Evolutionary Algorithms for Solving Multi-Objective...

Carlos A. Coello Coello, Gary B. Lamont and David A. Van Veldhuizen

Second Edition Evolutionary Algorithms for Solving Multi-Objective Problems

Genetic and Evolutionary Computation Series

Series Editors

David E. Goldberg Consulting Editor IlliGAL, Dept. of General Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801 USA Email: [email protected]

Medical Informatics

Stanford, CA 94305-5479 USA

Consulting Editor

Stanford University

John R. Koza

Email: [email protected]

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Markus Brameier, Wolfgang Banzhaf Linear Genetic Programming, 2007 ISBN 978-0-387-31029-9

Nikolay Y. Nikolaev, Hitoshi Iba Adaptive Learning of Polynomial Networks, 2006 ISBN 978-0-387-31239-2

Tetsuya Higuchi, Yong Liu, Xin Yao

ISBN 978-0-387-24386-3 Evolvable Hardware, 2006

David E. Goldberg The Design of Innovation: Lessons from and for Competent Genetic Algorithms, 2002 ISBN 978-1-4020-7098-3

Genetic Programming IV: Routine Human-Computer Machine Intelligence ISBN: 978-1-4020-7446-2 (hardcover), 2003; ISBN: 978-0-387-25067-0 (softcover), 2005

Carlos A. Coello Coello, David A. Van Veldhuizen, Gary B. Lamont Solving Multi-Objective Problems, 2002

ISBN: 978-0-306-Evolutionary Algorithms for

John R. Koza, Martin A. Keane, Matthew J. Streeter, William Mydlowec, Jessen Yu, Guido Lanza

46762-2

Lee Spector Automatic Quantum Computer Programming: A Genetic Programming Approach

4-1 (hardcover), 2004; ISBN 978-0-387-36496-4 (softcover), 2007 ISBN: 978-1-4020-789

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For a complete listing of books in this series, go to http://www.springer.com

Second Edition

Carlos A. Coello Coello Gary B. Lamont David A. Van Veldhuizen

Evolutionary Algorithms for Solving

Multi-Objective Problems

Gary B. Lamont Department of Electrical and Computer

Depto. de Computación Engineering

Col. San Pedro Zacatenco Air Force Institute of Technology México, D.F. 07360 MEXICO 2950 Hobson Way

WPAFB, Dayton, OH 45433-7765 [email protected]

402 Scott Dr., No. 3L3

Graduate School of Engineering

Carlos A. Coello Coello CINVESTAV-IPN

Av. Instituto Politécnico Nacional No. 2508

[email protected]

David A. Van Veldhuizen

Series Editors: David E. Goldberg John R. Koza Consulting Editor Consulting Editor IlliGAL, Dept. of General Engineering Medical Informatics University of Illinois at Urbana-Champaign Stanford University Urbana, IL 61801 USA Stanford, CA 94305-5479 USA [email protected] [email protected]

Library of Congress Control Number: 2007930239

ISBN 978-0-387-33254-3 e-ISBN 978-0-387-36797-2

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to our wives

Preface to the Second Edition

The response of the multiobjective optimization community to our first edi-tion in 2002 was extremely enthusiastic. Many have indicated their use of ourmonograph to gain insight to the interdisciplinary nature of multiobjective op-timization employing evolutionary algorithms. Others are appreciative for ourproviding them a foundation for associated contemporary multiobjective evo-lutionary algorithm (MOEA) research. We appreciate these warm commentsalong with readers’ suggestions for improvements. In that vein, we have sig-nificantly extended and modified our previous material using contemporaryliterature resulting in this new edition, which is extended into a textbook. Inaddition to new classroom exercises contained in each chapter, the MOEAdiscussion questions and possible research directions are updated.

