Applied Genetic Algorithm and Its Variants: Case Studies and New Developments 9819934273, 9789819934270

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Springer Tracts in Nature-Inspired Computing

Nilanjan Dey   Editor

Applied Genetic Algorithm and Its Variants Case Studies and New Developments

Springer Tracts in Nature-Inspired Computing Series Editors Xin-She Yang, School of Science and Technology, Middlesex University, London, UK Nilanjan Dey, Department of Computer Science and Engineering, Techno International New Town, Kolkata, India Simon Fong, Faculty of Science and Technology, University of Macau, Macau, Macao

The book series is aimed at providing an exchange platform for researchers to summarize the latest research and developments related to nature-inspired computing in the most general sense. It includes analysis of nature-inspired algorithms and techniques, inspiration from natural and biological systems, computational mechanisms and models that imitate them in various fields, and the applications to solve real-world problems in different disciplines. The book series addresses the most recent innovations and developments in nature-inspired computation, algorithms, models and methods, implementation, tools, architectures, frameworks, structures, applications associated with bio-inspired methodologies and other relevant areas. The book series covers the topics and fields of Nature-Inspired Computing, Bio-inspired Methods, Swarm Intelligence, Computational Intelligence, Evolutionary Computation, Nature-Inspired Algorithms, Neural Computing, Data Mining, Artificial Intelligence, Machine Learning, Theoretical Foundations and Analysis, and Multi-Agent Systems. In addition, case studies, implementation of methods and algorithms as well as applications in a diverse range of areas such as Bioinformatics, Big Data, Computer Science, Signal and Image Processing, Computer Vision, Biomedical and Health Science, Business Planning, Vehicle Routing and others are also an important part of this book series. The series publishes monographs, edited volumes and selected proceedings.

Nilanjan Dey Editor

Applied Genetic Algorithm and Its Variants Case Studies and New Developments

Editor Nilanjan Dey Department of Computer Science and Engineering Techno International New Town Kolkata, West Bengal, India

ISSN 2524-552X ISSN 2524-5538 (electronic) Springer Tracts in Nature-Inspired Computing ISBN 978-981-99-3427-0 ISBN 978-981-99-3428-7 (eBook) https://doi.org/10.1007/978-981-99-3428-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

A genetic algorithm is a search technique that takes inspiration from Charles Darwin’s theory of natural evolution. It mimics the process of natural selection, where the strongest individuals are chosen to produce the next generation, to find the best solution to a problem. This book highlights the concept and origin of the genetic algorithm and its variants. It includes several optimization-based applications using the flower pollination algorithm by covering different applications. The book includes ten chapters, where Goswami et al. in Chap. 1 aim to encourage scientists to use Genetic Algorithm (GA) and its new variations to tackle difficult engineering design problems. It provides an overview of GA’s fundamental principles, parametric configuration for GA, its different variations, and examples of GA usage in different fields. Then, Iglesias and Galvez in Chap. 2 offer a gentle description of the fundamentals and background of genetic algorithms. Then, it explores some of the most exciting recent advances regarding the application of genetic algorithms to challenging realworld problems in non-standard fields, such as reverse engineering for manufacturing (curve and surface reconstruction from point clouds), medicine and bioinformatics (cancer prediction, detection and diagnosis; cancer classification and treatment; and Covid-19), computer animation and video games (behavioral animation of NPCs for video games), and robotics (robot path planning). In Chap. 3, Türko˘glu and Ero˘glu present the use of genetic algorithms in route optimization problems and also reported one of the best practices of the genetic algorithm used in power transmission line route optimization. Furthermore, in Chap. 4, Tunca and Carbas reported that in a reinforced concrete structure, beams play a critical role as they transfer both the vertical loads and loads from the slabs to the columns/shear walls. This connection system forms a frame, where the beams support the slabs and connect the columns to one another. It is important to note that during an earthquake, beams and columns are the most vulnerable structural elements, and, therefore, designing the beams accurately and safely carries vital importance. To minimize the weight of the reinforced concrete beams, they need to be designed to resist maximum bending moments, especially under the influence of two-point and distributed vertical loadings, which is the main objective of this chapter.

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In Chap. 5, Ibrahim and Tawhid refined genetic algorithm of the straightforward to create the improved genetic algorithm called IGA for the dimensional crossover and mutation operations. IGA is tested using eight real-optimization applications of nonlinear equation systems and the CEC 2018 test functions. In Chap. 6, Boopathi D et al. propose innovative energy management for intelligent smart grid systems by utilizing a Genetic Algorithm (GA) tuned controller. The smart grid contains renewable energy sources and energy storage units. Renewable energy resources comprise wind turbine generators. The energy storage unit includes a fuel cell and battery energy storage system. In Chap. 7, Rabipour and Asadi reported how Genetic Algorithm can be deployed in predicting mental illness. They have taken Schizophrenia as a case study. In Chap. 8, Murugesan D et al. examine the performance comparisons of generally well-used cost functions (profitability) in automatic control of voltage and frequency of a single-area multisource power framework. The 40-story outrigger-braced structure is optimized using the genetic algorithm (GA) in terms of the structural weight and the location of the outrigger and truss belt system. In the next step, the 60-story diagrid building, which has a more complex topology than the first structure, is optimized in terms of weight and the diagrid angle using the bi-linear membership function applied on the fuzzified GA reported by Salar Farahmand-Tabar in Chap. 9. In the last chapter, Rajinikanth and Rama used a suitable scheme for analyzing underwater images with GA based on the optimally assigned gene and chromosome is described. The editor is thankful to the outstanding authors and referees for their contributions and outcomes. Their diligence and collaboration led to this remarkable book. The appreciation is also directed to the members of the Springer team for their support. Kolkata, West Bengal, India

Nilanjan Dey

Contents

1

Variants of Genetic Algorithms and Their Applications . . . . . . . . . . . Radha Debal Goswami, Sayan Chakraborty, and Bitan Misra

2

Genetic Algorithm Applications for Challenging Real-World Problems: Some Recent Advances and Future Trends . . . . . . . . . . . . Andrés Iglesias and Akemi Gálvez

3

Genetic Algorithm for Route Optimization . . . . . . . . . . . . . . . . . . . . . . Bahaeddin Türko˘glu and Hasan Ero˘glu

4

Design Weight Minimization of a Reinforced Concrete Beam Through Genetic Algorithm and Its Variants . . . . . . . . . . . . . . . . . . . . Osman Tunca and Serdar Carbas

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IGA: An Improved Genetic Algorithm for Real-Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Abdelmonem M. Ibrahim and Mohamed A. Tawhid

6

Application of Genetic Algorithm-Based Controllers in Wind Energy Systems for Smart Energy Management . . . . . . . . . . . . . . . . . 139 D. Boopathi, K. Jagatheesan, Sourav Samanta, B. Anand, and J. Jaya

7

Application of Genetic Algorithm in Predicting Mental Illness: A Case Study of Schizophrenia . . . . . . . . . . . . . . . . . . . . . . . . . . 161 S. Rabipour and Z. Asadi

8

Comparison of Biologically Inspired Algorithm with Socio-inspired Technique on Load Frequency Control of Multi-source Single-Area Power System . . . . . . . . . . . . . . . . . . . . . . . 185 D. Murugesan, K. Jagatheesan, Anand J. Kulkarni, Pritesh Shah, and Ravi Sekhar

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Contents

Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing and Topology Design of Real-Size Tall Building Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Salar Farahmand-Tabar

10 Evaluation of Underwater Images Using Genetic Algorithm-Monitored Preprocessing and Morphological Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Venkatesan Rajinikanth and Arulmozhi Rama

Editor and Contributors

About the Editor Nilanjan Dey is Associate Professor, Department of Computer Science and Engineering, Techno International New Town, Kolkata, India. He is Visiting Fellow of the University of Reading, UK. He was Honorary Visiting Scientist at Global Biomedical Technologies Inc., CA, USA (2012–2015). He was awarded his Ph.D. from Jadavpur University in 2015. He has authored/ edited more than 70 books with Elsevier, CRC Press, and Springer and published more than 300 papers. He is Editor-in-Chief of International Journal of Ambient Computing and Intelligence, IGI Global, and Associated Editor of International Journal of Information Technology, Springer. He is Series Co-editor of Springer Tracts in Nature-Inspired Computing, Springer, Series Co-editor of Advances in Ubiquitous Sensing Applications for Healthcare, Elsevier, and Series Editor of Computational Intelligence in Engineering Problem Solving and Intelligent Signal processing and data analysis, CRC. He is Fellow of IETE and Senior Member of IEEE.

Contributors B. Anand Hindusthan College of Engineering and Technology, Coimbatore, Tamil Nadu, India

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Z. Asadi Al-Ameen College of Pharmacy, Rajiv Gandhi University of Health Science (RGUHS), Bangalore, India D. Boopathi Paavai Engineering College, Puduchatram, Tamil Nadu, India Serdar Carbas Department of Civil Engineering, Faculty of Engineering, Karamanoglu Mehmetbey University, Karaman, Türkiye Sayan Chakraborty Department of CSE, Techno International New Town, Kolkata, India Hasan Ero˘glu Faculty of Engineering and Architecture, Department of ElectricalElectronics Engineering, Recep Tayyip Erdogan University, Rize, Turkey Salar Farahmand-Tabar Department of Civil Engineering Eng, Faculty of Engineering, University of Zanjan, Zanjan, Iran Akemi Gálvez Department of Applied Mathematics and Computational Sciences, E.T.S.I. Caminos, Canales y Puertos, University of Cantabria, Santander, Spain; Faculty of Pharmaceutical Sciences, Toho University, Funabashi, Japan Radha Debal Goswami Department of ECE, VIT, Vellore, India Abdelmonem M. Ibrahim Mathematics Department, Faculty of Science, AlAzhar University, Cairo, Egypt Andrés Iglesias Department of Applied Mathematics and Computational Sciences, E.T.S.I. Caminos, Canales y Puertos, University of Cantabria, Santander, Spain; Faculty of Pharmaceutical Sciences, Toho University, Funabashi, Japan K. Jagatheesan Paavai Engineering College, Puduchatram, Tamil Nadu, India J. Jaya Hindusthan College of Engineering and Technology, Coimbatore, Tamil Nadu, India Anand J. Kulkarni MIT World Peace University, Pune, India Bitan Misra Department of CSE, Techno International New Town, Kolkata, India D. Murugesan Paavai Engineering College, Puduchatram, Tamil Nadu, India S. Rabipour School of Medicine, Iran University of Medical Science, Tehran, Iran Venkatesan Rajinikanth Department of Computer Science and Engineering, Division of Research and Innovation, Saveetha School of Engineering, SIMATS, Chennai, India Arulmozhi Rama Department of Computer Science and Engineering, Division of Research and Innovation, Saveetha School of Engineering, SIMATS, Chennai, India Sourav Samanta University Institute of Technology, The University of Burdwan, Bardhaman, West Bengal, India

Editor and Contributors

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Ravi Sekhar Symbiosis Institute of Technology (SIT) Pune Campus, Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India Pritesh Shah Symbiosis Institute of Technology (SIT) Pune Campus, Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India Mohamed A. Tawhid Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada Osman Tunca Department of Civil Engineering, Faculty of Engineering, Karamanoglu Mehmetbey University, Karaman, Türkiye Bahaeddin Türko˘glu Faculty of Engineering, Department of Computer Engineering, Ni˘gde Ömer Halis Demir University, Ni˘gde, Turkey

Chapter 1

Variants of Genetic Algorithms and Their Applications Radha Debal Goswami, Sayan Chakraborty, and Bitan Misra

1 Introduction The majority of evolutionary algorithms are adaptive heuristic search algorithms known as genetic algorithms [1, 2] (GAs). Two primary aspects of GA include natural selection and genetics. To achieve the best solution in the search space, an intelligent use of a random search technique is incorporated along with previous data. This technique is often used to achieve the best solution to a search or optimization problem. GA simulates the method of natural selection, which demands that the fittest species can only survive, procreate, and pass on to the next generation by adapting themselves to the changes in the environment. To solve an issue, they essentially replicate the famous Darwin theory about “survival of the fittest” among individuals of successive generations. According to the “survival of the fittest” concept, organisms that are better adapted to their surroundings have a higher chance of surviving and procreating. Therefore, the algorithm which provides the best solution among various other available algorithms for a given set of complex problems is said to survive more for other sets of problems as well and the weak or inaccurate algorithms eventually fail to succeed. Genetic algorithms come in handy while providing outcomes to constrained and unconstrained optimization problems. Constrained optimization [3] problems require a function to be maximized or decreased while adhering to certain restrictions. Finding the maximum value or minimum value of a differentiable function of numerous variables over a predefined set is the goal of unconstrained optimization.

R. D. Goswami Department of ECE, VIT, Vellore, India S. Chakraborty Department of CSE, Techno International New Town, Kolkata, India B. Misra (B) Department of CSE, Techno International New Town, Kolkata, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_1

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The required computations can be made using a computer algebra system to handle the problems’ complexity. The algorithm does not only focus on computer science or mathematics specifically but also fields like economics. Economic models such as the cobweb model or the game theory equilibrium resolution problems can also be solved using this algorithm. With the tremendous growth in machine learning, artificial intelligence, and various neural network trends, more and more algorithms are being developed daily which is derived from the basics of genetic algorithm only. Biomedical industries in particular have seen an increase in demand post the COVID-19 pandemic and have also pointed out a deficiency in the availability of proper infrastructure in densely populated countries such as India, China, and Pakistan. Medical imaging is receiving ever greater attention in the computational study of lung illness with the introduction of deep learning technologies, with that genetic algorithm and its multiple variants are gaining more popularity.

2 Background An enormous number of potential solutions to an optimization issue is evolved toward better solutions in a genetic algorithm. In the past, solutions have been represented as binary strings containing 0 and 1; however, different types of encodings are also feasible. Each potential solution has a set of properties that may be changed and transformed, much like chromosomes. To understand genetic algorithms in detail, it is also important to understand their biological analogy. A chromosome is a threadlike structure comprising Deoxyribonucleic Acid (DNA) strands, the basic genetic information carrier of the human body. The chromosome is responsible for deciding the sex of the human body. A combination of the X chromosome and Y chromosome (two available sex chromosomes) decides if the resultant fetus is a male or female. The male contains the XY chromosome whereas the female is composed of the XX chromosome. When there is mating between the two sex, two possible scenarios occur. The female can only provide the X chromosome whereas the male sex chromosome plays a critical part in determining the sex of the infant. If it provides the X chromosome, a female is born, and if a Y chromosome is intertwined with the X, a male is born. This is a normal case where a healthy infant is born by the mating of their parents. Therefore, a candidate or a problem set in the case of an algorithm contains its properties which can be represented as the chromosomes and the outcome is the best possible outcome generated by the analysis of all the possible solutions. Father

X chromosome

Y chromosome

Mother (X chromosome)

XX (female)

XY (male)

This can also be explained in terms of genetic algorithms. Assume X and Y to be bits of data like 0 or 1. Then the same table can be represented in the following manner.

1 Variants of Genetic Algorithms and Their Applications Problem

1 (true)

3 0 (false)

1 (true)

11

10

0 (false)

01

00

The problem can only be solved if both the values are true (as shown in 11). Therefore, the problem computes all the possible solutions and provides the best outcome as the culmination of the entire set of solutions. A generation is an expression applied to represent the population in an individual iteration of evolution. It usually initiates with a randomly generated population of solutions. In each generation, the fitness value of each individual is assessed. Fitness typically represents the objective function in the optimization problem being addressed. The next generation is created by stochastically selecting the fittest people from the current population, recombining their genomes, and maybe introducing random mutations. The new set of candidate solutions is used in the next generation. The algorithm typically comes to an end when the population has reached a desirable level of fitness or the maximum iteration number is reached. Once the genetic presentation and objective function are established, a GA starts by creating candidate solutions and continues using operators namely mutation, crossover, inversion, and selection to improve the solution. Mutation, crossover, and inversion are types of disorders created in the chromosomes that may result in outcomes different from the normal norms. A lengthy DNA segment is altered in a chromosomal mutation. Sections or segments of DNA may be deleted, inserted, inverted, or moved as a result of these changes. When deleted sections join to other chromosomes, both the chromosomes lose the DNA and the one getting it may become disrupted. Some of the major types of chromosomal disorders are Down Syndrome, Turner Syndrome, Trisomy 13, and Trisomy 18. Some stages are considered during the development of the algorithm. It can be broken down into five major steps: . . . . .

Initial selection of solution set. Deciding fitness function. Selection. Crossover. Mutation.

The GA may evaluate the performance of each individual in the population using the objective function [4, 5]. The fitness function has to be carefully designed because it is the only link between the GA and the application. The fitness function needs to accurately represent the application regarding how the parameters are to be reduced. The selection operator chooses chromosomes from the present generation to be the parents of the following generation. The probability has to be evaluated of the selection of chromosome which is in turn evaluated using the given formula.

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Fig. 1 Block diagram of basic genetic algorithm

Ps(i ) = f (i )/

N Σ

f ( j)

(1)

J =1

f (i) represents the fitness value of the ith chromosome whereas Ps(i) denotes the probability the ith chromosome is selected from multiple pairs of parent chromosomes. Pairs of parents are chosen. The parent is chosen from the remaining chromosomes once one chromosome has been chosen, and the odds are then renormalized without the chosen chromosome [6]. Therefore, each pair consists of two unique chromosomes. Each chromosome has the potential to be in many pairs. Figure 1 demonstrates the basic flowchart of the selection and transfer of characteristics of the Genetic Algorithm. The GA’s main local search procedure is the crossover technique. For each set of parent pairs that the crossover/reproduction operator receives from the selection operator, it calculates two offspring. The next generation is made up of these descendants following mutation. Before the process begins, a chance of crossover is established, which controls whether each parent pair crosses over or reproduces. The upshot of reproduction is that the offspring pair is identical to the parent pair. To create the offspring pair, the crossover operation transforms the parent pair to binary notation and swaps bits at a crossover point that is chosen at random. A mutation is a worldwide search technique that is implemented in Genetic Algorithms. Before the algorithm is performed on each individual bit of each offspring chromosome to decide if it is to be inverted, a chance of mutation is once more specified [7]. The elitist operator guarantees that the GA won’t worsen as it develops. By skipping the chance of mutation and crossovers, the elitist operator passes on the best solution to the following generation. This ensures that the fittest chromosome will never lose fitness which is the fundamental concept of Darwin’s concept of survival of the fittest. The probabilities can represent the current state of the GA by computing the probability of crossover and mutation as a function of fitness. For each parent pair, a separate crossover probability is utilized, and for each chromosome within the same generation, a different mutation probability is employed. Using a high likelihood of mutation, one may then extract underperforming chromosomes with low relative fitness to be regarded more as global searches. This enables the GA to focus on

1 Variants of Genetic Algorithms and Their Applications

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chromosomes that perform better while utilizing the other to look for new minima or maxima.

3 Genetic Algorithms and Their Variants Throughout the years, multiple variants of GA have been developed. The variants have been incorporated to fix flaws and make the algorithm more universal in terms of usage on various types of problems.

3.1 Variants of GA Based on Operators and Chromosomes Depending on the recombination operators and representation of chromosomes, GA can be of two types, Real-Coded Genetic Algorithm (RCGA) and Binary-Coded Genetic Algorithm.

3.1.1

Binary Genetic Algorithm (BGA) and Binary-Coded Genetic Algorithm (BCGA)

The fundamental building components of the Binary Genetic Algorithm (BGA) are genes and chromosomes. The optimization parameters are encoded into a binary code string by the traditional binary GA. A binary bit is a gene in GA. A population in binary-coded genetic algorithms is nothing more than a collection of “chromosomes” that stand in for potential answers. By employing genetic operators, these chromosomes are changed or edited to generate a new generation. The problemsolving procedure is repeated several times or until no progress is made. The Binary GA solutions get better with time and iterations but it is assured to provide an answer which makes it a reliable algorithm to implement. Since deployment, the Binary GA has performed significantly well in noisy environments and noisy data. Data that is noisy is useless. The phrase has frequently been used interchangeably with faulty data. Nevertheless, it now refers to any data, including unstructured data, that cannot be effectively processed and analyzed by machines. Such data do not provide any constructive outcome but have a high chance of deviating the algorithm from providing the correct solution. One major edge this algorithm holds over the other optimization algorithms is its inherent ability for parallel computing and distributed processing. The fundamental problem with Binary GA is delayed convergence since it requires a lot of calculation time to convert a number’s actual value into its equivalent binary value or its binary value into its real value. When switching several bits is necessary to go to an adjacent solution, the progressive search in the solution space is hampered.

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The string length restricts the precision of the solution if fixed. Due to these very negatives of the binary algorithm, a new generation has evolved named as Real-Coded GA. One of the main differences between BGA and Classical Genetic Algorithm (CGA) is the way they represent the potential solutions in the population. In BGA, the potential solutions are represented as binary strings, with each bit representing a decision variable that can take on values of 0 or 1. In contrast, in CGA, the potential solutions are reported as real-valued vector decision variables. This difference in representation affects the way that the selection, crossover, and mutation operators are applied in algorithms. Another difference between BGA and CGA is the way they handle the encoding of potential solutions. In BGA, the binary encoding of the potential solutions can lead to issues with premature convergence, where the algorithm may converge to a local optimum instead of the global optimum. In contrast, in CGA, the real-valued encoding of the potential solutions allows for a more continuous search space, which can help avoid issues with premature convergence. BGA and CGA also differ in their search operators. In BGA, the crossover operator and mutation operators are implemented on the binary strings directly, while in CGA, these operators are applied to the real-valued decision variables. This difference in search operators affects the investigation and exploitation of the solution space, which can impact the efficiency and effectiveness of the algorithm. Another difference between BGA and CGA is their performance on different types of optimization problems. BGA is particularly suited for binary optimization problems, where the decision variables are binary (i.e., take on values of 0 or 1). In contrast, CGA is more suitable for continuous optimization problems, where the decision variables can take on any real value.

3.1.2

Real-Coded Genetic Algorithm (RCGA)

When working with continuous solution spaces with huge dimensions and a high level of numerical accuracy is required, real-coded genetic algorithms (RCGA) have several benefits over their binary-coded counterparts. In RCGA, each gene stands for one variable of the problem, and the length of the problem’s solution is maintained by maintaining the same chromosomal size. Therefore, unlike the binary approach, RCGA can handle vast domains without reducing precision. Additionally, RCGA has the ability to locally tune the solutions and integrate domain knowledge to escalate the performance of Genetic Algorithms (GA). However, the demand for population variety and the regular computation of fitness continue to nag at RCGA and might become exceedingly time-consuming. As a result, much like with its binary implementation, its intrinsic parallelism is hindered, and its application space is constrained by the performance bottleneck. The optimization parameters are represented discretely rather than directly by the binary GA. Encoding a real number will eventually result in a discretization mistake. For problems with actual optimization parameters, the encoding and decoding processes also increase the computing cost of the approach. Therefore, it is worthwhile to create a unique GA that operates directly on the actual optimization parameters.

1 Variants of Genetic Algorithms and Their Applications

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Based on the concept of Binary GA, Real-coded Genetic Algorithm’s selection operator is dependent on the fitness value. The selection operators used in Binary GA can be used in Real-coded GA as well. Although it requires re-adjustment of crossover and mutation operators, the crossover operators in Real-Coded can be of four types: single-point crossover, which is similar to binary-coded GAs operator; linear crossover, in which three solutions are generated from two parents, and among total 5 solutions, only two are chosen; blend crossover: given two parents, blend crossover randomly selects child in the predefined range; and simulated binary crossover which simulates single-point crossover operator of the binary-coded GAs. There are three types of mutation operators in RCGA such as the following: . Random Mutation: It develops solution randomly within the parameter range. . Alternative Random Mutation: It is similar to random mutation but the solution is generated within a range of the original solution. . Non-Uniform Mutation: In this mutation, the mutated solution is closer to the actual solution, as the illegally mutated gene values are evolved to make them adjustable within the allowed range. Real-coded genetic algorithms (GAs) interact directly with the variables in the issue without using any coding. The recombination operators used in real-coded and binary-coded GA implementations are a major distinction. Even though several real-coded crossover solutions were proposed, most of them were created intuitively and without much research. Based on the search characteristics of the singlepoint crossover operator applied in BGAs, a real-coded crossover operator has been created. As compared to current real-coded crossover implementations, the simulated binary crossover (SBX) operator has been proven to perform admirably in numerous test situations with continuous solution space. The real-coded GA with SBX includes nondominated sorting implementations and sharing function approaches to address multimodal and multi-objective issues, respectively. It has been found that RCGAs outperform BGAs in a variety of test issues, sometimes even better. The SBX operator has the benefit of being able to limit offspring solutions to any unpredictable degree of similarity to the parent solutions, negating the need for a distinct mating restriction method for improved performance. Hence, Real-Coded GA solved all the flaws observed in the Binary GA as well as made advancements to improve it as a whole.

3.2 Variants of GA Based on Crossover Genetic Algorithms can have three types of crossover methods, namely Single-point crossover, Mid-point crossover, and Blend crossover. Mid-point crossover and Blend crossover are applicable in the case of RCGA only; however, Single-point crossover can be applied in BGA and RCGA both. Based on these three crossovers, GA can be of three types Steady-State Genetic Algorithm (SSGA), Generational Genetic Algorithm (GGA), and Steady-Generational Genetic Algorithm (SGGA).

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R. D. Goswami et al.

Steady-State Genetic Algorithm (SSGA)

Steady-State Genetic Algorithm (SSGA) is a variant of the Genetic Algorithm that has been widely used in optimization. The Steady-State Genetic Algorithm (SSGA) is a variant of the Genetic Algorithm (GA) that was first introduced [8] in the early 1990s. The SSGA was developed as an alternative to the standard GA, which has been in use since the 1970s. The standard GA works by generating a new population of solutions at each generation by combining the fittest solutions from the previous generation. However, this approach can lead to premature convergence and a loss of diversity in the solution space. The SSGA was developed to address these issues by replacing only a small subset of the population at each generation. The concept of steady-state evolution has been around since the 1970s, but it was not applied to GA until the early 1990s. One of the first papers on SSGA was published by Lawrence Davis in 1991. In his paper, Davis introduced a variant of the GA [9, 10] that used a steady-state approach to generate and evolve the population of solutions. This approach was shown to be effective in solving a range of optimization problems, inclusive of the traveling salesman problem and the knapsack problem. One of the primary advantages of SSGA is its capability to converge to the optimal solution quickly while maintaining a diverse set of solutions. This is because SSGA focuses on improving a small subset of the population at each iteration, which allows it to adapt to changes in the solution space more quickly than the standard Genetic Algorithm. SSGA also allows for the identification of multiple optima in the solution space, which can be useful in problems where the optimal solution is not distinctive. SSGA has been applied in a wide range of optimization problems, including scheduling, routing, and inventory management. It is highly effective in these applications, often outperforming other optimization techniques. SSGA is particularly useful in problems where the solution space is complex and there are many local optima. In these cases, SSGA’s ability to maintain a diverse set of solutions allows it to search the solution space more effectively and reach a globally optimal solution. In each iteration step, a steady-state genetic algorithm (SSGA) chooses two individuals from the population pool using a selection method. As it is a steady state, there are no generations. In contradiction with the Simple Genetic Algorithm [11] here the two best individuals from the two parents and two children are kept in the population to maintain a uniform population size, as opposed to the Simple Genetic Algorithm, which adds the children of the selected parents into the population of next generation. By genetically modifying the chosen people, a new progeny is produced, which is then introduced into the population pool to replace a less-suited person. Therefore, during the following iteration step, the parents and children can coexist in the population pool. An SSGA has an advantage over a GGA in that it only performs one function evaluation per kid each cycle. Every cycle, a GGA is required to evaluate P (where P represents the size of the population) function evaluations. Hence, while comparing GGAs with SSGAs, only function evaluations are taken into account.

1 Variants of Genetic Algorithms and Their Applications

3.2.2

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Generational Genetic Algorithm (GGA)

The Generational Genetic Algorithm (GGA) is a popular variant of the Genetic Algorithm (GA) that is widely used in optimization. In GGA, a new population of solutions is obtained at each generation by selecting the fittest individuals from the previous generation and applying genetic operators such as crossover and mutation to generate a new solution. The population of new solutions is then assessed to determine their fitness, and the method is repeated for a predetermined number of generations or until a satisfactory solution is found. The concept of the GGA was first introduced in the late 1960s and early 1970s, but it was not until the 1980s that it gained widespread use in the field of optimization. One of the prime advantages of the GGA is its potential to converge to the optimal solution quickly, as it focuses on improving the fitness of the entire population at each generation. This approach ensures that the best solutions are retained while also allowing for the exploration of the solution space. The conventional Genetic Algorithm (GA) and Generational Genetic Algorithm (GGA) are both optimization techniques [12, 13] that apply the principles of natural selection and genetics to obtain solutions to complex optimization problems. However, there are some key differences between the two approaches. One of the main differences between the two approaches is in the way they generate new populations. In the conventional GA, a new population [14–16] is generated by selecting the fittest individuals from the current population and then applying genetic operators such as crossover and mutation to generate a new population. This process is repeated for a predetermined number of generations or until a satisfactory solution is obtained. In contrast, the GGA generates a new population at each generation by selecting the fittest individuals from the previous generation and applying genetic operators to create a new population. Another difference between the two approaches is in the selection mechanism. In the conventional GA, the selection mechanism [17, 18] is usually based on a fitness-proportionate selection mechanism, where individuals with higher fitness values are expected more to be selected for creating new solutions. In the GGA, different selection mechanisms can be used, including tournament selection and rank-based selection. GGA is particularly useful in problems where the solution space is simple and there are few local optima. In these cases, GGA’s ability to converge quickly allows it to find the globally optimal solution more efficiently than other optimization techniques. In recent years, there has been a renewed interest in GGA, particularly in the development of hybrid algorithms that combine GGA with other optimization techniques. One such example is the Multi-Objective Genetic Algorithm (MOGA), which uses GGA to optimize multiple objectives simultaneously. MOGA proves to be highly effective in a large number of applications, including engineering design and financial forecasting.

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3.3 Chaotic Genetic Algorithm (CGA) Chaotic Genetic Algorithm is a hybrid Genetic algorithm that is based on the chaotic system. It has been observed that chaotic [19] systems are often used in optimization instead of random processes. Chaos is a popular problem statement that can have a non-linear solution. It also helps to reflect the complexity of any framework or method. Despite being different from random processes, chaos itself is a random process and has similar features such as random variables, and randomness which is triggered by chaotic dynamics. Chaotic dynamics make chaotic systems different than a random process. Genetic algorithm uses a chaotic system to replace the random process to generate the initial population and apply crossover and mutation. Popular chaotic systems are used in CGA [19]. The chaotic systems used in CGA include Lorenz, Hennon map, Logistic map, etc. The main reason for choosing a chaotic system instead of random procedures in CGA is the ergodicity and pseudorandomness of chaotic structures. These two characteristics help to avoid local convergence in standard GA.

3.4 Adaptive Genetic Algorithm The driving force of Genetic Algorithm (GA) is the parameters that control the development of a solution pool. Previous studies have discovered that optimum parameter settings changes based on the problem statement and its fitness function. For example, a smooth, continuous, fitness function with one maxima is easily solved and doesn’t need any mutation. On the contrary, a complex fitness function with many local maxima requires a higher solution pool and may need mutation to find the optimal solution. Hence, it is evident that parameters in GA and their optimum settings depend on the fitness function and its landscape. To investigate this, Adaptive Genetic Algorithm was introduced. Adaptive Genetic Algorithm [18] is capable of real-time adaptation of control parameters of Genetic algorithm, such as mutation probability (pm), crossover probability (pc), and power scaling factor. This helps to maintain the consistent progression of fitness. It also helps to improve the convergence speed and degree of solution accuracy of standard Genetic algorithm. Instead of using fixed pm and pc values, AGA changes [20] the pm and pc values according to the fitness function which makes the algorithm more automated than manual.

3.5 Niching Genetic Algorithm In our ecosystem, every living being or animals compete and survive in different ways. Niche is the method that allows different species to survive in different conditions. Based on the Niching methods, the primary objective of NGA was to parallelly

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investigate multiple solutions in the population. To minimize the selection operator’s genetic drift Niching Genetic Algorithm [21] was introduced. A niche technology is primarily used to make an adjustment to fitness and deploy a replacement strategy while a new generation is being developed. This allows the individuals or existing species to adjust to the new environment. It also allows the diversity of evolution of the population and helps to obtain multiple global optimums at the same time. In NGA, the niche methods [21] that are usually applied consist of preselecting, crowding, and sharing. In NGA, niche methods are directly used on GA operators without affecting the encoding structure. Certain researchers have managed to successfully combine previously discussed AGA with Niche Genetic Algorithm to develop a hybrid mechanism known as Adaptive Niche Hierarchy Genetic Algorithm (ANHGA). In this work, they primarily changed the NGA by applying AGA on mutation probability which actually affected the encoding method.

3.6 Interactive Genetic Algorithm The natural evolutionary process to find the optimal solution is used to develop an interactive Genetic Algorithm. This special type of evolutionary algorithm doesn’t need continuous and differentiable objective functions. The main difference between IGA with the existing Genetic Algorithm is it is capable of identifying the fitness function based on the user’s preference. Thus, IGA [22] has managed to successfully establish its interest among researchers. It is primarily being applied to robot control, image processing and retrieval systems, voice and natural language processing, art creation, education, entertainment, and other areas. The main highlight of IGA is it capable of integrating human intelligence to obtain individual fitness based on user preferences. IGA is also capable of integrating [22] the requirement of products for customers and is adaptive to solve configuration problems. It can also help to develop a computer-aided design. Despite being such an advanced mechanism, IGA has several problems such as user fatigue and user preference which can lead to erroneous solutions due to the user’s mental and emotional conditions.

3.7 Saw-Tooth Genetic Algorithm The Saw-tooth Genetic Algorithm (SGA) is a variant of Genetic Algorithm (GA) which was developed by David E. Goldberg and Ling-Tim Wong in 1985. The algorithm is named “saw-tooth” because of the jagged pattern of its search trajectory. The SGA algorithm begins with randomly generated potential solutions, each of which is represented as a binary string. The individuals go through a selection, crossover, and mutation process, which imitates the biological evolution method. In the selection phase, the fittest from the population are chosen for reproduction. In the crossover phase, the genetic material of two selected individuals is combined to

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produce offspring. In the mutation phase, a small random change is incorporated into the offspring to expand the diversity of the population. The SGA algorithm is unique in that it employs an adaptive mutation rate. The mutation rate is a parameter that determines the probability of a mutation occurring during the mutation phase. In traditional GA [7, 23], the mutation rate is fixed throughout the entire optimization process. However, in SGA, the mutation rate is dynamically adjusted depending on the population fitness. If the population fitness is huge, then the rate of mutation is reduced to promote convergence toward the optimal solution. Conversely, if the population fitness is small, the rate of mutation is incremented to encourage exploration of the search space. A variety of techniques have been created to increase the computational efficiency and resilience of GAs. According to the statistics of each generation, a simple GA (standard GA) employs a constant population size to direct the evolution of a group of randomly chosen chromosomes through several iterations that are put through consecutive selection, crossover, and mutation. One of the key factors affecting the resilience and evaluation effectiveness of the GAs [24, 25] is population size. Large population sizes result in a significant increase in computing effort, while minor population sizes can cause an early convergence to suboptimal solutions. In the literature, several strategies have been put up to broaden the population’s variety and prevent early convergence. A Saw-tooth pattern uses a changeable population size with periodic reinitialization and a defined amplitude and period of fluctuation (saw-tooth GA). The population size drops linearly during each period, and new, randomly generated people are added to the population at the start of the following period. The SGA algorithm is widely applied in various domains, including engineering, finance, and biology. In engineering, SGA has been applied to optimize the design of complex systems, such as aircraft wings and automotive engines. In finance, SGA has been used to develop trading strategies that maximize profitability while minimizing risk. In biology, SGA has been applied to study the genetic basis of complex traits, such as disease susceptibility and drug response. One of the major advantages of SGA is its ability to efficiently investigate the solution space while converging toward the best optimal solution. The adaptive mutation rate allows the algorithm to maintain a balance between the survey and utilization of the search space, which is crucial for solving complex optimization problems. Additionally, SGA can handle highdimensional optimization problems, which are difficult to solve using traditional optimization techniques.

3.8 Differential Evolution Genetic Algorithm Differential Evolution Genetic Algorithm (DEGA) is a hybrid optimization algorithm that combines two different evolutionary [26] algorithms: Differential Evolution (DE) and Genetic Algorithm (GA). The DE algorithm was first proposed by Price and Storn in 1995. The GA algorithm was first proposed by John Holland

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in the 1960s. The DEGA algorithm depends on the principles of natural selection, mutation, and recombination, and it is used to solve complex optimization problems. The DE algorithm is a type of stochastic optimization algorithm that applies a population-based technique to find the optimal solution. In DE, potential candidate solutions are generated randomly, and then a process of mutation and crossover is applied to generate new potential solutions. The selection of the best candidates in the population is based on their fitness value, which evaluates the quality of each solution. The DEGA algorithm was first proposed by Rainer Storn in 1996. The DEGA algorithm combines the advantages of both DE and GA [27] algorithms. The DE algorithm has the advantage of being very efficient in exploring the solution space, while the GA algorithm [28, 29] has the advantage of being very effective in converging to the optimal solution. The goal of differential evolution (DE) is to use alternative differential operators in place of the genetic algorithm’s (GA) traditional crossover and mutation techniques. In the community of machine intelligence and cybernetics, the DE algorithm has lately gained a lot of popularity. It frequently outperforms GA [30, 31] or particle swarm optimization (PSO). The variation process, which permits investigating various areas of the solution space, and the selection process, which ensures utilization of the learned information about the fitness landscape, are the two primary processes that operate the evolution of a DE population, as they do in other evolutionary algorithms. The issue of early convergence to a particular unsatisfactory area of the search space affects DE. Additionally, the performance of classical DE degrades as the size of the search space rises, similar to other stochastic optimization strategies. In the DEGA algorithm, the mutation and crossover operations are employed to obtain new potential individuals, while the selection of the best individuals depends on the fitness function. The mutation operation is used to introduce random changes in the population, while the crossover operation is applied to combine the genetic material of two individuals to create a new individual. The DEGA algorithm has been successfully employed for an extensive range of optimization problems in various fields, including engineering, finance, and medicine. For example, the DEGA algorithm has been used to optimize the design of a wind turbine, optimize the allocation of financial assets, and optimize the treatment of cancer patients. In conclusion, Steady-State Genetic Algorithm [32, 33] is a powerful optimization technique. Its ability to converge quickly while maintaining a diverse set of solutions has made it a worthwhile tool for resolving complex optimization problems. SSGA is a useful technique in problems where the solution space is complex and there are many local optima.

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4 Applications The application of Genetic Algorithms spans various domains. It is not specific to a particular field and can cross boundaries. There has been a major trend and inflow of machine learning algorithms in recent times, and it can be analyzed that most of the algorithms are derivations [34, 35] of the Genetic Algorithm. In machine learning, genetic algorithms are mostly adaptive heuristics [36, 37] or search engine algorithms that offer answers for search and optimization issues. It is a strategy that uses natural selection to tackle both confined and unconstrained optimization issues. Complex problems that would have taken considerably longer to address manually may now be resolved fast. By taking into account all the limitations, they are one search optimization procedure that aids in locating the optimal solution. In contrast to previous algorithms, it also uses directed random search. The best answer is found by starting the process with a random initial cost function and then looking for the one with the lowest cost in the available space. When looking for answers in vast and complicated databases, it is helpful. Genetic Algorithms can be implemented while solving some of the classic problems like the traveling salesman problem. One of the practical combinatorial optimization issues that were resolved by genetic optimization was the traveling salesman problem (TSP). With the distance between two places and the routes that the salesperson must take into consideration, it aids in locating the best route on a given map. Several iterations produce child solutions that share the characteristics of parent solutions after each iteration. By doing this, we avoid receiving the answer just once and have the option to select the ideal route structure. It is used in real-time operations including production, shipping, and planning. Likewise, some of the other optimization problems include Job-shop Scheduling Problems and Vehicle Routing Problems. The vehicle routing problem is one of the generalizations of the aforementioned traveling salesperson problem. Finding the ideal weight of items to be transported via the ideal combination of delivery routes is assisted by genetic algorithms. If there are any limits, factors like depot points, wait times, and distance are taken into account when solving these tasks. Additionally, compared to simulated annealing techniques and tabu search, the genetic algorithm approach is competitive in terms of solution quality and speed. Genetic Algorithm’s application is not only limited to computer science or mathematics specifically. It is majorly used in the mechanical engineering domain as well for the optimization of designs in various models [4]. Mechanical engineers design CAD models on various software [37, 38] which need to be analyzed in order to prevent any design flaws or chances of failure. Some of the analyses done are structural in the case of solids or metallic structures, and aerodynamics like gaseous or fluids. Genetic algorithms analyze the designs based on the various parameters and figure out the area of maximum or minimum stress, drag, etc. Genetic algorithms optimization fitness function is adaptable to factors that arise from demands for a certain design.

