Cover
Title Page
Preface
About the Authors
List of Figures
1 Overview of Optimization
1. Summary
2. 1.1 Optimization
3. 1.2 Examples of the Formulation of Various Engineering Optimization Problems
4. 1.3 Conclusion
2 Introduction to Meta‐Heuristic and Evolutionary Algorithms
1. Summary
2. 2.1 Searching the Decision Space for Optimal Solutions
3. 2.2 Definition of Terms of Meta‐Heuristic and Evolutionary Algorithms
4. 2.3 Principles of Meta‐Heuristic and Evolutionary Algorithms
5. 2.4 Classification of Meta‐Heuristic and Evolutionary Algorithms
6. 2.5 Meta‐Heuristic and Evolutionary Algorithms in Discrete or Continuous Domains
7. 2.6 Generating Random Values of the Decision Variables
8. 2.7 Dealing with Constraints
9. 2.8 Fitness Function
10. 2.9 Selection of Solutions in Each Iteration
11. 2.10 Generating New Solutions
12. 2.11 The Best Solution in Each Algorithmic Iteration
13. 2.12 Termination Criteria
14. 2.13 General Algorithm
15. 2.14 Performance Evaluation of Meta‐Heuristic and Evolutionary Algorithms
16. 2.15 Search Strategies
17. 2.16 Conclusion
18. References
3 Pattern Search
1. Summary
2. 3.1 Introduction
3. 3.2 Pattern Search (PS) Fundamentals
4. 3.3 Generating an Initial Solution
5. 3.4 Generating Trial Solutions
6. 3.5 Updating the Mesh Size
7. 3.6 Termination Criteria
8. 3.7 User‐Defined Parameters of the PS
9. 3.8 Pseudocode of the PS
10. 3.9 Conclusion
11. References
4 Genetic Algorithm
1. Summary
2. 4.1 Introduction
3. 4.2 Mapping the Genetic Algorithm (GA) to Natural Evolution
4. 4.3 Creating an Initial Population
5. 4.4 Selection of Parents to Create a New Generation
6. 4.5 Population Diversity and Selective Pressure
7. 4.6 Reproduction
8. 4.7 Termination Criteria
9. 4.8 User‐Defined Parameters of the GA
10. 4.9 Pseudocode of the GA
11. 4.10 Conclusion
12. References
5 Simulated Annealing
1. Summary
2. 5.1 Introduction
3. 5.2 Mapping the Simulated Annealing (SA) Algorithm to the Physical Annealing Process
4. 5.3 Generating an Initial State
5. 5.4 Generating a New State
6. 5.5 Acceptance Function
7. 5.6 Thermal Equilibrium
8. 5.7 Temperature Reduction
9. 5.8 Termination Criteria
10. 5.9 User‐Defined Parameters of the SA
11. 5.10 Pseudocode of the SA
12. 5.11 Conclusion
13. References
6 Tabu Search
1. Summary
2. 6.1 Introduction
3. 6.2 Tabu Search (TS) Foundation
4. 6.3 Generating an Initial Searching Point
5. 6.4 Neighboring Points
6. 6.5 Tabu Lists
7. 6.6 Updating the Tabu List
8. 6.7 Attributive Memory
9. 6.8 Aspiration Criteria
10. 6.9 Intensification and Diversification Strategies
11. 6.10 Termination Criteria
12. 6.11 User‐Defined Parameters of the TS
13. 6.12 Pseudocode of the TS
14. 6.13 Conclusion
15. References
7 Ant Colony Optimization
1. Summary
2. 7.1 Introduction
3. 7.2 Mapping Ant Colony Optimization (ACO) to Ants’ Foraging Behavior
4. 7.3 Creating an Initial Population
5. 7.4 Allocating Pheromone to the Decision Space
6. 7.5 Generation of New Solutions
7. 7.6 Termination Criteria
8. 7.7 User‐Defined Parameters of the ACO
9. 7.8 Pseudocode of the ACO
10. 7.9 Conclusion