The first edition presented an organized variety of MOEA topics based onfundamental principles derived from single-objective evolutionary algorithm(EA) optimization and multiobjective problem (MOP) domains. Yet, manynew developments occurred in the intervening years. New MOEA structureswere proposed with new operators and therefore better search techniques.The explosion of successful MOEA applications continues to be reported inthe literature. Statistical testing methods for evaluating results now offersimproved analysis of comparative techniques, innovative metrics, and bettervisualization tools. The continuing development of MOEA activity in the-ory, algorithmic innovations, and MOEA practice calls for these new conceptsto be integrated into our generic MOEA text. Note that the continuing im-provement (speed, memory, etc.) of computer hardware provides computa-tional platforms that permit larger search spaces to be addressed at higherefficiencies using both serial and parallel processing. This phenomenon, inconjunction with user-friendly software interfacing tools, permits an increas-ing number of scientists and engineers to explore the use of MOEAs in theirparticular multiobjective problem domains.

With this new edition, we continue to provide an interdisciplinary com-puter science and computer engineering text that considers other academicfields such as operations research, industrial engineering, and management

VIII Preface to the Second Edition

science. Examples from all these disciplines, as well as all engineering areasin general, are discussed and addressed as to their fundamental unique prob-lem domain characteristics and their solutions using MOEAs. An expandedreference list is included with suggestions of further reading for both the stu-dent and practitioner. As in the previous edition, this book addresses MOEAdevelopment and applications issues through the following features:

• The text is meant to be both a textbook and a self-contained reference.The book provides all the necessary elements to guide a newcomer in thedesign, implementation, validation, and application of MOEAs in eitherthe classroom or the field.

• Researchers in the field benefit from the book’s comprehensive review ofstate-of-the-art concepts and discussions of open research topics.

• The book is also written for graduate students in computer science, com-puter engineering, operations research, management science, and otherscientific and engineering disciplines, who are interested in multiobjectiveoptimization using evolutionary algorithms.

• The book is also for professionals interested in developing practical applica-tions of evolutionary algorithms to real-world multiobjective optimizationproblems.

• Each chapter is complemented by discussion questions and several ideasmeant to trigger novel research paths. Supplementary reading is stronglysuggested for deepening MOEA understanding.

• Key features include MOEA classifications and explanations, MOEA ap-plications and techniques, MOEA test function suites, and MOEA perfor-mance measurements.

• We created a website for this book at:

http://www.cs.cinvestav.mx/~emoobook

which contains considerable material supporting this second edition. Thissite contains all the appendices of the book (which have been removedfrom the original monograph due to space limitations), as well as public-domain software, tutorial slides, and additional sources of contemporaryMOEA information.

This new synergistic text is markedly improved from the first edition. Newmaterial is integrated providing more detail, which leads to a realignment ofmaterial. Old chapters were modified and a new one was added. As before,the various features of MOEAs continue to be discussed in an innovativeand unique fashion, with detailed customized forms suggested for a varietyof applications. The flow of material in each chapter is intended to presenta natural and comprehensive development of MOEAs from basic concepts tocomplex applications.

Chapter 1 presents and motivates MOP and MOEA terminology and thenomenclature used in successive chapters including a lengthy discussion on the

Preface to the Second Edition IX

impact of computational limitations on finding the Pareto front along withinsight to MOP/MOEA building block (BB) concepts.

In Chapter 2, MOEA developmental history has proceeded in a numberof ways from aggregated forms of single-objective Evolutionary Algorithms(EAs) to true multiobjective approaches such as MOGA, MOMGA, NPGA,NSGA, NSGA-II, PAES, PESA, PESA-II, SPEA, SPEA2 and their exten-sions. Each MOEA is presented with historical and algorithmic insight. Beingaware of the many facets of historical multiobjective problem solving providesa foundational understanding of the discipline. Various MOEA techniques,operators, parameters and constructs are compared. Contemporary MOEAdevelopment emphasizes new MOP variable representation, and novel MOEAstructures and operators. In addition, constraint-handling techniques usedwith MOEAs are also discussed. A comprehensive comparison of contempo-rary MOEAs provides insight to an individual algorithm’s advantages anddisadvantages.

In Chapter 3, a new chapter, both coevolutionary MOEAs and hybridiza-tions of MOEAs with local search procedures (the so-called memetic MOEAs)are covered. A variety of MOEA implementations within each of these twotypes of approaches (i.e., coevolution and hybrids with local search mecha-nisms) are presented, summarized, categorized and analyzed.