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There are several cost function examples in the industrial sector [26, 39]. Finding the best choice of parameters for these functions presents a challenge based on the same. This task of employing the optimized [40] set of parameters to reduce the cost function is carried out using genetic optimization. It may be used to create the best production plans for products by taking into account dynamic factors like capacity, stocks, or material quality [41, 42]. Hence, every industry is implementing such algorithms based on their needs and demands in order to channel and streamline their entire manufacturing process of goods as well as cut costs as much as possible. Many imaging, diagnostic, monitoring, and therapeutic equipment in medicine is powered by electronic chips and computers. Software, which is based on algorithms, manages and controls these devices, which are made up of various hardware components. Radiologists must quickly examine and understand a significant volume of data produced by imaging modalities used in radiology [27, 43]. Computer-aided detection and diagnosis is a field of study that is constantly expanding. Its goal is to help radiologists analyze images more quickly and accurately by identifying, classifying, and segmenting the normal and abnormal patterns seen in various imaging modalities. These include ultrasound, computed tomography (CT) scans, X-rays, and magnetic resonance imaging (MRI). Consequently, edge detection becomes one of the essential components of automatic image processing methods [28, 44, 45]. For edge identification of pictures obtained using various imaging modalities, such as MRI, CT, and ultrasound, several studies have exploited the GAs. Feature selection in machine learning refers to the process of choosing a subset of pertinent characteristics to build a model by eliminating variables with little to no analytical significance. The selection of features is crucial since doing otherwise would make computation more difficult, expensive, and time-consuming while also decreasing the model’s accuracy. In addition, fewer features would make the model easier to understand and generalize while avoiding the issue of overfitting and the risk of failure due to missing data. In research to determine if a region of interest in mammograms is normal or has a mass, as well as to distinguish benign from malignant breast cancers in ultrasound images, GAs have been used for feature selection. One of the key advantages of GA [46] is that it can find solutions that may not be immediately apparent through other methods [23]. This is because GA uses a randomized approach to explore the solution space, which allows it to overcome local optima and find globally optimal solutions. GA is also highly flexible and can be used in a wide range of applications, including optimization, classification, clustering, and data mining. The application of GA in machine learning has led to significant advancements in the field. One of the most notable examples is the development of deep neural networks [24]. Deep learning algorithms are highly complex and require a large number of parameters to be optimized. GA provides an efficient way to search for the optimal combination of parameters, allowing deep neural networks to attain state-of-the-art performance in an extensive range of applications, comprising image recognition, natural language processing, and speech recognition [25]. Another area where GA has been highly successful is feature selection. Feature selection is the method of recognizing the most relevant features in a dataset for a given task. This is important because it can help to minimize the dimensionality of the problem and

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improve the accuracy of the model. GA can be used to identify the most important features by evaluating the fitness of different subsets of features. This approach is highly effective in a range of applications, including cancer diagnosis, credit scoring, and fraud detection. One of the most promising areas of research is in the development of multiobjective optimization techniques. Multi-objective optimization is the method of obtaining the optimal solution to a problem that involves multiple conflicting objectives. For example, in a classification problem, we may want to reduce the false positive rate while increasing the true positive rate. GA can be used to obtain a set of solutions that represents a trade-off between these objectives, known as the Pareto front. This approach is proven to be highly effective in a range of applications, including portfolio optimization, supply chain management, and energy management. Another area where GA is being used is in the development of hybrid algorithms. Hybrid algorithms combine different optimization techniques to overcome the limitations of each individual method. For example, GA can be combined with gradient descent to improve the rate of convergence of the algorithm. This approach is highly effective in a range of applications, including parameter tuning, optimization of black-box functions, and optimization of dynamic systems. In conclusion, Genetic Algorithm is a salient tool in the field of machine learning. Its potential to search the solution space in a randomized manner and find globally optimal solutions has led to significant advancements in a broad range of applications. Recent advancements in multi-objective optimization and hybrid algorithms have further expanded the scope of GA and made it an even more powerful tool for solving complex optimization problems.

5 Future Research Directions Engineering, health, and finance are just a few of the industries where genetic algorithms have been employed to address issues. They have several benefits over conventional algorithms, including the capacity to handle huge and difficult issues, the capacity to solve readily stated problems, and the capacity to discover several solutions to a single problem. The basics of GA are rooted deep in biology and hence it is easy to understand the concepts related to the algorithm. Due to its great degree of flexibility, GA may be used to solve issues that are challenging to define in terms of conventional mathematical concepts, such as gradients. Genetic Algorithm is also faster than other optimization algorithms based on the sheer speed of calculation. The inherent ability of GA is parallel and distributed computing which makes it faster than other optimizations. Various mechanical simulation software employs GA for this very reason. When the design gets complex or needs to be analyzed based on various real-world parameters and conditions, computational speed becomes a really important factor, faster software gets a competitive market edge over the others. GA undergoes a non-knowledge-based optimization process using the fitness function in order to trace out the best and most useful solution space. All these advantages combined,

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GA successfully calculates the global optimum without being falsely trapped within the local maxima which is a disadvantage for various problems planning on solving large-scale problems. The method’s drawback is that it relies on random sampling, which necessitates the definition of effective crossover and mutation processes in order to achieve excellent efficiency. If the GA’s operations are flawed, it will either be no better than a purely random method or so constrained that it will take a long time to find duplicate answers. If there is no background in stochastic sampling, it may not always be clear why good GA is difficult to learn. Tuning a GA can take a fair amount of time. It relies heavily on the crossover operation to avoid the local maxima problem which may result in more iterations and more time complexity in some cases. Hence even though the algorithm has been in existence for quite a while now and major research work has already been incorporated in order to make it near-perfect, some work is still required to optimize the crossover and mutation operation and apply it to complex problems and models.

6 Conclusions Even though the world has advanced and the times have changed, with new algorithms coming up every second, Genetic Algorithm (GA) provides the stepping stone for the advent of all such modern technologies. It is still superior in terms of performance in comparison to other modern techniques and its applications transcend domains and boundaries. It is applied in mechanical engineering, computer science engineering, mathematics, and any possible area which requires the optimization of a variable. The future prospects for genetic algorithms are bright, as they continue to gain popularity in various fields of study. Advancements in computing technology have made it possible to apply genetic algorithms to increasingly complex problems, leading to new and innovative solutions. In the field of machine learning, genetic algorithms are being used to develop more sophisticated models and improve the accuracy of predictions. In addition, genetic algorithms are being used to optimize systems in various fields, such as transportation, logistics, and finance. As more data becomes available, genetic algorithms will become even more effective in solving problems and providing previously unattainable insights. The potential for genetic algorithms is vast, and with continued research and development, they are likely to remain a valuable tool for solving complex problems in various fields of study. The advantage of GA overweighs its disadvantage while all the variants provide their own use cases. Research has been still going on for the development of GA and making it readily usable for various other domains which currently lack the facility.

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Chapter 2

Genetic Algorithm Applications for Challenging Real-World Problems: Some Recent Advances and Future Trends Andrés Iglesias and Akemi Gálvez

1 Introduction 1.1 Motivation Genetic algorithms (GA) are a class of metaheuristic methods based on the principles of natural evolution. Since their introduction from the work of J. Holland in the 70s, GA have been at the core of the evolutionary techniques and the most general field of metaheuristics. GA are one of the first and most popular methods in the field of evolutionary computing. Furthermore, they have traditionally been considered the most genuine representatives of this area of research. GA are very general and broad-spectrum methods, capable of solving problems in a wide variety of scientific domains. They are also relatively easy to understand and implement. Owing to these reasons, they have been the first point of contact with evolutionary computation and metaheuristics for many researchers and practitioners. Aiming at this type of readers, this chapter will offer a gentle description of the fundamentals and background of natural evolution and genetic algorithms in Sect. 2. Genetic algorithms have been traditionally used in a wide range of applications across many fields, including engineering, computer science, economics, and biology. Some classical applications of genetic algorithms are

A. Iglesias (B) · A. Gálvez Department of Applied Mathematics and Computational Sciences, E.T.S.I. Caminos, Canales y Puertos, University of Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain e-mail: [email protected] A. Gálvez e-mail: [email protected] Faculty of Pharmaceutical Sciences, Toho University, 2-2-1 Miyama, 274-8510 Funabashi, Japan © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_2

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• Optimization problems: Genetic algorithms are typically used to solve complex optimization problems, such as finding the optimal solution to a mathematical function or optimizing the design or functioning of a complex system [33, 34]. • Financial trading: Genetic algorithms are being used in the field of financial trading to optimize investment strategies. This involves using genetic algorithms to evolve trading strategies that are better suited to current market conditions. This trend, sparked by the massive amount of financial data and the impressive computational power currently available, has expanded to include forecasting, trading, cash-flow and portfolio management, option pricing, arbitrage, and many other tasks [3, 16, 25, 71]. • Game theory: Genetic algorithms can be used to model and analyze complex board games, such as chess, poker, checkers, shogi, or go. • Image and signal processing: Genetic algorithms can be used to optimize image and signal processing algorithms, such as edge detection, noise reduction, or image segmentation. • Bioinformatics: Genetic algorithms can be used to analyze and interpret biological data, such as gene expression profiles or protein structures. In addition to these rather standard applications, new exciting examples of the application of genetic algorithms are constantly appearing in the literature, making it difficult to follow some of the most recent developments in the field. This chapter aims at lessening this problem by describing some of the most exciting recent advances regarding the application of GA to challenging real-world problems in nonstandard fields.

1.2 Main Contributions Main contributions of this chapter are • Firstly, we provide our readers with a gentle description of the fundamentals and background of natural evolution, evolutionary computing, and genetic algorithms in Sect. 2. This description can be useful for pre-graduate and post-graduate students and junior researchers, as well as scholars and practitioners interested in the field. • Secondly, we describe some of the most exciting recent advances regarding the application of GA to challenging real-world problems in nonstandard fields. • Finally, we outline some of the future trends regarding the applications of genetic algorithms in other potential fields along with some current challenges for future work in the field.

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1.3 Structure of This Chapter The structure of this chapter is as follows: it begins with a gentle description of the fundamentals and background of natural evolution, evolutionary computing, and genetic algorithms in Sect. 2. Then, we describe some of the most exciting GA-based approaches reported in the literature in the fields of reverse engineering for manufacturing (Sect. 3.1), medicine and bioinformatics (Sect. 3.2), computer animation and video games (Sect. 3.3), and robotics (Sect. 3.4). These sections discuss how genetic algorithms can be applied to address these problems, including some variants and/or hybridizations with other approaches for better performance. Then, Sect. 4 discusses some future trends of genetic algorithms in other potential fields and outlines some potential challenges and developments of GA for the foreseeable future. The chapter closes with a brief discussion in Sect. 5 and the main conclusions of this work given in Sect. 6.

2 Genetic Algorithms: Fundamentals and Background 2.1 Natural Evolution and Evolutionary Computing Genetic algorithms are classical methods of a larger class of computational techniques globally known as evolutionary computing (EC). The trademark of this class is that all evolutionary computing methods take their inspiration from the processes and phenomena commonly observed in natural evolution. In the nineteenth century, scientists and scholars realized about the power of evolution as they noticed how species adapted to different ecological niches by developing specialized physiological modifications and/or behavioral strategies particularly tailored to improve their chances to survive and reproduce. These observations led to a general framework that is common to the natural evolution and all EC methods. Everything starts with a given environment, populated by individuals of a certain species striving for survival and reproduction. Each individual possesses a different value of fitness, a measure of their adaptation ability to the environmental conditions and be successful in reaching its goals. Note that the fitness is by no means an absolute value associated solely with the individual (i.e., it does not talk about how good an individual is), but strongly related to the environment, talking about how well an individual adapts to that environment. Therefore, the fitness is always relative to an environment: individuals highly adapted to one environment may be poorly adapted to another. Under this paradigm, it is not that an individual modifies its morphology to get better adapted to the environment, but the opposite. It is the environment itself which is responsible to select the individuals better adapted or fit to the environmental conditions. At this point, the Darwinian theory of evolution, developed by Darwin in his highly influential book in [32], offers a compelling explanation of the sources of the extraordinary biological diversity in nature. According to Darwin, populations

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evolve over generations through the process of natural selection, which favors the individuals with best adaptation to the environment. This competitive nature of the evolution can be explained by the fact that natural environments offer a limited amount of food and other resources, and hence can only host a limited number of individuals, which are therefore forced to compete for the resources and their own survival. The winners of this competition on the long term are the individuals better adapted to the environment, a process known as survival of the fittest. This term must be taken with care, because it does not mean that the best adapted individual always survives and reproduces. Randomness plays a decisive role in natural life, so it may happen that the fittest individual at a particular instance does not survive. Adaptation is never a guarantee of survival or success, so neither is fitness, but, on average, the fittest individuals have more chances than the others to do so. Natural selection is a fundamental principle of natural evolution but it alone cannot explain how changes occur in species. There is indeed another primary force for evolution, given by the variations among the members of a population, particularly at the phenotypic level. By phenotype we understand the set of observable and measurable features or characteristics of an organism, which can influence the individual’s response to the environment, and therefore their fitness. Each individual has a unique set of phenotypic traits that elicit a different response from the environment. If such a response is more favorable (i.e., the phenotypic traits provide the individual some kind of advantage with respect to others), the individual has more chances to survive and reproduce. In this way, favorable traits can be transmitted to the individual’s offspring (provided that such traits are heritable). Based on his own observations, Darwin claimed that favorable traits can arise by small random variations of existing traits by reproduction. An important observation is that this mechanism, called mutation, is not finalist, but driven by stochasticity. When new generations are generated by reproduction, new combinations of phenotypic traits can also be generated randomly. If they evaluate favorably, they survive and are passed on to their offspring. Repetitions of small changes over a large number of generations can lead to drastic changes in the phenotype of species, where the most favorable traits are preserved along the generations, as long as their environment does not change significantly. Phenotypic changes occur at a macroscopic level, but there are also changes at a microscopic level. They occur in the genotype, the set of genetic material present in all living organisms. This genetic material can be found in the form of genes and chromosomes. They are the functional units of inheritance of the phenotypic traits, which are encoded as chemical information. Hence, living organisms have a dual representation, given by the phenotype (observable features) and the genotype (the genetic material encoding the phenotype). The recombination (combination of genetic material of paternal and maternal features in offspring) is realized through reproduction, a process usually called crossover in EC. Interestingly, all variations in the phenotype are carried out either through mutations or combinations at the genotypic level. This contrasts with the natural selection, which occurs at the phenotypic level exclusively. Note, however, that the selection mechanism also affects the genotype, but implicitly rather than directly.

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Algorithm 1 Pseudocode scheme of a general evolutionary computing algorithm Require: Population of N individuals Fitness function F Termination condition T C BEGIN INITIALIZE population ⊿ Using random individuals as candidate solutions EVALUATION of each individual ⊿ Using the fitness function F repeat 1 SELECTION parents 2 CROSSOVER pairs of parents 3 MUTATION of resulting offspring 4 EVALUATION of resulting offspring ⊿ Using the fitness function F 5 SELECTION of the individuals for next generation until (T C is fulfilled) END

Once the underlying mechanisms of natural evolution were unraveled, they served as inspiration for artificial problem-solving, including optimization problems. The idea behind is to use the analogy between both fields, according to which the environment is identified with the problem to be solved, while the population is made up of individuals representing the candidate solutions of the problem, and the fitness function now measures the quality of these solutions. With all these ingredients, a typical evolutionary computing algorithm follows the general scheme shown in Algorithm 1. Note that the evolution is driven by two main mechanisms: on the one hand, two operators (mutation and crossover) are applied to increase the variability of the population, expanding the pool of individuals, and hence the generation of novel (and potentially better) candidate solutions. On the other hand, the selection operator increases the average quality of the population by preserving the best solutions for the next generations (usually but not necessarily through an elitist strategy) while the worst ones are removed from the population being replaced by the modified offspring. Note also that the operators do not work in a fully deterministic way; on the contrary, they typically depend on some random parameters, an additional procedure to increase the population diversity.

2.2 Genetic Algorithms Genetic algorithms are a type of evolutionary computing methods, arguably the most popular ones. The sources of GA can be traced back in the 70s, where John Holland conceived them as a way to study adaptive behavior in both natural and artificial systems [57]. This may sound surprising to many, as GA have been largely regarded as a formalism for optimization problems, owing to some early successful examples of application in this area, along with the popularity of the book in [50] during the 90s. In fact, GA have become one of the classical approaches for optimization, and they

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are often used for comparative purposes with new surging metaheuristic algorithms for optimization. Amazingly, the classical genetic algorithm approach was quite limited in comparison with many later variations described in the literature. Being an evolutionary method, all these GA variations follow the general EC scheme in Algorithm 1, but differ on the particular choices for the evolutionary operators and other features. For instance, the original GA (usually referred to as the canonical GA) uses a binary/bit string representation for the genome, where the individuals are represented as strings of 0s and 1s. Also, it uses a fitness proportional selection through the mechanism of roulette wheel, where the individual i with fitness f i is selected with probability fi pi = Σ N , one point crossover (where a random point of the genome is chosen j fj and the genome of the parents is switched at this point to yield two offsprings; this crossover is applied randomly with a given probability pc ), and uniform mutation in the genome, where all bits have the same chance of being flipped, given by a mutation probability pm , typically having low values. Also, the survival selection is performed in a generational fashion where the intermediary population obtained by mutation and crossover replaces the previous one entirely. Note that individuals of the previous generation can be preserved for the new generation without any modification, although the likelihood of this to happen is usually quite low (note that it depends on the values of pc and pm ). Note that the canonical GA does not incorporate elitism. Genetic algorithms have shown a remarkably good performance for many practical problems. However, it has also been noticed that the original GA might show some important drawbacks for complex optimization problems. In some instances, GA show a very slow convergence to the solutions. They can also get stuck in a local optimum while failing to reach the global one. In fact, a limiting factor of GA is that convergence to the global optima is not guaranteed. Considerable attention has been put on improving the performance of GA, and this canonical GA has been further extended in many different ways, leading to more advanced GA approaches. For instance, their genome representation has been generalized to floating point numbers, and different types of mutation, crossover, and selection procedures as well as elitist strategies have been included in the algorithm. Other features such as self-adaptation have also been added for better performance. Perhaps the most remarkable extensions are the multi-objective GA, specially tailored to deal with multiple (conflicting) objective functions, parallel GA, particularly aimed at decreasing the computational times and enhance the exploratory capabilities of standard GA, and other modifications aimed at preventing the population from premature convergence, where the method gets typically stuck too early in some local optimum. Strategies such as niching methods, fitness sharing, penalty methods, and others have been described in the literature to address these issues. A detailed analysis of these variations and modifications is out of the scope of this chapter and is not discussed here. Any interested reader is referred, for instance, to [19, 53, 86] for further details about the basic GA, its variations and extensions and implementation aspects.

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3 Genetic Algorithms for Real-World Problems: Recent Advances 3.1 Reverse Engineering for Manufacturing As a consequence of the current globalization process, manufacturers are urged to optimize their processes to obtain better and more competitive products at lower costs. In this process, design is gaining a central role in the product development life cycle. This process has been exacerbated by the current trend of mass customization, in which the customers are able to add differential, personalized features to the manufactured goods. As a result, industrial goods are produced in lesser amounts but with greater diversity of available designs and products. The CAD/CAM (computer-aided design/manufacturing) has become essential tools for many design and manufacturing processes [14, 40]. Nowadays, it is common to build different prototypes of products using either soft materials (foam, clay, plastics) or more rigid materials (wood, metal) as a key step of the initial conceptual design process. Such prototypes are used for interrogation about the visual aspect, shape, size, proportions, weight, and interaction with the model. Once the prototype is physically built, it is digitally stored through 3D scanning and other digitizing technologies. The resulting point clouds are fitted through mathematical curves and surfaces, thus yielding a digital model of the physical object which is easier to manipulate and cheaper to store than the cloud of points. Also, this digital model can be used even when the original object gets lost, damaged, or unavailable. This process of moving from the real object to its digital model is called reverse engineering and it is commonly used today in many manufacturing fields. This reverse engineering approach is favored by the increasing popularity of user-friendly computer-aided design systems along with affordable devices for additive layer manufacturing (3D printing) and other rapid prototyping techniques. Relevant examples of application of reverse engineering appear in medical imaging, where a dense point cloud of the outer surface of a volumetric object (e.g., an inner organ or a tumor) can be obtained by using non-invasive methods such as ultrasound imaging, magnetic resonance imaging (MRI), and so on. The main goal is to generate an accurate volumetric representation of the organ or tumor to be subsequently used for diagnosis and other medical purposes. Other examples from the medical field arise in biomedical engineering, where point clouds are obtained from patients through 3D scanning to obtain a digital model for manufacturing customized medical implants, such as orthopedic implants to replace a missing joint or bone. Other examples arise in the automotive industry (design and manufacturing of car bodies), aerospace industry (parts of plane fuselage, such as wings, rudders, spoilers, flaps, airfoils, and ailerons), shipbuilding industry (parts of ship hulls, such as bows, quarters, bilges, decks, keels, topsides, and waterlines), footwear industry (lasts, molds), home appliances, and many others. Further examples can be found in manufacturing (determination of trajectories for tool-path generation in computer numerically controlled (CNC) operations such as machining and milling and other

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manufacturing-related processes such as filing, turning, grinding, and pocketing); rapid prototyping and manufacturing, metrology and product quality assessment, archaeology, digital heritage (digital representation, storage, and analysis of historical and cultural remains and artifacts using handheld laser scanners and portable coordinate-measuring machines, total stations, and photogrammetry techniques to capture the geometry of historical buildings); and LIDAR technology combined with digital elevation models for ancient settlements, archaeological sites, and natural landscapes to mention a few. The problem of data fitting though curves/surfaces (also called curve/surface reconstruction) has received a lot of attention in the last few decades. However, in spite of intensive research work in the field, it is also a challenging and elusive problem. A big reason to explain it is the fact that the input data are typically affected by irregular sampling, some level of noise, and other artifacts (holes, gaps in data, etc.). As a consequence, the mathematical entities for data fitting are not necessarily differentiable (or even continuous), so the classical mathematical optimization techniques cannot be applied here, or produce unsatisfactory results. Due to these limitations, the scientific community in the field has shifted their attention to artificial intelligence and evolutionary techniques, such as the genetic algorithms. This approach makes full sense, as these methods are very general, they do not require strong knowledge about the problem at hand, and can be adapted to varying optimization scenarios with little (if any) modifications.

3.1.1

Curve Reconstruction

Given a set of noisy data points {Q j } j=1,...,m in either R2 or R3 , lying on an unknown ˆ curve C(t), the problem of curve reconstruction is to obtain a parametric curve C(t) ˆ j ), ∀ j = 1, . . . , m. approximating C(t) at the given points, i.e., Q j (= C(t j )) ≈ C(t Note that it can be written as an optimization problem by considering the fitness function given by m ( )2 Σ ˆ j) (1) Q j − C(t E= j=1

Of course, the goal is to minimize this fitness function. To do it, we can assume that ˆ can be formulated as a combination of functions: the approximating function C(t) ˆ = C(t)

n Σ

Ci φi (t)

(2)

i=0

where {Ci }i=0,...,n are 2D or 3D coefficients usually called poles and the functions {φi (t)}i=0,...,n (usually called blending functions) can be assumed to be linearly independent functions in the vector space of polynomial functions of degree ≤ n, which has dimension n + 1. Functions {φi (t)}i can be assumed to be valued on a finite

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interval [α, β], the same for all blending functions. Without losing generality, we can assume that [α, β] = [0, 1]. In this section all vectors are represented in bold. There are several options to be chosen as blending functions. They can be roughly classified into two groups: global-support and local-support functions. In former case, the support of blending functions is the domain [0, 1]. From a practical standpoint, this implies that these functions provide a global control of the shape of the fitting curve. Some examples of global-support functions are the polynomial basis functions ( ) n n t i (1 − t)n−i {φi (t)} = {t i }, the Bernstein polynomials {Bin (t)} where Bi (t) = i ( ) n n! and , the trigonometric basis, Hermite polynomial basis, radial = i! (n − i )! i basis, hyperbolic basis, and polyharmonic basis. In the latter case, the support of blending functions is a subset of the whole domain, and is typically expressed as piece-wise functions. Examples of functions with local support are the piece-wise B-spline basis, piece-wise linear, quadratic, sinusoidal basis, radial, Lagrange, and Harmut basis, among others. Minimization of the fitness function E is usually a nonlinear problem since Eq. (1) generally involves blending functions that are nonlinear in their parameters. An additional problem is how to obtain a proper parameterization of data points, i.e., to determine suitable values of the parameters t j associated with data points Q j . In general, it can be proved that when the parameterization is given, the minimization of E is a standard least-squares problem. However, if no parameterization is provided, the problem becomes a continuous nonlinear optimization problem [35, 40, 60].

3.1.2

Surface Reconstruction

Similar to the previous paragraphs, given a set of sample points X lying on an unknown surface V , the problem of surface reconstruction consists of constructing a surface S approximating V . This problem is often addressed through a least-squares scheme. The most common approach is to assume that S has a given functional structure depending on parameters to be obtained. To that purpose, we assume that S is a free-form parametric surface S(u, v) given by S(u, v) =

M Σ N Σ

Ci, j φi (u)ψ j (v)

(3)

i=0 j=0

where φi (u) and ψ j (v) are univariate blending functions, u and v are the surface parameters assumed to take values on the intervals [αl , βl ], for l = u, v, respectively; and coefficients Ci, j are the poles. Given the set {Qk }k=1,...,n k of 3D data points, we can compute, for each component of (xk , yk , z k ) of Qk , the minimization of the sum of squared errors with respect to the data:

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Errμ =

nk Σ k=1

⎛ ⎝μk −

M Σ N Σ

⎞2 Ciμj φi (u k )ψ j (vk )⎠ ;

μ = x, y, z

(4)

i=0 j=0 y

Coefficients Ci j = (Cixj , Ci j , Cizj ) are to be determined from the data points Qk = (xk , yk , z k ). Note that the component-wise minimization of such errors is equivalent to the minimization of the sum of the distances between data points and their associated points from S in R3 . Note that along with the coefficients, Ci j , the parameter values (u k , vk ) of the data points are also unknowns in our formulation. The functions φi (u), ψ j (v) are nonlinear, and so is the least-squares error minimization [129], involving a high number of unknowns for large data sets, a rather common situation in practice.

3.1.3

Genetic Algorithms for Curve and Surface Reconstruction

Intensive research done during the last few decades has shown that AI techniques can yield good results for the curve reconstruction problem [6, 31, 36, 39, 42, 56, 61, 63, 73]. But although these AI-based techniques represented a significant advance in the field, they still fail to provide a good solution to general problem. Recent results described in the literature show that metaheuristic techniques are very good tools to tackle this issue. Some of the first methods in the field with genetic algorithms formulated the original problem as a discrete one [109]. Yoshimoto and co-workers published a paper in 1999 describing the application of a discrete version of GA to the problem of computing the knots of B-spline curves [135]. However, these early works were affected by the discretization errors derived from the conversion process. The work in [136] overcomes this limitation through a real-code GA. A combination of GA and functional networks reporting good results can be found in [45]. A paper on the design of genetic shapes can be found in [125]. GA for free-form shape design are studied in [84] and for 2D shape optimization in [66]. A GA-based technique for curves and surfaces in the context of pattern recognition is described in [132]. However, none of these methods can handle features like discontinuities and cusps. The ultimate reason is that it is almost impossible to obtain identical values by using purely stochastic approaches. This problem was addressed in [43] by using a hybridization of genetic algorithms and particle swarm optimization. See also [29] for a general overview of the problem for both global-support curves and surfaces by using GA and particle swarm optimization. The work in [85] applies a GA to compute the parameter values of data points as knots of a cubic spline interpolating curve. GA have also been applied to the sub-problem of parameter optimization of B-spline curves in [72]. A hierarchical generic algorithm approach was applied to the problem of curve reconstruction with B-splines in [46]. An improved GA was used in [98] to compute the knots for B-

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spline contours from scattered data. A genetic algorithm approach was applied in [17] to generate a ductile cast iron microscale model using Bézier curves. Multi-objective GA have been applied in [79] to obtain a polygonal approximation of digital curves. This problem has also been addressed in [134], where three polygonal fitting approaches using GA were proposed: one to fit digital curves by minimization of the number of sides of a polygon with a fitting error less than a prescribed threshold, a second one minimizing the fitting error by looking for a polygon with a given number of sides, and a third one determining the fitting polygon without any condition automatically. Other examples of polygonal approximations with genetic algorithms can be found in [59]. Other multi-objective genetic algorithm approach for curve approximation can be found in [105]. In [54], multi-objective genetic algorithms are applied to shape optimization of plane airfoils. In [10], GA have been applied to airfoil shape optimization using Bézier curves. Parallel hierarchical GA have been applied to scattered data fitting using B-splines in [75]. There are fewer papers regarding surface reconstruction with genetic algorithms than those for curve reconstruction. Two early proposals are those in [69, 127], where variations of GA in the form of genetic programming and genetic evolution were used to address the problem. The work in [128] combines non-uniform rational B-splines with solid modeling using a hybrid GA scheme. The method is easy to understand and implement but fails to perform data point parameterization. As a result, it is limited to simple examples where a flat base surface can be used. Other method closely related to this approach is that in [18]. Another method is given in [123], where an MOEA (multi-objective evolutionary algorithm) is used to construct a smooth surface for several data sets. This approach is constrained in several regards: the method assumes a cubic degree and that the weights are always 1, so the surface is actually a simpler cubic non-rational B-spline. Furthermore, it assumes that the sampled points are uniformly spaced, thus preventing scattered data points from being reconstructed. Also, data points need to be projected on to a basis surface for evaluation within the MOEA. In [108], a simulated evolution method called SimE is used to fit data with NURBS. This technique is quite limited, because it does compute neither the knots nor the data point parameters. Moreover, it assumes that the order is equal to the number of control points, a very strong (and unrealistic in many cases) restriction aimed at computing easily the control points by direct application of the least-squares method. A GA-based method for obtaining quadric surfaces fitting a collection of data points is described in [26]. The work in [27] addresses the generation of an optimized tessellation from a point set in the context of reverse engineering with potential application to surface reconstruction. The method in [49] also obtains a polygonal mesh rather than a mathematical surface. In that paper, GA-based algorithms (in particular, evolutionary search with 1 + λ and soft selection) are applied to recover the shape of several shapes (a sphere, a fractal surface, and a head). However, the method is unable to recover examples for which a proper triangulation cannot be easily obtained. More recently, an example of the application of PSO to the reconstruction of Bézier surfaces is given in [41]. The challenging case of surface reconstruction with localsupport surfaces is addressed in [44] by using an iterative two-step genetic algorithm

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method to compute the data parameterization and the surface fitting, respectively. Finally, some other applications of genetic algorithms for computer-aided design (not necessarily to curve/surface reconstruction) can also be found in [101].

3.2 Medicine and Bioinformatics Medical sciences and bioinformatics are two fields where artificial intelligence has been applied more intensively in recent years. The extraordinary advances in medicine occurred in recent decades have pushed the limits of science toward new challenges that require the resolution of complex problems involving a massive amount of data, making computers essential tools for data management and analysis. As a consequence, fields such as artificial intelligence, machine learning, and data analytics have become absolutely essential for the progress of medical sciences, and new interdisciplinary fields, such as bioinformatics, at the crossroads of the domains of artificial intelligence, computational biology, machine learning, data mining, image processing, computer simulation, statistics, and others, are receiving attention in last years. In this context, this section explores some of the recent GA-based developments in these fields. Obviously, this description is by no means exhaustive. Doing so would probably require a whole book and even several volumes. Therefore, we will restrict our discussion here to some particular problems, and a quite limited number of references to keep the section at reasonable size. The interested reader is kindly suggested to see the table of contents of this volume for a larger description and many additional references about these topics in other chapters of this book.

3.2.1

Genetic Algorithms for Cancer Prediction, Detection, and Diagnosis

Cancer is a serious health problem as the second global cause of death. According to the World Cancer Research Fund International, cancer affected in 2020 to 18.1 million worldwide. The most common cancers in the world are breast and lung, which add up to 25% of the total. Currently, cancer is one of the most critical global health problems of the twenty-first century. Therefore, the development of efficient and reliable methods, tools and procedures for cancer prediction, and diagnosis and treatment is of capital importance. In this section, we describe some of the recent GA-based developments for these tasks. Genetic algorithms have been applied to cancer detection, prediction, and diagnosis for decades. In [91], a fuzzy-GA approach for diagnosis of breast cancer is introduced. The method combines evolutionary algorithms and fuzzy logic to generate diagnostic systems in an automatic way. The method exhibits high classification performance while involving a small set of rules, making it human-interpretable. The methods in [28, 120] apply the GA methodology to the diagnosis of breast cancer,

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the most common case of cancer for women worldwide. In most recent cases, GA are hybridized with other methods for better performance. For instance, the work in [93] combines wavelet analysis and GA for the detection of breast cancer from mammograms. The method in [77] combines linear discriminate analysis and GA for the characterization and classification of SERS spectra between patients of bladder cancer and controls. The method in [104] applies hybridization of GA and PSO for detection of breast cancer through microwave tomography. The method in [83] is based on the combination of GA and the k-Nearest Neighbors (k-NN) technique for efficient feature selection to reduce dimensionality in order to enhance a classifier for prognosis of lung cancer patients, the most common case of cancer for men worldwide. The work in [102] combines support vector machines (SVM) and GA for the diagnosis of breast cancer from infrared thermography images. Interested readers are referred to [20] for a recent survey of the existing literature and a comparative analysis of state-of-the-art techniques for cancer detection and prediction using genetic algorithms, along with some future challenges in the field.

3.2.2

Genetic Algorithms for Cancer Classification and Treatment

Genetic algorithms have been used intensively for cancer classification and treatment. A classification of breast cancer tissue through microarrays and GA is proposed in [37]. The combination of GA, SVM, and K-NN was proposed in [51] for feature selection in cancer classification. Also, a combination of GA and the wavelets technique is used in [90] for classification of mass spectrometry cancer data, and in [107] for microarray cancer classification. With the same purpose, a hybridization of GA with learning automata is used in [87]. A two-step hybrid gene selection combining GA and mutual information is proposed in [62] for classification of cancer data. The interplay between GA and neural networks is used in [2] for breast cancer classification from micro-RNA profiles. Other recent GA-based hybridization schemes for cancer classification can be found in [80, 92] using multilayer recursive feature elimination method (MGRFE) and AdaBoost, respectively. A very recent paper applies GA to identify biomarker genes for cancer diagnosis [111]. The GA computes the ideal genes using the Mahalanobis distance. Such genes are inputted into a GA classifier using SVM as a classification basis. Genetic algorithms have also been applied to cancer treatment. An early example is described in [96], where the Gompertz equation is used to model the tumor growth, whose size is to be minimized under the action of some drugs. GA are applied to obtain optimal drug regimes, which are encoded as chromosomes evolved during the GA to derive enhanced treatment strategies for patients. A more recent approach in the same line is described in [99]. In [4], a GA approach is introduced for the design of treatment plans for intensity-driven radiotherapy, where the method determines beam intensities to match the prescribed dose distribution. GA have been applied in [94, 95] to the scheduling of radiotherapy treatment and pre-treatment of cancer patients.

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Genetic Algorithms for the COVID-19

One of the most recent examples of applications of genetic algorithms to the medical sciences happened during the recent pandemics caused by the COVID-19. Soon after the first cases were publicly announced, the scientific community engaged in a massive research effort on the new source agent of the pandemic. Sophisticated artificial intelligence techniques have been applied to track the spread of the disease, help to characterize the new variants of the coronavirus, and develop vaccines and other drugs for its effective treatment. In this regard, the scientific literature offers an ample corpus of research about the use of GA in this process. Its extent is again too large to be included here exhaustively, so we limit ourselves to include some illustrative examples. One of the most important uses of GA for the COVID-19 was model estimation and data analysis. The paper in [7] applied GA to the estimation of epidemic curves of COVID-19 using public COVID-19 data sets to infer mathematical models accounting for the deceased, confirmed, and recovered cases for countries with many cases (e.g., Italy, USA, Spain, China) and on the global scale. The estimated epidemic curves were very close to the tendency of real data. The paper in [1] used GA to analyze the evolution of the epidemiological parameters of COVID-19 from series of data of deceased. Other important field of application of GA for COVID-19 was for prediction, detection, and diagnosis of the disease. The paper in [100] applied a combination of GA with an improved SEIR model for prediction of the epidemic trend in China. The paper in [82] applied GA to optimize convolutional networks for detection of COVID-19 from X-ray images of the chest. In [13], a computational architecture is comprised of three main components, a unsupervised feature extractor based on a convolutional autoencoder, a feature selector based on a multi-objective GA, and a binary classifier based on a set of support vector machines, which were applied for automatic detection of COVID-19 from medical images. In [112], a new algorithm called GABFCov-19 based on GA was introduced for identification of positive cases of the disease. The work in [38] combines GA and a machine learning classifier for the COVID-19 diagnosis from blood samples. The paper in [68] applies a GA-based image reconstruction method for the diagnosis of COVID-19 from X-ray images via digital holography. GA have also been applied to classification of COVID-19. The paper in [15] reports a hybrid intelligent model combining GA and neutrosophic logic for the classification of X-ray images of the chest of patients affected by COVID-19. With this method, a set of rules are generated from three values accounting for the degrees of the triplet (truth, indeterminacy, falsity) and GA are applied to obtain the optimal rules for performance enhancement of the classification system. In [24], GA and convolutional features are used for COVID-19 classification from computer tomography scanned images. Firstly, a CNN is applied for feature extraction from CT images, followed by an optimization of the hyperparameters. Then, GA are applied for feature selection. Other uses and applications of GA for the COVID-19 include the analysis and simulation of the variants of the disease and their survivability in [110], the enhancement

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of the social distancing measures to increase the social immunity against the disease in [52], the development of efficient procedures for the collection and transport of COVID-19 diagnostic samples in [118], a better understanding of the dynamic of the disease using GA and cellular automata in [48], and parametric fitting of COVID-19 SEIR models in [133], to mention just a few examples.

3.3 Computer Animation and Video Games Computer game is a field strongly related to artificial intelligence since the first steps of AI. The reasons are varied. On the one hand, computer games require (at some extent) abilities that are classically regarded as intelligent, such as reasoning, action planning, problem-solving, or decision-making. On the other hand, computer games involve several senses and cognitive skills, ranging from perception, attention, physical and mental coordination, visual understanding, language skills, social and interpersonal skills, and many others. Computer games also offer us the opportunity to recreate or simulate any real or virtual environment, and analyze the player’s reactions and their role on the virtual world or any sequence of actions while being in a safe and controlled environment. Therefore, it is understandable why computer games have been traditional benchmarks for new AI techniques.

3.3.1

Behavioral Animation of NPCs in Video Games

At the dawn of artificial intelligence, some AI researchers focused their attention on bard games, such as checkers, chess, poker, bridge, and others [55]. Later on, the attention shifted toward computer games [64]. At early times, videos games were mostly a competition between human players, followed soon by playing against the machine. Typically, a human player was confronted against one or several computercontrolled enemies, called bots or NPCs (non-player characters). Actions from these NPCs are not determined by any human player but the computer, so they need some kind of artificial intelligence (AI) dictating how they behave. For a long time, the AI of NPCs was based on scripts, thus yielding simple and repetitive behavioral patterns. Evolutionary computing techniques contributed to more sophisticated and increasingly realistic human-like behavioral routines.

3.3.2

Genetic Algorithms for Behavioral Animation of NPCs

Although the potential of genetic algorithms has not been fully exploited in video games yet, we can find some illustrative examples of its application in this field. In [30], GA were applied to improve the parameter tuning of bots for the popular first-person shooter game Counter Strike. In that game, the AI determining the bot’s behavior is based on a set of rules triggered according to the threshold values of some

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activation parameters dealing with the bot’s weapon selection, gameplay style, path preference, and even its initial aggressiveness. However, determining suitable values for such parameter thresholds (a process referred to as tweaking the bot in video games) is a time-consuming and tedious process, generally carried out in trial-anderror mode. In that game, the GA performed fine-tuning of the parameters related to bot weapon selection and aggressiveness. Their results showed that the bots trained with GA played statistically similar to those adjusted by humans with expertise about the game. They also showed that GA reduce the development time significantly, and provide valuable task automation, thus freeing the programmer to focus on other tasks. The video game City Conquest used a GA in a rather different way. In that game, the AI was not aimed at improving player’s experience, but along the game design stage. The game developers created an automated tool based on GA and then run a large number of simulated games between red and blue opponents, each associated to a population of scripts. These scripts receive a fitness value and evolve according to the GA operators to optimize the population. Then, the winning scripts are plugged into the game. This procedure allowed to tune and refine some units and buildings, tweaking several game parameters (e.g., resource costs, health, speed, damage, rate of fire, and others). In this way, the game AI controlled by GA acted as an automatic play-testing team in the sense that it identified dominant strategies and allowed the game design to evolve even before its release. In the mobile game Darwin’s Nightmare, a casual shooter for iPhone, iPad, and iPod Touch, evolutionary techniques are used for exploration of a large combinatorial space that defines the visual appearance and behavior of enemy crafts. These enemies are generated by varying a single 6-bit genotype, where each gene corresponds to a specific ability (e.g., shooting) or feature (e.g., speed). Whenever a new enemy is generated, some individuals are selected from a pool; then, parents are selected using tournament selection so the most skilled fighters are selected and their genome is recombined to create the next wave of enemies. It should be noticed that the tweaking can have significant effects on the gameplay, even after minor changes in some parameters, a phenomenon actually observed in many GA-based AI video games. The work in [88] applied GA to evolve efficient paths between checkpoints in a 3D landscape for a PC strategy game. The generated paths take into account conditions such as terrain elevation and spatial constraints such as obstacles. For instance, the proposed method attempts to minimize elevation changes, avoid enemies, and minimize 2D movement, removing unnecessary wandering motion. The resulting paths turned out to be more realistic compared to those obtained using a variant of memory-constrained A*. It was also observed that divide-and-conquer yields better paths than using a single large genotype, but at the expense of larger execution times. The work in [70] applied GA to a reduced version of the popular puzzle game Lemmings. In particular, the authors used GA to evolve a strategy for solving Lemming maps. They observed that propagating strategies learned at early executions on simpler maps increase the GA evolution speed toward a successful strategy when compared to having a random initial population. In particular, it was found that chro-

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mosomes initialized with the best strategy from the previous execution are more likely to evolve faster to a successful strategy, while populations initialized with null pairs offer more diversity in the final score, and chromosomes initialized randomly are most likely to be a disadvantage. Other examples of the application of GA to video games include the work in [65], where a new scoring system for GA was proposed to solve the MasterMind game, a classical game involves decoding a secret code of six possible colors in four slots and other variations. In [103], GA were applied to generate strategies for a war game with uncertain rules. The game consists of a naval war between two parties, red and blue, where the goal for the blue team is to break the blockade imposed by the red team, which plays the opponent role while the blue team is the player for which a strategy is being determined. They found that the combination of elitism and the count method yields better strategies than the other tested combinations. In [119], the GA are used to evolve neural controllers for the real-time strategy game Warcraft 3. In [8], GA are applied to optimize the fitness function of the dominos game of four sides. Four different strategies were evaluated, where the best is obtained as a combination of the other strategies. The best results give a mean value of 69.18%, better than that with other previous methods for this problem [9].

