data sets #02 #04

MO-VNS MO-ABC 0 out of 12 data sets -MO-GSA 0 out of 12 data sets -MO-FA 0 out of 12 data sets -NSGA-II 2 out of 12 data sets #03 #07SPEA2 0 out of 12 data sets -

MO-ABC MO-GSA 4 out of 12 data sets #01 #02 #05 #07MO-FA 6 out of 12 data sets #01 #02 #03 #04 #05 #06NSGA-II 3 out of 12 data sets #02 #04 #05SPEA2 2 out of 12 data sets #02 #04

MO-GSA MO-FA 3 out of 12 data sets #01 #02 #05NSGA-II 2 out of 12 data sets #02 #05SPEA2 1 out of 12 data sets #02

MO-FA NSGA-II 3 out of 12 data sets #02 #04 #05SPEA2 2 out of 12 data sets #02 #04

NSGA-II SPEA2 2 out of 12 data sets #02 #04

As we may observe in Figure 7.2, in general, all the algorithms perform better than the well-known approaches. Concretely, the multiobjective swarm proposals and the DEPT algorithmclearly obtain higher median of HV than both NSGA-II and SPEA2 in almost the twelve scenarios;however, we highlight the particular good performance of the swarm approaches when the amountof traffic is over 100 demands. Lastly, if we focus on the trajectory-based MOEA (MO-VNS), wefind that it works similar to the well-known MOEAs (NSGA-II and SPEA2).

An illustrative pairwise comparison among the five proposed multiobjective approaches by usingthe median hypervolume is presented in Figure 7.3 and Figure 7.4. We can notice that the MO-FAclearly overcomes the results achieved by the rest of MOEAs in those data sets with a large numberof demands. Furthermore, we may see the promising results of the other swarm MOEAs (MO-ABCand MO-GSA) when we compare them with the algorithms DEPT and MO-VNS.

On the other hand, we have made a comparison among the MOEAs by using the Set Coverageindicator. In Table 7.4, we present a pairwise comparison of MOEAs in order to determine thecoverage relation of each one with the rest.

As we can see, the algorithms DEPT and MO-ABC cover at least a half of the non-dominatedsolutions obtained by the rest of MOEAs. Besides, if we focus on the MO-GSA, we can observethat it only covers a quarter of the non-dominated solutions from MO-FA and nearly 40% of thenon-dominated solutions obtained by the DEPT algorithm. However, it is able to cover over 50%of the sets of non-dominated solutions achieved by the other MOEAs. As we may observe, thenon-dominated solutions achieved by MO-FA in each data set of the optical network COST239clearly cover the non-dominated solutions obtained by any other MOEA (100% in all cases).

Finally, if we check the coverage relation obtained by the well-known approaches NSGA-II andSPEA2, we realize that their values of coverage relation are lower than 50% in almost all cases.We may conclude that the MO-FA is the MOEA which obtains the best results in the COST239optical network, followed by the MO-ABC, MO-GSA, DEPT, NSGA-II, MO-VNS, and SPEA2.


Table 7.4. COST239 network. Comparison among the MOEAs by using the Set Coverage (SC) indicator,A B. Note that, SC represents the mean coverage of an algorithm A over an algorithm B in all thedata sets.

COST239A B #01 #02 #03 #04 #05 #06 #07 #08 #09 #10 #11 #12 SC

DEPT

MO-VNS 100% 100% 100% 100% 100% 100% 100% 100% 50% 40% 0% 75% 80.42%NSGA-II 100% 100% 100% 100% 100% 100% 100% 100% 50% 100% 0% 100% 87.50%SPEA2 100% 100% 66.67% 100% 100% 100% 100% 75% 33.33% 100% 75% 100% 87.50%MO-GSA 100% 100% 100% 100% 100% 100% 100% 75% 50% 0% 0% 25% 70.83%MO-ABC 100% 100% 0% 100% 100% 100% 100% 75% 25% 20% 40% 0% 63.33%MO-FA 100% 100% 0% 100% 100% 100% 100% 0% 0% 0% 0% 0% 50.00%

MO-VNS

DEPT 0% 0% 0% 0% 0% 0% 0% 0% 20% 25% 40% 0% 7.08%NSGA-II 0% 0% 0% 0% 0% 0% 100% 0% 0% 100% 0% 100% 25%SPEA2 0% 0% 33.33% 0% 100% 66.67% 100% 0% 0% 100% 100% 100% 50%MO-GSA 0% 0% 100% 0% 0% 50% 25% 0% 0% 0% 0% 0% 14.58%MO-ABC 0% 0% 0% 0% 0% 0% 25% 0% 25% 0% 40% 0% 7.50%MO-FA 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

NSGA-II

DEPT 0% 100% 0% 100% 100% 0% 0% 0% 40% 0% 20% 0% 30%MO-VNS 100% 100% 100% 100% 100% 100% 0% 100% 100% 0% 0% 0% 66.67%SPEA2 100% 100% 33.33% 100% 100% 100% 100% 75% 33.33% 75% 100% 50% 80.56%MO-GSA 0% 100% 100% 100% 100% 100% 0% 75% 50% 0% 0% 0% 52.08%MO-ABC 0% 100% 0% 100% 100% 0% 25% 75% 100% 0% 0% 0% 41.67%MO-FA 0% 100% 0% 100% 100% 0% 0% 0% 0% 0% 0% 0% 25%

SPEA2

DEPT 0% 100% 50% 100% 0% 0% 0% 33.33% 80% 0% 0% 0% 30.28%MO-VNS 100% 100% 50% 100% 0% 33.33% 0% 100% 100% 0% 0% 0% 48.61%NSGA-II 0% 100% 0% 100% 0% 0% 0% 66.67% 50% 0% 0% 33.33% 29.17%MO-GSA 0% 100% 50% 100% 0% 100% 0% 75% 50% 0% 0% 0% 39.58%MO-ABC 0% 100% 0% 100% 0% 0% 0% 75% 100% 0% 0% 0% 31.25%MO-FA 0% 100% 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% 16.67%

MO-GSA

DEPT 100% 100% 0% 0% 100% 0% 0% 0% 40% 25% 60% 50% 39.58%MO-VNS 100% 100% 0% 100% 100% 0% 100% 100% 100% 100% 100% 100% 83.33%NSGA-II 100% 100% 0% 0% 100% 0% 100% 0% 0% 100% 100% 100% 58.33%SPEA2 100% 100% 0% 0% 100% 0% 100% 0% 0% 100% 100% 100% 58.33%MO-ABC 100% 100% 0% 0% 100% 0% 50% 25% 100% 80% 100% 33.33% 57.36%MO-FA 100% 100% 0% 0% 100% 0% 0% 0% 0% 0% 0% 0% 25%

MO-ABC

DEPT 100% 100% 100% 100% 100% 100% 0% 0% 40% 25% 40% 50% 62.92%MO-VNS 100% 100% 100% 100% 100% 100% 0% 100% 50% 100% 50% 100% 83.33%NSGA-II 100% 100% 100% 100% 100% 100% 100% 0% 0% 100% 100% 100% 83.33%SPEA2 100% 100% 100% 100% 100% 100% 100% 0% 0% 100% 100% 100% 83.33%MO-GSA 100% 100% 100% 100% 100% 100% 75% 75% 0% 20% 0% 50% 68.33%MO-FA 100% 100% 100% 100% 100% 100% 0% 0% 0% 0% 0% 0% 50%

MO-FA

DEPT 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%MO-VNS 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%NSGA-II 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%SPEA2 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%MO-GSA 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%MO-ABC 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%


Figure 7.5. The optical network topology: National Science Foundation (NSF).

7.2.2 National Science Foundation (NSF) Network

In this subsection we present a comparison among the MOEAs by using the real-world opticalnetwork topology National Science Foundation (NSF), which consists of 14 nodes and 42 opticallinks.

The NSF physical topology is shown in Figure 7.5 and all specifications about those data setsused in this subsection are presented in Table 4.2.

Firstly, we present a comparison among the algorithms DEPT, MO-VNS, MO-ABC, MO-GSA,MO-FA, NSGA-II, and SPEA2 by using the HV indicator.

As we can see in Table 7.5, when the data sets contain a low number of demands (less than orequal to 30 demands), there are no statistical differences of HV among the MOEAs (see Table 7.6).That is to say, all the algorithms are able to obtain high quality sets of non-dominated solutions.

However, we may observe in Table 7.5 how the differences of HV among the MOEAs increasewhen the number of demands is higher. In addition, like in the COST239 network, we realize thatthe multiobjective swarm approaches obtain very promising results of HV in the 30 independentruns, highlighting the particular performance of the MO-FA, which clearly overcomes the value ofHV achieved by the rest of MOEAs in those data sets with a large number of demands (NSF#07

Table 7.5. NSF network. Comparison among the MOEAs by using the Hypervolume (HV) indicator.The notation used is HVIQR, where HV is the median hypervolume and IQR is the interquartile rangein 30 independent runs.


NSF#01 67.14%1.87e−05 67.14%3.89e−05 67.14%9.17e−05 67.14%9.54e−05 67.14%9.50e−06 67.14%4.11e−05 67.14%5.24e−05

NSF#02 72.67%6.34e−06 72.67%1.33e−05 72.67%7.16e−05 72.67%8.22e−05 72.67%3.70e−05 72.67%4.46e−05 72.67%1.89e−05

NSF#03 71.74%6.12e−03 71.70%9.28e−03 71.74%6.68e−03 70.16%4.64e−03 71.74%6.28e−03 71.68%5.75e−03 68.70%1.25e−03

NSF#04 72.67%7.43e−05 72.67%6.12e−05 72.67%4.55e−05 72.67%7.15e−06 72.67%1.19e−06 72.67%4.59e−05 72.67%3.71e−05

NSF#05 72.92%9.28e−04 72.90%5.99e−03 72.92%4.33e−03 72.92%7.33e−03 72.92%8.62e−04 72.92%4.74e−03 72.92%1.05e−03

NSF#06 67.01%8.23e−03 65.99%1.06e−03 68.00%5.86e−04 66.96%2.92e−03 68.00%6.97e−03 67.01%6.53e−03 65.01%3.70e−03

NSF#07 54.32%3.02e−05 54.32%1.50e−05 54.32%1.65e−05 54.32%3.00e−07 54.32%3.07e−05 52.79%2.55e−03 53.56%7.88e−03

NSF#08 58.33%4.01e−03 57.68%5.01e−03 57.37%7.19e−03 58.61%6.06e−03 66.25%7.67e−04 55.73%1.03e−03 55.71%5.52e−03

NSF#09 53.56%5.69e−03 52.29%6.19e−03 49.68%1.41e−03 52.46%7.00e−03 62.30%6.43e−03 51.58%7.96e−03 51.41%2.70e−03

NSF#10 47.51%6.42e−03 46.93%5.65e−03 45.01%8.16e−03 47.75%4.05e−03 60.01%1.83e−04 44.77%8.78e−03 44.18%6.30e−03

NSF#11 41.90%7.38e−03 38.79%9.09e−03 42.79%1.39e−03 38.85%5.57e−03 60.47%7.48e−04 38.95%7.08e−03 38.32%8.28e−03

NSF#12 41.74%7.50e−03 39.35%4.05e−03 42.21%8.81e−03 39.76%5.26e−04 60.62%8.25e−03 39.74%1.64e−03 39.25%9.63e−03

1167.

Solvingthe

RW

Aproblem

NSF#01NSF#02NSF#03NSF#04NSF#05NSF#06NSF#07NSF#08NSF#09NSF#10NSF#11NSF#1235 40 45 50 55 60 65 70 75

DEPT

NSG

A-II

SPEA2

Median Hypervolume (%)

(a):

DE

PT



MO

-VN

S N

SGA

-II SPEA

2

(b):

MO

-VN

S



MO

-ABC

NSG

A-II

SPEA2

(c):M

O-A

BC


Median Hypervolume (%) M

O-G

SA N

SGA

-II SPEA

2

(d):

MO

-GS

A



MO

-FA N

SGA

-II SPEA

2

(e):M

O-F

A

Fig

ure

7.6

.C

omparison

among

NSG

A-II,

SPE

A2

andeach

proposed

MO

EA

(DE

PT

,M

O-V

NS,

MO

-A

BC

,M

O-G

SA,

andM

O-FA

)by

usingthe

median

valueof

Hyp

ervolume

obtainedin

30indep

endentruns

with

theN

SFnetw

ork.

7.2C

omparative

Study117



DEPT

MO

-VN

S

(a):

DE

PT

vs

MO

-VN

S



DEPT

MO

-ABC

(b):

DE

PT

vs

MO

-AB

C



DEPT

MO

-GSA

(c):D

EP

Tvs

MO

-GS

A



DEPT

MO

-FA

(d):

DE

PT

vs

MO

-FA



MO

-VN

S M

O-A

BC

(e):M

O-V

NS

vs

MO

-AB

C

Fig

ure

7.7

.C

omparison

bypairs

ofM

OE

As

(DE

PT

,M

O-V

NS,

MO

-AB

C,

MO

-GSA

,and

MO

-FA)

byusing

them

edianvalue

ofH

ypervolum

eobtained

in30

independent

runsw

iththe

NSF

network

(1/2).

1187.

Solvingthe

RW

Aproblem



MO

-VN

S M

O-G

SA

(a):

MO

-VN

Svs

MO

-GS

A



MO

-VN

S M

O-FA

(b):

MO

-VN

Svs

MO

-FA



MO

-ABC

MO

-GSA

(c):M

O-A

BC

vs

MO

-GS

A



O-A

BC M

O-FA

(d):

MO

-AB

Cvs

MO

-FA



MO

-GSA

MO

-FA

(e):M

O-G

SA

vs

MO

-FA

Fig

ure

7.8

.C

omparison

bypairs

ofM

OE

As

(DE

PT

,M

O-V

NS,

MO

-AB

C,

MO

-GSA

,and

MO

-FA)

byusing

them

edianvalue

ofH

ypervolum

eobtained

in30

independent

runsw

iththe

NSF

network

(2/2).


Table 7.6. Statistical Analysis among the MOEAs in the NSF network. The table indicates in whichdata sets two algorithms have no statistically significant differences.


DEPT MO-VNS 5 out of 12 data sets #01 #02 #04 #05 #07MO-ABC 6 out of 12 data sets #01 #02 #03 #04 #05 #07MO-GSA 5 out of 12 data sets #01 #02 #04 #05 #07MO-FA 6 out of 12 data sets #01 #02 #03 #04 #05 #07NSGA-II 5 out of 12 data sets #01 #02 #04 #05 #06SPEA2 4 out of 12 data sets #01 #02 #04 #05

MO-VNS MO-ABC 5 out of 12 data sets #01 #02 #04 #05 #07MO-GSA 5 out of 12 data sets #01 #02 #04 #05 #07MO-FA 5 out of 12 data sets #01 #02 #04 #05 #07NSGA-II 4 out of 12 data sets #01 #02 #04 #05SPEA2 4 out of 12 data sets #01 #02 #04 #05

MO-ABC MO-GSA 5 out of 12 data sets #01 #02 #04 #05 #07MO-FA 6 out of 12 data sets #01 #02 #03 #04 #05 #07NSGA-II 5 out of 12 data sets #01 #02 #04 #05 #06SPEA2 4 out of 12 data sets #01 #02 #04 #05

MO-GSA MO-FA 5 out of 12 data sets #01 #02 #04 #05 #07NSGA-II 4 out of 12 data sets #01 #02 #04 #05SPEA2 4 out of 12 data sets #01 #02 #04 #05

MO-FA NSGA-II 4 out of 12 data sets #01 #02 #04 #05SPEA2 4 out of 12 data sets #01 #02 #04 #05

NSGA-II SPEA2 4 out of 12 data sets #01 #02 #04 #05

- NSF#12). Without taking into account the results obtained by the MO-FA, we realize that thesecond best algorithm is the multiobjective version of the Differential Evolution.

Like in the previous experiment, we present diverse comparisons among each one of the fiveproposed MOEAs and the well-known algorithms NSGA-II and SPEA2 in Figure 7.6 with the aimof demonstrating their goodness. In this way, an illustrative comparison of HV obtained by eachproposed MOEA in the 30 runs are reported in Figure 7.6, as well as the values of HV obtainedby the NSGA-II and SPEA2.

As we can see, all the algorithms are able to obtain results of HV with higher quality than thealgorithms NSGA-II and SPEA2. Contrary to the previous study with the COST239, in this case,the MO-VNS obtains better values of HV than the standard NSGA-II and SPEA2. Besides, werealize that the algorithms MO-FA and DEPT clearly overcomes these well-known approaches.

In Figure 7.7 and Figure 7.8, we illustrate a pairwise comparison among the five proposedmultiobjective approaches by using the median HV. As we expect from the previous results, theMO-FA overcomes the results achieved by all the other approaches in the vast majority of datasets. In addition, we can notice that the DEPT algorithm achieved higher values of HV than theMO-ABC and MO-GSA in the data sets with a large number of demands.

Next, we compare the MOEAs by using the SC indicator. As we can see in Table 7.7, allalgorithms obtain 100% of coverage relation with the rest of MOEAs in data sets with a low numberof demands; however it is not applicable to those data sets with a large number of demands. In thefirst place, we focus on the results obtained by the MO-FA and we realize that they are the bestones, it is able to cover all non-dominated solutions (100%) obtained by the other MOEAs. Then,we can check that the coverage relation among the algorithms DEPT, MO-ABC, and MO-GSAis similar. If we check the coverage relation of the MO-VNS, we may observe that it only coversthe non-dominated solutions of the NSGA-II and SPEA2. Finally, the reference approaches in themultiobjective domain (NSGA-II and SPEA2) obtain a poor coverage relation value, due to they


Table 7.7. NSF network. Comparison among the MOEAs by using the Set Coverage (SC) indicator,A B. Note that, SC represents the mean coverage of an algorithm A over an algorithm B in all thedata sets.

NSFA B #01 #02 #03 #04 #05 #06 #07 #08 #09 #10 #11 #12 SC

DEPT

MO-VNS 100% 100% 100% 100% 100% 100% 100% 66.67% 50% 100% 83.33% 75% 89.58%NSGA-II 100% 100% 100% 100% 100% 100% 100% 33.33% 100% 100% 71.43% 0% 83.73%SPEA2 100% 100% 100% 100% 100% 100% 100% 75% 100% 100% 100% 100% 97.92%MO-GSA 100% 100% 100% 100% 100% 100% 100% 0% 0% 0% 60% 0% 63.33%MO-ABC 100% 100% 100% 100% 100% 50% 100% 71.43% 100% 0% 0% 0% 68.45%MO-FA 100% 100% 100% 100% 100% 0% 100% 0% 0% 0% 0% 0% 50%

MO-VNS

DEPT 100% 100% 0% 100% 0% 0% 100% 40% 20% 50% 0% 0% 42.50%NSGA-II 100% 100% 50% 100% 0% 0% 100% 0% 100% 100% 0% 0% 54.17%SPEA2 100% 100% 100% 100% 0% 66.67% 100% 75% 100% 100% 100% 100% 86.81%MO-GSA 100% 100% 100% 100% 0% 66.67% 100% 0% 0% 0% 0% 0% 47.22%MO-ABC 100% 100% 0% 100% 0% 0% 100% 57.14% 100% 0% 0% 0% 46.43%MO-FA 100% 100% 0% 100% 0% 0% 100% 0% 0% 0% 0% 0% 33.33%

NSGA-II

DEPT 100% 100% 0% 100% 100% 0% 0% 40% 0% 0% 0% 0% 36.67%MO-VNS 100% 100% 66.67% 100% 100% 100% 0% 33.33% 0% 0% 100% 100% 66.67%SPEA2 100% 100% 100% 100% 100% 100% 0% 100% 100% 100% 100% 100% 91.67%MO-GSA 100% 100% 100% 100% 100% 100% 0% 0% 0% 0% 0% 50% 54.17%MO-ABC 100% 100% 0% 100% 100% 0% 0% 42.86% 100% 0% 0% 0% 45.24%MO-FA 100% 100% 0% 100% 100% 0% 0% 0% 0% 0% 0% 0% 33.33%

SPEA2

DEPT 100% 100% 0% 100% 100% 0% 0% 20% 0% 0% 0% 0% 35%MO-VNS 100% 100% 0% 100% 100% 33.33% 0% 16.67% 0% 0% 0% 0% 37.50%NSGA-II 100% 100% 0% 100% 100% 0% 100% 0% 0% 0% 0% 0% 41.67%MO-GSA 100% 100% 0% 100% 100% 33.33% 0% 0% 0% 0% 0% 0% 36.11%MO-ABC 100% 100% 0% 100% 100% 0% 0% 0% 100% 0% 0% 0% 41.67%MO-FA 100% 100% 0% 100% 100% 0% 0% 0% 0% 0% 0% 0% 33.33%

MO-GSA

DEPT 100% 100% 0% 100% 100% 0% 100% 100% 40% 100% 0% 0% 61.67%MO-VNS 100% 100% 0% 100% 100% 66.67% 100% 100% 100% 100% 100% 100% 88.89%NSGA-II 100% 100% 0% 100% 100% 0% 100% 100% 100% 100% 71.43% 100% 80.95%SPEA2 100% 100% 100% 100% 100% 66.67% 100% 100% 100% 100% 100% 100% 97.22%MO-ABC 100% 100% 0% 100% 100% 0% 100% 100% 100% 0% 0% 0% 58.33%MO-FA 100% 100% 0% 100% 100% 0% 100% 0% 0% 0% 0% 0% 41.67%

MO-ABC

DEPT 100% 100% 100% 100% 100% 100% 100% 20% 0% 0% 100% 100% 76.67%MO-VNS 100% 100% 100% 100% 100% 100% 100% 33.33% 0% 0% 100% 100% 77.78%NSGA-II 100% 100% 100% 100% 100% 100% 100% 0% 0% 100% 100% 80% 81.67%SPEA2 100% 100% 100% 100% 100% 100% 100% 75% 0% 100% 100% 100% 89.58%MO-GSA 100% 100% 100% 100% 100% 100% 100% 0% 0% 0% 100% 75% 72.92%MO-FA 100% 100% 100% 100% 100% 0% 100% 0% 0% 0% 0% 0% 50%

MO-FA



Figure 7.9. The optical network topology: Nippon Telegraph and Telephone (NTT).

cover 0% of the non-dominated solutions (with some exceptions) achieved by the other MOEAs inthe most complicated data sets.

For the NSF topology we conclude that the best multiobjective evolutionary algorithm is MO-FA. It obtains higher quality Pareto fronts than DEPT, MO-VNS, MO-ABC, MO-GSA, NSGA-II,and SPEA2. Unlike in the previous experiment, in this case the second best algorithm is theDEPT, followed by the MO-ABC, MO-GSA, MO-VNS, NSGA-II, and SPEA2.

7.2.3 Nippon Telegraph and Telephone (NTT) Network

The Nippon Telegraph and Telephone (NTT, Japan) is a large optical network which consists of55 nodes and 144 optical links, see Figure 7.9.

In Table 4.3, we present all the specifications about those data sets used in our experiments.We start comparing the MOEAs by using the HV indicator (see Table 7.8). Since the NTT

network topology is larger than the previous ones (COST239 and NSF), in Table 7.9 we can seehow the differences of HV among the MOEAs are, in general, only statistically not significant whenwe deal with scenarios in which the number of demands is low (NTT#01 - NTT#06).

In Table 7.8, we may see once again that the MO-FA presents better results of HV than the rest

Table 7.8. NTT network. Comparison among the MOEAs by using the Hypervolume (HV) indicator.The notation used is HVIQR, where HV is the median hypervolume and IQR is the interquartile rangein 30 independent runs.


NTT#01 69.55%4.74e−05 69.55%5.24e−05 69.55%7.02e−05 69.55%2.60e−06 69.55%3.61e−05 69.55%8.41e−06 69.55%8.23e−05

NTT#02 70.73%2.82e−03 70.70%4.40e−03 70.75%3.12e−03 69.73%2.31e−03 71.13%7.22e−03 70.75%4.34e−03 70.74%1.57e−03

NTT#03 64.49%2.13e−03 65.07%2.27e−03 66.98%6.24e−04 63.96%8.17e−03 67.47%1.35e−03 65.20%1.07e−03 64.12%6.81e−03

NTT#04 70.87%5.55e−05 70.87%3.23e−05 70.87%4.65e−06 70.87%2.52e−05 70.87%8.09e−05 70.87%6.16e−05 70.87%8.75e−05

NTT#05 69.42%7.45e−03 68.78%8.87e−03 69.42%4.83e−03 68.79%5.31e−03 70.58%4.21e−03 69.42%5.20e−03 69.42%9.66e−03

NTT#06 66.49%2.41e−03 63.90%8.05e−03 66.49%1.54e−03 65.21%4.28e−03 69.29%5.15e−03 66.49%7.65e−03 65.62%1.26e−03

NTT#07 46.61%2.27e−03 46.70%4.85e−05 46.70%8.74e−06 46.70%5.02e−05 46.70%7.94e−04 46.14%9.27e−03 44.34%8.90e−03

NTT#08 39.50%9.62e−03 41.34%8.08e−03 40.31%1.00e−02 41.35%5.40e−03 45.62%2.90e−04 39.03%4.00e−03 35.69%4.11e−03

NTT#09 44.54%5.11e−03 46.40%1.01e−03 44.95%3.97e−03 46.60%5.38e−03 56.09%8.55e−04 43.72%9.25e−03 44.12%4.51e−03

NTT#10 40.62%5.10e−03 41.08%9.51e−03 42.35%8.63e−03 41.30%1.32e−03 56.40%9.30e−04 39.53%4.96e−03 38.43%1.23e−03

NTT#11 40.15%1.21e−03 39.68%3.42e−03 40.54%1.20e−03 40.71%4.71e−03 56.34%2.60e−05 39.93%3.31e−03 39.27%5.05e−04

NTT#12 37.92%5.09e−03 37.78%6.64e−03 38.06%1.92e−04 38.06%3.84e−03 56.35%6.16e−04 36.57%1.96e−03 36.06%8.10e−04

1227.

Solvingthe

RW

Aproblem

NTT#01NTT#02NTT#03NTT#04NTT#05NTT#06NTT#07NTT#08NTT#09NTT#10NTT#11NTT#1230 35 40 45 50 55 60 65 70 75

DEPT

NSG

A-II

SPEA2


(a):

DE

PT



MO

-VN

S N

SGA

-II SPEA

2

(b):

MO

-VN

S



MO

-ABC

NSG

A-II

SPEA2

(c):M

O-A

BC



O-G

SA N

SGA

-II SPEA

2

(d):

MO

-GS

A



MO

-FA N

SGA

-II SPEA

2

(e):M

O-F

A

Fig

ure

7.1

0.

Com

parisonam

ongN

SGA

-II,SP

EA

2and

eachprop

osedM

OE

A(D

EP

T,

MO

-VN

S,M

O-

AB

C,

MO

-GSA

,and

MO

-FA)

byusing

them

edianvalue

ofH

ypervolum

eobtained

in30

independent

runsw

iththe

NT

Tnetw

ork.

7.2C

omparative

Study123



DEPT

MO

-VN

S

(a):

DE

PT

vs

MO

-VN

S



DEPT

MO

-ABC

(b):

DE

PT

vs

MO

-AB

C



DEPT

MO

-GSA

(c):D

EP

Tvs

MO

-GS

A



DEPT

MO

-FA

(d):

DE

PT

vs

MO

-FA



MO

-VN

S M

O-A

BC

(e):M

O-V

NS

vs

MO

-AB

C

Fig

ure

7.1

1.

Com

parisonby

pairsof

MO

EA

s(D

EP

T,

MO

-VN

S,M

O-A

BC

,M

O-G

SA,

andM

O-FA

)by

usingthe

median

valueof

Hyp

ervolume

obtainedin

30indep

endentruns

with

theN

TT

network

(1/2).

1247.

Solvingthe

RW

Aproblem



MO

-VN

S M

O-G

SA

(a):

MO

-VN

Svs

MO

-GS

A



MO

-VN

S M

O-FA

(b):

MO

-VN

Svs

MO

-FA



MO

-ABC

MO

-GSA

(c):M

O-A

BC

vs

MO

-GS

A



O-A

BC M

O-FA

(d):

MO

-AB

Cvs

MO

-FA



MO

-GSA

MO

-FA

(e):M

O-G

SA

vs

MO

-FA

Fig

ure

7.1

2.

Com

parisonby

pairsof

MO

EA

s(D

EP

T,

MO

-VN

S,M

O-A

BC

,M

O-G

SA,

andM

O-FA

)by

usingthe

median

valueof

Hyp

ervolume

obtainedin

30indep

endentruns

with

theN

TT

network

(2/2).


Table 7.9. Statistical Analysis among the MOEAs in the NTT network. The table indicates in whichdata sets two algorithms have no statistically significant differences.


DEPT MO-VNS 2 out of 12 data sets #01 #04MO-ABC 5 out of 12 data sets #01 #02 #04 #05 #06MO-GSA 2 out of 12 data sets #01 #04MO-FA 2 out of 12 data sets #01 #04NSGA-II 5 out of 12 data sets #01 #02 #04 #05 #06SPEA2 4 out of 12 data sets #01 #02 #04 #05

MO-VNS MO-ABC 3 out of 12 data sets #01 #04 #07MO-GSA 5 out of 12 data sets #01 #04 #05 #07 #08MO-FA 3 out of 12 data sets #01 #04 #07NSGA-II 2 out of 12 data sets #01 #04SPEA2 2 out of 12 data sets #01 #04

MO-ABC MO-GSA 4 out of 12 data sets #01 #04 #07 #12MO-FA 3 out of 12 data sets #01 #04 #07NSGA-II 5 out of 12 data sets #01 #02 #04 #05 #06SPEA2 4 out of 12 data sets #01 #02 #04 #05

MO-GSA MO-FA 3 out of 12 data sets #01 #04 #07NSGA-II 2 out of 12 data sets #01 #04SPEA2 2 out of 12 data sets #01 #04

MO-FA NSGA-II 2 out of 12 data sets #01 #04SPEA2 2 out of 12 data sets #01 #04

NSGA-II SPEA2 4 out of 12 data sets #01 #02 #04 #05

of MOEAs, including the standard multiobjective approaches NSGA-II and SPEA2. In addition,we notice that these differences of HV are more remarkable when we tackle a larger number ofdemands.

In Figure 7.10 we compare each multiobjective proposal with the well-known approaches NSGA-II and SPEA2. After analyzing the plots, we realize that, when we tackle a large optical network,our approaches obtain higher values of HV, concretely in the most complicated data sets.

Next, we present an illustrative pairwise comparison among the five multiobjective proposalswith the aim of studying the behaviour of each MOEA, see Figure 7.11 and Figure 7.12. As weexpect, the differences of HV between the MO-FA and any other MOEA are remarkable whenwe increase the number of demands. Furthermore, like in the previous experiments, the othermultiobjective swarm proposals (MO-ABC and MO-GSA) and the DEPT algorithm are able toobtain very promising results as well. Finally, we can see that the MO-VNS obtains similar valuesof HV to the MO-GSA.

Finally, in Table 7.10 we present a comparison by pairs of MOEAs by using the SC indicator. Aswe can see, all swarm proposals (MO-ABC, MO-GSA, and MO-FA) obtain a set of non-dominatedsolutions which is able to cover 100% of the non-dominated solutions obtained by the NSGA-IIand SPEA2 in all data sets. However, both NSGA-II and SPEA2 are able to cover less than 45%of the non-dominated solutions obtained by MO-ABC, MO-GSA, and MO-FA.

As occurred with COST239 and NSF, the multiobjective evolutionary algorithm based onfireflies behaviour (MO-FA) obtains better sets of non-dominated solutions (in terms of quality)than any of the other proposed MOEAs.


Table 7.10. NTT network. Comparison among the MOEAs by using the Set Coverage (SC) indicator,A B. Note that, SC represents the mean coverage of an algorithm A over an algorithm B in all thedata sets.

NTTA B #01 #02 #03 #04 #05 #06 #07 #08 #09 #10 #11 #12 SC

DEPT

MO-VNS 100% 100% 100% 100% 100% 50% 0% 0% 33.33% 75% 100% 100% 71.53%NSGA-II 100% 100% 100% 100% 0% 25% 100% 0% 100% 100% 100% 100% 77.08%SPEA2 100% 100% 100% 100% 0% 100% 100% 33.33% 100% 100% 100% 100% 86.11%MO-GSA 100% 100% 100% 100% 0% 0% 0% 0% 0% 0% 0% 0% 33.33%MO-ABC 100% 100% 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% 25%MO-FA 100% 0% 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% 16.67%

MO-VNS

DEPT 100% 33.33% 0% 100% 0% 0% 100% 100% 0% 0% 0% 20% 37.78%NSGA-II 100% 33.33% 0% 100% 0% 25% 100% 100% 0% 0% 0% 100% 46.53%SPEA2 100% 50% 50% 100% 0% 60% 100% 100% 0% 75% 100% 100% 69.58%MO-GSA 100% 33.33% 0% 100% 0% 0% 100% 50% 0% 0% 0% 0% 31.94%MO-ABC 100% 33.33% 0% 100% 0% 0% 100% 0% 0% 0% 0% 0% 27.78%MO-FA 100% 0% 0% 100% 0% 0% 100% 0% 0% 0% 0% 0% 25%

NSGA-II

DEPT 100% 100% 100% 100% 100% 100% 0% 50% 0% 0% 0% 0% 54.17%MO-VNS 100% 100% 100% 100% 100% 100% 0% 0% 33.33% 25% 50% 0% 59.03%SPEA2 100% 100% 100% 100% 100% 100% 100% 100% 66.67% 100% 100% 100% 97.22%MO-GSA 100% 100% 100% 100% 100% 0% 0% 0% 0% 0% 0% 0% 41.67%MO-ABC 100% 100% 0% 100% 100% 0% 0% 0% 0% 0% 0% 0% 33.33%MO-FA 100% 0% 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% 16.67%

SPEA2

DEPT 100% 66.67% 0% 100% 100% 0% 0% 0% 0% 0% 0% 0% 30.56%MO-VNS 100% 100% 100% 100% 100% 25% 0% 0% 33.33% 0% 0% 0% 46.53%NSGA-II 100% 66.67% 0% 100% 66.67% 0% 0% 0% 0% 0% 0% 0% 27.78%MO-GSA 100% 66.67% 0% 100% 66.67% 0% 0% 0% 0% 0% 0% 0% 27.78%MO-ABC 100% 66.67% 0% 100% 66.67% 0% 0% 0% 0% 0% 0% 0% 27.78%MO-FA 100% 0% 0% 100% 0% 0% 0% 0% 0% 0% 0% 0% 16.67%

MO-GSA

DEPT 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%MO-VNS 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%NSGA-II 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%SPEA2 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%MO-ABC 100% 100% 0% 100% 100% 0% 100% 100% 100% 0% 0% 33.33% 61.11%MO-FA 100% 0% 0% 100% 0% 0% 100% 0% 0% 0% 0% 0% 25%

MO-ABC

DEPT 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%MO-VNS 100% 100% 100% 100% 100% 100% 100% 66.67% 33.33% 100% 100% 100% 91.67%NSGA-II 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%SPEA2 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%MO-GSA 100% 100% 100% 100% 100% 100% 100% 50% 0% 100% 50% 75% 81.25%MO-FA 100% 0% 0% 100% 0% 0% 100% 0% 0% 0% 0% 0% 25%

MO-FA


7.3 Comparison with other works 127

7.2.4 Conclusions of the Comparative Study

In this section we have presented a comparative s tudy on different MOEAs for solving the Routingand Wavelength Assignment (RWA) problem.

We have evaluated five multiobjective metaheuristics: a multiobjective version of the Differen-tial Evolution (DEPT), a multiobjective trajectory-based approach based on the Variable Neigh-bourhood Search (MO-VNS), and three swarm intelligence algorithms based on the behavior ofhoney bees (Multiobjective Artificial Bee Colony, MO-ABC), on the law of gravity and mass inter-actions (Multiobjective Gravitational Search Algorithm, MO-GSA), and also on the flash patternof fireflies (Multiobjective Firefly Algorithm, MO-FA).

In order to study the goodness of these algorithms, we have evaluated their capabilities tosolve the RWA problem over three real-world optical networks, comparing their results with thoseobtained by two well-known multiobjective approaches: Fast Non-Dominated Sorting Genetic Al-gorithm (NSGA-II) and the Strength Pareto Evolutionary Algorithm 2 (SPEA2). The opticalnetworks tested in this study correspond with the Pan European Optical Network (COST239, Eu-rope, 11 nodes and 52 links), the National Science Foundation network (NSF, U.S.A., 14 nodes and42 links), and the Nippon Telegraph and Telephone network (NTT, Japan, 55 nodes and 144 links).For each optical network, we have designed twelve data sets with different number of demands,therefore a total of 36 data sets.

Our study has revealed that the usage of the proposed MOEAs for solving the RWA problemis very suitable. More concretely, there exist remarkable differences in the quality of the solutionsbetween our multiobjective approaches and the standard NSGA-II and SPEA2. As we may see inTable 7.11, the swarm approaches works much better (higher values of HV) because they generatenew individuals taking not only parts from its parents, but also from the rest of the population,increasing the richness of search as a result. We have also concluded that - for this particulartelecommunication problem - the use of swarm intelligence and the DEPT algorithm is speciallyadvisable when the number of demands increases. In addition, we have to highlight that theMO-FA is a very promising approach to deal with this telecommunication problem.

Table 7.11. Ranking of the MOEAs when tackling the RWA problem.

COST239 NSF NTT

1. MO-FA 72.33% 1. MO-FA 65.76% 1. MO-FA 61.36%2. MO-ABC 68.92% 2. DEPT 60.12% 2. MO-GSA 55.70%3. MO-GSA 68.67% 3. MO-ABC 59.71% 3. MO-ABC 55.58%4. DEPT 68.08% 4. MO-GSA 59.52% 4. MO-VNS 55.15%5. MO-VNS 66.87% 5. MO-VNS 59.37% 5. DEPT 55.07%6. NSGA-II 66.78% 6. NSGA-II 58.97% 6. NSGA-II 54.76%7. SPEA2 66.69% 7. SPEA2 58.46% 7. SPEA2 54.02%

7.3 Comparison with other works

In this section we present a comparison with other works. In [42], the authors make a comparisonbetween typical heuristics in the Telecommunication field and different varieties of MultiobjectiveAnt Colony Optimization algorithms (MOACO) for solving the RWA problem.

On the one hand, the typical heuristics presented in [42] are: 3SPFF (3-Shortest Path routing,First-Fit wavelength assignment), 3SPLU (3-Shortest Path routing, Least-Used wavelength assign-ment), 3SPMU (3-Shortest Path routing, Most-Used wavelength assignment), 3SPRR (3-Shortest


Path routing, Random wavelength assignment), SPFF (Shortest Path Dijkstra routing, First-Fitwavelength assignment), SPLU (Shortest Path Dijkstra routing, Least-Used wavelength assign-ment), SPMU (Shortest Path Dijkstra routing, Most-Used wavelength assignment), and SPRR(Shortest Path Dijkstra routing, Random wavelength assignment). On the other hand, the differ-ent varieties of MOACOs ([41] and [42]) are: BIANT (Bicriterion Ant), COMP (COMPETants),MOAQ (Multiple Objective Ant Q Algorithm), MOACS (Multi-Objective Ant Colony System),M3AS (Multiobjective Max-Min Ant System), MAS (Multiobjective Ant System), PACO (ParetoAnt Colony Optimization), and MOA (Multiobjective Omicron ACO).

With the aim of proving the goodness of the proposed multiobjective metaheuristics, we firstcompare the MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) and the Typical Heuris-tics [42]; then, we make a comparison between them and the MOACOs presented in [41] and [42].

In [42], the comparison is carry out by using the NTT network topology and the following setsof demands: NTT#02, NTT#03, NTT#04, and NTT#05; therefore, in these comparisons we haveused exactly the same topology and sets of demands. Unfortunately, the authors do not providethe necessary data to make a comparison with them using their proposed metrics. However, foreach set of demands, they provide the best Pareto front obtained by the best MOACO and bythe best Typical Heuristics. In this way, we can compare with them by using the Hypervolume(HV) and the Set Coverage (SC) indicators. Finally, with the aim of making a fair comparison,the stopping criterion in our MOEAs was established to 100 generations (like in [41] and [42]).

7.3.1 Comparison with Typical Heuristics

The typical heuristics tackles the static-RWA problem dividing it up into two subproblems: on theone hand solving the wavelength assignation, and on the other hand minimizing the number ofhops.

For solving the wavelength assignment subproblem, the authors propose the following heuristics:

• Random. The random wavelength assignment algorithm chooses one of the free availablewavelengths on all links randomly, with uniform distributions, to establish a connection.

• First-Fit (FF). This algorithm assumes that the wavelengths are arbitrarily ordered. Thefirst-fit algorithm checks the status of the wavelengths sequentially and chooses the firstavailable wavelength to establish a connection.

• Most-Used (MU). The free wavelengths that are used on the greatest number of links inthe network are chosen first to establish a connection.

Table 7.12. Summary of the Typical Heuristics published in the literature for the RWA problem.

Routing

Wavelength 3-shortest Shortest-PathAssignment Paths (3-SP) Dijsktra (SP)

First Fit (FF) 3SPFF SPFFLeast Used (LU) 3SPLU∗ SPLUMost Used (MU) 3SPMU SPMURandom (RR) 3SPRR SPRR∗ Best Typical Heuristics in all the data sets.


Table 7.13. Comparison between the best Typical Heuristics and the proposed MOEAs (DEPT, MO-VNS,MO-ABC, MO-GSA, and MO-FA) by using the HV indicator.

Best TypicalDEPT MO-VNS MO-ABC MO-GSA MO-FA Heuristics

NTT#02 70.73% 70.70% 70.75% 69.73% 71.13% 62.96%NTT#03 64.49% 65.07% 66.98% 63.96% 67.47% 63.18%NTT#04 70.87% 70.87% 70.87% 70.87% 70.87% 70.87%NTT#05 69.42% 68.78% 69.42% 68.79% 70.58% 66.81%

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Figure 7.13. Illustrative comparison between the proposed MOEAs and the best Typical Heuristics byusing the HV indicator.

• Least-Used (LU). The free wavelength that is used on the least number of links in thenetwork is chosen to establish a connection.

To minimize the number of hops, they make use of the following techniques:

• Shortest Path (SP) Dijkstra: this algorithm finds the shortest route from a given sourceto a destination in a graph. The route is a path whose cost is the least from a given sourceto the destination.

• K-Shortest Path (K-SP): K-shortest path algorithms find more than one route for eachsource and destination pair. K alternative paths provide flexibility in route selection.

Therefore, the authors in [42] presented a total of eight typical heuristics for solving the RWA,a summary of these heuristics is presented in Table 7.12. As is pointed in [42], the best approachis the 3SPLU in all the data sets tested; thus, we carry out the comparison with this heuristics.

On the one hand, we compare our multiobjective approaches and the best typical heuristics byusing the HV indicator. As we may observe in Table 7.13, the values of HV obtained by the bestheuristics are lower than those ones obtained by our five proposed MOEAs in almost all the cases.However, we realize that all of them obtain exactly the same value of HV in the easiest data set(NTT#04). In Figure 7.13, we present an illustrative comparison between the five MOEAs andthe best typical heuristics with the aim of visualizing easily the differences of HV among them.

Then, we present a comparison with the best typical heuristics by using the SC indicator. Aswe can see in Table 7.14, we only present the coverage relation between each algorithm and thebest typical heuristics and vice versa; however, we do not present the coverage relation amongour approaches because it was presented in the comparative study (see the previous section). The


Table 7.14. Comparison between the best Typical Heuristics and the proposed MOEAs (DEPT, MO-VNS,MO-ABC, MO-GSA, and MO-FA) by using the SC indicator.