Chapter 4 offers a detailed development of contemporary MOP test suitesranging from numerical functions (unconstrained and with side constraints)and generated functions to discrete NP -Complete problems and real-worldapplications. Our website contains the algebraic description as well as thePareto fronts (and, if generated by enumeration, the Pareto optimal set aswell) of many of the proposed test functions. This knowledge leads to anunderstanding and ability to select appropriate MOEA test suites based upona set of desired comparative characteristics.

MOEA performance comparisons are presented in Chapter 5 using manyof the test function suites discussed in Chapter 4. Also included is an exten-sive discussion of possible comparative metrics and presentation techniques.The selection of key algorithmic parameter values (population size, crossoverand mutation rates, etc.) is emphasized. A limited set of MOEA results arerelated to the design and analysis of efficient and effective MOEAs employingthese various MOP test suites and appropriate metrics. The chapter has beenexpanded to include new testing concepts such as attainment functions, elabo-rated dominance relations, and “quality” Pareto compliant indicator analysis.A wide spectrum of empirical testing and statistical analysis techniques areprovided for the MOEA user.

Although MOEA theory is still relatively limited, Chapter 6 presents acontemporary summary of known results. Topics addressed in this chapter in-clude MOEA convergence to the Pareto front, Pareto ranking, fitness sharing,mating restrictions, stability, running time analysis, and algorithmic complex-ity.

X Preface to the Second Edition

It is of course unrealistic to present every generic MOP application, thus,Chapter 7 attempts to group and classify the multitude of various contem-porary MOEA applications via representative examples. This limited com-pendium with an extensive reference listing provides the reader with a start-ing point for their own application and research. Specific MOEA operatorsas well as encodings adopted in many MOEA applications are integrated foralgorithmic understanding.

In Chapter 8, research and development of parallel MOEAs is classified andanalyzed. The three foundational paradigms (master-slave, island, and diffu-sion) are defined. Using these three structures, many contemporary MOEAparallel developments are algorithmically compared and analyzed in termsof advantages and disadvantages for different computational architectures.Some general observations about the current state of parallel and distributedMOEAs are also included.

Chapter 9 discusses and compares the two main schools of thought re-garding multi-criteria decision making (MCDM): Outranking approaches andMulti-Attribute Utility Theory (MAUT). Aspects such as the operational atti-tude of the Decision Maker (DM), the different stages at which preferences canbe incorporated, scalability, transitivity and group decision making are alsodiscussed. However, the main emphasis is in describing the most representativeresearch regarding preference articulation into MOEAs. This comprehensivereview includes brief descriptions of the approaches reported in the literatureas well as an analysis of their advantages and disadvantages.

Chapter 10 discusses multiobjective extensions of other search heuristics.The main techniques covered include Tabu search, scatter search, simulatedannealing, ant system, distributed reinforcement learning, artificial immunesystems, particle swarm optimization and differential evolution.

New examples are integrated throughout the second edition. New algo-rithms are addressed with special emphasis on the spectrum of MOEA oper-ators and how they are implemented in contemporary and historic MOEAs.Part of the focus is on classifying MOEAs as to implicit or explicit BB types.Other classification features such as probabilistic vs. stochastic are investi-gated. References are updated to include the current state-of-the-art MOEAsand applications.

Class exercises are integrated into all chapters for pedagogical purposes.Discussion questions within every chapter are updated and expanded. Thesuggested and focused research ideas from the first edition are brought up-to-date and continue to emphasize the current state-of-the-art horizon.

To profit from the book, one should have at least single-objective EAknowledge and experience. Also, some mathematical knowledge is appropri-ate in order to understand symbolic functions as well as theoretical MOEAaspects. This knowledge includes basic linear algebra, calculus, probabilityand statistics. This second edition may be used in the classroom at the seniorundergraduate or graduate level depending upon the instructor’s purpose. Asa class, we suggest that all material could fill a two semester course or with

Preface to the Second Edition XI

careful selection of topics, a one-semester course. Also, the material in therevised text can be effectively employed by practitioners in many fields.