3.4 Robotics 3.4.1

Path Planning

One of the most relevant fields of application of artificial intelligence techniques at large, and genetic algorithms in particular, is robotics. Several critical operations in robotics require highly sophisticated techniques and procedures, such as those based on artificial intelligence methods. A typical example is path planning, one of the most relevant tasks in robotics. In short, path planning aims to answer the question of how a robot can get where it is going. Path planning is applied to guide the robot to reach a particular target point. This task involves not only determining a trajectory but also planning the selection of all suitable sequences of action required to reach such a target point. This is generally achieved through a sequence of actions aimed at maintaining the robot motion from the source point to the target point through several intermediate states. Note that this requires several subtasks such as the continuous tracking of the robot states, and decision-making about the next actions to be executed according to the information available at the current step and the different criteria used for motion, such as moving according to the shortest distance measures to the target point measured through the Euclidean or other alternative distances. Interestingly, path planning is not always possible in advance. Proper algorithms for path planning usually require the generated path to fulfill some important conditions, such as to satisfy the shortest distance, to be collision-free and keep the robot far from potential obstacles, and require the shortest time and energy to reach the target point. The robot trajectory must ideally be smooth, without sudden turns and

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also satisfy all robot motion constraints. In general, there are four important criteria of consideration for path planning in robotics. They are optimization (to ensure the selected robot trajectory is the best path in terms of distance, traveling time, collision avoidance, energy, cost, safety, and other relevant factors), completeness (to ensure that the path planning algorithm will actually complete the planned trajectory and reach the target point), motion execution time, and accuracy. Sometimes, the robot has an incomplete knowledge (or no knowledge at all) about the environment where it is placed in. It is still possible, however, to design path planning algorithms that can handle with partially or unknown structured environments. This is a particular domain where artificial intelligence-based methods play a fundamental role. This problem has received significant attention of the scientific community and the list of papers devoted to this topic is actually overwhelming.

3.4.2

Genetic Algorithms for Path Planning of Robots

Regarding genetic algorithms applied to path planning, the list of papers is also very plentiful. A combination of genetic algorithms and adaptive fuzzy logic controllers for path planning by mobile robots is presented in [12], where a system of two Kinect cameras on the robot is used for path planning and execution in offline mode. Then, a GA is applied to generate a collision-free optimal path between the source and the target points of the mobile robot and a piece-wise cubic interpolation polynomial is applied to smooth the obtained path. The paper in [21] presents an implementation of a GA-based path planning for small-size robots designed for the RoboCup league, where the algorithm is adapted to the resource constraints of the micro-controllers in use. The generated path is updated to environmental changes, e.g., dynamic obstacles, until the robot arrives at its end point, as required to compete in that league. The papers in [22, 23] uses a GA with limited resolution of a simulated environment. The method is also limited by the fact that neither the robot geometry and its location nor the obstacle shapes or locations were actually considered. In [58], the proposed GA incorporates the domain knowledge into its evolutionary operators, enhanced through local search. The method in [67] uses a quad-tree structure to represent the environment and is comprised of two main steps: in the first, the method checks whether or not a chromosome can reach a goal position. In the positive case, the reached chromosome is transferred to the second step are then evolved. The method can obtain suboptimal robot paths for static and dynamic environments. In [74], an enhanced crossover operator is proposed to solve path planning for mobile robots in static environments using GA. The method in [78] uses domain knowledge-based genetic operators and is combined with fuzzy logic for self-adaptive tuning of the mutation and crossover probabilities in the GA. The paper in [117] introduces a cost function for GA path planning in grid-based 2D static environments. The cost function is made up of several terms which estimated the energy consumption according to the dynamics of an autonomous underwater vehicle. The authors in [89] consider a new environment representation supporting a more efficient approach for obstacles dilation. In their

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method, paths are represented as chromosomes with floating point variable-length genes, and the Minkowski sum is used for obstacle dilations. In addition, the CBPRM (clearance-based probabilistic roadmap method) method is used for population initialization of the GA. In [121], variable-length chromosomes are used to find a path in a discretized gridlike 2D environment, where each node in the grid is a gene, and hence the amount of genes is environment dependent. In [122], a novel mutation operator is proposed to prevent premature convergence. This operator checks for the free nodes near the mutation node simultaneously rather than selecting a node at a time randomly. In [126], a variable-length island parallel GA (IPGA) is used for path planning of wheeled non-holonomic robots where the path is a piece-wise curve made up of several η3 -spline segments, introduced previously in [97]. In [130], a GA-based adaptive fuzzy controller for smooth robot motion is proposed. The method relies on the knowledge from the fuzzy map to generate the fuzzy rule sets automatically and for adaptive parameter tuning of the membership functions. In [76], an improved GA is combined with numerical potential field for obstacle avoidance in dynamic environments. The work in [81] applies a double layer GA where the first layer is used to find a smooth path with static barriers and the evolution result from the first layer is injected as initial population for the second one, which also handles dynamic barriers. Finally, in [137], a step-spreading map (SSM) algorithm was used to instruct a multi-objective GA to construct a Pareto front for path planning with multi-objective optimization.

4 Genetic Algorithms for Real-World Problems: Future Trends and Current Challenges The emergence of new AI and machine learning (ML) approaches, such as convolutional deep neural networks (CNN), generative adversarial networks (GAN), and many others, seem to have outpaced the capabilities of genetic algorithms. However, a careful review of the literature shows us that, despite the years that have elapsed and the new developments in the field, GA are still widely used in many fields and are constantly finding new application niches. This trend is expected to continue in upcoming future. In fact, these new technologies within artificial intelligence have surprisingly led to a resurgence of genetic algorithms, which are being applied to problems such as the determination of certain parameters involved in deep learning methodologies or to evolve different neural network architectures. They are also widely used in hybridization with other metaheuristics such as simulated annealing, PSO, differential evolution, and with other techniques such as fuzzy logic, support vector machines, Q-learning, and others. Many experts believe that instead of GA being used in its purest form, they will be mostly used in combination with other complex AI and ML techniques.

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Other exciting current and future trend in genetic algorithms is the appearance of several variations and modifications of the standard GA methodology, with new extensions such as parallel GA and quantum GA gaining traction in the field. Most of these variations of GA have been developed to improve their performance and extend their applicability to new problem domains. Some of the most prominent examples are • Multi-objective genetic algorithms (MOGAs): MOGAs are designed to solve problems with multiple conflicting objectives, such as maximizing performance while minimizing cost. MOGAs generate a set of solutions that represent the trade-off between different objectives, known as the Pareto front. • Cultural genetic algorithms (CGAs): They combine traditional genetic algorithms with a cultural evolution process, which allows the population to learn and evolve cultural knowledge. CGAs have been applied to a variety of complex problems, such as social network analysis and online marketing. • Adaptive genetic algorithms (AGAs): Adaptive and self-adaptive genetic algorithms are extensions of the classical GAs that dynamically adjust their parameters during the search process using explicit or implicit knowledge or rules to improve its efficiency and effectiveness. AGAs have been used to solve problems with dynamic and uncertain environments, such as resource allocation and scheduling. • Estimation of distribution algorithms (EDAs): EDAs use probabilistic models to estimate the distribution of good solutions in the search space, which are then used to guide the search process. EDAs have been applied to a variety of problems, including combinatorial optimization, machine learning, and bioinformatics. • Memetic genetic algorithms (MGAs): MGAs combine genetic algorithms with local search heuristics, such as hill climbing or simulated annealing, or other metaheuristics, such as particle swarm optimization or differential evolution, to improve the quality of solutions. MGAs have been applied to a variety of optimization problems, including scheduling, routing, and engineering design. • Quantum genetic algorithms (QGAs): QGAs combine the classical procedures of genetic algorithms with the principles of quantum computation using quantum operations such as superposition and entanglement, which allows the algorithm to search multiple potential solutions simultaneously. Potential benefits of using QGAs include faster convergence to the optimal solution, improved accuracy, and the ability to handle high-dimensional search spaces. So far, they have been used for optimization problems, data mining, image processing, and financial forecasting. • Parallel genetic algorithms (PGAs): PGAs extend the classical genetic algorithms using parallel computing to solve complex problems by exploring a large solution space efficiently. PGAs can be used efficiently in distributed computing environments such as grid computing and cloud computing. Regarding some current challenges in the field, an open problem is the determination of enhanced crossover, mutation, and selection operators. Currently, the choice criteria for these operators is mostly empirical, based on a bulk of computer simulations, a process that is time-consuming and finally relies on the human expert

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for good performance. The potential use of self-adaptive mechanisms such as selforganizing GA might open new avenues for automatic and more efficient procedures to address this issue. Also, the issue of premature convergence is always a challenge in GA. Solving it would probably require to redefine the different steps and operators acting as the building blocks of GA. Several attempts have been investigated to alleviate the problem of premature convergence, but no methodology has achieved this goal in all its generality. Finally, application of GA to new real-world problems that are still unexplored or barely investigated is a tendency that will continue in the foreseeable future. It has been argued that GA could find a prominent role in new meta-learning architectures, where novel computational methodologies might be used to learn and address difficult problems that the human experts do not fully understand. Some intriguing and striking GA-based developments for motion animation and even to evolve virtual creatures [113–116] or generate novel 3D shapes through evolutionary genetic operators from an initial population of 3D models [131] are showing the new and exciting possibilities of GA to spark creativity in fields ranging from computer modeling and animation [11, 106, 124], computer-generated music and dancing [5, 47], synthesized painting, and evolutionary digital arts.

5 Discussion 5.1 Limitations of Genetic Algorithms In previous sections, we highlighted some advantageous features and many remarkable applications of genetic algorithms. However, it is worthwhile to mention that GAs also face several drawbacks and limitations that researchers need to be aware of. Some of the most relevant are • Local Optima: GAs are not guaranteed to find the global optimum solution. Instead, they may converge to a local optimum, which is the best solution in the neighborhood of the starting point, but not necessarily the best over the whole search space. Many recent GA approaches consider strategies to prevent the method being stuck in local optima. • Premature convergence: GAs may converge too early, meaning that the algorithm stops searching before finding the optimal solution. Premature convergence occurs when the population converges too quickly, leading to a loss of diversity in the population. Some mechanisms have been described in the literature to prevent premature convergence, but it is still a strong limitation of GAs for many applications. • Population size: The performance of genetic algorithms is strongly dependent on the population size. A small population size may not provide sufficient diversity for the algorithm to explore the search space effectively. On the other hand, a large population size may lead to high computational cost and memory require-

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ments. Determining the optimal population size is a great challenge of current GA approaches. Selection pressure: Selection pressure defines the extent at which the algorithm favors the selection of fitter individuals. If the selection pressure is too high, it may cause the algorithm to converge prematurely, while low selection pressure may cause the algorithm to converge slowly. Representation and encoding: The representation and encoding of the problem solution can greatly impact the performance of the genetic algorithm. Poorly designed encoding schemes can lead to suboptimal results. The first GA approaches used a binary string representation, but complex problems may require more sophisticated representation and encoding schemes. Complexity and computational load: Genetic algorithms can become very complex, computationally expensive, and time-consuming for large or complex problems. Recent extensions such as parallel GAs and quantum GAs were devised to tackle this issue. Interpretability and understandability: Genetic algorithms may produce results that are difficult to interpret or explain, which can make it challenging to understand why a particular solution was chosen.

5.2 Limitations of The Study in This Chapter The work in this chapter also comes with some limitations. On one hand, our analysis is restricted to some particular problems and fields of application while other topics and application domains have had to be overlooked in this chapter. Given the enormous amount of variations, applications, and publications on genetic algorithms, it would have been impossible to report all of them in the quite limited space of a single chapter. As a result, some exciting recent examples of application of GA in nonstandard fields have been omitted in this chapter. Some remarkable examples are • Generative design: Genetic algorithms are being used in the field of generative design to create complex and optimized designs for various products. This involves creating a population of potential designs and using genetic algorithms to evolve them into more optimized and efficient solutions. Generative design with genetic algorithms has been applied in a wide range of fields, including architecture, automotive design, product design, and aerospace engineering. • Decision-making in robotics: Genetic algorithms are being used in the field of robotics to optimize the behavior and decision-making of robots. This involves using genetic algorithms to evolve the algorithms that control the robot’s movements and decision-making processes. • Drug discovery: Genetic algorithms are being used in the field of drug discovery to optimize the design of new drugs. This involves creating a population of potential drug candidates, each represented as a string of values, which are evaluated based

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on different criteria (i.e., their binding affinity to a target receptor or their ability to penetrate cell membranes), and then using genetic algorithms to evolve them into more effective solutions. This process can also lead to the discovery of new drug candidates that might not have been identified using traditional methods. • Machine learning: Genetic algorithms can be used in conjunction with other machine learning algorithms to optimize their performance. For example, genetic algorithms can be used to optimize the weights and biases of a neural network. • Traffic flow optimization: Genetic algorithms are being used in the field of traffic engineering to optimize traffic flow. This involves using genetic algorithms to evolve traffic control strategies (for instance, adjusting traffic light timings, vehicle routing, speed limits, flow rules, and other variables) that minimize traffic congestion or travel times, maximize the safety of transportation systems, and improve overall traffic flow efficiency. This list is not exhaustive and many other examples can readily be added, a clear evidence of the great importance that genetic algorithms have achieved in current science and technology.

6 Conclusions In this chapter, we have discussed some of the most exciting recent applications of genetic algorithms to challenge real-world problems in several fields. We have shown that GA can be successfully applied to address such problems using different variations and modifications of the original GA procedures. We have also discussed some limitations of this approach, along with some current challenges and future trends in the field, with more than 130 bibliographic entries reporting valuable results in the field. Recent work shows that genetic algorithms are not just a matter of the past; on the contrary, new applications are continually emerging in different areas. In fact, we are witnessing a renewed hype in genetic algorithms as a consequence of their application to topics of recent interest and wide application, such as deep learning, in which genetic algorithms can play a key role in shaping their architecture and determine optimal operating modes and parameter values. At the same time, new modifications and extensions of genetic algorithms are being reported in the literature in response to current challenges and new needs. Researchers have already developed several techniques to address certain drawbacks and limitations, such as hybridization with other optimization techniques, population diversity maintenance, and advanced selection mechanisms. Reported work has shown that it is essential to carefully design and configure genetic algorithms to ensure optimal performance for a given problem. Regarding the foreseeable future of genetic algorithms, an important area of development will be the integration of GAs with other machine learning techniques, such as deep learning and reinforcement learning. This could allow for more sophisticated problem-solving and decision-making in complex environments. This process will

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likely involve advancements in both the theory and application of genetic algorithms. Another area of focus will be improving the efficiency of GA through the development of new selection, crossover, and mutation operators. This could lead to more effective optimization of complex problems with large search spaces. Other current issues to be addressed include the difficulty of parameter tuning, the risk of premature convergence, and the need for large amounts of computational resources. As such, ongoing research and development will be necessary to address these challenges and to ensure that genetic algorithms remain a valuable tool for solving complex problems in the future. Finally, new applications will be envisioned in fields where genetic algorithms have been sparely used to the date. Hybridization of GA with other metaheuristics and powerful extensions of GA, such as quantum GA, parallel GA, adaptive GA, and others, will open the door for new exciting avenues of research and development for the coming decades. Acknowledgements The authors would like to thank the financial support from the European Union’s Horizon 2020 research and innovation programme, Marie Sklodowska-Curie action, RISE program of the project PDE-GIR with grant agreement reference number 778035, and also from the Agencia Estatal de Investigación (AEI) of the Spanish Ministry of Science and Innovation, for the grant of the Computer Science National Program with reference number #PID2021-127073OB-I00 of the MCIN/AEI/10.13039/501100011033/FEDER, EU.

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Chapter 3

Genetic Algorithm for Route Optimization Bahaeddin Türko˘glu and Hasan Ero˘glu

1 Introduction Route optimization is a common problem in many fields, from logistics and transportation to supply chain management and even urban planning [1]. It involves finding the most efficient route between multiple points, while considering various constraints and objectives such as minimizing travel time, fuel consumption, and vehicle wear and tear. One approach to solving route optimization problems is through the use of genetic algorithms (GAs). GAs is an optimization algorithm inspired by the crossover and mutation processes, where the population of solutions evolves to find the best solution. GAs have been shown to be effective in many different optimization studies, including route optimization. In route optimization, GAs can be used to find the optimal sequence of stops for a vehicle or a group of vehicles, considering factors such as distance, time, and capacity constraints. By generating and evaluating multiple potential solutions and iteratively improving upon them, GAs can quickly converge on a near-optimal solution, even for complex problems with many variables. Overall, the use of genetic algorithms in route optimization can lead to significant improvements in efficiency, cost savings, and resource utilization, making it a valuable tool for businesses and organizations in a variety of industries. GA, which effectively uses biological evolution-based crossover and mutation mechanisms, is a successful algorithm type in search optimization applications in computing and artificial intelligence. In general, the GA ends when the maximum number of iterations is reached or a satisfactory convergence is reached for B. Türko˘glu (B) Faculty of Engineering, Department of Computer Engineering, Ni˘gde Ömer Halis Demir University, Ni˘gde, Turkey e-mail: [email protected] H. Ero˘glu Faculty of Engineering and Architecture, Department of Electrical-Electronics Engineering, Recep Tayyip Erdogan University, Rize, Turkey © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_3

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the population. Complex problems, which are usually constrained and unconstrained, are effectively solved by the genetic algorithm. Genetic algorithm is used in many fields for optimization purposes. In the context of route optimization, genetic algorithms can be preferred to find the most efficient route between multiple locations. Some benefits of using genetic algorithms in route optimization: • Exploration of a large search space: Genetic algorithms are capable of exploring a large search space in a short amount of time, which is useful when there are many possible routes to consider. • Flexibility: Genetic algorithms can be used with different types of optimization criteria and can be customized to fit specific requirements of the problem at hand. • Ability to handle constraints: Genetic algorithms can be designed to consider constraints such as road closures, one-way streets, or delivery time windows, which can make route optimization more challenging. • Genetic algorithms can converge to a solution very close to the optimum in acceptable short times depending on the problem type. Especially when compared to exhaustive search algorithms. • Ability to handle uncertainty: Genetic algorithms can be used to find robust solutions that are resilient to uncertainty or changing conditions, such as traffic congestion. Genetic algorithms can be a powerful tool for route optimization, offering flexibility, robustness, and efficiency in finding near-optimal solutions to complex routing problems. In the light of all these evaluations, this section has been written with the thought that this study, which includes GA-related route optimization examples and a comparative application of power transmission line optimization, will contribute greatly to the literature.

2 Related Works GA is used in many fields for optimization purposes. The subject of route optimization, which is one of these areas, is also a very important and wide-ranging field of study. There are important sub-working areas such as logistics [2–4], power transmission [5–13], network [14–16], and robot optimization [17–19]. In the study conducted by Qin et al. [4], the reliability of the model was tested by comparing samples with a numerical model and genetic algorithm. The relationship between GA-based optimization and cost and customer satisfaction has been revealed. In another study [20], the flexible time window model was used in the cold chain logistics optimization problem, and faster convergence was achieved by increasing the population diversity of the classical genetic algorithm. In some logistics optimization studies, Adaptive GA, an improved version of GA, has been proposed [21–23]. In reverse logistics, problem of the multi-stage reverse logistics network (m-rLNP)

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using minimization of the total transportation cost and fixed opening costs of disassembly centers and machining centers is proposed in [24]. The proposed method in the study is an improvement of GA and uses a modified crossover function, weight mapping crossover (WMX) with the hybrid priGA. On the other hand, some recent studies have been carried out for power transmission lines route optimization. In [25], investigation of the effectiveness of GA and ABC (Artificial Bee Colony) algorithm in determining optimal routes for power transmission lines. The use of improved GA and ABC algorithms has been shown to provide faster and more accurate results. The study shows that environmental factors are also considered in the determination of transmission line routes and the performance of the proposed algorithms give better results when compared to other methods shown comparatively. GA is also widely used in robot route optimization, which is another current topic. A new population of chromosomes was used in [26] with the help of dynamic population control mechanisms using polygamous cross mating. In the results of the study, it has been demonstrated comparatively that the proposed method gives efficient results. In addition, there are many successful GA applications in network optimization. The common points of these studies and many other studies in the literature are the development of functions such as crossover in order to make GA work more effectively. It is striking that there is no study in the literature in which the important aspects of all these studies were examined and a useful application example was given comparatively. Elimination of this deficiency is our most important motivation source in carrying out this paper. This chapter presents an application framework by analyzing detailed critical information on route optimization of GA. In this context, the types of route optimization studies have been diversified and presented with examples in the literature. In addition, power transmission line route optimization, which is one of these types, has been examined in detail. In this chapter, the working principle of GA is mentioned. Chapter 4 focuses on Parameter Selection in Genetic Algorithm. Chapter 5 provides information on GA applications in route optimization. An application of GA for power transmission line route optimization is given in Chap. 6. Finally, the study is concluded and future recommendations are given in Chap. 7.

3 Principle of Genetic Algorithm The exponentially increasing data volume and complexity of today’s problems increase the need for new methods to produce fast and easy solutions. There may be many solutions for some problems, but it is necessary to reach the most suitable solution quickly [27, 28]. In particular, the inability to solve NP problems in an acceptable time has led to the use of evolutionary computational strategies instead of brute-force algorithms [29]. GA, one of the best practices of evolutionary strategy, stand out with their success in solving problems in many different domains [30].

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Holland introduced GA in 1975, inspired by the evolutionary processes and the principles of selection in nature. Genetic Algorithms are simple but very powerful algorithms. The primary purpose of GA is to search for solutions to complex optimization problems involving many constraints by modeling evolutionary systems. As a working principle, GA is a search technique that tries to find solutions using random search techniques and is based on parameter encoding [31]. Genetic algorithms operating according to probability rules only need fitness functions. Instead of scanning the entire solution space sequentially, they search for the most efficient part of the solution space within an evolution strategy. In this way, they aim to solve the problem in a much shorter time by making a more effective search [32]. Another critical feature of genetic algorithms compared to standard techniques is that they work simultaneously by using a population of search agents instead of a single search agent. Thus, they are not stuck with local optimal solutions. GA has been successfully applied in numerous fields, such as route optimization [26], path planning [33], task scheduling [34], facility location [35], land consolidation [36], assembly line balancing [37], chip design [38], and traveling salesman problem [39]. The working steps of GA can be detailed as in Table 1. Genetic algorithms encode each solution in a solution space with a binary string of bits called chromosomes. Every solution has a fitness value. Instead of a single solution, it uses a set of candidate solutions called populations. Each iteration creates Table 1 Working steps of GA No

Step

Description

1

Genetic coding of the problem

All possible & candidate solutions in the search space are coded according to a coding technique

2

Creation of initial population

A random set of solutions is generated and chosen as the initial population

3

Evaluation of fitness function

A fitness function value is evaluated for each individual in the population, and these fitness values represent the solution quality of the individuals

4

Application of reproduction process

A group of individuals is randomly selected according to a selection strategy and stored for use to produce new individuals

5

Application of evolutionary operators

The fitness function values of the new individuals are evaluated and subjected to crossover and mutation processes

6

Creating the new generation

Throughout the initially determined number of iterations, selection, crossover, mutation, generation of new individuals, and fitness value calculation processes continue

7

Presenting the best individual as a best solution

When the iteration process is completed, the individual with the best fitness value represents the best solution

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the next generation from individuals in the population using genetic operators such as crossover and mutation. The new generation is not the same as its ancestor; instead, it may have a better or worse fitness value. Better solutions are chosen through strategies such as natural selection and elitism. The goal is to update the population throughout the iterations to include a generation with better fitness [40]. In the continuation of this chapter, the coding of the solutions, the creation of the initial population, the evaluation of the fitness values, the reproduction, crossover, mutation, and the creation of the new generation will be detailed.

3.1 Genetic Coding of the Problem In order to solve a problem through GA, it is first necessary to construct the solution’s parameters in the form of an array. All possible candidate solutions (chromosomes) are represented as a series of bits (gen) of the same length. Each candidate solution as an individual represents a solution in the solution space. In the most general sense, the process of converting problem-specific information into the form that genetic algorithms will operate is called parameter coding. There are various coding strategies in the literature, such as value coding, tree coding, binary coding, and permutation coding. For example, the permutation coding technique is used in the traveling salesman problem. Descriptions and examples of coding strategies are shown in Table 2.

3.2 Creation of the Initial Population A solution space containing all possible candidate solutions is created. The solution space created is called the population, each candidate solution is called an individual or chromosome, and the parameters that make up the individual are called genes. In continuous problems, the initial population is created by a function that generates random values. In binary problems, if the randomly generated number is less than 0.5, it is coded as 0; otherwise, it is coded as 1. When the number of individuals in the population is selected is small, iterations are completed faster, but the probability of getting stuck in local minimums increases. A large number of individuals (chromosomes) in the population (size) increases the quality of the solution, but iterations take longer to be completed.

3.3 Evaluation of Fitness Function The first step after creating a new generation is to calculate the fitness value of each individual in the population. Different individuals (solutions) should be compared,

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Table 2 Coding strategies Strategy

Description

Example

Binary coding

It is the most used coding method. It is suitable for simple, easy, and fast operations. Identification of chromosomes is constructed with a binary number system (0, 1)

Chromosome −1: 1,101,100,100,110,010 Chromosome −2: 1,101,111,000,011,010

Permutation coding

It is suitable for scheduling or Chromosome −1: 3 5 1 7 8 9 2 4 6 organizing problems. It solves problems Chromosome −2: 2 5 4 7 1 3 7 6 9 such as traveling salesman problems (TSP) or job scheduling

Value coding It is used in problems where complex data, such as real-world problems, are used. Values can be real numbers, characters, or objects. In this type of coding, it is necessary to develop new problem-specific crossover and mutation methods Tree coding

Chromosome −X: 1.269 8.453 3.165 7.256 Chromosome −Y: AJDKFDJLMNDFAMDJ Chromosome −Z: (forward), (left), (right), (left)

This method is used for problems with expressive solutions; each chromosome is a tree of an object. It is suitable for evolving programs or any other structure encoded as a tree. For example, it functions or commands in a programming language

LISP

and the good ones should be selected. ith the fitness value of the individual can be represented by f (x i ). If f (x 1 ) > f (x 2 ), then individual x1 is better than individual x2 . The graphical representation of the fitness function is shown in Fig. 1. A fitness function is designed for each problem. This function returns a numerical fitness value proportional to the performance of each candidate solution. This ratio is directly proportional to Maximization problems and inversely proportional to minimization problems. The fitness function value guides the selection of more suitable solutions in each generation. The higher the fitness value of an individual is, the higher the survival rate, reproduction, and representation in the next generation are.

3.4 Application of Reproduction Process In this process, individuals are ranked according to their fitness function values. The individuals who can transfer the best-inherited traits to the next generation are selected with a selection strategy. Methods such as tournament and roulette selection increase the probability of individuals with higher fitness values in the next

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Fig. 1 Graphical representation of fitness functions

generation. The reproduction process consists of selecting individuals, collecting the chosen individuals in a matching pool, and grouping the individuals in the collection into pairs. After the fitness values are calculated, a new generation is created from the current generation. The selection process is the stage of deciding which individuals should be involved in producing the next generation. The probability of an individual being selected is proportional to the fitness value. In summary, individuals above average fitness function value have a greater opportunity to reproduce. The population size is not allowed to change while these processes are being carried out. There are various selection strategies in the literature, such as the roulette wheel, tournament method, stochastic universal sampling, rank selection, truncation selection, fitness proportionate selection, reward-based selection, Boltzmann selection, and elitist selection [41]. Some of these methods are detailed in Table 3.

3.5 Application of the Crossover Operation It is the process of creating new offspring (chromosomes) by replacing some genes in the parent. Crossover is performed to increase the performance/potential of the current gene pool and to create a better generation than the previous generation. It is applied to pairs of individuals selected according to a determined crossover ratio.

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Table 3 Selection strategies Name

Description

Roulette wheel

– – – – – – – –

Chromosomes are selected according to their fitness function value Better chromosomes are those that have a greater chance of being selected All chromosomes in the population are placed on a roulette wheel The size of the location of the chromosome on the roulette wheel is proportional to the fitness of the chromosome The more suitable chromosome will have a larger part in the wheel This method has some disadvantages For example, if the fit of the best chromosome is 90% of the sum of all other chromosomes, the chances of selecting the other chromosomes will be very low To prevent this, sequential selection can be used

Sequential – In the sequential selection, the chromosome with the worst fit is given a value of selection 1, the next a value of 2, …, and the last one N (number of individuals) – Compared to roulette selection, individuals with a low fitness value increase the chance of being selected – However, this method may cause the solution to converge later Steady state selection

– In each generation, chromosomes with high fitness values are selected to produce the next generation – Chromosomes with low fitness value are removed, and these newly created chromosomes are put in their place – The rest of the population is also transferred to the next generation

The crossover operation is a process that directly affects the performance of GA. Two individuals are randomly selected from the new population obtained as a result of the reproduction process and are crossed over to each other. In the crossover operation, k integers in the range of 1 ≤ k ≤ N are selected, with the individual length N , and the crossover operation is applied to the individual according to this k value. The simplest crossover method is the one-point crossover. Both individuals must be in the same length in order to perform a one-point crossover. In the two-point crossover, the individual is cut from the two determined values, and the values in the opposite positions are swapped. Examples of single-point and two-points crossovers are shown in Table 4. Table 4 Crossover techniques

Crossover technique

Parents

Offspring

One-point

P1 = 101 | 0010

O1 = 101 | 1001

P2 = 011 | 1001

O2 = 011 | 0010

Two points

P1 = 1101 | 1,001,001| O1 = 1101 | 10,110 1,110,000 | 10,110 P2 = 1101| 1,110,000 | O2 = 1101 | 11,110 1,001,001 | 11,110

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Table 5 Mutation methods Mutation methods

Description

Example

Shifting method

It is the placement of a randomly determined gene in the same chromosome at a random location within the same sequence

P1 = AKDEMZIF , P1 = AEMZIKDF

Placement method

It is the random selection of a gene in the chromosome and placing it in another position within the same chromosome

P1 = AKDEMZIF , P1 = ADEMZKIF

Mutual exchange method

Two genes are randomly selected in the chromosome, and these two genes change places with each other

P1 = AKDEMZIF , P1 = AZDEMKIF

3.6 Application of the Mutation Operation After crossover, the population does not always contain all the encoded information and cannot be passed on to the next generation. As a new generation is created during iterations, some individuals may recur and reduce chromosome diversity. Therefore, the algorithm cannot produce a satisfactory solution after a while. To overcome this situation, mutation operation, an evolutionary operator capable of generating new chromosomes from existing chromosomes, is required. In GA, protection can be provided against the loss of a good individual due to the mutation process. In problems where the binary coding system is used, conversion from 0 to 1 or from 1 to 0 is achieved by mutation operation. Different mutation strategies are used in specific problems where the binary coding system is not used, such as the translation, insertion, and reciprocal exchange methods. Some of these methods are detailed in Table 5. The mutation process aims to provide and maintain genetic diversity regardless of the coding strategy. The mutation is applied to some of the genes to increase the chromosome diversity in the generation. After the crossover and mutation operations, the new generation may have better fitness values than their parents.

3.7 Creating a New Generation and Finishing the Algorithm The new generation is formed after reproduction, crossover, and mutation processes and becomes the next generation’s parents. During iterations, new generations are generated by evolutionary operators. This process continues until a predetermined number of iterations or a stopping criterion is continued. The stopping criterion can be the number of iterations or the targeted fitness value.

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Fig. 2 Flowchart of genetic algorithm

The flowchart of GA is shown in Fig. 2.

4 Parameter Selection in Genetic Algorithm The performance of GA is highly dependent on the hyperparameter values. Numerous studies have been carried out in the literature to determine the most optimal parameters. However, there is no universal set of parameters for all problems. These are population size, crossover and mutation probability, selection strategy, and generation range. These control parameters are detailed in the following chapter.

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4.1 Population Size Population size is one of the most critical parameters. When this parameter value is small, iterations proceed faster, but they are more likely to get stuck in local optima. The large population size value increases the time to reach the solution. In a study in the literature, a parameter size determination equation was proposed according to individual (chromosome) length [42]. In another study, it was stated that, regardless of the problem size, different experimental studies on a wide variety of benchmark functions obtained the optimum results when the population size was between twenty and thirty [43].

4.2 Crossover Probability The purpose of crossover is to combine the characteristics of good chromosomes in the parents to form better chromosomes. Chromosome pairs are crossed at the determined crossover probability. Increasing the crossover ratio increases population diversity but increases the probability that some good chromosomes will also degenerate. There have been many studies in the literature on the determination of the crossover ratio [44].

4.3 Mutation Probability The purpose of the mutation is to preserve the genetic diversity in the population. Mutation is carried out at the rate of mutation probability determined in the genes in the chromosomes. If the mutation rate increases too much, it turns the GA into a random search. However, the high mutation rate prevents declining genetic diversity. In the literature, studies have been carried out to adjust the mutation probability of this parameter using various methods, such as linear, adaptive, and fuzzy logic [45, 46].

4.4 Generation Range The ratio that depends on the number of new chromosomes added to each generation’s population determines the generation interval. This ratio represents how many chromosomes will be subjected to evolutionary operators. A higher value means that more chromosomes will undergo evolutionary transformation [47].

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4.5 Selection Strategies Various methods can be used when developing the previous generation with evolutionary operators. In the generational strategy, all individuals in the current generation are replaced by new individuals. Since the best individual in the population is updated while this process is taking place, there is a possibility that the best individual can be registered. Therefore, this strategy is used by hybridizing with the elitist strategy. In elitist technique, the best few individuals of each generation are passed on to the next generation without undergoing any change or exposure to the evolutionary operator. In this way, the best individuals are preserved for generations. In the equilibrium strategy, only the individuals who cannot improve themselves are replaced by new individuals. In other words, when new chromosomes join the population, the worst chromosomes are renewed [48].

5 GA Applications in Route Optimization Route optimization problems, which are solved by genetic algorithm, can be basically examined in four different categories. These are routing optimization for logistics, power line routing, robot route optimization, and network optimization [49]. The GA applications and their details for the solution of these different problems are given below.

5.1 Routing Optimization for Logistics The change in people’s quality of life and eating habits with technology has resulted in a rapid change in the cold chain logistics sector. One of the most important parameters of cold chain logistics is the cost. In order to minimize this important criterion, many researchers [2–4] have done a lot of work to determine the best among different solution alternatives. In addition to the cost, criteria such as customer satisfaction and carbon emissions should also be considered. First, Qin et al. [4] demonstrated the reliability of the model they established by making a sample comparison with a numerical model and genetic algorithm they used in their studies. With the GA-based optimization, it was determined that a small increase in the cost increased the average client satisfaction. It has also been found that increasing the carbon price reduces carbon emissions while reducing customer satisfaction. CEGA (cycle evolutionary genetic algorithm), which they proposed in the study, simulates the evolution and decay characteristics of standard GA based on catastrophic theory due to the early maturation problem and shows cyclical reciprocation characteristics. As a result of the study, it was stated that with the proposed GA-based new model, customer satisfaction increased by 30%.

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In another study [20], the cold chain logistics optimization problem was tried to be solved through the soft time window model. In this study, a modified GA is presented to increase the population diversity of the classical genetic algorithm and to achieve faster convergence. In this direction, the GA has been tried to be improved by making use of the uncertainty reasoning idea in SOA (seeker optimization algorithm), by fuzzing the fitness values of the chromosomes and increasing the search for new chromosomes in regions with high fitness values [50]. In the result of the study, the proposed improved GA shortens the transport distance and thus reduces the total cost. Adaptive GA has been proposed in some logistics optimization studies [21–23]. The basic logic in adaptive GA is to make the algorithm avoid the local optimum. Individuals that vary with the crossover operator may have worse fitness values. The mutation operator has an important task to move away from this local optimum. High probability in mutation will save individuals from local optimum to global optimum [23]. The mutation probability formula used for adaptive GA in this study [23] is as follows: Pm =

0.5 1 + e(G×(g−g0 ))

(1)

where G is the maximum iteration number, g is the current number of itteretion, g0 = G/2 is the variable point. Table 6 shows the average percent deviations from the optimum completion time produced by adaptive GA evaluation of 1000 and 5000 schedules, respectively, in an optimization project. This data shows that adaptive GA gives more effective results. Table 6 Average deviations (%) of different algorithms [22]

Algorithm

References

Iterations 1000

5000

SAGA (Self-adapting GA)

[23]

37.1

35.3

ALGA (Activity list GA)

[51]

39.3

36.7

SA (Simulated annealing)

[52]

42.8

37.6

PRGA (Priority rule GA)

[23]

39.9

38.4

AS (Adaptive sampling)

[53]

39.8

38.7

PSLFT (Parallel sampling LFT)

[54]

39.6

38.7

AS (Adaptive sampling)

[55]

41.3

40.4

PSGA (Problem space GA)

[56]

42.9

40.6

SSLFT (Serial sampling LFT)

[54]

42.8

41.8

RKGA (Random key GA)

[23]

45.8

42.2

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5.2 Power Line Route Optimization Power transmission line route determination studies are at the forefront of optimization studies using GA. As in all route determination studies, it is very important to determine the criteria affecting the route in ETL route determination studies. Many studies have been carried out on the determination of the criteria affecting the ETL route from past to present [5–13]. The most important issue in route determination studies is to determine the weight percentages of the criteria affecting the route relative to each other. The route direction is set es per the weight percentages taken by the criteria. In simple routing problems, the choice between criteria is easy for the human brain. However, it is very difficult to choose between criteria in complex problems. For this reason, many methods are used to determine the weight values of the criteria relative to each other [57, 58]. In [25] after the determination of the criteria, a Total Weighted Surface Raster Map (TWSRM) is created with the help of Geographical information systems (GIS). With the TWSRM, it is possible to collect the weighted data of the ETL route by bringing them together and evaluate them. To run the GA on the TWSRM created for ETL route determination, the map needs to be converted to numerical matrix form. For this purpose, first, the pixelbased map is converted to a “txt” file containing pixel values, coordinates, etc. This converted file is used as the input of the algorithm, and individual pixel values are stored with the help of loops. Thus, the map is transferred to the computer to be used in GA. In the genetic algorithm, the chromosomes that make up the population were created by coding the coordinates (x and y values) of the pixels on the map. X and Y values are encoded with the character “,” and inter-pixel with a space (“”) character. Thus, each route created is coded as a sequence value consisting of x, y values, and all operations in the genetic algorithm are done over these sequence values. Chromosomes consist of two series, the Map series (MS) and the Value series (VS). While X and Y coordinate information of the pixels on the map are kept in MS, the values of the pixels are kept in VS [25]. The creation of an example TWSRM and its conversion to series for processing with GA is shown in Fig. 3. In another study [59], an improved GA was used for solving TNEP (transmission network expansion planning) problem. Intensification phase of a Tabu Search algorithm was applied to overcome the shifting problem to a local optimum while creating the improved GA. A GA based method was used to determine the route of the power transmission lines of the urban distribution networks [60]. With the developed software, route optimization was carried out for a 10 kV power transmission line in a real city example. In the study, operators such as generation, mutation, crossover, and selection are used as given in the above example studies. The results obtained from the study show that the most suitable power transmission line route for projected or reconstructed urban power grids can be effectively optimized with GA.

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Fig. 3 Demonstrating the transformation of TWSRM for power transmission line route optimization with GA [25]

5.3 Robot Route Optimization The robot route finding problem is similar to the classical TSP (Traveling Salesman Problem). Similar to the functions used in TSP, determining the robot route and moving the robot are performed. In the example in Fig. 4a, five different robot stations are seen. In the problem, all points are requested to be visited as soon as possible, provided that these 5 points are visited only once. Figure 4b presents a typical best (optimal) route path and Fig. 4c presents a typical Worst (non-optimal) route path. In [26], an improved GA with was developed to be applied in robot route optimization. Polygamous crossover mating process was applied to the chromosomes in the selected cluster. A population of new chromosomes was created with the help

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Fig. 4 Sample robot routes, a possible points, b best route, c worst route

of dynamic population control mechanisms. These processes were stopped when the convergence to the global solution was achieved. The results of the study prove that the proposed algorithm gives better results than the existing methods. Similar GAbased approaches have been used in many studies [17–19] and it has been revealed that GA is an effective solution for robot route optimization.