A B NTT#02 NTT#03 NTT#04 NTT#05 SC

DEPT 100% 100% 100% 0% 75%MO-VNS Best 100% 100% 100% 0% 75%MO-ABC Typical 100% 100% 100% 0% 75%MO-GSA Heuristics 100% 100% 100% 0% 75%MO-FA 100% 100% 100% 100% 100%

DEPT 0% 0% 100% 0% 25%Best MO-VNS 0% 0% 100% 0% 25%Typical MO-ABC 0% 0% 100% 0% 25%Heuristics MO-GSA 0% 0% 100% 0% 25%

MO-FA 0% 0% 100% 0% 25%

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Figure 7.14. Sets of non-dominated solutions obtained by the best Typical Heuristics and the worstproposed MOEA (MO-GSA).

results of this comparison clearly points that our approaches clearly dominates the non-dominatedsolutions obtained by the best typical heuristics proposed in [42]. On the contrary, the best typicalheuristics does not cover any non-dominated solutions achieved by our MOEAs, except for thedata set NTT#04 (the easiest one, all the algorithms obtain identical value of HV). We highlightthe particular performance of the MO-FA in the NTT#05. In this way, it is able to dominate thetypical heuristics completely, whereas the coverage relation of the rest of MOEAs is 0% for thisdata set.


Finally, in Figure 7.14 we present the set of non-dominated solutions obtained by the besttypical heuristics (3SPLU) and the MOEA which obtains the lower value of HV in average (MO-GSA). As we may observe, the MO-GSA covers the 3SPLU heuristics in NTT#02 and NTT#03;however, we observe a tie in NTT#05. The same behaviour occurs with DEPT, MO-VNS, andMO-ABC. Note that, in this comparison we discard the NTT#04 due to both approaches obtainexactly the same set of non-dominated solutions.

On the whole, the proposed MOEAs work better than or equal to the best Typical Heuristicsproposed in [42], except for the MO-FA, which works better than the typical heuristics in all datasets. Therefore, we can say that our MOEAs overcome all of the eight heuristics proposed in [42].

7.3.2 Comparison with MOACOs

Ant Colony Optimization (ACO) is a metaheuristic inspired by the foraging behavior of ant colonies.The different MOACO algorithms implemented in [41] and [42] are summarized next:

• Multiple Objective Ant Q Algorithm (MOAQ). This algorithm maintains an ant colonyfor each objective. This way, in a problem with k objectives, it will use k colonies in chargeof optimizing each specific objective.

• Bicriterion Ant (BIANT). This algorithm uses two different pheromone matrices, one foreach objective. This way, it is expected that the different ants carry out searches in differentregions of the Pareto Front.

• Pareto Ant Colony Optimization (PACO). This algorithm is based on the use of severalk pheromone matrices, one for each objective. The best two ants for each objective updatethe corresponding pheromone matrix, carrying out an elitist update.

• Multi-Objective Ant Colony System (MOACS). This algorithm was implemented con-sidering two objectives; it uses a pheromone matrix and two different visibilities, one for eachobjective of the problem, together with a pseudo-random rule.

• Multiobjective Max-Min Ant System (M3AS). This algorithm extends the Max-MinAnt System to solve multiobjective problems. It uses a global pheromone matrix, whichmaintains pheromone information considering all objectives.

• COMPETants (COMP). This algorithm was used in bi-objectives problems, with twopheromone matrices and two visibilities.

Table 7.15. Summary of the MOACOs published in the literature for the RWA problem.

BIANT∗ Bicriterion AntCOMP COMPET AntsMOAQ Multiple Objective Ant Q AlgorithmMOACS Multiple Objective Ant Colony SystemM3AS Multiobjective Max-Min Ant SystemMAS Multiobjective Ant SystemPACO Pareto Ant Colony OptimizationMOA∗∗ Multiobjective Omicrom ACO∗ Best Typical MOACO in data set NTT#03∗∗ Best Typical MOACO in data sets NTT#02, NTT#04, and NTT#05


Table 7.16. Comparison between the best MOACO and the proposed MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the HV indicator.

DEPT MO-VNS MO-ABC MO-GSA MO-FA Best MOACO

NTT#02 70.73% 70.70% 70.75% 69.73% 71.13% 56.01%NTT#03 64.49% 65.07% 66.98% 63.96% 67.47% 57.52%NTT#04 70.87% 70.87% 70.87% 70.87% 70.87% 70.87%NTT#05 69.42% 68.78% 69.42% 68.79% 70.58% 64.79%

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Figure 7.15. Illustrative comparison between the proposed MOEAs and the best MOACO by using theHV indicator.

• Multiobjective Omicron ACO (MOA). The algorithm was implemented considering twoobjectives; it uses one pheromone matrix and two visibility, one for each objective. A Paretoset of non dominated solutions updates the pheromone levels.

• Multiobjective Ant System (MAS). It is a simple extension of the Ant System (AS) tomanage multiple objectives. It maintains one pheromone matrix with a visibility for eachobjective to be optimized; the ants are distributed in regions of the search space.

For further information about these multiobjective varieties of ACOs, please refer to [41] and[42]. In Table 7.15 we summarize the aforementioned MOACOs as well as showing the best MOACOproposed in [41] and [42] for each data set. As we can see, the BIANT algorithm is the MOACOwith best performance in data set NTT#03 and the MOA is the best one in data sets NTT#02,NTT#04, and NTT#05.

Like we did with the typical heuristics, in the first place we compare our five multiobjectiveproposals (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) with the best MOACOs by usingthe HV indicator (see Table 7.16). We can see that the differences of HV are remarkable in almostall data sets, except for data set NTT#04, where all the approaches obtain identical value of HV.In order to perceive these differences of HV among the approaches, in Figure 7.15, we illustratethe value of HV obtained by each algorithm.

A comparison between the best MOACO and each proposed MOEA by using the SC indicatoris presented in Table 7.17. Note that we only present the coverage relation between each algo-rithm and the best MOACO and vice versa. We do not present the coverage relation among ourapproaches because it was presented in the comparative study (see the previous section). In Table7.17 we may see the goodness of our proposed MOEAs. We realize that the sets of non-dominatedsolutions achieved by our MOEAs clearly cover the best MOACOs in all data sets (except for the


Table 7.17. Comparison between the best MOACO and the proposed MOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA) by using the SC indicator.

A B NTT#02 NTT#03 NTT#04 NTT#05 SC

DEPT 100% 100% 100% 50% 87.50%MO-VNS Best 100% 100% 100% 50% 87.50%MO-ABC MOACO 100% 100% 100% 50% 87.50%MO-GSA 100% 100% 100% 50% 87.50%MO-FA 100% 100% 100% 100% 100%

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MO-GSA 0% 0% 100% 0% 25%MO-FA 0% 0% 100% 0% 25%

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Figure 7.16. Sets of non-dominated solutions obtained by the best MOACO and the worst proposedMOEA (MO-GSA).

easiest data set, NTT#04; where all the approaches obtain identical value of HV).A final comparison between the best set of non-dominated solutions obtained for each data

set in [41] and [42], and the set of non-dominated solutions obtained by the MO-GSA (MOEAwith the lowest value of HV) is shown in Figure 7.16. As we may observe, the MO-GSA clearlydominates the best MOACO in NTT#02 and NTT#03; however, in NTT#05, there exist somenon-dominated solutions of the best MOACO that are not covered by the MO-GSA. Note that, inthis comparison we discard the NTT#04 because both approaches obtain the same set of solutions.


After performing an exhaustive comparison among our MOEAs and more than fifteen ap-proaches, we can conclude that the proposed MOEAs are very suitable for solving the RWA prob-lem.

7.4 Performance of the Parallel Approach

In this section we apply the parallel versions of the Differential Evolution with Pareto Tournaments(pDEPT) to the RWA problem with the aim of reducing the runtime of the algorithm, obtainingresults of identical quality in a reasonable amount of time. A detailed explanation of the parallelDEPT is presented in Chapter 6 (Section 6.8.1).

As we explained in Chapter 3, the performance of a parallel algorithm is commonly measuredby computing the speedup and efficiency. For further information about these metrics, please referto Section 3.3.5.

In this experiments, we perform 30 independent runs of each parallel version on a homogenouscluster which consists of 4 multi-core nodes, where each node is equipped with a 8 cores; thus, wehave a total of 32 cores. The stopping criterion was established to 100 generations. Furthermore,we have carried out the statistical analysis presented in Section 3.4 with the aim of determiningwhether the differences of speedup and efficiency are statistically significant with a confidence levelof 95% (p-value under 0.05).

Furthermore, to carry out these experiments, we have selected the corresponding first six sets ofdemands to the Nippon Telegraph and Telephone network: NTT#01-NTT#06, because it involvesthe use of a large network and those data sets tackled by other authors in the literature ([41] and[42]).

Since we have four 8-core nodes interconnected through the same data network, we presentdifferent comparisons among shared-memory (OpenMP), distributed-memory (MPI), and/or hy-brid shared/distributed-memory approaches (OpenMP+MPI). We therefore compare OpenMP orHybrid versions with the MPI version in different scenarios with 2, 4, 8, 16, and 32 cores.

The parameter configuration of the pDEPT algorithm is slightly different to the configurationpresented in Section 7.1, we increase the population size in order to reach 32 individuals (oneindividual per core). Therefore, the final configuration of the pDEPT in these experiments is:Population size (Ns)=32, Crossover probability (CR)=25%, Mutation rate (F )=50%, and Selectionscheme (S)=Best/1/Binomial.

In the first place, we run the sequential version of the DEPT algorithm in order to know theruntime of the algorithm when using a single-core. In Table 7.18 we present the mean runtime andthe standard deviation in 30 independent runs. As we may observe, the most complicated data setis NTT#03 in which runtime is 1229.81 seconds.

Table 7.18. Mean runtime and standard deviation for the sequential version of the DEPT algorithm in30 independent runs.

T1

NTT#01 582.051.27e+01

NTT#02 858.986.18e+00

NTT#03 1229.814.89e+00

NTT#04 591.342.15e+01

NTT#05 814.805.05e+00

NTT#06 955.324.82e+00

7.4 Performance of the Parallel Approach 135

Table 7.19. Mean runtime, speedup, and efficiency for the OpenMP and MPI versions of the pDEPTalgorithm with 2 cores in 30 independent runs. Note that we report the standard deviation of theruntime.

OpenMP MPI

T2 S2 E2 T2 S2 E2 SS∗

NTT#01 291.665.34e+00 1.996 99.78% 298.965.61e+00 1.947 97.35%

NTT#02 434.142.46e+00 1.979 98.93% 436.003.51e−01 1.970 98.51%

NTT#03 620.162.30e+00 1.983 99.15% 622.742.21e+01 1.975 98.74%

NTT#04 295.861.18e+01 1.999 99.93% 301.384.68e+00 1.962 98.11%

NTT#05 414.202.59e+00 1.967 98.36% 414.309.31e+00 1.967 98.34%

NTT#06 484.161.93e+00 1.973 98.66% 484.114.18e−01 1.973 98.67%

∗ SS: Are there statistically significant differences? Yes()/No()

Figure 7.17. Communication and computation time for the OpenMP and MPI versions of the pDEPTalgorithm with 2 cores.

We start the parallel experiments by comparing the performance of the OpenMP and MPIversions of the pDEPT algorithm with 2 cores.

As we may observe in Table 7.19, both approaches obtain very promising performance, over95% in all the cases; which leads to obtain the same quality results in nearly half of the sequentialruntime. However, according to the statistical analysis performed, the OpenMP version is slightlybetter than the MPI version in three out of the six data sets tested. Furthermore, in those datasets in which the OpenMP version is better than the MPI, its efficiency is close to the ideal (over99%).

Whereas the MPI version spends a few seconds in communication, the OpenMP does not spendany time in communications; see Figure 7.17. As we can see in Figure 7.17, we corroborate theslightly better parallel performance of the shared-memory version of the pDEPT algorithm.

In the next experiment, we set to 4 the number of cores in the multi-core system in order tocompare the OpenMP and MPI versions of the pDEPT.

If we focus on studying the speedup and efficiency of both approaches, we realize that, in thiscase, the OpenMP performs better than the MPI in all the data sets, see Table 7.20. As we maysee, the OpenMP reaches a mean efficiency near to 98%, whereas in the MPI version the efficiencyis around 94-95%.

In this experiments with 4 cores, we may observe that the communication time is higher thanwith only two MPI processes. In Figure 7.18(a), we present an illustrative comparison of the



OpenMP MPI

T4 S4 E4 T4 S4 E4 SS∗

NTT#01 148.424.07e+00 3.922 98.04% 153.831.22e−01 3.784 94.59%

NTT#02 219.071.48e+00 3.921 98.02% 225.732.69e−01 3.805 95.13%

NTT#03 313.141.37e+00 3.927 98.18% 323.967.88e−01 3.796 94.90%

NTT#04 152.864.64e+00 3.869 96.71% 155.267.65e−02 3.809 95.22%

NTT#05 208.471.19e+00 3.908 97.71% 214.954.57e+00 3.791 94.77%

NTT#06 243.641.48e+00 3.921 98.02% 252.151.88e−01 3.789 94.72%



OpenMP MPI

T8 S8 E8 T8 S8 E8 SS∗

NTT#01 75.611.81e+00 7.698 96.23% 81.178.24e−02 7.171 89.64%

NTT#02 109.565.58e−01 7.841 98.01% 116.002.77e+00 7.405 92.56%

NTT#03 156.215.34e−01 7.873 98.41% 163.835.46e+00 7.507 93.83%

NTT#04 76.203.05e+00 7.760 97.00% 81.509.89e−02 7.256 90.70%

NTT#05 104.015.15e−01 7.834 97.93% 111.081.67e−01 7.335 91.69%

NTT#06 122.366.70e−01 7.808 97.59% 129.609.98e−02 7.371 92.14%


(a): speedup (4 cores) (b): speedup (8 cores)

Figure 7.18. Communication and computation time for the OpenMP and MPI versions of the pDEPTalgorithm with 4 and 8 cores.

runtime of each parallel approach. We can see that the differences are clear in this case, as well asthe communication time worsens the performance of the MPI version.

On the whole, we may observe that the OpenMP version is able to run nearly four times faster,without losing quality in the non-dominated solutions obtained.

Now, we study the parallel performance of the shared-memory version and the distributed-


Table 7.22. Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions of thepDEPT algorithm with 16 cores in 30 independent runs. Note that we report the standard deviationof the runtime.

OpenMP+MPI MPI

T16 S16 E16 T16 S16 E16 SS∗

NTT#01 38.327.37e−01 15.189 94.93% 45.372.85e−02 12.829 80.18%

NTT#02 56.002.62e−01 15.339 95.87% 64.835.70e−02 13.250 82.81%

NTT#03 79.461.91e−01 15.477 96.73% 88.533.72e+00 13.891 86.82%

NTT#04 38.901.02e+00 15.203 95.02% 47.434.77e−02 12.467 77.92%

NTT#05 53.662.99e−01 15.185 94.90% 62.451.13e+00 13.047 81.54%

NTT#06 63.435.82e−01 15.060 94.13% 72.772.02e+00 13.128 82.05%


Figure 7.19. Communication and computation time for the OpenMP+MPI and MPI versions of thepDEPT algorithm with 16 cores.

memory version, with a higher number of cores (8 cores). In this experiment we employ a singlenode with 8 cores.

In Table 7.21 we report the mean rutime, speedup, and efficiency of both parallel approacheswhen performing 30 independent runs. As we may observe, in this case, the differences are remark-able. On the one hand, we can see that the speedup and the efficiency in the OpenMP versionremains constant (over 96%). However, in the MPI version, there exists a substantial decrease ofefficiency and speedup (around 90%), which is due to the time spent in communication among theeight MPI processes.

An illustrative comparison between the runtime of both parallel versions is shown in Figure7.18(b). As we may observe, the differences of runtime are clear in all the data sets.

Since the nodes in our system are equipped with 8 cores, we cannot use a pure OpenMPversion in the experiment with 16 cores. Therefore, we compare a hybrid OpenMP+MPI versionwith a pure MPI version when we use two interconnected nodes, a total of 16 cores. Note that, inthe hybrid version we have 2 MPI processes which are divided into 8 threads by using OpenMPdirectives.

In Table 7.22 we present the runtime, speedup, and efficiency when we use a hybrid version anda pure MPI version. As we can see, the MPI version spends much more time in communicationthan the hybrid version, in which only two MPI processes need to communicate between them. Inthis case, whereas the efficiency of the hybrid version is around 95% in all the cases, the efficiency in


Table 7.23. Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions of thepDEPT algorithm with 32 cores in 30 independent runs. Note that we report the standard deviationof the runtime.

OpenMP+MPI MPI

T32 S32 E32 T32 S32 E32 SS∗

NTT#01 19.674.26e−01 29.596 92.49% 29.597.06e−01 19.671 61.47%

NTT#02 28.766.30e−01 29.864 93.32% 38.651.08e+00 22.222 69.44%

NTT#03 40.451.04e−01 30.401 95.00% 50.221.86e+00 24.487 76.52%

NTT#04 20.046.52e−01 29.508 92.21% 29.814.51e−01 19.836 61.99%

NTT#05 27.481.38e−01 29.655 92.67% 37.035.28e−01 22.005 68.77%

NTT#06 33.551.09e+00 28.473 88.98% 42.491.14e+00 22.482 70.26%


Figure 7.20. Communication and computation time for the OpenMP+MPI and MPI versions of thepDEPT algorithm with 32 cores.

the pure MPI decreases to 80%. Figure 7.19 illustrates the amount of time spend by each parallelapproach in communication and computation. In this way, as we expected, the differences betweenthe hybrid version and the pure MPI version are clear.

The last experiments were performed by using the whole system, four multi-core systemsequipped with eight cores; thus, a total of 32 cores. Like in the previous experiment, we com-pare the parallel performance of the hybrid version with the pure MPI version.

As we can see in Table 7.23, the performance of the pure MPI version is much lower than thehybrid version in all data sets. In this case, we realize that, whereas the runtime of the hybridversion in the data set NTT#01 is 19.67s, the runtime of the MPI version is 29.59s, a difference of10s. Furthermore, the runtime in the best case for the hybrid version (NTT#03) is more than thirtytimes faster than the sequential version; however, in the best case of the MPI version (NTT#03)it is lower than 25 times faster.

In Figure 7.20 we may observe the substantial differences of runtime, as well as the amount oftime spend by each parallel approach in communications.

To summarize, in Table 7.24 we report the mean speedup and efficiency obtained by theOpenMP or OpenMP+MPI and by the pure MPI version in the six data sets. Note that, thepure OpenMP version may be considered a hybrid version in which we have 1 MPI process and 2,4, or 8 threads.

In Figure 7.21 we summarize the communication and computation time for the parallel versions


Figure 7.21. Summary of the communication and computation time for the OpenMP+MPI and MPI

versions of the pDEPT algorithm.

(a): speedup (b): efficiency

Figure 7.22. Summary of the mean speedup and efficiency obtained by the parallel versions of the pDEPTin all the data sets.

of the pDEPT.As we may observe in Table 7.24, the performance of the OpenMP+MPI version remains almost

constant independently of the number of cores; however, the efficiency of the pure MPI versiondecreases exponentially when the number of cores increases. In Figure 7.22, we illustrate thesedifferences of speedup and efficiency between the two parallel approaches.

On the whole, we can conclude that the hybrid version OpenMP+MPI of the pDEPT is verysuitable for solving NP-hard problems, such as the RWA problem. In this problem, we haveobtained a mean efficiency of 92.45% with 32 cores, which implies that we may obtain high qualityresults nearly 30 times faster.


Table 7.24. Summary of the mean speedup and efficiency for the OpenMP+MPI and MPI versions ofthe pDEPT algorithm in all the data sets.

OpenMP+MPI MPI

#Cores (c) Sc Ec Sc Ec

1 1 100% 1 100%2 1.983 99.14% 1.966 98.28%4 3.911 97.78% 3.796 94.89%8 7.802 97.53% 7.341 91.76%16 15.242 95.26% 13.102 81.89%32 29.583 92.45% 21.784 68.07%

8Solving the Traffic Grooming problem

In this chapter we apply diverse multiobjective evolutionary algorithms (MOEAs) for solving theTraffic Grooming problem.

These MOEAs are multiobjective variants of Differential Evolution (DE), Variable Neighbor-hood Search algorithm (VNS), Artificial Bee Colony Algorithm (ABC), Gravitational Search Al-gorithm (GSA), and Firefly Algorithm (FA); we will refer to them as DEPT, MO-VNS, MO-ABC,MO-GSA, and MO-FA, respectively. For a full description of these multiobjective algorithms pleaserefer to Chapter 6.

With the aim of proving the goodness of our proposals, we compare each MOEA with twostandard multiobjective approaches: the Non-Dominated Sorting Genetic Algorithm (NSGA-II[90]) and the Strength Pareto Evolutionary Algorithm 2 (SPEA2 [91]). Furthermore, in order tocheck the accuracy of the proposals, we make a comparison with several techniques published inthe literature that have dealt with the Traffic Grooming problem.

We start tuning the parameters of each MOEA with two optical networks and two small trafficmatrices, a total of 13 scenarios. Then, we study the performance of each MOEA when solvingfour optical networks with different sizes and different loads of traffic. In Chapter 5 we present afull description of the data sets used in the experiments.

Many authors have dealt with this telecommunication problem in the last years. In [63], theauthors proposed two efficient heuristics that have resulted to be reference methods for testing newapproaches, they are: Maximizing Single-hop Traffic (MST) and Maximizing Resource Utilization(MRU). A INtegrated Grooming PROCedure (INGPROC) based on an auxiliary graph model isreported in [66]. This problem has been also tackled by using the Clique Partitioning conceptin [55]. Other authors have solved this problem by using multiobjective optimization. A well-known Multiobjective Evolutionary Algorithm (MOEA) is presented in [79], the Strength ParetoEvolutionary Algorithm (SPEA). Therefore, we compare the best proposed MOEA with theseapproaches published in the literature.

Since Traffic Grooming is an NP-hard problem that has proven to be computationally in-tractable when the given topology contains a large number of nodes [63], one of the main moti-vations of this work is the proposal of a parallel multiobjective evolutionary algorithm based onthe Artificial Bee Colony algorithm [112] (MO-ABC), that not only solves the Traffic Groomingproblem in a reasonable time, but also in an efficient way.

141

142 8. Solving the Traffic Grooming problem

8.1 Parameter Tuning

In this section we describe the parameter adjustment (for the Traffic Grooming problem) of thefive multiobjective proposals: DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA. Furthermore,we tune the parameters of the standard NSGA-II and SPEA2. Finally, after the parameter tuningof each multiobjective approach, we present a brief comparison among them.

To adjust the parameters of the diferent MOEAs we have used the most widely tackled opticalnetworks in the literature [63]. The first one is a small network with six nodes (6-node), a capacity(C) per link of OC-48, and a traffic matrix with a total amount of traffic of 988 OC-1 units. Thesecond one is the large real-world National Science Foundation (NSF) network topology with 14nodes, a capacity of OC-192, and a total amount of requested traffic of 5724 OC-1 units. We havetested different scenarios over these topologies (varying the number of transceivers per node (T )and the number of wavelengths (W ) per link):

• 6-node: T=3,4,5,7 W=3 and T=3,4,5 W=4.

• NSF: T=3,4,5 W=3 and T=4,5,6 W=4.

Like in the RWA problem, to decide the best value of each parameter in each MOEA, wehave performed 30 independent runs and measured the Hypervolume (HV) of each run. In thisway, we compute the median HV in the 30 runs, and select the value with the highest median.Furthermore, we have conducted the statistical analysis presented in Section 3.4 with the aim ofensuring a statistical relevance. For further information about the data sets and their correspondingruntime, please refer to Chapter 5 and see Table 5.1 and Table 5.3. All the data sets were run byusing g++ (GCC) 4.4.5 on a 2.3GHz Intel PC with 1GB RAM.

In the Traffic Grooming problem, we use the Yen’s algorithm [10] to obtain the k shortest pathsbetween each pair of nodes (see Section 5.3). Therefore, we have used exactly the same value of kin all the approaches, k=10. In the following, we present the main parameters for each MOEA, aswell as the values tested. Note that, for each parameter in each approach, we have highlighted inbold the value with the highest median of HV (thus, the configuration used for that parameter).

• DEPT

– Population Size (Ns): 25, 50, 75, 100, 125, 150, 175, 200.

– Crossover probability (CR): 1%, 5%, 10%, 25%, 50%, 75%, 95%.

– Mutation rate (F ): 1%, 5%, 10%, 25%, 50%, 75%, 95%.

– Selection scheme (S):

1. Best/1/Exponential2. Rand/1/Exponential3. RandToBest/1/Exponential4. Best/2/Exponential5. Rand/2/Exponential6. Best/1/Binomial7. Rand/1/Binomial8. RandToBest/1/Binomial9. Best/2/Binomial

10. Rand/2/Binomial

8.1 Parameter Tuning 143

• MO-VNS

– Neighbourhood degrees (nmax): 5 degrees

– Mutation rate (F ): from 15% to 75% depending on the neighbourhood.

• MO-ABC


– Maximum Limit value (limitmax): 2, 5, 10, 15, 20, 25.

– Mutation rate (F ): 1%, 5%, 10%, 25%, 50%, 75%, 95%.

• MO-GSA


– Initial Gravitational Constant (G0): 101, 102, 103, 104, 105.

– Alpha (α): 2, 5, 20, 40, 60, 80, 100.

– Number of best agents (kbest): it starts with Ns best agents and decreases linearly toNs

4 throughout the execution.

• MO-FA


– Attractiveness (β0): 0.05, 0.1, 0.25, 0.5, 0.75, 1.

– Absorption coefficient (γ): 0.05, 0.1, 0.25, 0.5, 0.75, 1.

– Control parameter for exploration (α): 0.05, 0.1, 0.25, 0.5, 0.75, 1.

• NSGA-II


– Selection method: Binary tournament.



– Mutation probability (F ): 1%, 5%, 10%, 25%, 50%, 75%.

• SPEA2



– Crossover probability (CR): 1%, 5%, 10%, 25%, 50%, 75%, 95%.

– Mutation probability (F ): 1%, 5%, 10%, 25%, 50%, 75%.

After performing a parameter tuning of each algorithm, we obtain the best parameter configura-tion of each MOEA for solving the Traffic Grooming problem. In Table 8.1, we present the medianvalue of HV obtained by each MOEA with its corresponding best configuration. As we may observe,all the algorithms are well configured and obtain very promising results in all scenarios. However,we highlight the particular performance of those MOEAs based on swarm intelligence, such asMO-ABC, MO-GSA, and MO-FA (see Figure 8.1). The notation used int Table 8.1 is HVIQR,where HV is the median hypervolume and IQR is the interquartile range in 30 independent runs.


Table 8.1. Comparison among the MOEAs by using the Hypervolume (HV) indicator.

6-nodeT W DEPT MO-VNS MO-ABC MO-GSA MO-FA NSGA-II SPEA2

3 3 37.95%5.47e−03 38.16%9.50e−03 38.79%7.36e−03 37.76%8.28e−03 38.40%5.02e−03 36.16%2.93e−03 35.73%4.05e−03

4 3 48.89%8.28e−03 48.50%1.23e−04 49.93%8.87e−03 48.63%6.57e−03 49.07%4.90e−03 46.64%9.50e−03 45.88%6.31e−03

5 3 56.95%5.79e−03 55.52%8.60e−04 58.87%6.20e−03 57.11%1.25e−03 57.17%8.26e−03 54.41%3.43e−03 53.07%6.51e−03

7 3 64.81%2.69e−03 63.36%8.92e−03 69.08%2.04e−04 66.50%6.08e−03 66.38%3.93e−03 62.40%7.65e−03 60.68%4.00e−04

3 4 38.27%9.09e−03 38.20%4.68e−04 38.73%3.57e−03 37.83%2.68e−03 37.92%6.22e−04 36.20%7.11e−03 35.69%2.08e−03

4 4 49.02%5.11e−03 48.58%7.60e−03 50.01%8.38e−03 48.64%4.51e−03 48.60%6.05e−03 46.38%5.36e−03 45.95%6.27e−03

5 4 58.22%1.57e−03 56.56%2.92e−03 59.24%8.33e−03 57.85%6.22e−03 57.75%3.44e−03 55.82%7.12e−03 54.95%3.84e−03

NSFT W DEPT MO-VNS MO-ABC MO-GSA MO-FA NSGA-II SPEA2

3 3 29.10%5.65e−03 19.11%5.97e−03 29.92%7.14e−03 32.01%8.47e−03 26.89%8.18e−03 28.61%5.65e−03 28.36%9.77e−03

4 3 39.32%8.87e−03 32.65%8.69e−03 40.19%9.04e−03 42.48%3.70e−03 35.76%7.87e−03 38.55%2.62e−03 37.88%2.69e−03

5 3 48.78%4.38e−03 40.68%7.12e−03 48.87%3.70e−03 51.40%2.27e−03 44.09%4.64e−03 46.70%9.41e−03 46.32%1.63e−03

4 4 39.91%6.62e−04 26.04%4.07e−03 39.98%4.29e−03 42.24%1.58e−04 35.73%8.08e−03 38.35%6.05e−03 36.73%4.63e−03

5 4 48.03%5.00e−03 34.65%4.24e−03 49.09%7.36e−03 51.00%3.29e−03 44.29%1.19e−03 46.92%5.58e−03 46.22%9.31e−03

6 4 54.72%5.59e−04 42.57%8.93e−03 55.31%9.14e−03 57.61%4.35e−03 51.02%9.71e−03 53.34%8.21e−03 53.15%3.82e−03

3 4 5 6 7

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Figure 8.1. Illustrative comparison among the five proposed MOEAs and the well-known NSGA-II andSPEA2, by using the median HV in 30 independent runs


8.2 Comparative Study

This section is devoted to present a comparative study among diverse MOEAs when tacklingdifferent scenarios in the Traffic Grooming problem.

Since the aim of this section is to compare several MOEAs when dealing with the TrafficGrooming problem, we have used four network topologies with different number of nodes, and foreach one, a variety of traffic matrices with different amounts of traffic: small, medium, and large.

On the one hand, the four optical topologies tested are: The 6-node network (6-node, 6 nodesand 16 links), the European Optical Network (COST239, Europe, 11 nodes and 52 links), theNational Science Foundation network (NSF, USA, 14 nodes and 42 links), and the Nippon Telegraphand Telephone (NTT, Japan, 55 nodes and 144 links).

On the other hand, for each optical network, we have generated three traffic matrices (TM)or sets of low-speed traffic requests with small, medium, and large amount of traffic. Table 8.2reports the amount of traffic (in OC-1 units) for each optical network and traffic matrix.

Table 8.2. Amount of Traffic (in OC-1 units) and Runtime (in seconds) per Optical network

TM1 TM2 TM3 Runtime (s)

6-node 988 1976 3952 30COST239 3187 6272 12037 120NSF 5724 11448 22896 360NTT 77233 153307 309820 720

In this work, for assessing the performance of the MOEAs, we have used two well-knownmultiobjective indicators: the Hypervolume (HV) quality indicator [117] and the Set Coverage(SC) indicator [121]. For computing the HV is required to establish the points ideal and nadir. Inthis work, these points are computed for each scenario as:

ideal = ( Amount of T raffic, 1, 1 )nadir = ( 1, |N | ∗ T, maxHops )

where, maxHops corresponds with the maximum number of hops in the K shortest path; so,the value of maxHops is 6, 5, 9, and 18 for the network topologies 6-node, COST239, NSF, andNTT, respectively. For example, in COST239 (11 nodes) using the traffic matrix TM2 (6272 OC-1units) and two transceivers per node (T =2), the ideal and nadir points are (6272, 1, 1) and (1,2*11, 5), respectively.

For further details about the different scenarios and the network topologies, please refer toChapter 5. The number of independent runs of the MOEAs is 30 for each data set. All data setswere run using g++ (GCC) 4.4.5 on a 2.3GHz Intel PC with 1GB RAM.


Figure 8.2. The optical network topology: 6-node Network (6-node).

8.2.1 6-node Network

In the first place we start comparing the MOEAs by using the small 6-node network topology. Aswe may observe in Figure 8.2, this optical network consists of 6 nodes and 16 physical links whereeach physical link presents a capacity of OC-48 units.

In order to test the performance of the approaches, we have three different traffic matrices. Aswe can see in Table 8.2, the total amount of traffic for each one is: 988, 1976, and 3952 OC-1units. The available resources depend on the traffic matrix; therefore, we use a different numberof transceivers per node (T ) and a different number of wavelengths per link (W ), depending onthe amount of traffic (see Section 5.4).

In the first place, we start comparing the MOEAs by using the HV indicator. As we mayobserve in Table 8.3, Table 8.4, and Table 8.5, the MO-ABC algorithm obtains better values ofHV than the other MOEAs in almost all data sets.

We can see that the differences of HV are remarkable in those data sets with a low number ofwavelengths per link, that is to say, when the given resources are very limited.

In Figure 8.3, Figure 8.4, and Figure 8.5, we can also notice that the DEPT, MO-GSA, MO-ABC, and MO-FA clearly obtain higher quality results than the standard algorithms NSGA-II andSPEA2. Finally, we may see that the MO-VNS algorithm, the trajectory-based MOEA, obtainsthe lowest value of HV in almost all data sets; however, it performs better than the well-knownNSGA-II and SPEA2 with a small amount of traffic (TM1), see Figure 8.3.

An illustrative summary comparison among the proposed approaches is presented in Figure 8.6.In Figure 8.6 we can see the particular performance of the MO-ABC, as well as the clear differencesamong the rest of MOEAs. On the whole, the second and third best MOEAs are the MO-GSAand the DEPT algorithm respectively.

In Table 8.6, we may observe those data sets in which the differences of HV in 30 independentruns are statistically no significant. As we can see the results of HV obtained by the best MOEA(MO-ABC) are statistically significant in almost all data sets. Furthermore, we may say that theDEPT and the MO-GSA algorithms perform similar, that is to say, statistically no significantdifferences were found in 57 out of the 144 data sets tested for this small network topology.

A comparison using the SC indicator is reported in Table 8.7. As we can see, in Table 8.7 wepresent, for each pair of MOEAs, the average percentage of coverage taking into account W .

In the first place we can see that the best MOEA is the MO-ABC algorithm. On the one hand,we may observe that it covers above 90% of the non-dominated solutions obtained by the NSGA-II,SPEA2, MO-GSA, and MO-FA; as well as covering more than 70% of the front obtained by the


Table 8.3. 6-node network (TM1). Comparison among the MOEAs by using the Hypervolume (HV) indi-cator. The notation used is HVIQR, where HV is the median hypervolume and IQR is the interquartilerange in 30 independent runs.

Traffic Matrix 1T W DEPT MO-VNS MO-ABC MO-GSA MO-FA NSGA-II SPEA2

#1 1 1 13.63%8.62e−03 13.72%9.28e−03 13.76%4.43e−03 13.64%9.21e−03 13.67%2.39e−03 13.16%3.77e−03 13.00%7.54e−03

#2 2 1 26.05%6.16e−03 26.25%6.69e−03 26.57%3.93e−03 25.95%8.85e−04 25.90%6.12e−04 24.97%5.95e−04 24.63%8.96e−03

#3 3 1 34.96%1.32e−03 35.21%6.41e−03 36.90%6.46e−04 35.27%2.98e−03 35.42%3.23e−03 33.35%2.08e−03 32.33%7.27e−03

#4 4 1 39.29%6.91e−03 40.01%3.07e−03 42.73%8.49e−03 40.26%6.02e−03 41.28%7.38e−03 36.48%7.46e−03 36.58%2.49e−03

#5 5 1 41.80%1.27e−03 42.67%3.55e−03 46.15%1.71e−03 42.84%8.38e−03 45.00%1.82e−03 38.87%5.10e−03 38.79%6.28e−03

#6 6 1 43.45%3.00e−03 44.39%6.74e−03 48.44%3.58e−03 45.08%2.52e−03 47.45%5.28e−03 39.84%2.25e−03 40.24%3.03e−03

#7 7 1 44.61%1.28e−03 45.61%5.88e−03 50.00%2.05e−03 46.81%7.76e−03 49.18%1.25e−03 39.99%4.68e−03 41.26%2.14e−04

#8 8 1 45.48%9.67e−03 46.53%2.16e−03 51.17%4.42e−03 47.08%2.50e−05 50.47%6.15e−03 42.76%3.88e−03 42.01%8.93e−03

#9 9 1 46.88%5.22e−03 46.57%4.23e−03 52.11%4.30e−05 47.92%1.55e−03 48.18%8.86e−03 43.66%2.78e−03 42.37%8.69e−03

#10 10 1 47.44%6.95e−03 47.11%8.39e−03 52.84%9.67e−03 48.83%9.25e−03 48.79%7.24e−03 43.91%3.60e−03 42.83%9.39e−03

#11 11 1 47.90%1.43e−03 47.55%2.62e−03 53.43%3.96e−03 48.78%3.88e−03 49.29%1.06e−03 43.61%6.28e−03 43.20%2.17e−03

#12 12 1 48.28%4.70e−03 47.91%3.66e−03 53.90%7.65e−03 49.54%6.78e−03 49.71%6.27e−03 43.64%8.50e−03 43.52%1.35e−03

#13 1 2 13.63%3.91e−04 13.67%6.38e−03 13.77%5.94e−03 13.68%6.13e−03 13.65%5.72e−03 13.18%4.85e−04 12.91%7.94e−03

#14 2 2 26.21%6.38e−03 26.38%1.08e−03 26.70%8.34e−04 26.15%3.88e−03 26.31%1.25e−03 25.12%1.59e−03 24.79%1.72e−04

#15 3 2 38.02%1.99e−03 37.93%8.87e−03 38.75%4.72e−03 37.75%9.05e−03 37.40%9.23e−04 36.05%7.72e−03 35.50%1.09e−03

#16 4 2 47.03%4.80e−03 46.53%2.26e−03 48.49%6.89e−03 47.36%7.59e−04 47.33%9.88e−03 44.46%2.48e−03 43.66%6.27e−03

#17 5 2 52.60%1.63e−03 51.95%1.78e−04 54.47%5.98e−03 53.39%1.87e−03 53.31%2.29e−04 50.29%5.93e−03 49.29%2.64e−03

#18 6 2 56.41%9.93e−03 55.35%9.41e−03 58.49%9.48e−03 57.24%5.18e−03 57.16%5.39e−03 53.17%5.39e−03 52.51%3.50e−03

#19 7 2 58.97%6.97e−03 57.74%5.76e−04 61.18%9.01e−03 60.19%4.27e−03 59.95%3.86e−03 56.41%6.11e−03 54.79%1.05e−03

#20 8 2 60.87%2.22e−03 59.52%8.66e−03 63.29%3.27e−04 62.11%3.56e−03 62.03%8.67e−03 57.82%3.51e−03 56.48%9.13e−03

#21 9 2 62.30%4.78e−03 60.89%8.17e−03 65.01%5.67e−03 63.83%5.77e−03 63.63%8.51e−03 59.42%9.10e−03 57.79%8.80e−05

#22 10 2 63.44%7.33e−03 61.99%3.16e−03 66.20%8.79e−03 65.09%2.50e−03 64.91%6.90e−03 60.48%4.30e−03 58.84%3.75e−03

#23 11 2 64.37%8.48e−03 62.88%1.30e−03 67.19%8.59e−03 66.17%3.22e−03 65.95%7.16e−03 61.40%2.65e−03 59.69%2.54e−03

#24 12 2 65.14%2.29e−03 63.63%4.45e−03 68.09%6.06e−03 66.95%6.66e−03 66.82%7.03e−03 62.21%3.81e−03 60.40%6.56e−03

#25 1 3 13.66%2.13e−03 13.70%6.78e−03 13.77%7.43e−03 13.62%5.09e−03 13.66%4.75e−03 13.10%8.65e−03 12.90%7.40e−03

#26 2 3 26.47%1.64e−03 26.46%4.16e−03 26.73%8.36e−03 26.16%7.55e−03 26.09%6.90e−05 24.97%6.49e−03 24.73%7.79e−03

#27 3 3 37.95%5.12e−03 38.16%9.09e−03 38.79%9.27e−03 37.76%4.14e−03 38.40%6.72e−03 36.16%4.35e−03 35.73%3.44e−03

#28 4 3 48.89%9.61e−03 48.50%5.17e−03 49.93%3.36e−03 48.63%3.26e−03 49.07%1.47e−03 46.64%3.93e−03 45.88%6.01e−03

#29 5 3 56.95%9.02e−03 55.52%9.83e−03 58.87%9.09e−03 57.11%8.08e−03 57.17%4.46e−03 54.41%1.61e−03 53.07%8.39e−03

#30 6 3 61.40%4.78e−03 59.88%6.65e−03 64.89%5.15e−04 62.55%9.68e−03 62.33%6.63e−03 59.33%4.30e−03 57.45%6.49e−03

#31 7 3 64.81%1.24e−03 63.36%5.35e−03 69.08%2.04e−03 66.50%3.21e−03 66.38%7.85e−03 62.40%4.65e−03 60.68%8.84e−03

#32 8 3 67.35%7.52e−03 65.95%9.95e−03 72.44%9.89e−03 69.36%6.47e−03 69.41%8.36e−03 65.52%1.75e−03 63.09%4.26e−03

#33 9 3 69.22%2.07e−03 68.90%9.32e−03 74.58%9.25e−04 71.99%8.99e−03 72.19%7.67e−03 67.36%3.97e−03 65.04%7.07e−03

#34 10 3 70.80%6.52e−03 70.46%3.64e−03 76.78%6.49e−03 73.41%7.65e−03 74.08%4.62e−03 68.67%1.23e−03 66.54%2.53e−03

#35 11 3 72.09%9.89e−03 71.73%6.68e−04 78.27%5.66e−04 74.95%7.39e−03 75.62%6.81e−03 70.47%5.91e−03 67.74%1.24e−03

#36 12 3 73.16%1.31e−03 72.79%5.80e−03 79.80%5.11e−03 76.40%8.63e−03 76.91%2.85e−03 71.07%7.20e−03 68.78%6.68e−03

#37 1 4 13.57%1.16e−03 13.72%7.84e−03 13.77%7.22e−04 13.71%1.90e−03 13.70%8.99e−03 13.14%5.17e−03 13.05%2.92e−03

#38 2 4 26.31%8.10e−04 26.33%7.16e−03 26.77%4.71e−03 26.16%7.45e−03 26.32%3.13e−04 25.07%5.47e−03 24.71%9.97e−03

#39 3 4 38.27%3.71e−03 38.20%8.02e−03 38.73%3.07e−03 37.83%1.54e−03 37.92%7.49e−03 36.20%1.23e−03 35.69%7.90e−03

#40 4 4 49.02%9.95e−04 48.58%8.14e−04 50.01%9.08e−03 48.64%5.13e−03 48.60%8.60e−03 46.38%2.26e−03 45.95%6.56e−04

#41 5 4 58.22%7.41e−03 56.56%3.77e−03 59.24%5.10e−03 57.85%7.10e−03 57.75%6.37e−03 55.82%1.09e−03 54.95%5.62e−03

#42 6 4 64.54%3.85e−03 60.93%3.82e−04 65.72%6.99e−03 64.30%3.28e−03 64.19%8.52e−04 61.75%8.23e−04 60.87%5.04e−03

#43 7 4 69.04%3.86e−03 64.31%3.63e−03 70.33%2.31e−04 68.88%2.84e−03 68.75%5.60e−03 66.33%4.42e−03 64.89%1.23e−03

#44 8 4 72.36%9.67e−03 67.05%4.06e−03 73.62%8.79e−03 72.37%2.78e−03 72.15%8.11e−03 68.88%2.14e−03 67.87%7.50e−05

#45 9 4 74.93%4.80e−05 68.78%2.87e−03 76.22%5.35e−03 74.92%8.33e−03 74.78%4.75e−03 71.40%8.69e−03 70.17%2.77e−03

#46 10 4 76.98%1.90e−03 70.99%2.95e−03 78.35%5.55e−03 77.08%4.87e−03 76.87%8.56e−03 73.91%7.09e−03 72.00%4.45e−03

#47 11 4 78.64%7.80e−03 71.71%1.27e−03 80.11%9.47e−03 78.84%3.42e−03 78.58%7.77e−03 73.63%8.43e−03 73.50%7.38e−03

#48 12 4 80.03%4.75e−03 72.80%9.11e−03 81.37%5.90e−03 80.20%2.51e−03 79.99%4.33e−04 76.93%4.52e−03 74.74%7.97e−03