11. References
8 Particle Swarm Optimization
1. Summary
2. 8.1 Introduction
3. 8.2 Mapping Particle Swarm Optimization (PSO) to the Social Behavior of Some Animals
4. 8.3 Creating an Initial Population of Particles
5. 8.4 The Individual and Global Best Positions
6. 8.5 Velocities of Particles
7. 8.6 Updating the Positions of Particles
8. 8.7 Termination Criteria
9. 8.8 User‐Defined Parameters of the PSO
10. 8.9 Pseudocode of the PSO
11. 8.10 Conclusion
12. References
9 Differential Evolution
1. Summary
2. 9.1 Introduction
3. 9.2 Differential Evolution (DE) Fundamentals
4. 9.3 Creating an Initial Population
5. 9.4 Generating Trial Solutions
6. 9.5 Greedy Criteria
7. 9.6 Termination Criteria
8. 9.7 User‐Defined Parameters of the DE
9. 9.8 Pseudocode of the DE
10. 9.9 Conclusion
11. References
10 Harmony Search
1. Summary
2. 10.1 Introduction
3. 10.2 Inspiration of the Harmony Search (HS)
4. 10.3 Initializing the Harmony Memory
5. 10.4 Generating New Harmonies (Solutions)
6. 10.5 Updating the Harmony Memory
7. 10.6 Termination Criteria
8. 10.7 User‐Defined Parameters of the HS
9. 10.8 Pseudocode of the HS
10. 10.9 Conclusion
11. References
11 Shuffled Frog‐Leaping Algorithm
1. Summary
2. 11.1 Introduction
3. 11.2 Mapping Memetic Evolution of Frogs to the Shuffled Frog Leaping Algorithm (SFLA)
4. 11.3 Creating an Initial Population
5. 11.4 Classifying Frogs into Memeplexes
6. 11.5 Frog Leaping
7. 11.6 Shuffling Process
8. 11.7 Termination Criteria
9. 11.8 User‐Defined Parameters of the SFLA
10. 11.9 Pseudocode of the SFLA
11. 11.10 Conclusion
12. References
12 Honey‐Bee Mating Optimization
1. Summary
2. 12.1 Introduction
3. 12.2 Mapping Honey‐Bee Mating Optimization (HBMO) to the Honey‐Bee Colony Structure
4. 12.3 Creating an Initial Population
5. 12.4 The Queen
6. 12.5 Drone Selection
7. 12.6 Brood (New Solution) Production
8. 12.7 Improving Broods (New Solutions) by Workers
9. 12.8 Termination Criteria
10. 12.9 User‐Defined Parameters of the HBMO
11. 12.10 Pseudocode of the HBMO
12. 12.11 Conclusion
13. References
13 Invasive Weed Optimization
1. Summary
2. 13.1 Introduction
3. 13.2 Mapping Invasive Weed Optimization (IWO) to Weeds’ Biology
4. 13.3 Creating an Initial Population
5. 13.4 Reproduction
6. 13.5 The Spread of Seeds
7. 13.6 Eliminating Weeds with Low Fitness
8. 13.7 Termination Criteria
9. 13.8 User‐Defined Parameters of the IWO
10. 13.9 Pseudocode of the IWO
11. 13.10 Conclusion
12. References
14 Central Force Optimization
1. Summary
2. 14.1 Introduction
3. 14.2 Mapping Central Force Optimization (CFO) to Newton’s Gravitational Law
4. 14.3 Initializing the Position of Probes
5. 14.4 Calculation of Accelerations
6. 14.5 Movement of Probes
7. 14.6 Modification of Deviated Probes
8. 14.7 Termination Criteria
9. 14.8 User‐Defined Parameters of the CFO
10. 14.9 Pseudocode of the CFO
11. 14.10 Conclusion
12. References
15 Biogeography‐Based Optimization
1. Summary
2. 15.1 Introduction
3. 15.2 Mapping Biogeography‐Based Optimization (BBO) to Biogeography Concepts
4. 15.3 Creating an Initial Population