In support of this text, one can find up-to-date MOEA reference listingsof journal papers, conference papers, MOP software, and MOEA software atthe Evolutionary Multiobjective Optimization (EMOO) Repository internetweb site http://delta.cs.cinvestav.mx/~ccoello/EMOO. This site is con-tinually updated to support the MOEA community and our text. If you havea contribution, please send it to [email protected].

Creating a book such as this requires the efforts of many people. Theauthors thank Matthew Johnson, Michael Putney, Jesse Zydallis, Tony Kadro-vach, Giovani Gomez-Estrada, Dragan Cvetkovic, Jose Alfredo Lopez, NareliCruz-Cortes, Gregorio Toscano-Pulido, Luis Gerardo de la Fraga, and manyothers for their assistance in generating computational results and reviewingvarious aspects of the material. We also thank all those researchers who sentus some of their research papers and theses to enrich the material containedin this edition.

We express our sincere appreciation to Professors David E. Goldberg andJohn R. Koza for including this book as a volume in their Genetic and Evo-lutionary Computation book series, published by Springer.

Also, it has been a pleasure working with Springer’s professional editor-ial and production staff. We particularly thank Melissa Fearon and ValerieSchofield for their prompt and kind assistance at all times during the devel-opment of this book.

We also want to thank other primary MOEA researchers not only fortheir innovative papers but for various conversations providing more in-sight to developing better algorithms. Such individuals include David Corne,Tomoyuki Hiroyasu, Kalyanmoy Deb, Marco Laumanns, Jurgen Branke,Sanaz Mostaghim, Nirupam Chakraborti, Alfredo G. Hernandez-Dıaz, JulianMolina, Rafael Caballero, Peter Fleming, Carlos Fonseca, Xavier Gandibleux,Yaochu Jin, Kay Chen Tan, Jeffrey Horn, Hisao Ishibuchi, Piero Bonis-sone, Jonathan Fieldsend, Marco Farina, Arturo Hernandez-Aguirre, Lyn-don While, Evan J. Hughes, Rajeev Kumar, Shigeru Obayashi, Joshua D.Knowles, J. David Schaffer, Ian Parmee, El-Ghazali Talbi, Hernan Aguirre,Oliver Schutze, Lothar Thiele, and Eckart Zitzler.

The authors also express their gratitude to Antonio Nebro, Enrique Alba,Margarita Reyes-Sierra, Luis V. Santana-Quintero, Ricardo Landa-Becerra,Mario A. Ramırez-Morales, Emanuel Tellez-Enrıquez, Richard Day, CharlesHaag, and Mark Kleeman for their valuable help at different stages of thedevelopment of this second edition. Without their help, this book would hadnever been finished. Carlos A. Coello Coello also states that his contributionto this book was developed using the computing facilities of the Departmentof Computer Science of the Centro de Investigacion y de Estudios Avanzadosfrom the Instituto Politecnico Nacional (CINVESTAV-IPN) with support pro-vided by CONACyT (the Mexican council of science and technology) to thefirst author through project no. 45683-Y, which was also greatly appreciated.

XII Preface to the Second Edition

Last but not least, we owe a debt of gratitude to our wives for their en-couragement, understanding, and exemplary patience.

We hope that the new edition continues to represent not only a compre-hensive introduction to MOEAs, but also the contemporary state-of-the-artin MOEA structures, applications, testing and theory.

Carlos A. Coello CoelloGary B. Lamont

David A. Van VeldhuizenSpring 2007

Foreword to the Second Edition

Researchers and practitioners alike are increasingly turning to search, opti-mization, and machine-learning procedures based on natural selection andnatural genetics to solve problems across the spectrum of human endeavor.These genetic algorithms and techniques of evolutionary computation aresolving problems and inventing new hardware and software that rival hu-man designs. The Springer Series on Genetic and Evolutionary Computationpublishes research monographs, edited collections, and graduate-level texts inthis rapidly growing field. Primary areas of coverage include the theory, imple-mentation, and application of genetic algorithms (GAs), evolution strategies(ESs), evolutionary programming (EP), learning classifier systems (LCSs) andother variants of genetic and evolutionary computation (GEC). The series alsopublishes texts in related fields such as artificial life, adaptive behavior, artifi-cial immune systems, agent-based systems, neural computing, fuzzy systems,and quantum computing as long as GEC techniques are part of or inspirationfor the system being described.