5.4 Network Route Optimization There are few samples for WSN (wireless sensor networks), such as military, commercial, and medical. Certain parameters need to be developed for WSN. These are life of the network and the energy consumed for routing, which are important for any application [14]. WSNs use batteries and mobile energy sources, and these batteries run out of energy after a certain period of time. Therefore, optimum use and management of energy is very important [15]. In [16], studies on GA applications in wireless networks are given in detail. In this study, two categorizations were made as Multi-objective GA (MOGA) and Single-objective GA (SOGA). Mobile ad-hoc network (MANET), Wireless area network (WAN), Local area network (LAN), wireless sensor networks (WSN), Wireless mesh network (WMN), and cellular network

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Table 7 GA studies for wireless network optimization [16] References

Network type

Objectives

Technique used

[61]

MANET

SOGA

Integer-coded, variable sized chromosomes

[62]

MANET

MOGA

Multi-populational GA

[63]

LANs

SOGA

GA with integer code for random distributions

[64]

WAN

SOGA

Unbiased coding based GA

[65]

MANET

SOGA

Memory scheme based GA

[66]

WMNs

SOGA

GA used for ILP formulation

[67]

WSN

MOGA

ILP—GA

[68]

CN

MOGA

SGA (Sequential GA)

(CN) are used as network type. In addition, many studies have been presented in eight different categories as a type of network optimization with GA. These categories are Quality of service (QoS), load bandwidth determination, channel routing and detection, localization, and wireless AP placement. As can be seen in Table 7, the routing category in wireless networks with GA has been carried out with many studies. Studies show that GA is an effective optimizer that is widely used for WSNs.

6 GA for Power Transmission Line Route Optimization: An Application An initial population of GA needs to be created using TWSRM, as an example which is given in Sect. 4.2. In classical population generation of GA, random genes from the start gene to the ending gene are first selected and inserted into the chromosome. Each chromosome is made up of many genes that represent the “nrows ” and “ncols ” values of the text file matrix shown in Fig. 3. Gene addition is done until the ending gene is reached. Therefore, the chromosome does not have a standard length. Every genes must be chosen from around the existing genes so that the route is not interrupted. This operation, called “break-off control”, is performed in new gene selection. The production function ends if the target gene is reached. However, in this random selection, the route sometimes loops around itself, causing infinite loops. Therefore, classical population generation of GA needs to be developed. Preventing the algorithm from entering these endless loops slows the algorithm down considerably. In order not to enter these endless loops, four main regions shown in Fig. 5 have been defined. The key regions in Fig. 5 are determined by the position of the start and end genes. This main region determination makes the population generation of GA more selective and rapid and is important in direction selection and angle detection of the subregions. Then, in the selection of the new gene, one of the constrained aspects around the existing gene is chosen. This selection is made by looking at the direction

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Fig. 5 Possible directions in the divided regions for the generation function of GA

between the starter and target gene. Thus, the sub-region is selected. Subregions are used to avoid endless loops in reaching the target gene. “Gene repetition control”, which is done to prevent a gene from being re-selected, is done in every gene selection. All these enhancements to the production function are called “Intelligent direction sensing” (IDS). With the IDS generation function, the production of unnecessary route parts is reduced, and the speed of the production function is increased. All controls used in power transmission line route optimization with GA are re-selection, infinite loop, main-sub zone, direction, start–end gene, break-off, and maximum allowed loop. Since the random function used in gene production repeats in large cycle numbers, a new random function has been developed. As a result of these repetitions, the candidate route (chromosome) does not expand its direction and does not find alternative routes and goes towards the average of the restricted directions. For example, if the

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Fig. 6 Representation of classical and developed random function routes on workspaces

constrained directions are assumed to be 0, 1, 2, 3, 4 in the first or sixteenth subregion in, the route chooses different directions and goes towards direction 2, which is the average of the directions in a narrow area. In the developed random function, as shown in the example routes in Fig. 6, the route goes in one direction (for example 1) using different directions, then turns to another direction (for example 3) and uses different directions with different selection possibilities. Thus, the use of larger areas of the working area is ensured. With the developed random function, the new routes cover the entire study area, increase the richness of the population, and thus make the algorithm more robust. The fitness value of ith route is expressed in Eq. 2: Fitnessi =

N Σ

VSMSn,0 ,MSn,1

(2)

n=0

where N is the genes number in the chromosome. After the population is created, the total weight values of the routes in the population are determined by summing the pixel values stored in matrix format according to their x and y addresses. After the values of all the routes in the population are determined, the total weight values of the routes are calculated to make the RW (roulette wheel) selection. Each route is specified inversely proportional to the magnitude of the weight value (passing difficulty value). With this selection, the probability of choosing routes with a high weight value, that is, routes where power transmission line route crossing may be difficult, decreases. After the roulette wheel selection is made, the selected routes are graded according to their weight values from smallest to largest. Then, the crossover function is started. The crossover operation is done by the exchange of the route segments between the two common pixels in the created routes. Thus, a new route with less cost is

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created by changing the transit route of a route. In this study, each of the routes obtained by using the roulette wheel selection is compared with each other, and the crossover process is performed. The mutation process means replacing certain parts of chromosomes with new ones. In this process, break control is crucial for the start and end points of the mutation should not break the integrity of the first chosen route. The parameters used in the study are as follows: PS = 2 ×

(3)

n rows × n cols 4

(4)

n rows × n cols 4 × (n rows + n cols )

(5)

MNI = MR =

n rows × n cols n rows + n cols

where PS is the population number, MNI is the maximum iteration number, and MR is the mutation rate. The maximum iteration number is accepted as the convergence criterion of the algorithm. Flowchart of the algorithm for power line route optimization is shown in Fig. 7. By running the algorithms many times, the optimal population number was accepted as 100 and the maximum number of cycles between 5000 and 10000. Two different routes obtained as a result of the study are shown in Fig. 8. When the routes in Fig. 8 are examined in detail, the similarity of the routes shows the accuracy of the programs written through the Genetic and Dijkstra algorithms. Although Dijkstra’s algorithm is fast, it has a disadvantage that the turning angles of the resulting routes are sharp. The most important disadvantage of the GA used in this study is its slow operation. Although the GA works slower than the Dijkstra algorithm, it can find routes with less cost and smaller line break angles. Due to the random generation characteristic of GA, the most suitable solution can be found in the first run as well as in a very long time. The convergence curve for GA is presented in Fig. 9. Table 8 shows the fitness values of the routes in Fig. 8. According to this table, GA has reached a better fitness value than Dijkstra’s algorithm. Since the cost of the route is important in long power transmission lines, routes with lower fitness values found by the algorithms may be preferred even though their time performance is lower. Therefore, GA should be used as an effective alternative in power transmission line routing studies.

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Fig. 7 Flowchart of the algorithm for power line route optimization

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Fig. 8 Two different routes obtained through GA and Dijkstra algorithms

Fig. 9 Convergence of GA

Table 8 Fitness value of the algorithms

Fitness Value GA

Dijkstra

1511

1517

7 Discussions As a result of the investigations made in this study, it has been presented that GA are widely used in route optimization problems due to their ability to find solutions close to the optimum in a reasonable time. However, like any other optimization algorithm,

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GAs have their own set of advantages when it comes to route optimization. Some of these are: • Global search: GAs are well-suited for finding the global optimum in complex, high-dimensional search spaces. This is because they use a population-based approach, which explores multiple regions of the search space simultaneously. This means that genetic algorithms can find better solutions than other search algorithms that are based on local search. • Scalability: GAs can handle large and complex problems with a large number of variables. This is because the algorithm is parallelizable, and each member of the population can be evaluated independently. As the complexity of the problem increases, genetic algorithms can still find solutions in a an acceptable time. • Flexibility: GAs are flexible and can be adapted to different types of route optimization problems. They can handle both static and dynamic problems, and can be modified to incorporate various constraints such as time windows, vehicle capacity, and multiple objectives. • No gradient information required: GAs do not require any gradient information, making them suitable for non-linear optimization problems. This is an advantage over gradient-based methods, which can be difficult to apply in non-linear problems. • Exploration vs. exploitation: GAs strike a balance between exploration and exploitation, which is important in route optimization problems. Exploration is necessary to search for new solutions, while exploitation is needed to improve on the current best solution. Genetic algorithms achieve this balance by using selection, crossover, and mutation operators to generate new solutions from the current population. Also, GA has some disadvantages. Here are some of the disadvantages of using genetic algorithms in route optimization: • Computational Complexity: Genetic algorithms can be computationally expensive and time-consuming, particularly for large-scale optimization problems. As the size of the problem increases, the search space grows exponentially, leading to longer execution times and memory requirements. • Fitness Function Design: The fitness function used in a GA is critical to the quality of the solutions generated. Designing an effective fitness function for route optimization problems can be challenging, particularly when the problem involves multiple objectives or constraints. • Premature Convergence: Genetic algorithms are prone to premature convergence, where the algorithm converges to a suboptimal solution too early and fails to explore the search space fully. This can happen when the population size is chosen too small, the mutation rate is too low, or the crossover operator is not appropriate. • Difficulty in Handling Dynamic Environments: Genetic algorithms are not well suited to handling dynamic environments where the problem parameters change over time. This is because GAs rely on an initial population of solutions that may become obsolete as the problem changes.

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• Lack of Diversity: Genetic algorithms can suffer from a lack of diversity in the population, particularly if the initial population is poorly chosen or the genetic operators are not well designed. Without sufficient diversity, the algorithm may focus on the local optimum and drift away from the overall optimum. In summary, while genetic algorithms are a powerful tool for route optimization problems, they are not without limitations. These disadvantages must be considered when using GAs for route optimization, and alternative methods may be more appropriate in some situations.

8 Conclusion and Future Works Real-world problems have many constraints that make them difficult to solve in a reasonable time. The best route optimization we encounter in many areas is one of these problems. These problems are inherently NP problems that cannot be solved in polynomial time. Solution spaces increase exponentially with the number of features [69]. General algorithms are especially well suited for multidimensional, complex optimization problems. Since it contains evolutionary strategies that update many dimensions simultaneously, it has a high ability to get rid of local minimums. GAs are also powerful tools in route optimization. By mimicking the process of natural selection and evolution, GAs can efficiently search through a large space of possible solutions to find the optimal route. This approach has been used in many real-world problems, from vehicle routing to logistics and supply chain management, and has proven to be an effective and flexible solution. However, as with any optimization technique, the performance of GAs is mainly dependent on the problem at hand and the quality of the input data. Additionally, the implementation of genetic algorithms requires expertise in both optimization techniques and computer programming. Overall, genetic algorithms are valuable tools in route optimization, and their use will likely continue to grow as computational power and optimization techniques advance. In this book chapter, a study has been carried out on using genetic algorithms in route optimization. Studies in literature have been categorized under different headings. The solution of power transmission lines route optimization problem, which is one of the best practices of the subject, with the genetic algorithm is discussed. The use, advantages and disadvantages of the genetic algorithm in route optimization have been revealed. In future studies, problems of genetic algorithms in different areas that have not been solved before can be examined. Genetic algorithms can be developed through different strategies, such as quantum behavior and intelligent search, and their performance in existing problems can be increased. There are several promising areas for future research on genetic algorithms in route optimization. One potential direction

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is to explore the integration of genetic algorithms with other optimization techniques such as simulated annealing or ant colony optimization. This could lead to more robust and efficient optimization solutions. Another area for future work is to investigate the application of genetic algorithms to dynamic route optimization problems. Real-world routing problems often involve changing conditions such as traffic congestion or delivery time constraints, and genetic algorithms could potentially adapt to these changes in real-time to find optimal solutions. On the other hand, there is a need to explore the effectiveness of genetic algorithms in large-scale optimization problems involving a high number of vehicles or locations. The scalability of genetic algorithms can be challenging, and developing novel optimization techniques and parallelization strategies could help to overcome these challenges. Also, investigation of the role of human input and domain knowledge in the optimization process is needed. While genetic algorithms can effectively search through a large space of possible solutions, incorporating human insights and knowledge could potentially lead to more efficient and effective solutions. In conclusion, the future of genetic algorithms in route optimization is promising, and continued research and development in this area can lead to significant improvements in the efficiency and effectiveness of transportation and logistics operations.

References 1. Dey N, Ashour A, Bhattacharyya S. Applied nature-inspired computing: algorithms and case studies, p 275. https://books.google.com/books/about/Applied_Nature_Inspired_Computing_ Algori.html?hl=tr&id=eY2oDwAAQBAJ. Accessed 04 Apr 2023 2. Zhang Y, Hua G, Cheng TCE, Zhang J (2020) Cold chain distribution: how to deal with node and arc time windows? Ann Oper Res 291(1–2):1127–1151. https://doi.org/10.1007/s10479018-3071-0 3. Wang S, Tao F, Shi Y (2018) Optimization of location–routing problem for cold chain logistics considering carbon footprint. Int J Environ Res Public Health 15(1). https://doi.org/10.3390/ ijerph15010086 4. Qin G, Tao F, Li L (2019) A vehicle routing optimization problem for cold chain logistics considering customer satisfaction and carbon emissions. Int J Environ Res Public Health 16(4):576. https://doi.org/10.3390/ijerph16040576 5. Houston G, Johnson C (2006) EPRI-GTC overhead electric transmission line siting methodology. Georgia. Accessed 14 Feb 2023. https://www.epri.com/research/products/000000000 001013080 6. EroØlu H, Aydin M (2015) Optimization of electrical power transmission lines’ routing using AHP, fuzzy AHP, and GIS. Turk J Electr Eng Comput Sci 23(5):1418–1430. https://doi.org/ 10.3906/elk-1211-59 7. Li S, Coit DW, Felder F (2016) Stochastic optimization for electric power generation expansion planning with discrete climate change scenarios. Electr Power Syst Res 140:401–412. https:// doi.org/10.1016/j.epsr.2016.05.037 8. Ravadanegh SN, Roshanagh RG (2014) On optimal multistage electric power distribution networks expansion planning. Int J Electr Power Energy Syst 54:487–497. https://doi.org/10. 1016/j.ijepes.2013.07.008 9. Manríquez F, Sauma E, Aguado J, de la Torre S, Contreras J (2020) The impact of electric vehicle charging schemes in power system expansion planning. Appl Energy 262:114527. https://doi.org/10.1016/j.apenergy.2020.114527

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Chapter 4

Design Weight Minimization of a Reinforced Concrete Beam Through Genetic Algorithm and Its Variants Osman Tunca and Serdar Carbas

1 Introduction There are three main materials steel, wood, and reinforced concrete widely utilized in the design of structures nowadays. Among these, the reinforced concrete is the most popular structural material with its cost and various advantages [1, 2]. The reinforced concrete consists of the concrete and the steel. Although the concrete is strong in compression stress, it is weak under the tension stress. The steel has the same strong behavior under the tension and the compression. Therefore, the concrete is supported by steel rebars in regions where the concrete has tensile stress [3]. Thus, the reinforced concrete is popularly emerged as a structural material. However, its calculation is more complex than other structural materials because it is a composite material [4]. Especially in hyper static reinforced concrete design problems, some quantities should be preselected before calculating reinforced concrete structural members. Moreover, all designs in structural engineering are based on three main rules as safety, economy and aesthetic [5]. For this reason, it is not enough for the attained reinforced concrete design to be able to carry loads alone [6]. In addition, it is aimed that the resulting design should be economical. Besides, the resulting design solution must also fulfill the obligations in the relevant practical code specifications [7]. When all these criteria are evaluated, it is understood how complex the reinforced concrete design problems are and that solution methods based on trial and error methods cannot be suitable for the design of the reinforced concrete structure [5]. Therefore, there are many iterative methods used in solving such problems in the literature [8]. Among these, the optimization is most popular iterative solution method O. Tunca (B) · S. Carbas Department of Civil Engineering, Faculty of Engineering, Karamanoglu Mehmetbey University, Karaman, Türkiye e-mail: [email protected] S. Carbas e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_4

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in structural design [9–12]. The elements of an optimization process contain objective function or multi-objective functions, design variables, and design constraints [13]. This definition overlaps with many design problems in structural engineering [10, 14, 15]. Thus, there are many studies in which optimization is used in solving structural design problems in the literature [16–18]. Optimization is divided into two main branches as stochastic and deterministic [19, 20]. The deterministic optimization includes quadratic mathematical functions. So. it is not possible to apply this method to all kinds of structural design problems. However, the stochastic optimization methods can be applied to most of the structural design problems because of their simple internal structure [21]. Yet, it is not claimed that the attained designs via stochastic optimizers are global optimum because of their probabilistic feature[22]. Additionally, the finding the best result for an optimization algorithm is just as important as how long time it takes to find that result. Moreover, the performance of the optimization algorithm can be varying different optimization problems [23]. So, there are various stochastic optimization algorithm in the literature. The standard genetic algorithms (sGAs) are one of the most recent stochastic optimization methods. It was developed by Goldberg and Holland in 1988 inspiring from the theory of evolutionary [24]. sGAs are successfully utilized on many structural engineering design problems [25–28]. But the researchers are not satisfied with the algorithmic performance of the sGAs. The different stochastic algorithms have been developed inspiring natural phenomena [29, 30]. Moreover, the new variants of both sGAs and other stochastic search algorithms have been developed to improve their algorithmic performance, and new stochastic algorithms continue to be generated today. Among these, Bekiroglu et al. present an adaptive version of the sGAs (aGAs) [31] to minimize the weight of the steel truss structures, Togan and Daloglu offer an improved version of sGAs (iGAs1 ) [32] to optimize transmission tower and truss structures, and Tian et al. study on another improved version of sGAs (iGAs2 ) [33] for optimization design of a three-dimensional braided composite joint. These researchers have claimed to attain well algorithmic performance via proposed sGAs variants. The design optimization of a reinforced concrete beam with different sizes to attain minimum structural weight utilizing gray wolf (GW) and backtracking search optimization (BSO) algorithms is handled here as a reference study to evaluate the algorithmic performance [34]. The width and height of reinforced concrete beam, diameters of the longitudinal steel bars, and the steel stirrups in the beam are taken as design variables. The provision rules of Turkish Requirements for Design and Construction of Reinforced Concrete Structures (TS500) [35] and Turkish Building Earthquake Code-2018 [36] are taken into consideration as structural design constraints. In addition to the design criteria in the regulations, geometric constraints are also considered. In reinforced concrete construction applications, the beams usually carry the moment and transmit the shear forces [37]. The design examples in the reference study are subjected to three different distances between two-point loading. In this study, a recently optimized reinforced concrete beam design problem as stated in the reference study [34] is taken into consideration once again to obtain

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minimum structural design weight originally using sGAs and its three different variants. Thus, algorithmic performances of gray wolf [38] and backtracking search [39] optimization algorithms can be comprised of the sGAs and their three different variants handled in this study. Besides, all attained optimal results are statistically discussed in finding the minimum weighted beam design. The fundamental organization of this study is itemized as follows: • In Sect. 1, an introduction section including the principles of the study with the importance of the content is described. • In Sect. 2, the principles of the related previous background studies are summarized. • In Sect. 3, the fundamentals of the reinforced concrete beam design are retailed. • In Sect. 4, the standard genetic algorithms and their variants utilized as optimizers of this study are defined. • In Sect. 5, the handled design optimization problem, which is a reinforced concrete design problem, and attained optimum designs are discussed. • In Sect. 6, the attained main concluding remarks of this study are presented.

2 Related Previous Background Studies In recent years, the structural design optimization of a reinforced concrete beam has been considered by enviable studies in the structural engineering field. Some of these studies aim at minimum cost, while in others, minimum CO2 emission is purposed. Both are indirectly related with usage quantities of the steel and concrete. In reinforced concrete structure design, the diameter of the steel rebars is selected from the ready lists. So, considering directly the reinforcement area as a design variable is not feasible for fabrication. Furthermore, the required reinforcement area can be easily computed for singly reinforced beams. Thus, addition of reinforcement area to design variables is not productive for this type of optimization problems. The practical fundamentals should be implemented according to practice code specifications Additionally, some geometric fabrication conditions should be checked such as that the attained reinforcements should fit into the width of the beam. In this chapter, by considering all these aspects, the minimum weighted reinforced concrete beams are optimized in detail. In this context, the principles of the related previous background studies are summarized as follows: Alsakka et al. study on the design of reinforced concrete structures regarding the contribution of cement production on earth’s greenhouse emissions. Here, depth, width, and reinforcement ratio are taken as design variables. They attain a notable reduction in CO2 emission and cost [40]. The design of reinforced concrete continuous beam is handled in the study of Zang and Wang [41]. Single-objective optimization problem focuses on CO2 emission. The sGAs is taken as optimizer. They suggest carbon reduction strategies. And, they indicate the significance of emission factor variations in the optimum design of reinforced concrete continuous beams.

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Abdelmgeed et al. investigate the optimum cost design of reinforced concrete beams utilizing an artificial bee colony (ABC) algorithm [42]. The width and depth of the beam, diameter of flexural bars, number of flexural bars, diameter of bars for shear reinforcement, and its number are considered as design variables. They indicate that the attained optimum costs strongly depend on the relative cost ratios between concrete, steel, and the formwork material and their costs. Bektas et al. minimize the cost and CO2 emission of the reinforced concrete frames [43]. They utilize the time-history analysis to determine the design loads of the frame. The design rules of the ACI-318 building code are satisfied. They claim that CO2 emission minimization is greatly different from the optimum cost design results for reinforced concrete frame design. Yucel et al. generate sustainable models of rectangular and T-sections reinforced concrete structures via multi-objective three different metaheuristics [11]. The study in which the concrete class is also taken as a design variable proves that C25 and C30 concrete classes are more convenient in cost and CO2 emission. Pierott et al. propose a computation model to minimize the construction cost of reinforced continuous and simply supported concrete beams via sGAs [44]. The finite element analysis software is utilized to obtain this model. The characteristic strength of concrete, cross-sectional area, and reinforcement are the design variables of the optimization problem. In their study, up to 14% cheaper designs have been attained from the comparison of the studies they reference.

3 Design of the Reinforced Concrete Beam The design structural weight minimization of a reinforced concrete beam in structural engineering has crucial importance. Because both the dead load on the beam and the earthquake load acting on the structure are directly related with total structural weight. In Fig. 1, the fundamentals of design problem of a reinforced concrete beam are depicted. Here, some selections should be made during the design of the reinforced concrete beam. These selecting quantities are so-called design variables of the optimum design problem. The optimization process begins through the random selection of the height and width of the reinforced concrete beam. After calculating the required reinforcement areas, the reinforcement diameters to be used in the design are selected. So, the candidate design vector I T consists of four design variables as shown in Eq. 1. Here, I 1 and I 2 represent the width and the height of reinforced concrete beam, and I 3 and I 4 are diameters of the longitudinal rebars and the closed stirrups. ] [ I T = I 1, I 2, I 3, I 4

(1)

The minimum structural weight of the beam equals the multiplication of volumes and unit weights of reinforcing steel and concrete as in Eq. 2. Here, the length and cross-sectional area of reinforcements are multiplied to attain the volume of

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Fig. 1 The reinforced concrete beam

reinforcements. ( ) πθ2 πθ2 L l + ns s L s W = γc hbw L b + γr n 4 4

(2)

In Eq. 2, W denotes the minimum structural weight of the beam. The γ r and γ c represent the unit weight of the reinforcement and concrete. The beam width and height are represented by bw and h. Length (L) of the beam, longitudinal rebars, and stirrups are indexed with b, l,, l, and s, respectively. The diameters of the longitudinal rebars and stirrups are represented with Ø and Øs , and a total number of these are symbolized with n and ns , respectively. As in the abovementioned, the first selection process in reinforced concrete beam design consists of the determination of the width and the height of beam’s concrete portions. The cross section of the concrete should neither be insufficient nor uneconomical. In the reinforced concrete design, it is assumed that the concrete cannot bear tension forces. It carries just the compressive stresses which are formed in the cross section. The distribution of compressive stress along beam cross section is parabolic as seen in Fig. 2a. To facilitate calculation, it is transformed into a rectangular prism model with the help of k 1 and k 3 factors as shown in Fig. 2b. The determination of neutral axis location in the beam is indispensable to compute the moment bearing capacity. In the rectangular prism distribution model, the length of the reduced neutral axis from the top of the beam is symbolized by k 1 c. And, it is calculated by utilizing Eq. 3. / k1 c = d −

d2 −

2Md k3 f cd bw

(3)

The k 1 c is the border of the effective compression area at the same time. The M d is the external design moment, and d is the effective height of the reinforced concrete beam. The design strength of concrete ( f cd ) is converted to transform the rectangular

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Fig. 2 The distribution of compressive stress beam cross-section: a Real distribution, b uniform distribution

prism distribution model by multiplying the k 3 factor. In beam cross section, values of the longitudinal tension and compression force are equal because the mentioned beam is stable. And, the concrete cannot bear tension force in assumption. So, all of the tension forces should be carried by steel reinforcements. The required area of longitudinal steel reinforcement can be easily calculated through this information via Eq. 4. As =

Md ( f yd d −

k1 c 2

)

(4)

After the calculation of As , the designers should select the diameter of the longitudinal rebars. Thus, the number of rebars can be calculated utilizing Eq. 5. Here, the number of longitudinal rebars is represented as n. The expression of Eq. 5 includes int function because the attained value must be an integer. ( n = int

As πθ 2 4

) +1

(5)

The calculations of the length of the longitudinal rebars are also required to determine the weight of the reinforced concrete beam. Figure 1 is used for this calculation. Besides longitudinal rebars, there are steel stirrups in the reinforced concrete beam. So, the beam is strengthened against the shearing effect. In stirrup design,

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the beam is fictionally divided into two regions. These are the confinement region and the remaining region. In the confinement region, the distance between stirrups is decreased by half. The length of the confinement region is considered as two times of the height of the reinforced concrete beam as illustrated in Fig. 1. Here, the area of cross section of the reinforced concrete beam should be enough for designing steel stirrups. So, beam cross section should be controlled via Eq. 6. Vd_ max = 0.85bw h

√

f ck

(6)

Here, V d_max represents the maximum shear force for which the section can be designed. And, f ck is the characteristic strength of the concrete. After checking process, the critic shear force is calculated utilizing Eq. 7. The critic shear force means that shear bearing capacity of the beam in unreinforced condition. Vcr = 0.65 f ctd bw d

(7)

In Eq. 7, V cr and f ctd are the critic shear force and design tension strength of concrete. The remainder of the shear force on the section is carried by the steel stirrups. Thus, the remainder of the shear force should be calculated by Eq. 8. Vw = Vd − 0.8Vcr

(8)

Here, V w is the shear design force on the steel stirrups. Then, the selection of the diameter of the strips is necessary. The distance between two stirrups is calculated regarding with the selected diameter of strips and the requirements of structural specifications. In the confinement region, Eq. 9. is operated to calculate the distance between stirrups. ⎛ ⎜ ⎜ Sc = min⎜ ⎝

Asw f ywd d Vw h 4

8θ 150 mm

⎞ ⎟ ⎟ ⎟ ⎠

(9)

Here, S c is the distance between consecutive two stirrups in the confinement region. In the remaining region, it is indexed with o and symbolized with S o . Moreover, the requirements of structural specifications are changed as in Eq. 10. ⎛

Asw f ywd d Vw

⎜ So = min⎝ Vd ≤ 3Vcr ⇒ So = Vd > 3Vcr ⇒ So =

⎞ d 2 d 4

⎟ ⎠

(10)

After the diameter design of the stirrups, the total number of these and the length of any one should be determined to calculate the minimum structural weight of the beam. Equation 11 is operated for the calculation of the total number of steel stirrups.

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( n c = int

) ( ) 4d L − 4d + 1 + int +1 Sc Sc

(11)

The length of a steel stirrup is determined by means of Fig. 3. Here, the doublearmed steel stirrups are considered. The hook length is calculated by multiplying the diameter of a stirrup with 6. However, the minimum hook length of a stirrup should be 8 cm. On condition of that attained hook length is smaller than 8 cm, it is taken as 8 cm. In addition to the correct mechanical calculations in a design, the applicability of the design is also important. Lastly, that attained reinforcements fit into the width of the beam shown in Fig. 4. should be checked by Eq. 12. bw ≥ 2Cc, + 2θs + nθ + (n − 1)Cc

(12)

In Fig. 4. and Eq. 12, C c is the clear distance between two longitudinal rebars and Cc, is the depth of concrete cover, respectively. Fig. 3 The settlement of steel stirrups in reinforced concrete beam

Fig. 4 The settlement of reinforcements

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4 Genetic Algorithms The Genetic Algorithms (GAs) is a solution method generated by taking the alternation of generations in nature as an example. GAs is a search algorithm based on natural selection and natural genetic mechanisms (based on the evolution thesis of genetics), which can be summarized as the survival of the strong in the natural system, first developed by Charles Darwin [45].

4.1 Alternation of Generations in Nature The GAs is a stochastic search algorithm that imitates some natural phenomena. It mimics the process of biological evolution. According to theory, natural populations are updated from generation to generation according to the principles of natural selection and survival of the fittest, as outlined in Charles Darwin’s Origin of Species [45]. In nature, struggles between individuals for scarce resources such as food, water, shelter, or mates result in the superiority of highly adaptable or fit individuals over weaker ones. The well-adapted individuals continue to live and reproduce by producing more offspring. The low-adaptive individuals have fewer children, perhaps even no children at all. This results in the distribution of the genes of suitable individuals into more individuals in the next generation. The combinations of genes from different ancestors can sometimes result in superior offspring with greater fitness than both ancestors. Thus, species become more and more suitable for the environmental conditions in which they live. For example, some rabbits are quicker and more argus-eyed than others in a rabbit population. These rabbits are less likely to be hunted by foxes. Thus, most of them survive. Some of the other rabbits also live because they are lucky. The living rabbit population continues to breed. Since the majority of the remaining rabbits are faster and argus-eyed rabbits that can get rid of the foxes, the average of the baby rabbits is faster and argus-eyed than the previous population [46].

4.2 General Structure of Standard Genetic Algorithms Genetic Algorithms (GAs) were firstly encoded by Goldberg and Holland in 1988 [24] inspiring the alternation of generations in nature as mentioned in the previous section. The brief pseudocode of the standard genetic algorithms (sGAs) is illustrated in Fig. 5. According to Fig. 5, two main rules of the sGAs can be described as follows: (i) The gender of individuals in the population is not considered. (ii) All process is based on randomization and this is operated by generating random numbers between 0 and 1.

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Fig. 5 Pseudocode of sGAs

4.2.1

Formation of sGAs Population

A solution group/population is randomly generated considering an initially defined search space via Eq. 13. The population size is an important factor here. A large population size requires a lot of computational processing. So, the solution process takes a lot of time. A small number of populations increases the probability of achieving a local optimum. A successful solution vector is much more likely to become dominant within a small population. Therefore, small solution groups lose their diversity very quickly. Since it is not known that the aimed chromosomes are placed in the design space, the population (N) and the size number (d) randomly distribute population into search space. ) ( xi, j = xmin j + ϕ xmax j − xmin j

(13)

Here, i and j represent the integer numbers 1–N and 1–d, respectively. Thus, x i,j is explained as the jth genes of the ith chromosomes. x minj and x maxj are the least and the highest values. The ϕ takes random values between 1 and 0.

4.2.2

Selection Strategies

The selection process is one of the main parts of the sGAs. A minimum of two different chromosomes should be selected to operate the crossover process. The mating probabilities of the strongest members of the population are higher than others in nature. Thus, a roulette wheel is utilized in sGAs. In the roulette wheel, the probability of individuals being selected is determined in proportion to their fitness values. However, there are numerous selectors in the literature. Any of these can be

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Fig. 6 The selectors used in the sGAs [47]

accepted as a selection strategy considering the type of optimization problem. Some of the selectors can be listed in Fig. 6 [47].

4.2.3

Crossover Strategies

After the selection of parents, in case of the probability of crossover is bigger than a new generated arbitrary number within the range of 0–1, the crossover process is initialized. Here, the main target is that generate children’s chromosomes using the parent’s chromosomes. As in selection, there are a lot of crossover methods to produce children’s chromosomes as in Fig. 7 [47].

4.2.4

Mutation Strategies

The mutation is rarely seen in nature. In some cases, a small change occurs on the chromosomes of the new child, which compose of the parent chromosomes. Thus, the genes are obtained that the parents also do not have. In the sGAs, the mutation process serves to avoid local optima. The exploration capacity of the sGAs is boosted via this process. In literature, some utilized mutation methods are illustrated in Fig. 8 [47].

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Fig. 7 The crossover methods used in the sGAs [47]

Fig. 8 The mutation methods used in the sGAs [47]

4.3 Variants of Genetic Algorithms The optimization algorithms are used in solutions of wide-scale optimization problems. The algorithmic performances of these are varied in different fields. So, there

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are several approaches to instrumentally operate these. As mentioned in previous subsections, there are numerous techniques for selection, crossover, and mutation strategy. Moreover, some researchers investigate various updated, improved, and adaptive versions of sGAs. This is because the most important disadvantage of sGAs is that as in the structural optimization problems it takes a long time to reach the optimum result due to the fact that the design problem is not linear, it is quite complex and the design variables are much. In addition, sGAs may frequently encounter the problem of premature convergence in reaching the global optimum. Therefore, in this study, an adaptive and two improved versions of the sGAs are handled as optimizers to minimize the structural weight of the simple supported reinforced concrete beam design problem which was previously optimized via Gray Wolf (GW) and Backtracking Search Optimization (BSO) algorithms in reference study [34].

4.3.1

Adaptive Genetic Algorithms (aGAs)

Bekiro˘glu et al. [31] study on an adaptive version of sGAs as a new alternative to minimum weighted design of 72-bar space truss and 120-bar space truss [31]. They suggest that the probability of mutation and the probability of crossover should not be constant during the optimization process. Besides, they adapt these by utilizing Eqs. 14 and 15. (

0.5( f max − f min )/( f max − f ave ) ⇐ ( f min ≥ f ave ) 1.0 ⇐ ( f min < f ave ) ( 0.5( f max − f )/( f max − f ave ) ⇐ ( f ≥ f ave ) pm = 1.0 ⇐ ( f < f ave )

pc =

(14)

(15)

Here, f is the fitness of a chromosomes, f min , f max , and f ave are minimum, maximum, and average fitness values of current population, respectively.

4.3.2

Improved Genetic Algorithms (iGAs1 )

To˘gan and Dalo˘glu [32] develop an improved genetic algorithms (iGAs1 ), and they tested it on the design optimization of 10-bar truss, 25-bar space truss, 120-bar space truss, 200-bar truss, 240-bar roof truss, and 244-bar truss steel transmission tower structures [32]. When the attained results are investigated, the proposed improved version of genetic algorithms yields better solution than harmony search (HS) algorithm and sGAs. The proposed modification formulations on probabilities of crossover and mutation are given as following equations. ( pc =

( f max − f low )/( f max − f ave ) ⇐ ( f low ≥ f ave ) 1.0 ⇐ ( f low < f ave )

(16)

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( pm =

0.5( f max − f )/( f max − f ave ) ⇐ ( f ≥ f ave ) ( f ave − f )/( f ave − f min ) ⇐ ( f < f ave )

(17)

Here, the f low is the lower fitness value of each parent.

4.3.3

Improved Genetic Algorithms (iGAs2 )

Tian et al. [33] present an improved version of sGAs to optimize 3D braided joints with the help of its finite element model [33]. They suggested a different dynamic algorithm parameter approach. The process initializing with twice constants for crossover and mutation probabilities. This process is operated according to the following equations. ( pc = ( pm =

pc1 − ( pc1 − pc2 )( f − f ave )/( f max − f ave ) ⇐ ( f ≥ f ave ) pc1 ⇐ ( f < f ave )

(18)

pm1 − ( pm1 − pm2 )( f − f ave )/( f max − f ave ) ⇐ ( f ≥ f ave ) pm1 ⇐ ( f < f ave )

(19)

Here, pc1 and pc2 are two constant values which assigned to algorithm in the initial step. They represent the probabilities of the crossover. The pm1 and pm2 represent the probabilities of mutation. In the reference study, while the pc1 and pc2 are taken as 0.8 and 0.1, the pm1 and pm2 are considered as 0.25 and 0.001, respectively.

5 Design Examples As design examples, the reinforced concrete beam taken from the reference study [34] is optimally designed via sGAs and its abovementioned three different variants (aGAs, iGAs1 , and iGAs2 ) for three different distances between two-point loading. This distance is symbolized with a as illustrated in Fig. 9. All design examples have a 6 m total span length. In the application of reinforced concrete structures, the beams are fixed on columns, and columns behave as supports. In all design examples, the width of columns/supports are considered as 40 cm. The material properties of steel longitudinal rebars and steel stirrups are taken as the same. The yield strength of steel is taken as 420 MPa. The same quantity is taken 25 MPa for concrete. The unit weights of both materials are considered as 24 kN/m3 and 76.98 kN/m3 , respectively. The simple supported reinforced concrete beam is exposed to uniformly distributed design load of 25 kN/m and two-point design loads of 100 kN as illustrated in Fig. 9. The locations of two-point loading represent the joint of beam-to-beam connection in any reinforced concrete structure. The distance between the abovementioned two

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points can vary from structure to structure. Thus, the a distance in Fig. 9. (distance between two-point loads) is variously considered as 2, 3, and 4 m. Thus, three different design examples occurred are attained for the same reinforced concrete beams. A reinforced concrete beam in a structure carries its own load beside the external loads. Moreover, there is a direct relation between the magnitude of the earthquake load and the total structural weight. So, during an optimization process, minimum weighted structural design is aimed as a main objective. The optimum design solution of a simple supported reinforced concrete beam cannot manually be calculated since the selections of some quantities by trial and error directly affect the design during this calculation. So, all selections during the calculation have to be considered as design variables. The height (h) and the width (bw ) of reinforced concrete beam are often estimated by a designer before initializing the calculation. Additionally, after the calculation of the areas of longitudinal rebars (Ø), the diameter of these should be selected, as well. Similarly, the diameter of the steel stirrup (Øs ) should be selected during the design of the reinforced concrete beam. Eventually, these abovementioned parameters are considered as design variables of simple supported reinforced concrete beam design problem. The structural practice code requirements mentioned in Turkish Requirements for Design and Construction of Reinforced Concrete Structures [35] and Turkish Building Earthquake Code [36] are considered as structural design constraints. Moreover, the geometric constraint is added to check whether the steel reinforcements fit into the beam width. The pc and pm parameters of sGAs are taken as 0.9 and 0.1, respectively. The population sizes are taken as 20, and maximum iteration number of each optimization process is fixed as 500 to make a valid comparison with reference study [34]. Minimum weighted structural designs of simple supported reinforced concrete beams are attained via sGAs, aGAs, iGAs1 , and iGAs2 , respectively. The optimum results of these and the GW and BSO algorithms announced in the reference study are tabulated in Table 1. Although the geometric constraint related to steel rebars compromises with the width of the beam is appeared as dominant, any violations of the design constraints are not detected in attained optimum designs. Each optimizer is operated on each design example five independent times. Thus, statistic

Fig. 9 The static structural illustration of the simple supported reinforced concrete beams

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evaluations can be made on optimal design results. The best design history graphs for each design example are given in Fig. 10. In the simple supported reinforced concrete beam with 2 m distances between two-point loading design example, when the sGAs yield 19.742 kN minimum design weight, the aGAs, iGAs1 , and iGAs2 attain 19.409 kN, 19.415 kN, and 19.511 kN design weights, respectively. While sGAs need 384 iterations for finding minimum weight, the aGAs, iGAs1 , and iGAs2 require 451, 462, and 422 iterations, respectively. Yet, in the reference study, GW and BSO algorithms find the same design weight as 17.371 kN in 377th and 221st iterations. So, that means BSO algorithm converges 73.76% more rapidly than the sGAs, and the best minimum design weight of which is 13.64% heavier than the yielded weight announced in the reference study. In the simple supported reinforced concrete beam with 3 m distances between two-point loading design example, while the sGAs obtain the minimum weight of 18.270 kN, the aGAs, iGAs1, and iGAs2 yield optimal design weights of 17.916 kN, 17.553 kN, and 17.549 kN, respectively. Thus, best minimum design weight is computed via iGAs2 . Yet, this value attained approximately 16 kN in the reference study. However, when the iteration numbers are examined, it is seen that the iGAs2 reaches the optimum design 78.26 and 45.34% faster than the GW and BSO algorithms in the reference study. Also, in the simple supported reinforced concrete beam with 4 m distances between two-point loading design example, the sGAs, aGAs, iGAs1, and iGAs2 attain 16.469 kN, 15.935 kN, 15.904 kN, and 16.329 kN minimum weighted designs in 477th, 490th, 494th, and 476th iteration, respectively. Both optimization algorithms of GW and BSO announced in the reference study obtain the same design weight as 14.534 kN in 315th and 350th iterations, respectively. So, it can be clearly observed from Fig. 10 that the best minimum design weight attained for this design case in this study is 9.43% more heavier than those announced in the reference study. Furthermore, the minimum design weight is obtained 51.11% more fast than the best one among sGAs and their variants. Moreover, the statistical findings are also tabulated in Table 1, and the box-plot diagrams are illustrated in Fig. 11. The worst, average, and the standard deviation values are given to demonstrate the scattering of the attained minimum design weights via five different initial populations. The standard deviations and the box-plots lead to an idea about the robustness of the utilized optimization algorithms. The statistical all data argues that the novel stochastic search algorithms, GW and BSO, illustrate robust and more stable performance than sGAs and its three variants of aGAs, iGAs1, and iGAs2 in finding the minimum weighted design of the simple supported reinforced concrete beams.