5. 15.4 Migration Process
6. 15.5 Mutation
7. 15.6 Termination Criteria
8. 15.7 User‐Defined Parameters of the BBO
9. 15.8 Pseudocode of the BBO
10. 15.9 Conclusion
11. References
16 Firefly Algorithm
1. Summary
2. 16.1 Introduction
3. 16.2 Mapping the Firefly Algorithm (FA) to the Flashing Characteristics of Fireflies
4. 16.3 Creating an Initial Population
5. 16.4 Attractiveness
6. 16.5 Distance and Movement
7. 16.6 Termination Criteria
8. 16.7 User‐Defined Parameters of the FA
9. 16.8 Pseudocode of the FA
10. 16.9 Conclusion
11. References
17 Gravity Search Algorithm
1. Summary
2. 17.1 Introduction
3. 17.2 Mapping the Gravity Search Algorithm (GSA) to the Law of Gravity
4. 17.3 Creating an Initial Population
5. 17.4 Evaluation of Particle Masses
6. 17.5 Updating Velocities and Positions
7. 17.6 Updating Newton’s Gravitational Factor
8. 17.7 Termination Criteria
9. 17.8 User‐Defined Parameters of the GSA
10. 17.9 Pseudocode of the GSA
11. 17.10 Conclusion
12. References
18 Bat Algorithm
1. Summary
2. 18.1 Introduction
3. 18.2 Mapping the Bat Algorithm (BA) to the Behavior of Microbats
4. 18.3 Creating an Initial Population
5. 18.4 Movement of Virtual Bats
6. 18.5 Local Search and Random Flying
7. 18.6 Loudness and Pulse Emission
8. 18.7 Termination Criteria
9. 18.8 User‐Defined Parameters of the BA
10. 18.9 Pseudocode of the BA
11. 18.10 Conclusion
12. References
19 Plant Propagation Algorithm
1. Summary
2. 19.1 Introduction
3. 19.2 Mapping the Natural Process to the Planet Propagation Algorithm (PPA)
4. 19.3 Creating an Initial Population of Plants
5. 19.4 Normalizing the Fitness Function
6. 19.5 Propagation
7. 19.6 Elimination of Extra Solutions
8. 19.7 Termination Criteria
9. 19.8 User‐Defined Parameters of the PPA
10. 19.9 Pseudocode of the PPA
11. 19.10 Conclusion
12. References
20 Water Cycle Algorithm
1. Summary
2. 20.1 Introduction
3. 20.2 Mapping the Water Cycle Algorithm (WCA) to the Water Cycle
4. 20.3 Creating an Initial Population
5. 20.4 Classification of Raindrops
6. 20.5 Streams Flowing to the Rivers or Sea
7. 20.6 Evaporation
8. 20.7 Raining Process
9. 20.8 Termination Criteria
10. 20.9 User‐Defined Parameters of the WCA
11. 20.10 Pseudocode of the WCA
12. 20.11 Conclusion
13. References
21 Symbiotic Organisms Search
1. Summary
2. 21.1 Introduction
3. 21.2 Mapping Symbiotic Relations to the Symbiotic Organisms Search (SOS)
4. 21.3 Creating an Initial Ecosystem
5. 21.4 Mutualism
6. 21.5 Commensalism
7. 21.6 Parasitism
8. 21.7 Termination Criteria
9. 21.8 Pseudocode of the SOS
10. 21.9 Conclusion
11. References
22 Comprehensive Evolutionary Algorithm
1. Summary
2. 22.1 Introduction
3. 22.2 Fundamentals of the Comprehensive Evolutionary Algorithm (CEA)
4. 22.3 Generating an Initial Population of Solutions
5. 22.4 Selection
6. 22.5 Reproduction
7. 22.6 Roles of Operators
8. 22.7 Input Data to the CEA
9. 22.8 Termination Criteria
10. 22.9 Pseudocode of the CEA
11. 22.10 Conclusion
12. References
Wiley Series in Operations Research and Management Science
Index
End User License Agreement