This is the second (revised and extended) edition of an encyclopedic vol-ume on the use of the algorithms of genetic and evolutionary computationfor the solution of multi-objective problems. Multi-objective evolutionary al-gorithms (MOEAs) are now even more popular than in 2002, when the firstedition of this book was published. Researchers and practitioners remain tofind an irresistible match between the population available in most geneticand evolutionary algorithms and the need in multi-objective problems to ap-proximate the Pareto trade-off curve or surface.

The authors have kept the remarkable job that distinguished the first edi-tion in collecting, organizing, and interpreting the burgeoning literature ofMOEAs in a form that should be welcomed by novices and old hands alike.The volume starts with an extraordinarily thorough introduction, includingshort vignettes and photographs of many of the pioneers of multi-objectiveoptimization. It continues with as complete a discussion of the many vari-eties of MOEAs as appears anywhere in the literature. This second editionnow adds a new chapter fully devoted to coevolutionary and memetic (i.e.,

XIV Foreword to the Second Edition

hybrids with local search mechanisms) MOEAs. A discussion of MOEA testsuites surveys the important topic of test landscapes and is followed with im-portant chapters on empirical testing and MOEA theory. Such chapters havebeen considerably extended with respect to the first edition, adding materialon state-of-the-art test functions and performance measures (and their lim-itations), as well as the new developments on the theoretical foundations ofMOEAs. Practitioners will especially welcome the thorough survey of real-world MOEA applications, which clearly indicates the growing interest in thisfield. There is also an ample discussion on parallelization, and a thoroughreview of mechanisms to incorporate user’s preferences in a MOEA (an areacalled multi-criteria decision making). The final chapter of special topics dis-cusses multi-objective extensions of other methods in soft computation suchas simulated annealing, ant colony optimization, and artificial immune sys-tems. These chapters have also been considerably extended and refurbishedto reflect the many new developments that have arisen in this field since thepublication of the first edition of this book. With about 200 extra pages, aconsiderable number of new problems and research ideas at the end of eachchapter and additional supporting material available through a website, thissecond edition aims to be adopted as a textbook, while preserving much of itsmonograph nature.

If you enjoyed the first edition of this book, then you will certainly benefiteven more from this second edition. If you still do not know this book, then, Iurge you to run—don’t walk—to your nearest on-line or off-line book purveyorand click, signal, or otherwise buy this important addition to our literature.

David E. GoldbergConsulting EditorUniversity of Illinois at [email protected], Illinois, USAMay 2007

Contents

1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Single-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . 41.2.2 The Multiobjective Optimization Problem . . . . . . . . . . . . 51.2.3 Multiobjective Optimization Problem . . . . . . . . . . . . . . . . 71.2.4 Definition of MOEA Progress . . . . . . . . . . . . . . . . . . . . . . . 141.2.5 Computational Domain Impact . . . . . . . . . . . . . . . . . . . . . 141.2.6 Pareto Epsilon Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.7 Decision Maker Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 General Optimization Algorithm Overview. . . . . . . . . . . . . . . . . . 211.5 EA Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 Origins of Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . 29

1.6.1 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . 301.6.2 Early Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.7 Classifying Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.7.1 A priori Preference Articulation . . . . . . . . . . . . . . . . . . . . . 321.7.2 A Posteriori Preference Articulation . . . . . . . . . . . . . . . . . 461.7.3 Progressive Preference Articulation . . . . . . . . . . . . . . . . . . 47

1.8 Using Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.8.1 Pareto Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.8.2 MOEA Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Further Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2 MOP Evolutionary Algorithm Approaches . . . . . . . . . . . . . . . . . 612.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2 MOEA Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.2.1 A Priori Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

XVI Contents

2.2.2 Progressive Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.2.3 A Posteriori Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.2.4 Generic MOEA Goals and Operator Design . . . . . . . . . . . 77