#49 2 2 14.53%4.59e−03 14.32%1.51e−03 14.53%3.01e−03 14.52%3.71e−03 14.53%9.08e−03 14.39%1.24e−03 14.22%4.69e−04

#50 4 2 28.32%9.35e−03 28.05%6.55e−03 28.62%7.23e−03 28.26%3.89e−03 28.02%2.21e−04 27.53%3.09e−03 26.90%5.63e−03

#51 6 2 36.69%4.51e−03 36.55%6.55e−03 38.15%8.01e−03 36.93%4.82e−03 36.26%1.51e−03 35.22%4.36e−03 34.81%5.30e−03

#52 8 2 40.19%1.74e−03 40.47%4.17e−03 43.26%5.51e−03 41.06%7.93e−03 40.37%3.88e−03 39.11%1.07e−03 38.65%7.31e−03

#53 10 2 42.48%8.19e−03 43.21%7.99e−03 45.90%3.39e−03 43.61%1.29e−03 42.79%7.43e−03 40.99%4.54e−03 40.79%8.98e−03

#54 12 2 43.99%2.94e−03 44.80%2.09e−03 47.49%4.29e−03 45.14%9.47e−03 44.39%4.56e−03 43.20%6.49e−03 42.20%4.04e−03

#55 14 2 45.07%6.91e−03 45.93%6.62e−03 48.34%2.60e−03 46.50%7.55e−04 45.53%9.26e−04 44.08%9.21e−03 43.21%3.52e−03

#56 16 2 45.87%3.35e−03 46.78%5.28e−03 50.51%5.45e−03 47.49%7.07e−03 46.39%8.99e−03 44.71%8.04e−03 43.96%5.77e−03

#57 18 2 47.02%7.84e−03 46.35%5.41e−03 50.63%2.20e−03 48.08%4.45e−04 46.83%5.72e−03 45.22%4.84e−03 44.75%9.58e−03

#58 20 2 47.54%2.19e−03 46.85%7.70e−03 51.35%2.97e−04 48.58%7.43e−03 47.36%4.47e−03 45.92%4.01e−04 45.23%1.57e−04

#59 22 2 47.95%8.95e−03 47.25%2.13e−03 51.38%1.44e−04 49.12%7.51e−03 47.78%8.50e−03 45.58%5.57e−03 45.63%1.10e−03

#60 24 2 48.30%3.33e−03 47.58%4.72e−03 52.94%1.25e−03 49.32%6.59e−03 48.14%3.52e−03 46.41%4.47e−03 45.95%3.25e−03

#61 2 4 14.53%6.84e−04 14.25%2.51e−03 14.53%4.99e−03 14.52%1.69e−03 14.53%4.29e−03 14.32%3.73e−03 14.14%5.57e−03

#62 4 4 28.72%1.31e−03 28.30%6.52e−04 28.90%8.28e−03 28.62%7.06e−04 28.45%7.39e−03 27.90%2.77e−03 27.63%9.72e−03

#63 6 4 41.26%6.94e−03 40.19%7.77e−03 41.92%3.44e−03 41.12%3.59e−03 40.44%2.40e−05 39.77%8.70e−03 39.17%3.72e−03

#64 8 4 49.77%5.05e−03 47.98%7.93e−03 50.77%7.03e−03 49.87%9.44e−03 48.99%1.95e−03 48.45%7.55e−03 47.48%2.12e−03

#65 10 4 54.94%8.20e−03 53.03%8.09e−03 56.10%6.37e−03 55.13%4.60e−03 54.41%3.56e−03 53.37%6.83e−03 52.19%3.39e−03

#66 12 4 58.24%8.72e−04 55.99%5.17e−03 59.55%6.23e−03 58.55%2.24e−03 57.88%9.43e−03 56.47%4.14e−03 55.30%7.45e−04

#67 14 4 60.59%6.40e−03 58.04%2.89e−03 61.99%8.86e−03 61.00%7.03e−03 60.35%7.07e−03 58.80%9.20e−05 57.51%4.61e−03

#68 16 4 62.34%7.99e−03 59.66%3.76e−04 63.76%3.56e−03 62.89%6.36e−03 62.19%5.91e−03 60.69%7.82e−03 59.16%7.64e−03

#69 18 4 63.71%9.69e−03 60.88%5.08e−03 65.22%5.02e−03 64.25%1.56e−03 63.62%8.86e−04 62.44%9.37e−03 60.44%5.84e−03

#70 20 4 64.79%7.79e−03 61.85%7.37e−04 66.26%4.51e−03 65.42%8.67e−03 64.77%2.29e−03 63.12%5.16e−04 61.46%2.84e−03

#71 22 4 65.68%6.94e−03 62.65%5.91e−03 67.23%5.96e−03 66.34%4.77e−03 65.70%8.74e−03 63.95%6.40e−03 62.30%9.34e−03

#72 24 4 66.42%5.00e−03 63.31%4.51e−03 68.14%1.73e−03 67.09%7.98e−03 66.48%5.72e−03 64.06%8.24e−03 62.99%2.68e−03

#73 2 6 14.53%7.16e−03 14.32%9.30e−03 14.53%6.65e−04 14.52%2.99e−03 14.53%2.48e−04 14.33%7.31e−03 14.21%7.18e−03

#74 4 6 28.69%7.13e−03 28.28%9.42e−03 28.93%5.74e−03 28.63%4.58e−03 28.54%5.19e−03 27.88%3.70e−03 27.52%6.23e−03

#75 6 6 41.65%7.83e−03 40.97%4.09e−03 42.45%7.61e−03 41.44%5.37e−03 40.52%9.88e−03 40.04%7.74e−04 39.35%4.44e−03

#76 8 6 52.80%1.42e−03 50.97%9.61e−04 53.97%1.63e−04 52.56%7.72e−03 51.29%5.05e−04 50.65%1.33e−03 49.99%3.42e−03

#77 10 6 60.20%5.64e−03 57.07%1.23e−03 61.79%4.34e−03 60.18%5.50e−03 58.28%1.81e−03 57.95%4.06e−03 57.03%3.87e−03

#78 12 6 64.36%7.65e−03 61.19%3.49e−03 67.26%3.17e−03 65.08%7.88e−03 63.09%6.85e−03 62.54%4.19e−03 61.50%1.24e−04

#79 14 6 67.78%7.24e−03 64.26%4.64e−03 70.71%8.01e−03 68.65%9.42e−03 66.59%8.42e−03 66.23%3.15e−03 64.87%6.06e−03

#80 16 6 70.34%1.26e−03 66.36%1.09e−04 73.81%8.44e−03 71.40%1.19e−03 69.21%3.30e−03 68.60%2.80e−03 67.39%7.22e−03

#81 18 6 73.04%9.23e−03 68.48%9.65e−03 75.91%6.13e−03 73.49%3.32e−03 71.49%7.69e−03 70.48%9.95e−03 69.33%7.55e−04

#82 20 6 74.61%6.89e−03 69.80%5.50e−03 77.80%9.18e−03 75.10%3.99e−03 73.14%2.31e−03 72.02%9.13e−03 70.87%6.72e−03

#83 22 6 75.89%5.57e−03 70.88%6.68e−03 79.01%4.55e−03 76.60%5.83e−03 74.49%5.40e−03 72.95%5.15e−03 72.12%3.47e−03

#84 24 6 76.96%7.56e−03 71.77%9.55e−03 80.27%5.86e−03 77.74%8.51e−03 75.61%9.81e−04 74.09%5.84e−03 73.16%3.95e−03

#85 2 8 14.52%7.05e−03 14.32%7.79e−03 14.53%2.61e−03 14.52%1.57e−03 14.53%4.73e−03 14.37%9.89e−04 14.21%7.80e−03

#86 4 8 28.67%7.77e−03 28.35%3.09e−03 28.93%5.92e−03 28.64%3.02e−03 28.58%3.23e−03 27.75%6.54e−03 27.20%1.67e−03

#87 6 8 41.66%5.65e−03 41.01%5.63e−03 42.39%7.51e−03 41.47%5.18e−03 40.74%1.93e−03 39.97%4.74e−04 39.44%8.71e−03

#88 8 8 52.96%8.21e−04 51.26%4.35e−03 53.98%4.09e−03 52.80%6.79e−03 51.26%4.29e−03 50.85%9.18e−03 50.35%7.65e−04

#89 10 8 61.32%2.07e−03 57.07%8.69e−03 62.55%8.25e−04 61.30%8.54e−03 59.60%6.26e−03 59.51%2.12e−03 58.74%9.67e−03

#90 12 8 67.09%7.00e−03 61.96%4.97e−03 68.19%7.48e−03 66.98%9.59e−04 64.96%3.49e−03 64.95%2.34e−03 63.51%6.46e−03

#91 14 8 71.10%4.14e−03 64.95%2.63e−03 72.19%3.58e−04 70.93%4.84e−03 69.02%9.95e−03 69.29%2.02e−04 67.16%6.38e−03

#92 16 8 74.10%6.27e−03 67.60%7.03e−03 75.08%2.10e−03 73.96%3.34e−03 72.09%1.12e−03 72.26%9.21e−03 69.88%3.36e−03

#93 18 8 76.42%1.98e−03 69.66%8.30e−03 77.54%5.66e−03 76.32%1.32e−03 74.47%7.73e−03 74.77%3.23e−03 72.00%5.22e−03

#94 20 8 78.27%1.47e−03 71.31%2.20e−03 79.41%5.25e−04 78.25%1.48e−03 76.37%6.15e−03 76.31%4.77e−03 73.69%3.13e−03

#95 22 8 79.79%6.35e−03 72.65%1.99e−03 80.98%9.10e−03 79.76%7.61e−03 77.93%3.10e−03 78.03%5.58e−03 75.07%3.30e−05

#96 24 8 81.05%1.82e−04 73.77%4.35e−03 82.37%1.43e−03 80.97%4.27e−03 79.22%8.25e−03 79.10%3.84e−03 76.21%8.19e−03




#97 3 3 10.75%5.58e−03 10.58%6.32e−03 10.90%9.22e−03 10.85%4.01e−03 10.87%7.60e−05 10.40%6.66e−03 10.25%2.57e−03

#98 6 3 21.02%9.75e−03 20.81%5.98e−03 21.52%1.43e−03 21.20%3.96e−04 21.15%9.97e−03 20.52%7.40e−03 20.14%9.20e−03

#99 9 3 28.08%5.77e−03 27.37%8.00e−03 29.27%8.28e−03 28.25%6.05e−03 27.82%1.41e−03 26.64%3.37e−03 26.68%3.31e−03

#100 12 3 31.34%9.48e−03 30.69%4.72e−03 33.25%7.87e−03 31.52%3.66e−03 31.11%1.86e−03 29.92%9.09e−03 29.22%8.17e−03

#101 15 3 33.32%9.83e−03 32.73%6.04e−03 35.31%8.11e−03 33.63%6.83e−03 33.09%4.34e−03 31.59%1.84e−03 31.01%6.82e−03

#102 18 3 34.63%1.07e−03 34.05%3.17e−03 36.73%6.74e−03 35.10%7.71e−03 34.41%4.72e−03 33.19%8.83e−03 32.19%3.27e−04

#103 21 3 35.56%4.49e−03 34.95%1.47e−03 37.96%7.30e−04 36.12%2.58e−03 35.35%7.54e−03 33.86%7.08e−03 33.04%4.89e−03

#104 24 3 36.26%7.94e−03 35.63%4.61e−03 39.13%1.10e−05 36.94%5.25e−03 36.05%8.43e−03 34.67%5.44e−04 33.67%1.00e−02

#105 27 3 36.63%6.59e−03 36.82%8.47e−03 39.51%4.06e−03 37.49%2.72e−03 36.97%3.12e−03 34.62%3.53e−03 34.40%4.42e−03

#106 30 3 37.05%7.94e−04 37.33%2.19e−03 40.39%5.53e−03 37.88%8.25e−03 37.42%3.76e−03 35.41%6.13e−03 34.78%8.34e−03

#107 33 3 37.40%6.63e−03 37.63%7.35e−03 40.11%8.73e−03 38.11%6.83e−03 37.79%4.04e−03 35.49%7.94e−04 35.09%7.24e−03

#108 36 3 37.69%2.36e−03 37.94%1.82e−03 41.82%3.57e−03 38.56%9.67e−03 38.10%1.09e−04 36.14%2.06e−03 35.34%2.56e−03

#109 3 6 10.79%8.60e−03 10.38%1.24e−03 10.90%2.70e−03 10.84%2.59e−03 10.88%6.75e−03 10.51%1.08e−03 10.30%6.48e−03

#110 6 6 21.13%6.67e−03 20.57%5.19e−03 21.54%8.38e−04 21.29%2.95e−03 21.24%1.90e−04 20.69%5.06e−03 20.30%8.60e−03

#111 9 6 31.33%2.19e−04 30.89%1.02e−03 31.97%3.44e−03 31.31%6.21e−03 30.92%8.12e−03 30.30%8.73e−03 29.96%9.07e−03

#112 12 6 39.66%6.94e−03 38.18%7.52e−03 40.52%5.05e−03 39.65%3.70e−04 38.36%1.26e−03 38.14%8.56e−03 37.95%2.11e−03

#113 15 6 44.06%6.70e−03 41.72%9.73e−03 45.69%3.03e−03 44.77%6.19e−03 43.47%7.12e−03 43.50%1.05e−03 42.86%6.61e−03

#114 18 6 47.58%4.90e−03 44.28%7.37e−03 49.23%5.76e−03 48.17%9.07e−03 47.16%1.89e−03 46.41%7.45e−03 46.03%2.47e−03

#115 21 6 49.88%2.64e−04 46.11%3.09e−03 51.66%5.85e−03 50.54%1.94e−03 49.57%4.30e−03 48.70%5.32e−03 48.20%6.45e−03

#116 24 6 51.60%3.32e−03 47.47%5.89e−03 53.55%8.20e−03 52.35%3.84e−03 51.37%4.70e−03 50.33%5.98e−03 49.83%1.49e−03

#117 27 6 52.94%4.69e−03 48.53%2.42e−03 55.05%6.94e−03 53.73%2.86e−03 52.76%5.75e−04 51.87%8.18e−03 51.09%5.32e−03

#118 30 6 54.00%1.39e−03 49.38%7.10e−03 56.20%1.50e−03 54.80%7.19e−03 53.88%5.74e−03 53.02%1.98e−03 52.10%7.37e−03

#119 33 6 54.88%6.21e−04 50.07%4.42e−03 56.91%5.55e−04 55.73%2.48e−03 54.79%3.83e−04 53.66%6.58e−03 52.93%1.42e−03

#120 36 6 55.60%7.06e−03 50.65%1.68e−03 57.77%9.51e−03 56.54%9.77e−03 55.56%8.14e−03 54.16%9.51e−03 53.61%6.14e−03

#121 3 9 10.70%4.92e−03 10.52%8.08e−03 10.90%3.90e−03 10.84%7.23e−03 10.87%5.82e−03 10.59%3.87e−04 10.46%8.78e−03

#122 6 9 21.18%3.40e−03 20.76%4.24e−03 21.56%5.00e−05 21.29%3.13e−03 21.49%6.51e−03 20.67%6.35e−03 20.37%6.16e−03

#123 9 9 31.35%8.60e−03 30.97%6.21e−04 32.08%3.12e−03 31.49%6.92e−03 30.92%1.09e−03 30.59%3.92e−03 29.92%9.31e−04

#124 12 9 41.24%4.17e−03 39.90%6.05e−03 42.14%3.69e−03 41.24%4.49e−04 40.17%4.74e−03 39.84%7.52e−03 39.35%6.89e−03

#125 15 9 48.55%9.23e−03 45.45%4.18e−03 49.72%3.29e−03 48.62%8.74e−03 47.09%2.18e−03 46.92%7.12e−03 46.27%4.48e−03

#126 18 9 53.41%5.20e−03 49.03%2.83e−03 54.67%3.15e−03 53.54%2.01e−03 52.12%8.29e−03 51.79%6.86e−03 51.21%2.09e−03

#127 21 9 56.86%1.39e−04 51.58%5.40e−03 58.35%2.38e−03 57.07%3.84e−03 55.56%2.60e−03 55.12%3.68e−03 54.52%7.01e−03

#128 24 9 59.44%2.52e−03 53.49%3.76e−03 60.87%7.41e−03 59.61%4.89e−03 58.14%7.95e−04 57.49%5.12e−03 57.18%7.84e−03

#129 27 9 61.22%9.29e−03 55.67%2.24e−03 62.97%5.44e−03 61.71%6.67e−03 60.21%1.08e−03 59.70%1.11e−04 58.64%1.11e−03

#130 30 9 62.79%9.09e−03 56.88%2.35e−03 64.83%2.60e−03 63.37%8.28e−03 61.86%6.23e−03 61.18%6.76e−03 60.11%5.33e−03

#131 33 9 64.07%5.02e−03 57.87%3.66e−03 66.18%1.29e−04 64.71%2.91e−03 63.21%1.32e−03 62.27%2.72e−03 61.31%9.90e−03

#132 36 9 65.14%1.50e−03 58.69%5.62e−03 67.30%6.32e−03 65.74%4.02e−03 64.34%5.16e−03 63.34%4.29e−03 62.29%8.41e−03

#133 3 12 10.81%7.22e−03 10.48%6.88e−03 10.90%9.69e−03 10.86%2.63e−04 10.85%3.78e−03 10.58%5.90e−03 10.31%3.42e−04

#134 6 12 21.14%2.38e−03 20.79%3.82e−03 21.57%4.62e−04 21.28%8.34e−03 21.32%7.60e−03 20.59%1.40e−03 20.30%5.56e−03

#135 9 12 31.41%5.35e−03 31.09%5.36e−03 32.00%4.84e−03 31.49%9.32e−03 31.02%1.39e−03 30.59%3.08e−03 30.18%3.19e−03

#136 12 12 41.28%4.47e−03 39.77%7.54e−03 42.10%2.20e−03 41.32%3.71e−03 40.17%9.91e−03 40.16%5.84e−03 39.44%9.37e−03

#137 15 12 50.39%3.57e−03 46.93%1.19e−03 51.38%5.90e−03 50.32%2.06e−03 48.30%9.75e−03 48.72%4.66e−03 47.86%5.91e−03

#138 18 12 56.56%7.51e−03 52.15%1.15e−03 58.05%5.07e−03 56.92%8.71e−03 54.86%2.50e−03 55.09%1.09e−03 54.23%4.75e−03

#139 21 12 60.95%1.81e−03 55.24%3.50e−03 63.44%9.54e−03 61.49%5.29e−03 59.42%9.35e−03 59.77%7.51e−03 58.43%1.27e−03

#140 24 12 64.24%2.44e−03 57.56%2.17e−03 66.65%1.16e−03 64.95%2.84e−03 62.83%3.56e−03 62.85%7.26e−03 61.55%3.53e−03

#141 27 12 66.79%7.03e−03 59.36%4.58e−03 69.41%6.30e−05 67.70%3.46e−03 65.48%3.12e−03 65.62%2.97e−03 63.99%9.42e−03

#142 30 12 68.83%1.36e−03 60.80%9.73e−03 71.56%6.27e−04 69.86%6.22e−03 67.60%4.96e−03 67.58%5.11e−03 65.93%6.17e−03

#143 33 12 70.49%5.24e−03 61.97%9.50e−03 73.43%7.15e−03 71.71%6.83e−03 69.33%7.07e−03 69.16%3.39e−03 67.51%4.96e−03

#144 36 12 71.88%4.60e−03 62.95%8.94e−03 75.17%2.10e−03 73.12%9.56e−03 70.77%6.94e−03 70.99%3.46e−03 68.83%6.72e−03


(a): DEPT (b): MO-VNS

(c): MO-ABC (d): MO-GSA

(e): MO-FA

Figure 8.3. 6-node network (TM1). Comparison among NSGA-II, SPEA2 and each proposed MOEA byusing the HV indicator. Note that, each point represents the mean of the medians of HV reported inTable 8.3 for W =1,2,3,4.




(e): MO-FA



(a): Traffic Matrix 1 (TM1) (b): Detail of TM1

(c): Traffic Matrix 2 (TM2) (d): Detail of TM2

(e): Traffic Matrix 3 (TM3) (f): Detail of TM3

Figure 8.6. 6-node network. Illustrative summary of the performance of each proposed MOEA by usingthe HV indicator. Note that, each point represents the mean of the medians of HV reported in Table8.3 (TM1), Table 8.4 (TM2), and Table 8.5 (TM3) for the different values of W .


Table 8.6. 6-node network. Statistical Analysis among the MOEAs in the 6-node network. The tableindicates in which data sets two algorithms have no statistically significant differences.


DEPT MO-VNS 13 out of 144 data sets #1 #13 #14 #15 #25 #26 #37 #38 #39 #51#97 #105 #121

NSGA-II 3 out of 144 data sets #49 #85 #121SPEA2 0 out of 144 data sets -MO-GSA 57 out of 144 data sets #1 #2 #13 #14 #25 #27 #29 #37 #38 #43

#44 #45 #46 #47 #48 #49 #50 #61 #62 #63#64 #65 #73 #74 #77 #85 #86 #87 #88 #89#90 #91 #92 #93 #94 #95 #96 #97 #98 #99#100 #109 #110 #111 #112 #121 #122 #123 #124 #125#126 #128 #133 #134 #135 #136 #137

MO-ABC 12 out of 144 data sets #1 #13 #25 #49 #61 #62 #73 #85 #97 #109#121 #133

MO-FA 39 out of 144 data sets #1 #2 #13 #14 #25#28 #37 #38 #45 #46 #47 #48 #49 #52 #57#58 #59 #60 #61 #68 #69 #70 #71 #72 #73#74 #85 #86 #97 #98 #109 #110 #117 #118 #119#120 #121 #133 #134

MO-VNS NSGA-II 13 out of 144 data sets #49 #61 #73 #85 #97 #109 #110 #112 #121 #122#124 #133 #134

SPEA2 9 out of 144 data sets #42 #49 #61 #73 #77 #85 #109 #121 #133MO-GSA 10 out of 144 data sets #1 #3 #5 #13 #15 #25 #28 #37 #38 #40MO-ABC 4 out of 144 data sets #1 #13 #25 #37MO-FA 19 out of 144 data sets #1 #13 #14 #25 #37 #38 #40 #50 #52 #62

#88 #105 #106 #107 #108 #111 #112 #123 #135

NSGA-II SPEA2 16 out of 144 data sets #1 #4 #5 #12 #25 #37 #47 #49 #59 #61#73 #85 #97 #99 #112 #121

MO-GSA 3 out of 144 data sets #49 #73 #85MO-ABC 2 out of 144 data sets #49 #85MO-FA 15 out of 144 data sets #49 #85 #89 #90 #92 #94 #95 #96 #113 #125

#136 #140 #141 #142 #143

SPEA2 MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 0 out of 144 data sets -

MO-GSA MO-ABC 11 out of 144 data sets #1 #13 #25 #37 #49 #73 #85 #97 #109 #121#133

MO-FA 43 out of 144 data sets #1 #2 #3 #10 #12 #13 #14 #16 #17 #18#20 #21 #22 #24 #25 #26 #29 #31 #32 #33#37 #38 #39 #40 #41 #42 #43 #45 #49 #61#62 #73 #74 #85 #86 #97 #98 #109 #110 #121#122 #133 #134

MO-ABC MO-FA 13 out of 144 data sets #1 #13 #25 #37 #49 #61 #73 #85 #97 #109#121 #122 #133

DEPT and the MO-VNS. On the other hand, we can see that the NSGA-II and the SPEA2 areable to cover only 4.27% and 2.97% of the non-dominated solutions achieved by the MO-ABCalgorithm.

Furthermore, the other approaches (DEPT, MO-VNS, MO-GSA, and MO-FA) only cover36.66%, 33.38%, 19.28%, and 16.45% of the Pareto front obtained by the MO-ABC, respectively.

Finally, we can also highlight the good performance of the other approaches (DEPT, MO-VNS,MO-GSA, and MO-FA) which are able to cover a high percentage of the solutions obtained by thestandard NSGA-II and SPEA2.


Table 8.7. 6-node network. Comparison among the MOEAs by using the Set Coverage (SC) indicator,A B. Note that, SC represents the mean coverage of an algorithm A over an algorithm B in all thedata sets.

TM 1 (988 OC-1 units) TM2 (1976 OC-1 units) TM3 (3952 OC-1 units)A B W=1 W=2 W=3 W=4 W=2 W=4 W=6 W=8 W=3 W=6 W=9 W=12 SC

DEPT

MO-VNS 53.60% 60.32% 57.30% 51.92% 54.32% 64.78% 64.24% 72.15% 53.89% 54.50% 51.77% 60.02% 58.23%MO-ABC 41.93% 47.13% 37.71% 34.31% 41.16% 38.15% 30.69% 27.54% 43.51% 32.97% 35.34% 29.50% 36.66%MO-GSA 78.17% 76.61% 73.54% 69.82% 74.16% 74.77% 67.92% 66.71% 78.83% 66.52% 71.72% 62.04% 71.73%MO-FA 83.19% 73.71% 65.95% 63.91% 88.16% 76.84% 87.49% 89.56% 85.36% 92.98% 88.42% 88.78% 82.03%NSGA-II 97.34% 98.13% 94.25% 95.52% 94.40% 94.79% 93.12% 88.61% 98.74% 93.01% 94.74% 90.41% 94.42%SPEA2 96.11% 97.49% 98.54% 93.21% 96.26% 97.47% 91.61% 96.43% 97.21% 93.64% 94.77% 90.38% 95.26%

MO-VNS

DEPT 66.23% 47.63% 49.83% 50.67% 66.08% 40.65% 38.36% 29.06% 59.46% 45.75% 37.51% 33.83% 47.09%MO-ABC 49.16% 44.78% 39.49% 36.99% 38.91% 25.59% 24.62% 20.67% 41.46% 30.75% 24.59% 23.53% 33.38%MO-GSA 82.93% 71.66% 70.30% 62.45% 76.02% 60.04% 56.61% 40.10% 71.41% 53.15% 45.18% 45.35% 61.27%MO-FA 82.56% 70.66% 59.71% 61.42% 81.80% 59.08% 68.50% 62.31% 76.20% 75.54% 71.19% 69.20% 69.85%NSGA-II 97.83% 92.58% 84.48% 81.96% 90.52% 83.99% 75.87% 58.65% 88.98% 73.85% 67.98% 66.63% 80.28%SPEA2 95.39% 95.08% 89.44% 83.54% 89.97% 87.57% 77.88% 66.40% 88.55% 75.64% 74.34% 70.07% 82.82%

MO-ABC

DEPT 73.24% 62.57% 67.07% 76.45% 80.65% 71.87% 77.02% 79.77% 81.14% 76.64% 67.67% 74.74% 74.07%MO-VNS 68.61% 64.72% 67.64% 66.68% 76.45% 79.97% 79.61% 85.16% 69.17% 64.72% 69.32% 67.50% 71.63%MO-GSA 93.80% 87.52% 90.67% 89.98% 92.20% 92.85% 92.41% 91.27% 91.09% 88.31% 87.67% 86.75% 90.38%MO-FA 90.66% 80.78% 75.22% 80.68% 96.23% 86.76% 97.28% 98.82% 94.59% 98.15% 97.44% 97.44% 91.17%NSGA-II 98.34% 98.98% 97.80% 99.42% 99.58% 99.17% 99.36% 98.78% 99.49% 98.82% 98.07% 98.84% 98.89%SPEA2 97.16% 99.85% 98.98% 97.33% 98.29% 99.69% 98.92% 99.65% 99.48% 98.32% 99.28% 98.06% 98.75%

MO-GSA

DEPT 35.44% 27.70% 30.03% 32.25% 42.17% 30.63% 35.32% 38.42% 40.86% 43.86% 32.26% 40.04% 35.75%MO-VNS 27.31% 32.45% 25.44% 31.58% 34.41% 35.43% 38.47% 51.65% 34.00% 34.21% 36.18% 39.82% 35.08%MO-ABC 18.69% 20.78% 16.35% 18.23% 23.40% 16.50% 15.07% 15.33% 27.64% 21.32% 19.88% 18.20% 19.28%MO-FA 68.02% 51.09% 49.97% 50.47% 75.23% 59.27% 79.14% 81.80% 80.18% 91.26% 87.47% 89.89% 71.98%NSGA-II 91.73% 91.73% 90.26% 89.80% 91.90% 91.00% 89.86% 88.32% 94.59% 92.93% 90.05% 90.34% 91.04%SPEA2 84.73% 93.46% 93.06% 84.39% 87.90% 90.24% 90.93% 89.61% 94.08% 91.01% 92.62% 90.44% 90.21%

MO-FA

DEPT 25.94% 24.24% 30.44% 38.82% 27.84% 29.58% 18.36% 16.04% 27.39% 12.72% 12.75% 11.82% 22.99%MO-VNS 23.86% 31.60% 32.64% 33.52% 27.33% 41.22% 24.68% 28.30% 26.23% 15.44% 14.65% 13.70% 26.10%MO-ABC 20.08% 22.37% 24.98% 22.85% 18.37% 23.14% 12.62% 10.70% 19.57% 7.93% 8.24% 6.50% 16.45%MO-GSA 41.06% 48.60% 46.82% 55.32% 39.94% 45.78% 23.99% 21.23% 44.33% 12.73% 16.58% 15.31% 34.31%NSGA-II 79.61% 82.84% 77.56% 84.81% 72.06% 72.05% 53.26% 44.39% 78.23% 36.57% 39.54% 38.34% 63.27%SPEA2 78.59% 87.63% 81.54% 84.87% 71.28% 80.27% 61.21% 51.72% 80.08% 41.17% 43.74% 43.09% 67.10%

NSGA-II

DEPT 8.24% 4.24% 7.13% 6.35% 14.61% 7.49% 9.32% 12.67% 8.84% 11.32% 6.91% 8.24% 8.78%MO-VNS 7.62% 7.08% 8.85% 9.99% 14.07% 8.83% 11.79% 23.74% 10.82% 10.20% 10.43% 12.66% 11.34%MO-ABC 5.82% 3.81% 5.27% 4.24% 8.16% 5.24% 3.90% 4.74% 3.28% 2.63% 2.54% 1.64% 4.27%MO-GSA 12.44% 9.68% 14.12% 11.89% 18.32% 13.15% 11.12% 13.90% 13.22% 10.10% 11.06% 8.81% 12.32%MO-FA 22.27% 13.61% 18.85% 13.65% 34.54% 26.26% 39.56% 48.74% 35.60% 60.41% 52.88% 54.71% 35.09%SPEA2 51.74% 57.45% 57.34% 47.08% 53.83% 56.94% 54.22% 57.70% 65.06% 52.75% 54.30% 54.53% 55.25%

SPEA2

DEPT 5.66% 2.50% 3.18% 3.40% 8.11% 5.71% 6.80% 5.00% 5.21% 6.23% 3.14% 4.30% 4.94%MO-VNS 5.49% 4.04% 5.06% 7.16% 11.94% 8.10% 9.55% 17.14% 9.16% 11.22% 5.91% 7.89% 8.55%MO-ABC 4.22% 2.08% 2.41% 2.87% 6.52% 4.86% 3.60% 2.80% 2.16% 1.81% 1.44% 0.86% 2.97%MO-GSA 11.07% 4.72% 5.07% 8.50% 12.43% 8.24% 7.99% 5.60% 7.62% 6.50% 5.24% 3.71% 7.23%MO-FA 17.23% 6.65% 12.18% 8.31% 25.58% 17.06% 29.37% 30.02% 20.27% 47.69% 34.52% 38.62% 23.96%NSGA-II 39.74% 26.29% 31.68% 36.52% 41.60% 31.33% 30.39% 28.16% 35.95% 36.00% 28.11% 30.30% 33.01%


Figure 8.7. The optical network topology: European Optical Network (COST239).

8.2.2 European Optical Network

The second optical network in the comparison is the European Optical Network (COST239). InFigure 8.7, we can see that this optical network consists of 11 nodes and 52 physical links. Fur-thermore, the capacity of each physical link is OC-96.

For this optical network topology we have also tested the MOEAs with different loads of traffic.As we can see in Table 8.2, for each matrix, TM1, TM2, and TM3, the total amount of trafficis 3187, 6272, and 12037 OC-1 units; respectively. In addition, we have used exactly the samenumber of transceivers (T ) and available wavelength channels per link (W ) as in the 6-node foreach traffic matrix (see Section 5.4); however, the capacity of the links is higher (OC-96).

In the first place, we compare the approaches by using the hypervolume indicator. In Table8.8, Table 8.9, and Table 8.10, we present the median value of HV obtained by each algorithm foreach data set.

As we can observe, like occurred in the 6-node network, the multiobjective approach thatachieves higher values of HV in all test instances is the MO-ABC algorithm.

In order to prove the goodness of each proposed MOEA (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA), diverse isolated comparisons are shown in Figure 8.8, Figure 8.9, and Figure8.10. In these illustrative comparisons, each point represents the mean of the medians of HVreported in Table 8.8 (TM1), Table 8.9 (TM2), and Table 8.10 (TM3) for the different values ofW .

As we may observe in Figure 8.8 (TM1), in general, all the algorithms perform better than thewell-known approaches. Concretely, the algorithms MO-ABC, MO-GSA, and DEPT clearly obtainhigher median of HV than both NSGA-II and SPEA2. In Figure 8.9 and Figure 8.10 (TM2 andTM3), we can see that the multiobjective approaches MO-VNS and MO-FA do not obtain as goodperformance as the well-known NSGA-II and SPEA2 when T is greater than 8 transceivers pernode. However, the other three MOEAs (MO-ABC, MO-GSA, and DEPT) overcome the medianHV of these standard approaches in all the scenarios.

In Figure 8.11, we can notice the great average performance of this algorithm (MO-ABC) basedon the behaviour of honey bees when dealing with this network topologies using different amountof traffic. The second best approaches are the MO-GSA and DEPT algorithms, which obtain verysimilar results.

We may also observe in Figure 8.11 how the performance of the trajectory-based MOEA (MO-VNS) decreases when the amount of traffic increases. We can notice that the MO-VNS obtains


Table 8.8. COST239 network (TM1). Comparison among the MOEAs by using the Hypervolume(HV) indicator. The notation used is HVIQR, where HV is the median hypervolume and IQR is theinterquartile range in 30 independent runs.


#1 1 1 10.23%4.52e−03 9.97%8.37e−03 11.05%3.04e−03 10.18%4.97e−03 10.77%7.38e−03 9.62%6.91e−03 9.26%9.47e−04

#2 2 1 18.52%5.10e−03 18.18%3.22e−03 20.28%2.31e−03 18.79%8.19e−03 19.06%4.81e−03 17.81%9.04e−03 17.53%7.35e−03

#3 3 1 27.12%4.29e−04 25.94%8.23e−04 29.02%7.96e−03 27.05%9.60e−03 26.93%4.97e−03 25.77%2.78e−03 25.03%5.18e−03

#4 4 1 34.28%8.93e−03 33.72%9.16e−03 37.12%1.50e−03 34.36%5.90e−03 33.54%2.69e−03 32.39%6.75e−03 32.04%2.23e−04

#5 5 1 38.46%8.84e−04 38.25%5.82e−03 43.00%9.04e−03 39.21%5.54e−03 37.94%9.21e−03 36.80%7.89e−03 36.19%8.65e−03

#6 6 1 42.00%1.44e−04 41.36%7.81e−03 46.84%8.50e−05 42.48%6.96e−03 40.73%7.29e−04 39.28%4.85e−03 38.71%8.73e−03

#7 7 1 43.86%7.62e−03 43.07%3.03e−04 49.96%5.37e−03 44.48%5.47e−03 43.88%9.16e−03 41.69%4.27e−03 40.94%2.87e−03

#8 8 1 45.33%7.48e−03 44.95%2.65e−03 52.90%9.39e−04 46.31%5.53e−03 45.73%3.17e−03 42.48%9.81e−03 42.39%2.25e−03

#9 9 1 45.21%6.48e−03 46.78%2.70e−03 53.74%9.57e−03 47.21%1.95e−03 45.27%9.17e−03 43.21%1.70e−03 43.47%4.09e−03

#10 10 1 46.03%9.49e−03 47.72%7.90e−03 54.75%9.47e−03 48.62%8.11e−03 46.28%2.11e−03 44.93%2.14e−03 44.35%7.39e−03

#11 11 1 46.70%5.87e−04 48.52%4.70e−05 55.55%1.07e−03 49.31%7.97e−03 47.11%6.90e−03 45.31%7.30e−03 45.05%1.09e−03

#12 12 1 47.26%4.30e−03 49.49%3.68e−03 56.43%3.69e−03 49.95%8.37e−03 47.80%4.23e−03 46.30%8.07e−03 45.65%1.57e−03

#13 1 2 9.68%6.02e−03 9.87%5.86e−03 11.13%7.97e−03 10.16%4.52e−03 10.74%4.72e−03 9.49%8.70e−05 9.34%7.21e−03

#14 2 2 18.35%6.97e−03 17.80%7.89e−03 20.30%8.11e−03 18.78%5.50e−03 19.17%7.10e−03 17.91%3.41e−03 17.51%2.04e−03

#15 3 2 27.57%3.09e−03 25.53%4.42e−03 29.12%6.51e−03 27.34%1.76e−03 26.56%2.28e−03 25.81%4.75e−03 25.53%8.02e−03

#16 4 2 36.11%5.99e−03 34.17%7.94e−03 37.89%3.33e−03 35.84%2.56e−03 33.57%6.91e−03 33.96%8.21e−03 33.00%1.14e−03

#17 5 2 43.33%5.22e−03 42.28%6.52e−03 46.11%2.42e−03 43.78%9.80e−05 41.32%9.03e−03 41.75%6.99e−03 40.95%9.77e−03

#18 6 2 50.46%2.57e−03 48.37%8.36e−03 52.37%3.51e−03 49.94%1.27e−03 47.17%6.71e−03 47.81%5.67e−03 47.42%7.62e−03

#19 7 2 54.62%2.52e−03 53.47%1.37e−03 57.00%1.60e−03 54.73%6.45e−03 51.43%5.92e−03 52.49%5.15e−03 52.12%6.64e−03

#20 8 2 58.38%1.11e−03 57.31%5.17e−04 60.38%6.43e−03 58.31%5.87e−03 55.46%9.51e−03 56.08%2.79e−03 55.66%6.69e−03

#21 9 2 60.97%1.93e−03 59.01%7.24e−03 63.36%9.50e−03 61.08%8.55e−03 58.09%3.93e−03 58.95%3.72e−03 58.41%4.27e−03

#22 10 2 63.52%5.17e−03 61.11%6.29e−03 65.38%6.84e−03 63.36%1.11e−03 60.75%2.91e−03 61.07%7.83e−03 60.66%8.42e−03

#23 11 2 65.30%9.59e−03 62.89%1.03e−03 67.05%1.72e−03 65.03%6.47e−03 62.68%5.71e−03 63.21%3.60e−03 62.44%9.63e−03

#24 12 2 66.77%3.98e−04 64.12%5.97e−03 68.81%5.29e−03 66.69%5.77e−03 64.28%3.65e−03 64.76%4.56e−03 63.98%7.21e−03

#25 1 3 10.16%1.80e−03 9.60%6.41e−04 11.09%1.33e−03 10.17%7.94e−03 10.72%8.21e−03 9.55%5.93e−03 9.41%5.95e−03

#26 2 3 19.11%7.36e−03 17.88%5.43e−03 20.29%8.25e−03 18.79%7.43e−03 19.32%5.85e−03 17.86%7.53e−04 17.49%3.37e−03

#27 3 3 27.38%9.02e−04 25.99%7.51e−03 29.25%8.37e−03 27.26%1.02e−03 26.58%6.79e−03 26.01%6.81e−03 25.76%6.72e−03

#28 4 3 35.94%9.62e−03 33.75%7.26e−03 37.88%4.47e−03 35.79%3.02e−03 34.10%9.83e−03 34.14%3.36e−03 33.70%5.18e−03

#29 5 3 44.14%4.76e−03 41.72%7.50e−03 45.95%1.45e−03 43.80%2.46e−03 41.23%3.00e−06 41.79%2.47e−03 41.20%2.35e−03

#30 6 3 49.62%6.35e−03 48.70%1.28e−03 52.18%5.44e−03 49.93%6.20e−05 47.57%6.74e−03 47.70%2.54e−03 47.68%5.03e−03

#31 7 3 54.94%1.37e−03 53.04%3.17e−03 57.23%8.03e−03 54.61%6.14e−03 51.99%8.58e−03 52.50%3.53e−03 51.86%7.30e−03

#32 8 3 58.57%2.58e−03 56.38%6.78e−03 60.49%2.31e−03 58.15%5.15e−03 56.14%4.92e−03 55.92%9.87e−03 55.50%3.27e−03

#33 9 3 60.94%4.20e−03 60.28%2.98e−03 63.40%1.41e−04 60.88%8.90e−03 59.05%6.08e−03 58.89%9.21e−03 58.69%5.98e−03

#34 10 3 63.11%5.06e−03 62.41%4.95e−03 65.54%8.56e−03 63.26%3.84e−03 60.92%1.52e−03 61.03%2.81e−03 60.66%9.01e−03

#35 11 3 64.96%8.03e−03 64.33%1.67e−03 67.13%6.03e−03 65.11%6.55e−03 62.84%1.90e−03 62.76%1.62e−03 62.53%7.16e−04

#36 12 3 66.50%7.63e−03 65.89%4.66e−03 68.80%5.67e−03 66.56%8.52e−03 64.43%3.77e−03 64.55%6.09e−03 64.09%1.30e−03

#37 1 4 9.96%9.20e−03 9.92%6.11e−03 11.16%2.49e−03 10.18%2.76e−03 10.90%9.36e−03 9.63%1.98e−03 9.36%7.02e−03

#38 2 4 18.94%5.34e−03 17.75%1.33e−03 20.27%7.77e−03 18.79%8.12e−03 19.26%7.42e−03 17.86%5.61e−03 17.29%2.79e−03

#39 3 4 27.41%1.90e−03 26.35%1.17e−03 29.12%7.51e−03 27.27%9.81e−03 26.64%9.74e−03 25.89%1.49e−03 25.31%4.93e−03

#40 4 4 35.59%4.19e−03 34.17%8.52e−03 37.83%4.41e−03 35.87%6.81e−03 33.66%4.16e−03 34.19%5.09e−03 33.63%3.11e−03

#41 5 4 43.51%5.83e−03 43.11%1.15e−03 46.00%6.01e−03 43.86%6.18e−03 40.98%3.49e−03 41.81%9.60e−04 41.23%6.73e−03

#42 6 4 50.46%7.74e−03 49.17%8.96e−03 52.39%5.83e−03 49.96%2.59e−03 47.05%3.76e−03 47.63%1.71e−03 47.33%5.16e−03

#43 7 4 54.90%1.73e−03 54.29%5.14e−03 56.90%6.24e−03 54.64%8.21e−03 51.71%1.13e−03 52.35%3.18e−04 51.74%6.77e−03

#44 8 4 58.61%2.63e−04 57.99%6.00e−03 60.58%6.94e−03 58.20%8.25e−03 55.99%3.93e−03 55.90%2.65e−04 55.68%8.93e−03

#45 9 4 61.42%4.22e−03 60.15%6.30e−03 63.44%4.36e−03 60.94%3.13e−03 58.37%8.20e−03 59.02%6.02e−03 58.53%9.28e−03

#46 10 4 63.12%2.56e−03 62.55%6.98e−03 65.50%8.61e−03 63.20%1.93e−03 60.72%7.05e−03 60.96%1.02e−03 60.81%1.06e−03

#47 11 4 64.95%6.96e−03 64.29%6.00e−03 67.35%5.67e−04 65.07%8.86e−03 62.64%4.97e−03 63.03%1.96e−03 62.80%2.57e−03

#48 12 4 66.48%5.22e−03 65.75%9.96e−03 69.05%2.96e−03 66.62%7.38e−03 64.24%7.54e−03 64.46%2.25e−03 64.32%3.50e−03




#49 2 2 13.22%7.86e−03 12.53%1.21e−04 14.44%4.40e−03 13.43%1.50e−03 13.79%6.38e−03 12.78%5.67e−03 12.48%8.59e−03