List of Tables

Chapter 02
1. Table 2.1 The input data of the example problems presented in Chapter 1.
2. Table 2.2 The decision variables of the example problems presented in Chapter 1.
3. Table 2.3 The state variables of the example problems presented in Chapter 1.
4. Table 2.4 The objective function of the example problems presented in Chapter 1.
5. Table 2.5 The constraints of the example problems presented in Chapter 1.
6. Table 2.6 Recommended reporting of the solutions calculated in k runs of an algorithm that solves a minimization problem.
Chapter 03
1. Table 3.1 The characteristics of the PS.
Chapter 04
1. Table 4.1 The characteristics of the GA.
Chapter 05
1. Table 5.1 The characteristics of the SA.
Chapter 06
1. Table 6.1 The characteristics of the TS.
Chapter 07
1. Table 7.1 The characteristics of the ACO.
Chapter 08
1. Table 8.1 The characteristics of the PSO.
Chapter 09
1. Table 9.1 The characteristics of the DE.
Chapter 10
1. Table 10.1 The characteristics of the HS.
Chapter 11
1. Table 11.1 The characteristics of the SFLA.
Chapter 12
1. Table 12.1 The characteristics of the HBMO.
Chapter 13
1. Table 13.1 The characteristics of the IWO.
Chapter 14
1. Table 14.1 The characteristics of the CFO.
Chapter 15
1. Table 15.1 The characteristics of the BBO.
Chapter 16
1. Table 16.1 The characteristics of the FA.
Chapter 17
1. Table 17.1 The characteristics of the GSA.
Chapter 18
1. Table 18.1 The characteristics of the BA.
Chapter 19
1. Table 19.1 The characteristics of the PPA.
Chapter 20
1. Table 20.1 The characteristics of the WCA.
Chapter 21
1. Table 21.1 The characteristics of the SOS.
Chapter 22
1. Table 22.1 The characteristics of the CEA.
2. Table 22.2 Types of crossover operators in the CEA.
3. Table 22.3 Types of mutation operators in the CEA.
4. Table 22.4 List of the parameters of the CEA.