6 Conclusions In this chapter, the simple supported reinforced concrete beam design problems, which have recently been optimized by backtracking search optimization (BSO) algorithm and gray wolf (GW) optimizer as announced in the reference study, is

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Table 1 The attained final minimum weighted designs Optimizer 2 m bw (mm)

sGA [24] aGA [31] iGA1 [32] iGA2 [33] BS [39] GW [38] 284

304

298

310

252

252

h (mm)

474

435

444

429

468

468

Øs

8

8

8

8

8

8

Ø

30

34

34

36

40

40

Weight (kN)

19.742

19.409

19.415

19.511

17.317

17.317

No. of iteration

384

451

462

422

221

377

21.159

21.123

21.502

20.265

17.394

Worst weight (kN) 23.872 Std. dev

1.718

0.898

0.712

0.799

1.307

0.035

Mean weight (kN)

20.960

20.080

20.674

20.297

18.718

17.379

289

292

258

291

251

250

3 m bw (mm) h (mm)

431

418

464

410

433

434

Øs

8

8

8

8

8

8

Ø

36

38

38

32

38

38

Weight (kN)

18.270

17.916

17.553

17.549

15.992

15.966

No. of iteration

479

464

480

161

287

234

20.162

18.775

19.558

17.189

15.966

Worst weight (kN) 21.197 Std. dev

1.219

0.843

0.499

0.757

0.578

0.000

Mean weight (kN)

19.404

19.266

18.248

18.516

16.395

15.966

306

295

255

275

250

250

4 m bw (mm) h (mm)

365

366

424

403

394

394

Øs

8

8

8

8

8

8

Ø

36

32

32

28

38

38

Weight (kN)

16.469

15.935

15.904

16.329

14.534

14.534

No. of iteration

477

490

494

476

350

315

17.881

17.589

17.663

15.596

14.534

Worst weight (kN) 19.254 Std. dev

1.243

0.830

0.693

0.491

0.435

0.000

Mean weight (kN)

17.745

16.672

16.762

17.050

14.916

14.534

re-handled via sGAs and its three different variants of aGAs, iGAs1 , and iGAs2 . To represent beam-to-beam connections in different positions, the handled simple supported reinforced concrete beams with 2, 3, and 4 m distances between two-point loads are taken into consideration. The main purpose of design examples is to attain the minimum weighted reinforced concrete beam design. The design weight of the steel stirrups was included in the computation of the weight of the longitudinal rebars and concrete to obtain the total weight of the reinforced concrete beams. Since the fabrication conditions are also considered, all design variables are taken as discrete. Moreover, the geometric constraint is added for the same purpose. Additionally, the structural practice code requirements of Turkish Requirements for Design and

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Fig. 10 The design history graphs of reinforced concrete beams having, a 2 m; b 3 m; and c 4 m between two-point loads

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Fig. 11 The box-plot diagrams of the reinforced concrete beams having, a 2 m; b 3 m; and c 4 m between two-point loads

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Construction of Reinforced Concrete Structures [35], and Turkish Building Earthquake Code [36] are also treated as structural design constraints. Thus, three different design examples are optimized utilizing sGAs and its abovementioned three variants. The main conception of the utilized variants of sGAs is based on the converting the constant algorithm parameters to dynamic algorithmic parameters. Thus, the algorithmic performances of sGAs and their variants are comprehensively compared. All optimization algorithms compute each design example five independent times to statistically examine the attained results. The obtained main conclusions of this study can be briefly given as following: • The stochastic-based algorithms can effectively be utilized as a design tool to attain a minimum structural weight of reinforced concrete beams. • All the modified versions of sGAs operated in this chapter perform more robust than sGAs. • In the design examples with 3 m distances between two-point loading design example, the best minimum weight is computed via iGAs2 . Yet, this value attained approximately 16 kN in the reference study via both GW and BSO algorithms. However, when the iteration numbers are examined, it is seen that the iGAs2 reaches the optimum design 78.26 and 45.34% faster than the GW and BSO algorithms. Except this, in all other design attempts, the GW and BSO algorithms present better algorithmic performances. • Finally, the design history graphs indicate that sGAs and their various variants have a strong exploration capacity. As its exploitation capacity is increased, more successful results are attained in the simple supported reinforced concrete design optimization problems.

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25. Deb K, Gulati S (2001) Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37:447–465. https://doi.org/10.1016/S0168-874X(00)00057-3 26. Vu-Huu T, Pham-Van S, Pham Q-H, Cuong-Le T (2023) An improved bat algorithms for optimization design of truss structures. Structures 47:2240–2258. https://doi.org/10.1016/j.ist ruc.2022.12.033 27. Aydın Z (2022) Size, layout and tendon profile optimization of prestressed steel trusses using Jaya algorithm. Structures 40:284–294. https://doi.org/10.1016/j.istruc.2022.04.014 28. Liu D, Wang Z, Pan J, Zheng Y, Hu Z (2022) Optimum design of nonlinear semi-rigid steel frame based on performance-price ratio via genetic algorithm. J Build Eng 61:105287. https:/ /doi.org/10.1016/j.jobe.2022.105287 29. Dehghani M, Montazeri Z, Trojovská E, Trojovský P (2023) Coati optimization algorithm: a new bio-inspired metaheuristic algorithm for solving optimization problems. Knowl-Based Syst 259:110011. https://doi.org/10.1016/j.knosys.2022.110011 30. Kaveh M, Mesgari MS, Saeidian B (2023) Orchard algorithm (OA): a new meta-heuristic algorithm for solving discrete and continuous optimization problems. Math Comput Simul S0378475422005146. https://doi.org/10.1016/j.matcom.2022.12.027 31. Bekiro˘glu S, Dede T, Ayvaz Y (2009) Implementation of different encoding types on structural optimization based on adaptive genetic algorithm. Finite Elem Anal Des 45:826–835. https:// doi.org/10.1016/j.finel.2009.06.019 32. To˘gan V, Dalo˘glu AT (2008) An improved genetic algorithm with initial population strategy and self-adaptive member grouping. Comput Struct 86:1204–1218. https://doi.org/10.1016/j. compstruc.2007.11.006 33. Tian Z, Yan Y, Hong Y, Guo F, Ye J, Li J (2018) Improved genetic algorithm for optimization design of a three-dimensional braided composite joint. Compos Struct 206:668–680. https:// doi.org/10.1016/j.compstruct.2018.08.051 34. Tunca O, Carba¸s S (2022) Minimum weight design of reinforced concrete beams utilizing grey wolf and backtracking search optimization algorithms. Chall J Struct Mech 13:72–79. https:// doi.org/10.20528/cjcrl.2022.02.003 35. TS500 (2000) Requirements for design and construction of reinforced concrete structures, Standard TS500, Ankara, Turkey 36. TBEC (2018) Turkish building earthquake code specifications for design of buildings under seismic effects, Ankara, Turkey 37. Liao W, Liu K, Ma C, Liang J (2023) Experimental study on consistency of the impact performance between composite beams and reinforced concrete beams. Compos Struct 308:116677. https://doi.org/10.1016/j.compstruct.2023.116677 38. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007 39. Civicioglu P (2013) Backtracking search optimization algorithm for numerical optimization problems. Appl Math Comput 219:8121–8144. https://doi.org/10.1016/j.amc.2013.02.017 40. Alsakka F, Haddad A, Ezzedine F, Salami G, Dabaghi M, Hamzeh F (2023) Generative design for more economical and environmentally sustainable reinforced concrete structures. J Clean Prod 387:135829. https://doi.org/10.1016/j.jclepro.2022.135829 41. Zhang X, Wang F (2023) Influence of parameter uncertainty on the low-carbon design optimization of reinforced concrete continuous beams. Struct Concr 24:855–871. https://doi.org/ 10.1002/suco.202100903 42. Abdelmgeed FA, Ghallah G, Hamoda A (2022) Optimum cost design of reinforced concrete beams using artificial bee colony algorithm. Int J Adv Struct Geotech Eng 06:110–129. https:/ /doi.org/10.21608/asge.2022.274739 43. Bekda¸s G, Nigdeli SM, Kim S, Geem ZW (2022) Modified harmony search algorithm-based optimization for eco-friendly reinforced concrete frames. Sustainability 14:3361. https://doi. org/10.3390/su14063361 44. Pierott R, Hammad AWA, Haddad A, Garcia S, Falcón G (2021) A mathematical optimisation model for the design and detailing of reinforced concrete beams. Eng Struct 245:112861. https:/ /doi.org/10.1016/j.engstruct.2021.112861

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Chapter 5

IGA: An Improved Genetic Algorithm for Real-Optimization Problem Abdelmonem M. Ibrahim and Mohamed A. Tawhid

1 Introduction Many engineering and scientific models are known to be based on the system of nonlinear equations (NLSs), and it is crucial to their effective solution for these fields to advance. NLSs can arise directly in some applications or indirectly through converting real-world models into NLSs [1]. Theoretically, it can be challenging to develop an efficient and reliable solution for the NLSs. While solving a single nonlinear equation is simple, doing it for a group of nonlinear equations is challenging. One of the most studied issues in science and engineering, resource scheduling, including data fitting, vehicle route planning, and scheduling, is how to solve a set of NLSs [2–4]. To deal with NLSs problems, various numerical techniques, including the Newtontype method and iterative and recursive techniques [5–7], have been proposed. Such techniques have various shortcomings, even if they significantly improve performance when solving NLSs. For instance, most of them use single-point methods, which cannot find multiple roots in a single run. Moreover, the initial guess selection affects how well numerical methods work. It is challenging to locate the exact roots using the numerical method when an initial guess is quite far from the optimal situation. Nonetheless, choosing the right first guess is a simple task. The researchers used population-based methodologies to solve a range of optimization problems as a result of these restrictions. The collective behavior that evolved from social insects cooperating according to a minimal set of norms can be compared to population-based techniques. Fish school, birds flock, and bugs A. M. Ibrahim Mathematics Department, Faculty of Science, Al-Azhar University, Assiut Branch, Cairo, Egypt e-mail: [email protected] M. A. Tawhid (B) Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC V2C 0C8, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_5

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swarm are examples of prominent algorithms used in population-based techniques. Population-based techniques are also regarded as computer models that replicate natural swarm processes. Many population-based approaches have, up until this point, been put forth in the literature and effectively used to address optimization issues [8, 9]. Researchers have looked at free-derivative algorithms (population-based methods) such as Genetic Algorithm (GA) [10], Particle Swarm Optimization (PSO) [11, 12], Whale Optimization Algorithm (WOA) [13], Artificial Rabbits Optimization (ARO) [14], Bacterial Foraging Optimizer (BF) [15], Cat Swarm Optimization (CSO) [16], Sine-Cosine Algorithm (SCA) [17], Cuckoo Search Algorithm (CS) [18], Grasshopper Optimization Algorithm (GOA) [19], Gradient-Based Optimizer (GBO) [20], and Artificial Rabbits Optimization (ARO) [14]. GA has been broadly utilized in optimization in recent years because of its superb global search abilities and highly parallel processing capacities [21, 22]. Nevertheless, there are combats between global optimum and convergence speed in some optimization problems, which causes slow convergence in GA. Since solving nonlinear systems is not simple, numerous numerical methods and optimization strategies have been used over time [23, 24]. To solve the NLS problem, proposing an improved GA (IGA) is one of the effectual approaches. In this chapter, the genetic algorithm’s crossover operator and mutation operator are improved to enhance the convergence efficiency and precision of the algorithm without affecting the effectiveness of the IGA on most optimization problems. The efficacy of the IGA is also verified through many comparison experiments and applications in the field of nonlinear system applications. The main contributions of this chapter are as follows: • Two mutation operators are proposed to improve the global search in the stable case of local optimal by enhancing the fundamental mutation of the IGA, which traverses random genes in the offspring. The first mutation operator is the dimensional mutation operator with two different mutation factors and the second mutation operator is the suggested manual/standard mutation. • Using two different pairs of weight/mutation factors, we construct the dimensional crossover operator between the genes from selected parents and the genes obtained via the dimensional mutation operator, enhancing exploration and exploitation across the search phase. The population is divided depending on the crossover probability to apply the crossover operator in crossover operation on the portion of chromosomes randomly chosen at each iteration, and the remaining portion is for mutation. • GA is applied on 30 CEC’18 benchmark problems and eight nonlinear systems to test and check its performance by comparing it with standard continuous GA and other algorithms such as Particle Swarm Optimization (PSO) [11, 12], Whale Optimization Algorithm (WOA) [13], Artificial Rabbits Optimization (ARO) [14], Adaptive Opposition Slime Mould Algorithm (AOSMA) [25], Improved Cuckoo Optimization Algorithm (iCOA) [26], and Improved Real-coded Genetic Algorithm (IRGA) [27].

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The organization of this chapter is as follows: Sect. 2 presents the related work in the area of solving the nonlinear equations, Section 3 presents our proposed improved approach, IGA. The experimental results of the suggested strategy IGA compared with other techniques are shown and discussed in Sects. 4 and 5. Further works are then included in Sect. 6 to wrap up this chapter.

2 Related Work In this section, the prominent related research being carried out in the area of nonlinear equation systems is discussed. El-Shorbagy et al. [1] presented a hybrid grasshopper optimization algorithm (GOA) with GA. The hybrid approach integrates the merits of both GOA and GA, where GOA’s exploitability and GOA’s exploration potential are combined. Furthermore, they use the technique of changing the initial intervals of the variables. Venkatesh and Deepak [28] introduced a new variant of the GA developed to handle multivariable, multiobjective, and very high search space optimization problems like the solving system of nonlinear equations. It is an integer-coded GA with conventional crossover and mutation. However, an inverse algorithm varies its search space by varying its digit length on every cycle and does a fine search followed by a coarse search. Gong et al. [29] presented a comprehensive survey of the most recent development of the transformation techniques in nonlinear equations. They also reviewed solving nonlinear equations (NEs) with intelligent optimization algorithms. They tested the performance comparison of several state-of-the-art algorithms on benchmark functions. They tried to give a full image of the challenges and possible open issues for solving nonlinear equations. Weifeng et al. [30] proposed a two-phase evolutionary algorithm to find multiple solutions of a nonlinear equation system. It transforms a nonlinear equation system into a multimodal optimization problem. In phase one of the proposed algorithm, they proposed a strategy that combines a multiobjective optimization technique and a niching technique to maintain population diversity. Phase two consists of detection and local search methods to encourage convergence. The detection method finds several promising subregions, and the local search method locates the corresponding optimal solutions in each promising subregion. Zuowen et al. [31] examined their proposed method on 30 NESs. They introduced a generic framework of memetic niching-based evolutionary algorithms. They listed four main features of the framework: (i) the numerical method for an NES is integrated into an EA to obtain highly accurate roots; (ii) the niching technique is employed to improve the diversity of the population; (iii) different roots of the NESs are located simultaneously in a single run; and (iv) different numerical methods and different niching techniques can be used in the framework. Xinming et al. [32] investigated the ability of the cuckoo search algorithm to solve nonlinear equations. They evaluated the usefulness of the niche cuckoo search algorithm based on the fitness-

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sharing principle, where a Niche strategy is introduced to enhance the efficiency of the cuckoo search algorithm. Several numerical methods have recently been produced to solve NEs such as Raj and Arora [33] presented the iterative methods of fifth- and eighth-order convergence for solving systems of nonlinear equations. Fifth-order method comprises two steps, namely, Newton’s and Newton-like steps. It requires evaluating two functions, two first derivatives and one matrix inversion in each iteration. The eighth-order method comprises three steps, of which the first two steps are that of the proposed fifth-order method, whereas the third is Newton-like. This method requires one extra function evaluation in addition to the evaluations of the fifth-order method. Abubakar and Kumam [34] proposed an algorithm for solving a system of nonlinear equations. The idea combines the descent Dai-Liao method and the hyperplane projection method. They used the monotonicity and Lipschitz continuity assumptions and proved that the proposed method is globally convergent. As we observe that most of the related work employed independent techniques for solving nonlinear equations while using various evaluation metrics. Furthermore, they used different techniques to improve the ability of their proposed methods to solve such problems, especially the methods that combine with GA. In contrast, our proposed method used different techniques depending on combining the mutation and crossover operators, increasing the efficiency in finding the optimal solution.

3 Improve Genetic Algorithm The goal of a genetic algorithm is to find the global optimal solution by preserving the best chromosomes in the population through fitness function selection, and the requirement for obtaining the best individuals is to ensure population diversity in order to ensure the algorithm’s quick convergence and successfully prevent reaching the local optimal solution. In order to accurately find the global optimal solution, the genetic algorithm heavily relies on selecting crossover and mutation probabilities. Scientifically sound probability parameters can help genetic algorithms work better globally by effectively preventing the algorithm’s premature occurrence. In light of the problems mentioned above, this chapter makes some improvements to the conventional genetic algorithm, focusing on the crossover and mutation operations in the improved GA algorithm and using the random parent selection instead of using other selection techniques such as roulette wheel, in order to bring the algorithm closer to the population’s actual situation and improve the efficiency of screening out excellent individuals. Figure 1 displays the proposed method’s flowchart. The genetic algorithm is improved in this research in the following two ways.

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Fig. 1 Flowchart of a proposed IGA

3.1 Adaptive Crossover and Mutation The crossover operation is the fundamental component of a genetic algorithm that simulates individual gene recombination. More top-notch new individuals are regularly produced by crossover operations, broadening the search space covered by the algorithm and ensuring the genetic algorithm’s superior search performance. The choice of its value is crucial because it is the single index of crossover operation intensity. The algorithm is likely to become slow and ineffective, which will have a detrimental effect on the algorithm’s overall effectiveness if the crossover chance is too low. The algorithm will continue to raise its search intensity if the crossover chance is too high, but its overall effectiveness will suffer. The crossover operator enhances the quality of the offspring. The crossover method chooses the two parents at random, reducing repetitions and accelerating convergence. Using different pairs of weight/mutation factors, we create the dimensional crossover operator in IGA between the selected parents on one side and the other of the chromosomes obtained from the dimensional mutation operator to enhance exploration and exploitation throughout the search phase. The population is divided depending on the crossover probability to apply the crossover operator in crossover conduct on the portion of chromosomes randomly chosen at each iteration. The remaining portion is for mutation. In order to improve the crossover operator, this research uses an adaptive crossover probability to change the crossover probability pc and employing continually. An additional crossover parameter alpha is employed to identify the genes chosen from

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the selected parents to produce a new offspring. In this study, the number of crossover offspring, denoted by nc, is as follows: nc = 2 × round(pc × N/2),

(1)

where the population size is N , and the probability of a crossover is pc. The proposed updating of the offspring depends primarily on the dimensional mutation operator and selecting the two parents randomly from the current population. Real-valued search spaces commonly implement the mutation by giving each element of the vector a regularly distributed random value. Self-adaptation frequently controls the step size or mutation strength. Here, we employ two steps of varying sizes ε1,2 Eqs. 2 and 3 that are based on the population’s best chromosome so far. The parents are chosen at random from the current population Pr 1 , Pr 2 , and two of weight/mutation factors μ and δ. ε1 = μ(P ∗ − Pr2 (t)), ∗

(2)

ε2 = δ(P − Pr1 (t)),

(3)

ch m1 (t + 1) = Pr1 + ε1 ,

(4)

ch m2 (t + 1) = Pr2 + ε2 ,

(5)

where μ and δ indicate that the mutation factors are used in order to enhance the exploration and exploitation of the search by improving the quality of the new offspring ch 1,2 (Eq. 7). ch m1 and ch m2 are the mutated children, and t is the current iteration. It is worth mentioning that the number of the generated children in each iteration is taken based on the crossover probability, which means the iterative overpopulation will be nc/2. Here, we set δ = max(0.3, rand(0, 1)), while the other factor is affected by the number of iterations. This parameter is adapted from [14] as follows: | ⎡ ( )2 ⎤|| | t−1 | | T ⎦| , μ = ||sin(2 × pi × rand) × ⎣e1 − e | | |

(6)

where T denotes the number of iterations that can be conducted. The crossover operator will then be used to generate the new offspring. Based on the crossover percentage parameter μ, the new children can be generated as follows: { j ch i (t

+ 1) =

j

ch m1 (t + 1), rand(0, 1) ≤ α j otherwise. Pi (t),

(7)

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem

{ j ch i+1 (t

+ 1) =

111

j

ch m2 (t + 1), rand(0, 1) ≤ α; j otherwise. Pi+1 (t),

(8)

Algorithm 1 illustrates the phases of the suggested modified algorithm IGA.

Algorithm 1: Improved Genetic Algorithm (IGA) Initialize population with N chromosomes; Set crossover probability pc; Crossover rate for each solution α; The mutation rate for each solution pm; iteration counter t = 1; Determine crossover Offsprings number nc based on Eq. 1; while (not tr mmax or find the optimal chromosome) do compute the objective function of each chromosome; Replace the existing population with a newly created population based on fitness; → ∈ [1, N ]; Select random numbers of chromosomes to be mutated − nm for i ← 1 to N step 2 (Half chromosomes in the population) do if i in nm | i + 1 in nm then % Apply normal mutation; Apply mutation on (i )th and (i + 1)th offspring with mutation rate according to Eq. 9; else Update the mutation factor μ based on Eq. 6; for j ← 1 to D (All genes in current chromosome) do % Apply mutation and crossover operations with crossover rate to the selected pair; Pick a random pair of parents from the current population (Pr1 /= Pr2 ); Apply the mutation operator according to Eqs. 4 and 5; % Updating the first child Eq. 7; if rand ≤ α then Update the current gene of ith chromosome based on the mutation vector j (ch m1 ); else Update the current gene from the current ith chromosome; end % Updating the second child Eq. 8; if rand ≤ α then Update the current gene of (i + 1)th chromosome based on the mutation j vector (ch m2 ); else Update the current gene from the current (i + 1)th chromosome; end end end end Increment the current iteration t by 1; end Set the best objective value as the best chromosome;

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3.1.1

Normal Mutation

Although mutation can keep the population diverse, it cannot ensure that the population and the algorithm will follow the same convergence trend or that the speed of convergence will not be impacted. A normal mutation is suggested to preserve the equilibrium between population diversity and convergence speed and enhance algorithm performance. To attain the nm of mutated individuals, the normal mutated individuals are ran→ = randperm(N, (N−nc)/2). Once domly chosen from the current population as − nm the individuals have been chosen for mutation, choose a number of genes to be changed based on the mutation probability pm, which is set to c = ⎡pm × D⏋, 1 ≤ c ≤ D (the ceiling function is employed here to guarantee that at least one chromosomal gene will be changed). Then pick c of the genes from the current chro− → mosome (i or i + 1 severally) at random and set them in C . The new normal mutated offspring ch i and ch i+1 will be updated based on the following equation: { j ch i or i+1 (t

+ 1) =

− → j Pi or i+1 (t) + [0.1 × (u − l) × r andn], j ∈ C ; j Pi or i+1 (t), otherwise,

(9)

where the top and lower boundaries are u and l, respectively, r andn is a random scalar drawn from the standard normal distribution. Note, if current chromosomes i and i + 1 match the chosen random chromosomes (based on the crossover probability pc), the mutation is applied to them (see Algorithm 1).

3.2 Selection Operator Once the population has reached the same size as before the reproduction, crossover, and mutation processes, the algorithm compares the values of each individual in the new generation to those of the corresponding individuals in the previous generation. If it fails, the previous generation individual is kept, and the new generation individual is ignored; if it succeeds, the previous one is replaced with the new one. This method protects the best individuals from mutation, crossover, and reproduction impacts. It is a critical guarantee for the convergence of the improved genetic algorithm.

4 Computational Evaluation To demonstrate the computational strength of our approach, this section offers a thorough analysis of IGA. By comparing the performance of the IGA algorithm with basic GA and other methods from the literature, two computer simulations are conducted to assess the usefulness of the IGA algorithm. The first examination uses CEC

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2018 benchmarks [35]. This experiment demonstrates IGA’s superiority compared to other state-of-the-art optimization algorithms in terms of accuracy and speed of convergence. Eight real nonlinear systems from the field of chemical engineering and others are used in the second experiment. The parameter settings of the proposed algorithm IGA list as crossover probability pc = 0.9, crossover rate for each solution α = 0.8, and mutation probability pm = 0.2. While in the continuous basic GA, the parameter setting lists as crossover probability pc = 0.9 and mutation probability pm = 0.1. However, the parameter choices for the comparative methods are obtained from the articles in which they originally appeared. In terms of the minimum, average, std (standard deviation), and maximum objective values, the outcome results are contrasted with those of other algorithms. Additionally, the obtained results are statistically analyzed using Friedman and Wilcoxon’s rank-sum tests, two non-parametric statistical methods. These tests are run at the 5% level of significance [36]. To avoid transitiveness between the results, it is required to compare using the Friedman test since there are multiple algorithms involved in [36]. The experiments are carried out using MATLAB 2019a, and they are powered by an Intel Pentium G3220 processor running at 3.0 GHz, 8 GB of RAM, and Windows 10 with 64-bit support.

4.1 Unconstrained CEC 2018 The first computer experiment using 30 benchmark CEC 2018 functions with 30 dimensions is shown in this subsection [35]. Four categories can be defined of these functions: (1) unimodal functions (f1–f3); (2) simple multimodal functions (f4–f10); (3) hybrid functions (f11–f20); and (4) composition functions (f21–f30). Further detailed explanations of these functions may be found at [35]. IGA is used to test the performance of CEC 2018 by comparing it with standard continuous GA and other algorithms from the literature, such as Particle Swarm Optimization (PSO) [11, 12], Whale Optimization Algorithm (WOA) [13], Artificial Rabbits Optimization (ARO) [14], Adaptive Opposition Slime Mould Algorithm (AOSMA) [25], Improved Cuckoo Optimization Algorithm (iCOA) [26], and Improved Real-coded Genetic Algorithm (IRGA) [27].

4.1.1

Results and Discussion

This subsection uses 30 benchmark functions from CEC 2018 to test the performance of the suggested technique. Additionally, performance evaluation criteria include the lowest mean, standard deviation, maximum fitness scores, and convergence graphs. In the search range [–100,100] D , IGA is compared with the other optimization methods based on 30-dimensional (30D) versions of the CEC 2018 benchmark functions. There are 30 individuals in the population, and there can be a maximum of (103 , D × 103 ) evaluations of fitness, where D is 30 (dimensional function) [35].

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Tables 1, 2, 3 and 4 show the outcomes of 30 separate runs of each method. The efficiency of the proposed method is shown in the obtained results, and the performance between other metaheuristic algorithms becomes more competitive. It is important to note that the author of the benchmark CEC 2018 eliminated function f2; hence it is listed as non-applicable (N/A) in the results tables. Tables 5 and 6 display the outcomes of Friedman ranks and Wilcoxon signed-rank tests, respectively. The Wilcoxon signed-rank test shows the significance between the two algorithms. There are three h signs in Table 6. The “1” sign indicates that the null hypothesis has been disproved and that the first method performs better than the second. The null hypothesis is rejected, and the first method is inferior to the second one when the sign “−1” is present. The null hypothesis is accepted, as shown by the sign “0”, and the first algorithm matches the second one. In Table 1, IGA effectively solves most functions of the CEC’18 benchmark according to the best fitness values. In unimodal functions, only IGA obtains the optimal solution for f1. Nevertheless, in multimodal, IRGA finds the best solutions for five functions out of seven, including the best solution for function six, while suggested IGA finds the best solutions for the remaining two functions. In the hybrid functions, the IGA surpasses all other algorithms in this group’s functions for locating the optimal solution. The algorithms IGA, GA, and IRGA are competitive in identifying the optimum solution for the composition functions group. The proposed IGA outperforms in scoring five out of 10, followed by IRGA with three and GA with two. Conversely, the remaining algorithms are out of competition in finding the best solution for all groups (Table 1). The results in Table 2 show the best average fitness values that various algorithms successfully achieve. IGA and ARO achieve the best outcomes for the f1 and f3 unimodal functions class, respectively. For multimodal problems, IGA achieves the best solutions for functions f4, f5, f7, f9 and f10, while IRGA achieves high precision for functions f6 and f8. For nine out of ten functions in hybrid problems, IGA outperforms other algorithms; however, for remained function, f15, ARO achieves the best results. In a comparable situation, IGA outperforms the other algorithms in the composition functions group for 8 of the ten functions. However, it fails to get the best results for functions f22 and f28, achieved by ARO and IRGA, respectively. On the other hand, the algorithms IGA, ARO, and IRGA have the best results, as can be seen from the study results in Table 2, whereas the remaining algorithms are not in competition. The results shown in Table 3 demonstrate the best standard deviation values that various algorithms can produce for the benchmark functions CEC 2018. IGA and ARO get the best results for the f1 and f3 unimodal function classes, respectively. IGA produces the greatest results for functions f5, f7, f8, f9, and f10 in multimodal problems, whereas AOSMA produces the best results for f4 and f6 for IRGA. IGA surpasses other algorithms for five of the ten functions in hybrid problems; for functions f15 and f17, IRGA produces the best results, and GA, iCOA, and AOSMA get the best for f13, f16, and f20, respectively. In addition, IGA surpasses the other algorithms in the composition functions group for 8 of the ten functions. However, it cannot compete with PSO and IRGA for the performance of functions f22 and

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Table 1 Obtained best results for CEC’18 Fun.

IGA

GA

PSO

WOA

ARO

f1

100

f2

N/A

12674.54 1.5E+10

2.3E+10

100.8136 103.9341 100.0205 109.2091

N/A

N/A

f3

381.0604 435.4105 17160.38 91667.14 300.0765 300.2591 4552.521 1615.069

N/A

f4

404.1909 404.2994 2271.785 4876.542 400.2831 458.5878 404.4052 400.0048

f5

535.8185 547.1928 647.6561 858.1719 574.6218 569.6633 574.6217 524.8740

f6

600.0207 600.2157 621.8359 666.6070 600.0129 605.7919 605.6157 600

f7

765.1107 794.1961 898.4311 1347.323 842.0013 842.7935 813.7098 760.6130

f8

832.8336 839.8560 923.6559 1076.374 851.7378 877.6923 875.6167 837.5634

f9

903.1736 922.4416 4243.867 6805.130 1698.249 1592.664 1427.960 933.2469

f10

2387.700 2767.217 4675.788 7304.957 3071.487 3331.332 2874.697 2164.787

f11

1108.304 1117.688 1584.675 6612.171 1138.831 1192.240 1188.425 1121.437

f12

8502.293 105412.8 3.58E+08 1.44E+09 15914.37 305786.7 350662.9 56030.08

f13

1333.428 1679.609 391612.9 37153099 1447.684 53048.05 22465.26 1576.885

f14

1452.930 2944.155 14117.27 103004.7 1936.975 5611.753 1952.672 2280.703

f15

1515.337 1646.140 52961.06 4.95E+06 1616.792 6462.434 8164.595 1540.006

f16

1620.795 2140.820 2641.284 4017.780 2037.645 2116.533 2169.002 1853.999

f17

1740.264 1822.457 1919.048 2362.849 1898.948 1931.352 1832.874 1894.296

f18

3081.515 28464.86 125557.3 3.80E+06 8371.934 37313.62 37763.21 31014.21

f19

1913.050 1944.455 488969.6 2.51E+07 1960.008 2761.920 58391.82 1930.584

f20

2007.175 2078.764 2195.316 2472.938 2181.251 2254.948 2322.762 2046.436

f21

2333.126 2360.122 2454.085 2621.888 2355.578 2364.168 2364.580 2340.171

f22

2304.983 2300.698 5970.642 7169.358 2300.000 2300.023 2300.000 2300

f23

2689.565 2697.385 2897.791 3159.396 2720.476 2724.856 2696.287 2692.716

f24

2864.632 2839.433 3062.918 3293.895 2891.681 2879.443 2904.605 2861.228

f25

2875.113 2883.721 3456.960 3469.873 2883.962 2883.583 2883.449 2884.459

f26

3705.512 2900.480 7247.971 8519.057 2800.000 4412.929 2800.000 2800

f27

3200.006 3191.313 3295.683 3500.072 3219.165 3205.426 3221.719 3213.547

f28

3300.005 3196.504 4249.752 4402.735 3100.007 3104.009 3200.086 3100

f29

3138.861 3390.129 4096.191 5049.278 3436.909 3630.649 3528.890 3380.381

f30

3218.280 6251.456 640008.2 1.85E+07 5378.112 35631.40 604242.8 5844.700

N/A

AOSMA N/A

iCOA N/A

IRGA N/A

f30. On the other hand, the WOA algorithm produces the lowest outcomes in this criterion. Table 4 gives the highest fitness values over 30 functions. IGA exhibits a minor decline in some functions but maintains top quality for the majority of the functions. IGA still outperforms the best result for function f1 in the unimodal group. Intriguingly, IGA continues to perform at the highest level while the other algorithms significantly suffer in finding the best results in the multimodal functions. The results in Table 4 demonstrate that IGA has the highest performance in the hybrid functions,

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Table 2 Obtained average results for CEC’18 Fun.

IGA

GA

f1

100.0067 26757.68 3.40E+10 3.88E+10 3278.449 6824.622 3232.592 3240.774

f2

N/A

f3

1149.966 927.0988 137347.8 205861.9 301.0320 306.4747 18854.22 11013.66

f4

437.2612 489.9587 7882.051 9285.227 483.4509 488.8959 496.2607 464.9077

f5

561.5230 605.2153 723.9585 914.9991 617.5048 619.0499 645.8779 564.3766

f6

600.1623 600.5579 649.2077 685.1185 603.0286 619.8477 629.4945 600.0000

f7

790.8504 846.9298 1159.194 1453.845 945.1634 932.2137 856.3995 801.3444

f8

867.2480 893.4562 1017.758 1151.476 904.1387 917.1914 930.5711 860.7005

f9

946.0611 1028.171 8737.702 11437.41 2494.200 4228.696 2796.082 1148.806

f10

3438.363 3999.221 6023.050 8342.095 4149.916 4686.919 4661.230 3790.109

f11

1123.433 1162.376 6589.156 12524.62 1202.000 1324.930 1269.129 1181.808

f12

58843.69 1.81E+06 3.91E+09 4.46E+09 113936.4 2.34E+06 6.61E+06 5.03E+05

f13

14977.10 18237.33 2.83E+09 6.14E+08 16632.85 138793.4 125557.4 17521.48

f14

1770.870 27268.68 1.02E+06 4.93E+06 6490.784 39380.89 17663.12 143861.5

f15

8111.000 6730.157 2.95E+08 2.60E+08 4316.115 29880.57 57477.12 5005.187

f16

2399.145 2731.492 4036.495 5130.340 2563.357 2695.219 2668.473 2545.568

f17

2023.014 2244.242 2872.877 3182.292 2192.020 2371.289 2175.456 2129.140

f18

41851.95 132766.2 1.95E+07 4.56E+07 94859.10 298911.6 209215.5 792990.9

f19

3003.070 7205.702 2.51E+08 2.57E+08 6223.528 20176.55 491015.2 8746.003

f20

2331.212 2482.332 2707.196 3057.677 2475.705 2563.571 2583.094 2449.910

f21

2359.131 2399.963 2550.551 2723.555 2395.241 2421.606 2425.008 2367.200

f22

4936.422 2772.561 7476.840 9650.689 2541.960 5306.779 5514.414 3344.023

f23

2720.790 2770.144 3072.797 3425.653 2785.156 2771.134 2834.785 2734.425

f24

2897.334 2970.272 3266.268 3585.282 2959.830 2944.161 2993.481 2913.917

f25

2877.687 2898.622 4327.502 4292.513 2899.995 2889.342 2906.969 2893.669

f26

4181.838 4766.838 8643.655 10666.69 4954.360 5049.623 4961.722 4631.189

f27

3200.006 3232.497 3435.524 3941.639 3259.532 3226.987 3285.826 3243.164

f28

3300.006 3216.320 6780.673 5962.301 3191.158 3223.301 3240.057 3135.624

f29

3447.169 3685.643 5092.057 7212.865 3745.958 4071.221 3991.938 3743.509

f30

5953.313 10060.33 2.85E+08 1.79E+08 8993.375 118995.7 3.09E+06 7987.740

N/A

PSO N/A

WOA N/A

ARO N/A

AOSMA N/A

iCOA N/A

IRGA N/A

whereas ARO and IRGA receive the best outcomes for one function. Once more, IGA outperforms the overall technique in the composition functions by seven out of ten functions, as opposed to two for IRGA and one for ARO. Overall, the results above demonstrate that the proposed IGA is efficient compared to the best, mean, standard deviation, and worst obtained values. Figure 2 displays the number of functions for each set of functions; the algorithms are more effective than others, with success rates in getting the best average values for each set of problems (unimodal, simple multimodal, hybrid and composition).

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem

117

Table 3 Obtained standard deviation results for CEC’18 Fun.

IGA

GA

f1

0.019299 11100.05 9.30E+09 8.90E+09 4417.389 7693.653 4216.648 4608.911

f2

N/A

f3

613.1347 417.0102 84996.01 55308.61 1.015566 10.04621 10686.95 8334.680

f4

28.48943 26.25670 4393.201 2250.064 36.98555 12.14071 25.97572 34.77793

f5

16.66660 23.89876 42.83968 39.62402 26.08864 30.24957 35.91751 21.94245

f6

0.105883 0.240417 14.44757 11.28176 3.534633 9.735847 13.59044 0.000257

f7

14.37221 29.33550 168.5295 65.02461 61.71514 62.69985 30.52995 19.97905

f8

16.54977 25.68319 40.61047 37.70208 23.98204 28.64634 26.57641 19.51102

f9

50.57575 83.73445 2388.952 1965.347 565.4424 1053.817 965.6822 392.3124

f10

534.5632 593.0457 677.9840 573.3158 545.8639 641.0712 640.2090 641.6783

f11

9.076582 34.01536 8366.376 3580.501 46.31639 75.79403 60.70443 40.55465

f12

50917.67 1074820

f13

19186.62 14851.66 2.54E+09 6.20E+08 14989.95 54899.84 71577.68 17343.71

f14

607.1732 25583.54 2645939

f15

20021.54 8199.221 5.74E+08 2.23E+08 4645.755 18731.60 38998.86 4519.788

f16

323.3272 285.9177 654.5932 1004.604 328.2568 332.9962 280.6756 330.9381

f17

190.3090 206.4303 481.6862 555.4873 186.4549 240.0442 219.4529 162.4441

f18

71624.78 84619.77 2.85E+07 4.29E+07 105696.2 199931.2 180738.3 769954.9

f19

3482.061 5772.165 4.91E+08 2.94E+08 4634.763 20145.09 290635.3 9030.823

f20

184.2350 175.4945 224.5032 240.8965 172.5811 160.4605 165.1629 223.4691

f21

14.34565 23.27309 50.99427 62.81180 24.86249 31.82907 31.17478 17.83683

f22

966.0672 1238.878 782.5551 809.3075 931.1007 1703.330 1589.216 1633.295

f23

22.14013 31.99710 101.2641 164.7313 28.99110 25.29732 51.18448 27.54936

f24

24.02289 64.49686 79.51696 208.4993 42.92057 29.94532 50.52967 33.47538

f25

1.668485 19.41379 816.2134 422.0066 17.92416 8.804158 22.08693 13.87211

f26

271.8316 1089.067 1010.733 1297.041 1471.107 520.8453 1042.988 477.8147

f27

0.000301 16.99973 115.0778 338.2526 31.35599 15.46979 49.52543 18.63095

f28

0.000376 17.79658 1331.158 978.5608 39.12735 43.26599 22.79151 57.51112

f29

158.2418 246.9800 599.0019 1484.900 201.8036 229.6968 218.5489 216.4098

f30

8384.877 3195.132 5.06E+08 1.59E+08 2662.486 69223.58 1.52E+06 2254.027

N/A

PSO N/A

WOA N/A

ARO N/A

AOSMA N/A

iCOA N/A

IRGA N/A

2.47E+09 2.24E+09 76379.59 1.63E+06 7.39E+06 469277.8 4.80E+06 4105.565 23579.21 16319.51 244530.7

In Fig. 2, IGA provides more effective results than other algorithms. These results show that IGA is a consistent algorithm with balanced search criteria because the improved mechanisms of mutation and crossover perform the adaption.

118

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Table 4 Obtained worst results for CEC’18 Fun.

IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

f1

100.0950 53295.3

5.18E+10 5.16E+10 20295.57 20941.76 16996.41 19576.59

f2

N/A

N/A

f3

2749.203 2217.801 314100.5 293814.7 303.6055 349.3049 49443.30 38195.31

N/A

N/A

N/A

N/A

N/A

N/A

f4

478.0313 517.1132 18604.89 14804.59 516.8789 516.8470 521.6193 517.1610

f5

591.8606 663.1947 813.8299 1005.146 678.0977 715.9125 740.7782 620.3894

f6

600.4379 601.0873 684.2609 710.4800 612.1346 644.9167 670.2448 600.0014

f7

824.9953 919.0762 1685.851 1572.282 1063.686 1079.626 931.1187 846.3053

f8

899.4955 936.3384 1083.990 1229.311 939.2937 996.7077 981.0814 930.2104

f9

1101.351 1241.198 12947.55 16004.33 3949.208 6098.583 4812.063 3038.541

f10

4375.134 5049.600 7455.92

f11

1153.807 1229.370 34081.76 20129.22 1280.369 1470.282 1414.950 1262.459

9721.709 5314.207 5958.930 5654.929 5230.492

f12

197553.3 4670648

1.06E+10 1.07E+10 349779.1 5.72E+06 3.45E+07 1.89E+06

f13

60298.25 62549.18 1.07E+10 2.66E+09 56441.68 271965.4 255222.8 62301.84

f14

4544.743 92495.06 10503691 20159067 19190.9

f15

110302.7 40096.16 1.86E+09 7.24E+08 24406.95 68626.19 169303.3 20308.07

86964.32 65477.64 1127890

f16

3077.251 3386.912 5519.516 7268.744 3458.123 3373.247 3235.600 3365.817

f17

2395.296 2594.288 3800.916 4873.908 2542.251 2785.504 2715.708 2489.654

f18

371310.3 363254.9 1.22E+08 1.52E+08 586399.8 1.11E+06 699283.4 2898334

f19

16961.47 24743.80 1.83E+09 1.36E+09 20294.05 57434.83 1.18E+06 32953.74

f20

2722.543 2809.452 3191.709 3537.421 2946.648 2848.293 2930.727 2966.584

f21

2388.598 2450.091 2630.666 2848.285 2448.677 2497.966 2476.559 2405.708

f22

6466.786 6497.125 8703.105 11234.85 6512.723 7768.429 7352.952 6197.623

f23

2770.070 2823.414 3408.768 3830.680 2832.659 2814.096 2925.342 2803.322

f24

2974.995 3090.180 3414.435 4383.922 3064.356 3015.016 3102.540 2988.019

f25

2879.561 2942.055 6750.006 5306.806 2939.137 2920.266 2957.229 2948.138

f26

4629.018 6371.955 12204.16 14174.56 6892.828 6373.931 6252.802 5827.038

f27

3200.007 3266.004 3767.047 4828.461 3372.406 3266.121 3426.321 3278.756

f28

3300.007 3261.379 8813.070 8438.312 3258.807 3334.556 3298.036 3260.135

f29

3741.462 4356.305 6555.104 11074.36 4182.908 4617.116 4455.735 4209.986

f30

46787.18 20110.28 1.52E+09 7.31E+08 16041.25 313976.7 6.06E+06 15046.23

4.2 Non-parametric and Convergence History Testing Table 5 presents the obtained results of the Friedman rank test with a confidence level of 0.95% (α = 0.05). It shows that IGA is superior to other competing approaches in its ability to identify the optimal results based on the proposed mutation and crossover improvement. IGA outperforms other algorithms in all categories for 24 out of 29 functions. IGA surpasses the alternative techniques throughout the board

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem

119

Table 5 Friedman rank test obtained for CEC’18 Fun.

p -value

IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

f1

1.42E-35

1.00

5.83

7.37

7.63

3.53

4.03

3.17

3.43

f2

−

−

−

−

−

−

−

−

−

f3

8.47E-40

3.73

3.30

7.27

7.67

1.17

1.83

5.77

5.27

f4

7.53E-29

1.93

4.03

7.43

7.57

3.87

4.03

4.70

2.43

f5

4.65E-35

1.53

3.73

6.90

8.00

4.33

4.27

5.43

1.80

f6

1.31E-39

2.20

3.30

6.77

8.00

3.53

5.27

5.93

1.00

f7

3.93E-37

1.40

3.47

6.87

7.97

5.43

5.37

3.63

1.87

f8

7.34E-34

2.13

3.37

6.97

8.00

3.83

4.87

5.20

1.63

f9

2.01E-38

1.17

2.23

7.07

7.83

4.57

5.70

4.77

2.67

f10

1.97E-28

2.03

2.93

6.70

8.00

3.60

4.90

4.93

2.90

f11

7.73E-37

1.17

2.60

7.13

7.87

3.60

5.63

4.93

3.07

f12

1.63E-37

1.37

4.50

7.43

7.57

1.87

4.73

5.50

3.03

f13

1.35E-34

2.10

2.77

7.70

7.30

2.73

5.47

5.33

2.60

f14

3.64E-32

1.03

4.07

6.37

7.87

2.57

4.93

3.60

5.57

f15

1.96E-34

2.13

3.00

7.20

7.77

2.60

5.07

5.70

2.53

f16

6.12E-24

2.67

4.30

6.90

7.93

3.37

3.93

3.83

3.07

f17

6.48E-22

2.60

4.13

6.83

7.60

3.40

4.87

3.33

3.23

f18

1.24E-31

1.43

3.27

7.00

7.67

2.57

4.77

3.83

5.47

f19

1.34E-35

1.27

3.33

7.47

7.53

3.10

4.23

6.00

3.07

f20

1.03E-18

2.17

3.90

5.90

7.67

3.63

4.57

4.83

3.33

f21

2.04E-34

1.47

3.93

6.93

8.00

3.57

4.93

5.10

2.07

f22

2.20E-29

4.20

2.83

6.83

7.97

1.97

4.73

4.80

2.67

f23

1.36E-33

1.73

3.67

7.00

8.00

4.27

3.73

5.50

2.10

f24

8.32E-31

1.80

4.40

7.00

8.00

4.00

3.57

4.87

2.37

f25

4.28E-32

1.00

4.03

7.43

7.57

4.30

3.07

4.60

4.00

f26

2.96E-27

1.83

3.63

7.07

7.93

4.03

4.03

4.17

3.30

f27

3.50E-34

1.03

3.30

6.87

8.00

4.63

2.90

5.30

3.97

f28

3.08E-36

5.97

3.20

7.70

7.30

2.33

3.43

4.57

1.50

f29

2.57E-31

1.53

2.93

7.00

7.93

3.43

5.07

4.73

3.37

f30

4.12E-38

1.33

3.30

7.37

7.63

2.90

4.97

6.00

2.50

2.00

3.48

7.04

7.79

3.40

4.46

4.89

2.94

Avg.

for the hybrid group’s functions, IRGA is the second by three functions, and ARO is the third by the remaining two functions. The last row of Table 5 includes the average rank over 29 functions, which shows how effective and reliable the proposed approach IGA is. As mentioned before, in this study, the rank sum test, a variant of the Wilcoxon test, is used to compare two paired groups. Table 6 gives the obtained results by the Wilcoxon rank-sum test with a confidence level of 0.95% (α = 0.05) and shows that IGA outperforms other competing approaches in almost all CEC 2018 functions. IGA outperforms basic GA for 24 functions while the fit is just short in three, as indicated by the symbol ‘1’ count, which denotes the significant difference between both

120

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Table 6 Wilcoxon rank-sum test obtained for CEC’18 IGA versus

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

Fun.

p

h

p

h

p

h

p

h

p

h

p

h

p

h

f1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

f2

−

−

−

−

−

−

−

−

−

−

−

−

−

−

f3

3.33E-02

– 1

1.73E-06

1

1.73E-06

1

1.73E-06

– 1

1.73E-06

– 1

1.73E-06

1

1.92E-06

1

f4

8.47E-06

1

1.73E-06

1

1.73E-06

1

1.74E-04

1

1.92E-06

1

4.29E-06

1

1.66E-02

1

f5

5.75E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

2.35E-06

1

1.73E-06

1

6.14E-01

0

f6

2.88E-06

1

1.73E-06

1

1.73E-06

1

8.47E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

– 1

f7

2.60E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

3.50E-02

1

f8

3.59E-04

1

1.73E-06

1

1.73E-06

1

3.72E-05

1

2.88E-06

1

2.13E-06

1

8.22E-02

0

f9

3.72E-05

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.97E-05

1

f10

3.16E-03

1

1.73E-06

1

1.73E-06

1

1.60E-04

1

9.32E-06

1

3.18E-06

1

4.07E-02

1

f11

1.97E-05

1

1.73E-06

1

1.73E-06

1

1.92E-06

1

1.73E-06

1

1.73E-06

1

2.35E-06

1

f12

1.73E-06

1

1.73E-06

1

1.73E-06

1

3.85E-03

1

1.73E-06

1

1.73E-06

1

2.13E-06

1

f13

3.18E-01

0

1.73E-06

1

1.73E-06

1

4.41E-01

0

1.92E-06

1

2.88E-06

1

1.85E-01

0

f14

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

2.35E-06

1

1.73E-06

1

f15

1.25E-01

0

1.73E-06

1

1.73E-06

1

6.14E-01

0

4.86E-05

1

3.52E-06

1

7.19E-01

0

f16

1.15E-04

1

1.92E-06

1

1.73E-06

1

8.59E-02

0

6.04E-03

1

1.48E-03

1

1.02E-01

0

f17

1.29E-03

1

2.88E-06

1

1.73E-06

1

6.84E-03

1

1.74E-04

1

2.70E-02

1

5.45E-02

0

f18

3.32E-04

1

1.73E-06

1

1.73E-06

1

4.90E-04

1

2.13E-06

1

4.86E-05

1

1.73E-06

1

f19

3.72E-05

1

1.73E-06

1

1.73E-06

1

9.71E-05

1

9.32E-06

1

1.73E-06

1

2.41E-04

1

f20

6.04E-03

1

2.88E-06

1

1.73E-06

1

1.38E-03

1

1.25E-04

1

9.71E-05

1

6.27E-02

0

f21

2.60E-06

1

1.73E-06

1

1.73E-06

1

1.36E-05

1

1.73E-06

1

1.73E-06

1

1.11E-01

0

f22

4.29E-06

– 1

1.92E-06

1

1.73E-06

1

1.64E-05

– 1

2.45E-01

0

8.22E-02

0

7.71E-04

– 1

f23

1.97E-05

1

1.73E-06

1

1.73E-06

1

4.29E-06

1

4.29E-06

1

2.13E-06

1

8.59E-02

0

f24

5.31E-05

1

1.73E-06

1

1.73E-06

1

8.47E-06

1

2.84E-05

1

1.73E-06

1

5.45E-02

0

f25

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

f26

9.27E-03

1

1.73E-06

1

1.73E-06

1

6.42E-03

1

1.92E-06

1

4.99E-03

1

9.71E-05

1

f27

1.92E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

1.73E-06

1

f28

1.73E-06

– 1

1.73E-06

1

1.73E-06

1

1.73E-06

– 1

2.35E-06

– 1

1.73E-06

– 1

1.73E-06

– 1

f29

2.41E-04

1

1.73E-06

1

1.73E-06

1

5.75E-06

1

1.73E-06

1

2.35E-06

1

3.11E-05

1

f30

3.32E-04

1

1.73E-06

1

1.73E-06

1

3.32E-04

1

1.92E-06

1

1.73E-06

1

4.90E-04

1

Best (=1)

24

29

29

23

26

27

16

Equal (=0)

2

0

0

3

1

1

10

Worst (=– 1)

3

0

0

3

2

1

3

algorithms. The results appear more competitive with the IRGA algorithm, where GA wins in 16 functions, fails in three, and performs adequately in 10; however, the Friedman test gives preference to the proposed method GA for those ten functions 5. Finally, the results illustrate that IGA outperforms GA and other algorithms in finding the best solutions.

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem

121

Fig. 2 The number of functions where the algorithms achieve successful average fitness values according to the function groups

Moreover, over CEC’18 functions, the convergence speeds of proposed and comparable methods are contrasted. Figures 3, 4 and 5, display each plotted convergence graph (objective function values vs iterations) separately. It is crucial to remember that only the odd functions are displayed here in relation to the size number of functions in the CEC’18 benchmark. Figures 3, 4 and5 indicate that the suggested IGA converges far faster for most functions than competing techniques. Compared to the other approaches, the improved algorithm has a higher convergence rate, uses less computational power, and uses less memory.

5 Nonlinear Optimization Test In this subsection, the performance and effectiveness of improving technique for GA, IGA, is investigated using IGA on eight various nonlinear systems of equations benchmark problems from the real world. In addition, the standard GA method and other existing algorithms are compared in this experiment with our IGA algorithm.

5.1 Problem Statement Consider a nonlinear equation system with n components and m nonlinear equations in their generic form.

122

A. M. Ibrahim and M. A. Tawhid

1012

108 106

IGA GA PSO WOA ARO AOSMA iCOA IRGA

1010

Fitness

1010

Fitness

1012

IGA GA PSO WOA ARO AOSMA iCOA IRGA

108 106 104

104 102 100

F1 101

102

102 100

103

F3 101

1200

IGA GA PSO WOA ARO AOSMA iCOA IRGA

Fitness

1000 900 800

3500

103

IGA GA PSO WOA ARO AOSMA iCOA IRGA

3000 2500

Fitness

1100

102

Iterations

Iterations

2000 1500

700 1000

600 0

10

1

10

3

10

0

101

108

IGA GA PSO WOA ARO AOSMA iCOA IRGA

104

102

103

Iterations

Iterations

105

Fitness

10

2

IGA GA PSO WOA ARO AOSMA iCOA IRGA

107

Fitness

10

F7

F5

106 105 104

103 100

F9 10

1

10

2

10

3

103 100

F11

Iterations

101

102

103

Iterations

Fig. 3 The convergence history of the odd functions of CEC’18 (cont’d)

f 1 ( p1 , p2 , p3 , . . . , pn ) = 0 f 2 ( p1 , p2 , p3 , . . . , pn ) = 0 .. . f m ( p1 , p2 , p3 , . . . , pn ) = 0.

(10)

Obtaining the global minimum of the merit function is comparable to solving the system of nonlinear equations. Researchers have employed numerous merit functions

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem 1012

IGA GA PSO WOA ARO AOSMA iCOA IRGA

Fitness

108

106

6

104 100

104

F13 101

102

F15

100

103

101

106

1010

IGA GA PSO WOA ARO AOSMA iCOA IRGA

Fitness

105

103

IGA GA PSO WOA ARO AOSMA iCOA IRGA

108

104

106

104

F17

103 100

10

1

10

2

10

102 100

3

F19 101

4400

IGA GA PSO WOA ARO AOSMA iCOA IRGA

2900 2800 2700

4000 3800

2600

3600 3400 3200

2500

3000

F21

2400 10

103

IGA GA PSO WOA ARO AOSMA iCOA IRGA

4200

Fitness

3000

102

Iterations

Iterations

Fitness

102

Iterations

Iterations

Fitness

Fitness

108

10

1010

IGA GA PSO WOA ARO AOSMA iCOA IRGA

1010

123

0

2800

10

1

10

2

10

3

Iterations

100

F23 101

102

103

Iterations

Fig. 4 The convergence history of the odd functions of CEC’18 (cont’d)

to turn m nonlinear equations into optimization problems [37–39] as follows. ⎧ ⎪ ⎨ Find: ⎪ ⎩ Min:

P = ( p1 / , p2 , . . . , pn ) T , P ∈ R n ; n Σ f i2 , F(P) =

(11)

i=1

where R n is the solution region. The problem with solving the system of nonlinear Eq. (10) is equivalent to finding global optimal Finding the global optimal solution to the unconstrained optimization problem (i.e., resolving the lowest values of the

A. M. Ibrahim and M. A. Tawhid 6500

IGA GA PSO WOA ARO AOSMA iCOA IRGA

104

IGA GA PSO WOA ARO AOSMA iCOA IRGA

6000 5500

Fitness

Fitness

124

5000 4500 4000 3500

F25 100

101

102

103

F27

100

101

102

103

Iterations

Iterations 106

IGA GA PSO WOA ARO AOSMA iCOA IRGA

Fitness

105

104

103 100

F29 101

102

103

Iterations

Fig. 5 The convergence history of odd functions of CEC’18

fitness function or merit function) is the same as the challenge presented by the system of nonlinear equations problem Eq. (10).

5.2 Results and Discussion The proposed algorithm IGA’s performance is assessed in this experiment by solving eight real nonlinear equation systems. Table 7 lists details about the shared systems. The outcomes are compared with those of the basic GA algorithm and other metaheuristics algorithms such as PSO [12], WOA [13], ARO [14], AOSMA [25], iCOA [26], and IRGA [27]. Fifty consecutive algorithm runs are performed with a maximum function evaluation of 160 × 103 and population size 20 in every system. The statistical results for each system and algorithm’s minimum, best, maximum, worst, and standard deviation values are presented in Tables 8, 9, 10, 11, 12, 13 and 14. It is worth mentioning that in this experiment, the Eq. (11) is used as an objective function in this experiment. The outcomes in Tables 8, 9, 10, 11, 12, 13 and 14 are proposed as minimal (Fmin ), average (Favg ), standard deviation (Std) and maximum (Fmax ) values. Moreover, Table 16 displays the success number (SN) for each test system of the competing algorithms. Successful run requirements are defined when

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem Table 7 Best results obtained by different methods for Sys1 System Problem no. ⎧ 2 p1 + p2 + p3 + p4 + p5 = 6, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p1 + 2 p2 + p3 + p4 + p5 = 6, Sys 1 p1 + p2 + 2 p3 + p4 + p5 = 6, ⎪ ⎪ ⎪ p1 + p2 + p3 + 2 p4 + p5 = 6, ⎪ ⎪ ⎩ p1 p2 p3 p4 p5 = 1,

Sys 2

Sys 3

Sys 4

Sys 5

Sys 6

⎧ ⎪ p1 + 41 p22 p4 p6 + 0.75 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p2 + 0.405 exp(1 + p1 p2 ) − 1.405 = 0, ⎪ ⎨ p − 1 p p + 1.5 = 0, 3 2 4 6 ⎪ p4 − 0.605 exp(1 − x32 ) − 0.395 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ p5 − 21 p2 p6 + 1.5 = 0, ⎪ ⎪ ⎩ p6 − p1 p5 = 0. ) ( Σ pi − cos 2 pi − 4j=1 p j = 0, 1 ≤ i ≤ 4. ⎧ ⎪ p12 + p32 = 1, ⎪ ⎪ ⎪ ⎪ p22 + p42 = 1, ⎪ ⎪ ⎪ ⎨ p p3 + p p3 = c , 6 4 1 5 3 3 + p p3 = c , ⎪ p p 6 2 2 5 1 ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ p5 p1 p3 + p6 p2 p4 = c3 , ⎪ ⎩ p5 p3 p12 + p6 p4 p22 = c4 , ⎧ p1 − 0.25428722 − 0.18324757 p4 p3 p9 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ p2 − 0.37842197 − 0.16275449 p1 p10 p6 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ p3 − 0.27162577 − 0.16955071 p1 p2 p10 = 0, ⎪ ⎪ ⎪ ⎪ p4 − 0.19807914 − 0.15585316 p7 p1 p6 = 0, ⎪ ⎪ ⎪ ⎨ p − 0.44166728 − 0.19950920 p p p = 0, 7 6 3 5 ⎪ − 0.14654113 − 0.18922793 p p5 p10 = 0, p 6 8 ⎪ ⎪ ⎪ ⎪ ⎪ p7 − 0.42937161 − 0.21180486 p2 p5 p8 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ p8 − 0.07056438 − 0.17081208 p1 p7 p6 = 0, ⎪ ⎪ ⎪ ⎪ p9 − 0.34504906 − 0.19612740 p10 p6 p8 = 0, ⎪ ⎪ ⎩ p10 − 0.42651102 − 0.21466544 p4 p8 p1 = 0, ⎧ 2 2 p ⎪ i + pi+1 − 1 = 0, ⎪ ⎪ ⎪ ⎪ ⎨ α1i p1 p3 + α2i p1 p4 + α3i p2 p3 + α4i p2 p4 +α5i p2 p7 + α6i p5 p8 + a7i p6 p7 + α8i p6 p8 ⎪ ⎪ ⎪ +α9i p1 + a10i p2 + a11i p3 + α12i p4 + α13i p5 ⎪ ⎪ ⎩ +α14i p6 + α15i p7 + α16i p8 + α17i = 0.

125

Description The five-dimensional instance of Brown’s virtually linear system [37, 40], where −10 ≤ pi ≤ 10, i = 1, . . . , 5. The two best options for this system are P ∗ = (1, 1, 1, 1, 1)T and P ∗ = (0.916 . . . , 0.916 . . . , 0.916 . . . , 0.916 . . . , 1.418 . . .)T . This six-dimensional system is described in [40, 41]. Where −2 ≤ pi ≤ 2, i = 1, . . . , 6. P ∗ = (−1, 1, −1, 1, −1, 1)T is the system’s exact solution.

The four nonlinear equations’ system is given in [42, 43]. Examples of applications for neurophysiology are provided in [44, 45]. This example is used to evaluate the performance of our algorithm. There are six nonlinear equations in this example. This problem is known as 10-dimensional of interval arithmetic benchmark, where −2 ≤ pi ≤ 2 [37].

This test problem is chosen to be the inverse position problem for a six-revolute joint problem application [46, 47]. Where −10 ≤ pi ≤ 10, i = 1, . . . , 4, and the coefficients α ji , 1 ≤ i ≤ 4 and 1 ≤ j ≤ 17 can be found in [37]. (continued)

126

A. M. Ibrahim and M. A. Tawhid

Table 7 (continued) System no.

Sys 7

Sys 8

Problem

Description

⎧ ⎪ p2 + 2 p6 + p9 + 2 p10 − 10−5 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ p3 + p8 − 3 × 10−5 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ p1 + p3 + 2 p5 + 2 p8 + p9 + p10 − 5 × 10−5 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ p4 + 2 p7 − 10−5 = 0, ⎪ ⎪ ⎪ ⎨ 0.5140437 × 10−7 p − p 2 = 0,

The combustion problem [37] is stated as a system made up of ten nonlinear equations that took place at a temperature of 3000 ◦ C, where −20 ≤ pi ≤ 20.

5

1

⎪ 0.1006932 × 10−6 p6 − 2 p22 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.7816278 × 10−15 p7 − p42 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 0.1496236 × 10−6 p8 − p1 p3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 0.6194411 × 10−7 p9 − p1 p2 = 0, ⎪ ⎪ ⎩ 0.2089296 × 10−14 p10 − p1 p22 = 0. ⎧ ⎪ A = bh − (b − 2t)(h − 2t), ⎪ ⎨ 3 (b−2t)(h−2t)3 I y = bh , 12 − 12 ⎪ ⎪ 2 ⎩ 2t (h−t) (b−t)2 . In = h+b−2t

Dimensions of the rectangular girder section for thin walls [37, 45, 47] is considered as the benchmark problem. where b denotes the section’s width, h denotes its height, and t denotes its thickness. A = 165, I y = 9369 and In = 6835, 0 ≤ pi ≤ 25.

the fitness function trends towards 0 (i.e. the exact solution is found). The number of solutions found out of all the trials that are deemed effective is known as a success number (50 in this study). The obtained numerical results show the effectiveness of the proposed method and demonstrate that IGA outperforms comparable algorithms in the majority of systems in terms of the lowest, average, Std and maximum of the objective function values. As a further assessment of IGA’s effectiveness, Figs. 6, 7, 8, 9, 10, 11, 12 and 13 show the algorithm convergence history. Based on the graph for each nonlinear system, the graphical analysis results of the ANOVA tests for the best values of 30 separate runs have been compared. Below are the outcomes of nonlinear equation systems, along with an explanation of the results. From Table 8, IGA performs better than all other algorithms, including classical GA, in the first system (Sys 1) of nonlinear equations. Additionally, only IGA can find the best solution for this system (See the number of success rates in Table 16). In Fig. 6, IGA converges faster than all other algorithms, supported by the ANOVA test, and demonstrates IGA’s stability. In contrast, the performance of the other metaheuristics, except IRGA, either yields the lowest results after IGA or appears to be similar to or even worse. Table 9 demonstrates that IGA delivers the best results (exact solution) for the system (Sys 2) in terms of minimum and maximum values, whereas ARO obtains the best average and standard deviation outcomes. In this system, the results are

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem

127

Table 8 Best results obtained by different methods for Sys1 p1

IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

0.916355

1

1.073639

0.920509

1

0.999991

0.422179

0.999618

p2

0.916355

1

1.108488

1.119761

1

0.999991

0.986689

0.999608

p3

0.916355

1

1.078072

0.894877

1

0.999991

0.341989

0.999627

p4

0.916355

1

1.074583

1.091884

1

0.999991

0.499092

0.999608

p5

1.418227

1

0.609342

0.969365

1

0.999991

0.231025

1.001850

f1

0

5.80E-09

0.017763

–0.083094

0

–5.33E-05

–3.096848

–7.12E-05

f2

0

1.87E-08

0.052613

0.116158

0

–5.33E-05

–2.532337

–8.10E-05

f3

0

–2.24E-09

0.022197

–0.108727

0

–5.33E-05

–3.177038

–6.24E-05

f4

0

1.57E-09

0.018708

0.088281

0

–5.33E-05

–3.019935

–8.11E-05

f5

0

–4.08E-08

–0.159885

–0.023706

–1.11E-15

–4.45E-05

–0.983574

3.09E-04

Fmin

0

4.53E-08

0.171726

0.201430

1.11E-15

1.16E-04

9.10E-06

3.43E-04

Favg

1.55E-17

0.104779

0.563323

0.491657

3.65E-05

1.18E-03

0.034957

7.42E-03

Std

1.10E-16

0.418934

0.437063

0.384396

1.49E-04

5.48E-04

0.246960

4.64E-03

Fmax

7.77E-16

1.746301

1.764584

1.747609

7.97E-04

2.44E-03

1.746301

1.74E-02

105

1.8 1.6 1.4 1.2

10-5 10-10 10-15 10-20 100

fitness

Fitness

100

IGA GA PSO WOA ARO AOSMA iCOA IRGA

1 0.8 0.6 0.4 0.2 0

101

102

A

IG

Iterations

GA

O PS

A WO

O PS

WO

O AR

A SM

A iCO

IR

MA

A iCO

GA IR

AO

GA

Fig. 6 The convergence history and ANOVA test for system 1 102

2.5 2

100 10-1 10

-2

10-3 100

fitness

Fitness

10

3

1

IGA GA PSO WOA ARO AOSMA iCOA IRGA

1.5 1 0.5 0

101

102

Iterations

A

IG

GA

Fig. 7 The convergence history and ANOVA test for system 2

A

O

AR

S AO

128

A. M. Ibrahim and M. A. Tawhid

Table 9 Best results obtained by different methods for Sys2 IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

p1

–1

–1.043200

–1.085148

–1.319093

–1

–1.001078

0.828187

–1.003750

p2

1

–0.550936

1.277096

1.352978

0.999999

1.000163

0.944965

1.002935

p3

–1

0.431933

–0.663204

–1.099068

–1

–0.997858

0.201859

–0.995741

p4

1

1.759663

1.680226

1.021047

0.999999

1.002889

0.847009

1.005548

p5

–1

–2.104871

–0.800220

–0.624825

–1

–0.999577

0.027342

–0.997825

p6

1

2.195799

0.645232

0.966347

0.999999

1.000796

0.128006

1.001872

f1

0

4.84E-07

0.106901

–0.117549

1.31E-07

–7.51E-05

1.602391

–4.11E-04

f2

0

–1.95E-06

0.147448

0.132761

–5.82E-08

–3.40E-04

1.947828

2.32E-04

f3

0

–1.94E-07

0.294728

–0.092410

–1.38E-07

2.98E-04

1.647648

5.44E-04

f4

0

5.77E-07

0.225899

0.134636

–1.13E-07

2.95E-04

–1.13E+00 3.84E-04

f5

0

1.50E-06

0.287769

0.221452

6.59E-08

–5.59E-05

1.466861

–2.31E-04

f6

0

–8.78E-07

–0.223126

0.142145

–6.61E-08

1.41E-04

0.105362

3.05E-04

Fmin

0

2.73E-06

0.551053

0.356870

2.47E-07

5.66E-04

5.15E-05

9.02E-04

Favg

2.99E-02

0.046613

1.403969

1.241737

6.76E-03

3.48E-02

0.022823

0.081900

Std

8.65E-02

0.109934

0.405852

0.476593

4.05E-02

8.48E-02

0.100155

0.166480

Fmax

0.286452

0.373753

2.517476

3.076523

0.286452

0.286587

0.708277

0.560490

Table 10 Best results obtained by different methods for Sys3 IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

p1

0.514933

0.899454

–0.810255

0.995503

–0.226139

0.514926

0.725634

0.899454

p2

0.514933

–0.226139

0.957700

–0.811203

–0.226139

0.514926

0.911006

0.899454

p3

0.514933

–0.226139

1.031011

0.979001

0.899454

0.514926

0.761631

–0.22614

p4

0.514933

0.899454

0.965039

0.980290

0.899454

0.514926

0.910381

–0.22614

f1

0

–2.05E-13

2.22E-03

7.12E-03

0

–2.09E-05

1.008315

1.02E-14

f2

0

1.60E-13

–1.64E-02

1.08E-04

0

–2.09E-05

0.826949

1.13E-14

f3

0

–2.49E-14

3.43E-02

–3.83E-03

0

–2.09E-05

0.974582

–4.22E-15

f4

0

1.52E-13

–1.23E-02

–3.01E-03

0

–2.09E-05

0.82757

–2.64E-15

Fmin

0

3.02E-13

4.00E-02

8.63E-03

0

4.18E-05

2.38E-06

1.60E-14

Favg

3.13E-17

6.04E-12

0.155539

0.152038

1.26E-07

6.96E-04

8.40E-06

3.27E-09

Std

4.97E-17

6.71E-12

0.092285

0.083919

8.81E-07

3.81E-04

3.40E-06

6.98E-09

Fmax

1.24E-16

4.31E-11

0.306717

0.389472

6.23E-06

1.47E-03

1.68E-05

2.71E-08

more evenly distributed, and the performance of the competing methods declines (see Fig. 7). Nonetheless, the convergence history and ANOVA test of IGA and GA in Fig. 7 seem comparable, with ARO having a slight advantage. Our suggested algorithm IGA, together with other algorithms, solves System 3, and the most well-known result of this case study is provided in Table 10. Only IGA and ARO achieve the best outcomes for this system in Table 10, but IGA has the best results for average, worst, and standard deviation. IGA receives two precise solutions for this system. The superiority of the proposed approach is illustrated by the convergence history and ANOVA test for system 3 in Fig. 8.

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem 100

129

0.4 0.35 0.3 0.25

10-10

10-15

10-20 100

fitness

Fitness

10-5

IGA GA PSO WOA ARO AOSMA iCOA IRGA

0.2 0.15 0.1 0.05 0

101

102

A IG

Iterations

GA

O PS

A WO

A O SM AR AO

A iCO

GA IR

Fig. 8 The convergence history and ANOVA test for system 3 Table 11 Best results obtained by different methods for Sys4 IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

p1

–0.993464

0.906110

0.800855

0.313885

–0.114369

0.650386

0.707630

IRGA 0.922797

p2

–0.993464

–0.200788

–0.673739

–0.003311

–0.657691

0.650366

0.323311

0.268291

p3

–0.114149

–0.423042

–0.598228

–0.949464

0.993438

0.759601

0.758643

–0.385287

p4

–0.114149

0.979635

0.737930

0.999986

–0.753287

0.759614

0.308150

0.963338

p5

0.323131

–8.34E-09

–2.92E-02

–0.005778

9.1E-163

0.833743

0.623833

–4.16E-20

p6

–0.323131

7.19E-10

–7.39E-02

–0.006454

1.2E-162

–0.83374

0.300512

–1.26E-18

f1

0

4.19E-10

–7.54E-04

6.47E-06

0

–5.08E-06

0.076280

0

f2

0

-5.59E-09

–1.53E-03

–1.68E-05

0

–1.19E-05

–0.800514

0

f3

0

1.31E-09

–2.34E-02

–1.51E-03

3.9E-163

–1.53E-05

0.281177

–1.13E-18

f4

0

–6.21E-09

7.61E-03

–1.79E-04

–3.3E-163

2.37E-05

0.231204

–5.70E-20

f5

0

–1.49E-09

1.87E-02

–1.61E-03

–5.4E-163

2.05E-06

0.263294

–3.20E-19

f6

0

2.93E-09

–1.36E-02

5.40E-04

–3.7E-163

1.47E-05

0.246663

–7.39E-20

Fmin

0

9.08E-09

3.38E-02

2.28E-03

0

3.44E-05

1.19E-05

1.17E-18

Favg

3.4E-141

3.84E-07

0.406578

0.370319

1.39E-11

7.40E-04

5.15E-03

2.77E-03

Std

2.4E-140

1.45E-06

0.274353

0.250899

9.84E-11

8.11E-04

2.34E-02

1.87E-02

Fmax

1.7E-139

9.89E-06

0.997877

0.850276

6.96E-10

0.003852

0.154192

0.132533

Table 11 contrasts the proposed algorithm’s most well-known solutions with various results from existing algorithms for system 4. Equation 1 is used in this case study to provide distinct optimal results utilizing the IGA and ARO approaches. However, the success number for each method can be found in Table 16. Moreover, IGA surpasses the other algorithms regarding the average, worst and standard deviation. Figure 9 provides additional support for this conclusion by describing how the algorithms for this system performed and demonstrating the superiority of the proposed IGA. Table 12 compares the proposed algorithm’s best-known solutions with various results from other algorithms for system 5. Moreover, IGA outperforms other algorithms across the board (best, average, worst and Std). By discussing how the algo-

130

A. M. Ibrahim and M. A. Tawhid 100

1

0.8

10-100

10-150 100

fitness

Fitness

10-50 IGA GA PSO WOA ARO AOSMA iCOA IRGA

0.6

0.4

0.2

0

101

102

A

IG

Iterations

GA

O PS

A

WO

O AR

A SM

AO

A iCO

GA IR

Fig. 9 The convergence history and ANOVA test for system 4 Table 12 Best results obtained by different methods for Sys5 IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

p1

0.257833

0.257833

0.291560

0.237330

0.257833

0.258163

0.482618

0.257833

p2

0.381097

0.381097

0.383139

0.323021

0.381097

0.381937

0.691733

0.381097

p3

0.278745

0.278745

0.220396

0.300952

0.278745

0.277788

0.941050

0.278745

p4

0.200669

0.200669

0.239071

0.217162

0.200669

0.200036

0.170365

0.200669

p5

0.445251

0.445251

0.334536

0.445510

0.445251

0.445250

0.115881

0.445251

p6

0.149184

0.149184

0.168790

0.170502

0.149184

0.148995

0.063075

0.149184

p7

0.432010

0.432010

0.455214

0.402625

0.432010

0.432847

0.034437

0.432010

p8

0.073403

0.073403

0.158298

0.022716

0.073403

0.074437

0.889376

0.073403

p9

0.345967

0.345967

0.333983

0.387729

0.345967

0.345616

0.176326

0.345967

p10

0.427326

0.427326

0.448546

0.437179

0.427326

0.425766

0.208352

0.427326

f1

–2.13E-17

–4.45E-07

3.40E-02

–2.16E-02

6.59E-17

3.56E-04

2.23E-01

–1.91E-17

f2

–1.65E-17

2.53E-07

1.12E-03

–5.83E-02

3.13E-15

8.49E-04

3.12E-01

–1.34E-17

f3

–2.43E-17

–1.63E-07

–5.97E-02

2.36E-02

–2.60E-18

–9.56E-04

6.58E-01

–2.43E-17

f4

9.97E-18

2.35E-07

3.75E-02

1.65E-02

–2.08E-17

–6.38E-04

–2.79E-02

–9.76E-17

f5

2.13E-17

–7.93E-07

–1.11E-01

–2.79E-04

1.78E-17

8.72E-06

–3.26E-01

–8.54E-17

f6

1.30E-18

–2.78E-07

1.78E-02

2.31E-02

7.24E-17

–2.16E-04

–8.75E-02

–1.64E-16

f7

5.20E-18

1.79E-07

2.15E-02

–2.74E-02

–1.86E-17

7.94E-04

–4.10E-01

–4.94E-17

f8

–5.20E-18

4.60E-07

8.39E-02

–5.06E-02

3.38E-17

1.03E-03

8.19E-01

–1.52E-17

f9

1.31E-17

–3.20E-07

–1.34E-02

4.23E-02

2.28E-15

–3.59E-04

–1.71E-01

1.43E-17

f 10

–2.50E-17

–2.98E-07

1.97E-02

1.04E-02

1.70E-15

–1.57E-03

–2.34E-01

–2.43E-17

Fmin

5.21E-17

1.22E-06

1.64E-01

1.02E-01

4.23E-15

2.55E-03

3.09E-05

2.20E-16

Favg

9.58E-17

2.76E-06

3.27E-01

3.39E-01

1.61E-06

4.67E-03

5.78E-05

6.63E-14

Std

1.03E-16

8.09E-07

8.36E-02

8.11E-02

2.48E-06

1.08E-03

1.34E-05

1.24E-13

Fmax

6.73E-16

4.64E-06

5.63E-01

5.43E-01

9.56E-06

7.65E-03

9.16E-05

5.53E-13

rithms for this system performed and showcasing the superiority of the suggested IGA, Fig. 10 further supports this conclusion. The top results from many methods, including IGA, GA, and others for system 6, are displayed in Table 13. While ARO performs best regarding the standard deviation

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem

131

100 0.5

10-10

10-15

10-20 100

0.4

fitness

Fitness

10-5

IGA GA PSO WOA ARO AOSMA iCOA IRGA

0.3 0.2 0.1 0

101

102

A

IG

Iterations

GA

O PS

A

WO

O AR

MA

S AO

A iCO

GA IR

Fig. 10 The convergence history and ANOVA test for system 5 Table 13 Best results obtained by different methods for Sys6 IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

p1

–0.420221

–0.929003

–0.961389

–0.264657

–0.981976

–0.416234

0.773182

IRGA 0.772568

p2

0.907422

–0.370030

–0.482897

0.884219

0.189008

0.909817

0.762907

0.634777

p3

0.420221

–0.929068

–0.805459

0.175588

–0.981976

–0.413856

0.425984

0.772687

p4

0.907422

0.369740

0.371942

0.749390

–0.189005

0.910917

0.619435

0.634728

p5

0.420221

–0.929139

0.014565

0.581163

–0.981976

0.411871

0.108541

–0.772741

p6

–0.258313

–0.525617

–0.331645

–0.724753

0.261984

–0.259388

0.185098

0.500573

p7

–0.610318

1.726148

1.068668

–0.534962

0.741520

–0.586784

0.013499

0.018272

p8

–0.832092

0.168353

–0.734970

–0.888138

–1.449989

–0.999521

0.992697

–1.200890

f1

0

–3.14E-05

1.57E-01

–1.48E-01

–6.74E-08

1.02E-03

1.80E-01

–1.97E-04

f2

0

8.93E-05

–1.18E-01

–1.87E-01

4.81E-07

–9.56E-04

–2.37E-01

–1.27E-05

f3

0

–1.25E-04

–2.13E-01

–4.08E-01

–4.96E-07

1.05E-03

–4.35E-01

–7.40E-05

f4

0

6.69E-06

–8.61E-01

–1.01E-01

–6.70E-08

–5.93E-04

–6.05E-01

8.50E-06

f5

–1.39E-17

2.38E-06

–2.56E-01

5.32E-01

1.23E-07

2.91E-04

8.11E-01

2.69E-04

f6

9.19E-17

–1.50E-05

–6.14E-02

–4.39E-01

5.68E-08

5.42E-04

–7.92E-01

3.19E-04

f7

0

–7.58E-06

–1.39E-01

–1.08E-01

6.76E-08

–1.11E-04

2.37E+00

7.09E-05

f8

0

9.40E-06

–1.26E-01

5.37E-02

2.54E-08

1.83E-04

3.61E+00

7.79E-05

Fmin

9.30E-17

1.58E-04

9.65E-01

8.51E-01

7.14E-07

1.95E-03

2.55E-04

4.80E-04

Favg

5.16E-03

2.49E-02

2.01E+00

1.76E+00

7.83E-03

4.49E-02

2.13E-02

5.65E-02

Std

3.65E-02

5.92E-02

7.70E-01

5.25E-01

1.23E-02

4.49E-02

4.77E-02

7.87E-02

Fmax

0.257816

0.257816

4.729937

3.085307

0.062968

0.258934

0.238144

0.258032

and worst values, IGA finds the best solution and the system’s lowest average fitness values. IGA is quite competitive with the ARO algorithm and converges more quickly than the GA approach, as seen in Fig. 11. For the nonlinear system 7, Table 14 displays the optimal outcome from several solutions derived using IGA and other techniques. Our method finds many nondominant solutions for this problem and ultimately outperforms all other algorithms to produce the best result. By outperforming the other competing approaches in terms

132

A. M. Ibrahim and M. A. Tawhid

102 4.5 4

3

100

10-2

fitness

Fitness

3.5

IGA GA PSO WOA ARO AOSMA iCOA IRGA

100

2.5 2 1.5 1 0.5 0

101

102

A

IG

Iterations

GA

O

PS

A WO

O AR

SM AO

A

A iCO

GA IR

Fig. 11 The convergence history and ANOVA test for system 6 Table 14 Best results obtained by different methods for Sys7 IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

p1

0.671554

0.164432

0.208515

0.641926

0.164432

0.162463

0.620897

0.164432

p2

0.740955

–0.986388

–0.975316

0.759106

–0.986388

–0.987340

0.661936

–0.986388

p3

0.951893

0.718453

0.719375

0.929696

–0.947064

0.717122

0.872120

–0.947064

p4

–0.306431

–0.695576

–0.691753

–0.348460

–0.321046

–0.696646

0.720892

–0.321046

p5

0.963811

–0.997964

–0.991171

–0.983210

0.998233

0.998601

0.180732

0.998233

p6

0.266587

0.063774

–0.132563

–0.182457

0.059418

0.061007

0.445688

–0.059418

p7

0.404641

–0.527809

–0.506085

0.383828

0.411033

–0.530057

0.080835

0.411033

p8

–0.914475

–0.849363

0.860576

0.929696

–0.911620

–0.848785

0.022688

0.911620

f1

0

–1.82E-07

1.03E-02

2.17E-02

0.00E+00

–1.24E-03

–1.23E+00 –1.71E-12

f2

0

2.03E-08

–1.02E-03

3.07E-02

0.00E+00

2.64E-03

–4.95E-01

1.91E-09

f3

–3.90E-18

–5.30E-08

–4.41E-02

6.36E-02

8.67E-19

1.68E-03

2.35E-01

–2.03E-10

f4

–5.55E-17

–4.05E-08

-3.11E-02

2.66E-02

6.66E-16

1.29E-03

2.09E-02

–3.96E-10

f5

0

–1.59E-08

–5.28E-03

–1.17E-02

2.22E-16

1.23E-03

–1.76E-01

–8.23E-10

f6

0

3.91E-08

–3.98E-03

–1.42E-02

0.00E+00

–4.20E-04

2.80E-01

–6.96E-10

f7

0

–1.60E-07

–7.75E-06

–7.08E-06

2.22E-16

9.26E-04

–7.69E-01

–1.55E-11

f8

0

1.22E-07

–3.29E-03

1.17E-02

0.00E+00

1.40E-03

–9.93E-01

–8.21E-12

Fmin

5.56E-17

2.83E-07

5.54E-02

8.16E-02

7.36E-16

4.18E-03

1.46E-05

2.24E-09

Favg

1.62E-16

9.21E-07

4.18E-01

4.32E-01

2.81E-05

2.65E-02

9.22E-03

8.93E-08

Std

1.37E-16

4.65E-07

1.88E-01

1.82E-01

6.77E-05

8.94E-02

6.49E-02

1.40E-07

Fmax

8.72E-16

2.13E-06

7.77E-01

7.54E-01

3.62E-04

0.459583

0.459121

7.84E-07

of the best, average, worst, and standard deviation results, IGA surpasses them. As seen from Fig. 12, one should also look at the convergence curve and the ANOVA plot to determine how well the IGA performs. The best outcomes from this system, IGA, and other algorithms are displayed in Table 15. The nonlinear system 8 has a number of solutions [37]. None of the other methods identified the system’s ideal solution; only IGA and ARO can do so (The number of successes can be found in Table 16). Also, it can be seen in the

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem 100

133

0.8 0.7 0.6

10-10

10-15

0.5

fitness

Fitness

10-5 IGA GA PSO WOA ARO AOSMA iCOA IRGA

100

0.4 0.3 0.2 0.1 0

101

102

A

IG

Iterations

GA

O

PS

A WO

O AR

SM AO

A

A iCO

GA IR

Fig. 12 The convergence history and ANOVA test for system 7 Table 15 Best results obtained by different methods for Sys8 IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

p1

12.25652

12.26484

9.057908

12.26327

12.25652

14.40943

0.321636

12.26601

p2

22.89494

22.52301

23.18105

23.08968

22.89494

20.08871

0.357895

23.12719

p3

2.789818

3.028786

13.02549

2.677748

2.789818

7.234547

0.900494

2.656717

f1

0

9.035688

–3.796830

–4.348767

0

124.8022

–167.0197

–5.173163

f2

0

–0.316905

0.088551

0.310781

0

366.5628

–9369.369

0.028229

f3

0

0.074697

–0.076608

–0.374878

0

–690.3304

–6835.158

0.009768

Fmin

0

9.041552

3.798635

4.375944

0

791.5175

0.007779

5.173249

Favg

1.17E-13

116.9119

210.8050

161.0242

0.298191

3185.163

21.94184

112.3195

Std

3.20E-13

60.93919

549.4405

82.53490

2.108526

1368.196

44.00345

68.10639

Fmax

1.50E-12

210.8354

3968.454

288.0275

14.90953

6395.614

143.2582

246.0997

105 6000 5000 4000

10-5

10-10

10-15 100

fitness

Fitness

10

0

IGA GA PSO WOA ARO AOSMA iCOA IRGA

3000 2000 1000 0

101

102

Iterations

A IG

GA

O PS

A WO

A O SM AR AO

A

iCO

GA IR

Fig. 13 The convergence history and ANOVA test for system 8

convergence and ANOVA plot in Fig. 13 IGA’s curve is faster than others, and the results are focused on the optimal solution, resulting in higher precision than other methods.