2.3 Structures of Various MOEAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.3.1 Multi-Objective Genetic Algorithm (MOGA) . . . . . . . . . 882.3.2 Nondominated Sorting Genetic Algorithm (NSGA) . . . . 912.3.3 Niched-Pareto Genetic Algorithm (NPGA) . . . . . . . . . . . 942.3.4 Pareto Archived Evolution Strategy (PAES) . . . . . . . . . . 952.3.5 Strength Pareto Evolutionary Algorithm (SPEA) . . . . . . 972.3.6 Multiobjective Messy Genetic Algorithm (MOMGA) . . . 992.3.7 Pareto Envelope-based Selection Algorithm (PESA) . . . 1012.3.8 The Micro-Genetic Algorithm for Multiobjective

Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.3.9 Multiobjective Struggle GA (MOSGA) . . . . . . . . . . . . . . . 1052.3.10 Orthogonal Multi-Objective Evolutionary Algorithm

(OMOEA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.3.11 General Multiobjective Evolutionary Algorithm

(GENMOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.3.12 Criticism to Pareto sampling techniques . . . . . . . . . . . . . . 111

2.4 Constraint-Handling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.5 Critical MOEA Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.5.1 MOEA Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.5.2 MOEA Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.5.3 MOEA Fitness Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.5.4 MOEA Chromosomal Representations . . . . . . . . . . . . . . . 1172.5.5 MOEA Problem Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 119

2.6 MOEA Design Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121


3 MOEA Local Search and Coevolution . . . . . . . . . . . . . . . . . . . . . 1313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.2 MOEA Local Search Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.2.1 Hybrid MOEA Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 1343.2.2 Comments on Hybrid MOEA Techniques . . . . . . . . . . . . . 143

3.3 MOEA Coevolutionary Techniques . . . . . . . . . . . . . . . . . . . . . . . . 1443.4 Coevolution and Symbiosis in EAs . . . . . . . . . . . . . . . . . . . . . . . . . 147

3.4.1 Coevolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 1473.4.2 Cooperative Coevolutionary Genetic Algorithms . . . . . . . 1493.4.3 Symbiogenetic Coevolution . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.5 Coevolution and Symbiosis in MOEAs . . . . . . . . . . . . . . . . . . . . . 1523.5.1 Elitist Recombinative MOGA with Coevolutionary

Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.5.2 Parmee’s Co-Evolutionary MOEA . . . . . . . . . . . . . . . . . . . 154

Contents XVII

3.5.3 Genetic Symbiosis Algorithm . . . . . . . . . . . . . . . . . . . . . . . 1553.5.4 Interactive GA with Co-evolving Weighting Factors . . . . 1573.5.5 Multiobjective Co-operative Co-evolutionary GA . . . . . . 1583.5.6 Lohn’s Coevolutionary Genetic Algorithm . . . . . . . . . . . . 1593.5.7 Distributed Cooperative Coevolutionary Algorithm . . . . 1613.5.8 Coello’s Coevolutionary MOEA . . . . . . . . . . . . . . . . . . . . . 1633.5.9 Nondominated Sorting Cooperative Coevolutionary GA 165

3.6 Applying Coevolutionary MOEAs . . . . . . . . . . . . . . . . . . . . . . . . . 1653.6.1 Coevolving Multiple MOEAs . . . . . . . . . . . . . . . . . . . . . . . 1663.6.2 Coevolving MOEAs with other Search Algorithms . . . . . 1673.6.3 Coevolving Density Estimators . . . . . . . . . . . . . . . . . . . . . . 1673.6.4 Coevolving Target Solutions . . . . . . . . . . . . . . . . . . . . . . . . 1673.6.5 Coevolving Competing Populations . . . . . . . . . . . . . . . . . . 168

3.7 Final Comments on Coevolutionary MOEAs . . . . . . . . . . . . . . . . 168


4 MOEA Test Suites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.2 MOEA Test Function Suite Issues . . . . . . . . . . . . . . . . . . . . . . . . . 1764.3 MOP Domain Feature Classification . . . . . . . . . . . . . . . . . . . . . . . 179

4.3.1 Unconstrained Numeric MOEA Test Functions . . . . . . . . 1824.3.2 Side-Constrained Numeric MOEA Test Functions . . . . . 1874.3.3 MOP Test Function Generators . . . . . . . . . . . . . . . . . . . . . 193