List of Illustrations

Chapter 01
1. Figure 1.1 Decision space of a constrained two‐dimensional optimization problem.
2. Figure 1.2 Schematic of global and local optimums in a one‐dimensional maximizing optimization problem.
3. Figure 1.3 Different types of decision spaces: (a) maximization problem with single‐modal surface and one global optimum; (b) maximization problem with multimodal surface that has one global optimum.
4. Figure 1.4 Demonstration of near optima in a one‐dimensional maximizing optimization problem.
5. Figure 1.5 Compound gear train made of four gears.
6. Figure 1.6 Schematic of a two‐bar truss.
7. Figure 1.7 Schematic of a hydropower dam.
Chapter 02
1. Figure 2.1 Sampling grid on a two‐dimensional decision space.
2. Figure 2.2 General schematic of a simple algorithm; K denotes the counter of iterations.
3. Figure 2.3 Diagram depicting the relation between a simulation model and an optimization algorithm.
4. Figure 2.4 The main components of the optimization by meta‐heuristic and evolutionary algorithms.
5. Figure 2.5 Different solutions in a two‐dimensional decision space.
6. Figure 2.6 Selection probability of a set of solutions 1–10 of a hypothetical maximization problem.
7. Figure 2.7 The flowchart of the general algorithm.
8. Figure 2.8 Convergence history of an optimization algorithm toward the best solution in a minimization problem.
9. Figure 2.9 Convergence of an optimization algorithm in which the best solution is not always transferred to the next iteration during the search in a minimization problem.
10. Figure 2.10 Convergence of different runs of an optimization algorithm toward near‐optimal solutions of a minimization problem.
11. Figure 2.11 Convergence of different runs of an optimization algorithm in a minimization problem.
Chapter 03
1. Figure 3.1 The flowchart of the PS.
2. Figure 3.2 Meshes generated by the GPS and MADS methods in a two‐dimensional decision space.
Chapter 04
1. Figure 4.1 The flowchart of the GA.
2. Figure 4.2 Demonstration of a roulette wheel. F = Fitness function value; P = probability of selecting a solution.
3. Figure 4.3 Ranking chromosomes (or solutions) according to the fitness function (F) in a maximizing problem.
4. Figure 4.4 The process of constituting a new generation from the previous generation.
5. Figure 4.5 Different approaches of crossover: (a) one‐point crossover, (b) two‐point crossover, and (c) uniform crossover.
6. Figure 4.6 An example of the mutation operator.
Chapter 05
1. Figure 5.1 The flowchart of the SA.
Chapter 06
1. Figure 6.1 The flowchart of the TS.
2. Figure 6.2 The decision space of a two‐dimensional optimization problem.
3. Figure 6.3 Illustration of steps based on recency‐based memory in a two‐dimensional decision space.
Chapter 07
1. Figure 7.1 A double‐bridge experiment with pathways of unequal length.
2. Figure 7.2 The flowchart of the ACO.
3. Figure 7.3 Representation of a solution in the ACO; there are M such solutions.
Chapter 08
1. Figure 8.1 The flowchart of the PSO.
2. Figure 8.2 Concepts of the best individual position in a two‐dimensional maximization problem.
3. Figure 8.3 Concept of the global best position in a maximization problem.
Chapter 09
1. Figure 9.1 The flowchart of the DE.
Chapter 10
1. Figure 10.1 The flowchart of the HS.
2. Figure 10.2 Generating a new solution based on memory strategy.
Chapter 11
1. Figure 11.1 The flowchart of the SFLA.
2. Figure 11.2 Sorting frogs according to the fitness function F(X) in a maximizing problem.
3. Figure 11.3 Assigning frogs to different memeplexes; Z = number of memeplexes and Y = number of frogs assigned to each memeplex.
4. Figure 11.4 The representation of a memeplex and a submemeplex within the entire population.
Chapter 12
1. Figure 12.1 The flowchart of the HBMO algorithm.
2. Figure 12.2 Determining the queen and trial solutions according to the fitness function F(X).
3. Figure 12.3 Different crossover approaches: (a) one‐point crossover, (b) two‐point crossover, and (c) uniform crossover.
4. Figure 12.4 Performance of mutation operator.
Chapter 13
1. Figure 13.1 The flowchart of the IWO.
2. Figure 13.2 Number of seeds for each weed with respect to the fitness value.
Chapter 14
1. Figure 14.1 The flowchart of the CFO.
2. Figure 14.2 Distribution of initial probes in the CFO algorithm.
Chapter 15
1. Figure 15.1 Species immigration and emigration pattern in a habitat.
2. Figure 15.2 The flowchart of the BBO.
3. Figure 15.3 Comparison of two solutions for one problem.
4. Figure 15.4 Ranking habitats (solutions) according to their fitness values in a minimization problem.
Chapter 16
1. Figure 16.1 The flowchart of the FA.
Chapter 17
1. Figure 17.1 Gravity force between different particles; Force₁ is the resultant force on Mass₁.
2. Figure 17.2 The flowchart of the GSA.
Chapter 18
1. Figure 18.1 The flowchart of the BA.
Chapter 19
1. Figure 19.1 The flowchart of the PPA.
Chapter 20
1. Figure 20.1 The flowchart of the WCA for a minimization problem.
2. Figure 20.2 Classification of raindrops and relations between the sea, rivers, and streams according to their fitness values (F(X)) for a minimization problem where M = 10 and R = 3.
3. Figure 20.3 Schematic of stream flowing toward a river at distance d: X: existing stream; X′: new stream.
Chapter 21
1. Figure 21.1 The flowchart of the SOS.
Chapter 22
1. Figure 22.1 The flowchart of the CEA.
2. Figure 22.2 Illustration of a roulette wheel.
3. Figure 22.3 Types of one‐point cut crossover in the CEA: (a) Type 1 (new solution (1)) and type 2 (new solution (2)), (b) Type 3 (new solution (1)) and type 4 (new solution (2)), (c) Type 5 (new solution (1)) and type 6 (new solution (2)), (d) Type 7 (new solution (1)) and type 8 (new solution (2)), and (e) Type 9 (new solution (1)) and type 10 (new solution (2)).
4. Figure 22.4 Types of two‐point cut crossover in the CEA: (a) Type 11 (new solution (1)) and type 12 (new solution (2)), (b) Type 13 (new solution (1)) and type 14 (new solution (2)), (c) Type 15 (new solution (1)) and type 16 (new solution (2)), (d) Type 17 (new solution (1)) and type 18 (new solution (2)), and (e) Type 19 (new solution (1)) and type 20 (new solution (2)).
5. Figure 22.5 Types of overall crossover in CEA: (a) Type 21, (b) Type 22, and (c) Type 23.