134

A. M. Ibrahim and M. A. Tawhid

Table 16 Success number of obtained the exact solutions by different methods Fun. Sys1

Sys2

Sys3

Sys4

Sys8

IGA

GA

PSO

WOA

ARO

AOSMA

iCOA

IRGA

SN

49

0

0

0

0

0

0

0

Evmin

15620

160000

160000

160000

160000

160000

160000

160000

Evavg

21590

160000

160000

160000

160000

160000

160000

160000

SN

25

0

0

0

0

0

0

0

Evmin

27900

160000

160000

160000

160000

160000

160000

160000

Evavg

95897.6

160000

160000

160000

160000

160000

160000

160000

SN

35

0

0

0

18

0

0

0

Evmin

8820

160000

160000

160000

9600

160000

160000

160000

Evavg

55240.4

160000

160000

160000

106961.6

160000

160000

160000

SN

48

0

0

0

32

0

0

0

Evmin

22380

160000

160000

160000

55600

160000

160000

160000

Evavg

96859.6

160000

160000

160000

108241.2

160000

160000

160000

SN

31

0

0

0

27

0

0

0

Evmin

8880

160000

160000

160000

18880

160000

160000

160000

Evavg

68119.2

160000

160000

160000

87415.6

160000

160000

160000

5.3 Success Rate Five nonlinear systems out of the eight have the optimum solution with a fitness value of “Zero”. The success rate for each system attained through various techniques is shown in Table 16. In addition, the minimum Evmin and average Evavg function evaluation are listed in Table 16. While the other algorithms in systems 1 and 2 do not succeed in finding the optimum solutions in any of the 50 runs, IGA is the only algorithm that does so in 49% and 25%, respectively, of the 50 separate runs that it achieves. According to our algorithm IGA, the percentage of times an exact solution is obtained in systems 3, 4, and 9 is 35%, 48%, and 31%, respectively. While ARO achieves the exact solutions for the same systems 3, 4, and 9 by 18%, 32%, and 27%, respectively. In terms of best and average function evaluations, it is clear that the fitness evaluation’s numbers are very small compared to those of the ARO method. In contrast, none of the other algorithms can identify a shared system’s precise solution. Based on the explanations mentioned above, it can be concluded that the improved IGA is more effective and capable of locating the exact solution.

6 Conclusions and Future Work Avoiding local optimum traps is a crucial objective of mutation in genetic algorithms. A population will naturally congregate around the fittest model when fitness reaches a plateau, resulting in a population that is mainly genetically homogeneous. Small

5 IGA: An Improved Genetic Algorithm for Real-Optimization Problem

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mutations try to keep the quest for Global Optimums from descending into these plateaus. In essence, evolutionary valleys are what optimal traps are, and occasionally minor mutations cannot get out of them. In these cases, it was necessary to modify the mutation operator, leading to the implementation of both mutation approaches based on the IGA’s dimensional and standard mutation operators in a new generation. In order to prevent fitness decay in the population’s upper percentile, keeping an elite selection was not the ideal option; thus, we used the corresponding individuals’ selection on their fitness. Identifying a use case in which the suggested improved genetic algorithm would be the best option is challenging. IGA is typically simpler to implement and uses fewer resources when dealing with low-dimensional problems. However, algorithmic models and adaptation are frequently required when dealing with larger-scale problems because of the complexity of fitness modelling and the high computational demand to find a solution. In this chapter, an improved GA, IGA, is suggested to solve two problems: unconstrained optimization problems and nonlinear systems. In IGA, the crossover and mutation operation of the simple genetic algorithm is improved. Our suggested algorithm is applied to eight nonlinear systems and 30 CEC’18 bench test problems to demonstrate the capability of the IGA and compare it with other algorithms. The computational results demonstrate that our proposed algorithm, IGA, outperforms other compared algorithms in solving unconstrained optimization problems and nonlinear systems. Two nonparametric statistical tests are carried out for the two problems to demonstrate how well IGA performs in comparison to other compared algorithms: Unconstrained optimization functions and nonlinear equation systems. In future work, our suggested algorithm, IGA, will be utilized to solve various problems such as bilevel programming [48], large-scale problems and molecular potential energy function [49, 50], multi-objective problems [51–53], job-shop scheduling problem [54], and constrained and engineering optimization problems [49], travel salesman problems [55, 56], feature selection problems [57–60]. We investigate further improvement for IGA by combining chaos theory [43, 61] and hybridization IGA with other algorithms [39, 41, 46, 48, 62]. Acknowledgements The Natural Sciences and Engineering Research Council of Canada (NSERC) has contributed to funding the second author’s research.

References 1. El-Shorbagy MA, El-Refaey AM (2020) Hybridization of grasshopper optimization algorithm with genetic algorithm for solving system of non-linear equations. IEEE Access 8:220944– 220961 2. Barbashov BM, Nesterenko VV, Chervyakov AM (1982) General solutions of nonlinear equations in the geometric theory of the relativistic string. Commun Math Phys 84:471–481 3. Holstad A (1999) Numerical solution of nonlinear equations in chemical speciation calculations. Comput Geosci 3:229–257

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4. Michael Bartholomew-Biggs (2008) Nonlinear optimization with engineering applications, vol 19. Springer Science & Business Media 5. Friedlander A, Gomes-Ruggiero MA, Kozakevich DN, Mario Martínez J, Augusta Santos S (1997) Solving nonlinear systems of equations by means of quasi-neston methods with a nonmonotone stratgy. Optim Methods Softw 8(1):25–51 6. Candelario G, Cordero A, Torregrosa JR, Vassileva MP (2023) Generalized conformable fractional newton-type method for solving nonlinear systems. Num Algorithms 7. Ji Y, Kang Z, Zhang C (2021) Two-stage gradient-based recursive estimation for nonlinear models by using the data filtering. Int J Control Autom Syst 19(8):2706–2715 8. Dey N, Ashour AS, Bhattacharyya S (2020) Applied nature-inspired computing: algorithms and case studies. Springer Tracts in Nature-Inspired Computing, Springer Singapore 9. Dey N (2017) Advancements in applied metaheuristic computing. In: Advances in data mining and database management (2327-1981). IGI Global 10. Holland J (1992) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. MIT Press, Cambridge, Mass 11. Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, vol 4. IEEE Publications, pp 1942–1948 12. Marini F, Walczak B (2015) Particle swarm optimization (PSO). A tutorial. Chemom Intell Lab Syst 149:153–165 13. Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67 14. Wang L, Cao Q, Zhang Z, Mirjalili S, Zhao W (2022) Artificial rabbits optimization: a new bio-inspired meta-heuristic algorithm for solving engineering optimization problems. Eng Appl Artif Intell 114:105082 15. Zhao W, Wang L (2016) An effective bacterial foraging optimizer for global optimization. Inf Sci 329:719–735 (Special issue on Discovery Science) 16. Chu S-C, Tsai P-W, Pan J-S (2006) Cat swarm optimization. In Yang Q, Webb G (eds) PRICAI 2006: trends in artificial intelligence. Springer, Berlin, Heidelberg, pp 854–858 17. Mirjalili S (2016) Sca: A sine cosine algorithm for solving optimization problems. KnowlBased Syst 96:120–133 18. Joshi AS, Kulkarni O, Kakandikar GM, Nandedkar VM (2017) Cuckoo search optimizationa review. Mater Today: Proc 4(8):7262–7269; International conference on advancements in aeromechanical materials for manufacturing (ICAAMM-2016): organized by MLR Institute of Technology, Hyderabad, Telangana, India 19. Meraihi Y, Gabis AB, Mirjalili S, Ramdane-Cherif A (2021) Grasshopper optimization algorithm: theory, variants, and applications. IEEE Access 9:50001–50024 20. Ahmadianfar I, Bozorg-Haddad O, Chu X (2020) Gradient-based optimizer: a new metaheuristic optimization algorithm. Inf Sci 540:131–159 21. Lambora A, Gupta K, Chopra K (2019) Genetic algorithm-a literature review. In: 2019 international conference on machine learning, big data, cloud and parallel computing (COMITCon). IEEE, pp 380–384 22. Alhijawi B, Awajan A (2023) Genetic algorithms: theory, genetic operators, solutions, and applications. Evol Intell 1–12 23. Broyden CG (1965) A class of methods for solving nonlinear simultaneous equations. Math Comput 19(92):577–593 24. Crina G, Ajith A (2008) A new approach for solving non-linear equations system. IEEE Trans Syst Man Cybern 38(3):698–714 25. Naik MK, Panda R, Abraham A (2021) Adaptive opposition slime mould algorithm. Soft Comput 25(22):14297–14313 26. Mahdi A, Asgarali B, Davoud A (2016) Improved cuckoo optimization algorithm for solving systems of nonlinear equations. J Supercomput 72(3):1246–1269 27. Das AK, Pratihar DK (2021) Solving engineering optimization problems using an improved real-coded genetic algorithm (IRGA) with directional mutation and crossover. Soft Comput 25(7):5455–5481

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28. Venkatesh SS, Mishra D (2021) Variable search space converging genetic algorithm for solving system of non-linear equations. J Intell Syst 30(1):142–164 29. Gong W, Liao Z, Mi X, Wang L, Guo Y (2021) Nonlinear equations solving with intelligent optimization algorithms: a survey. Complex Syst Model Simul 1(1):15–32 30. Gao W, Li G, Zhang Q, Luo Y, Wang Z (2021) Solving nonlinear equation systems by a two-phase evolutionary algorithm. IEEE Trans Syst Man Cybern Syst 51(9):5652–5663 31. Liao Z, Gong W, Wang L (2020) Memetic niching-based evolutionary algorithms for solving nonlinear equation system. Expert Syst Appl 149:113261 32. Xinming Z, Qian W, Youhua F (2017) Applying modified cuckoo search algorithm for solving systems of nonlinear equations. Neural Comput Appl 31(2):553–576 33. Sharma JR, Arora H (2016) Improved newton-like methods for solving systems of nonlinear equations. SeMA J 74(2):147–163 34. Abubakar AB, Kumam P (2018) A descent dai-liao conjugate gradient method for nonlinear equations. Num Algorithms 81(1):197–210 35. Awad NH, Ali MZ, Liang JJ, Qu BY, Suganthan PN (2016) Problem definitions and evaluation criteria for the CEC 2017 special session and competition on single objective bound constrained real-parameter numerical optimization. In: Technical report. Nanyang Technological University Singapore, pp 1–34 36. Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol Comput 1(1):3–18 37. Turguta O, Turgutb M, Cobana M (2014) Chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations. Comput Math Appl 68(4):508–530 38. Alikhani Koupaei J, Hosseini SMM (2015) A new hybrid algorithm based on chaotic maps for solving systems of nonlinear equations. Chaos Solitons Fractals 81:233–245 39. Ibrahim AM, Tawhid MA (2018) A hybridization of differential evolution and monarch butterfly optimization for solving systems of nonlinear equations. J Comput Des Eng 6(3):354–367 40. Ibrahim AM, Tawhid MA (2017) Conjugate direction DE algorithm for solving systems of nonlinear equations. Appl Math Inf Sci 11(2):339–352 41. Tawhid Mohamed A, Ibrahim Abdelmonem M (2020) A hybridization of grey wolf optimizer and differential evolution for solving nonlinear systems. Evol Syst 11(1):65–87 42. Sharma JR, Arora H (2013) On efficient weighted-newton methods for solving systems of nonlinear equations. Appl Math Comput 222:497–506 43. Tawhid MA, Ibrahim AM (2022) Improved SALP swarm algorithm combined with chaos. Math Comput Simul 202:113–148 44. Tawhid MA, Ibrahim AM (2021) Solving nonlinear systems and unconstrained optimization problems by hybridizing whale optimization algorithm and flower pollination algorithm. Math Comput Simul 190:1342–1369 45. Abdollahi M, Isazadeh A, Abdollahi D (2013) Imperialist competitive algorithm for solving systems of nonlinear equations. Comput Math Appl 65(12):1894–1908 46. Ibrahim Abdelmonem M, Tawhid Mohamed A (2019) A hybridization of cuckoo search and particle swarm optimization for solving nonlinear systems. Evol Intell 12(4):541–561 47. Jaberipour M, Khorram E, Karimi B (2011) Particle swarm algorithm for solving systems of nonlinear equations. Comput Math Appl 62(2):566–576 48. Tawhid MA, Paluck G (2021) Solving linear bilevel programming via particle swarm algorithm with heuristic pattern search. Inf Sci Lett 6(1):1 49. Ali AF, Tawhid MA (2016) Hybrid simulated annealing and pattern search method for solving minimax and integer programming problems. Pacific J Optim 12(1):151–184 50. Tawhid MA, Ali AF (2017) A hybrid social spider optimization and genetic algorithm for minimizing molecular potential energy function. Soft Comput 21(21):6499–6514 51. Tawhid MA, Savsani V (2019) Multi-objective sine-cosine algorithm (MO-SCA) for multiobjective engineering design problems. Neural Comput Appl 31(2):915–929 52. Savsani V, Tawhid MA (2017) Non-dominated sorting moth flame optimization (NS-MFO) for multi-objective problems. Eng Appl Artif Intell 63:20–32

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53. Tawhid MA, Savsani V (2018) A novel multi-objective optimization algorithm based on artificial algae for multi-objective engineering design problems. Appl Intell 48(10):3762–3781 54. Ibrahim AM, Tawhid MA (2022) An improved artificial algae algorithm integrated with differential evolution for job-shop scheduling problem. J Intell Manuf 1–16 55. Tawhid MA, Savsani P (2019) Discrete sine-cosine algorithm (DSCA) with local search for solving traveling salesman problem. Arab J Sci Eng 44(4):3669–3679 56. Savsani P, Tawhid MA (2018) Discrete heat transfer search for solving travelling salesman problem. Math Found Comput 1(3):265 57. Wang J, Zhang Q, Hedar A, Ibrahim AM (2014) A rough set approach to feature selection based on scatter search metaheuristic. J Syst Sci Complex 27(1):157–168 58. Ibrahim AM, Tawhid MA, Ward RK (2020) A binary water wave optimization for feature selection. Int J Approx Reason 120:74–91 59. Tawhid MA, Ibrahim AM (2020) Feature selection based on rough set approach, wrapper approach, and binary whale optimization algorithm. Int J Mach Learn Cybern 11(3):573–602 60. Ibrahim AM, Tawhid MA A new hybrid binary algorithm of bat algorithm and differential evolution for feature selection and classification. In: Applications of bat algorithm and its variants. Springer, pp 1–18 61. Ibrahim AM, Tawhid MA (2022) Chaotic electromagnetic field optimization. Artif Intell Rev 1–42 62. Tawhid MA, Ibrahim AM (2023) An efficient hybrid swarm intelligence optimization algorithm for solving nonlinear systems and clustering problems. Soft Comput 1–29

Chapter 6

Application of Genetic Algorithm-Based Controllers in Wind Energy Systems for Smart Energy Management D. Boopathi , K. Jagatheesan , Sourav Samanta , B. Anand , and J. Jaya

1 Introduction In recent days the power demand raises because the need for power is increasing rapidly. Electric power generates in all possible ways. At present more than 65% of the electric power is generated from non-renewable energy sources, i.e., coal, and fossil fuels. This creates many problems for the environment in terms of pollution. To protect the environment from pollution, also the shortage of fossil fuels turns the direction of electric power generation toward renewable energy sources. Mostly solar and wind energy sources are utilized for electric power generation. These sources can be harvested only on the specified climate conditions, also these kinds of renewable energy sources are providing non-linear outputs [1]. When introducing renewable energy sources to the power system there are few notable issues are raised, i.e., frequency deviation and voltage fluctuation. The frequency deviation problem is handled by the Load frequency control (LFC) scheme. The LFC scheme is introduced in the power system with the support of an auxiliary controller to minimize the steady-state error. The performance of the secondary controller is enhanced by an optimized gain parameter of the controller. Many researchers are handled the LFC with different controllers and optimization techniques as discussed in the literature review.

D. Boopathi (B) · K. Jagatheesan Paavai Engineering College, Puduchatram, Tamil Nadu, India e-mail: [email protected] S. Samanta University Institute of Technology, The University of Burdwan, Bardhaman, West Bengal, India B. Anand · J. Jaya Hindusthan College of Engineering and Technology, Coimbatore, Tamil Nadu, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_6

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PID controller based on a genetic algorithm is employed as a secondary controller for a hybrid power production that consists of thermal, solar, wind, geothermal, and electrical vehicles (EV) [2], the reliability of the proposed controller test by robustness analysis. A five-area interlinked power system is developed and analyzed for LFC by introducing GA-PID controller, the improvement of the proposed controller is provided by comparing it with a conventional PID controller [3]. A comparative analysis for LFC of multi-source power system conducted by [4], with GA, Bacteria Foraging Algorithm (BFA), and Particle Swarm Optimization (PSO) based PID controller. Renewable energy source-based power system performance was examined by GA-PID controller, and GA-PI controller response was compared with the proposed controller for supremacy [5]. An intelligent PID controller is tuned GA, and PSO, also the gain parameters are optimized by conventional method [6]. The response comparison provided that GA-PID controller is better than other technique. The genetic algorithm utilized to enhance the controller performance of the fuzzy logic controller (FLC), the PID controller is assigned as a secondary controller for the interlinked two-area thermal power system [7]. By the result comparison between GA-based FLC, PID, and conventional PID, the FLC controller performs well against frequency deviation. Double chains quantum-based genetic algorithm tuned PID controller was employed for the frequency oscillation for an interlinked power system, and the response of the GA-PID controller is compared with another popular optimized controller [8]. GA-PID controller has been involved to stabilize the oscillation of the system frequency in a thermal power system [9], authority of the GA-PID controller is shown by the result comparison with FLC. A multiple-source interlink system is involved for LFC by implementing a genetic algorithm-based slid mode controller, also the results of the proposed controller, conventional PID, and a few elder research papers to prove the improvement of the proposed controller [10]. A multi-objective genetic algorithm-based Proportional integral (PI) controller involved for LFC of a grid-connected thermal system, the performance of the GA was analyzed in the system with/without wind turbine [11]. Frequency stabilization for an interconnected multiple-source system has been studied, by adopting a GA-PID controller as an auxiliary controller [12]. The improvement of the GA-PID controller was verified with the conventional PID controller response. The region of convergence for a hybrid power system was studied with the support of the genetic algorithm-based cascade (PI-PD) controller developed, and the superiority is validated by a conventional PID controller [13]. GA-PID controller performance in a two-area system was studied, also the Differential evolution (DE) algorithm response was compared to show the improvement [14]. A wind energy-integrated microgrid was utilized to study the performance of the GA for LFC over Particle swarm optimization (PSO) [15]. Multi-objective diversity genetic algorithm optimized the parameters of the PI and PID controllers for a three-area power network [16], the response comparison provided GA-PID controller is better for LFC. To improve the power quality of a Photovoltaic (PV) power generating system GA was utilized [17]. A decentralized PI controller parameter is adjected by GA for the LFC of a conventional power system, and the proposed method is provided better result than conventional methods [18].

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A multi-microgrid power system constructed with wind farms, PV, fuel cell, and energy storage systems were designed and tested for LFC, by implementing a genetic algorithm based on a cascade (PI + I + PD) controller [19]. To demonstrate the dominance of the suggested controller, its response was compared to that of standard PI and PID controllers. Multi-objective genetic algorithm (MOGA) tuned PI controller has been utilized for the system stability of a three-area power network, the behavior of the proposed method is verified with few literature results and robust tests [20]. The author in [21] utilized GA to tune the Integral Proportional (IP) controller which is developed for an interconnected thermal power network as a secondary controller. The performance of the proposed controller was tested with conventional IP and PI controllers. For the LFC of three-area power grid networks, the GA-PID controller is utilized as an auxiliary controller [22]. To determine the robust margin, the robust analysis of the suggested controller is undertaken using Kharitonov’s theorem. The genetic algorithm was also utilized by many researchers for the LFC study, Mayfly algorithm-based PID controller was utilized as a secondary controller for multiple-source systems, to demonstrate the improvement of the proposed controller GA used in [23]. A comparative study is conducted by the author in [24], for the LFC of an interconnected power network. In the study, PSO, GA, and firefly algorithms were used. A standalone power system is investigated by PSO-PID controller for LFC, for validation GA, and is used in [25]. The cuckoo search algorithm was utilized for three-area power system LFC, for the efficiency of the proposed technique GA was used by the author [26] Multi-verse optimization (MVO) algorithm-based 3Degree of PID controller designed for a multiple area thermal power network, also the improvement of the MVO-3DoFPID controller justified by result comparison with GA [27]. To improve the performance of the induction generator-based hybrid wind farm the genetic algorithm used a PI controller designed by [28], for a double-feed induction generator [29]. GA is utilized for the LFC of a standalone power system [30]. The wind power plant is included as a part of a power network [31]. Genetic algorithm is not only used in the field of LFC, it is utilized in various fields like image process, medical factories, construction, and the field of economics. In [32] GA is utilized for the image filtering process. The Medline text data mining process utilized the GA [33]. To find the structural failure of the RC structure a multi-objective GA was employed [34]. To maximize the profit of a business the GA-based approach is helped a lot of the people [35]. To detect the water quality, multi objective based GA is utlized by the author in [36]. The detailed review provides the application of the genetic algorithm in a wide range in the field of electric power generation. The literature review supported the determination of the exact research gap. From the analysis, the significance of the proposed work is given below. A comparative review is reported in Table 1.

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Table 1 Literature review summary System configuration

Controller

Previous studies

References

Hybrid system—thermal, solar, wind, geothermal, and EV

GA-PID

Robustness analysis

[2]

Five area—thermal

GA-PID

Conventional PID controller

[3]

Three area—wind, thermal, PV

GA-PID

Compared with GA-PI controller

[5]

Thermal

GA-PID

Conventional, fuzzy, and PSO-PID controller

[6]

Two area—thermal, hydro, and PV

GA-PID

Conventional, fuzzy—PID controller

[7]

Thermal with GDB and GRC

GA-PID

Fuzzy controller

[9]

Two area—thermal, hydro

GA-PID

Fuzzy logic controller

[12]

Multi-source—thermal, hydro, gas

GA-PID

GA-PI

[20]

Two are—thermal

GA-IP

IP, PI controller

[21]

wind, BEES

PSO-PID

GA-PID

[24]

Wind, BESS, FC

ACO-PID

GA-PID, PSO-PID

[37]

1.1 Research Gap Analysis In this chapter, a standalone wind power system is undergone to the LFC using a GA tune PID controller. The review process helped to find the actual research gap in the field of LFC using wind power plants. In the previous experiments, the researchers utilized the wind power plant as a part of the interlinked power network. But the recent days the zero-carbon scheme is implemented by most countries, so electricity production also supposes to move toward wind power plants. The LFC investigation in a standalone wind power plant is much needed one. This gap is found and analyzed in this chapter.

1.2 Novelty of the Chapter A standalone wind power plant investigated for LFC is a new idea in this chapter. Also, the large scale of energy storage systems is included with the standalone wind power plant to support the secondary controller against frequency fluctuations.

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1.3 Organization of the Chapter In Sect. 1, the literature review and introduction of the research work are discussed. The proposed power system model development and the functional blocks are explained in Sect. 2. Section 3 deals with the need and design of a secondary controller for the proposed model. Genetic algorithm behavior, advantages, and the flow chart of tuning secondary controller are discussed in this Sect. 4. The performance analysis of the wind power plant with/without energy storage systems (BESS and FC) is studied in Sect. 5. The conclusion of the chapter is discussed in Sect. 6. As per the above structure, the chapter is constructed.

1.4 Contribution of the Work • A zero-carban renewable energy (wind) based power generating system is developed. • The energy storage systems are implemented in the proposed system in a wide range. • A tributary controller (PID) is developed to execute the LFC scheme in the proposed system. • Genetic algorithm successfully implemented to optimize the controller parameters (gain value). • The performance of the GA is analyzed by different scenarios (with/without energy storage).

2 Investigated Power System A pure (zero carbon) renewable energy source-based power generating system (wind turbine generator) is designed in this chapter for a detailed analysis of LFC. The wind turbine generator is equipped with energy storage technologies, such as fuel cell and battery energy storage systems [1]. When renewable energy sources are introduced in power generation energy storage units are a must to operate the load smoothly. Because renewable energy source-based power generations are mostly providing non-linear outputs. The nominal parameters of the investigated model are given in Annexure 1. The overall Simulink model is shown in Fig. 4.

2.1 Wind Turbine Generator A wind turbine generator is a device that converts kinetic energy from the wind into electrical energy. At the top of the tower, the wind turns blades connected to

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Fig. 1 Block diagram of wind turbine generator Fig. 2 Block diagram of battery energy storage system

a shaft. The shaft is connected to a generator and spins it, which in turn creates electricity for the generator. Wind energy is also called as zero-carbon source for electric power generation because it would pollute the environment in any of the formats. The designed model of the wind turbine is given in Fig. 1 [37].

2.2 Battery Energy Storage System Energy storage systems play a crucial role in meeting the demand for electrical energy, especially in renewable source-based power systems. The battery stores electrical energy inside in the form of electrochemical. The battery has a dual role in the power system, during the minimum loading it will be loaded, and during maximum loading, it delivers power to the system. The block diagram of a battery is shown in Fig. 2 [38].

2.3 Fuel Cell The electrochemical energy of a fuel, such as a hydrogen, is transformed into electrical energy via an electrochemical device known as a fuel cell. Cathode and anode electrodes are often separated by an electrolyte in fuel cells. Fuels like hydrogen are divided into protons and electrons when they are injected into the anode. While the electrons go via an external circuit, creating an electric current, the protons move through the electrolyte to the cathode. Protons and electrons mix with oxygen at the cathode to produce water, which releases energy in the process. A block diagram of the fuel cell is given in Fig. 3 [39, 40] (Fig. 4).

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Fig. 3 Block diagram of fuel cell

Fig. 4 Overall proposed simulink power system model

3 Secondary Controller Design A secondary controller is essential to implement the load frequency control scheme in the power system. In general, a primary controller is integrated with the power system to manage the unbalance between generation and load. But the primary controller is not much better for controlling action during unexpected loading situations. The PID controller is most popular in industrial automation and control systems. PID controller always calculates the difference between the desired set values and measured values. The difference between the values is taken as error signal e(t). The controller goes to minimize the error by adjusting the process control inputs according to the controller gain parameters. A PID controller is designed by integrating three different controllers (P, I, and D), and each controller has its controlling action. The basic structure of the classical PID controller is shown in Fig. 5. The transfer function of the PID controller is in Eq. 1 [41, 42]. U (s) Ki = Kp + + s Kd E(s) s where Kp

Controller gain value (Proportional)

(1)

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Fig. 5 Structure of classical PID controller

Ki Kd U(s) E(s)

Controller gain values (Integral) Controller gain value (Derivative) Actuating signal Error signal.

While tuning the controller parameters the most important is the cost function. In this chapter, the genetic algorithm will be used to optimize the controller gain parameters. The ITAE objective function is utilized to minimize the steady-state error. The mathematical expression of the ITAE cost function is given in Eq. 2. Area control error is “ACE” and simulation time denotes “t” in Eq. 2 [42]. ∫ JITAE =

t · |ACE|dt

(2)

4 Genetic Algorithm The genetic algorithm is the most famous optimization method for solving complex problems, it’s working with the principle of genetic selection. It is a statistical approach, based on heredity and evolution. GA was introduced by John Holland in 1975. In the first step, the random populations are assigned to each generation. In parallel, the fitness value of each generation is calculated. Based on the minimum fitness value the new populations are converted. In the conversation process three steps are involved (i) Genetic operator selection (ii) Crossover (iii) Mutation [2]. Detail flow chart of GA associated with PID controller is given in Fig. 6 [3]. GA has a few advantages listed as following • Independent of variable utilization. • No requirement for derivative parameters. • Variables are used in a very complicated manner.

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Fig. 6 Functional flow chart of GA associated with PID controller

• GA may utilize a variety of data, including numerical, experimental, and analytical data functions. GA is perfectly utilized for the secondary controller gain tuning, and the tuned gain parameters are reported in Table 2.

148 Table 2 The GA-tuned secondary controller gain values

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System/tuned gain values

KP

KI

KD

Wind_PID

0.0724

0.9965

0.0830

Wind_BESS_PID

0.9960

0.9022

0.2712

Wind_FC_PID

0.5274

0.9968

0.0260

Wind_FC_BESS_PID

0.9771

0.9359

0.0065

5 Performance Analysis The designed power system model includes wind, BESS, and FC has been analyzed under a simulation environment with one percentage load disturbance. The evolutionary algorithm was perfectly applied to maximize the secondary controller’s gain values. In this section performance of the genetic algorithm is analyzed in detail under various working environments of the proposed power system. In case 1, GAtuned PID controller performance in wind power plant and BESS-integrated wind power plant is discussed. In case 2, the FC unit is introduced with the wind power plant and performance has been discussed. Case 3, both BESS and FC integrated wind power plant performance is analyzed.

5.1 Case 1: Performance Investigation with/without BESS in Wind Power Plant Genetic algorithm-optimized PID controllers are utilized to perform the LFC in the power system (WPP). The control behavior of the GA-PID controller is analyzed between with/without BESS in the WPP. The graphical response comparison of frequency deviation is given in Fig. 7, the ACE comparison is shown in Fig. 8, and the numerical values from the response comparison such as settling time (T s ), peak overshoot (POs), peak undershoot (PUs) is reported in Table 3. It is observed from Figs. 7 and 8, Table 3 that the GA-PID controller has performed well during the emergency loading conditions for controlling oscillations in the system frequency. When the BESS is integrated with the power system, initially it consumes power from the system to store in it. At initial a small peak occurs in the system frequency, even though BESS acted as a load at the beginning GA-PID controller minimized the steady-state error and bring back to the predetermined value at 65 s. The control behavior of GA-PID controller in terms of TS wind power plant with/without BESS is compared by a column chart shown in Fig. 9.

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Fig. 7 Frequency variation comparison between with/without BESS in wind power plant

Fig. 8 ACE comparison between with/without BEES in wind power plant Table 3 Frequency variation time domain specific parameters with/without BESS in wind power plant

System/controlled parameters

TS (s)

POS (Hz)

PUS (Hz)

Wind

60

0

0.125

Wind with BESS

65

0.01

0.180

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Fig. 9 Ts comparison between with/without BEES in wind power plant

5.2 Case 2: Performance Investigation with/without FC in Wind Power Plant In this section the response analysis of wind power plants with/without fuel cells, using GA-PID controller during the emergency loading conditions is discussed. The result comparisons are shown in Figs. 10 and 11, respectively, frequency variation and ACE. Also, the numerical values in Figs. 10 and 11 are provided in Table 4. When the fuel cell is introduced into the proposed system, it is supported by the GA-PID controller for controlling the oscillation and minimizing the steadystate error. GA-PID controller settles the oscillation of system frequency at the 60 s without the support of the FC in the WPP during unexpected power demand variation. After integrating FC with the WPP, the error is minimized and brought into the predetermined value of the system frequency 2 s advance (58 s). Figure 12 shows the TS comparison of the wind power plant with/without FC.

5.3 Case 3: Performance Investigation with/without FC in Wind Power Plant Sections 5.1 and 5.2 explained the performance of the BESS and FC with wind power plant for frequency stabilization. In this section, both BESS and FC are included with the wind power plant. BESS and FC are having separate behavior with a wind power plant, and that performance is discussed. The frequency variation and ACE response

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Fig. 10 Frequency variation comparison between with/without FC in wind power plant

Fig. 11 ACE comparison between with/without FC in wind power plant Table 4 Frequency variation time domain specific parameters with/without FC in wind power plant

System/controlled parameters

TS (s)

POS (Hz)

PUS (Hz)

Wind

60

0

0.125

Wind with FC

58

0

0.130

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Fig. 12 Ts comparison between with/without FC in wind power plant

comparisons are shown in Figs. 13 and 14, respectively. Table 5 having the numerical parameters from Figs. 13 and 14. The analysis from Fig. 10 and Table 4, when the proposed power system incorporates two different energy storage units (BESS and FC). The control action more difficult than the other two working environments is in Sects. 5.1 and 5.2. The two different energy storage units have different behavior during the execution of LFC

Fig. 13 Frequency variation comparison between with/without BESS & FC in wind power plant

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Fig. 14 ACE comparison between with/without BEES and FC in wind power plant

Table 5 Frequency variation time domain specific parameters of with/without BESS and FC in wind power plant

System/controlled parameters

TS (s)

POS (Hz)

PUS (Hz)

Wind

60

0

0.125

Wind with BESS & FC

82

0.04

0.12

in the proposed power system. Because of the different behavior controlling tasks is somewhat curial. But the GA-PID controller takes care of the problem and settles the variation in the system frequency at 82 s. For the deep analysis, a column chart is plotted for settling time with/without energy storage units. Figure 15 shows the comparison between with/without BEES and FC in wind power plants.

5.4 Performance Investigation with/without BESS & FC in Wind Power Plant The proposed wind power plant is integrated with energy storage units (BESS and FC), and its performance was analyzed under various working environment in Sects. 5.1, 5.2, and 5.3. Overall response comparison of del F and ACE are shown in Figs. 16 and 17 accordingly. GA-PID controller performed well under various working scenario of the proposed power system model (wind turbine generator). The controlling behavior in all the scenarios is appreciable. The overall steady of the chapter is shown in Fig. 18.

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Fig. 15 Ts comparison between with/without BEES and FC in wind power plant

Fig. 16 Frequency variation comparison with/without energy storage units in wind power plant

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Fig. 17 ACE comparison with/without energy storage units in wind power plant Fig. 18 Overall study of the chapter

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6 Conclusion In this chapter, a standalone wind power plant was investigated for LFC, and the study extended to different working scenarios. The genetic algorithm is utilized to optimize the secondary controller (PID) gain parameters. The results were as follows: • The wind power plant without an energy storage unit is investigated by GAPID controller, and the results were compared with the same system with BESS. During the analysis, the BESS incorporated system takes 5 more seconds than without BESS. • In case 2, the response of the proposed power plant without ESU was compared with the fuel cell integrated power plant. Because of the FC support the GA-PID controller settled the oscillation at 58 s. It’s 2 s advances than without FC. • While integrated BESS and FC unit with the proposed power system was executed. The oscillation is the controller at 85 s. It’s more than without ESUs, because of the two non-linear energy storage units. The proposed GA-PID controller performed well in all the different working environments and the controlling action is better in terms of minimized settling time (TS).

7 Challenges and Limitations While considering or integrating the wind power plants in the power system, there are a few challenges and limitations. They are as follows [43, 44]: • Power output—Daily wind velocities cannot be predicted with high precision, and the wind frequently varies from minute to minute and hour to hour. So, we cannot expect for accurate output power. • Flickers—Wind turbines can generate irregular output power, which can cause flickers in the power system network. • Power quality—The location and intermittent nature of wind turbines can result in power quality issues such as voltage drops, frequency fluctuations, and a low power factor. Wind turbines, particularly inductive devices, have a propensity to absorb reactive power from the system, resulting in a low power factor. If wind turbines consume an excessive amount of reactive power, the system can become unstable.