4.4 Generic Scalable MOP Test Problems . . . . . . . . . . . . . . . . . . . . . . 1994.4.1 Okabe’s Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.4.2 Huband’s Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

4.5 Combinatorial MOEA Test Functions . . . . . . . . . . . . . . . . . . . . . . 2204.6 Real-World MOEA Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2224.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228


5 MOEA Testing and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2335.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2335.2 MOEA Experiments: Motivation and Objectives . . . . . . . . . . . . . 2355.3 Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

5.3.1 MOP Pareto Front Determination . . . . . . . . . . . . . . . . . . . 2365.3.2 MOEA Algorithms Testing . . . . . . . . . . . . . . . . . . . . . . . . . 2385.3.3 Key MOEA Algorithmic Parameters . . . . . . . . . . . . . . . . . 239

5.4 MOEA Experimental Measurements . . . . . . . . . . . . . . . . . . . . . . . 2435.4.1 Selection of MOEA Comparison Measures . . . . . . . . . . . . 2455.4.2 Generic Attainment Function . . . . . . . . . . . . . . . . . . . . . . . 2455.4.3 Dominance Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2505.4.4 Primary Quality Indicators . . . . . . . . . . . . . . . . . . . . . . . . . 254

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5.4.5 Other MOEA Quality Indicators . . . . . . . . . . . . . . . . . . . . 2635.4.6 MOEA Experimental Metrics Summary . . . . . . . . . . . . . . 267

5.5 MOEA Statistical Testing Approaches . . . . . . . . . . . . . . . . . . . . . 2685.5.1 Statistical Testing Techniques . . . . . . . . . . . . . . . . . . . . . . . 2685.5.2 Non-Parametric Statistics (Analysis of Variance) . . . . . . 2705.5.3 Methods for Presentation of MOEA Results . . . . . . . . . . 2725.5.4 Visualization of Test Results . . . . . . . . . . . . . . . . . . . . . . . . 272

5.6 Software Support of MOEA Testing. . . . . . . . . . . . . . . . . . . . . . . . 2735.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276


6 MOEA Theory and Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2836.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2836.2 Pareto-Related Theoretical Contributions . . . . . . . . . . . . . . . . . . . 284

6.2.1 Partially Ordered Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2846.2.2 MOEA Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

6.3 MOEA Theoretical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3006.3.1 Fitness Landscapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3006.3.2 Fitness Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3056.3.3 Pareto Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3076.3.4 Pareto Niching and Fitness Sharing . . . . . . . . . . . . . . . . . . 3106.3.5 Recombination Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3146.3.6 Mating Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3156.3.7 Solution Stability and Robustness . . . . . . . . . . . . . . . . . . . 3176.3.8 MOEA Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3176.3.9 MOEA Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.3.10 Running Time Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3206.3.11 MOEA Computational “Cost” . . . . . . . . . . . . . . . . . . . . . . 3266.3.12 NFL-Theorem for Multiobjective Optimization

Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3266.3.13 Alternative Definitions of Optimality . . . . . . . . . . . . . . . . 3276.3.14 Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333


7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3397.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3397.2 Engineering Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

7.2.1 Environmental, Naval and Hydraulic Engineering . . . . . . 3407.2.2 Electrical and Electronics Engineering . . . . . . . . . . . . . . . 3477.2.3 Telecommunications and Network Optimization . . . . . . . 3567.2.4 Robotics and Control Engineering . . . . . . . . . . . . . . . . . . . 3607.2.5 Structural and Mechanical Engineering . . . . . . . . . . . . . . . 369

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7.2.6 Civil and Construction Engineering . . . . . . . . . . . . . . . . . . 3767.2.7 Transport Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3777.2.8 Aeronautical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

7.3 Scientific Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3887.3.1 Geography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3887.3.2 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3897.3.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.3.4 Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3937.3.5 Ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3967.3.6 Computer Science and Computer Engineering . . . . . . . . . 397

7.4 Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4077.4.1 Design and Manufacture . . . . . . . . . . . . . . . . . . . . . . . . . . . 4087.4.2 Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4167.4.3 Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4247.4.4 Grouping and Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