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Library of Congress Cataloguing‐in‐Publication Data

Names: Bozorg‐Haddad, Omid, 1974– author. | Solgi, Mohammad, 1989– author. | Loaiciga, Hugo A., author.
Title: Meta‐heuristic and evolutionary algorithms for engineering optimization / Omid Bozorg‐Haddad, Mohammad Solgi, Hugo A. Loáiciga.
Description: Hoboken, NJ : John Wiley & Sons, 2017. | Series: Wiley series in operations research and management science | Includes bibliographical references and index. |
Identifiers: LCCN 2017017765 (print) | LCCN 2017026412 (ebook) | ISBN 9781119387077 (pdf) | ISBN 9781119387060 (epub) | ISBN 9781119386995 (cloth)
Subjects: LCSH: Mathematical optimization. | Engineering design–Mathematics.
Classification: LCC QA402.5 (ebook) | LCC QA402.5 .B695 2017 (print) | DDC 620/.0042015196–dc23
LC record available at https://lccn.loc.gov/2017017765

Cover Design: Wiley
Cover Images: (Top Image) © Georgina198/Gettyimages;(Bottom Image) © RomanOkopny/Gettyimagess

Preface

Engineers search for designs of new systems that perform optimally and are cost effective or for the optimal operation and rehabilitation of existing systems. It turns out that design and operation usually involve the calibration of models that describe physical systems. The tasks of design, operation, and model calibration can be approached systematically by the application of optimization. Optimization is defined as the selection of the best elements or actions from a set of feasible alternatives. More precisely, optimization consists of finding the set of variables that produces the best values of objective functions in which the feasible domain of the variables is restricted by constraints.

Meta‐heuristic and evolutionary algorithms, many of which are inspired by natural systems, are optimization methods commonly employed to calculate good approximate solutions to optimization problems that are difficult or impossible to solve with other optimization techniques such as linear programming, nonlinear programming, integer programming, and dynamic programming. Meta‐heuristic and evolutionary algorithms are problem‐independent methods of wide applicability that have been proven effective in solving a wide range of real‐world and complex engineering problems. Meta‐heuristic and evolutionary algorithms have become popular methods for solving real‐world and complex engineering optimization problems.

Yet, in spite of meta‐heuristic and evolutionary algorithms’ frequent application, there is not at present a reference that presents and explains them in a clear, systematic, and comprehensive manner. There are several bibliographical sources dealing with engineering optimization and the application of meta‐heuristic and evolutionary algorithms. However, their focus is largely on the results of application of these algorithms and less on their basic concepts on which they are founded. In view of this, it appears that a comprehensive, unified, and insightful overview of these algorithms is timely and would be welcome by those who seek to learn the principles and ways to apply meta‐heuristic and evolutionary algorithms.