#50 4 2 24.29%2.16e−03 23.00%9.87e−03 26.09%4.44e−03 24.54%9.56e−03 23.33%7.81e−03 23.49%5.69e−03 23.22%8.12e−03

#51 6 2 35.16%8.33e−03 33.77%8.55e−03 36.59%9.37e−03 34.85%7.29e−03 32.56%3.92e−03 33.21%1.11e−03 32.78%1.78e−03

#52 8 2 42.28%3.56e−03 40.50%2.72e−03 44.72%1.21e−03 42.16%3.09e−03 39.62%6.45e−03 40.46%7.14e−03 39.74%8.47e−03

#53 10 2 46.09%8.56e−03 44.32%3.74e−03 50.03%2.05e−03 46.61%5.53e−03 43.53%4.84e−03 44.24%7.01e−03 43.37%8.26e−03

#54 12 2 49.16%8.94e−04 46.47%8.10e−03 53.46%2.18e−03 49.24%3.38e−03 46.43%4.32e−03 46.98%3.14e−03 46.19%4.73e−03

#55 14 2 51.06%6.49e−03 48.78%3.46e−03 55.48%4.65e−03 51.49%3.29e−03 48.50%5.57e−03 48.39%1.16e−03 48.04%1.45e−03

#56 16 2 52.49%3.54e−03 50.25%3.62e−03 57.12%1.20e−05 52.87%7.05e−03 50.05%7.06e−03 49.81%9.40e−03 49.29%4.92e−03

#57 18 2 53.24%2.15e−03 50.63%7.08e−03 58.78%3.37e−03 54.41%4.69e−04 50.67%5.84e−03 51.09%5.57e−04 50.48%8.61e−03

#58 20 2 54.10%1.30e−03 51.51%2.11e−03 60.10%1.18e−03 55.03%5.78e−03 51.60%7.53e−03 51.63%7.40e−03 51.30%8.15e−03

#59 22 2 54.81%8.79e−03 52.22%5.17e−03 60.77%1.95e−03 55.91%8.18e−03 52.36%6.41e−03 52.32%3.87e−03 52.02%1.30e−03

#60 24 2 55.39%3.08e−03 52.82%9.51e−03 61.56%5.47e−03 56.57%8.41e−04 52.99%4.02e−03 53.13%5.90e−03 52.58%6.89e−03

#61 2 4 13.41%1.26e−03 12.23%5.59e−03 14.40%7.73e−04 13.42%6.43e−03 13.98%3.70e−05 12.76%7.71e−04 12.36%1.46e−03

#62 4 4 24.80%5.65e−03 22.73%9.41e−03 26.07%3.64e−04 24.64%6.92e−03 23.52%9.08e−03 23.53%2.50e−04 23.13%4.79e−03

#63 6 4 35.31%4.51e−03 34.04%2.01e−03 37.26%1.72e−03 35.52%6.87e−03 32.79%3.90e−03 33.95%4.95e−03 33.32%4.48e−04

#64 8 4 45.68%3.53e−03 43.62%4.59e−03 47.01%6.00e−03 45.44%9.39e−03 42.55%9.38e−03 43.76%2.05e−03 42.47%5.06e−03

#65 10 4 52.77%1.68e−03 49.68%3.47e−03 54.13%2.79e−04 52.43%4.37e−03 49.31%8.75e−03 50.68%9.85e−03 49.61%3.76e−03

#66 12 4 57.60%4.40e−03 53.26%4.55e−03 58.72%4.51e−03 57.14%6.49e−03 54.36%1.97e−03 55.66%3.91e−03 54.22%5.77e−03

#67 14 4 60.73%7.24e−03 56.24%9.52e−03 62.27%6.93e−03 60.50%9.35e−03 58.01%3.82e−03 59.02%8.52e−03 57.78%9.14e−03

#68 16 4 63.27%7.74e−03 58.21%5.92e−03 64.77%2.58e−03 63.12%7.40e−05 60.71%6.48e−03 61.57%8.84e−03 60.45%2.23e−03

#69 18 4 65.24%3.01e−03 60.52%2.32e−03 66.90%3.29e−03 65.06%4.51e−03 62.81%9.79e−03 63.44%2.71e−03 62.61%2.09e−03

#70 20 4 66.81%1.15e−03 61.78%9.39e−03 68.22%4.21e−03 66.65%4.90e−05 64.48%2.37e−03 65.01%1.07e−03 64.08%4.77e−03

#71 22 4 68.10%3.86e−03 62.82%7.15e−03 69.58%8.18e−03 67.99%9.91e−04 65.85%8.94e−03 66.42%4.75e−03 65.32%5.57e−03

#72 24 4 69.17%1.54e−03 63.68%8.42e−03 70.62%8.84e−03 69.05%4.32e−03 67.00%8.69e−03 67.40%7.40e−05 66.59%8.14e−03

#73 2 6 13.23%5.77e−03 12.32%5.35e−04 14.42%9.84e−03 13.41%1.29e−03 13.89%5.11e−03 12.78%7.98e−03 12.44%2.44e−03

#74 4 6 23.37%7.35e−03 23.70%4.82e−03 26.08%9.09e−03 24.67%7.43e−03 24.02%6.69e−03 23.34%9.25e−03 22.97%6.11e−03

#75 6 6 35.67%4.88e−03 34.58%4.72e−03 37.16%9.82e−03 35.44%6.02e−03 33.33%7.01e−03 33.95%6.76e−03 33.30%8.58e−03

#76 8 6 45.35%5.48e−04 43.06%5.58e−03 47.18%2.40e−03 45.47%8.07e−03 42.49%3.52e−03 43.79%7.71e−03 42.45%1.31e−03

#77 10 6 52.77%3.62e−03 49.48%8.59e−03 54.09%7.82e−03 52.37%6.22e−03 49.44%2.40e−03 50.87%5.98e−04 48.74%4.88e−03

#78 12 6 57.26%1.34e−03 53.61%5.94e−03 58.91%9.52e−03 57.20%1.20e−03 54.36%4.29e−03 55.67%9.79e−03 53.92%7.91e−03

#79 14 6 60.82%4.76e−03 56.34%2.34e−03 62.14%1.52e−03 60.60%8.91e−03 57.85%5.29e−04 59.01%4.03e−03 58.08%1.22e−03

#80 16 6 63.35%3.01e−03 58.37%3.40e−03 64.78%8.69e−03 63.13%1.97e−03 60.54%4.64e−03 61.63%2.94e−03 60.84%3.96e−03

#81 18 6 65.30%4.35e−03 60.75%3.90e−03 66.86%7.83e−03 65.14%4.84e−03 62.38%5.22e−03 63.55%1.64e−03 62.73%4.36e−03

#82 20 6 66.89%1.20e−05 62.05%9.57e−03 68.31%1.79e−03 66.76%3.40e−04 64.04%4.68e−03 65.03%4.18e−03 64.37%6.97e−03

#83 22 6 68.18%9.47e−03 63.12%3.59e−03 69.44%7.45e−03 68.07%5.77e−03 65.40%9.94e−03 66.47%2.51e−03 65.74%9.73e−03

#84 24 6 69.26%1.18e−03 64.00%4.76e−03 70.66%9.90e−03 68.99%4.75e−04 66.53%9.98e−03 67.32%4.23e−03 66.89%6.57e−03

#85 2 8 13.25%4.64e−03 12.58%1.16e−03 14.46%7.40e−03 13.44%5.74e−04 13.71%1.57e−03 12.78%9.05e−03 12.54%9.96e−04

#86 4 8 24.56%6.58e−03 23.78%5.08e−03 26.12%5.69e−03 24.59%4.19e−03 23.91%8.65e−03 23.65%5.38e−03 23.07%8.19e−03

#87 6 8 35.40%9.38e−03 34.64%9.02e−03 37.15%3.41e−04 35.53%3.88e−03 33.27%3.68e−03 34.00%2.66e−03 33.19%2.80e−03

#88 8 8 45.39%2.67e−03 43.93%8.27e−03 47.25%4.46e−03 45.46%5.46e−03 42.19%5.88e−03 43.75%7.87e−03 42.43%3.02e−03

#89 10 8 52.44%1.34e−03 49.17%1.94e−03 54.15%6.47e−03 52.42%1.38e−03 49.33%4.55e−03 50.56%3.31e−03 49.28%7.54e−03

#90 12 8 57.15%1.22e−03 53.05%9.02e−04 58.83%1.04e−03 57.23%8.02e−03 54.16%7.85e−03 55.41%4.72e−04 54.03%1.98e−03

#91 14 8 61.19%9.68e−03 55.58%6.01e−03 62.54%3.33e−03 60.58%7.08e−03 57.61%3.08e−03 58.80%1.36e−03 58.06%9.42e−03

#92 16 8 63.73%2.58e−03 57.56%1.46e−03 64.73%7.86e−03 63.10%3.10e−03 60.29%3.64e−03 61.55%1.07e−03 60.81%1.96e−03

#93 18 8 65.70%8.80e−03 59.12%7.45e−03 66.83%9.16e−03 65.13%8.65e−03 62.37%5.94e−03 63.69%4.43e−03 62.78%1.47e−03

#94 20 8 67.28%9.89e−03 60.36%5.62e−03 68.40%7.56e−03 66.77%5.37e−03 64.05%7.45e−03 65.07%2.49e−03 64.39%2.29e−03

#95 22 8 68.56%5.63e−03 62.52%7.44e−03 69.60%9.76e−04 67.94%9.81e−03 65.42%4.98e−03 66.42%8.46e−03 65.72%3.83e−03

#96 24 8 69.64%7.33e−03 63.36%2.02e−03 70.83%1.90e−03 69.03%1.59e−04 66.56%6.65e−04 67.39%2.73e−03 66.88%5.51e−03




#97 3 3 11.79%9.45e−03 9.60%7.68e−03 12.54%7.08e−03 11.87%6.21e−03 11.64%6.58e−03 11.31%2.50e−05 10.95%8.60e−05

#98 6 3 21.88%4.08e−03 19.74%3.06e−03 23.37%7.09e−03 22.22%4.20e−03 20.81%2.25e−04 21.15%6.84e−03 20.78%6.08e−03

#99 9 3 31.34%5.14e−03 29.46%3.40e−03 33.13%4.71e−03 31.52%6.17e−03 29.46%5.59e−03 30.19%8.30e−04 29.18%5.99e−03

#100 12 3 37.13%6.04e−03 35.23%3.39e−03 39.37%5.42e−03 37.68%2.54e−03 35.28%3.05e−03 35.96%5.75e−03 35.38%3.38e−03

#101 15 3 40.78%8.43e−04 39.09%3.08e−03 43.36%5.18e−03 41.09%6.07e−03 39.18%4.69e−03 39.05%4.24e−03 38.55%6.19e−03

#102 18 3 43.05%1.25e−04 41.47%6.14e−03 45.82%9.53e−03 43.62%2.29e−03 41.66%4.56e−03 41.43%9.22e−03 40.71%5.16e−03

#103 21 3 44.62%1.56e−03 43.21%4.94e−03 47.72%9.64e−03 45.18%4.47e−03 43.43%7.51e−03 43.11%2.73e−03 42.27%6.75e−03

#104 24 3 45.80%5.05e−03 44.44%6.04e−03 49.10%2.04e−03 46.53%6.47e−03 44.75%1.22e−03 44.35%4.07e−03 43.46%9.05e−03

#105 27 3 47.30%3.60e−03 44.51%5.23e−03 50.04%3.05e−03 47.51%1.02e−04 45.49%3.73e−04 45.28%5.66e−04 44.64%3.57e−03

#106 30 3 48.06%5.05e−03 45.14%1.18e−03 50.91%3.82e−03 48.27%2.48e−03 46.28%4.64e−03 45.65%7.08e−03 45.43%8.54e−03

#107 33 3 48.68%1.42e−03 45.65%9.97e−03 51.31%8.43e−03 48.88%2.07e−03 46.93%8.58e−03 46.06%4.50e−03 46.09%7.60e−03

#108 36 3 49.20%5.92e−04 46.10%2.12e−03 52.00%9.59e−03 49.27%9.08e−03 47.47%4.52e−03 46.97%9.41e−03 46.60%9.74e−03

#109 3 6 11.48%5.39e−03 10.01%4.00e−03 12.47%3.36e−03 11.85%8.43e−04 11.72%5.39e−03 11.36%4.88e−03 11.01%6.76e−03

#110 6 6 22.01%4.21e−03 20.28%3.96e−03 23.42%5.42e−03 22.23%6.99e−03 21.00%1.45e−03 21.05%8.61e−03 20.66%8.87e−03

#111 9 6 32.16%6.78e−03 30.03%5.11e−03 33.70%8.06e−03 32.23%6.42e−03 30.09%2.42e−03 30.87%6.83e−03 29.49%5.78e−03

#112 12 6 41.34%3.85e−03 38.48%8.12e−03 43.08%8.91e−03 41.42%5.54e−03 38.98%2.99e−03 39.78%7.74e−03 38.32%8.72e−03

#113 15 6 48.86%2.17e−03 42.74%5.93e−04 50.73%5.14e−03 49.09%4.93e−03 46.22%4.31e−03 47.29%4.41e−03 45.62%6.93e−03

#114 18 6 53.79%9.67e−03 47.45%7.07e−03 55.97%7.58e−03 54.16%8.69e−03 51.59%4.91e−03 52.45%8.21e−03 51.07%7.03e−03

#115 21 6 57.81%6.47e−03 49.89%7.22e−03 59.32%7.55e−03 57.84%2.10e−03 55.33%1.32e−03 56.02%4.35e−04 54.79%2.95e−03

#116 24 6 60.53%8.68e−03 52.22%8.46e−03 62.06%1.35e−03 60.55%1.76e−03 58.14%8.57e−03 58.80%2.88e−03 57.53%6.38e−03

#117 27 6 62.63%9.58e−03 53.52%4.03e−03 64.12%2.48e−03 62.67%2.44e−03 60.32%5.48e−03 60.82%6.55e−03 59.71%9.64e−03

#118 30 6 64.32%4.36e−03 54.55%1.12e−03 65.74%8.42e−03 64.33%9.72e−03 62.07%5.18e−03 62.61%2.55e−03 61.45%4.72e−03

#119 33 6 65.69%8.05e−03 55.40%2.66e−03 67.15%8.68e−03 65.80%1.78e−03 63.49%3.18e−03 64.06%5.84e−03 62.70%9.63e−04

#120 36 6 66.84%2.27e−03 56.11%5.16e−03 68.24%3.00e−04 66.93%1.70e−03 64.68%1.84e−03 64.98%2.15e−03 64.13%3.13e−03

#121 3 9 11.33%8.90e−04 9.59%8.02e−03 12.52%6.53e−03 11.85%1.10e−03 11.61%6.18e−03 11.37%9.61e−03 11.08%4.71e−03

#122 6 9 21.72%2.70e−03 20.48%1.72e−03 23.38%2.93e−03 22.22%7.38e−03 20.91%7.04e−03 21.29%5.67e−03 20.50%5.30e−03

#123 9 9 31.80%8.76e−03 30.16%4.92e−03 33.68%3.35e−03 32.13%1.22e−03 30.18%2.72e−03 30.84%2.68e−03 29.83%6.70e−03

#124 12 9 41.55%5.27e−03 38.14%1.52e−03 43.00%6.89e−04 41.42%4.17e−04 39.11%7.88e−03 39.79%4.17e−03 38.55%7.35e−04

#125 15 9 49.29%5.92e−04 42.63%7.85e−03 50.68%5.14e−03 49.12%4.72e−03 46.31%8.38e−03 47.29%7.09e−03 45.13%1.63e−03

#126 18 9 53.95%4.89e−03 47.52%6.64e−03 55.82%4.10e−03 54.24%4.10e−05 51.94%4.78e−03 52.69%5.52e−03 50.99%1.85e−03

#127 21 9 57.63%2.28e−03 50.07%4.04e−03 59.41%8.75e−03 57.91%3.56e−03 56.01%2.20e−03 56.21%1.52e−03 54.79%4.97e−03

#128 24 9 60.37%3.91e−03 51.97%9.64e−03 62.02%6.31e−03 60.67%8.26e−03 58.87%1.20e−03 58.88%2.71e−03 57.73%7.96e−03

#129 27 9 62.65%8.30e−03 55.10%5.19e−03 64.15%7.94e−03 62.72%5.65e−03 60.45%5.07e−03 60.98%4.27e−03 60.15%3.57e−03

#130 30 9 64.34%9.85e−03 56.32%5.84e−03 65.97%4.62e−03 64.49%1.58e−03 62.19%2.49e−03 62.72%9.83e−03 62.04%1.65e−03

#131 33 9 65.73%2.52e−03 57.31%3.53e−03 67.35%7.48e−03 65.93%8.90e−03 63.63%1.34e−04 64.26%6.26e−03 63.42%8.82e−03

#132 36 9 66.89%5.11e−03 58.14%5.80e−03 68.40%1.51e−03 67.02%6.27e−03 64.83%8.49e−04 65.41%8.68e−03 64.73%7.17e−03

#133 3 12 11.57%5.02e−04 9.79%7.58e−03 12.48%2.24e−04 11.88%4.18e−03 11.57%5.18e−03 11.28%4.65e−03 10.89%3.90e−03

#134 6 12 21.87%6.74e−03 19.12%8.28e−03 23.43%9.21e−03 22.23%5.85e−03 20.72%6.65e−03 21.31%4.77e−03 20.69%1.92e−03

#135 9 12 32.39%7.18e−03 29.88%4.13e−03 33.77%1.17e−03 32.23%6.91e−03 30.07%1.96e−04 30.86%3.99e−03 29.87%5.40e−03

#136 12 12 41.08%3.49e−03 38.01%4.57e−03 43.07%7.64e−03 41.53%8.37e−03 38.85%7.94e−03 39.74%5.51e−03 38.15%9.66e−03

#137 15 12 48.88%7.15e−03 43.17%1.52e−03 50.63%1.78e−03 49.22%3.54e−03 46.26%9.11e−03 47.35%4.81e−03 45.88%6.96e−03

#138 18 12 53.93%8.41e−03 48.11%3.60e−03 55.63%7.35e−03 54.23%8.03e−03 51.81%8.57e−04 52.45%6.17e−03 50.91%4.21e−03

#139 21 12 57.92%2.75e−04 50.83%2.36e−03 59.44%9.61e−03 57.84%5.57e−03 55.64%1.64e−03 56.10%2.03e−03 54.01%4.44e−03

#140 24 12 60.67%2.81e−03 52.51%6.42e−03 62.19%6.95e−03 60.64%8.29e−03 58.50%6.14e−04 58.97%7.25e−03 56.90%6.98e−03

#141 27 12 62.81%4.81e−03 53.82%8.18e−04 64.09%7.55e−03 62.77%6.66e−03 60.72%9.29e−03 60.91%8.31e−03 59.48%1.01e−03

#142 30 12 64.52%9.66e−03 54.86%6.72e−04 65.78%3.10e−03 64.49%3.56e−03 62.49%1.50e−03 62.89%5.65e−03 61.33%4.09e−03

#143 33 12 65.92%5.72e−04 55.71%5.90e−03 67.13%8.92e−03 65.88%9.38e−03 63.94%9.86e−03 64.32%6.36e−03 63.24%7.97e−03

#144 36 12 67.08%5.57e−03 56.42%5.55e−03 68.40%7.64e−03 67.03%4.47e−03 65.15%2.34e−03 65.34%2.37e−03 63.25%4.00e−04




(e): MO-FA

Figure 8.8. COST239 network (TM1). Comparison among NSGA-II, SPEA2 and each proposed MOEAby using the HV indicator. Note that, each point represents the mean of the medians of HV reportedin Table 8.8 for W =1,2,3,4.




(e): MO-FA






Figure 8.11. COST239 network. Illustrative summary of the performance of each proposed MOEA byusing the HV indicator. Note that, each point represents the mean of the medians of HV reported inTable 8.8 (TM1), Table 8.9 (TM2), and Table 8.10 (TM3) for the different values of W .


Table 8.11. Statistical Analysis among the MOEAs in the COST239 network. The table indicates inwhich data sets two algorithms have no statistically significant differences.


DEPT MO-VNS 2 out of 144 data sets #13 #37NSGA-II 4 out of 144 data sets #13 #74 #109 #121SPEA2 0 out of 144 data sets -MO-GSA 65 out of 144 data sets #1 #19 #20 #21 #22 #24 #25 #27 #28 #3

#33 #34 #35 #36 #38 #39 #4 #46 #47 #48#52 #54 #61 #62 #68 #69 #70 #71 #72 #73#76 #78 #81 #82 #83 #85 #86 #87 #88 #89#90 #97 #99 #107 #108 #111 #112 #115 #116 #117#118 #119 #120 #124 #125 #129 #130 #132 #135 #139#140 #141 #142 #143 #144

MO-ABC 0 out of 144 data sets -MO-FA 5 out of 144 data sets #3 #7 #9 #97 #133

MO-VNS NSGA-II 22 out of 144 data sets #14 #21 #22 #25 #26 #27 #29 #3 #38 #40#52 #53 #58 #59 #63 #64 #86 #88 #101 #102#103 #104

SPEA2 17 out of 144 data sets #15 #24 #25 #28 #49 #57 #61 #65 #73 #85#89 #100 #105 #112 #122 #135 #136

MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 18 out of 144 data sets #100 #101 #102 #111 #123 #135 #24 #4 #54 #56

#57 #58 #59 #60 #77 #86 #89 #99

NSGA-II SPEA2 8 out of 144 data sets #30 #107 #8 #25 #48 #46 #13 #33MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 18 out of 144 data sets #28 #30 #33 #34 #35 #36 #44 #50 #55 #58

#59 #60 #62 #101 #110 #128 #141 #144

SPEA2 MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 27 out of 144 data sets #100 #130 #132 #134 #20 #22 #29 #30 #31 #40

#43 #45 #46 #47 #48 #50 #52 #53 #57 #64#66 #75 #76 #87 #89 #90 #98

MO-GSA MO-ABC 0 out of 144 data sets -MO-FA 2 out of 144 data sets #3 #109

MO-ABC MO-FA 0 out of 144 data sets -

competitive values of HV when the load of traffic is low; however, when the amount of traffic ismedium or large, the rest of algorithms obtains clearly better results than the MO-VNS algorithm.

In order to prove that the differences of HV among the MOEAs are statistically significant, inTable 8.11 we compare the differences of HV by pair of MOEAs, reporting those data sets in whichthe differences of HV between two algorithms are not statistically significant.

As we may observe, in this case, the MO-ABC obtains differences of HV statistically significantwith any other MOEA in the 30 independent runs. Like in the small network, the performance ofthe DEPT is quite similar to the performance of the MO-GSA in a high number of data sets. Wecan also notice the similar behaviour of the NSGA-II and SPEA2 in some of the scenarios.

Finally, we compare the MOEAs by using the SC indicator. In Table 8.12, we present acomparison by pairs in which we indicate the percentage of non-dominated solutions obtained byan algorithm A that cover the non-dominated solutions of an algorithm B.

In this comparison, we report, for each pair of MOEAs, the average percentage of coveragefor each W . As we can see, the set of non-dominated solutions of the MO-ABC covers a highpercentage of the solutions that belong to the Pareto front achieved by the other approaches.

Furthermore, whereas the DEPT and MO-VNS cover less than 30% of the solutions obtained bythe MO-ABC algorithm, the other approaches (NSGA-II, SPEA2, MO-GSA, and MO-FA) coverless than 10%.


Table 8.12. COST239 network. Comparison among the MOEAs by using the Set Coverage (SC) indicator,A B. Note that, SC represents the mean coverage of an algorithm A over an algorithm B in all thedata sets.

TM1 (3187 OC-1 units) TM2 (6272 OC-1 units) TM3 (12037 OC-1 units)A B W=1 W=2 W=3 W=4 W=2 W=4 W=6 W=8 W=3 W=6 W=9 W=12 SC

DEPT


MO-VNS


MO-ABC


MO-GSA


MO-FA


NSGA-II


SPEA2



Figure 8.12. The optical network topology: National Science Foundation (NSF).

8.2.3 National Science Foundation Network

The National Science Foundation (NSF) is the third network topology considered in this study. Itconsists of 14 nodes and 42 physical links - the capacity of each link is OC-192. Other specificationsof this optical network topology are presented in Chapter 5 (see Section 5.4).

In the same way as with the previous optical networks, for the NSF network we test the per-formance of the MOEAs when solving the Traffic Grooming problem with three different amountsof traffic: 5724, 11448, and 22896 OC-1 units. In the data sets, the available resources (T and W )depend on the amount of traffic.

Firstly, we compare the quality of the non-dominated solutions obtained by the MOEAs byusing the HV indicator. In Table 8.13, Table 8.14, and Table 8.15, we present the median of 30independent runs obtained by each MOEA using different number of transceivers per node andwavelengths per link in each traffic matrix.

As we can see, unlike in 6-node and COST239 networks, in this case, the algorithm with thehighest value of HV in all data sets is the MO-GSA.

In addition, we may observe that, like in the other topologies, the performance of the MO-VNSdecreases as the amount of traffic is increased.

Like in the previous experiment, we present diverse comparisons among each one of the fiveproposed MOEAs and the well-known algorithms NSGA-II and SPEA2 with the aim of illustratingthe differences with these standard MOEAs.

In this way, an illustrative comparison of HV obtained by each proposed MOEA in the 30 runs,as well as the values of HV obtained by the NSGA-II and SPEA2 are reported in Figure 8.13,Figure 8.14, and Figure 8.15.

As we can see, it is clear to see that the MO-GSA, MO-ABC, and DEPT algorithms clearlyovercome the results of the standard approaches in all the cases (TM1, TM2, and TM3). However,we may observe that the NSGA-II and SPEA2 obtain higher quality results than the trajectory-based MO-VNS algorithm and the MO-FA, independently of the amount of traffic considered (TM1,TM2, or TM3).

In Figure 8.16, we can notice the remarkable differences of HV obtained by the MO-GSAregarding the rest of MOEAs. We may also observe that the MO-ABC algorithm is the secondbest approach, that is to say, it obtains very promising results but lower than those obtained bythe MO-GSA.

The results of the statistical analysis carried out by pair of MOEAs are reported in Table 8.16.


Table 8.13. NSF network (TM1). Comparison among the MOEAs by using the Hypervolume (HV) indi-cator. The notation used is HVIQR, where HV is the median hypervolume and IQR is the interquartilerange in 30 independent runs.


#1 1 1 9.65%3.48e−03 8.32%4.79e−04 10.24%3.17e−03 11.34%8.47e−03 10.44%3.13e−03 9.63%9.87e−03 9.46%3.37e−03

#2 2 1 19.89%7.36e−04 19.58%5.71e−03 20.31%6.85e−03 22.12%8.13e−03 19.06%7.70e−03 19.53%8.24e−03 19.27%8.83e−03

#3 3 1 26.27%4.17e−03 25.70%7.17e−03 27.41%9.70e−03 30.17%1.04e−03 24.09%7.53e−03 25.50%8.29e−03 25.46%7.60e−03

#4 4 1 28.28%6.75e−03 28.30%3.12e−04 31.10%6.50e−03 34.57%8.47e−04 26.85%9.29e−03 28.06%6.94e−03 28.41%3.35e−03

#5 5 1 31.01%7.84e−03 30.10%8.81e−03 32.57%4.50e−03 36.59%7.25e−03 28.93%4.05e−03 30.12%4.10e−04 29.56%2.89e−03

#6 6 1 32.10%8.14e−03 31.08%4.85e−03 35.76%7.51e−04 38.73%2.86e−03 29.94%8.43e−03 31.08%1.10e−03 30.66%9.96e−03

#7 7 1 32.88%6.86e−03 31.78%3.99e−03 36.32%3.54e−03 39.70%1.33e−03 30.65%5.86e−03 31.99%3.81e−03 31.45%4.27e−03

#8 8 1 33.45%5.35e−04 32.31%4.23e−03 36.28%5.10e−03 42.37%7.18e−03 31.19%1.44e−03 32.58%9.44e−03 32.04%9.41e−03

#9 9 1 33.10%5.51e−03 32.15%3.26e−03 36.37%9.22e−03 42.94%4.57e−04 31.23%2.91e−03 33.39%3.95e−03 32.82%5.76e−03

#10 10 1 33.45%8.41e−03 32.47%5.59e−04 36.77%9.64e−03 41.96%8.57e−03 31.56%8.25e−03 32.93%7.86e−03 33.18%1.63e−03

#11 11 1 33.74%8.16e−03 32.72%4.86e−03 37.63%7.48e−03 42.85%6.10e−03 31.83%1.64e−03 34.38%2.74e−03 33.48%9.72e−03

#12 12 1 33.98%7.98e−03 32.94%7.19e−03 38.77%1.29e−03 43.05%5.21e−03 32.06%1.59e−03 33.81%4.60e−05 33.75%6.69e−03

#13 1 2 9.40%9.60e−03 8.11%3.65e−03 10.03%8.44e−03 11.24%6.35e−03 10.32%7.21e−03 9.42%7.92e−04 9.38%3.29e−03

#14 2 2 18.89%4.46e−03 14.11%4.58e−03 19.87%7.85e−03 21.61%7.77e−03 18.64%7.68e−03 18.97%6.77e−03 18.77%6.03e−03

#15 3 2 28.65%2.89e−04 25.24%2.32e−03 29.83%1.79e−03 32.21%7.58e−03 27.02%1.98e−03 28.76%7.84e−03 28.60%3.64e−03

#16 4 2 37.14%2.68e−03 35.13%4.42e−03 38.77%9.62e−03 41.68%2.29e−03 34.62%7.48e−03 37.21%4.83e−03 36.71%4.44e−03

#17 5 2 43.78%9.25e−03 41.33%8.62e−03 45.02%7.34e−03 48.70%3.45e−03 39.48%7.35e−03 42.26%9.21e−03 41.76%2.39e−03

#18 6 2 46.84%4.49e−03 43.95%5.33e−03 50.93%6.22e−04 53.37%3.68e−03 42.68%1.07e−03 45.22%9.50e−03 44.86%3.72e−04

#19 7 2 49.04%7.48e−03 46.90%1.08e−03 51.07%4.40e−03 56.41%4.29e−03 45.58%3.76e−03 47.74%8.96e−03 47.96%5.83e−03

#20 8 2 51.18%8.87e−03 47.89%6.73e−03 53.69%5.85e−03 59.67%8.60e−03 47.29%7.95e−03 49.45%3.33e−03 49.57%6.19e−03

#21 9 2 52.96%7.14e−03 48.99%2.77e−03 56.11%1.32e−03 60.85%6.73e−03 48.66%8.16e−03 51.16%6.57e−04 50.78%6.89e−03

#22 10 2 54.11%6.84e−03 49.89%1.03e−03 57.11%6.64e−03 63.68%7.20e−03 49.76%4.59e−03 52.23%6.46e−03 52.28%1.33e−03

#23 11 2 55.06%4.07e−03 50.61%5.53e−03 58.42%9.06e−03 62.88%4.69e−03 50.37%5.16e−03 52.81%8.79e−03 53.18%6.97e−03

#24 12 2 55.84%1.83e−03 51.21%2.02e−03 59.66%8.53e−03 65.02%4.78e−03 51.10%9.04e−03 53.74%2.50e−03 53.94%9.02e−04

#25 1 3 9.31%6.10e−03 7.41%8.73e−03 10.00%6.45e−03 11.19%3.02e−03 10.22%7.45e−03 9.45%2.39e−03 9.38%5.22e−03

#26 2 3 17.81%1.38e−03 12.53%1.15e−03 19.91%1.57e−03 21.51%3.67e−03 18.66%1.53e−03 19.00%1.42e−03 18.77%5.57e−04

#27 3 3 29.10%2.23e−03 19.11%7.66e−03 29.92%5.05e−04 32.01%3.98e−03 26.89%2.73e−03 28.61%3.86e−03 28.36%3.38e−03

#28 4 3 39.32%7.13e−03 32.65%1.92e−03 40.19%4.84e−03 42.48%3.75e−03 35.76%3.64e−03 38.55%5.37e−04 37.88%7.66e−03

#29 5 3 48.78%6.70e−04 40.68%2.90e−03 48.87%4.34e−03 51.40%1.24e−03 44.09%7.54e−03 46.70%1.41e−03 46.32%1.89e−03

#30 6 3 54.62%4.92e−03 48.45%6.98e−03 55.22%7.02e−03 58.06%9.74e−03 49.72%9.89e−03 52.84%2.27e−03 52.63%7.45e−04

#31 7 3 58.76%9.25e−03 53.64%7.53e−04 59.92%4.03e−03 62.96%7.54e−03 54.02%2.06e−03 57.21%7.03e−04 56.32%6.00e−03

#32 8 3 61.17%6.27e−03 57.77%8.52e−03 63.65%3.82e−03 66.14%4.82e−03 57.31%2.66e−03 60.69%6.05e−03 59.90%6.22e−03

#33 9 3 64.39%6.26e−03 60.67%1.85e−03 66.24%9.34e−03 68.87%6.11e−03 59.98%2.57e−03 63.00%8.93e−03 62.26%6.95e−03

#34 10 3 66.76%4.07e−03 62.56%7.97e−04 68.32%8.98e−03 71.03%6.36e−03 61.54%7.81e−03 64.68%1.78e−03 63.64%5.06e−03

#35 11 3 68.38%4.11e−03 61.52%4.44e−03 69.50%5.98e−03 73.11%7.34e−03 63.20%1.71e−03 65.81%3.26e−03 65.59%6.63e−03

#36 12 3 69.72%1.80e−04 62.48%3.32e−03 71.52%6.36e−03 74.32%7.33e−03 64.52%6.10e−05 67.52%2.27e−03 66.97%7.64e−03

#37 1 4 9.50%7.33e−03 7.53%1.71e−03 10.12%1.74e−03 11.17%7.68e−03 10.29%6.97e−03 9.44%3.88e−03 9.38%8.69e−04

#38 2 4 18.66%6.06e−03 13.43%8.86e−03 19.80%1.57e−03 21.51%4.59e−03 18.59%8.35e−04 18.93%5.47e−03 18.71%9.52e−03

#39 3 4 28.76%8.90e−03 20.22%1.77e−03 29.90%3.05e−03 31.94%3.34e−03 26.76%1.44e−03 28.54%4.23e−03 28.13%9.59e−03

#40 4 4 39.91%2.81e−03 26.04%9.64e−03 39.98%9.37e−03 42.24%8.07e−03 35.73%6.52e−03 38.35%8.57e−03 36.73%5.95e−03

#41 5 4 48.03%8.17e−03 34.65%1.58e−03 49.09%4.64e−03 51.00%6.46e−03 44.29%5.80e−03 46.92%8.36e−03 46.22%8.29e−03

#42 6 4 54.72%3.82e−03 42.57%2.72e−03 55.31%8.35e−03 57.61%2.59e−03 51.02%7.58e−03 53.34%6.51e−03 53.15%2.80e−03

#43 7 4 58.93%7.59e−03 50.37%8.23e−03 60.22%1.65e−03 62.13%9.32e−04 56.15%2.90e−05 58.05%1.77e−03 58.29%3.72e−04

#44 8 4 63.07%5.87e−03 54.21%4.30e−03 64.05%4.79e−03 65.88%8.26e−03 60.08%9.26e−03 62.06%8.56e−03 62.02%9.25e−04

#45 9 4 65.82%5.94e−04 58.83%9.60e−04 66.75%6.84e−03 68.94%9.31e−03 62.86%6.03e−03 64.73%3.57e−03 64.79%4.85e−03

#46 10 4 68.53%9.20e−05 60.89%1.91e−03 68.95%4.48e−03 70.81%7.98e−03 65.46%5.45e−03 67.04%9.90e−03 67.22%7.52e−03

#47 11 4 70.09%6.76e−03 63.89%1.70e−04 70.97%8.78e−03 72.72%2.36e−03 67.54%8.57e−03 69.03%5.19e−03 68.83%8.04e−03

#48 12 4 72.09%4.13e−03 65.86%3.16e−03 72.54%6.68e−03 74.22%1.08e−03 69.21%6.59e−03 70.61%5.01e−03 70.74%8.21e−03




#49 2 2 12.50%4.78e−03 9.19%7.93e−03 13.35%1.36e−03 14.47%2.89e−03 12.66%6.75e−03 12.83%5.47e−03 12.53%2.69e−03

#50 4 2 24.12%5.25e−03 21.81%4.61e−03 24.87%1.79e−03 26.61%1.76e−03 22.33%5.02e−03 23.75%8.88e−03 23.62%2.79e−04

#51 6 2 29.88%1.27e−03 28.25%3.09e−04 31.27%8.69e−03 34.16%3.39e−03 27.52%5.18e−03 28.90%8.41e−03 28.71%4.02e−03

#52 8 2 31.79%7.35e−03 29.87%6.92e−03 34.03%7.37e−03 37.55%7.74e−04 29.74%2.21e−03 31.68%3.24e−03 31.09%7.95e−03

#53 10 2 33.22%7.36e−03 31.52%1.43e−03 36.36%9.86e−03 39.35%1.27e−03 31.19%3.47e−03 33.16%4.55e−03 32.82%7.25e−04

#54 12 2 34.17%7.89e−03 32.33%5.93e−03 36.80%5.46e−03 40.65%8.76e−04 32.16%6.17e−03 34.12%1.64e−03 33.44%5.81e−03

#55 14 2 34.84%5.37e−03 32.90%7.08e−03 37.64%1.13e−03 42.16%5.35e−03 32.85%5.61e−03 34.39%7.49e−03 34.17%8.38e−03

#56 16 2 35.35%8.17e−03 33.33%1.57e−03 38.98%2.19e−03 42.13%9.60e−03 33.36%8.51e−03 35.16%1.78e−03 34.69%9.99e−03

#57 18 2 36.83%2.10e−03 33.26%7.05e−03 40.28%9.77e−03 42.09%3.98e−03 34.20%1.76e−03 35.69%8.75e−03 35.75%1.10e−03

#58 20 2 37.17%4.14e−03 33.52%5.00e−03 39.81%2.30e−03 43.41%9.55e−03 34.53%2.38e−03 35.64%9.73e−03 36.08%4.73e−03

#59 22 2 37.45%4.43e−03 33.72%3.64e−03 39.58%3.78e−03 43.02%5.39e−03 34.79%2.50e−03 36.05%9.49e−03 36.35%5.74e−03

#60 24 2 37.68%6.11e−03 33.90%9.95e−03 40.04%8.97e−04 45.50%2.94e−04 35.01%3.52e−03 36.18%8.87e−03 36.58%2.70e−03

#61 2 4 12.51%1.95e−03 8.78%8.05e−03 13.43%3.39e−03 14.48%3.42e−03 12.71%1.00e−03 12.87%7.39e−03 12.73%4.46e−04

#62 4 4 25.55%8.40e−03 17.63%2.73e−03 26.07%9.90e−05 27.40%8.15e−03 23.41%4.65e−04 25.06%6.56e−03 24.07%4.00e−03

#63 6 4 37.41%2.04e−03 29.65%1.70e−04 38.04%9.26e−03 39.77%7.09e−03 34.55%8.25e−03 36.58%3.91e−03 35.17%3.53e−03

#64 8 4 45.44%4.20e−03 39.31%9.15e−03 46.58%9.18e−03 48.87%2.09e−03 42.15%7.57e−03 44.22%7.87e−03 42.54%8.73e−03

#65 10 4 49.85%8.97e−03 44.95%9.28e−03 51.44%8.56e−03 54.00%5.59e−03 47.02%4.77e−03 48.98%3.18e−03 48.36%5.97e−04

#66 12 4 53.16%5.35e−03 47.36%5.35e−04 54.89%7.14e−03 57.65%4.38e−03 49.86%6.08e−04 51.42%4.94e−03 51.33%3.49e−03

#67 14 4 55.28%2.46e−03 48.85%7.93e−03 57.26%8.31e−03 60.50%9.47e−03 52.07%9.31e−04 53.72%4.01e−03 53.17%4.81e−03

#68 16 4 56.87%7.86e−03 49.96%2.06e−03 59.36%6.14e−03 62.40%4.55e−03 53.64%6.56e−03 55.10%6.48e−03 54.97%6.00e−03

#69 18 4 58.11%4.61e−03 50.83%1.25e−03 60.16%1.03e−04 63.45%7.08e−04 54.87%7.81e−03 56.76%5.03e−03 56.13%7.92e−03

#70 20 4 59.09%7.87e−03 51.52%2.57e−04 61.33%8.25e−03 65.69%6.11e−03 55.85%7.81e−03 57.21%9.99e−03 57.16%3.34e−03

#71 22 4 59.90%4.02e−04 52.09%4.13e−04 61.99%4.13e−03 66.52%4.27e−03 56.65%8.80e−03 57.88%6.09e−03 58.08%7.98e−03

#72 24 4 60.57%4.49e−03 52.56%7.11e−04 65.44%8.87e−03 66.21%9.54e−03 57.32%9.34e−03 59.41%5.76e−03 58.57%7.48e−03

#73 2 6 12.80%4.14e−03 9.09%1.11e−03 13.35%5.81e−03 14.45%5.39e−03 12.70%7.47e−03 12.90%4.18e−03 12.63%5.04e−03

#74 4 6 24.83%5.25e−03 18.69%8.01e−04 26.07%3.64e−03 27.38%5.59e−03 23.33%8.00e−03 25.01%6.31e−03 23.65%1.89e−03

#75 6 6 38.23%1.45e−04 31.69%3.86e−03 38.80%3.57e−03 40.12%1.86e−03 35.30%6.15e−03 37.14%4.22e−03 36.27%9.40e−03

#76 8 6 49.66%7.99e−03 41.88%4.77e−03 49.80%1.69e−03 51.39%6.42e−03 45.82%1.45e−03 47.86%8.62e−03 47.45%9.06e−03

#77 10 6 56.75%9.85e−03 50.62%7.37e−03 57.39%4.11e−03 58.88%4.23e−03 53.01%4.20e−03 55.11%1.44e−03 53.91%4.23e−03

#78 12 6 61.48%3.30e−03 53.63%2.40e−03 62.17%8.38e−03 63.93%3.70e−03 57.74%2.39e−03 59.47%9.19e−03 59.92%2.51e−04

#79 14 6 65.50%9.91e−04 55.67%1.06e−04 65.54%2.52e−03 67.50%8.81e−04 61.15%2.18e−03 63.27%2.31e−03 62.76%5.91e−04

#80 16 6 68.08%3.67e−03 57.25%8.13e−03 68.91%5.44e−04 70.36%3.07e−03 63.73%7.15e−03 65.62%6.30e−03 65.44%3.03e−03

#81 18 6 69.36%3.38e−03 58.87%3.98e−03 70.40%7.10e−03 72.25%6.39e−03 65.83%1.08e−03 67.72%4.12e−04 67.77%4.36e−03

#82 20 6 70.93%8.43e−03 59.87%1.80e−03 72.45%6.41e−03 73.90%9.80e−03 67.44%7.94e−03 69.36%4.24e−03 69.12%1.95e−03

#83 22 6 72.54%1.89e−04 60.68%2.43e−03 73.59%1.13e−03 75.47%3.60e−03 68.77%3.93e−03 70.63%8.58e−03 70.70%3.09e−03

#84 24 6 73.27%8.06e−03 61.36%8.32e−03 74.83%9.18e−03 76.58%3.36e−03 69.87%4.71e−03 71.16%3.11e−03 71.64%9.07e−03

#85 2 8 12.59%6.95e−03 8.73%4.76e−03 13.37%4.53e−03 14.55%1.77e−03 12.72%1.50e−03 12.92%5.94e−03 12.71%4.89e−03

#86 4 8 25.63%3.28e−03 17.06%5.51e−03 26.10%7.48e−03 27.38%8.27e−03 23.27%6.74e−03 25.03%3.09e−03 24.14%8.41e−03

#87 6 8 38.68%8.61e−03 29.52%5.06e−03 39.06%3.78e−03 40.27%9.98e−03 35.27%9.04e−03 37.23%2.96e−04 35.03%3.86e−03

#88 8 8 49.33%8.84e−03 41.62%6.50e−03 50.13%9.48e−03 51.50%9.51e−03 45.98%5.30e−03 47.88%3.69e−03 47.41%4.88e−03