7.5 Miscellaneous Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4287.5.1 Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4287.5.2 Classification and Prediction . . . . . . . . . . . . . . . . . . . . . . . . 430

7.6 Future Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4347.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435


8 MOEA Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4438.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4438.2 pMOEA Fundamental Background . . . . . . . . . . . . . . . . . . . . . . . . 445

8.2.1 pMOEA Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4458.2.2 pMOEA Motivation and Issues . . . . . . . . . . . . . . . . . . . . . . 446

8.3 pMOEA Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4508.3.1 Master-Slave pMOEA Model . . . . . . . . . . . . . . . . . . . . . . . . 4528.3.2 Island pMOEA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4558.3.3 Diffusion pMOEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4588.3.4 Hierarchical Hybrid pMOEA Models . . . . . . . . . . . . . . . . . 459

8.4 pMOEAs From the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4608.4.1 Master-Slave pMOEAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4608.4.2 Island pMOEAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4658.4.3 Diffusion pMOEAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

8.5 pMOEA Analyses and Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4758.5.1 pMOEA Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4768.5.2 pMOEA Suitability Issues . . . . . . . . . . . . . . . . . . . . . . . . . . 4768.5.3 pMOEA Hardware and Software Architecture Issues . . . 4778.5.4 pMOEA Test Function Issues . . . . . . . . . . . . . . . . . . . . . . . 4808.5.5 pMOEA Metric/Parameter Issues . . . . . . . . . . . . . . . . . . . 484

8.6 pMOEA Development Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4888.6.1 pMOEA Creation Options . . . . . . . . . . . . . . . . . . . . . . . . . . 490

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8.6.2 Master-Slave Implementation Issues . . . . . . . . . . . . . . . . . 4918.6.3 Island Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . 4938.6.4 Diffusion Implementation Issues . . . . . . . . . . . . . . . . . . . . . 4998.6.5 Parallel Niching Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5008.6.6 Parallel Archiving Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 5028.6.7 pMOEA Theory Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

8.7 A “Generic” pMOEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5038.7.1 Engineering a pMOEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5048.7.2 “Genericizing” a pMOEA . . . . . . . . . . . . . . . . . . . . . . . . . . 507

8.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507


9 Multi-Criteria Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5159.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5159.2 Multi-Criteria Decision Making. . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

9.2.1 Operational Attitude of the Decision Maker . . . . . . . . . . . 5179.2.2 When to Get the Preference Information? . . . . . . . . . . . . 518

9.3 Incorporation of Preferences in MOEAs . . . . . . . . . . . . . . . . . . . . 5209.3.1 Definition of Desired Goals . . . . . . . . . . . . . . . . . . . . . . . . . 5229.3.2 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5269.3.3 Preference Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5289.3.4 Outranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5319.3.5 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5339.3.6 Compromise Programming . . . . . . . . . . . . . . . . . . . . . . . . . 535

9.4 Issues Deserving Attention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5369.4.1 Preserving Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5379.4.2 Transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5379.4.3 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5379.4.4 Group Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5379.4.5 Other important issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540


10 Alternative Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54710.2 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

10.2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54810.2.2 Multiobjective Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55010.2.3 Advantages and Disadvantages of Simulated Annealing . 556

10.3 Tabu Search and Scatter Search . . . . . . . . . . . . . . . . . . . . . . . . . . . 55710.3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55810.3.2 Multiobjective Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

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10.3.3 Advantages and Disadvantages of Tabu Search andScatter Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

10.4 Ant System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57210.4.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57210.4.2 Multiobjective Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57510.4.3 Advantages and Disadvantages of the Ant System . . . . . 581

10.5 Distributed Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . 58210.5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58210.5.2 Advantages and Disadvantages of Distributed

Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58310.6 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

10.6.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58410.6.2 Multiobjective Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58510.6.3 Advantages and Disadvantages of Particle Swarm

Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59310.7 Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

10.7.1 Multiobjective Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59610.7.2 Advantages and Disadvantages of Differential Evolution 604

10.8 Artificial Immune Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60410.8.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60510.8.2 Multiobjective Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60610.8.3 Advantages and Disadvantages of Artificial Immune

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61110.9 Other Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

10.9.1 Cultural Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61210.9.2 Cooperative Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

10.10Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616


Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

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