This book fills the cited gap by presenting the best‐known meta‐heuristic and evolutionary algorithms, those whose performance has been tested in many engineering domains. Chapter 1 provides an overview of optimization and illustration of its application to engineering problems in various specialties. Chapter 2 presents an introduction to meta‐heuristic and evolutionary algorithms and their relation to engineering problems. Chapters 3–22 are dedicated to pattern search (PS), genetic algorithm (GA), simulated annealing (SA), tabu search (TS), ant colony optimization (ACO), particle swarm optimization (PSO), differential evolution (DE), harmony search (HS), shuffled frog‐leaping algorithm (SFLA), honey‐bee mating optimization (HBMO), invasive weed optimization (IWO), central force optimization (CFO), biogeography‐based optimization (BBO), firefly algorithm (FA), gravity search algorithm (GSA), bat algorithm (BA), plant propagation algorithm (PPA), water cycle algorithm (WCA), symbiotic organisms search (SOS), and comprehensive evolutionary algorithm (CEA), respectively. The order of the chapters corresponds to the order of chronological appearance of the various algorithms, with the most recent ones receiving the larger chapter numbers. Each chapter describes a specific algorithm and starts with a brief literature review of its development and subsequent modification since the time of inception. This is followed by the presentation of the basic concept on which the algorithm is based and the mathematical statement of the algorithm. The workings of the algorithm are subsequently described in detail. Each chapter closes with a pseudocode of the algorithm that constitutes an insightful and sufficient guideline for coding the algorithm to solve specific applications.

Several of the algorithms reviewed in this book were developed decades ago, and some have experienced modifications and hybridization with other algorithms. This presentation is concerned primarily with the original version of each algorithm, yet it provides references that are concerned with modifications to the algorithms.

This book was written for graduate students, researchers, educators, and practitioners with interests in the field of engineering optimization. The format and contents chosen are intended to satisfy the needs of beginners and experts seeking a unifying, complete, and clear presentation of meta‐heuristic and evolutionary algorithms.