#89 10 8 57.29%8.13e−03 50.97%2.05e−03 57.57%9.80e−03 59.10%5.23e−03 53.76%7.90e−03 55.46%5.94e−03 54.84%2.63e−03

#90 12 8 62.41%7.46e−03 55.75%4.94e−03 62.74%7.75e−04 64.08%2.58e−04 59.27%9.97e−03 60.87%5.54e−03 59.82%6.47e−03

#91 14 8 66.13%1.20e−03 57.44%2.69e−03 66.61%7.96e−03 67.71%1.42e−03 63.14%4.78e−03 64.78%7.89e−03 63.82%7.13e−03

#92 16 8 68.90%8.57e−03 59.22%3.72e−04 69.50%7.02e−03 70.60%9.20e−04 66.03%2.32e−03 67.43%5.79e−03 66.97%4.51e−03

#93 18 8 71.05%4.21e−03 60.59%4.52e−03 71.46%4.01e−04 72.76%5.74e−03 68.30%2.92e−03 69.73%2.36e−03 69.19%3.74e−03

#94 20 8 72.76%9.37e−03 61.69%4.25e−03 73.17%3.08e−03 74.22%5.53e−04 70.10%5.66e−03 71.40%8.46e−03 71.00%1.62e−03

#95 22 8 74.17%3.30e−03 62.59%2.97e−03 74.92%4.14e−03 75.81%3.11e−03 71.58%8.91e−03 72.87%4.62e−03 72.54%6.94e−03

#96 24 8 75.30%2.60e−03 63.34%3.75e−03 75.86%5.81e−03 76.89%6.75e−03 72.82%6.27e−03 74.09%2.81e−03 73.40%6.15e−03




#97 3 3 12.82%6.02e−03 8.34%5.43e−03 13.71%7.80e−03 14.56%8.31e−03 12.32%5.47e−03 13.11%8.55e−03 12.86%3.59e−03

#98 6 3 23.83%3.15e−03 20.64%8.54e−03 24.09%9.49e−03 25.55%8.65e−03 21.77%2.27e−03 23.32%8.87e−03 22.50%2.85e−03

#99 9 3 28.02%2.69e−03 26.17%5.91e−03 29.48%8.19e−03 31.04%5.17e−03 26.15%7.14e−03 27.57%2.14e−03 27.53%6.31e−03

#100 12 3 30.42%3.60e−03 27.76%7.10e−03 31.70%7.83e−03 33.54%1.01e−03 28.17%4.12e−03 29.67%2.95e−03 29.64%5.92e−03

#101 15 3 31.68%9.66e−03 28.67%3.87e−03 32.81%5.46e−03 35.11%2.33e−03 29.44%8.39e−03 30.57%6.54e−03 30.99%5.83e−03

#102 18 3 32.52%6.66e−03 29.27%8.60e−03 34.35%4.00e−06 36.62%6.18e−03 30.24%4.23e−03 31.60%4.06e−03 31.76%7.40e−03

#103 21 3 33.12%8.70e−03 29.70%8.10e−05 34.70%4.73e−03 36.89%7.43e−03 30.84%4.23e−03 32.32%5.77e−03 32.40%3.00e−03

#104 24 3 33.57%2.01e−04 30.03%8.41e−03 35.06%6.64e−03 37.84%1.42e−03 31.29%8.69e−04 32.96%6.88e−04 32.87%3.26e−03

#105 27 3 33.96%8.49e−04 29.88%2.37e−03 35.75%1.37e−03 37.61%8.71e−03 31.65%8.67e−03 33.15%1.99e−03 32.63%3.30e−03

#106 30 3 34.25%6.36e−03 30.07%6.33e−03 35.92%2.13e−04 38.27%4.76e−03 31.93%5.59e−03 33.65%3.05e−03 32.92%3.35e−03

#107 33 3 34.48%2.79e−03 30.23%5.48e−03 36.54%6.26e−03 38.44%9.63e−03 32.16%8.24e−03 33.80%1.87e−03 33.15%9.61e−03

#108 36 3 34.67%1.25e−03 30.36%4.94e−03 36.33%9.08e−03 38.61%8.19e−03 32.34%1.27e−03 33.62%8.14e−03 33.58%5.97e−03

#109 3 6 12.64%3.64e−03 8.24%8.33e−03 13.75%3.80e−03 14.59%7.31e−03 12.28%5.33e−03 13.14%2.57e−03 12.98%9.05e−04

#110 6 6 25.21%1.52e−03 17.35%1.69e−03 26.01%8.54e−03 27.08%9.51e−03 23.64%3.70e−03 24.89%9.32e−03 23.84%4.39e−03

#111 9 6 36.37%3.43e−03 30.81%7.89e−03 36.82%9.98e−03 38.16%7.92e−03 33.77%7.72e−03 35.36%8.94e−03 33.47%6.42e−03

#112 12 6 43.07%9.43e−03 37.25%1.95e−04 43.44%5.47e−03 45.35%9.33e−03 40.23%7.32e−03 41.73%4.56e−03 41.04%5.40e−03

#113 15 6 46.95%6.09e−03 39.35%8.97e−03 47.54%4.13e−03 49.90%7.33e−03 43.75%4.09e−03 45.76%9.87e−03 45.10%2.26e−03

#114 18 6 49.34%3.21e−03 40.75%5.66e−03 49.92%1.31e−03 52.27%1.85e−03 46.18%3.36e−03 48.05%6.28e−03 47.64%9.92e−03

#115 21 6 51.05%4.82e−03 41.74%1.52e−03 52.26%4.79e−03 54.26%1.73e−03 47.92%2.60e−03 49.41%2.28e−03 49.49%6.92e−03

#116 24 6 52.33%9.10e−04 42.48%4.16e−03 53.98%4.64e−03 56.38%3.19e−03 49.23%1.68e−03 51.13%9.05e−03 50.67%3.63e−03

#117 27 6 53.32%4.20e−03 43.06%3.21e−03 55.89%7.52e−03 57.12%8.99e−03 50.24%1.19e−03 51.78%7.83e−03 51.85%6.00e−03

#118 30 6 54.12%4.59e−03 43.52%7.80e−03 55.88%3.23e−03 57.96%6.21e−03 51.05%2.52e−03 52.88%4.73e−03 52.74%6.96e−03

#119 33 6 54.77%1.77e−03 43.90%3.79e−03 56.26%8.29e−03 58.32%2.42e−04 51.72%2.74e−03 53.25%6.14e−03 53.32%7.09e−03

#120 36 6 55.31%8.10e−03 44.22%2.72e−03 57.13%3.54e−03 59.05%1.60e−03 52.27%2.03e−03 54.29%7.47e−03 53.93%8.89e−03

#121 3 9 13.17%9.96e−03 7.81%3.92e−03 13.69%7.47e−03 14.54%3.86e−03 12.41%3.93e−03 13.12%5.52e−03 12.59%2.74e−03

#122 6 9 25.52%8.52e−03 19.12%9.52e−03 25.97%1.12e−03 27.07%5.03e−03 23.55%1.86e−03 24.92%7.17e−03 24.08%8.69e−03

#123 9 9 37.75%6.71e−03 32.06%4.89e−03 38.07%6.02e−03 39.21%4.19e−03 34.99%5.00e−03 36.76%3.29e−03 34.80%1.10e−03

#124 12 9 47.74%3.00e−03 37.87%1.04e−03 48.50%8.20e−03 49.66%1.85e−03 44.83%3.51e−03 46.54%1.53e−03 45.13%2.18e−03

#125 15 9 54.76%6.66e−03 43.20%7.26e−03 55.55%5.24e−03 56.71%1.37e−03 51.39%6.79e−03 53.52%3.34e−03 52.33%5.37e−03

#126 18 9 58.71%4.07e−03 45.02%2.76e−03 59.65%5.89e−03 61.23%3.00e−04 55.76%2.39e−03 57.61%9.99e−03 56.76%2.03e−03

#127 21 9 61.73%2.25e−03 46.31%3.27e−03 62.95%5.12e−03 64.89%6.02e−03 58.79%3.81e−03 60.65%5.62e−03 60.07%9.25e−03

#128 24 9 64.00%1.94e−03 47.29%3.94e−03 65.35%1.07e−03 67.03%6.93e−03 61.06%1.74e−03 62.95%7.50e−03 62.33%6.99e−04

#129 27 9 66.16%8.63e−03 47.82%9.64e−03 67.10%2.74e−03 69.12%5.82e−03 62.81%1.61e−03 64.71%1.45e−04 64.33%4.77e−03

#130 30 9 67.55%8.40e−05 48.42%6.20e−03 68.57%6.22e−03 70.62%3.20e−03 64.23%7.70e−04 66.05%9.11e−03 65.84%1.45e−04

#131 33 9 68.70%7.62e−03 48.91%5.09e−03 69.92%6.44e−03 71.99%4.02e−03 65.38%8.93e−03 67.21%6.12e−03 66.93%1.09e−03

#132 36 9 69.65%9.66e−03 49.32%5.21e−03 70.73%2.53e−03 73.46%7.83e−03 66.35%9.55e−04 68.11%5.30e−03 68.01%4.45e−03

#133 3 12 13.03%2.28e−03 8.13%1.72e−03 13.71%6.89e−03 14.58%2.15e−03 12.28%8.16e−04 13.13%5.09e−03 13.00%9.94e−03

#134 6 12 25.50%1.37e−03 15.94%8.22e−04 25.93%2.04e−04 27.10%8.97e−03 23.58%7.71e−03 25.03%1.15e−03 24.12%5.70e−03

#135 9 12 37.66%9.60e−03 31.17%2.61e−03 38.01%8.38e−03 39.27%3.82e−03 35.12%2.94e−03 36.67%8.12e−03 35.08%8.14e−03

#136 12 12 48.80%8.77e−03 40.36%3.96e−03 49.00%7.04e−03 50.22%7.59e−03 45.56%8.57e−03 47.45%7.17e−03 44.72%3.72e−03

#137 15 12 56.74%1.16e−03 41.69%5.32e−03 56.95%9.39e−04 57.95%9.19e−03 53.45%1.51e−03 55.24%7.72e−03 53.06%7.24e−03

#138 18 12 61.85%7.25e−03 43.36%8.80e−03 62.08%8.15e−03 63.26%4.12e−03 58.69%8.27e−03 60.49%6.52e−03 58.16%1.17e−03

#139 21 12 65.49%8.52e−03 44.55%4.99e−03 65.82%1.09e−03 66.86%6.21e−03 62.50%7.91e−03 64.24%3.84e−03 62.61%7.44e−03

#140 24 12 68.22%4.59e−03 45.44%5.44e−03 68.56%1.43e−03 69.75%2.60e−03 65.37%3.29e−03 67.02%6.72e−03 65.74%1.02e−03

#141 27 12 70.34%7.25e−03 46.13%2.50e−04 70.73%3.70e−04 71.74%4.92e−04 67.52%9.37e−03 69.25%6.16e−03 67.98%4.77e−03

#142 30 12 72.03%5.04e−03 46.69%9.24e−03 72.56%4.14e−04 73.45%4.31e−03 69.34%7.06e−03 70.93%8.78e−03 69.62%5.27e−03

#143 33 12 73.42%9.34e−03 47.14%5.31e−03 73.86%7.62e−03 74.87%2.11e−03 70.77%7.48e−03 72.42%4.78e−03 69.59%1.25e−03

#144 36 12 74.57%9.22e−03 47.52%8.25e−03 75.09%3.21e−03 76.07%8.37e−03 71.98%2.66e−03 73.47%5.90e−03 70.66%6.32e−03




(e): MO-FA

Figure 8.13. NSF network (TM1). Comparison among NSGA-II, SPEA2 and each proposed MOEA byusing the HV indicator. Note that, each point represents the mean of the medians of HV reported inTable 8.13 for W =1,2,3,4.




(e): MO-FA






Figure 8.16. NSF network. Illustrative summary of the performance of each proposed MOEA by usingthe HV indicator. Note that, each point represents the mean of the medians of HV reported in Table8.13 (TM1), Table 8.14 (TM2), and Table 8.15 (TM3) for the different values of W .


Table 8.16. Statistical Analysis among the MOEAs in the NSF network. The table indicates in whichdata sets two algorithms have no statistically significant differences.


DEPT MO-VNS 1 out of 144 data sets #4NSGA-II 16 out of 144 data sets #1 #12 #13 #14 #15 #16 #25 #37 #52 #53

#54 #56 #73 #74 #121 #133SPEA2 13 out of 144 data sets #1 #3 #4 #13 #14 #15 #25 #37 #49 #73

#85 #97 #133MO-GSA 0 out of 144 data sets -MO-ABC 4 out of 144 data sets #29 #40 #76 #79MO-FA 4 out of 144 data sets #38 #49 #73 #85

MO-VNS NSGA-II 3 out of 144 data sets #2 #5 #6SPEA2 1 out of 144 data sets #4MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 7 out of 144 data sets #22 #24 #52 #54 #55 #56 #99

NSGA-II SPEA2 39 out of 144 data sets #1 #3 #12 #13 #14 #15 #20 #22 #25 #37#42 #44 #45 #46 #47 #48 #50 #51 #57 #61#66 #68 #70 #80 #81 #83 #99 #100 #102 #103#104 #108 #109 #115 #117 #118 #119 #132 #133

MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 4 out of 144 data sets #49 #61 #73 #85

SPEA2 MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 12 out of 144 data sets #14 #26 #38 #49 #61 #73 #85 #110 #121 #123

#135 #139

MO-GSA MO-ABC 0 out of 144 data sets -MO-FA 0 out of 144 data sets -

MO-ABC MO-FA 0 out of 144 data sets #37

As we can see, for this network topology, the differences of HV among the approaches arestatistically significant in the majority of the data sets.

We can highlight the statistical results for the MO-GSA, where in all data sets, the differencesof HV are statistically significant. However, we notice that the well-known MOEAs (NSGA-II andSPEA2) perform similar with this 14-node network.

Finally, we compare the MOEAs by pairs, by using the SC indicator. In Table 8.17, we can seethe clear dominance of the MO-GSA over the rest of approaches.

In this case, the MO-GSA is able to cover around 90% of the non-dominated solutions obtainedby the NSGA-II, SPEA-2, MO-ABC, and MO-FA; as well as covering a great number of solutionsobtained by the DEPT and MO-VNS algorithms. However, the DEPT, MO-VNS, NSGA-II, SPEA-2, MO-ABC, and MO-FA only cover a low percentage of the solutions that belong to the Paretofront obtained by the MO-GSA.


Table 8.17. NSF network. Comparison among the MOEAs by using the Set Coverage (SC) indicator,A B. Note that, SC represents the mean coverage of an algorithm A over an algorithm B in all thedata sets.


DEPT


MO-VNS


MO-ABC


MO-GSA


MO-FA


NSGA-II


SPEA2



Figure 8.17. The optical network topology: Nippon Telegraph and Telephone (NTT).

8.2.4 Nippon Telegraph and Telephone Network

The last optical network is the largest one, the Nippon Telegraph and Telephone (NTT), whichconsists of 55 nodes and 144 physical links - the capacity of each link is OC-768 (see Figure 8.17).

Different traffic matrices have been tested for this optical network. Like in the other topologies,we have a small, a medium, and a large traffic matrix; where the amount of traffic of each oneis 77233, 153307, and 309820 OC-1 units; respectively. For this network we have used differentscenarios, each scenario with different number of transceivers per node and number of wavelengthsper link, see Section 5.4.

In the first place, we compare the MOEAs by using the HV indicator. In Table 8.18, Table 8.19,and Table 8.20 we present the median value of HV obtained by each algorithm at each scenario.

As we may observe, the algorithm that obtains higher values of HV in almost all data sets isthe MO-GSA, as occurred in the NSF network, and the second best algorithm is the MO-ABCalgorithm.

In Figure 8.18, Figure 8.19, and Figure 8.20 we compare each of the multiobjective proposalswith the well-known MOEAs NSGA-II and SPEA2. After analyzing the plots, we realize that, theMO-GSA and the MO-ABC clearly overcome the results of the NSGA-II and SPEA2.

We can also notice that the DEPT algorithm is very competitive, obtaining better results thanthe NSGA-II and slightly lower than the SPEA2 in some cases. In the case of the MO-VNS andMO-FA, we can see that, whereas the MO-FA performs similar to the NSGA-II and slightly worsethan the SPEA2, the MO-VNS is clearly worse than these well-known approaches.

In Figure 8.21, we present an illustrative comparison among the approaches by using the averageHV for each number of wavelengths per link. On the one hand, we can notice that the algorithmscould be sorted by quality as MO-GSA, MO-ABC, DEPT, MO-FA, and MO-VNS, when dealingwith this network.

On the other hand, we can say that the differences of HV between the trajectory-based MOEA(MO-VNS) and the rest of algorithms are remarkable in all scenarios. Concretely, we can see ahigh difference of HV between the best and the worst algorithm (MO-GSA and MO-VNS) in alldata sets.

In order to demonstrate that the differences of HV among the approaches are statisticallysignificant, we have carried out the statistical analysis that is shown in Section 3.4.

In Table 8.21, we present the results of this analysis. As we may see, the SPEA2 and the DEPTalgorithm obtain almost identical values of HV in many data sets. However, the differences of HV


Table 8.18. NTT network (TM1). Comparison among the MOEAs by using the Hypervolume (HV) indi-cator. The notation used is HVIQR, where HV is the median hypervolume and IQR is the interquartilerange in 30 independent runs.


#1 1 1 2.22%3.48e−03 2.00%4.79e−04 2.24%3.17e−03 2.37%8.47e−03 1.95%3.13e−03 2.06%9.87e−03 2.05%3.37e−03

#2 2 1 6.23%7.36e−04 4.57%5.71e−03 6.56%6.85e−03 7.01%8.13e−03 5.43%7.70e−03 5.72%8.24e−03 6.04%8.83e−03

#3 3 1 8.22%4.17e−03 5.24%7.17e−03 9.06%9.70e−03 9.54%1.04e−03 7.31%7.53e−03 7.65%8.29e−03 8.13%7.60e−03

#4 4 1 9.18%6.75e−03 5.53%3.12e−04 10.18%6.50e−03 11.19%8.47e−04 8.18%9.29e−03 8.39%6.94e−03 9.03%3.35e−03

#5 5 1 9.58%7.84e−03 5.70%8.81e−03 10.86%4.50e−03 11.53%7.25e−03 8.53%4.05e−03 8.79%4.10e−04 9.62%2.89e−03

#6 6 1 9.56%8.14e−03 6.02%4.85e−03 11.13%7.51e−04 11.80%2.86e−03 8.92%8.43e−03 9.26%1.10e−03 10.11%9.96e−03

#7 7 1 9.77%6.86e−03 6.11%3.99e−03 11.34%3.54e−03 13.05%1.33e−03 9.15%5.86e−03 9.11%3.81e−03 10.34%4.27e−03

#8 8 1 9.92%5.35e−04 6.17%4.23e−03 12.59%5.10e−03 12.51%7.18e−03 9.28%1.44e−03 8.98%9.44e−03 10.52%9.41e−03

#9 9 1 10.21%5.51e−03 6.00%3.26e−03 11.83%9.22e−03 13.06%4.57e−04 9.16%2.91e−03 9.64%3.95e−03 10.41%5.76e−03

#10 10 1 10.31%8.41e−03 6.04%5.59e−04 12.40%9.64e−03 13.46%8.57e−03 9.26%8.25e−03 9.90%7.86e−03 10.51%1.63e−03

#11 11 1 10.39%8.16e−03 6.07%4.86e−03 11.93%7.48e−03 12.88%6.10e−03 9.33%1.64e−03 9.50%2.74e−03 10.60%9.72e−03

#12 12 1 10.46%7.98e−03 6.10%7.19e−03 12.34%1.29e−03 13.09%5.21e−03 9.39%1.59e−03 9.80%4.60e−05 10.67%6.69e−03

#13 1 2 1.74%9.60e−03 1.47%3.65e−03 1.88%8.44e−03 1.99%6.35e−03 1.59%7.21e−03 1.62%7.92e−04 1.71%3.29e−03

#14 2 2 7.08%4.46e−03 5.97%4.58e−03 7.23%7.85e−03 7.50%7.77e−03 6.29%7.68e−03 6.60%6.77e−03 6.82%6.03e−03

#15 3 2 12.43%2.89e−04 9.09%2.32e−03 12.90%1.79e−03 13.39%7.58e−03 11.32%1.98e−03 11.70%7.84e−03 12.14%3.64e−03

#16 4 2 15.61%2.68e−03 10.75%4.42e−03 16.62%9.62e−03 17.69%2.29e−03 14.59%7.48e−03 14.98%4.83e−03 15.78%4.44e−03

#17 5 2 17.66%9.25e−03 12.16%8.62e−03 19.12%7.34e−03 19.65%3.45e−03 16.34%7.35e−03 16.81%9.21e−03 17.47%2.39e−03

#18 6 2 18.87%4.49e−03 11.23%5.33e−03 20.66%6.22e−04 21.33%3.68e−03 17.35%1.07e−03 17.60%9.50e−03 18.85%3.72e−04

#19 7 2 19.63%7.48e−03 13.31%1.08e−03 21.00%4.40e−03 22.15%4.29e−03 18.26%3.76e−03 18.66%8.96e−03 19.45%5.83e−03

#20 8 2 20.48%8.87e−03 11.85%6.73e−03 21.82%5.85e−03 22.75%8.60e−03 18.93%7.95e−03 19.23%3.33e−03 20.36%6.19e−03

#21 9 2 20.48%7.14e−03 12.02%2.77e−03 22.47%1.32e−03 23.44%6.73e−03 19.47%8.16e−03 19.53%6.57e−04 20.78%6.89e−03

#22 10 2 20.83%6.84e−03 12.17%1.03e−03 23.12%6.64e−03 23.83%7.20e−03 19.73%4.59e−03 19.87%6.46e−03 21.18%1.33e−03

#23 11 2 21.12%4.07e−03 12.28%5.53e−03 23.27%9.06e−03 24.34%4.69e−03 20.03%5.16e−03 20.42%8.79e−03 21.47%6.97e−03

#24 12 2 21.36%1.83e−03 12.38%2.02e−03 23.85%8.53e−03 24.37%4.78e−03 20.27%9.04e−03 20.37%2.50e−03 21.75%9.02e−04

#25 1 3 1.41%6.10e−03 1.05%8.73e−03 1.67%6.45e−03 1.82%3.02e−03 1.48%7.45e−03 1.41%2.39e−03 1.53%5.22e−03

#26 2 3 6.71%1.38e−03 5.65%1.15e−03 6.91%1.57e−03 7.20%3.67e−03 6.01%1.53e−03 6.28%1.42e−03 6.49%5.57e−04

#27 3 3 13.72%2.23e−03 11.60%7.66e−03 14.03%5.05e−04 14.57%3.98e−03 12.49%2.73e−03 12.97%3.86e−03 13.43%3.38e−03

#28 4 3 18.78%7.13e−03 15.00%1.92e−03 19.49%4.84e−03 19.94%3.75e−03 17.38%3.64e−03 17.85%5.37e−04 18.69%7.66e−03

#29 5 3 22.25%6.70e−04 16.28%2.90e−03 22.95%4.34e−03 23.63%1.24e−03 20.84%7.54e−03 21.05%1.41e−03 22.16%1.89e−03

#30 6 3 24.40%4.92e−03 17.90%6.98e−03 25.53%7.02e−03 26.41%9.74e−03 23.01%9.89e−03 23.19%2.27e−03 24.35%7.45e−04

#31 7 3 25.57%9.25e−03 20.15%7.53e−04 27.24%4.03e−03 28.25%7.54e−03 24.30%2.06e−03 24.61%7.03e−04 25.77%6.00e−03

#32 8 3 27.04%6.27e−03 15.75%8.52e−03 28.52%3.82e−03 29.39%4.82e−03 25.13%2.66e−03 25.59%6.05e−03 26.91%6.22e−03

#33 9 3 27.70%6.26e−03 18.78%1.85e−03 29.52%9.34e−03 29.96%6.11e−03 25.83%2.57e−03 26.56%8.93e−03 27.91%6.95e−03

#34 10 3 28.24%4.07e−03 19.10%7.97e−04 29.70%8.98e−03 31.09%6.36e−03 26.62%7.81e−03 27.38%1.78e−03 28.58%5.06e−03

#35 11 3 28.91%4.11e−03 19.36%4.44e−03 30.37%5.98e−03 31.85%7.34e−03 27.16%1.71e−03 27.89%3.26e−03 29.24%6.63e−03

#36 12 3 29.36%1.80e−04 19.57%3.32e−03 31.32%6.36e−03 32.30%7.33e−03 27.61%6.10e−05 28.00%2.27e−03 29.70%7.64e−03

#37 1 4 1.34%7.33e−03 0.80%1.71e−03 1.61%1.74e−03 1.87%7.68e−03 1.40%6.97e−03 1.22%3.88e−03 1.44%8.69e−04

#38 2 4 5.80%6.06e−03 4.58%8.86e−03 6.55%1.57e−03 6.81%4.59e−03 5.50%8.35e−04 5.90%5.47e−03 6.14%9.52e−03

#39 3 4 13.88%8.90e−03 11.71%1.77e−03 14.40%3.05e−03 14.75%3.34e−03 12.62%1.44e−03 13.22%4.23e−03 13.53%9.59e−03

#40 4 4 20.39%2.81e−03 16.69%9.64e−03 20.82%9.37e−03 21.22%8.07e−03 18.69%6.52e−03 19.20%8.57e−03 19.84%5.95e−03

#41 5 4 24.53%8.17e−03 19.32%1.58e−03 25.36%4.64e−03 25.90%6.46e−03 23.05%5.80e−03 23.50%8.36e−03 24.34%8.29e−03

#42 6 4 27.71%3.82e−03 21.68%2.72e−03 28.65%8.35e−03 29.45%2.59e−03 26.24%7.58e−03 26.39%6.51e−03 27.08%2.80e−03

#43 7 4 30.23%7.59e−03 22.83%8.23e−03 31.41%1.65e−03 31.66%9.32e−04 28.24%2.90e−05 28.67%1.77e−03 29.95%3.72e−04

#44 8 4 31.63%5.87e−03 22.62%4.30e−03 33.00%4.79e−03 33.49%8.26e−03 30.00%9.26e−03 29.90%8.56e−03 31.40%9.25e−04

#45 9 4 32.10%5.94e−04 21.96%9.60e−04 34.25%6.84e−03 35.13%9.31e−03 31.08%6.03e−03 31.29%3.57e−03 32.60%4.85e−03

#46 10 4 33.23%9.20e−05 25.60%1.91e−03 35.17%4.48e−03 35.67%7.98e−03 32.01%5.45e−03 32.21%9.90e−03 33.36%7.52e−03

#47 11 4 33.95%6.76e−03 24.32%1.70e−04 35.94%8.78e−03 36.92%2.36e−03 32.74%8.57e−03 33.37%5.19e−03 34.67%8.04e−03

#48 12 4 34.87%4.13e−03 25.32%3.16e−03 36.83%6.68e−03 37.74%1.08e−03 33.41%6.59e−03 33.85%5.01e−03 35.13%8.21e−03




#49 2 2 6.73%4.78e−03 5.79%7.93e−03 6.86%1.36e−03 7.08%2.89e−03 6.06%6.75e−03 6.37%5.47e−03 6.38%2.69e−03

#50 4 2 12.82%5.25e−03 8.67%4.61e−03 13.08%1.79e−03 13.60%1.76e−03 11.63%5.02e−03 12.00%8.88e−03 12.54%2.79e−04

#51 6 2 14.59%1.27e−03 10.08%3.09e−04 15.80%8.69e−03 15.99%3.39e−03 13.88%5.18e−03 14.13%8.41e−03 14.76%4.02e−03

#52 8 2 15.49%7.35e−03 9.82%6.92e−03 16.63%7.37e−03 17.34%7.74e−04 14.63%2.21e−03 15.16%3.24e−03 15.81%7.95e−03

#53 10 2 16.28%7.36e−03 10.06%1.43e−03 17.27%9.86e−03 17.99%1.27e−03 15.21%3.47e−03 15.64%4.55e−03 16.29%7.25e−04

#54 12 2 16.67%7.89e−03 10.23%5.93e−03 17.61%5.46e−03 18.46%8.76e−04 15.60%6.17e−03 15.96%1.64e−03 16.72%5.81e−03

#55 14 2 16.95%5.37e−03 10.34%7.08e−03 18.04%1.13e−03 19.01%5.35e−03 15.87%5.61e−03 16.10%7.49e−03 17.00%8.38e−03

#56 16 2 17.16%8.17e−03 10.43%1.57e−03 18.28%2.19e−03 19.14%9.60e−03 16.08%8.51e−03 16.40%1.78e−03 17.23%9.99e−03

#57 18 2 17.11%2.10e−03 10.60%7.05e−03 19.08%9.77e−03 19.22%3.98e−03 16.45%1.76e−03 16.55%8.75e−03 17.27%1.10e−03

#58 20 2 17.23%4.14e−03 10.65%5.00e−03 18.48%2.30e−03 19.36%9.55e−03 16.58%2.38e−03 16.85%9.73e−03 17.41%4.73e−03

#59 22 2 17.34%4.43e−03 10.70%3.64e−03 18.60%3.78e−03 19.44%5.39e−03 16.69%2.50e−03 17.04%9.49e−03 17.49%5.74e−03

#60 24 2 17.43%6.11e−03 10.74%9.95e−03 18.93%8.97e−04 19.85%2.94e−04 16.78%3.52e−03 17.04%8.87e−03 17.59%2.70e−03

#61 2 4 5.42%1.95e−03 4.12%8.05e−03 6.21%3.39e−03 6.43%3.42e−03 5.20%1.00e−03 5.50%7.39e−03 5.65%4.46e−04

#62 4 4 14.65%8.40e−03 12.41%2.73e−03 15.21%9.90e−05 15.56%8.15e−03 13.81%4.65e−04 14.27%6.56e−03 14.31%4.00e−03

#63 6 4 19.84%2.04e−03 15.09%1.70e−04 20.49%9.26e−03 21.02%7.09e−03 18.85%8.25e−03 19.01%3.91e−03 19.61%3.53e−03

#64 8 4 22.64%4.20e−03 16.05%9.15e−03 23.49%9.18e−03 24.06%2.09e−03 21.59%7.57e−03 21.77%7.87e−03 22.36%8.73e−03

#65 10 4 23.81%8.97e−03 17.06%9.28e−03 25.09%8.56e−03 25.65%5.59e−03 23.17%4.77e−03 23.10%3.18e−03 23.90%5.97e−04

#66 12 4 25.28%5.35e−03 17.34%5.35e−04 25.97%7.14e−03 26.70%4.38e−03 24.05%6.08e−04 24.22%4.94e−03 25.23%3.49e−03

#67 14 4 25.93%2.46e−03 17.72%7.93e−03 26.96%8.31e−03 27.43%9.47e−03 24.78%9.31e−04 24.88%4.01e−03 25.75%4.81e−03

#68 16 4 26.41%7.86e−03 17.99%2.06e−03 27.29%6.14e−03 28.12%4.55e−03 25.33%6.56e−03 25.42%6.48e−03 26.37%6.00e−03

#69 18 4 26.83%4.61e−03 18.21%1.25e−03 27.82%1.03e−04 28.88%7.08e−04 25.76%7.81e−03 25.78%5.03e−03 26.83%7.92e−03

#70 20 4 27.17%7.87e−03 18.38%2.57e−04 28.16%8.25e−03 28.97%6.11e−03 26.11%7.81e−03 26.10%9.99e−03 27.19%3.34e−03

#71 22 4 27.44%4.02e−04 18.52%4.13e−04 28.53%4.13e−03 29.35%4.27e−03 26.39%8.80e−03 26.33%6.09e−03 27.44%7.98e−03

#72 24 4 27.67%4.49e−03 18.64%7.11e−04 28.66%8.87e−03 29.45%9.54e−03 26.62%9.34e−03 26.57%5.76e−03 27.63%7.48e−03

#73 2 6 4.80%4.14e−03 2.83%1.11e−03 6.08%5.81e−03 5.88%5.39e−03 4.45%7.47e−03 4.71%4.18e−03 5.02%5.04e−03

#74 4 6 14.24%5.25e−03 10.68%8.01e−04 15.12%3.64e−03 15.39%5.59e−03 13.48%8.00e−03 13.91%6.31e−03 13.05%1.89e−03

#75 6 6 21.73%1.45e−04 18.00%3.86e−03 22.44%3.57e−03 22.79%1.86e−03 20.70%6.15e−03 20.90%4.22e−03 21.21%9.40e−03

#76 8 6 26.30%7.99e−03 20.45%4.77e−03 26.86%1.69e−03 27.32%6.42e−03 25.05%1.45e−03 25.38%8.62e−03 25.71%9.06e−03

#77 10 6 28.83%9.85e−03 21.65%7.37e−03 29.89%4.11e−03 30.16%4.23e−03 27.73%4.20e−03 27.76%1.44e−03 28.59%4.23e−03

#78 12 6 30.38%3.30e−03 19.88%2.40e−03 31.57%8.38e−03 31.91%3.70e−03 29.35%2.39e−03 29.11%9.19e−03 30.42%2.51e−04

#79 14 6 31.63%9.91e−04 20.35%1.06e−04 32.77%2.52e−03 33.25%8.81e−04 30.64%2.18e−03 30.59%2.31e−03 31.58%5.91e−04

#80 16 6 32.56%3.67e−03 20.73%8.13e−03 33.52%5.44e−04 34.07%3.07e−03 31.55%7.15e−03 31.74%6.30e−03 32.43%3.03e−03

#81 18 6 33.36%3.38e−03 24.50%3.98e−03 34.22%7.10e−03 34.83%6.39e−03 31.99%1.08e−03 32.13%4.12e−04 33.22%4.36e−03

#82 20 6 33.93%8.43e−03 24.83%1.80e−03 34.90%6.41e−03 35.60%9.80e−03 32.56%7.94e−03 32.62%4.24e−03 33.84%1.95e−03

#83 22 6 34.39%1.89e−04 25.09%2.43e−03 35.39%1.13e−03 36.17%3.60e−03 33.00%3.93e−03 33.21%8.58e−03 34.28%3.09e−03

#84 24 6 34.78%8.06e−03 25.31%8.32e−03 35.77%9.18e−03 36.53%3.36e−03 33.38%4.71e−03 33.44%3.11e−03 34.71%9.07e−03

#85 2 8 4.67%6.95e−03 2.14%4.76e−03 5.37%4.53e−03 6.30%1.77e−03 4.12%1.50e−03 3.75%5.94e−03 4.91%4.89e−03

#86 4 8 13.14%3.28e−03 8.22%5.51e−03 14.52%7.48e−03 14.90%8.27e−03 12.53%6.74e−03 13.07%3.09e−03 12.28%8.41e−03

#87 6 8 22.38%8.61e−03 16.19%5.06e−03 23.08%3.78e−03 23.52%9.98e−03 21.08%9.04e−03 21.47%2.96e−04 21.46%3.86e−03

#88 8 8 28.26%8.84e−03 21.68%6.50e−03 28.77%9.48e−03 29.15%9.51e−03 26.77%5.30e−03 27.12%3.69e−03 27.51%4.88e−03

#89 10 8 31.74%8.13e−03 24.59%2.05e−03 32.65%9.80e−03 33.26%5.23e−03 30.49%7.90e−03 30.81%5.94e−03 31.58%2.63e−03

#90 12 8 34.07%7.46e−03 25.38%4.94e−03 35.09%7.75e−04 35.65%2.58e−04 33.00%9.97e−03 33.02%5.54e−03 34.09%6.47e−03

#91 14 8 35.73%1.20e−03 26.78%2.69e−03 36.96%7.96e−03 37.56%1.42e−03 34.60%4.78e−03 34.57%7.89e−03 35.61%7.13e−03

#92 16 8 37.00%8.57e−03 27.43%3.72e−04 38.28%7.02e−03 38.91%9.20e−04 35.91%2.32e−03 35.92%5.79e−03 37.11%4.51e−03

#93 18 8 38.12%4.21e−03 28.01%4.52e−03 39.12%4.01e−04 40.12%5.74e−03 36.90%2.92e−03 36.83%2.36e−03 37.18%3.74e−03

#94 20 8 38.82%9.37e−03 28.48%4.25e−03 40.08%3.08e−03 40.83%5.53e−04 37.84%5.66e−03 37.39%8.46e−03 38.79%1.62e−03

#95 22 8 39.47%3.30e−03 28.86%2.97e−03 40.77%4.14e−03 41.28%3.11e−03 38.52%8.91e−03 38.04%4.62e−03 39.58%6.94e−03

#96 24 8 40.02%2.60e−03 29.18%3.75e−03 41.47%5.81e−03 42.04%6.75e−03 39.09%6.27e−03 38.64%2.81e−03 39.97%6.15e−03




#97 3 3 6.92%6.02e−03 5.88%5.43e−03 7.06%7.80e−03 7.23%8.31e−03 6.41%5.47e−03 6.58%8.55e−03 6.53%3.59e−03

#98 6 3 11.49%3.15e−03 8.45%8.54e−03 11.85%9.49e−03 12.21%8.65e−03 10.89%2.27e−03 10.97%8.87e−03 11.26%2.85e−03

#99 9 3 12.91%2.69e−03 9.32%5.91e−03 13.45%8.19e−03 13.91%5.17e−03 12.32%7.14e−03 12.44%2.14e−03 12.77%6.31e−03

#100 12 3 13.66%3.60e−03 9.30%7.10e−03 14.34%7.83e−03 14.93%1.01e−03 13.16%4.12e−03 13.24%2.95e−03 13.66%5.92e−03

#101 15 3 14.10%9.66e−03 9.52%3.87e−03 14.87%5.46e−03 15.31%2.33e−03 13.62%8.39e−03 13.55%6.54e−03 14.15%5.83e−03

#102 18 3 14.40%6.66e−03 9.67%8.60e−03 15.25%4.00e−06 15.64%6.18e−03 13.92%4.23e−03 13.94%4.06e−03 14.45%7.40e−03

#103 21 3 14.61%8.70e−03 9.77%8.10e−05 15.48%4.73e−03 15.97%7.43e−03 14.13%4.23e−03 14.17%5.77e−03 14.65%3.00e−03

#104 24 3 14.77%2.01e−04 9.85%8.41e−03 15.70%6.64e−03 16.00%1.42e−03 14.29%8.69e−04 14.34%6.88e−04 14.85%3.26e−03

#105 27 3 14.74%8.49e−04 9.16%2.37e−03 15.81%1.37e−03 16.41%8.71e−03 14.35%8.67e−03 14.33%1.99e−03 14.89%3.30e−03

#106 30 3 14.84%6.36e−03 9.20%6.33e−03 15.82%2.13e−04 16.30%4.76e−03 14.44%5.59e−03 14.31%3.05e−03 14.99%3.35e−03

#107 33 3 14.92%2.79e−03 9.23%5.48e−03 16.01%6.26e−03 16.38%9.63e−03 14.53%8.24e−03 14.62%1.87e−03 15.10%9.61e−03

#108 36 3 14.98%1.25e−03 9.26%4.94e−03 16.69%9.08e−03 16.56%8.19e−03 14.59%1.27e−03 14.56%8.14e−03 15.15%5.97e−03

#109 3 6 5.37%3.64e−03 3.77%8.33e−03 6.37%3.80e−03 6.59%7.31e−03 5.34%5.33e−03 5.63%2.57e−03 5.58%9.05e−04

#110 6 6 13.59%1.52e−03 10.91%1.69e−03 13.87%8.54e−03 14.09%9.51e−03 12.75%3.70e−03 12.94%9.32e−03 12.97%4.39e−03

#111 9 6 17.42%3.43e−03 12.27%7.89e−03 17.97%9.98e−03 18.30%7.92e−03 16.73%7.72e−03 16.86%8.94e−03 17.11%6.42e−03

#112 12 6 19.32%9.43e−03 12.56%1.95e−04 19.99%5.47e−03 20.48%9.33e−03 18.65%7.32e−03 18.34%4.56e−03 19.13%5.40e−03

#113 15 6 20.26%6.09e−03 12.96%8.97e−03 21.14%4.13e−03 21.66%7.33e−03 19.56%4.09e−03 19.67%9.87e−03 20.35%2.26e−03

#114 18 6 21.01%3.21e−03 13.24%5.66e−03 22.00%1.31e−03 22.60%1.85e−03 20.31%3.36e−03 20.52%6.28e−03 21.20%9.92e−03

#115 21 6 21.53%4.82e−03 13.45%1.52e−03 22.41%4.79e−03 23.37%1.73e−03 20.84%2.60e−03 20.65%2.28e−03 21.70%6.92e−03

#116 24 6 21.91%9.10e−04 13.60%4.16e−03 23.14%4.64e−03 23.54%3.19e−03 21.24%1.68e−03 21.25%9.05e−03 22.14%3.63e−03

#117 27 6 22.21%4.20e−03 13.72%3.21e−03 23.34%7.52e−03 23.77%8.99e−03 21.54%1.19e−03 21.31%7.83e−03 22.39%6.00e−03

#118 30 6 22.45%4.59e−03 13.81%7.80e−03 23.74%3.23e−03 24.19%6.21e−03 21.79%2.52e−03 21.91%4.73e−03 22.69%6.96e−03

#119 33 6 22.65%1.77e−03 13.89%3.79e−03 23.68%8.29e−03 24.34%2.42e−04 21.99%2.74e−03 21.94%6.14e−03 22.90%7.09e−03

#120 36 6 22.81%8.10e−03 13.95%2.72e−03 23.96%3.54e−03 24.60%1.60e−03 22.24%2.03e−03 22.07%7.47e−03 23.13%8.89e−03

#121 3 9 4.93%9.96e−03 2.34%3.92e−03 5.94%7.47e−03 6.25%3.86e−03 4.58%3.93e−03 4.83%5.52e−03 5.42%2.74e−03

#122 6 9 13.20%8.52e−03 9.16%9.52e−03 13.86%1.12e−03 14.11%5.03e−03 12.65%1.86e−03 12.89%7.17e−03 12.02%8.69e−03

#123 9 9 19.01%6.71e−03 14.25%4.89e−03 19.94%6.02e−03 20.23%4.19e−03 18.55%5.00e−03 18.60%3.29e−03 18.91%1.10e−03

#124 12 9 22.42%3.00e−03 15.13%1.04e−03 23.18%8.20e−03 23.73%1.85e−03 21.66%3.51e−03 21.89%1.53e−03 22.30%2.18e−03

#125 15 9 24.39%6.66e−03 15.43%7.26e−03 25.20%5.24e−03 25.63%1.37e−03 23.41%6.79e−03 23.58%3.34e−03 24.22%5.37e−03

#126 18 9 25.76%4.07e−03 15.86%2.76e−03 26.51%5.89e−03 27.04%3.00e−04 24.89%2.39e−03 24.83%9.99e−03 25.57%2.03e−03

#127 21 9 26.51%2.25e−03 16.16%3.27e−03 27.35%5.12e−03 27.90%6.02e−03 25.81%3.81e−03 25.52%5.62e−03 26.46%9.25e−03

#128 24 9 27.19%1.94e−03 16.39%3.94e−03 28.33%1.07e−03 28.71%6.93e−03 26.50%1.74e−03 26.31%7.50e−03 27.10%6.99e−04

#129 27 9 27.87%8.63e−03 18.18%9.64e−03 28.62%2.74e−03 29.33%5.82e−03 27.01%1.61e−03 26.82%1.45e−04 26.89%4.77e−03

#130 30 9 28.28%8.40e−05 18.35%6.20e−03 29.03%6.22e−03 29.55%3.20e−03 27.43%7.70e−04 27.23%9.11e−03 28.19%1.45e−04

#131 33 9 28.62%7.62e−03 18.50%5.09e−03 29.67%6.44e−03 30.07%4.02e−03 27.77%8.93e−03 27.44%6.12e−03 28.52%1.09e−03

#132 36 9 28.90%9.66e−03 18.62%5.21e−03 29.70%2.53e−03 30.61%7.83e−03 28.07%9.55e−04 27.79%5.30e−03 28.81%4.45e−03

#133 3 12 4.39%2.28e−03 1.63%1.72e−03 5.77%6.89e−03 6.18%2.15e−03 4.25%8.16e−04 4.60%5.09e−03 5.35%9.94e−03

#134 6 12 12.43%1.37e−03 7.06%8.22e−04 13.48%2.04e−04 13.77%8.97e−03 11.84%7.71e−03 12.03%1.15e−03 11.20%5.70e−03

#135 9 12 19.82%9.60e−03 14.50%2.61e−03 20.80%8.38e−03 20.95%3.82e−03 19.19%2.94e−03 18.95%8.12e−03 19.20%8.14e−03

#136 12 12 24.42%8.77e−03 17.66%3.96e−03 25.13%7.04e−03 25.46%7.59e−03 23.53%8.57e−03 23.60%7.17e−03 23.78%3.72e−03

#137 15 12 27.03%1.16e−03 18.84%5.32e−03 28.00%9.39e−04 28.17%9.19e−03 26.18%1.51e−03 26.03%7.72e−03 26.48%7.24e−03

#138 18 12 28.60%7.25e−03 19.77%8.80e−03 29.77%8.15e−03 30.08%4.12e−03 27.74%8.27e−03 27.90%6.52e−03 28.54%1.17e−03

#139 21 12 30.19%8.52e−03 20.39%4.99e−03 30.96%1.09e−03 31.33%6.21e−03 29.34%7.91e−03 29.04%3.84e−03 29.84%7.44e−03

#140 24 12 31.02%4.59e−03 20.85%5.44e−03 31.96%1.43e−03 32.51%2.60e−03 30.27%3.29e−03 30.10%6.72e−03 30.93%1.02e−03

#141 27 12 31.77%7.25e−03 21.20%2.50e−04 32.76%3.70e−04 33.28%4.92e−04 31.02%9.37e−03 30.44%6.16e−03 31.64%4.77e−03

#142 30 12 32.35%5.04e−03 21.49%9.24e−03 33.78%4.14e−04 33.85%4.31e−03 31.65%7.06e−03 31.17%8.78e−03 32.24%5.27e−03

#143 33 12 32.82%9.34e−03 21.72%5.31e−03 34.20%7.62e−03 34.36%2.11e−03 32.16%7.48e−03 31.74%4.78e−03 32.82%1.25e−03

#144 36 12 33.22%9.22e−03 21.92%8.25e−03 34.24%3.21e−03 34.72%8.37e−03 32.58%2.66e−03 31.73%5.90e−03 33.26%6.32e−03




(e): MO-FA

Figure 8.18. NTT network (TM1). Comparison among NSGA-II, SPEA2 and each proposed MOEA byusing the HV indicator. Note that, each point represents the mean of the medians of HV reported inTable 8.18 for W =1,2,3,4.