Omid Bozorg‐Haddad
Mohammad Solgi
Hugo A. Loáiciga

List of Figures

Figure 1.1	Decision space of a constrained two‐dimensional optimization problem.
Figure 1.2	Schematic of global and local optimums in a one‐dimensional maximizing optimization problem.
Figure 1.3	Different types of decision spaces: (a) maximization problem with single‐modal surface and one global optimum; (b) maximization problem with multimodal surface that has one global optimum.
Figure 1.4	Demonstration of near optima in a one‐dimensional maximizing optimization problem.
Figure 1.5	Compound gear train made of four gears.
Figure 1.6	Schematic of a two‐bar truss.
Figure 1.7	Schematic of a hydropower dam.
Figure 2.1	Sampling grid on a two‐dimensional decision space.
Figure 2.2	General schematic of a simple algorithm; K denotes the counter of iterations.
Figure 2.3	Diagram depicting the relation between a simulation model and an optimization algorithm.
Figure 2.4	The main components of the optimization by meta‐heuristic and evolutionary algorithms.
Figure 2.5	Different solutions in a two‐dimensional decision space.
Figure 2.6	Selection probability of a set of solutions 1–10 of a hypothetical maximization problem.
Figure 2.7	The flowchart of the general algorithm.
Figure 2.8	Convergence history of an optimization algorithm toward the best solution in a minimization problem.
Figure 2.9	Convergence of an optimization algorithm in which the best solution is not always transferred to the next iteration during the search in a minimization problem.
Figure 2.10	Convergence of different runs of an optimization algorithm toward near‐optimal solutions of a minimization problem.
Figure 2.11	Convergence of different runs of an optimization algorithm in a minimization problem.
Figure 3.1	The flowchart of the PS.
Figure 3.2	Meshes generated by the GPS and MADS methods in a two‐dimensional decision space.
Figure 4.1	The flowchart of the GA.
Figure 4.2	Demonstration of a roulette wheel. F = Fitness function value; P = probability of selecting a solution.
Figure 4.3	Ranking chromosomes (or solutions) according to the fitness function (F) in a maximizing problem.
Figure 4.4	The process of constituting a new generation from the previous generation.
Figure 4.5	Different approaches of crossover: (a) one‐point crossover, (b) two‐point crossover, and (c) uniform crossover.
Figure 4.6	An example of the mutation operator.62Figure 5.1The flowchart of the SA.
Figure 6.1	The flowchart of the TS.
Figure 6.2	The decision space of a two‐dimensional optimization problem.
Figure 6.3	Illustration of steps based on recency‐based memory in a two‐dimensional decision space.
Figure 7.1	A double‐bridge experiment with pathways of unequal length.
Figure 7.2	The flowchart of the ACO.
Figure 7.3	Representation of a solution in the ACO; there are M such solutions.
Figure 8.1	The flowchart of the PSO.
Figure 8.2	Concepts of the best individual position in a two‐dimensional maximization problem.
Figure 8.3	Concept of the global best position in a maximization problem.
Figure 9.1	The flowchart of the DE.
Figure 10.1	The flowchart of the HS.
Figure 10.2	Generating a new solution based on memory strategy.
Figure 11.1	The flowchart of the SFLA.
Figure 11.2	Sorting frogs according to the fitness function F(X) in a maximizing problem.
Figure 11.3	Assigning frogs to different memeplexes; Z = number of memeplexes and Y = number of frogs assigned to each memeplex.
Figure 11.4	The representation of a memeplex and a submemeplex within the entire population.
Figure 12.1	The flowchart of the HBMO algorithm.
Figure 12.2	Determining the queen and trial solutions according to the fitness function F(X).
Figure 12.3	Different crossover approaches: (a) one‐point crossover, (b) two‐point crossover, and (c) uniform crossover.
Figure 12.4	Performance of mutation operator.
Figure 13.1	The flowchart of the IWO.
Figure 13.2	Number of seeds for each weed with respect to the fitness value.
Figure 14.1	The flowchart of the CFO.
Figure 14.2	Distribution of initial probes in the CFO algorithm.
Figure 15.1	Species immigration and emigration pattern in a habitat.
Figure 15.2	The flowchart of the BBO.
Figure 15.3	Comparison of two solutions for one problem.
Figure 15.4	Ranking habitats (solutions) according to their fitness values in a minimization problem.
Figure 16.1	The flowchart of the FA.
Figure 17.1	Gravity force between different particles; Force1 is the resultant force on Mass1.
Figure 17.2	The flowchart of the GSA.
Figure 18.1	The flowchart of the BA.
Figure 19.1	The flowchart of the PPA.
Figure 20.1	The flowchart of the WCA for a minimization problem.
Figure 20.2	Classification of raindrops and relations between the sea, rivers, and streams according to their fitness values (F(X)) for a minimization problem where M = 10 and R = 3.
Figure 20.3	Schematic of stream flowing toward a river at distance d: X: existing stream; X′: new stream.
Figure 21.1	The flowchart of the SOS.
Figure 22.1	The flowchart of the CEA.
Figure 22.2	Illustration of a roulette wheel.
Figure 22.3	Types of one‐point cut crossover in the CEA: (a) Type 1 (new solution (1)) and type 2 (new solution (2)), (b) Type 3 (new solution (1)) and type 4 (new solution (2)), (c) Type 5 (new solution (1)) and type 6 (new solution (2)), (d) Type 7 (new solution (1)) and type 8 (new solution (2)), and (e) Type 9 (new solution (1)) and type 10 (new solution (2)).
Figure 22.4	Types of two‐point cut crossover in the CEA: (a) Type 11 (new solution (1)) and type 12 (new solution (2)), (b) Type 13 (new solution (1)) and type 14 (new solution (2)), (c) Type 15 (new solution (1)) and type 16 (new solution (2)), (d) Type 17 (new solution (1)) and type 18 (new solution (2)), and (e) Type 19 (new solution (1)) and type 20 (new solution (2)).
Figure 22.5	Types of overall crossover in CEA: (a) Type 21, (b) Type 22, and (c) Type 23.

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