(e): MO-FA






Figure 8.21. NTT network. Illustrative summary of the performance of each proposed MOEA by usingthe HV indicator. Note that, each point represents the mean of the medians of HV reported in Table8.18 (TM1), Table 8.19 (TM2), and Table 8.20 (TM3) for the different values of W .


Table 8.21. Statistical Analysis among the MOEAs in the NTT network. The table indicates in whichdata sets two algorithms have no statistically significant differences.


DEPT MO-VNS 0 out of 144 data sets -NSGA-II 8 out of 144 data sets #13 #25 #37 #38 #61 #73 #86 #121SPEA2 57 out of 144 data sets #3 #4 #5 #13 #18 #20 #25 #28 #29 #30

#32 #37 #46 #53 #54 #55 #56 #65 #66 #68#69 #70 #71 #72 #78 #79 #80 #81 #82 #83#84 #90 #91 #92 #94 #95 #96 #99 #100 #101#102 #103 #104 #113 #123 #124 #127 #128 #130 #131#132 #138 #140 #141 #142 #143 #144

MO-GSA 1 out of 144 data sets #1MO-ABC 4 out of 144 data sets #1 #13 #49 #97MO-FA 4 out of 144 data sets #25 #37 #109 #133

MO-VNS NSGA-II 2 out of 144 data sets #1 #13SPEA2 1 out of 144 data sets #1MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 2 out of 144 data sets #1 #13

NSGA-II SPEA2 11 out of 144 data sets #1 #13 #25 #49 #61 #62 #87 #97 #109 #110#129

MO-GSA 0 out of 144 data sets -MO-ABC 0 out of 144 data sets -MO-FA 44 out of 144 data sets #1 #7 #13 #21 #22 #24 #25 #44 #57 #65

#67 #68 #69 #70 #71 #72 #77 #79 #81 #82#84 #90 #91 #92 #93 #98 #99 #100 #101 #102#103 #104 #105 #106 #107 #108 #111 #113 #116 #118#119 #123 #126 #136

SPEA2 MO-GSA 0 out of 144 data sets -MO-ABC 1 out of 144 data sets #25MO-FA 7 out of 144 data sets #1 #13 #25 #37 #97 #129 #135

MO-GSA MO-ABC 8 out of 144 data sets #1 #8 #13 #25 #57 #108 #135 #142MO-FA 0 out of 144 data sets -

MO-ABC MO-FA 0 out of 144 data sets -

between the MO-GSA and the rest of MOEAs are statistically significant in almost all data sets,with the exception of the MO-ABC algorithm where in 8 out of the 144 scenarios, the differencesare statistically not significant.

The second comparison among the MOEAs is based on the SC indicator, which purpose is toreport the percentage of coverage by pair of algorithms. In Table 8.22, we present the averageresults of this comparison.

As we can see, the MO-GSA clearly covers the majority of the non-dominated solutions obtainedby the rest of MOEAs; however, the rest of MOEAs only cover a few solutions of the Pareto frontobtained by the MO-GSA.


Table 8.22. NTT network. Comparison among the MOEAs by using the Set Coverage (SC) indicator,A B. Note that, SC represents the mean coverage of an algorithm A over an algorithm B in all thedata sets.


DEPT


MO-VNS


NSGA-II


SPEA2


MO-GSA


MO-ABC


MO-FA



Table 8.23. Ranking of the MOEAs in the 6-node and COST239 optical network topologies when tacklingthe TG problem.

6-node Network

Traffic Matrix 1 Traffic Matrix 2 Traffic Matrix 3(988 OC-1 units) (1976 OC-1 units) (3952 OC-1 units)

1. MO-ABC 53.70% 1. MO-ABC 54.83% 1. MO-ABC 45.09%2. MO-FA 52.16% 2. MO-GSA 53.43% 2. MO-GSA 43.87%3. MO-GSA 51.90% 3. DEPT 53.06% 3. DEPT 43.41%4. DEPT 51.11% 4. MO-FA 52.33% 4. MO-FA 42.88%5. MO-VNS 49.94% 5. NSGA-II 51.47% 5. NSGA-II 42.19%6. NSGA-II 48.62% 6. MO-VNS 50.68% 6. SPEA2 41.48%7. SPEA2 47.65% 7. SPEA2 50.43% 7. MO-VNS 40.51%

COST239 Network


1. MO-ABC 46.84% 1. MO-ABC 52.08% 1. MO-ABC 48.27%2. MO-GSA 44.21% 2. MO-GSA 50.01% 2. MO-GSA 46.72%3. DEPT 43.97% 3. DEPT 50.00% 3. DEPT 46.52%4. MO-VNS 43.11% 4. NSGA-II 48.27% 4. NSGA-II 45.08%5. MO-FA 42.57% 5. MO-FA 47.66% 5. MO-FA 44.74%6. NSGA-II 42.17% 6. SPEA2 47.48% 6. SPEA2 44.04%7. SPEA2 41.75% 7. MO-VNS 46.72% 7. MO-VNS 41.33%

8.2.5 Conclusions of the Comparative Study

To conclude this performance study, we summarize some conclusions for each approach tested inthe study.

As we have seen, the DEPT algorithm is a competitive algorithm, it is the third best algorithmin almost all network topologies. However, the trajectory-based algorithm (MO-VNS) is not a goodchoice for this problem due to the lack of quality in their non-dominated solutions, particularlywhen it deals with medium or large networks, and also when deals with small networks but mediumor large loads of traffic.

The non-dominated solutions obtained by the well-known MOEAs (NSGA-II and SPEA2) haveresulted to be dominated by several of the proposed algorithms in all scenarios tested.

It seems that some of the algorithms based on swarm intelligence (MO-GSA and MO-ABC)obtain high quality results when solving the Traffic Grooming problem.

On the one hand, the algorithm based on the behaviour of honey bees, the MO-ABC algorithm,seems to be a very promising algorithm when dealing with small networks, such as 6-node andCOST239; however, its quality slightly decreases when dealing with large optical networks. In thesame way, the MO-FA performs better with small networks than with large ones.

On the other hand, the MO-GSA is a very efficient MOEA when the number of nodes isincreased, according to the previous sections. Furthermore, this algorithm based on the law ofgravity and mass interactions obtains the second best results in the small network topologies.

To sum up, in Table 8.23 and Table 8.24, we present a ranking of the seven MOEAs whendealing with the different mesh optical networks. Note that in this ranking the MOEAs are sortedby the median value of HV in the 144 data sets of each network. As we can see, the MO-ABCobtains a higher value of HV (around 2%) than the second MOEA (MO-GSA) in the three trafficmatrices of the topologies 6-node and COST239.

We realize that in the NSF network the order of MOEAs is the same independently on the


Table 8.24. Ranking of the MOEAs in the NSF and NTT optical network topologies when tackling theTG problem.

NSF Network


1. MO-GSA 47.23% 1. MO-GSA 51.36% 1. MO-GSA 48.44%2. MO-ABC 44.19% 2. MO-ABC 49.22% 2. MO-ABC 46.83%3. DEPT 42.44% 3. DEPT 47.84% 3. DEPT 45.85%4. NSGA-II 41.52% 4. NSGA-II 46.70% 4. NSGA-II 44.89%5. SPEA2 41.24% 5. SPEA2 46.25% 5. SPEA2 44.09%6. MO-FA 39.73% 6. MO-FA 45.15% 6. MO-FA 43.30%7. MO-VNS 37.89% 7. MO-VNS 41.14% 7. MO-VNS 35.10%

NTT Network


1. MO-GSA 19.60% 1. MO-GSA 25.53% 1. MO-GSA 21.71%2. MO-ABC 18.91% 2. MO-ABC 24.94% 2. MO-ABC 21.29%3. SPEA2 17.71% 3. DEPT 23.94% 3. DEPT 20.38%4. DEPT 17.65% 4. SPEA2 23.81% 4. SPEA2 20.28%5. NSGA-II 16.83% 5. NSGA-II 23.09% 5. MO-FA 19.74%6. MO-FA 16.52% 6. MO-FA 22.96% 6. NSGA-II 19.69%7. MO-VNS 12.25% 7. MO-VNS 16.89% 7. MO-VNS 13.38%

amount of traffic used, where the MO-GSA is the most promising approach. Finally, we canobserve that it also happened in the large NTT network topology, where the best MOEA is theMO-GSA, followed by the MO-ABC.

Finally, in Figure 8.22, we present the best set of non-dominated solutions obtained by the bestand the worst algorithm at each network and each traffic matrix, according to Table 8.23 and Table8.24. Depending on the traffic matrix: small, medium, or large; the set of non-dominated solutionsshowed in Figure 8.22 corresponds with the scenarios T=12 W=4 (Traffic Matrix 1), T=24 W=8(Traffic Matrix 2), or T=36 W=12 (Traffic Matrix 3); respectively. As we may observe, the setof non-dominated solutions obtained by the best approach at each scenario is able to cover muchmore objective space, leading to a higher value of HV.

In general, seeing Figure 8.22, we realize that the non-dominated solutions obtained by thebest algorithms (MO-ABC or MO-GSA), are better distributed in the objective space than thenon-dominated solutions achieved by the worst multiobjective approach. Concretely, if we focus onthose plots with the largest traffic matrices (TM3), we may observe that the area of the objectivespace with a low value of throughput (f1) is explored by the best and the worst approach; however,the area with a higher value of throughput is only explored by the best approach.

8.3 Comparison with other works

In this subsection we present a variety of comparisons with other approaches that have tackledthe Traffic Grooming problem. The final objective is to demonstrate the good performance of ourapproaches compared with other published by other authors. In these comparisons, we have usedour best metaheuristics (MO-ABC) and the 6-node optical network topology in order to make afair comparison with other authors (using the same traffic matrix, TM1). In order to comparethe approaches, we present an illustrative relationship between the network throughput and the


(a): 6-n TM1 T=12 W=4 (b): 6-n TM2 T=24 W=8 (c): 6-n TM3 T=36 W=12

(d): COST239 TM1 T=12 W=4 (e): COST239 TM2 T=24 W=8 (f): COST239 TM3 T=36 W=12

(g): NSF TM1 T=12 W=4 (h): NSF TM2 T=24 W=8 (i): NSF TM3 T=36 W=12

(j): NTT TM1 T=12 W=4 (k): NTT TM2 T=24 W=8 (l): NTT TM3 T=36 W=12

Figure 8.22. Best set of non-dominated solutions obtained by the best (∗) and the worst () approachat each scenario.


number of wavelengths per link for the algorithms under study. Note that, for the multiobjectiveMO-ABC, we have considered the highest value of throughput obtained in each data set.

In [63], the authors proposed two heuristics for solving the Traffic Grooming problem, butoptimizing only one objective: traffic throughput. These heuristics are:

• Maximizing Single-Hop Traffic (MST). This heuristics attempts to establish lightpaths be-tween source-destination pairs with highest T raffic(s, d) values, subject to constraints onthe number of transceivers at the two end nodes and the availability of a wavelength inthe path connecting the two end nodes. The connection requests between s and d will becarried on the newly established lightpath whenever possible. If there is enough capacity inthe network, every connection will traverse a single lightpath hop. Otherwise, the algorithmwill try to carry the blocked connection requests using available spare capacity of the virtualtopology.

• Maximizing Resource Utilization (MRU). This heuristics tries to establish lightpaths betweenthe node pairs with maximum resource utilization values. When no lightpath can be set up,the remaining blocked traffic requests will be routed on the virtual topology based on theirconnection resource utilization value.

In Figure 8.23, we present a comparison between the heuristics proposed in [63] and MO-ABC.In Figure 8.23(a), we keep constant the number of wavelengths per link, W =3, and we vary thenumber of transceivers per node. As we can see, our MOEA obtains higher values of throughput inall data sets. Furthermore, we may note that the difference in successfully routed traffic requestsbetween the heuristics (MST and MRU) and MO-ABC, comes out at 36 OC-1 units of throughputwith 5 and 7 transceivers per node.

If we increase the number of wavelengths per link (W =4, see Figure 8.23(b)), we can see that,again, our metaheuristics obtains better performance than MST and MRU. We may also see thatin data set T=5,W=4, all heuristics obtain maximum throughput (OC-988).

Zhu et al. [66] present an INtegrated Grooming PROCedure (INGPROC) to solve the trafficgrooming problem. It uses an Integrated Grooming Algorithm based on an Auxiliary Graph (IGA-BAG) model, to accommodate connection request demands, maximizing only one objective: traffic

(a): W=3 (b): W=4

Figure 8.23. Maximum throughput comparison among MO-ABC, MST and MRU


(a): W=3 (b): W=4

Figure 8.24. Maximum throughput comparison between MO-ABC and INGPROC (with different traffic-request-selection schemes)

throughput. For INGPROC, the authors propose in [66] the following traffic-request-selectionschemes for static traffic grooming:

• Least Cost First (LCF). Chooses the most cost-effective traffic request under the currentnetwork state and routes it. The cost of a traffic request is the weight of the shortest pathfor routing the traffic on the corresponding auxiliary graph divided by the amount of traffic,which is computed as the granularity multiplied by the units of traffic.

• Maximum Utilization First (MUF). Selects the connection with the highest utilization, whichis defined as the total number of requests divided by the number of hops from the source tothe destination on the physical topology.

• Maximum Amount First (MAF). Selects the connection with the largest number of demandsand routes it.

A throughput comparison between INGPROC (with different traffic selection schemes) andMO-ABC is shown in Figure 8.24. On the one hand, in Figure 8.24(a), we present a comparisonamong the heuristics with 3 wavelengths per link and varying the number of transceivers per node.We may observe that the best traffic-request-selection scheme for INGPROC is LCF, which obtainshigher value of throughput than MUF and MAF in almost all data sets (except on T=7,W=3, inwhich the three schemes obtain equal throughput). However, our approach obtains better resultsthan INGPROC with the LCF scheme in all data sets.

On the other hand, in Figure 8.24(b), we have compared the heuristics by establishing fourwavelengths per link (W =4). As we can see, the best traffic selection scheme for INGPROC isLCF, as occurred with W = 3 (see Figure 8.24(a)). Furthermore, we can see that the MO-ABCalgorithm obtains very promising values of throughput compared with the proposed INGPROC.

Prathombutr et al. [79] propose the well-known Strength Pareto Evolutionary Algorithm(SPEA) for maximizing traffic throughput, minimizing the number of lightpaths established, andminimizing the average propagation delay of the lightpaths. They also formulate the traffic groom-ing problem both allowing and not allowing wavelength conversion at all nodes of the network, so


(a): W=3 (b): W=4

Figure 8.25. Maximum throughput comparison between MO-ABC and SPEA

(a): W=3 (b): W=4

Figure 8.26. Maximum throughput comparison between MO-ABC and TGCP

in order to make a fair comparison, we make a comparison only with the SPEA version with nowavelength conversion.

In Figure 8.25, we present a direct comparison between SPEA and our proposed metaheuristics(MO-ABC). In the first place, we make a comparison by using three wavelengths per link (seeFigure 8.25(a)). As we can observe, the two multiobjective approaches obtain similar values ofthroughput in all data sets.

Next, in Figure 8.25(b), we compare the approaches with W =4. For these particular data sets,the performance of MO-ABC is slightly higher than the performance obtained by SPEA in almostall data sets (both obtain the maximum throughput in data set T=5,W=4 ).

Finally, De et al. [55] propose an algorithm to handle general multi-hop static Traffic Groomingbased on the Clique Partitioning (TGCP) concept. This approach works in two steps in order tomaximize the traffic throughput. In the first step, it constructs a virtual topology using clique-partitioning based heuristics and traffic grooming is taken into account during partitioning. In the


second step, as many connection requests as possible are routed on the virtual topology using asingle lightpath. The blocked connection requests are routed using multiple lightpaths.

As we have done with other approaches, in Figure 8.26 we present a comparison between ourbest MOEA (MO-ABC) and the TGCP algorithm proposed in [55].

If we establish W = 3 (see Figure 8.26(a)), we can see that the performance of MO-ABC ishigher in all data sets. Finally, if we focus on Figure 8.26(b) (W =4), we can observe how theMO-ABC has a similar behaviour to TGCP in terms of throughput.

Having made comparisons with a variety of approaches published in the literature, we canconclude that our multiobjective proposal obtains, in general, better results.

8.4 Performance of the Parallel Approach

In order to reduce the runtime, in this section we apply the parallel versions of the MultiobjectiveArtificial Bee Colony (pMOABC) presented in Section 6.8.2 to the Traffic Grooming problem. Inthis way, we are able to obtain results of identical quality in a reasonable amount of time. Adetailed explanation of the pMOABC approach is presented in Chapter 6 (Section 6.8.2).

As we explained in Chapter 3, the performance of a parallel algorithm is commonly measuredby computing the speedup and efficiency. For further information about these metrics, please referto Section 3.3.5.

We have used two different optical network topologies. The first one is a small network withsix nodes (6-node), a capacity (C) per link of OC-48, and a traffic matrix with a total amountof traffic of 988 OC-1 units (TM1). The second one is the large real-world National ScienceFoundation (NSF) network topology with 14 nodes, a capacity of OC-192, and a total amountof requested traffic of 5724 OC-1 units. We have tested different scenarios over these topologies,varying the number of transceivers per node (T ) and the number of wavelengths (W ) per link:

• 6-node Network: T=3,4,5,7 W=3 and T=3,4,5 W=4.

• NSF Network: T=3,4,5 W=3 and T=4,5,6 W=4.

The network topologies, the traffic matrices, and the instances were explained in Chapter 5.For further information about them, please refer to Section 5.4.

Like in Chapter 7, we perform 30 independent runs of each parallel version on a homogenouscluster which consists of 4 multi-core nodes, where each node is equipped with a 8 cores; thus, wehave a total of 32 cores. The stopping criterion was established to 3000 (6-node) and 7500 (NSF)generations. Furthermore, we have carried out the statistical analysis presented in Section 3.4 withthe aim of determining whether the differences of speedup and efficiency are statistically significantwith a confidence level of 95% (p-value under 0.05).

Since we have four multi-core nodes interconnected through the same data network, we presentdifferent comparisons among shared-memory (OpenMP), distributed-memory (MPI), and/or hy-brid shared/distributed-memory approaches (OpenMP+MPI). We therefore compare OpenMP orHybrid versions with the MPI version in different scenarios with 2, 4, 8, 16, and 32 cores.

The parameter configuration of the pMOABC algorithm is presented in Section 8.1. Therefore,the configuration of the pMOABC in these experiments is: Population size (Ns)=100, MaximumLimit value (limitmax)=5, and Mutation rate (F )=25%.

In the first place, we run the sequential version of the MO-ABC algorithm in order to know theruntime of the algorithm when using a single-core. In Table 8.25 we present the mean runtime andthe standard deviation in 30 independent runs. As we may observe, there exist differences among


Table 8.25. Mean runtime and standard deviation for the sequential version of the MO-ABC algorithmin 30 independent runs.

6-node NSF

T W T1 T W T1

3 3 54.669.72e−01 3 3 384.171.89e+00

4 3 56.195.83e−01 4 3 448.521.69e+00

5 3 55.795.41e−01 5 3 484.726.37e+00

7 3 55.754.05e−01 4 4 504.111.89e+00

3 4 54.649.12e−01 5 4 580.787.17e−01

4 4 56.272.21e−01 6 4 620.766.52e+00

5 4 55.321.74e−01

the scenarios in the NSF topology whereas in the 6-node network the runtime remains almostconstant.

Then, we measure the performance of the pMOABC by using OpenMP and MPI versions with2 cores.

In Table 8.26, we can see that both approaches obtain very promising efficiency, over 95% in the6-node scenarios as well as in the NSF scenarios; which leads to obtain the same quality results innearly half of the sequential runtime. However, we may observe that the OpenMP version achievesbetter performance, in terms of speedup and efficiency, than the MPI version in all the scenarios,there exists a difference of 2% of efficiency.

In addition, the MPI version spends a few seconds in communication, whereas the OpenMPdoes not need communication (message passing) among threads; therefore, no communicationtime is spent. In Figure 8.27, we compare the runtime of the two parallel versions, including thecommunication time and the computation time.

Now, in Table 8.27, we study the parallel performance of the shared-memory version and thedistributed-memory version, with a higher number of cores (4 cores).

As we can see in Table 8.27, the differences between the parallel approaches are remarkable. On

Table 8.26. Mean runtime, speedup, and efficiency for the OpenMP and MPI versions of the pMOABCalgorithm with 2 cores in 30 independent runs. Note that we report the standard deviation of theruntime.

OpenMP MPI

T W T2 S2 E2 T2 S2 E2 SS∗

6-no

de

3 3 27.522.11e−01 1.986 99.30% 28.344.27e−01 1.929 96.43%

4 3 28.193.71e−01 1.993 99.65% 28.665.82e−02 1.960 98.02%

5 3 27.982.88e−01 1.994 99.69% 28.673.19e−02 1.946 97.29%

7 3 28.133.26e−02 1.982 99.11% 28.723.61e−01 1.942 97.08%

3 4 27.494.84e−01 1.988 99.38% 28.253.41e−01 1.934 96.71%

4 4 28.331.70e−03 1.987 99.33% 28.896.65e−03 1.948 97.39%

5 4 27.804.05e−01 1.989 99.47% 28.502.05e−01 1.941 97.05%

NSF

3 3 194.222.65e+00 1.978 98.90% 196.769.60e−02 1.953 97.63%

4 3 226.356.87e−01 1.981 99.07% 231.271.65e+00 1.939 96.97%

5 3 244.513.38e+00 1.982 99.12% 249.775.00e+00 1.941 97.03%

4 4 253.701.85e+00 1.987 99.35% 258.362.51e−01 1.951 97.56%

5 4 293.711.35e+00 1.977 98.87% 305.194.80e+00 1.903 95.15%

6 4 312.109.30e−01 1.989 99.45% 322.603.80e+00 1.924 96.21%



(a): 6-node network (b): NSF network

Figure 8.27. Communication and computation time for the OpenMP and MPI versions of the pMO-ABCalgorithm with 2 cores.

the one hand, we can see that the OpenMP obtains nearly ideal efficiency in the 6-node network,whereas the efficiency of the MPI version decreases to 90%. On the other hand, we can see thatin the second network topology, the efficiency of the OpenMP version slightly decreases; however,it keeps better than the efficiency of the distributed-memory version.

Like in the previous experiment, the communication and computation time of each parallelapproach is presented in Figure 8.28.

In the next experiment, we use a single multi-core system; thus, we use the maximum numberof shared-memory cores (8 cores) in order to compare the OpenMP and MPI versions of thepMOABC.


OpenMP MPI


6-no

de

3 3 13.953.81e−02 3.919 97.98% 15.101.24e−02 3.620 90.50%

4 3 14.074.60e−02 3.994 99.86% 15.251.59e−01 3.686 92.14%

5 3 14.101.92e−01 3.956 98.89% 15.151.59e−01 3.682 92.04%

7 3 14.126.39e−02 3.948 98.71% 15.172.24e−01 3.675 91.87%

3 4 13.962.43e−01 3.914 97.85% 15.161.88e−01 3.604 90.10%

4 4 14.286.48e−02 3.941 98.53% 15.461.91e−01 3.639 90.98%

5 4 14.105.29e−02 3.923 98.08% 15.112.25e−01 3.662 91.55%

NSF

3 3 100.312.91e−01 3.830 95.74% 103.133.43e−02 3.725 93.12%

4 3 115.111.96e+00 3.896 97.41% 120.211.38e+00 3.731 93.28%

5 3 127.451.43e+00 3.803 95.08% 129.601.26e+00 3.740 93.51%

4 4 131.731.40e+00 3.827 95.67% 136.942.01e−01 3.681 92.03%

5 4 153.001.57e+00 3.796 94.90% 160.871.38e−01 3.610 90.26%

6 4 162.092.15e−02 3.830 95.75% 170.441.42e+00 3.642 91.05%





If we focus on studying the speedup and efficiency of both approaches, we realize that theOpenMP performs much better than the MPI in all the data sets, see Table 8.28. As we may see,the OpenMP reaches a mean efficiency in the range 92-96%, whereas the efficiency of the MPIpMOABC is around 80%.

In this case, we may observe that the communication time in the MPI version is higher thanwith two and four MPI processes, leading to a partial loss of efficiency. In Figure 8.29, we presentan illustrative comparison of the runtime of each parallel approach.

To sum up, we may observe that, whereas the OpenMP version is able to run nearly eight timesfaster, the MPI version is only around 6.5 times faster.


OpenMP MPI


6-no

de

3 3 7.082.22e−02 7.717 96.46% 8.427.20e−02 6.493 81.16%

4 3 7.447.03e−02 7.554 94.43% 8.692.01e−02 6.466 80.83%

5 3 7.291.56e−02 7.656 95.70% 8.459.33e−02 6.600 82.50%

7 3 7.344.28e−02 7.596 94.95% 8.431.16e−01 6.611 82.64%

3 4 7.127.00e−02 7.674 95.93% 8.591.09e−01 6.358 79.47%

4 4 7.281.65e−02 7.725 96.56% 8.713.74e−02 6.461 80.76%

5 4 7.221.53e−02 7.657 95.71% 8.601.24e−01 6.434 80.42%

NSF

3 3 50.621.19e+00 7.589 94.86% 58.248.45e−01 6.596 82.46%

4 3 59.449.01e−02 7.546 94.32% 68.657.84e−01 6.533 81.66%

5 3 65.171.96e−01 7.438 92.98% 74.826.98e−01 6.478 80.98%

4 4 68.401.21e+00 7.370 92.13% 77.286.14e−01 6.523 81.54%

5 4 78.161.27e−01 7.431 92.89% 90.106.71e−01 6.446 80.57%

6 4 83.857.23e−01 7.404 92.54% 94.914.50e−01 6.541 81.76%





The nodes of our system consists on 4 nodes with 8 cores per node; therefore, we need to usea hybrid version to exploit 2 nodes, a total of 16 cores. We compare a hybrid version with thepure MPI version in a system with two nodes interconnected through the same data network. Inthe hybrid version we have 2 MPI processes where each process exploits separately each multi-coresystem by using OpenMP directives.

The runtime, speedup, efficiency of the two parallel approaches are presented in Table 8.29. Aswe may see, the efficiency of the hybrid version is very promising (around 90%), whereas in thecase of the MPI version, the efficiency depends on the scenario and ranges between 64-76%, with68% of average efficiency.

Table 8.29. Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions of thepMOABC algorithm with 16 cores in 30 independent runs. Note that we report the standard deviationof the runtime.

OpenMP+MPI MPI

T W T16 S16 E16 T16 S16 E16 SS∗

6-no

de

3 3 3.849.88e−04 14.223 88.89% 5.333.15e−02 10.261 64.13%

4 3 3.934.50e−02 14.312 89.45% 5.476.40e−03 10.264 64.15%

5 3 3.963.91e−02 14.096 88.10% 5.362.22e−02 10.400 65.00%

7 3 3.955.65e−02 14.129 88.31% 5.366.12e−02 10.406 65.04%

3 4 3.822.47e−02 14.297 89.35% 5.244.78e−02 10.426 65.16%

4 4 3.893.96e−02 14.465 90.41% 5.313.66e−02 10.604 66.27%

5 4 3.904.26e−02 14.198 88.74% 5.251.78e−02 10.546 65.91%

NSF

3 3 26.485.82e−01 14.508 90.68% 31.444.69e−01 12.218 76.36%

4 3 30.981.24e−01 14.479 90.49% 38.093.56e−01 11.776 73.60%

5 3 33.265.42e−01 14.575 91.10% 42.124.89e−01 11.508 71.92%

4 4 34.611.41e−02 14.565 91.03% 43.733.64e−01 11.528 72.05%

5 4 39.821.19e−02 14.585 91.16% 53.974.11e−01 10.761 67.25%

6 4 42.332.33e−01 14.664 91.65% 56.133.38e−01 11.059 69.12%




Figure 8.30. Communication and computation time for the OpenMP+MPI and MPI versions of thepMO-ABC algorithm with 16 cores.

This remarkable difference of performance between the hybrid version and the pure MPI versionis due to the communication time among processes, see Figure 8.30. On the one hand, the hybridversion only requires communication between two processes; therefore, the communication timesare short. On contrast, the pure MPI pMOABC requires communications among the 16 processes,leading to a waste of time in communications.

Finally, in the last experiment, we use the whole multi-core system, four multi-core nodesequipped with eight cores, a total of 32 cores. Like in the previous experiment, we compare theparallel performance of the hybrid version with the pure MPI version.

In Table 8.30 we compare the parallel performance of the two approaches, hybrid and MPI.

Table 8.30. Mean runtime, speedup, and efficiency for the OpenMP+MPI and MPI versions of thepMOABC algorithm with 32 cores in 30 independent runs. Note that we report the standard deviationof the runtime.

OpenMP+MPI MPI

T W T32 S32 E32 T32 S32 E32 SS∗

6-no

de

3 3 2.122.25e−02 25.806 80.65% 2.872.87e−02 19.044 59.51%

4 3 2.181.68e−02 25.779 80.56% 2.948.54e−03 19.136 59.80%

5 3 2.179.21e−03 25.749 80.47% 2.931.82e−03 19.031 59.47%

7 3 2.151.55e−02 25.936 81.05% 2.925.83e−03 19.117 59.74%

3 4 2.101.77e−02 26.052 81.41% 2.977.60e−03 18.400 57.50%

4 4 2.164.81e−04 26.043 81.38% 2.956.74e−03 19.073 59.60%

5 4 2.127.43e−03 26.036 81.36% 2.992.30e−02 18.494 57.80%

NSF

3 3 14.522.40e−01 26.450 82.66% 19.934.41e−03 19.280 60.25%

4 3 17.492.32e−01 25.642 80.13% 22.851.77e−01 19.626 61.33%

5 3 18.871.51e−01 25.683 80.26% 25.292.06e−01 19.164 59.89%

4 4 19.703.63e−03 25.589 79.96% 25.907.52e−02 19.464 60.82%

5 4 22.622.68e−01 25.680 80.25% 30.022.13e−01 19.348 60.46%

6 4 24.072.60e−01 25.788 80.59% 31.741.55e−01 19.557 61.11%




Figure 8.31. Communication and computation time for the OpenMP+MPI and MPI versions of thepMO-ABC algorithm with 32 cores.

If we focus on comparing the efficiency, we realize that, in general, the efficiency of the hybridOpenMP+MPI pMOABC is over 80%, whereas in the MPI is below 60%. In this way, we can seethat, in average, the hybrid version obtains the same quality results than the sequential versionnearly 26 times faster with 32 cores; that is to say, in the scenario NSF topology with T=3 andW=3, whereas the sequential runtime is 384.17 seconds, the runtime in the hybrid OpenMP+MPIis 14.52 seconds.

In Figure 8.31 we may observe the substantial differences of runtime, as well as the amount oftime spend by each parallel approach in communications.

To conclude, we summarize in Table 8.31 the mean speedup and efficiency obtained by theOpenMP version or OpenMP+MPI version and by the MPI version in the two network topologies(6-node and NSF). Note that, the pure OpenMP version may be considered a hybrid version inwhich we have 1 MPI process and 2, 4, or 8 threads.

In addition, in Figure 8.32, we summarize the mean runtime (communication and computationtime) for the parallel approaches in order to provide a global view of the performance of each

Table 8.31. Summary of the mean speedup and efficiency for the OpenMP+MPI and MPI versions ofthe pMO-ABC algorithm in all the data sets.

6-node network NSF network

OpenMP+MPI MPI OpenMP+MPI MPI

#Cores (c) Sc Ec Sc Ec #Cores (c) Sc Ec Sc Ec

1 1 100% 1 100% 1 1 100% 1 100%2 1.988 99.42% 1.943 97.14% 2 1.983 99.13% 1.935 96.76%4 3.942 98.56% 3.652 91.31% 4 3.830 95.76% 3.688 92.21%8 7.654 95.68% 6.489 81.11% 8 7.463 93.29% 6.520 81.50%16 14.246 89.04% 10.415 65.10% 16 14.563 91.02% 11.475 71.72%32 25.915 80.98% 18.899 59.06% 32 25.805 80.64% 19.407 60.65%


(a): 6-node network

(b): NSF network

Figure 8.32. Summary of the communication and computation time for the OpenMP+MPI and MPI

versions of the pMO-ABC algorithm.

approach with different number of cores.As we can see in Table 8.31, the performance of the parallel versions of the MO-ABC remains

almost constant independently of the optical network. Furthermore, we may observe how theefficiency of the MPI version decreases exponentially when the number of cores increases. In Figure8.33 and Figure 8.34, we provide an illustrative comparison of speedup and efficiency between thetwo parallel approaches.

To sum up, we can say that the OpenMP+MPI version is a good approach for solving efficientlythe Traffic Grooming problem in a reasonable amount of time. Moreover, this parallel approach isnot only applicable to this tele communication problem but also to many real-world multiobjectiveoptimization problems.



Figure 8.33. Summary of the mean speedup and efficiency obtained by the parallel versions of theMO-ABC in the 6-node.


Figure 8.34. Summary of the mean speedup and efficiency obtained by the parallel versions of theMO-ABC in the NSF.

9MOEA/D-NBI for 3-objective optimization problems

In the Multiobjective Optimization problem (MOOP) [88] defined in Equation (9.1), the objectivefunctions may have different scales, leading to a neglecting of one or more objective functionsas a result. In the last decades, diverse multiobjective evolutionary algorithms (MOEAs) haveproposed the normalization of the objective space with the aim of solving this drawback. However,in the case of more than two objectives, a set of uniformly distributed solutions in the normalizedobjective space may not be uniformly distributed in the original objective space.

Among the most popular methods that use normalization in order to well distribute the so-lutions in the objective space are: Pareto Archived Evolution Strategy (PAES [89]), Fast Non-Dominated Sorting Genetic Algorithm (NSGA-II [90]), Strength Pareto Evolutionary Algorithm2 (SPEA2 [91]), or Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D[95]). The algorithms PAES, NSGA-II, and SPEA2 are based on Pareto dominance; whereas,the MOEA/D algorithm decomposes a multiobjective problem into a number of single-objectiveproblems, which are defined by a scalarizing function with different weight vectors.

In the MOEA/D, the two commonly-used aggregation techniques are the weighted Tchebycheffapproach and the weighted sum approach. However, both aggregation methods are really sensitiveto scales of objectives, which is crucial in many real-world MOOP where the objective functionsare in very different scales. In order to improve the performance and effectiveness of the classicalMOEA/D in real-world optimization problems, in this chapter we propose an improved version ofthis MOEA based on decomposition which uses the advantage of the Normal Boundary Intersec-tion (NBI) approach [149] and the Tchebycheff approach (MOEA/D-NBI). Unlike in the originalMOEA/D, in the proposed MOEA/D-NBI, all the subproblems have the same weight vector anduse a predefined set of reference points to ensure diversity in the Pareto Optimal set.

As we have seen throughout this document, the Wavelength Division Multiplexing (WDM) isa technology that aims to make the most of optical networks by dividing each single optical fiberlink into several wavelengths of light (λ) or channels. Each channel operates in the range of Gbps;unfortunately, the requirements of the vast majority of the current traffic connection requests area few Mbps, causing a waste of bandwidth at each channel, as a result.

We can solve this drawback by equipping each optical node with an access station for multiplex-ing or grooming several low-speed requests onto one single channel. This problem of multiplexingor grooming low-speed requests is known in the literature as the Traffic Grooming problem [63].This telecommunication problem consists of three subproblems: lightpath routing, wavelength as-

201

202 9. MOEA/D-NBI for 3-objective optimization problems

signment, and traffic routing. Therefore, we have to optimize the total throughput of the network,the number of lightpaths established, and the average propagation delay. As we may observe, it isa 3-objective optimization problem and their three objectives are commonly in very different scales.In order to handle the Traffic Grooming problem, in this chapter we propose the use of indirectencoding using a construction heuristics within the MOEA/D-NBI.

In our proposed method, we use a two-phase heuristics for constructing a solution to theproblem. In the first phase, we establish several lightpaths on the physical topology with the aimof generating a virtual topology. The second phase is devoted to routing the set of low-speed trafficrequests on the virtual topology. For constructing the virtual topology, we use a permutationvector σ (control parameter). Then, by using a Simulated Annealing (SA [97]) procedure, weoptimize the routing of low-speed traffic request on the given virtual topology. Therefore, weemploy multiobjective optimization for optimizing σ, and also the control parameters of the SAprocedure: cooling rate and neighbourhood size.

9.1 MOEA/D with the NBI-style Tchebycheff approach for3-objective Optimization problems

Let consider a generic minimization 3-objective optimization problem as:

minimize F (x) = (f1(x), f2(x), f3(x))subject to x ∈ Ω

(9.1)

where Ω is the decision space, F : Ω → R3 consists of the real-valued objective functions, andR3 is the objective space.

Let u, v ∈ R3, u is said to dominate v (u ≺ v), if and only if ui ≤ vi for every i ∈ 1, 2, 3 anduj < vj for at least one index j ∈ 1, 2, 3. For any two solutions x and y, x is said to dominatey if F (x) dominates F (y). Therefore, we can denote any solution x∗ as Pareto optimal or non-dominated solution if there exists no solution x in Ω such that F (x) dominates F (x∗). The set ofall the non-dominated solutions is called Pareto Optimal set or simply Pareto set. Correspondingly,the objective vectors of all solutions in the Pareto set are denoted as Pareto Optimal front.

The objective functions may have different scales, leading to a neglecting of one or more ob-jective functions as a result. In the literature, diverse multiobjective evolutionary algorithms(MOEAs) have proposed the normalization of the objective space with the aim of solving thisdrawback. However, in the case of more than two objectives, a set of uniformly distributed solu-tions in the normalized objective space (Figure 9.1(c)) may not be uniformly distributed in theoriginal objective space (Figure 9.1(d)). In Figure 9.1(f) an uniform distribution is presented forthe problem given in Figure 9.1(a) and Figure 9.1(b). A normalization of the distribution shownin Figure 9.1(f) is presented in Figure 9.1(e).

In Figure 9.1, the COV measure and Mesh ratio indicate if the solutions are uniformly dis-tributed [150]. Given any set of n points zin

i=1, the minimum distance between zi and any of theother points is γi = minj 6=i |zi − zj|. Therefore, we define the COV measure as:

COV measure =

√

n

∑ni=1 γ2

i

(∑n

i=1 γi)2(9.2)

and the Mesh ratio as:Mesh ratio =

maxi=1,...,n γi

mini=1,...,n γi(9.3)

9.1 MOEA/D with the NBI-style Tchebycheff approach for 3-objective Optimization problems 203

(a): 3D representation (b): 2D representation (c): Normalized Objective Space

(d): Real Objective Space (e): Normalized Objective Space (f): Real Objective Space

Figure 9.1. A set of uniformly distributed solutions in the normalized objective space may not beuniformly distributed in the original (real) objective space.

For a perfectly uniform mesh, γ1 = γ2 = . . . = γn, and the COV measure=0 and the Meshratio=1. In both metrics, the smaller the value is, the more uniform is the mesh. For furtherinformation, please refer to [150].

The Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D) [95] is apopulation-based evolutionary algorithm which uses an aggregation method to decompose a MOOPinto Ns (population size) single-objective optimization subproblems, optimizing each one by usingthe information from the optimization of neighbouring subproblems. In this way, MOEA/D pro-vides a general framework which allows the application of any single objective optimization tech-nique to optimize each subproblem, including local search procedures which is the major advantageof MOEA/D.

The two most commonly-used decomposition methods are: Weighted Sum Approach and WeightedTchebycheff Approach [151]. On the one hand, the Weighted Sum approach works out well onconvex MOOPs; however, it fails on non-convex MOOPs. On the other hand, the TchebycheffApproach is able to deal with convex and non-convex MOOPs. Unfortunately, both approachesare very sensitive to the scale of objectives.

In [149], a direction-based decomposition method independent of the relative scales of differentobjectives functions was proposed. This method is known as Normal Boundary Intersection (NBI)and attempts to find the intersection points between the Pareto front and a number of straightlines, which are defined by a normal vector and a set of uniformly-distributed points in the ConvexHull of Individual Minima (CHIM), see Figure 9.2.

According to [152], the NBI method cannot be easily used as decomposition approach withinthe MOEA/D due to its several constraints; therefore, in this paper we take the advantages ofthe NBI approach and the Tchebycheff approach and we use the NBI-Tchebycheff approach fordecomposing a 3-objective optimization problem, see equation (9.1):


Figure 9.2. Distribution of ri on the plane Π (CHIM) which contains the points F 1, F 2, and F 3

Let F 1 = (F 11 , F 1

2 , F 13 ), F 2 = (F 2

1 , F 22 , F 2

3 ), and F 3 = (F 31 , F 3

2 , F 33 ) the three extreme points

of the Pareto front of the 3-objective optimization problem in equation (9.1) and r1, r2, . . . , rNsa set of Ns reference points evenly distributed on the plane Π (CHIM) which contains the pointsF 1, F 2, and F 3. Then, we can decompose the 3-objective optimization problem into Ns singleobjective minimization subproblems, where the i-th one optimizes the function:

g(x|ri, n) = max n1(f1(x) − ri1),

n2(f2(x) − ri2),

n3(f3(x) − ri3)

(9.4)

Note that, in equation (9.4), n = (n1, n2, n3) is the normal vector to the plane Π. Note that,in this decomposition method, it is required to know the coordinates of the extreme points (F 1,F 2, and F 3); however, it is possible to use approximate points to the extreme points.

Once defined the NBI-Tchebycheff decomposition approach, the MOEA/D-NBI procedure worksas follows:

Input:

- 3-objective Optimization Problem (see equation (9.1));


- Neighbourhodd size (τ);

- Crowding Distance (α);

- Increment of the CHIM (φ).

Output:

- Set of non-dominated solutions: F (x1), . . ., F (xNs).



Step 1.2) Estimate the extreme points of the CHIM. Set F k to be the point among F (x1), . . . , F (xk)with smallest value of fk, k = 1, 2, 3. Then, use the DistributeReferencePoints(F 1,F 2,F 3,α,φ)procedure in order to obtain the set of Ns reference points (R = r1, . . . , rNs) dis-tributed uniformly on the plane delimited by the extreme points.


Step 1.3) Generate an initial population with Ns random individuals x1, x2, . . . , xNs .

Step 1.4) Compute the Neighbourhood. Calculate the Euclidean distance between any tworeference points in r1, . . . , rNs and then obtain the τ closest reference points to eachreference point. For each i = 1, . . . , Ns, set B(i) = i1, . . . , iτ, where ri

1, . . . , riτ are the

τ closest reference points to ri.

Step 2) Update: For each i ∈ 1, . . . , Ns, do

Step 2.1) Reproduction. Select randomly two indexes a,b from B(i), and we generate anew solution xc from xa and xb by using genetic operators. In case of any value inchromosome of xc is out of the boundary, repair it by reseting the value in chromosomewith a randomly value inside the boundary.

Step 2.2) Update neighboring solutions. For each index j ∈ B(i), if g(tn)(xc|rj , n) ≤g(tn)(xj |rj , n), then replace xj by xc.

Step 2.3) Update the Pareto set. Update the set of non-dominated solutions with xc.Remove from the set all solutions dominated by xc and add xc to the set only if noother solution dominates it.

Step 3) Stopping Criterion: If the stopping criterion is satisfied, then stop and output thePareto set, otherwise, go to Step 2.

In the MOEA/D-NBI, the DistributeReferencePoints procedure is used to uniformly distributethe reference points in the CHIM, it works in the following way:

Input:

- Extreme points of the Pareto front (F 1, F 2, F 3);

- Crowding Distance (α);

- Increment of the CHIM (φ).

Output:

- Set of Reference points (R = r1, . . . , rNs).

Step 1) Set R as empty.

Step 2) Calculate the initial points X , Y , and Z to satisfy d(X, Y ) = φ ∗ d(F 1, F 2), d(X, Z) =φ ∗ d(F 1, F 3), and d(Y, Z) = φ ∗ d(F 2, F 3).

Step 3) Let XY , XZ, and Y Z be the segments of the plane Π (CHIM), where d(X, Y ) ≥d(X, Z) ≥ d(Y, Z). Then, select the two largest segments: XY and XZ.

Step 4) Let α be the crowding distance among the reference points, calculate the number ofdivisions in each segment: NDXY = d(X,Y )

α and NDXZ = d(X,Z)α . In case of d(X, Y ) ≤ α,

then stop and output R.

Step 5) For m = 0 to NDXZ , do

Step 5.1) Obtain the point A = X + m ∗ Y −XNDXY

.

Step 5.2) Obtain the point B = X + m ∗ Z−XNDXZ

.


(a): Distribution 1 (b): Distribution 2

(c): Final distribution

Figure 9.3. Illustrative example of the DistributeReferencePoints procedure.

Step 5.3) Compute the number of divisions in the segment AB: NDAB = d(A,B)α .

Step 5.4) For n = 0 to NDAB, add the reference point A + n ∗ B−ANDAB

to R.

Step 6) Set X = X + NDXZ ∗Y −X

NDXYand go to Step 3.

As we may observe, the crowding distance (α) established is critical and determines the numberof individuals or subproblems (Ns) in the MOEA/D-NBI.

The main drawback of the NBI method is that for more than two objective functions (2D), theextreme points are not obtainable in all cases. Thus, for solving this issue, we can increment thesize of the CHIM (φ), enhancing the accuracy of the algorithm as a result. Note that, φ is alwaysequal to or greater than 1.

An illustrative example of this procedure is shown in Figure 9.3. As we can observe, we dis-tribute in different phases the reference points in the CHIM. Note that, in the partial distributionsof Figure 9.3, the labels X , Y , and Z are set according to the length of the segments. Furthermore,we illustrate the movement of the label X with an arrow.

9.1.1 Testing the MOEA/D-NBI with benchmark functions

We use three-objective UF8 [153], UF10 [153], DTLZ1 [154], and DTLZ2 [154] problems with sixdecision variables, which have identical range of values for each objective. In this work, we havemodified the problems by multiplying each objective function by different constant values (β1, β2,and β3), i.e., f ′

1=β1 ∗ f1, f ′2=β2 ∗ f2, and f ′

3=β3 ∗ f3. In Table 9.1 we present the constant valuesused for generating different scenarios.

The algorithms tested in this section are: MOEA/D, MOEA/D-NBI, and the Fast Non-Dominated Sorting Genetic Algorithm with (NSGA-II) and without normalization (NSGA-II∗)


Table 9.1. Test instances and parameters of MOEA/D-NBI for generating the reference points

MOEA/D-NBIβ1 β2 β3 α (Ref. Points)3 6 1 0.15 103110 100 0.1 0.87 103625 625 0.04 3.5 103550 2500 0.02 9.2 1038100 10000 0.01 27.1 1035

in the crowding distance assignment procedure. As in [155], all the algorithms use the samereproduction operators: a Differential Evolution (DE) operator and a polynomial mutation pro-cedure. In this section, the reproduction configuration for MOEA/D-NBI, MOEA/D, NSGA-II∗,and NSGA-II is the same as in [155]; for further information, please refer to [155].

In this section, the population size and weighted vectors (Ns) in MOEA/D are Ns = Cm−1H+m−1 =

C3−144+3−1 = 1035, where H = 44 and the number of objectives (m) is 3. The population size in

NSGA-II and NSGA-II∗ is set to 1035. In MOEA/D-NBI, the population size and number ofreference points depend on the crowding distance (α). In Table 9.1, we present the different valuesof α used for generating around 1035 reference points. Note that, we have increased the size theCHIM according to different values of φ, depending on the problem: φ=1.35 (UF8, UF10, andDTLZ2) and φ=1.1 (DTLZ1).

Table 9.2. IGD-metric values of the non-dominated solutions found by MOEA/D, MOEA/D-NBI, NSGA-II∗, and NSGA-II on UF8, UF9, DTLZ1, and DTLZ2 using different scales. The notation used forpointing the statistically non-significant differences is the following: (†) NSGA-II∗ and NSGA-II.

UF8 MOEA/D MOEA/D-NBI NSGA-II∗ NSGA-IIβ1 β2 β3 Mean Min Std Mean Min Std Mean Min Std Mean Min Std3 6 1 0.0731 0.0723 5.64E-04 0.0680 0.0674 2.73E-04 0.0907 0.0852 3.37E-03 0.0856 0.0808 3.20E-03 †10 100 0.1 5.8526 4.5085 1.58E+00 0.5485 0.5421 2.49E-03 1.5648 1.2551 2.50E-01 0.6427 0.6025 2.99E-0225 625 0.04 30.5286 30.2759 1.17E-01 2.1578 2.1165 1.93E-02 6.0919 4.6785 7.56E-01 2.5359 2.4178 1.00E-0150 2500 0.02 676.460 676.204 1.31E-01 6.0885 6.0057 4.77E-02 14.8495 12.8050 1.72E+00 7.1175 6.7701 2.24E-01100 10000 0.01 3360.67 3355.94 9.62E+00 17.7088 17.2516 2.88E-01 35.1636 31.7062 4.09E+00 19.8576 18.9601 5.02E-01

UF10 MOEA/D MOEA/D-NBI NSGA-II∗ NSGA-IIβ1 β2 β3 Mean Min Std Mean Min Std Mean Min Std Mean Min Std

3 6 1 0.1255 0.0876 4.18E-02 0.1072 0.0734 4.03E-02 0.6451 0.4559 9.05E-02 0.6415 0.4267 1.01E-01 †10 100 0.1 6.7707 3.8248 2.52E+00 1.2952 0.7935 2.66E-01 4.2797 2.8328 4.88E-01 3.5430 2.8810 3.60E-0125 625 0.04 118.636 36.3595 5.09E+01 4.5958 3.6804 4.53E-01 12.9185 11.5619 6.66E-01 10.7989 9.0911 7.10E-0150 2500 0.02 812.029 713.074 3.80E+01 12.4166 10.6620 1.13E+00 26.9508 25.3114 7.44E-01 23.9552 21.0282 1.12E+00100 10000 0.01 3438.79 3384.38 1.76E+01 28.5876 24.0121 2.60E+00 55.8033 51.8341 1.14E+00 51.8465 47.1860 1.67E+00

DTLZ1 MOEA/D MOEA/D-NBI NSGA-II∗ NSGA-IIβ1 β2 β3 Mean Min Std Mean Min Std Mean Min Std Mean Min Std3 6 1 0.1020 0.1018 8.09E-05 0.0381 0.0381 3.27E-06 0.0576 0.0557 6.92E-04 0.0538 0.0526 9.68E-04 †10 100 0.1 3.6193 3.6129 4.78E-03 0.2823 0.2822 4.22E-05 0.4728 0.4412 2.14E-02 0.3719 0.3639 6.71E-0325 625 0.04 59.9860 59.5383 1.63E-01 1.1591 1.1588 2.20E-04 1.6745 1.6111 3.31E-02 1.4805 1.4470 2.56E-0250 2500 0.02 689.979 689.456 2.02E-01 3.4382 3.4369 5.33E-04 4.5812 4.4775 7.82E-02 4.2192 4.1184 6.80E-02100 10000 0.01 3358.91 3357.10 2.74E+00 10.3471 10.3442 1.71E-03 12.9227 12.4959 2.10E-01 12.0884 11.7786 1.61E-01

DTLZ2 MOEA/D MOEA/D-NBI NSGA-II∗ NSGA-IIβ1 β2 β3 Mean Min Std Mean Min Std Mean Min Std Mean Min Std3 6 1 0.1652 0.1253 3.03E-02 0.1879 0.1393 3.06E-02 0.3621 0.2890 4.99E-02 0.2827 0.2405 2.45E-0210 100 0.1 6.7281 3.9863 2.25E+00 1.4518 1.2432 1.69E-01 4.0338 3.2065 4.05E-01 1.7178 1.3388 1.98E-0125 625 0.04 85.8282 31.7071 4.13E+01 4.8134 4.1169 4.39E-01 12.6187 11.0251 6.16E-01 6.4745 5.2801 6.48E-0150 2500 0.02 796.717 716.460 4.13E+01 12.2057 11.0997 7.63E-01 26.3213 24.7476 7.74E-01 16.6258 14.1951 1.28E+00100 10000 0.01 3432.48 3380.07 1.86E+01 29.1780 26.8154 1.53E+00 54.8874 52.8720 9.74E-01 40.5386 35.8703 2.24E+00


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All MOEAs were run using g++ (GCC) 4.4.5 on a 2.3GHz Intel PC with 1GB RAM. Eachalgorithm is run 30 times independently for each test instance by using the same initial populationand stopping after a given number of generations (2000 generations).

The Inverted Generational Distance (IGD) [156] metrics measures both the convergence as wellas the diversity among solutions. In Table 9.2, for each algorithm, we report the average, minimum,and standard deviation of IGD metrics. As we may observe, the MOEA/D with Tchebycheffapproach is very sensitive to the scale of objectives. Its performance worsens as the differenceof scale in the objective functions increases. Both NSGA-II and NSGA-II∗ obtain better resultsthan the simple MOEA/D. Although in some cases the NSGA-II works better than NSGA-II∗, wemay observe that in UF10 and DTLZ1, their performance is almost the same. Finally, we cansee that the performance of the MOEA/D-NBI is really promising, which mean that it is able todeal with this kind of problems properly. After a statistical analysis by pair of algorithms, thedifferences were statistically not significant in some scenarios between NSGA-II and NSGA-II∗

with a confidence level of 95% (see Table 9.2).

Since the results reported in Table 9.2 for the MOEA/D are not competitive enough withthe rest of algorithms; in Figure 9.4, Figure 9.5, Figure 9.6, and Figure 9.7, we only present theevolution of mean IGD values obtained by the MOEA/D-NBI, NSGA-II and NSGA-II∗ versusthe number of generations. We can notice that, while the NSGA-II and NSGA-II∗ do not evolveproperly in all the cases, the evolution of the MOEA/D-NBI is always clear.


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9.2 Indirect Encoding and Construction Heuristics for the

Traffic Grooming problem

In this paper, we propose a two-phase construction heuristics for the Traffic Grooming problem.In the first phase, we set up several lightpaths on the physical topology with the aim of generatinga virtual topology. Then, in the second phase, we route a given set of low-speed traffic requests onthe virtual topology by using a Simulated Annealing (SA [97]) procedure.

In the first place, we start defining the representation of the chromosome, where we can findthe following elements:

• Order of the Lightpaths σ = σ1, σ2, . . . , σM, where M = (|N | − 1) ∗ (|N | − 1) ∗ T . Notethat, σi where 1 ≤ i ≤ M , contains the following information: source node and destinationnode.

• Cooling Rate (cRate).

• Mutation Rate (mRate). In the SA algorithm, a neighbour solution is obtained by using amutation procedure where mRate indicates the mutation rate.

In this way, the reproduction operators used (crossover and mutation procedures) for σ (per-mutation representation) and for cRate and mRate (real values) are different.

On the one hand, for σ, we use a Partially Mapped Crossover (PMX) proposed by Whitley in[157]. In the mutation procedure, we swap the position of M*F lightpaths (swap mutation), whereF is a parameter that defines the mutation rate.

9.2 Indirect Encoding and Construction Heuristics for the Traffic Grooming problem 211

On the other hand, for cRate and mRate, the crossover and mutation are based on the DEoperator. Given two individuals xa and xb, the new value of cRate or mRate in the child solutionxc is computed as:

xc =

xa + F × (xa − xb) with probability CR;xa with probability 1-CR.

(9.5)

where CR and F are the crossover probability and the mutation rate, respectively.In order to generate a new random individual, the procedure works as follows:

ALGORITHM 1: GenerateRandomIndividual()

Input: empty.

Output:

- Order of the Lightpaths (σ);

- Cooling Rate (cRate);

- Mutation Rate (mRate);

- Virtual Topology (V );

- Set of Requests successfully established (S).

Step 1) Initialize the order of Lightpaths (σ) randomly. Firstly, for each pair of nodes s, d ∈ N ,add to σ a total of T lightpaths if and only if s 6= d and

∑

x∈1,3,12, and 48 Λxsd > 0. Then,

shuffle σ randomly.

Step 2) Initialize the Cooling Rate (cRate) randomly between [0.001, 0.3]. In the SA procedure,we consider an initial temperature tinit = 100 and a tend = 0.001; therefore, the SA willperform between two and ten iterations.

Step 3) Initialize the Mutation Rate (mRate) randomly between [0.05, 0.5].

Step 4) Construct the solution by using the ConstructSolution procedure. This procedure returnthe Virtual Topology (V ) established and the set of requests successfully routed (S); therefore,

(V, S) = ConstructSolution (σ, cRate, mRate) (9.6)

Step 5) Stop and return σ, cRate, mRate, V , and S.

As we may observe, the main step in the generation of a random individual is the Construct-Solution procedure. This heuristics consists of two phases, the Virtual Topology construction andthe routing of the low-speed traffic requests on that virtual topology. Basically, this procedureworks as follows:

ALGORITHM 2: ConstructSolution

Input:




- Mutation Rate (mRate).

Output:



Step 1) Construct V by using the VTopology procedure; thus,

V = V T opology (σ, K − spaths) (9.7)

Note that, K-spaths is a set which contains, for each pair of nodes s, d ∈ N the K shortespaths. It is only computed once, at the begining of the MOEA.

Step 2) Route the set of low-speed traffic requests (Λ) on V by using the simmulated annealing(SA) procedure in order to obtain S; therefore,

S = SA (Λ, V, cRate, mRate) (9.8)

Step 3) Stop and output V and S.

The two phases in the ConstructSolution procedure corresponds with the procedures VTopologyand SA. On the one hand, the VTopology procedure constructs a virtual topology in the followingway:

ALGORITHM 3: VTopology

Input:


- Set of K shortest paths (K-spaths).

Output:

- Virtual Topology (V ).

For each i ∈ 1, . . . , M, do

Step 1) Extract the source (si) and destination (di) nodes corresponding to σi.

Step 2) If there exist an available transmmiter at node si and an available receiver at node di, goto Step 3; otherwise, go to Step 4.

Step 3) For each path k in K-spathssi ditry to assign the first available wavelength (W) by using

the well-known heuristics FirstFit (FF). If a path is succesfully assigned to a wavelength W,then set V w

si di+= 1.

Step 4) Remove σi from σ.

End of for

Step 5) Stop and output V .

9.2 Indirect Encoding and Construction Heuristics for the Traffic Grooming problem 213

In the second phase we use a metaheuristics based on trajectory for routing the low-speed con-nection requests on a given virtual topology (V ). This metaheuristics is the well-known SimulatedAnnealing (SA), which works as follows:

ALGORITHM 4: SA

Input:

- Set of low-speed traffic requests (Λ);



- Mutation Rate (mRate).

Output:


Step 1) Initialization

Step 1.1) Initialize the order of low-speed requests randomly. First of all, for each pair ofnodes s, d ∈ N and granularity x ∈ 1, 3, 12, and 48, add to δ0 a total of Λx

sd requestsif and only if s 6= d and Λx

sd > 0. Then, shuffle the initial solution δ0 randomly.

Step 1.2) Initialize the best solution δ∗. Set δ∗=δ0.

Step 1.3) Establish the initial (tinit) and final temperature (tend), to 100 and 0.001, respec-tively.

Step 1.4) For each pair of nodes s, d ∈ V , where s 6= d, compute the K shortest virtualpaths from s to d, obtaining K-svpaths.

Step 2) Update

Step 2.1) Generate a neighbour solution δ′ by mutating δ0 with a mutation rate equal tomRate.

Step 2.2) Compute the difference of throughput (∆) between δ′ and δ0:

∆ = Routing(δ′, K-svpaths)−Routing(δ0, K-svpaths) (9.9)

Step 2.3) Check if δ′ replace δ0,

If ∆ ≥ 0 then, set δ0 = δ′.

else if random(0, 1) < e−∆

tinit then, set δ0 = δ′.

Step 2.4) If Routing(δ0, K-svpaths) > Routing(δ∗, K-svpaths) then, set δ∗=δ0.

Step 2.5) Set tinit ∗ = cRate.

Step 3) Go to Step 2 if tinit > tend; otherwise, stop and return the set of requests successfullyestablished (S) corresponding to the best solution (δ∗).


In the SA procedure, we try to maximize the number of low-speed traffic requests successfullyrouted on a given virtual topology. As we may observe, each solution is represented by a permuta-tion of the possible requests (δ); therefore, we try to find the best order of requests that optimizesthe total throughput. The throughput of a solution is calculated by using the Routing procedure.In this procedure, to set up a connection, we first try to use a single-hop routing, if it is not possible,we try a multi-hop routing. If the request is not successfully routed, it is dismissed.

In SA, to obtain a neighbour solution, we mutate the initial solution by using a swap mutation.Thus, let L be the total number of low-speed requests in δ0, we swap the position of L*mRaterequests.

9.3 MOEA/D-NBI for the Traffic Grooming problem

In this section we present several comparisons between the MOEA/D-NBI and other methods fordifferent scenarios in the Traffic Grooming problem.

We start describing some considerations. Then, we divide the experiments in three categories.In the first one, with the aim of proving the effectiveness of the MOEA/D-NBI framework in a real-world optimization problem with 3 objective functions in very different scale, we start comparingit with the well-known NSGA-II (NSGA-II∗∗). Both algorithms use the chromosome encoding,construction heuristics, and genetic operator that appear in Section 9.2.

In order to demonstrate the goodness of using an indirect encoding within the new MOEA/D-NBI framework, we compare the performance of the MOEA/D-NBI with the other five proposedMOEAs (DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA).

Finally, we present a single-objective comparison between the MOEA/D-NBI and several ap-proaches published in the literature by other authors.

9.3.1 Previous Considerations

In this section we describe the methodology followed for comparing different methods with ourproposal (MOEA/D-NBI). Therefore, with the aim of making a comprehensive performance com-parison, we have carefully selected some scenarios with different features: size of the network,amount of traffic, and available resources.

In the first place, we have chosen the same optical network topologies as in Chapter 8: a six nodenetwork (6-node), the European Optical network (COST239), the National Science Foundationnetwork (NSF), and the Nippon Telegraph and Telephone network (NTT).

Like in the previous experiments, for each optical network, we have generated three trafficmatrices (TM) or sets of low-speed traffic requests with small, medium, and large amount oftraffic. In Table 9.3, we re-show the amount of traffic (in OC-1 units) for each optical network andtraffic matrix.

Table 9.3. Amount of Traffic (in OC-1 units) and Runtime (in seconds) per Optical network

TM1 TM2 TM3 Runtime (s)

6-node 988 1976 3952 30COST239 3187 6272 12037 120NSF 5724 11448 22896 360NTT 77233 153307 309820 720

9.3 MOEA/D-NBI for the Traffic Grooming problem 215

Finally, we propose the use of different available resources per node and per fiber link. In thisway, for each traffic matrix, we use different number of transceivers per node (T) and availablewavelengths per optical fiber link (W). The scenarios tested in these experiments are:

• TM1: T=3, 4, 5, 6, 7 and W=3, 4

• TM2: T=6, 8, 10, 12, 14 and W=4, 6

• TM3: T=9, 12, 15, 18, 21 and W=6, 9

On the whole, we compare the algorithms in four optical network topologies, each one withthree different sized traffic matrices, and each traffic matrix is tested in 10 scenarios with differentavailable resources; thus, a total of 120 different scenarios.

Since we are dealing with a MOOP, the performance of the algorithms should be measuredwith multiobjective indicators. In this section, we have also used the well-known indicators: Hy-pervolume (HV) [117] and Set Coverage (SC) [121].

The number of independent runs of the algorithms is 30 for each scenario, where the stoppingcriterion is based on the runtime and depend on the topology (see Table 9.3). The approaches wererun by using g++ (GCC) 4.4.5 on a 2.3GHz Intel PC with 1GB RAM. Note that, the machineand the stopping criterion are exactly the same as in Chapter 8.

In order to ensure a certain level of statistical reliability, we have performed the statisticalanalysis presented in Section 3.4. Like in the other experiments of this thesis, the confidence levelused is 5%; so, the differences are unlikely to have occurred by chance with a probability of 95%.

9.3.2 Comparison with NSGA-II∗∗

In the first comparison, we study the performance of our multiobjective approach based on decom-position and the well-known NSGA-II (NSGA-II∗∗). In this comparison, both algorithms use thesame chromosome encoding, construction heuristics, and genetic operator (see Section 9.2). Theconfiguration used for the algorithms is shown in Table 9.4.

Table 9.4. Configuration of MOEA/D-NBI and NSGA-II∗∗

NSGA-II∗∗ MOEA/D-NBI

Ns 100 Ns ∼100F 25% F 25%CR 90% CR 90%Selection Binary Tournament α 0.02*Amount of Traffic

φ 1.1

As we can see in Table 9.4, the parameter configurations of the algorithms are identical. Aswe mentioned in Section 9.1, the number of reference points, number of subproblems, or simplypopulation size (Ns) in the MOEA/D-NBI depends on the crowding distance (α). In this work,we use different values of α depending on: amount of traffic by 0.02 (which implies around 100individuals). Note that, to solve the main drawback of the NBI method mentioned in Section 9.1,in this work, we increase the size of the CHIM in our experiments according to φ.

We start comparing both algorithms by using the median value of HV in the 30 independentruns (see Table 9.5). In Figure 9.8 and Figure 9.9, we present an illustrative comparison betweenthe approaches in order to easily visualize the differences of HV. Note that, in the plots of Figure


Table 9.5. Comparison between the MOEA/D-NBI and the NSGA-II∗∗ by using the Hypervolume(HV) indicator. The notation used is HVIQR, where HV is the median hypervolume and IQR is theinterquartile range in 30 independent runs.

6-node NetworkTM1 TM2 TM3

T W MOEA/D-NBI NSGA-II∗∗ T W MOEA/D-NBI NSGA-II∗∗ T W MOEA/D-NBI NSGA-II∗∗

3 3 39.91%8.55E-03 36.48%9.47E-03 6 4 42.68%7.72E-03 40.58%4.48E-03 9 6 39.83%6.52E-03 31.24%1.84E-03

4 3 51.67%1.94E-03 46.97%6.98E-03 8 4 51.85%1.37E-03 49.63%9.78E-03 12 6 47.76%5.33E-03 39.57%3.51E-03

5 3 61.00%3.33E-03 55.88%6.19E-03 10 4 57.29%1.30E-03 54.72%4.58E-03 15 6 53.46%4.82E-03 44.65%3.80E-04

6 3 67.05%5.98E-03 61.54%7.21E-03 12 4 60.86%9.76E-03 58.21%1.51E-03 18 6 57.02%6.01E-03 48.56%5.40E-03

7 3 71.62%1.36E-03 65.59%2.22E-03 14 4 63.41%5.31E-03 60.87%4.75E-03 21 6 59.75%3.54E-03 50.96%3.03E-03

3 4 39.92%4.03E-03 36.50%1.10E-04 6 6 42.95%8.70E-03 40.77%6.05E-03 9 9 40.32%1.19E-03 31.30%3.30E-03

4 4 51.72%7.31E-03 47.05%3.35E-03 8 6 55.24%5.60E-05 51.72%7.27E-03 12 9 49.41%1.48E-03 39.89%3.55E-03

5 4 61.06%5.09E-03 56.13%2.42E-03 10 6 63.58%2.41E-04 59.82%4.48E-03 15 9 57.56%9.79E-03 47.80%5.25E-03

6 4 67.50%7.10E-03 62.86%5.64E-03 12 6 69.35%2.33E-03 64.82%3.52E-04 18 9 62.87%7.31E-03 53.09%7.07E-04

7 4 72.38%6.45E-03 67.58%2.44E-03 14 6 73.36%5.55E-03 68.51%4.84E-03 21 9 66.60%8.50E-04 56.49%7.30E-03

COST239 NetworkTM1 TM2 TM3


3 3 32.14%3.57E-04 26.48%8.33E-03 6 4 40.41%8.28E-03 34.69%3.33E-03 9 6 36.14%2.63E-03 31.82%3.20E-03

4 3 40.30%8.33E-03 34.07%8.34E-03 8 4 50.08%2.23E-03 44.17%9.35E-03 12 6 45.55%5.14E-03 40.79%4.74E-03

5 3 47.33%5.00E-03 40.82%4.66E-03 10 4 57.39%7.21E-03 51.59%1.30E-03 15 6 53.56%5.63E-03 48.69%3.25E-03

6 3 54.53%6.03E-03 47.21%5.49E-03 12 4 62.32%6.08E-03 56.62%3.04E-03 18 6 58.88%5.27E-03 54.17%8.30E-04

7 3 59.70%7.25E-03 52.34%2.56E-03 14 4 66.17%7.44E-03 60.35%3.86E-03 21 6 62.53%9.77E-03 57.78%4.63E-03

3 4 32.32%3.42E-03 26.52%7.03E-03 6 6 40.48%6.17E-03 34.93%1.64E-03 9 9 36.11%3.44E-03 31.79%3.62E-03

4 4 40.39%9.28E-03 34.17%9.51E-03 8 6 50.01%8.73E-03 44.46%7.40E-05 12 9 45.47%4.58E-03 41.18%6.86E-03

5 4 47.34%4.88E-03 40.98%4.51E-03 10 6 57.47%2.77E-03 51.66%3.76E-03 15 9 53.47%2.99E-03 48.75%4.98E-03

6 4 54.61%5.98E-03 47.03%2.19E-03 12 6 62.48%4.28E-03 56.59%5.95E-03 18 9 58.89%4.44E-03 53.90%7.59E-03

7 4 59.70%5.86E-03 51.45%7.40E-03 14 6 66.13%6.18E-03 60.27%6.64E-03 21 9 62.53%8.42E-03 57.92%3.54E-04

NSF NetworkTM1 TM2 TM3


3 3 35.06%9.92E-03 29.12%7.12E-03 6 4 42.03%4.90E-03 35.64%5.13E-03 9 6 41.84%5.77E-03 37.07%7.61E-04

4 3 44.43%3.85E-03 37.24%6.42E-03 8 4 51.57%4.60E-03 44.55%7.03E-03 12 6 50.24%3.56E-03 44.77%6.85E-03

5 3 53.17%3.08E-03 45.38%1.78E-03 10 4 57.53%5.84E-03 49.65%6.31E-03 15 6 55.16%9.65E-03 49.23%4.49E-03

6 3 60.10%4.32E-03 51.95%6.76E-04 12 4 61.72%7.36E-03 53.46%2.36E-03 18 6 58.80%6.76E-03 52.47%9.41E-03

7 3 65.34%2.94E-03 56.90%3.19E-03 14 4 64.46%7.39E-03 55.36%7.39E-03 21 6 61.19%7.38E-03 54.60%7.43E-03

3 4 35.18%4.18E-03 29.11%3.41E-03 6 6 41.95%2.47E-03 35.80%4.43E-03 9 9 42.08%6.85E-03 37.47%2.14E-03

4 4 44.27%7.95E-03 37.33%5.93E-03 8 6 53.03%7.77E-03 46.20%8.17E-03 12 9 52.66%3.95E-03 47.84%5.30E-03

5 4 53.08%2.90E-04 44.79%9.33E-03 10 6 61.19%2.64E-03 54.39%2.23E-03 15 9 60.17%9.93E-03 55.31%6.01E-03

6 4 60.01%8.48E-03 51.73%5.31E-03 12 6 67.02%8.13E-03 59.71%8.16E-03 18 9 65.66%1.72E-03 60.29%4.59E-03

7 4 65.12%6.17E-03 56.94%6.14E-03 14 6 70.43%3.43E-03 63.40%8.93E-04 21 9 68.58%4.85E-03 63.90%5.59E-03

NTT NetworkTM1 TM2 TM3


3 3 16.15%5.64E-03 13.62%6.02E-03 6 4 22.57%9.53E-04 19.93%6.98E-03 9 6 18.81%4.73E-03 16.65%7.13E-03

4 3 22.27%7.48E-03 18.84%8.55E-03 8 4 25.66%9.02E-03 22.52%7.50E-03 12 6 20.97%8.06E-03 18.42%5.88E-03

5 3 26.18%4.39E-03 22.04%1.11E-04 10 4 27.75%7.03E-03 23.96%1.46E-04 15 6 22.21%5.36E-03 19.30%8.70E-05

6 3 28.63%3.70E-03 24.20%1.14E-03 12 4 28.98%8.20E-03 25.15%3.68E-03 18 6 22.95%8.37E-03 20.07%5.47E-03

7 3 30.54%3.33E-03 25.76%2.90E-03 14 4 29.78%3.07E-03 25.78%1.71E-03 21 6 23.68%2.23E-04 20.74%9.70E-03

3 4 16.55%5.29E-03 13.79%9.57E-04 6 6 23.94%3.40E-03 21.48%7.91E-03 9 9 20.56%7.55E-03 18.61%8.86E-03

4 4 23.44%9.31E-03 20.57%3.59E-04 8 6 28.90%1.52E-03 25.77%4.95E-03 12 9 23.93%1.74E-03 21.46%3.01E-03

5 4 28.43%1.08E-03 25.00%3.65E-03 10 6 32.08%3.81E-03 28.48%5.13E-03 15 9 26.15%2.97E-03 23.18%7.79E-03

6 4 32.00%6.34E-03 28.33%9.96E-03 12 6 33.97%5.70E-03 30.04%6.85E-03 18 9 27.27%1.52E-03 24.27%2.85E-04

7 4 34.71%1.17E-03 30.58%5.50E-04 14 6 35.59%2.85E-03 31.37%4.83E-03 21 9 28.34%7.01E-03 25.20%7.45E-03


(a): 6-node (TM1) (b): 6-node (TM2)

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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MOEA/D-NBI NSGA-II**

(c): 6-node (TM3) (d): COST239 (TM1)

(e): COST239 (TM2) (f): COST239 (TM3)

Figure 9.8. Comparison between MOEA/D-NBI and NSGA-II∗∗ by using the HV indicator. Note that,each point represents the mean of the medians of HV reported in Table 9.5 for W =3,4 (TM1),W =4,6 (TM2), and W =6,9 (TM3) – (PART 1/2).


(a): NSF (TM1) (b): NSF (TM2)

(c): NSF (TM3) (d): NTT (TM1)

(e): NTT (TM2) (f): NTT (TM3)

Figure 9.9. Comparison between MOEA/D-NBI and NSGA-II∗∗ by using the HV indicator. Note that,each point represents the mean of the medians of HV reported in Table 9.5 for W =3,4 (TM1),W =4,6 (TM2), and W =6,9 (TM3) – (PART 2/2).


Table 9.6. Set Coverage (AB)

A NSGA-II∗∗ MOEA/D-NBIB MOEA/D-NBI NSGA-II∗∗

6-nodeTM1 2.13% 99.86%TM2 1.78% 100%TM3 2.22% 100%

COST239TM1 0.04% 100%TM2 0.10% 100%TM3 0.21% 100%

NSFTM1 0.16% 100%TM2 0.05% 100%TM3 0.09% 100%

NTTTM1 0.08% 98.30%TM2 0.11% 96.60%TM3 0.01% 97.10%

SC 0.58% 99.32%

9.8 and Figure 9.9, we show the performance of the algorithms varying the number of availabletransceivers (T), but for each T the value of HV reported is the mean obtained with the twonumbers of wavelength tested.

In Table 9.5, we present the median values of HV obtained by the algorithms in each scenario.As we may observe, the MOEA/D-NBI obtains higher values of HV in all scenarios. After conduct-ing the statistical analysis showed in Section 3.4, we conclude that the differences of HV betweenMOEA/D-NBI and NSGA-II∗∗ were statistically significant in all the cases.

Secondly, we compare the algorithms by using the Set Coverage (SC) indicator. It measuresthe fraction of non-dominated solutions achieved by an algorithm B; which are covered by thenon-dominated solutions achieved by an algorithm A (AB).

In Table 9.6 we present the result of comparing the MOEA/D-NBI and the NSGA-II withthe SC indicator. As was expected, the MOEA/D-NBI is able to cover the vast majority of thenon-dominated solutions obtained by the NSGA-II∗∗ in all scenarios. We can conclude that, inaverage, the non-dominated solutions obtained by the NSGA-II∗∗ cover a very low percentage (nearto 0%) of the solutions obtained by the MOEA/D-NBI, whereas the mean percentage of coverageof MOEA/D-NBI is nearly 100%, that is to say, it is able to cover almost all the non-dominatedsolutions of the NSGA-II∗∗.

9.3.3 Comparison with the proposed MOEAs

As a second comparison, we present a comparison between the MOEA/D-NBI approach and theother five MOEAs proposed in Chapter 8 for solving the Traffic Grooming problem: DifferentialEvolution with Pareto Tournaments (DEPT), Multiobjective Variable Neighbourhood Search (MO-VNS), Multiobjective Artificial Bee Colony Algorithm (MO-ABC), Multiobjective GravitationalSearch Algorithm (MO-GSA), and Multiobjective Firefly Algorithm (MO-FA).

All of the aforementioned MOEAs use the same chromosome encoding, detailed in Section5.3. On the one hand, the configuration used for the MOEA/D-NBI is the same that we usein the previous section (see Table 9.4), and, on the other hand, the configurations of the otherfive multiobjective algorithms are described in Section 8.1. For further information regarding the


Table 9.7. 6-node network. Comparison between the MOEA/D-NBI and the five proposed algorithms interms of HV.

6-node network

TM1T W DEPT MO-VNS MO-ABC MO-GSA MO-FA MOEA/D-NBI

3 3 37.95% 38.16% 38.79% 37.76% 38.40% 39.91%4 3 48.89% 48.50% 49.93% 48.63% 49.07% 51.67%5 3 56.95% 55.52% 58.87% 57.11% 57.17% 61.00%6 3 61.40% 59.88% 64.89% 62.55% 62.33% 67.05%7 3 64.81% 63.36% 69.08% 66.50% 66.38% 71.62%3 4 38.27% 38.20% 38.73% 37.83% 37.92% 39.92%4 4 49.02% 48.58% 50.01% 48.64% 48.60% 51.72%5 4 58.22% 56.56% 59.24% 57.85% 57.75% 61.06%6 4 64.54% 60.93% 65.72% 64.30% 64.19% 67.50%7 4 69.04% 64.31% 70.33% 68.88% 68.75% 72.38%


6 4 41.26% 40.19% 41.92% 41.12% 40.44% 42.68%8 4 49.77% 47.98% 50.77% 49.87% 48.99% 51.85%10 4 54.94% 53.03% 56.10% 55.13% 54.41% 57.29%12 4 58.24% 55.99% 59.55% 58.55% 57.88% 60.86%14 4 60.59% 58.04% 61.99% 61.00% 60.35% 63.41%6 6 41.65% 40.97% 42.45% 41.44% 40.52% 42.95%8 6 52.80% 50.97% 53.97% 52.56% 51.29% 55.24%10 6 60.20% 57.07% 61.79% 60.18% 58.28% 63.58%12 6 64.36% 61.19% 67.26% 65.08% 63.09% 69.35%14 6 67.78% 64.26% 70.71% 68.65% 66.59% 73.36%


9 6 31.33% 30.89% 31.97% 31.31% 30.92% 39.83%12 6 39.66% 38.18% 40.52% 39.65% 38.36% 47.76%15 6 44.06% 41.72% 45.69% 44.77% 43.47% 53.46%18 6 47.58% 44.28% 49.23% 48.17% 47.16% 57.02%21 6 49.88% 46.11% 51.66% 50.54% 49.57% 59.75%9 9 31.35% 30.97% 32.08% 31.49% 30.92% 40.32%12 9 41.24% 39.90% 42.14% 41.24% 40.17% 49.41%15 9 48.55% 45.45% 49.72% 48.62% 47.09% 57.56%18 9 53.41% 49.03% 54.67% 53.54% 52.12% 62.87%21 9 56.86% 51.58% 58.35% 57.07% 55.56% 66.60%

chromosome encoding used or the parameters of the algorithms, please refer to the correspondingsections.

Like in the previous section, we compare the algorithms by using the well-known HV and SCindicators. We first measure the quality of the non-dominated solutions achieved by each approachby using the HV indicator; and, then, we compare by pairs of MOEAs by using the SC indicator.

In Table 9.7, Table 9.8, Table 9.9, and Table 9.10, we present the numerical results of HV.We can see that, in the 6-node and COST239 topologies (Table 9.7 and Table 9.8), the

MOEA/D-NBI obtains the higher values of HV in all the scenarios. Therefore, the new approachovercomes clearly the results of the previous best MOEA, the MO-ABC algorithm.

According to Chapter 8, the best MOEA in the topologies NSF and NTT is the MO-GSA;however, we realize that now the MOEA/D-NBI overcomes the results of the MO-GSA.

After performing the statistical study of these results, we can affirm that the differences arestatistically significant in almost all scenarios, we just found three scenarios (out of 120) in which


Table 9.8. COST239 network. Comparison between the MOEA/D-NBI and the five proposed algorithmsin terms of HV.

COST239 network


3 3 27.38% 25.99% 29.25% 27.26% 26.58% 32.14%4 3 35.94% 33.75% 37.88% 35.79% 34.10% 40.30%5 3 44.14% 41.72% 45.95% 43.80% 41.23% 47.33%6 3 49.62% 48.70% 52.18% 49.93% 47.57% 54.53%7 3 54.94% 53.04% 57.23% 54.61% 51.99% 59.70%3 4 27.41% 26.35% 29.12% 27.27% 26.64% 32.32%4 4 35.59% 34.17% 37.83% 35.87% 33.66% 40.39%5 4 43.51% 43.11% 46.00% 43.86% 40.98% 47.34%6 4 50.46% 49.17% 52.39% 49.96% 47.05% 54.61%7 4 54.90% 54.29% 56.90% 54.64% 51.71% 59.70%


6 4 35.31% 34.04% 37.26% 35.52% 32.79% 40.41%8 4 45.68% 43.62% 47.01% 45.44% 42.55% 50.08%10 4 52.77% 49.68% 54.13% 52.43% 49.31% 57.39%12 4 57.60% 53.26% 58.72% 57.14% 54.36% 62.32%14 4 60.73% 56.24% 62.27% 60.50% 58.01% 66.17%6 6 35.67% 34.58% 37.16% 35.44% 33.33% 40.48%8 6 45.35% 43.06% 47.18% 45.47% 42.49% 50.01%10 6 52.77% 49.48% 54.09% 52.37% 49.44% 57.47%12 6 57.26% 53.61% 58.91% 57.20% 54.36% 62.48%14 6 60.82% 56.34% 62.14% 60.60% 57.85% 66.13%


9 6 32.16% 30.03% 33.70% 32.23% 30.09% 36.14%12 6 41.34% 38.48% 43.08% 41.42% 38.98% 45.55%15 6 48.86% 42.74% 50.73% 49.09% 46.22% 53.56%18 6 53.79% 47.45% 55.97% 54.16% 51.59% 58.88%21 6 57.81% 49.89% 59.32% 57.84% 55.33% 62.53%9 9 31.80% 30.16% 33.68% 32.13% 30.18% 36.11%12 9 41.55% 38.14% 43.00% 41.42% 39.11% 45.47%15 9 49.29% 42.63% 50.68% 49.12% 46.31% 53.47%18 9 53.95% 47.52% 55.82% 54.24% 51.94% 58.89%21 9 57.63% 50.07% 59.41% 57.91% 56.01% 62.53%


Table 9.9. NSF network. Comparison between the MOEA/D-NBI and the five proposed algorithms interms of HV.

NSF network


3 3 29.10% 19.11% 29.92% 32.01% 26.89% 35.06%4 3 39.32% 32.65% 40.19% 42.48% 35.76% 44.43%5 3 48.78% 40.68% 48.87% 51.40% 44.09% 53.17%6 3 54.62% 48.45% 55.22% 58.06% 49.72% 60.10%7 3 58.76% 53.64% 59.92% 62.96% 54.02% 65.34%3 4 28.76% 20.22% 29.90% 31.94% 26.76% 35.18%4 4 39.91% 26.04% 39.98% 42.24% 35.73% 44.27%5 4 48.03% 34.65% 49.09% 51.00% 44.29% 53.08%6 4 54.72% 42.57% 55.31% 57.61% 51.02% 60.01%7 4 58.93% 50.37% 60.22% 62.13% 56.15% 65.12%


6 4 37.41% 29.65% 38.04% 39.77% 34.55% 42.03%8 4 45.44% 39.31% 46.58% 48.87% 42.15% 51.57%10 4 49.85% 44.95% 51.44% 54.00% 47.02% 57.53%12 4 53.16% 47.36% 54.89% 57.65% 49.86% 61.72%14 4 55.28% 48.85% 57.26% 60.50% 52.07% 64.46%6 6 38.23% 31.69% 38.80% 40.12% 35.30% 41.95%8 6 49.66% 41.88% 49.80% 51.39% 45.82% 53.03%10 6 56.75% 50.62% 57.39% 58.88% 53.01% 61.19%12 6 61.48% 53.63% 62.17% 63.93% 57.74% 67.02%14 6 65.50% 55.67% 65.54% 67.50% 61.15% 70.43%


9 6 36.37% 30.81% 36.82% 38.16% 33.77% 41.84%12 6 43.07% 37.25% 43.44% 45.35% 40.23% 50.24%15 6 46.95% 39.35% 47.54% 49.90% 43.75% 55.16%18 6 49.34% 40.75% 49.92% 52.27% 46.18% 58.80%21 6 51.05% 41.74% 52.26% 54.26% 47.92% 61.19%9 9 37.75% 32.06% 38.07% 39.21% 34.99% 42.08%12 9 47.74% 37.87% 48.50% 49.66% 44.83% 52.66%15 9 54.76% 43.20% 55.55% 56.71% 51.39% 60.17%18 9 58.71% 45.02% 59.65% 61.23% 55.76% 65.66%21 9 61.73% 46.31% 62.95% 64.89% 58.79% 68.58%


Table 9.10. NTT network. Comparison between the MOEA/D-NBI and the five proposed algorithms interms of HV. The notation used for pointing the statistically non-significant differences is the following:(†) MOEA/D-NBI and MO-GSA.

NTT network


3 3 13.72% 11.60% 14.57% 14.03% 12.49% 16.15%4 3 18.78% 15.00% 19.94% 19.49% 17.38% 22.27%5 3 22.25% 16.28% 23.63% 22.95% 20.84% 26.18%6 3 24.40% 17.90% 26.41% 25.53% 23.01% 28.63%7 3 25.57% 20.15% 28.25% 27.24% 24.30% 30.54%3 4 13.88% 11.71% 14.75% 14.40% 12.62% 16.55%4 4 20.39% 16.69% 21.22% 20.82% 18.69% 23.44%5 4 24.53% 19.32% 25.90% 25.36% 23.05% 28.43%6 4 27.71% 21.68% 29.45% 28.65% 26.24% 32.00%7 4 30.23% 22.83% 31.66% 31.41% 28.24% 34.71%


6 4 19.84% 15.09% 21.02% 20.49% 18.85% 22.57%8 4 22.64% 16.05% 24.06% 23.49% 21.59% 25.66%10 4 23.81% 17.06% 25.65% 25.09% 23.17% 27.75%12 4 25.28% 17.34% 26.70% 25.97% 24.05% 28.98%14 4 25.93% 17.72% 27.43% 26.96% 24.78% 29.78%6 6 21.73% 18.00% 22.79% 22.44% 20.70% 23.94%8 6 26.30% 20.45% 27.32% 26.86% 25.05% 28.90%10 6 28.83% 21.65% 30.16% 29.89% 27.73% 32.08%12 6 30.38% 19.88% 31.91% 31.57% 29.35% 33.97%14 6 31.63% 20.35% 33.25% 32.77% 30.64% 35.59%


9 6 17.42% 12.27% 18.30% 17.97% 16.73% 18.81%12 6 19.32% 12.56% 20.48% 19.99% 18.65% 20.97%15 6 20.26% 12.96% 21.66% 21.14% 19.56% 22.21%18 6 21.01% 13.24% 22.60% 22.00% 20.31% 22.95%21 6 21.53% 13.45% 23.37% 22.41% 20.84% 23.68%†

9 9 19.01% 14.25% 20.23% 19.94% 18.55% 20.56%12 9 22.42% 15.13% 23.73% 23.18% 21.66% 23.93%†

15 9 24.39% 15.43% 25.63% 25.20% 23.41% 26.15%18 9 25.76% 15.86% 27.04% 26.51% 24.89% 27.27%†

21 9 26.51% 16.16% 27.90% 27.35% 25.81% 28.34%


(a): 6-node (TM1) (b): 6-node (TM2)

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DEPT MO-VNS MO-ABC MO-GSA MO-FA MOEA/D-NBI

(c): 6-node (TM3) (d): COST239 (TM1)

(e): COST239 (TM2) (f): COST239 (TM3)

Figure 9.10. Comparison among DEPT, MO-VNS, MO-ABC, MO-GSA, MO-FA, and MOEA/D-NBIby using the HV indicator. Note that, each point represents the mean of the medians of HV reportedin Table 9.7 (6-node), Table 9.8 (COST239), Table 9.9 (NSF), and Table 9.10 (NTT) for W =3,4(TM1), W =4,6 (TM2), and W =6,9 (TM3) – (PART 1/2).


(a): NSF (TM1) (b): NSF (TM2)

(c): NSF (TM3) (d): NTT (TM1)

(e): NTT (TM2) (f): NTT (TM3)

Figure 9.11. Comparison among DEPT, MO-VNS, MO-ABC, MO-GSA, MO-FA, and MOEA/D-NBIby using the HV indicator. Note that, each point represents the mean of the medians of HV reportedin Table 9.7 (6-node), Table 9.8 (COST239), Table 9.9 (NSF), and Table 9.10 (NTT) for W =3,4(TM1), W =4,6 (TM2), and W =6,9 (TM3) – (PART 2/2).


Table 9.11. Summary of the Set Coverage indicator among the diverse MOEAs. We present the average,worst, and best coverage relation.

A Average Worst Best ∗ ∗ ∗B ∗ ∗ ∗ Average Worst Best

6-nodeTM1 3.88% 7.70% 1.09% 99.86% 98.99% 100%TM2 2.29% 3.91% 0.71% 99.86% 99.01% 100%TM3 1.79% 4.61% 0.00% 99.99% 99.95% 100%

COST239TM1 1.79% 4.17% 0.04% 84.50% 69.09% 94.99%TM2 0.04% 0.17% 0.00% 98.92% 96.41% 100%TM3 0.08% 0.34% 0.00% 97.50% 87.76% 100%

NSFTM1 7.99% 14.53% 1.69% 68.32% 37.78% 86.87%TM2 5.23% 12.55% 0.34% 74.91% 50.31% 91.81%TM3 0.01% 0.06% 0.00% 99.95% 99.77% 100%

NTTTM1 0.38% 1.63% 0.00% 95.34% 86.11% 99.90%TM2 1.08% 2.68% 0.02% 81.95% 69.17% 94.55%TM3 2.63% 6.23% 0.18% 51.52% 36.19% 78.59%

SC 2.26% 4.88% 0.34% 87.72% 77.54% 95.56%∗ MOEA/D-NBI

the differences of HV between the MO-GSA and MOEA/D-NBI are not statistically significant.Note that, in Table 9.7, Table 9.8, Table 9.9, and Table 9.10, we only present the results of thestatistical test between the MOEA/D-NBI and the rest of MOEAs.

In Figure 9.10 and Figure 9.11, we compare the MOEAs by using the HV indicator like inFigure 9.8 and Figure 9.9. We present an illustrative view of the behaviour of the MOEAs. Notethat in Figure 9.10 and Figure 9.11, each point represents the mean of the medians of HV reportedin Table 9.7 (6-node), Table 9.8 (COST239), Table 9.9 (NSF), and Table 9.10 (NTT) for W =3,4(TM1), W =4,6 (TM2), and W =6,9 (TM3).

As we may observe, the differences of HV between the MOEA/D-NBI and the other MOEAs areclear in all the test instances tested, highlighting the particular performance of the MOEA/D-NBIwhen we tackle the small 6-node network with the largest amount of traffic (Figure 9.10(c)).

Now, we present a comparison among the multiobjective approaches by using the SC indicator.In Table 9.11, we summarize the best, worst, and average percentage of solutions achieved bythe MOEAs (DEPT, MO-VNS, MO-GSA, MO-ABC, and MO-FA) which are covered by the non-dominated solutions obtained by the MOEA/D-NBI approach. As we can observe, in average,the MOEA/D-NBI covers 87.72% of the solutions of the other MOEAs, whereas its set of non-dominated solution is only covered 2.26%. In the worst case, the MOEA/D-NBI is able to cover77.54% of the solutions, whereas 4.88% of its non-dominated solutions are covered.

Finally, in Figure 9.12, we present the best set of non-dominated solutions obtained by theMOEA/D-NBI and by the best proposed MOEA at scenario. Note that, the best MOEAs arethe MO-ABC (6-node and COST239) and the MO-GSA (NSF and NTT), see Table 8.23 andTable 8.24 in Section 8.2.5. Depending on the traffic matrix: small, medium, or large; the set ofnon-dominated solutions showed in Figure 9.12 corresponds with the scenarios T=7 W=4, T=14W=6, or T=21 W=9; respectively. As we may observe in Figure 9.12, the sets of non-dominatedsolutions obtained by the MOEA/D-NBI are clearly better (in terms of quality) than those setsof non-dominated solutions achieved by the best approaches (MO-ABC and MO-GSA) presentedin Section 8.2.5 (Figure 8.22). In addition, we would like to highlight the good exploration of theMOEA/D-NBI of the objective space.


(a): 6-n TM1 T=7 W=4 (b): 6-n TM2 T=14 W=6 (c): 6-n TM3 T=21 W=9

(d): COST239 TM1 T=7 W=4 (e): COST239 TM2 T=14 W=6 (f): COST239 TM3 T=21 W=9

(g): NSF TM1 T=7 W=4 (h): NSF TM2 T=14 W=6 (i): NSF TM3 T=21 W=9

(j): NTT TM1 T=7 W=4 (k): NTT TM2 T=14 W=6 (l): NTT TM3 T=21 W=9

Figure 9.12. Best set of non-dominated solutions obtained by the MOEA/D-NBI (∗) and the bestproposed MOEA () at each scenario.


9.3.4 Comparison with other works

There exist diverse authors that have tackled the Traffic Grooming by using diverse approaches.In this section, with the aim of ensuring the goodness of the MOEA/D-NBI approach, we presenta single-objective comparison with these heuristics and metaheuristics published in the literature.

In the following, we briefly describe each one with the aim of understanding how they work,for further information about them, please refer to the corresponding references:

• Maximizing Single-Hop Traffic (MST) [63]. It always attempts to set up lightpaths betweenthose source and destination nodes with highest amount of pending traffic. Each connectionwill be routed on the newly established lightpath whenever possible (single-hop). If thereis no enough available capacity, it will be routed through several concatenated lightpaths(multi-hop).

• Maximizing Resource Utilization (MRU) [63]. It tries to establish lightpaths between thosesource and destination nodes with maximum resource utilization values. If there is no enoughavailable resources, the connection requests are routed on the existing lightpaths taking intoaccount their connection resource utilization values as well.

• INtegrated Grooming PROCedure (INGPROC) [66]. It uses an Integrated Grooming Algo-rithm based on an Auxiliary Graph (IGABAG) model, to accommodate the low-speed trafficrequests. For the INGPROC, the authors propose the following three traffic-request-selectionschemes:

– Least Cost First (LCF). It always tries to route the most cost-effective traffic requests.The cost of each connection is computed as: the weight of the path for routing theconnection on the auxiliary graph, divided by its amount of pending traffic.

– Maximum Utilization First (MUF). It selects those low-speed traffic requests with thehighest utilization. The utilization of a request is computed as the amount of traffic di-vided by the number of hops in the physical path from the source node to the destinationnode.

– Maximum Amount First (MAF). It tries to route, in the first place, those pairs of nodeswith the largest number of demands.

• Strength Pareto Evolutionary Algorithm (SPEA) [79]. This well-known MOEA was proposedby Prathombutr et al. for optimizing simultaneously the total traffic throughput, the numberof lightpaths established, and the average propagation delay of the lightpaths. However, theydo not present enough information for including a multiobjective comparison.

• Traffic Grooming based on Clique Partitioning (TGCP) [55]. It works in two steps. It startsconstructing a virtual topology using a clique-partitioning based heuristics. Then, it tries toestablish as many connection requests as possible on the virtual topology by using a singlelightpath (single-hop). When it is not possible, the connection is routed by using multiplelightpaths (multi-hop).

To compare the MOEA/D-NBI with these approaches, we present an illustrative relationshipbetween the maximum value of throughput (f1) and the number of available wavelengths perlink (W ) for the algorithms under study. Note that, we consider the highest value of throughputobtained in each scenario by the MOEA/D-NBI, as in [79] for the SPEA.


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In Figure 9.13, we compare the maximum number of successfully routed traffic requests obtainedby the aforementioned methods and by the MOEA/D-NBI in the 6-node network with the firsttraffic matrix (TM1), and W =3. On the one hand, in Figure 9.13(a) and Figure 9.13(b), we mayobserve that the MOEA/D-NBI performs much better than the classical MST, MRU, or INGPROC(LCF, MUF, MAF) with diverse number of transceivers per node (T ) used. On the other hand,it performs better than or equal to the well-known SPEA and the TGCP approach as well, seeFigure 9.13(a).

A comparison using the 6-node topology, TM1, and W =4 is shown in Figure 9.149.14. InFigure 9.14(a) and Figure 9.14(b), we can see that the heuristics MST, MRU, and INGPROCobtain lower values of throughput than MOEA/D-NBI with T =3 and T = 4. If we compare theMOEA/D-NBI, the TGCP, and SPEA with T=3,4 (see Figure 9.14(c)), we realize that withT =3 the TGCP obtains slightly higher value of throughput than the MOEA/D-NBI, whereas ourapproach is always better than SPEA. Finally, with T =5, all approaches obtained the maximumvalue of throughput (988 OC-1 units).


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9.4 Conclusions

In this chapter we present an innovative multiobjective framework based on decomposition forsolving real-world 3-objective optimization problems with objectives in very different scales. Thisalgorithm is a modified version of the MOEA/D which uses a decomposition method that takes theadvantages of the NBI approach and the Tchebycheff approach (MOEA/D-NBI) for decomposingany 3-objective optimization problem into a finite number of single-objective problems, optimizingeach one by using the information from the optimization of neighbouring solutions.

In order to test the accuracy of this multiobjective framework, we have applied it to a real-world Telecommunication problem, the Traffic Grooming problem. This problem is a 3-objectiveoptimization problem where the objectives are commonly in very different scales.

Furthermore, we propose a new construction heuristics for generating a single solution to thisTelecommunication problem. This new construction heuristics requires certain control parameterswhich largely determine the performance of the generated solutions. In this way, we use indirect

9.4 Conclusions 231

encoding in the proposed framework (MOEA/D-NBI) to optimize the control parameters of thisconstruction heuristics.

We present several experiments in a wide range of scenarios in order to prove the effectiveness ofthe MOEA/D-NBI framework. We start comparing it with the well-known NSGA-II where both ap-proaches use identical chromosome encoding, construction heuristics, and genetic operator. Then,we compare the MOEA/D-NBI with the other five proposed MOEAs for the Traffic Groomingproblem: DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA. Finally, we make a single-objectivecomparison between the MOEA/D-NBI and diverse heuristics and metaheuristics published in theliterature by other authors.

To sum up, it seems that the use of multiobjective optimization based on decomposition foroptimizing the control parameters of a construction heuristics, leads to very promising results in thisreal-world telecommunication problem with objective functions in very different scales. Actually,the proposed framework with indirect encoding may be used in many real-world optimizationproblems.

10Conclusions and future work

In this chapter we summarize the main conclusions obtained from the set of experiments and workpresented in this thesis. Furthermore, since the number of plausible lines of future work is large,we briefly describe the main ones.

10.1 Conclusions

In this PhD thesis we have tackled two real-world optimization problems related to the Telecommu-nication field. In these problems, the number of objective functions is greater than one; therefore,we face two multiobjective optimization problems. Since the two problems are NP-hard, we haveapplied Evolutionary Computation and Multiobjective Optimization jointly in order to solve themproperly.

Concretely, we have focused on optimizing Optical Network problems, because this kind ofnetworks has attracted much more attention from the Telecommunication industry in the lastdecade. The two problems tackled are: Routing and Wavelength Assignment problem and theTraffic Grooming problem.

After linking the two problems to the evolutionary algorithms domain, we have developed atotal of five different Multiobjective Evolutionary Algorithms (MOEAs) with the aim of solvingthese MOOPs efficiently. These MOEAs are listed as follows: Differential Evolution with ParetoTournaments (DEPT), Multiobjective Variable Neighbourhood Search (MO-VNS), MultiobjectiveArtificial Bee Colony Algorithm (MO-ABC), Multiobjective Gravitational Search Algorithm (MO-GSA), and Multiobjective Firefly Algorithm (MO-FA). Furthermore, we have designed two parallelalgorithms: pDEPT and pMOABC; for exploiting diverse multi-core systems in the same intercon-necting data network. The goal of the parallel approaches is to speed the runtime of the algorithms,obtaining identical high-quality solutions than the sequential model in a reasonable amount of time.

Firstly, we start summarizing the main conclusions extracted from the RWA problem. A com-parative study on different MOEAs for solving the RWA problem has been carried out in this thesis.In this comparative study, we have evaluated the five aforementioned multiobjective metaheuristics:DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA.

In this way, we have analyzed their capabilities to solve the RWA problem over three real-worldoptical networks, comparing their results with those obtained by two well-known multiobjective

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approaches: Fast Non-Dominated Sorting Genetic Algorithm (NSGA-II) and the Strength ParetoEvolutionary Algorithm 2 (SPEA2).

The optical networks tested in this study correspond with the Pan European Optical Network(COST239, Europe, 11 nodes and 52 links), the National Science Foundation network (NSF, U.S.A.,14 nodes and 42 links), and the Nippon Telegraph and Telephone network (NTT, Japan, 55 nodesand 144 links). For each optical network, we have designed twelve data sets with different numberof demands, therefore a total of 36 data sets.

Our study has revealed that the usage of these MOEAs for solving the RWA problem is verysuitable. More concretely, we have noticed remarkable differences in the quality of the solutionsbetween our multiobjective approaches and the standard multiobjective approaches (NSGA-II andSPEA2). From this study, we conclude that the MO-FA obtains very promising results whenfacing this Telecommunication problem, specially when the number of demands increases. Inaddition, since the other multiobjective proposals based on swarm intelligence also obtain suchgood results too, we may affirm that the swarm approaches work much better because they generatenew individuals taking not only parts from its parents, but also from the rest of the population,increasing the richness of search as a result. In addition, we have to highlight that the MO-FA isa very promising approach to deal with this telecommunication problem.

Then, we perform a comparison between the five proposed MOEAs and a great variety of heuris-tics and metaheuristics published in the literature for solving the RWA problem. In first place, in[42], the authors suggest eight Typical Heuristics in the Telecommunication field, formed by com-bining two Routing methods (3-Shortest-Paths and Shortest-Path-Dijsktra) and four WavelengthAssignment techniques (First Fit, Least Used, Most Used, and Random). Secondly, in [41] and [42],eight varieties of Multiobjective Ant Colony Optimization algorithms (MOACOs) are presented.After performing an exhaustive comparison among the MOEAs and more than fifteen approachespublished by other authors in the literature, we conclude that our MOEAs are able to overcomethe results of these methods.

To finish with the RWA problem, we have applied the parallel versions of the DifferentialEvolution with Pareto Tournaments (pDEPT) to this problem in order to reduce runtime of thealgorithm, obtaining results of identical quality in a reasonable amount of time. In this experimentswe have used four 8-core nodes interconnected through the same data network. We present dif-ferent comparisons among shared-memory (OpenMP), distributed-memory (MPI), and/or hybridshared/distributed-memory approaches (OpenMP+MPI). After studying the performance in differ-ent scenarios with 2, 4, 8, 16, and 32 cores, we can conclude that the hybrid version OpenMP+MPIof the pDEPT is very suitable for solving NP-hard problems, such as the RWA problem. In thisproblem, we have obtained a mean efficiency of 92.45% with 32 cores, which implies that we mayobtain high quality results nearly 30 times faster.

Secondly, we continue summarizing the main conclusions extracted from the Traffic Groomingproblem. Like in the RWA problem, we have presented a comparative study on different MOEAsfor solving the Traffic Grooming problem. The MOEAs involved on this comparative study arethe five proposed MOEAs: DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA.

Since the aim of this section was to compare several MOEAs when dealing with the TrafficGrooming problem, we have used four network topologies with different number of nodes, andfor each one, a variety of traffic matrices with different amounts of traffic (small, medium, andlarge). The four optical topologies tested are: The 6-node network (6-node, 6 nodes and 16 links),the European Optical Network (COST239, Europe, 11 nodes and 52 links), the National ScienceFoundation network (NSF, USA, 14 nodes and 42 links), and the Nippon Telegraph and Telephonenetwork (NTT, Japan, 55 nodes and 144 links). For each topology and each traffic matrix, we have

10.1 Conclusions 235

tested diverse scenarios with different available resources. On the whole, in this comparative study,we have used four optical network topologies, each one with three different sized traffic matrices,and each traffic matrix is tested in 48 scenarios with different available resources; thus, a total of576 different scenarios.

After performing a comprehensive comparison between our approaches and the two well-knownMOEAs (NSGA-II and SPEA2), we present some conclusions for each algorithm. In the firstplace, we have noticed that the DEPT algorithm is a competitive algorithm, it is the third bestalgorithm in almost all network topologies. However, the trajectory-based algorithm (MO-VNS)is not a good choice for this problem due to the lack of quality in their non-dominated solutions,particularly when it deals with medium or large networks, and also when deals with small networksbut medium or large loads of traffic. The non-dominated solutions obtained by the well-knownMOEAs (NSGA-II and SPEA2) have resulted to be dominated by several of the algorithms in allscenarios tested. It seems that some of the algorithms based on swarm intelligence obtain highquality results when solving the Traffic Grooming problem. On the one hand, the algorithm basedon the behaviour of honey bees, the MO-ABC algorithm, seems to be a very promising algorithmwhen dealing with small networks, such as 6-node and COST239; however, its quality slightlydecreases when dealing with large optical networks. In the same way, the MO-FA performs betterwith small networks than with large ones. On the other hand, the MO-GSA is a very efficientMOEA when the number of nodes is increased, according to the previous sections. Furthermore,this algorithm based on the law of gravity and mass interactions obtains the second best results inthe small network topologies.

Furthermore, we have made a comparison between the best MOEA in the 6-node network (MO-ABC) and several approaches published in the literature, such as Maximizing Single-Hop Traffic(MST), Maximizing Resource Utilization (MRU), INtegrated Grooming PROCedure (INGPROC)with three traffic-request-selection schemes (LCF, MUF, and MAF), the Strength Pareto Evolu-tionary Algorithm, and the Traffic Grooming based on Clique Partitioning (TGCP); concludingthat the results achieved by the MO-ABC algorithm overcome those ones obtained by other ap-proaches published in the literature. This may be due to the fact that the main feature of theMO-ABC algorithm is the collective behaviour of several decentralized and self-organized swarmsthat work together to explore efficiently the search space.

Like in the RWA problem, we have applied a parallel MOEA to this problem in order to reduceruntime of the algorithm, obtaining results of identical quality in a reasonable amount of time.In this case, we have used the second parallel approach, the pMOABC in four 8-core nodes inter-connected through the same data network (a total of 32 cores). In these experiments, we presentdifferent comparisons among different versions of the pMOABC for shared-memory (OpenMP),distributed-memory (MPI), and/or hybrid shared/distributed-memory (OpenMP+MPI). In thisway, after a performance study with different scenarios (2, 4, 8, 16, and 32 cores), we can concludethat the hybrid version OpenMP+MPI of the pMOABC is really promising for solving the TrafficGrooming problem (an NP-hard problem) in a reasonable amount of time for the industry, as it isable to solve (in average) the problem 26 times faster with 32 cores than with a single-core system.

From the positive results obtained in this thesis, it seems reasonable to think that the multiob-jective proposals of this manuscript could be applied not only in other Telecommunication problemsthat involve routing ([158], [159]), but also in diverse Engineering multiobjective problems ([160],[161]).

Finally, we propose a new indirect encoding model using a construction heuristics for solvingthe Traffic Grooming problem. In this method, we use a two-phase metaheuristics for constructinga solution to the problem. Furthermore, we formally define a new multiobjective framework based

236 10. Conclusions and future work

on decomposition for solving real-world 3-objective optimization problems with objectives in verydifferent scales.

After several experiments in a wide range of scenarios in order to prove the effectiveness of theMOEA/D-NBI framework, we start comparing it with the well-known NSGA-II where both ap-proaches use identical chromosome encoding, construction heuristics, and genetic operator. Then,we compare the MOEA/D-NBI with the other five proposed MOEAs for the Traffic Groomingproblem: DEPT, MO-VNS, MO-ABC, MO-GSA, and MO-FA. Finally, we make a single-objectivecomparison between the MOEA/D-NBI and diverse heuristics and metaheuristics published inthe literature by other authors. In all the comparisons, the MOEA/D-NBI has obtained goodresults. Furthermore, we can conclude that the use of the MOEA/D-NBI for optimizing the con-trol parameters of a construction heuristics, leads to very promising results in this real-worldtelecommunication problem with objective functions in very different scales.

10.2 Future work

Many are the lines of future work extracted from this PhD thesis. In the following, we summarizethe main ones in order to provide new ideas for future researches.

In the first place, in this work we have proposed five MOEAs for the RWA problem and theTraffic Grooming, as well as two parallel versions of the DEPT and the MO-ABC for solving themin a reasonable amount of time. In this way, a immediate line of research would be the use of thisparallel approaches to obtain higher quality results in both problems. Another idea would be theuse of a parallel team of metaheuristics which consists of the five MOEAs working together forsolving these telecommunication problems. It is well-known that commonly the algorithms exploredifferent areas of the search space; therefore, it is reasonable to think that the use of a cooperativeparallel metaheuristic leads to much better results.

In the second place, in this thesis we tackle the static version of the RWA problem and theTraffic grooming problem; thus, a possible line of future work is to use dynamic traffic patternsand prove the effectiveness and goodness of the five MOEAs proposed in this work.

We think there is further room to research in the normalization process within the MOEAs,it is a promising research area. In this work, we present a preliminary version that significantlyimproves the results obtained by other frameworks which use normalization; however, further ex-periments and more benchmarks with different characteristics (concave, convex, discrete, continue,constrained, unconstrained...) are needed to prove the effectiveness of this proposal.

Finally, the MOEA/D-NBI seems to be very suitable to be parallelized due to its nature: itdecomposes a MOOP into a finite number of single-objective subproblems; so, we may assign toeach core a single-objective subproblem.

11Scientific Achievements

In this chapter we present the papers that have been published as a result of the research workdeveloped throughout this PhD Thesis. Directly related to the topic of this thesis (see Section 11.1),we have obtained a total of 22 publications, 4 publications in different international journals and18 publications in conferences (international or national). In addition, 5 publications partiallyrelated to the main topic of this thesis have bee obtained (see Section 11.2).

All in all, a total of 27 publications have been achieved through these years, 5 publicationsin different international journals and 22 publications in diverse conferences. Finally, we includeother scientific achievements: International stays, reviewer of International Journals, participationin International Conferences, participation in Research Projects, and collaborations with otherinstitutions.

11.1 Publications related to this PhD Thesis

The publications obtained from this PhD thesis ensure a certain quality in the research work car-ried out throughout the last three years and a half. All of these publications have been publishedin international or national peer-reviewed conferences with worldwide renown, as well as in in-ternational peer-reviewed journals indexed by the Institute for Scientific Information (ISI) in theJournal Citation Reports (JCR). Next, we enumerate the publications obtained which are directlyrelated to the topic of this thesis.

A total of 4 publications in international journals indexed by the ISI in the JCR:

J1. Multiobjective Metaheuristics for Traffic Grooming in Optical Networks. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. IEEETransactions on Evolutionary Computation, IEEE, 2012, pp.1-17, ISSN:1089-778X. (ImpactFactor = 3.341 in 2011, 1st Quartile – Q1).

J2. A Comparative Study on Multiobjective Swarm Intelligence for the Routing and WavelengthAssignment Problem. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. IEEE Transactions on Systems, Man, and Cybernetics - PartC: Applications and Reviews, Volume 42, Issue 6, IEEE, 2012, pp.1644-1655, ISSN:1094-6977.(Impact Factor = 2.009 in 2011, 1st Quartile – Q1).

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J3. A Multiobjective Approach based on Artificial Bee Colony for the Static Routing and Wave-length Assignment Problem. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A.Gómez-Pulido, Juan M. Sánchez-Pérez. Soft Computing, Volume 17, Issue 2, Springer, 2012,pp.199-211, ISSN:1432-7643. (Impact Factor = 1.880 in 2011, 1st Quartile – Q1).

J4. Applying MOEAs to Solve the Static Routing and Wavelength Assignment Problem in OpticalWDM Networks. Álvaro Rubio-Largo and Miguel A. Vega-Rodríguez. Engineering Ap-plications of Artificial Intelligence, Volume XX, Issue XX, Pergamon-Elsevier Science, 2013,pp.1-31, ISSN:0952-1976. (Impact Factor = 1.665 in 2011, 1st Quartile – Q1).

A total of 18 publications in international (12) and national (6) conferences:

C1. MOEA/D for Traffic Grooming in WDM Optical Networks. Álvaro Rubio-Largo, QingfuZhang, and Miguel A. Vega-Rodríguez. GECCO 2013 Proceedings, Association for Comput-ing Machinery (ACM), Amsterdam, The Netherlands, 2013, pp.1-8 (pending). ISBN:(pending).

C2. Routing Low-speed Traffic Requests onto High-speed Lightpaths by using a Multiobjective Fire-fly Algorithm. Applications of Evolutionary Computation. Lecture Notes in Computer Sci-ence (LNCS), Vol. 7835. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. Springer-Verlag, 2013, pp.12-21. ISBN:978-3-642-37191-2.(LNCS Conference).

C3. A Multiobjective Approach Based on the Law of Gravity and Mass Interactions for OptimizingNetworks. Evolutionary Computation in Combinatorial Optimisation. Lecture Notes inComputer Science (LNCS), Vol. 7832. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez,Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. Springer-Verlag, 2013, pp.13-24. ISBN:978-3-642-37197-4. (LNCS Conference).

C4. Using a Multiobjective OpenMP+MPI DE for the Static RWA Problem. Computer AidedSystems Theory. Lecture Notes in Computer Science (LNCS), Vol. 6927. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. Springer-Verlag, 2012, pp.224-231. ISBN:978-3-642-27548-7. (LNCS Conference).

C5. A Multiobjective Gravitational Search Algorithm Applied to the Static Routing and Wave-length Assignment Problem. Applications of Evolutionary Computation. Lecture Notes inComputer Science (LNCS), Vol. 6625. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez,Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. Springer-Verlag, 2011, pp.41-50. ISBN:978-3-642-20519-4. (LNCS Conference).

C6. Tackling the Static RWA Problem by Using a Multiobjective Artificial Bee Colony Algorithm.Advances in Computational Intelligence. Lecture Notes in Computer Science (LNCS), Vol.6692. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, JuanM. Sánchez-Pérez. Springer-Verlag, 2011, pp.364-371. ISBN:978-3-642-21497-4. (LNCSConference).

C7. A Parallel Multiobjective Artificial Bee Colony Algorithm for Dealing with the Traffic Groom-ing Problem. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido,Juan M. Sánchez-Pérez. 14th IEEE International Conference on High Performance Comput-ing and Communications, IEEE Computer Society, 2012, pp.46-53. ISBN:978-0-7695-4749-7.(IEEE Conference).

11.1 Publications related to this PhD Thesis 239

C8. A Differential Evolution with Pareto Tournaments for solving the Routing and WavelengthAssignment Problem in WDM Networks. Álvaro Rubio-Largo, Miguel A. Vega-Rodriguez,Juan A. Gomez-Pulido, Juan M. Sanchez-Perez. Proceedings of the 2010 IEEE Congress onEvolutionary Computation (CEC 2010), IEEE Computer Society, 2010, pp.129-136. ISBN:978-1-4244-6910-9. (IEEE Conference).

C9. Improving Optical WDM Networks by using a Multi-core version of Differential Evolutionwith Pareto Tournaments. Distributed Computing and Artificial Intelligence. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. Springer-Verlag, 2010, pp.629-636. ISBN:978-3-642-14882-8. (International Conference).

C10. Solving the Routing and Wavelength Assignment Problem in WDM Networks by using a Multi-objective Variable Neighborhood Search Algorithm. Soft Computing Models in Industrial andEnvironmental Applications. Advances in Intelligent and Soft Computing, Vol. 73. ÁlvaroRubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez.Springer-Verlag, 2010, pp.47-54. ISBN:978-3-642-13160-8. (International Conference).

C11. A Multiobjective Hybrid Parallel Approach based on Swarm Intelligence for Optimizing Op-tical Networks. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido,Juan M. Sánchez-Pérez. Computer Aided Systems Theory - Extended Abstracts, A. Que-sada, J.C. Rodriguez, R. Moreno jr., R. Moreno (Eds.). IUCTC. Universidad de Las Palmasde Gran Canaria, 2013, pp.236-238. ISBN:978-84-695-6971-9. (International Conference).

C12. A Hybrid OpenMP/MPI Differential Evolution for Solving the RWA Problem. ÁlvaroRubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez.Computer Aided Systems Theory - Extended Abstracts, A. Quesada, J.C. Rodriguez, R.Moreno jr., R. Moreno (Eds.). IUCTC. Universidad de Las Palmas de Gran Canaria, 2011,pp.236-238. ISBN:978-84-693-9560-8. (International Conference).

C13. Algoritmo Multiobjetivo Inspirado en el Comportamiento de las Luciérnagas para Resolver elProblema RWA. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido,Juan M. Sánchez-Pérez. Actas del VIII Congreso Español sobre Metaheurísticas, AlgoritmosEvolutivos y Bioinspirados (MAEB 2012), J. A. Gámez, J. M. Puerta, F. Parreño y L. de laOssa (Eds.), Universidad de Castilla-La Mancha, 2012, pp.367-374. ISBN:978-84-615-6931-1.(National Conference).

C14. Optimización de Redes Ópticas WDM mediante Paralelismo e Inteligencia Colectiva. ÁlvaroRubio-Largo, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez.Actas de las XXIII Jornadas de Paralelismo (JP 2012), Servicio de Publicaciones. UniversidadMiguel Hernández, 2012, pp.135-140. ISBN:978-84-695-4471-6. (National Conference).

C15. Evolución Diferencial OpenMP+MPI en Redes Ópticas WDM. Álvaro Rubio-Largo, MiguelA. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. Actas de las XXII Jor-nadas de Paralelismo (JP 2011), Servicio de Publicaciones. Universidad de La Laguna, 2011,pp.15-20. ISBN:978-84-694-1791-1. (National Conference).

C16. Algoritmo Evolución Diferencial con Torneos de Pareto para Resolver el Problema de En-rutamiento y Asignación de Longitudes de Onda en Redes Ópticas WDM. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Juan Antonio Gómez-Pulido, Juan Manuel Sánchez-Pérez.Actas de las VIII Jornadas de Aplicaciones y Transferencia Tecnológica de la Inteligencia


Artificial, TTIA 2010 (AEPIA), Ibergarceta Publicaciones S.L., 2010, pp.45-54. ISBN:978-84-92812-57-8. (National Conference).

C17. Búsqueda de Entorno Variable Multiobjetivo para Resolver el Problema de Enrutamiento yAsignación de Longitudes de Onda en Redes Ópticas WDM. Álvaro Rubio-Largo, MiguelA. Vega-Rodríguez, Juan Antonio Gómez-Pulido, Juan Manuel Sánchez-Pérez. Actas deMAEB 2010. VII Congreso Español sobre Metaheurísticas, Algoritmos Evolutivos y Bioin-spirados, Ibergarceta Publicaciones S.L., 2010, pp.3-10. ISBN:978-84-92812-58-5. (NationalConference).

C18. Utilizando una versión Multi-núcleo de la Evolución Diferencial con Torneos de Pareto paraMejorar las Redes Ópticas WDM. Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, JuanA. Gómez-Pulido, Juan M. Sánchez- Pérez. Actas de las XXI Jornadas de Paralelismo (JP2010), Ibergarceta Publicaciones S.L., 2010, pp.43-50. ISBN:978-84-92812-49-3. (NationalConference).

11.2 Other Publications

On the other hand, we have obtained another 5 publications partially related to the main topicof this PhD thesis. This publications come from early research works, or simply from collabora-tion with other institutes or researchers. All in all, these publications were not only essential inthe coursework stage, but also in the research stage of the PhD candidate; therefore, we shouldenumerate them.

A total of 1 publications in international journals indexed by the ISI in the JCR:

J5. Solving the Reporting Cells Problem by Using a Parallel Team of Evolutionary Algorithms.David L. González-Álvarez, Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez, Sónia M.Almeida-Luz, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. Logic Journal of the IGPL,Vol. 20, Issue 4, Oxford University Press, 2012, pp.722-731, ISSN:1367-0751. (Impact Factor= 0.913 in 2011, 1st Quartile – Q1).

A total of 4 publications in international (2) and national (2) conferences:

C19. MO-ABC/DE - Multiobjective Artificial Bee Colony with Differential Evolution for Uncon-strained Multiobjective Optimization. Álvaro Rubio-Largo, David L. González-Álvarez,Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. 13th IEEEInternational Symposium on Computational Intelligence and Informatics, IEEE ComputerSociety, 2012, pp.157-162. ISBN:978-1-4673-5206-2. (IEEE Conference).

C20. A Parallel Cooperative Evolutionary Strategy for Solving the Reporting Cells Problem. SoftComputing Models in Industrial and Environmental Applications. Advances in Intelligentand Soft Computing, Vol. 73. Álvaro Rubio-Largo, David L. González-Álvarez, MiguelA. Vega-Rodríguez, Sónia M. Almeida-Luz, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez.Springer-Verlag, 2010, pp.71-78. ISBN:978-3-642-13160-8. (International Conference).

C21. Optimización y Paralelización de Software de Monte Carlo para el Estudio de Nano-DispositivosSemiconductores. Sara M. Rubio-Largo, Álvaro Rubio-Largo, Miguel A. Vega-Rodríguez,Jesús E. Velázquez-Pérez. Actas de las XXIII Jornadas de Paralelismo (JP 2012), Servicio dePublicaciones. Universidad Miguel Hernández, 2012, pp.158-162. ISBN:978-84-695-4471-6.(National Conference).

11.3 Other scientific achievements 241

C22. Resolución del Problema Reporting Cells mediante Computación Clúster y un Equipo Paralelode Algoritmos Evolutivos. David L. González-Álvarez, Álvaro Rubio-Largo, Miguel A.Vega-Rodríguez, Sónia Almeida-Luz, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. XXJornadas de Paralelismo (JP 2009), Servizo de Publicacións, Universidade da Coruña, 2009,pp:69-74. ISBN:84-9749-346-8. (National Conference).

11.3 Other scientific achievements

In this section we describe other scientific merits achieved throughout these years, such as: In-ternational stays, reviewer of International Journals, participation in International Conferences,participation in Research Projects, and collaborations with other institutions.

11.3.1 International Stays

Álvaro Rubio-Largo visited the School of Computer Science and Electronic Engineering at theUniversity of Essex (Colchester, UK) from 01/Sept/2012 to 01/Dec/2012.

During his visit, Álvaro Rubio-Largo worked on Multiobjective Optimization algorithms. Moreconcretely, he developed a new Multiobjective Evolutionary Algorithm based on Decomposition(MOEA/D-NBI) for solving real-world multiobjective optimization problems in which the objectivefunctions are in very different scales. This algorithm (MOEA/D-NBI) was applied to the TrafficGrooming problem.

11.3.2 Reviewer of International Journals

Álvaro Rubio-Largo has served as reviewer of diverse International Journals indexed by the ISI inthe JCR, such as:

1. Mathematics and Computers in Simulation.ISSN: 0378-4754 and Impact Factor = 0.738 in the year 2011.June 2011 – present.

2. Journal of Systems Architecture: Embedded Software Design.ISSN: 1383-7621 and Impact Factor = 0.444 in the year 2011.January 2012 – present.

3. Soft Computing.ISSN: 1432-7643 and Impact Factor = 1.880 in the year 2011.From: 2012 – present.

11.3.3 Participation in International Conferences

Álvaro Rubio-Largo has served as Session Chair or formed part of the program committee in thefollowing international conferences:

1. Session Chair in the 14th IEEE International Conference on High Performance Computingand CommunicationsThis conference is the 14th edition of the highly successful International Conference onHigh Performance and Communications (HPCC-12). It provides a forum for engineers and


scientists in academia, industry, and government to address the resulting profound challengesand to present and discuss their new ideas, research results, applications and experience on allaspects of high performance computing and communications. IEEE HPCC-2012 is sponsoredby IEEE, IEEE Computer Society, and IEEE Technical Committee on Scalable Computing(TCSC). HPCC-2012 was held in Liverpool, UK (25-27 June 2012).

2. Program Committee in the PBio 2013: International Workshop on Parallelism in Bioinfor-matics (EuroMPI 2013)EuroMPI is the preeminent meeting for users, developers and researchers to interact and dis-cuss new developments and applications of message-passing parallel computing, in particularin and related to the Message Passing Interface (MPI). The annual meeting has a long, richtradition, and the 20th European MPI Users’ Group Meeting will again be a lively forumfor discussion of everything related to usage and implementation of MPI and other parallelprogramming interfaces. EuroMPI is organized in cooperation with ACM SIGHPC (SpecialInterest Group on High Performance Computing), and the proceedings will be published inthe ACM Digital Library. PBio 2013 will be held in Madrid, Spain (15-18 September 2013).

11.3.4 Participation in Research Projects

Álvaro Rubio-Largo has actively participated in the following research projects as researcher:

• MSTAR TIN2008-06491-C04-04 (2009 – 2011)

– Title: Multiobjective Metaheuristics and Parallelism in Communications (MSTAR)

– Financial entity: Ministerio de Ciencia e Innovación (Spain)

– Participant entities: University of Extremadura, University of Málaga, University of LaLaguna, and University of Carlos III (Madrid)

– Duration: 3 years (2009 – 2011)

– Budget: 110,110e

– Main researcher : M. A. Vega-Rodríguez (UNEX node) and E. Alba (coordinate project)

– Number of researchers: 9 (UNEX node) and 39 (coordinate project)

– Summary: This project was aimed at innovating in multiple fronts of multiobjective opti-mization (MO). For this purpose, we achieved a relatively ambitious set of contributionsat the end of the project. First, we advanced in fundamental research by developingnew multiobjective models for algorithms, differential evolution, swarm intelligence andother procedures capable of solving problems of realistic dimension and complexity. Sec-ond, the problems tackled were not be limited to typical instances drawn from standardbenchmarks, but instead we also selected real problems in the communications field toperform an applied research. This way, the benefits will lie in both methodology andreal applications. Therefore, we proposed to solve problems chosen in this field, anddoing it by new multiobjective proposals, improving the efficiency and effectiveness withrespect to the present state of the art in the mentioned domain; we aimed at showingthat the contributed techniques were not only appealing in theory, but also effective anduseful for society.

• BIO TIN2012-30685 (2013 – present)

11.3 Other scientific achievements 243

– Title: BIO: Multiobjective Optimization and Parallelism in Bioinformatics (BIO)

– Financial entity: Ministerio de Ciencia e Innovación (Spain)

– Participant entities: University of Extremadura

– Duration: 3 years (2013 – 2015)

– Budget: 107,160e

– Main researcher : Miguel A. Vega-Rodríguez (coordinate project)

– Number of researchers: 12

– Summary: For this purpose, we will achieve a relatively ambitious set of contributionsat the end of the project, spreading the advances to multiple research niches (informat-ics, operations research, algorithmics, specialists and applications...), thus increasingthe international impact. First, we plan to advance in fundamental research by devel-oping new multiobjective models for algorithms such as differential evolution, variableneighbourhood search, artificial bee colony, firefly algorithm, gravitational search algo-rithm... and other procedures capable of solving problems of realistic dimension andcomplexity. In conclusion, we aim at improving, creating and disseminating advancedMO algorithms. Furthermore, we will study combinations with other techniques (hy-bridization) using also problem-aware operations. Second, we want to advance in theuse of new technologies and research lines that are presently hot topics at an interna-tional level, but from a multiobjective perspective in our case. Third, the problemstackled will not be the typical instances drawn from benchmarks, but instead we willaddress real-world problems from the hot domain of bioinformatics (applied research).This way, the benefits will lie in both methodology and real applications. The goal isto improve the efficiency and effectiveness of the solutions to bioinformatics problemswith respect to the present state of the art.

11.3.5 Collaboration with other Institutions

Throughout these three years and a half, Álvaro Rubio-Largo has punctually worked in collabora-tion with the following institutions:


Álvaro Rubio-Largo worked in collaboration with the School ofTechnology and Management (Polytechnic Institute of Leiria,Leiria, Portugal) in order to solve the Reporting cells problemby using Evolutionary Computation and Parallelism jointly.

Álvaro Rubio-Largo worked in collaboration with the Departmentof Applied Physics (University of Salamanca, Spain) for optimiz-ing and parallelizing a software based on the Monte Carlo modelfor semiconductor nano-devices.

Álvaro Rubio-Largo worked in collaboration with the School ofComputer Science and Electronic Engineering (University of Es-sex, UK) in order to explore new research fields in MultiobjectiveOptimization and Evolutionary Algorithms.

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