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Appendix [37]

Wind power plant Kp1

1.25

Kp2

1.4

Tp1

0.6S

Tp2

0.041S

KW

0.125

Battery energy storage system KBESS

−1/300

TBESS

0.1S

Fuel cell KFC

1/100

TFC

4S

Rotating mass KPS

120

TPS

20S

R

2.4

B

0.045

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Chapter 7

Application of Genetic Algorithm in Predicting Mental Illness: A Case Study of Schizophrenia S. Rabipour and Z. Asadi

1 Introduction Today, social vitality is one of the most important needs of society because citizens, especially residents of metropolises, have less time to think about themselves and their needs and are highly prone to mood disorders. People with mental illnesses are largely ignored around the world [1, 2]. Disorders such as depression, schizophrenia, and bipolar make up 1.84%, 0.60%, and 0.33% of the total disability-adjusted life years, respectively [3]. Mental health problems can lead to serious consequences such as depression, self-harm, especially for students who are not physically and mentally mature [4]. Depression, if not diagnosed and treated on time, can lead to suicide. Or have a negative effect on the process of physical diseases such as stroke, cancer, and increase medical costs [5]. Bipolar disorder is a common and recurring chronic disease that has a direct impact on the patient’s health and cannot be cured in two thirds of patients. According to the definition, mood is a penetrating emotional feeling that deeply affects a person’s perception and attitude towards himself, others, and the surrounding environment in general [6]. Mood may be normal or abnormal. A normal person experiences a wide range of moods and has a corresponding set of emotional manifestations. This person is able to control his morals and emotions. Mood disorders are clinical conditions that are associated with mood disturbance, lack of feeling of control over mood, and mental experience of extreme distress [7–10]. According to the definition of the World Health Organization, mood disorders, such as major depressive disorder and bipolar disorder, include recurrent and chronic mood disorders, therefore, patients with mood disorders suffer from severe S. Rabipour (B) School of Medicine, Iran University of Medical Science, Tehran, Iran e-mail: [email protected] Z. Asadi Al-Ameen College of Pharmacy, Rajiv Gandhi University of Health Science (RGUHS), Bangalore, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_7

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pressure, functional impairment, and increased risk of suicide [11]. Delay in diagnosis and in most cases misdiagnosis of mood disorders due to the similarity of their symptoms prevents effective treatment of patients. Predicting the onset and course of mood and anxiety disorders is very difficult despite its high importance. For example, bipolar disorder cannot be distinguished from major depressive disorder with high accuracy [12]. Diagnosis is more difficult for patients with more than one type of mental disorder [13–16]. If the disease is correctly diagnosed, cognitive-behavioral psychotherapy, along with usual treatments, is effective in reducing the symptoms of mood disorders in patients, and the use of a team approach based on the cooperation of a psychologist and a psychiatrist can have better treatment results [17–19]. Also, social skill training can provide the basis for reducing ineffective interactions by improving the interpersonal relationships of patients and thus reducing negative emotions and social adjustment of people [20]. Due to the complexity of correctly diagnosing the type of illness of sick people, easier and more accurate diagnostic tools are needed [21]. In this regard, the use of data mining science in health engineering research and patient information monitoring is increasing [22]. Machine learning is a well-known method for analysing big data that was invented by Alan Turing and Arthur Samuel in the 1950s and has been widely used in various fields, including the medical field, since the end of the 20th century [23]. Also, artificial intelligence has been widely used in the field of diagnosing mental disorders due to its high ability to automate and work with massive data and strong inference ability. With the expansion of social networks and the emergence of big data, a lot of data about mental patients is available to psychologists [24]. These data are a rich source to help psychologists in diagnosing mental disorders. Data can be collected from various sources such as wearable devices and smartphones. Analysing and organizing these data manually is difficult, while artificial intelligence can help psychologists diagnose mental illnesses based on images, audio, or text by monitoring data and making inferences and predictions [25]. Among artificial intelligence methods, two approaches, machine learning and deep learning, have shown good results in the evaluation and diagnosis of mental disorders. Machine learning can be considered as an important and fundamental part in big data analysis and can play a role in solving health care problems such as early disease diagnosis, real-time patient monitoring, patient care, and treatment enhancement [26]. Compared to association testing, prediction modeling is better in line with the goals of precision psychiatry [15], and it opens up the possibility of employing supervised machine learning, a group of techniques that uses data to learn the link between predictors and response [16]. Generalization refers to prediction in new individuals who were not included in the model training, and machine learning can detect non-linear relationships. It also prioritizes generalization over drawing inferences about a population from a sample, and it may hasten the development of precision psychiatry by improving prediction from both genetic and non-genetic factors [17]. There has been considerable interest in the use of polygenic risk scores as a tool for prediction in psychiatry [13–17]. In schizophrenia, polygenic risk scores currently explain around 8% of the variance in liability in samples of European ancestry, and

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achieve moderate discrimination between cases and controls (0.72 area under the receiver operator characteristic curve) [18]. Variance explained by polygenic risk scores in samples of non-European ancestry is generally lower as the genome-wide association studies used to calculate polygenic risk scores are based predominantly on European samples [21]. Polygenic risk scores alone in schizophrenia do not have clinical utility [3, 6]; useful prognostic models typically have area under the receiver operator characteristic curve over 0.8 [11]. Combining polygenic risk scores with other predictors is a natural progression and has proved fruitful in both schizophrenia [12] and outside psychiatry [13, 17], with most research to date using linear models rather than more flexible machine learning approaches. In schizophrenia, polygenic risk scores alone have no therapeutic usefulness [9]; effective prognostic models often have area under the receiver operator characteristic curve more than 0.8 [6]. Combining polygenic risk scores with other variables is a logical development that has been beneficial in both schizophrenia [11] and beyond psychiatry [5, 7], with the majority of studies to date employing linear models rather than more flexible machine learning techniques. From the very beginning machine learning usages for schizophrenia encompassed the prediction of clozapine response or weight changes as a result of medication using neural networks [7, 8], where machine learning models using single nucleotide polymorphisms combined with demographic and lifestyle data outperformed logistic regression algorithms. Recent research bringing together neuroimaging imagery with single nucleotide polymorphisms that are using machine learning has either not compared combined predictions to those from genetic or non-genetic data alone [11], has not found improved prediction from combined data types over only genetic or non-genetic predictors [17], or has found no added value from combining data types [9]. The prospective advantage of machine learning over conventional statistical methods is unknown, as machine learning predictions of performance may be excessively enthusiastic [15–21]. Additionally, researchers have previously identified pervasive high risk of bias in genetic-only machine learning models in psychiatry, in addition to an absence of compared to ANN-genetic algorithm or analysis of interference by population composition [20]. Researchers wanted to evaluate machine learning and ANN-genetic algorithm, assess the relative relevance of indicators, and explore model forecasts for connection with schizophrenia-related features. One of the sub-branches of Soft Computing is Genetic Algorithm (GA), which models the natural evolution of organisms. Like other branches of Soft Computing, Genetic Algorithm is rooted in nature. This method is an imitation of the evolution process using algorithms [27]. It is a computer. Neural networks are parallel processing systems that are used to detect very complex patterns among data. Each artificial neural network is an information processing system that has some common characteristics with biological neural networks. In other words, each network contains sets of neurons. It is with a special arrangement [28]. The main part of a neural network is the neurons and the communication lines between them. The multi-layered neural network consists of an input layer, one or more intermediate layers and an output layer. The layer that receives the input information is called It is

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called the input layer. This layer actually does something on the inputs. The output of the network is produced from its output layer. The rest of the layers are called hidden layers because they have no connection with the outside environment. The structure of a network is the way to connect different layers [29]. Schizophrenia is a chronic mental illness that there is a 1% risk of developing this disease in a person’s life, and approximately 9.2% of the population is affected at any time. Schizophrenia is more common in people with poor socio-economic status and has a lower age of onset [30]. According to research, more schizophrenic people are born in winter. This situation suggests that environmental factors such as viral infections or nutritional factors can contribute to the onset of the disease. Schizophrenia can lead to family separation and children are adopted as adopted children instead of living with their biological parents. The high frequency of schizophrenia in adopted children who are far from their biological parents shows the influence of genetic factors in the occurrence of this disease [31]. In other words, hereditary factors and environmental factors are considered as effective factors in schizophrenia [32]. Therefore, in this study, the two factors of father’s age and mother’s age as hereditary factors, the age of the patient’s father and mother at the time of their birth were normalized and considered as input to the software. Considering the role of environmental factors in the occurrence of the disease, this factor was considered as an effective factor in the disease. All the investigated patients lived in the same city and were in the same economic range (middle and poor), therefore, the same number was considered for all samples, and by using the genetic algorithm and artificial neural network technique, the prediction software Thinks pro Version 1.05) was trained. Also, in this paper authors address earlier limitations with error optimization by training neural network with optimized data from genetic algorithm system and measuring how well predictions are described by population structure.

2 Theoretical Framework 2.1 Machine Learning in Medicine In the early days of medicine, clinicians thought in a group, colleagues work together in order to solve problems that were extremely complex for a single individual, currently medicine has become even more complex, there are many more therapies, drugs, examinations, and processing such data exceed the comprehension capacity of the human mind [33], hence it is necessary to look for a new tool with the capacity to integrate huge amounts of data, recognize patterns and create models that allow solving human limitations, reduce medical burden, speed up care, provide more personalized management and reduce resources [34].

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Machine learning is already a reality in many areas and with very good results, such as finance, marketing and social sciences to predict behavior. However, in medicine its use is still limited by challenges inherent in the development of the models, as well as legal, ethical and philosophical challenges [35]. Despite these conflicts, the complexity of medicine makes this one of the fields that can benefit the most from these techniques, which is why in recent years different models and algorithms have been incorporated into medical processes, such as in detection of breast cancer comparing mammograms [36], detection of skin cancer with evaluation of images [37] and detection of cardiac pathologies analyzing echocardiograms [38]. Given the growing boom in machine learning techniques in the field of medicine, this section aims to provide a brief introduction to machine learning, exemplify its possible uses in the health area, and expose the challenges still to be resolved for its use implementation. It is expected that at the end of this writing the reader will obtain a basic knowledge on the subject and encourage interest in the development, evaluation and incorporation of these technologies in medicine. Machine learning, also known as automatic learning or machine learning, is a concept coined in 1959 by Arthur Samuel [39, 40]. Samuel pioneered the use of machine learning by programming a computer in the game of checkers that was capable of comparing the best move options, as well as remembering all positions from previous matches allowing the machine to learn and improve with each game [41]. In his article in the artificial intelligence section, Samuel describes machine learning as a technology that uses statistical techniques and computational algorithms to provide computers with the ability to learn, that is, to improve their performance in a specific task after processing enough data and without explicit external instructions provided by the programmer [42]. Machine learning is recognized as a branch of artificial intelligence because it learns by analyzing data and subsequently generates its own decision through algorithms, that is, its reasoning in solving tasks involves characteristics of human intelligence [43]. Machine learning techniques are part of the field of computer science that involves the evolution of pattern recognition systems allowing computers to learn from mistakes and predict outcomes [44, 45]. These tools are based on sets of mathematical procedures that describe the relationships between variables known as: algorithms [46, 47]. Through machine learning, computers have the ability to analyze data and create algorithms that allow the computer to learn and create predictions without explicit rules being programmed [39, 45]. Unlike conventional statistical methods where learning is through rules, in machine learning computers learn by example and the more exposure to data the tool improves and learns from experience [45, 46]. Sometimes the boundary between conventional statistical methods and machine learning can be confusing, however, although traditional statistical techniques such as linear and logistic regression are capable of creating predictions, their objective is merely to make inferences about the relationships between variables, instead the methods Machine learning techniques seek to understand the relationships between variables [42].

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Most machine learning methods can be classified into three categories based on the type of learning technique used: supervised, unsupervised, and reinforcement learning [47]. . Supervised learning: initially the machine is trained, providing inputs or features (inputs or features) that are associated with a known result or label (outputs or label) determined by human experts [49]. The goal of these algorithms is to learn general rules that map inputs to outputs [46]. . In medicine, a model can be used to relate characteristics of skin lesions (for example: size, color, shape) with a certain result (continuing with the example, classify as: benign or malignant lesion). In supervised learning, the computer gains experience with millions of pieces of data provided, learns the patterns, and then a new record is introduced to test the predictive ability of the model [45]. . Unsupervised learning: unlike the previous one, in this case no previous information is provided to the system, but rather large amounts of unlabeled data are introduced and the system is in charge of finding hidden trends or patterns and separating the information into groups of automatically [47]. For example, in medicine, images of brain tumors can be provided and the system is responsible for separating the images into groups according to the patterns observed [49]. . Reinforcement learning: This third variant of machine learning introduces the system into a dynamic environment, labeled as well as unlabeled data is provided, and the system interacts with the environment and receives negative or positive rewards (feedback) according to its actions which allows it to be perfected, developing better characterizations and classifications [49]; therefore, the computer “learns” without receiving explicit instructions. The ones just mentioned are the main categories of machine learning, and they are sub-classified into different types of machine learning depending on the goal you want to achieve. Among the supervised learning techniques are: decision trees, K-nearest neighbors, support vector machine, naive bayes, linear regression, neutral networks, among others; and in unsupervised learning there are: hierarchical clustering, k-means clustering, linear discriminant analysis [49]. The aforementioned are examples of subcategories, however, developing each of these techniques is not part of the objectives of this article. Among them, a method that has been of great interest is: deep learning, this tool has quickly gained attention and provides excellent results [43]. Models based on deep learning use artificial neural networks of multiple layers, the layers are connected by a set of neurons forming a network that allows extremely complex relationships between features and labels to be learned inspired by the human neurological network these models have exceeded the capacity human in some fields such as: image classification [48]. The understanding of these technologies is definitely not a familiar topic for clinicians, the preparation of automated learning techniques has an origin in statistics and programming language, so the route to put it into operation will only be explained superficially a machine learning system for a specific problem. . Preprocessing: Initially, data quality is checked, noisy data is reduced, irrelevant features are removed, and inconsistency issues are repaired.

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It is a complex job which prepares the large database that will later be processed by the corresponding model [49]. For example, a Stanford Hospital echocardiography database was created containing images, medical reports, and clinical information; in the initial phase, all unnecessary data was removed, such as: information related to image acquisition, identification information, and any other data that was not from the imaging sector [38]. Calculation of characteristics: in this step the important characteristics for decision making are extracted. Each variable [49] is assigned or represented by a numerical value, however, the way in which the characteristics are calculated is beyond the scope of this writing. For example, in radiology, characteristics such as: shape, texture, contours are extracted, and numerical values that represent the visual content of a radiological image are calculated [43]. Feature selection: A large number of features can be extracted in the previous step, however, having too many features can prevent learning the true basis of a decision [49], so it is important to reduce the number of features to be used in In the training process, in this step features are selected that will generate better predictions and truly have an impact on the model, without losing valuable information [49]. Training, testing and validation: once the data set is reduced and organized in a suitable format it is possible to start the training [45]. The type of sampling is decided and the database is divided, a subgroup of data will be used for training, another to test the performance of the algorithms and another for the validation process [49]. Select learning algorithm: choose different machine learning algorithms and compare results with each other, selecting the trained model that best suits the requirements. And finally, the selected model must be used in new cases, evaluating its functioning [45].

. Forecast Machine learning has the potential to predict the susceptibility of a subject to suffer from a disease, recurrence of a disease, life expectancy [34]. This type of tool would make it possible to offer patients precision medicine [45]. One of the fields that has most studied the use of machine learning in prognostic prediction is oncology. It has been reported that the use of machine learning methods improved cancer prognosis predictions by 15–25% [34]. Machine learning prognostic studies in cancer use genomic, clinical, histological, imaging, demographic, or epidemiological data [49]. For example, machine learning algorithms have been used in studies to predict the recurrence of stage IV colorectal cancer after tumor resection [50]. Other research predicts disease risk, for example, Ghorbani et al. describe how they trained a conventional neural network model on millions of patient echocardiogram images to identify cardiac structures and estimate cardiac function and predict factors of risk [38]. . Screening and diagnosis The Institute of Medicine, now known as the National Academy of Medicine published a report known as “To Err is Human”; This paper concludes that at least

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once in life a patient will be misdiagnosed [47], so an automatic learning technique capable of screening patients, staging patients by risk, and assisting physicians in decision-making would be of great help daily help from the doctor [34]. Machine learning algorithmic systems could be of help to medical personnel in scenarios where the diagnosis is not clear, or simply to streamline health systems, it is intended to use the capacity of machine learning systems to identify probable diagnoses through integration of data available electronically for each patient [43]. Advances in machine learning promise applications in pattern recognition in medical images, so they have great potential in fields such as radiology and pathology. Studies demonstrate the ability to train machine learning models to support clinical practice in evaluating images of complete pathology slides in prostate cancer, basal cell carcinoma, and breast cancer metastasis to axillary lymph nodes [51]. Likewise, in the field of radiology, studies demonstrate the potential of deep learning models in mammography screening by reducing the workload of radiologists and improving specificity without compromising sensitivity [36]. Other promising research is in the field of dermatology, where skin lesions have been classified using conventional neural networks by training the models directly with images and comparing their performance with certified dermatologists [37]. It is even intended to make applications available to people that they can use on a device such as a smartphone and that by means of a photograph the need to refer to a specialist is evaluated. . Treatments Machine learning models can learn to identify the best treatment for a patient, based on factual information [33]. Likewise, machine learning tools can improve the discovery and development of new drugs [52]. For example, new machine learning techniques are applied in virtual detection (computational field that helps in experimental studies to detect unknown interactions in new drugs), however, today the prediction performance continues to be low [53]. . Reduction of clinical burden and access to the health system The availability of electronically stored clinical information has become an important tool in health systems [54]. Having virtually stored medical data within reach opens the possibility of simplifying medical care, reducing costs and diagnostic time [53]. For example, Wall et al. developed a model capable of evaluating and diagnosing patients with autism. This study envisions applying the model in order to preliminarily capture patients and subsequently refer them to the doctor, aiming to alleviate clinical work and make the treatment more efficient system [34]. Machine learning algorithms are not yet integrated into health systems, however, there are automated learning projects to extend the scope of access to medical services [47]. Novel studies highlight the possibility of using machine learning models to provide medical assessment from an electronic device [49]. Although these technologies are not yet within the medical reach, there is no doubt about their potential to improve access to services, reduce unnecessary consultations, provide user convenience, reduce workload and reduce costs [50].

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Limitations of Machine learning in medicine: . High quality data collection A key challenge in building machine learning systems is the need to include diverse and representative data, the more access to high-quality information that fits the intentions of the model, the greater the accuracy [51]. It has been shown that not only large volumes of data are necessary, but also high-quality data, so it is essential to properly choose the most informative subset of features to train the model [49]. For example, if the system is trained with a data set that is mostly white and with little representation of African Americans, depending on the pathology, the model could generate erroneous predictions [51]. The training of machine learning systems is necessary to involve skilled clinicians and objective studies that distinguish between valuable data. Information fed into the model should be carefully monitored for the risk of reproducing human biases due to retrospective learning. Ensure that the system learns objective patterns, novel therapies instead of learning according to medical practices with subjective bases [47]. . Translation of results One of the strengths of machine learning is the ability to recognize patterns beyond human reach among the vastness of data provided, but at the same time this feature is not entirely miraculous, since the patterns identified may not represent the needs of the systems health and/or users. It may even be that learning systems overlook variables that are indeed modifiable [50]. A term widely mentioned in machine learning is: “black box” or “black box” which refers to the complexity of the functions and dimensionality of machine learning systems that often makes it difficult to interpret results, limiting their usefulness [46]. Researchers continue to work on solving this problem in order to find models that yield significant and modifiable results [46]. . Regulations, legislation and ethical problems Machine learning algorithms require a large amount of data to refine their models, so having access to a large number of medical records is of great importance, hence the regulations regarding who will have access to this information and regulations for the protection of the confidentiality are of the utmost importance. For these reasons, different countries already have data protection systems, for example, in Europe the European General Data Protection Regulation (GDPR) is created, which is in charge of ensuring that the subjects under study allow the use of their data through informed consent and that they know their rights [53]. In the United States, a similar institution is known as the Health Insurance Portability and Accountability Act (HIPAA) [39]. A robust supervision structure, legal regulation and local guidelines are required for each stage; from consent for the use of data during development, to rules of use and monitoring of machine learning systems [41]. In particular, the United States Food and Drug Administration (FDA) is working on appropriate regulatory pathways that establish standards for the validation of machine learning models. For example,

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the FDA has begun a precertification pilot program to evaluate medical software, based on five criteria of excellence [50]. Additionally, there are ethical challenges, it is feared that both doctors and future users of machine learning systems blindly trust the decision of a model. However, automation models are intended to be an assistance tool; one must be aware that these techniques have their limitations [55]. . Role of the doctor Artificial intelligence and techniques such as machine learning are gradually being introduced into the healthcare system [54]. In other times, when it was not possible to understand what was happening inside a patient, instruments such as stethoscopes, ophthalmoscopes, X-ray techniques, CT, and MRI were invented; today, when each patient owns a number of tests, diagnoses and treatments, it is even more necessary to create a tool that provides support to the doctor, compensating for his limitations. However, the result provided by software continues to be a suggestion; the final indication must be issued by a qualified professional [47]. Machine learning is emerging as a support for doctors and for the health system in general, but the medical hand is still necessary in every decision about a patient’s health. For this reason, various papers reiterate that machine learning is not expected to replace doctors, but rather to be a tool that helps improve workflow and diagnostic accuracy [54]. Machine learning is a tool that is widely under investigation in the area of medicine. During the search of the bibliographic material, a great variety of studies were found that apply machine learning models to predict prognoses and provide diagnoses. Especially medical activities that require repetitive work and manipulation of a lot of data are promising areas for the use of these technologies. However, despite the promising scope of machine learning, there are still great challenges, among them: difficulty of access to high-quality data, ethical and legal conflicts. Today there are already data protection policies, however, detailed regulations are still being worked on to regulate the entire process of development, validation and supervision of these systems. We believe that artificial intelligence is a very valuable tool that will change the way medicine is practiced in the coming years, so health professionals must prepare, learn and adapt.

2.2 Schizophrenia A person suffering from schizophrenia experiences changes in his behavior, perception and thinking, as a result of which he distances himself from reality, this state is also called psychosis. Unfortunately, this disorder carries a lot of stigma both for the affected person and for his family [55]. For example, a misconception about people with this disease is that these people are dangerous. Remember that a person with schizophrenia can rarely be dangerous, especially people who have proper treatment

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and support. Another fact about sufferers is that they have natural intelligence and are not mentally retarded [56]. Regarding the difference between schizophrenia and schizophrenia, it should be said that schizophrenia is the same as schizophrenia. Schizophrenia usually starts between the ages of 15 and 25, and that’s why it is also called youthful insanity [57]. The prevalence of schizophrenia is the same in men and women and is about 1% of the general population [58]. In the majority of cases, schizophrenia is a recurrent or persistent disorder. A small percentage of patients may have only one or more episodes of psychosis during their lifetime. During the onset of this disorder, a person may avoid others or even become depressed and anxious. Sometimes he may have unusual opinions and experiences. It should be noted here that the earlier the symptoms of schizophrenia are identified and treated, the better the results will be in the future [59–64]. Delusions [11]: Delusions are ideas that are far from reality, but the affected person strongly believes in them and has no doubt that they are not real. Such as hallucinations and delusions, during which a person thinks that others are trying to harass him, or thinks that people are following him, or that they have installed cameras or tape recorders in his home and workplace. Other losses include [6]: loss of control, loss of spreading thoughts, loss of attribution, loss of betrayal, loss of grand secretary, loss of guilt, etc. In loss of control, the affected person believes that he is not in control and that another force controls his behavior and thoughts. In thought broadcast delusion, the sufferer thinks that his thoughts are being broadcast from media such as radio and television. In the loss of attribution, the affected person attributes the behaviors and actions around him to himself, for example, he thinks that the TV news anchor is talking to him, or if two people are talking in a low voice in a group, he imagines that he is talking about him [17] they hit in case of betrayal, even though the person’s life partner is completely faithful, the patient believes that he is cheating on him. In grand secretarial delusion, the sick person believes that he has special talents and abilities that make him superior to other people, while in reality it is not so and he is an ordinary person. In the loss of guilt, while the person is not guilty, he strongly believes that he has committed a wrong and a sin [21]. Illusion: Illusion is the perception of one of the five senses without having an external and real origin, the most common of which is auditory illusion. It means that a person hears sounds that others cannot hear. Thinking disorder: its manifestation is in the form of irrelevant and unintelligible speech [65]. Lack of motivation: the affected person has no motivation to do daily activities [66]. Inability to express emotions: It is one of the symptoms of schizophrenia, during which the affected person has a masked face in response to various emotions and sometimes even expresses emotions disproportionately. For example, he laughs in response to a sad event and vice versa [67]. Social isolation: the affected person sometimes becomes very withdrawn and does not appear in the community for months or even years. This itself can be due to loss and damage or the lack of social skills in a person.

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Lack of insight: Almost all sufferers do not have insight into their illness because they think all their delusions and illusions are real, which is a very important reason why they refuse to receive treatment. And sometimes forced hospitalization is the only way to treat these people [68]. The stages of schizophrenia include: the early stages of the disease where psychosis has not yet occurred, but the person may suffer from depression, anxiety or obsession. Then, over time, hallucinations and delusions appear and may become chronic if not treated. Schizophrenia can sometimes be alleviated with treatment or even without treatment, and after a while its symptoms flare up again. Types of schizophrenia include: Paranoid: The most common type of this disease is paranoia, which is associated with hallucinations and organized delusions. Especially auditory hallucinations are often seen in them. Losses such as: suspicion, grand secretary, loss of jealousy and thinking that they are wanted, etc. [69]. Catatonic: They have movement disorders, they show excess in their behavior and their behavior is strange and sudden. Sometimes they have extreme irritability and restlessness, but sometimes they get nervous and slow in their movements and are indifferent to their surroundings. Even when they are hungry, they do not eat and cannot be fed. This type is very rare [9, 15]. Disturbed: They have chaotic and disproportionate behaviors, their facial expressions do not fit with their speech, and they often act out. Sometimes they laugh hard for no reason and are indifferent to cleanliness. Emotional: they have intense and sometimes violent movements [70]. Uncertain: They have various symptoms of schizophrenia, but their behaviour changes regularly and they cannot be placed in a specific category. Their symptoms include isolation, unemployment, and changes in behavior. If this condition is not accompanied by hallucinations, it may be very difficult to diagnose [71]. Residual: It means that they have experienced severe symptoms of schizophrenia before, but now they have milder symptoms that are left over from that severe attack and are not very serious [17]. In this disorder, like all psychiatric disorders, there is not only one responsible factor, but a set of factors and factors are responsible for the emergence of this disorder. What are the factors that cause schizophrenia? Genetics: One of the predisposing factors for schizophrenia is heredity. While, as mentioned before, the prevalence of this disorder in the general population is 1%, but if one of the parents has schizophrenia, this probability increases to 10% in their child [71]. Biochemical factors: disturbance in the balance of brain chemical messengers can cause schizophrenia [20]. The most important biochemical messenger of the brain in this disorder is dopamine. Antipsychotic drugs, which are the main treatment for schizophrenia, exert their therapeutic effects by affecting dopamine. Another example of the role of biochemical disorder in the development of schizophrenia is problems during pregnancy or childbirth, which cause structural damage to the brain of the fetus or newborn [72].

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Stress: stressful events can accelerate the manifestation of schizophrenia. This definitely happens in susceptible people. Alcohol and other substances: Regarding the relationship between schizophrenia and drugs, it should be said that alcohol and drugs, especially hashish and amphetamines, can be the detector of psychosis in people prone to schizophrenia. Also, these substances play a very important role in the relapse of schizophrenia [73]. The most important schizophrenia test for the diagnosis and treatment of schizophrenia is the clinical interview by a psychiatrist. However, in a person suspected of psychosis and especially in the beginning of schizophrenia disorder, a complete evaluation of the patient including a complete blood checkup, brain scan and brain map, brain imaging, urine screening test to detect drug use and also psychological tests is necessary [74]. Schizophrenia is one of the most debilitating psychiatric disorders. If, as mentioned before, the affected person becomes isolated and loses his professional and social performance and relationships. This brings a heavy burden to the caregivers, the patient’s family, and the society as well. People with schizophrenia are more exposed to alcohol, drugs and smoking, these factors endanger their health more and increase their problems. The probability of suicide in affected people is higher than in the general population [75, 76]. Lack of insight into the disease causes nonacceptance of treatment, discontinuation of medication during treatment, frequent relapses and multiple hospitalizations. The most important treatment for schizophrenia is drug therapy. The main ones are drugs that act on the dopamine pathway and receptors in the brain. What is the drug for schizophrenia? These drugs are divided into two categories: first-generation or older antipsychotics and second-generation or newer antipsychotics. First-generation drugs such as haloperidol, perphenazine, thiothixene, chlorpromazine, and thioridazine, and second-generation or newer drugs include: olanzapine, risperidone, quetiapine, aripiprazole, and clozapine [11, 15, 72]. Treatment with second generation drugs is usually started due to less side effects. The choice of medicine depends on the type of symptoms, age, sex, weight and possible side effects [77]. One of the important aspects of schizophrenia treatment is psychoeducation or family and individual education about the disorder and its treatment. In cases where there is a risk of suicide or aggression or symptoms resistant to outpatient treatment, hospitalization is necessary. In severe cases, you can sometimes use electrical convulsion therapy or shock therapy [13]. Also, in patients who do not accept treatment and have a lack of insight, it is possible to use injectable and long-acting types of antipsychotic drugs. These drugs are injected monthly to the affected person and maintain their effect for one month [21]. Schizophrenia, like other medical diseases such as diabetes, may require long-term treatment and sometimes for the rest of the life. Non-pharmacological treatments, including psychotherapy and occupational therapy, have a special role in the recovery of these patients. Also, in people who use drugs and alcohol, their addiction treatment by an expert addiction therapist is necessary and necessary [17].

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Regarding the lifespan of schizophrenic patients, it should be said that schizophrenia usually does not reduce the lifespan of people because it has nothing to do with vital organs such as the heart, lungs, etc. But schizophrenia comorbidities may affect a person’s health and longevity [18]. For example, more alcohol and smoking in these people, as well as isolation and lack of motivation, which prevents a person from taking care of his health and hygiene. People with schizophrenia may engage in high-risk behaviors such as injecting drug use and unprotected sex, which can increase the risk of deadly viruses such as HIV [18, 60]. The risk of suicide among people with schizophrenia is higher than that of the general population, and this makes the average life expectancy of people with schizophrenia less than that of the general population [72]. In addition, second-generation antipsychotic drugs may have side effects such as obesity, increased blood lipids, and metabolic syndrome, which are risk factors for cardiovascular diseases. Of course, if there are regular referrals to a psychiatrist and adherence to treatment among patients, the mentioned side effects are controlled in such a way that the benefits of taking the drug are more than its harms [17]. So people with schizophrenia need more medical and health attention [77].

3 Methodology In this study, data were used from a survey that was conducted in a four-month interval in one of Tehran’s psychiatric centres. These data were obtained from 189 patients with schizophrenia and included the age of parents of people with schizophrenia at the time of their birth, as well as the age of onset of the disease in people with schizophrenia [40]. The present research was conducted using the artificial neural network model by Thinks pro analysis software with the following specifications. 1. 2. 3. 4.

Feed forward multilayer network Error back propagation learning algorithm Sigmoid driving function Error function: mean square error.

The software uses two data series to train the network, the training series and the test series. In other words, the data used for the training series is separated from the data of the test series [41]. In this study, 4 input parameters and 1 output parameters have been defined, which are shown in (Fig. 1). Father’s age, mother’s age, environmental factors, and the age of affected people as input and the age of schizophrenia in children as the necessary output to construct an artificial neural model is considered. Due to the use of the sigmoid transformation function, all inputs of the network must be normalized so that the network is able to recognize, separate, classify data from each other and finally learn the correct patterns. In order to normalize from Eq. (1) was used [42].

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Fig. 1 A general diagram and PI of the prediction model

xnormalized =

x − xmin xmax − xmax

(1)

In the above equation, x normalized is the normalized data, x is the non-normalized data, x min is the minimum data, and x max is the maximum data in the data set. The data includes father’s age, mother’s age, environmental factors and the age of children’s disease. All the data after normalization by STATISTICA analysis software (spss 15.0 for windows version 2.0), were separated into two parts of network training and network testing. In this study, 189 data were used according to the age of people. 40% of the data (76 data) were used as testing and 60% of the data (113 data) were used as training. Each series has a real input and output, and finally the network is able to predict the output, and a network with a more suitable prediction is one that has the least difference between the actual output and the predicted output. Network training algorithm, back propagation and transfer function, bilinear sigmoid, error calculation method, absolute average, initial noise 0.001 and the learning rate of the first, second and output hidden layers are 0.4, 0.03 and 0.01, respectively. The momentum of the first, second and output hidden layers was chosen as 0.02, 0.01 and 0.00. The statistics of network training and test records are randomly selected using a computer program. Thinks pro software (Version 1.05) was used for data analysis.

4 Results and Discussion In this research, a number of 189 patients with schizophrenia were examined in terms of the age of the parents at birth and the economic and social conditions of the affected person’s family. Out of 189 examined patients, 121 were men and 68 were

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Fig. 2 Predicted age of onset of schizophrenia

women, and their average age was 39.66. The average age of fathers at birth was 33.08 and the average age of mothers at birth was 28.13. Examining the economic situation of the patients showed that 96.09% were in the poor and middle socioeconomic range and only 3.91% did not have any worries and stress of financing themselves. Considering the effect of genetic factors and environmental factors in the development of schizophrenia, the artificial neural network was trained with the genetic algorithm model, after 60,000 repetitions, the average error of training and testing and the maximum error of training and testing were respectively equal to 0.00634895. 0.00864354, 0.01986321 and 0.01986328 were obtained. The neural network predicted the age of onset of schizophrenia with acceptable accuracy. In Fig. 2, the graph related to the age of schizophrenia predicted by the genetic algorithm model was compared with the age of people with schizophrenia in all 189 samples. The results show that 96.65% of people have the same age as predicted by the artificial neural network technique. This finding shows the capability of this technique in predicting the age of onset of schizophrenia by using the effective factors in the occurrence of the disease. In today’s world, due to the lack of certainty, the decision-making process is very difficult. In this way, various tools have been created to help decision makers. In this regard, the existence of predictive tools helps to estimate the age of onset in some diseases by using a constructed model. In the field of artificial intelligence, genetic algorithm is an innovative method that imitates the process of natural evolution. Genetic algorithms are related to a larger group of evolutionary algorithms that provide solutions to solve problems using techniques inspired by natural evolution such as inheritance, mutation, selection and crossing over. A computer simulation of evolution was first performed in 1954 by a mathematician named Nils Aall Barricelli in Princeton, New Jersey. The publication of Nils’ paper was not noticed at first until in 1957 [46], Australian Alex Fraser published several papers on simulating the artificial selection of organisms with several different loci of control for a measurable trait. After that, in the early 1960s, computer simulations of evolution were popularized

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by biologists. Artificial evolution was widely recognized as an optimization method by the work of Ingo Rechenberg and Hans-Paul Schwefel. Rechenberg’s group was able to solve complex technical problems through evolutionary strategies. In recent years, artificial neural networks have been recognized as a powerful diagnostic tool for malignant lesions [47]. Artificial neural networks, based on the human nervous system, provide a method for modeling physical phenomena, the most important features of which are as follows: high processing speed due to the parallel structure, the ability to learn the model instead of simulation, the ability to generalize the results, and the lack of the need for physical justification of the problem. In the last decade, a large number of articles related to artificial neural network and medical sciences have been published. In 1994, Steen Paul M used artificial neural network to predict, analyze and create a model for patients with neuropathic pain symptoms using artificial neural network. Steen et al. were able to increase the accuracy of clinical diagnosis from 68 to 69% using this technique [48]. Bryce T. J et al. in 1998 predicted survival in head and neck squamous cell carcinoma. The result of their work showed the high power of artificial neural network in prediction compared to conventional methods [49]. In a clinical trial on bladder cancer, Parekattil and his colleagues succeeded in identifying patients who needed cystoscopy using an artificial neural network. They were also people with bladder cancer and those with muscle invasive disease were identified in the same way [50]. Naguib RN and colleagues used an artificial neural network to predict the involvement of lymph nodes after breast cancer using clinical and pathological information. The result showed that neural networks are able to provide strong indicators related to the condition Lymph nodes are only used to measure the primary breast tumor [51]. Gletsos M et al. used neural network and genetic algorithm to classify liver lesions identified by computed tomography images. Finally, Gletsos M and colleagues compared the two techniques used, which showed that the genetic algorithm method improves the classification [52]. Michael Behrman and colleagues in 2007 used artificial neural network to classify patients with neuropathic symptoms [53]. The result indicated the high capability of this technique in data analysis to classify neuropathic diseases. In 2009, Çınar et al. succeeded in early diagnosis of prostate cancer using artificial neural network technique. Considering that artificial neural networks have shown their effectiveness in predicting the trend of time series and time-spatial series [54, 78, 79]. Therefore, in this study, artificial neural network was used in predicting the age of onset of schizophrenia and building a suitable model to predict the age of onset of this disease according to the factors affecting it. Figure 3 shows the relationship between observed age and predicted age in people with schizophrenia. The y-axis represents the age observed in schizophrenic people, and the x-axis represents the age predicted by the genetic algorithm model and artificial neural network. The line y = x in the center of the diagram indicates the best possible situation. The R2 coefficient specifies the amount of error that exists in the interface shown in the picture.

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Fig. 3 A graph examining the relationship between predicted age and observed age in schizophrenic people

The R2 coefficient is a number between zero and one. The closer this number is to 5, the lower the error percentage. It is advised as normal procedure in the field of ANN-genetic algorithm to examine predictions made by models to see whether or not other elements that are relevant (such as confounders or consequential variables) can explain model predictions [80]. The cross-validation had been employed to carry out that in based on populations regulates that weren’t utilized to train or evaluating prejudice. This emphasized established connections with schizophrenia, such as fluid intelligence and processing speed (as measured by digit symbol substitution), in addition to BMI, social deprivation, and smoking habits. The distinctions among the modeling techniques depicted in Fig. 2 could prove crucial for clinical applications because of the variety in how algorithms weight the input information, which causes predictions from different methods of modelling to exhibit diversity in their correlation with variables that affect outcomes [81]. This emphasizes the value of going beyond straightforward scalar summaries of model performance and evaluating prediction of factors relevant to result. The concentration on prediction limits the use of covariates, and hence it is important to exercise caution when interpreting these results because a greater R2 might be partially accounted for by factors other than those in the beta regression model. The method identifies which factors, not schizophrenia itself, are linked to the theoretical predictions of schizophrenia; the low R2 for psychiatric phenotypes simply indicates they account for little variation in model forecasts, and doesn’t consequently conflict with referred to genetic and phenotypic correlations between characteristics [82]. Our findings also imply that present de-confounding techniques, and this decline off key elements from indicators, do not entirely eliminate the effects of population structure from the final predictions, especially when unaltered non-genetic variables are included in models; alternative de-confounding techniques might be necessary for

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ANN-genetic algorithm [81–85]. By utilizing a distinct subsample from the Iranian Data centers, this research also provides as a basic test of generalization. Although it might be perfect for results to be reproduced in a completely attached dataset, reliability of results throughout classifiers, metrics, and resampling of controls, consistency of results with expected direction of effects for associations with schizophrenia and schizophrenia-related traits, and the use of low risk of bias strategies such as nested cross-validation all contribute to increased certainty in the potential for generalization of models [86]. The scale of the entire Iranian Data centers dataset presents challenges for highly computationally complicated ANN-genetic algorithm algorithms. Researchers demonstrate that the embedded case-control research is a successful strategy for implementing ANN-genetic algorithm techniques to huge populations with a low computational burden, with discrimination that remains stable throughout sampling parts and an adequate number of remaining controls accessible for assessing predictions. Furthermore, we demonstrate how low error in modeling procedures can alleviate concerns about overstated performance estimates [87, 88]. Participate prejudice in the Iranian Data centers indicates that the sample generally is less socioeconomically deficient, healthier, and more inclined to consist of female as well as Tehrani population rather than the Iranian population portfolio in its entirety [89], while people in the most severe kinds of schizophrenia could be not represented or completely absent. Evaluation prejudices can lead the magnitude of effects to vary if calculated in a sample that is more accurately representative, with possible implications for prejudice and the calibration which require more diminution of parameters, recalibration, or retraining prior to applying to the wider population or a medical target sample [90].

5 Conclusion The result of this study shows that 98.43% of people have the same age as predicted by the artificial neural network technique. This finding shows the capability of this technique in predicting the age of onset of schizophrenia by using the effective factors in the occurrence of the disease. Therefore, by using genetic algorithm, it is possible to predict the age of onset of schizophrenia in children. Predicting the time of disease occurrence can always be considered as an effective factor in the treatment and prevention of mental diseases. Given that the ability of the genetic algorithm to predict the age of onset of the disease using factors affecting the disease was proven. By making a suitable model, the age range that is at risk of contracting the disease in the society can be identified and treated for that age group.

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Chapter 8

Comparison of Biologically Inspired Algorithm with Socio-inspired Technique on Load Frequency Control of Multi-source Single-Area Power System D. Murugesan, K. Jagatheesan, Anand J. Kulkarni, Pritesh Shah, and Ravi Sekhar

1 Introduction The nature-inspired technique is a reliable, powerful, and rule-based method, well adapted for arithmetic optimization issues, particularly in handling non-constraints as well as constraints in designing a power system model for load frequency control [1–5]. A few soft computing techniques are population-based algorithms has been created by following the collective performance of fish schooling, ant colony, honey bees, and more, For example, Particle Swarm Optimization (PSO) [6], Ant Colony Optimization (ACO) [7], flower pollination [8], firefly algorithm [9], genetic algorithm [10], etc. To tune the decision variables of the proportional–integral regulator for an issue in multi-source with multi-controllers in a single-area power system, Banaja Mohanty et al. in a research paper [11] used the differential evolution algorithm. Murugesan et al. [12] have introduced cohort intelligence optimization to D. Murugesan (B) · K. Jagatheesan Paavai Engineering College, Puduchatram, Tamil Nadu 637018, India e-mail: [email protected] A. J. Kulkarni MIT World Peace University, 124 Paud Road, Pune 411038, India P. Shah · R. Sekhar Symbiosis Institute of Technology (SIT) Pune Campus, Symbiosis International (Deemed University) (SIU), Pune 412115, Maharashtra, India e-mail: [email protected] R. Sekhar e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_8

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control the frequency irregularity for isolated and integrated power plants. Nikhil Pallwal et al. [13] have applied the Grey Wolf Optimizer algorithm in multiple sources of one area power network. Sulaiman et al. [14] applied the hybrid teaching– learning-based optimization technique on two-area power framework along with nonlinear processing on load frequency control. Acharyulu et al. [15] used a fruit fly algorithm for multi-source and multi-area power systems. Naidu et al. [16] introduced a multiobjective technique using Artificial Bee Colony (ABC) to tune the PID controllers for frequency irregularity control. The main motivation of this article is to address the importance of multi-controller performance over the single controller for the purpose of controlling irregularity of system frequency from the electrical power system point of view. In addition, The ability of nature-inspired algorithm (Cohort intelligence, TLBO) with bioinspired algorithm performance (GA, DE) has been compared in terms of solution cost (computation time) and time domain parameters (oscillation control and rapid settling behavior). Further, the flow of soft computing techniques is also included in terms of pseudo-code to the researcher to bring new advancement in soft computing. In this article, bio-inspired algorithm (GA and DE algorithms) and socio-inspired algorithm (TLBO and cohort intelligence) have been provided for tuning of PID regulator for the LFC in multi-source (thermal, hydro, and gas power framework) single-area environment. This nature-inspired algorithm is provided and investigated with one percentage step load disturbance. LFC is a fundamental essential function in recent power quality improvement. The ideal outcomes in the power framework have the need of coordinating the generation and demands of consumers in terms of negligible amounts of losses. if the usage of power increased rapidly from its standard pre-determined values, the system gets oscillation in its standard range of frequency. The fundamental point of LFC is to keep up the frequency irregularity of the framework at or extremely near a standard limit. This article is partitioned in the direction of five areas. The first category manages the introduction. Next, in the second segment, a brief explanation of the selected bio-inspired and socio-inspired methods is presented. The fourth segment manages the simulation performance with bar chart analysis. Finally, the fifth segment deals with the conclusion. The main highlight of this book chapter is mentioned below. i. To inspect a combination of various generating systems for frequency irregularity investigation, thermal power plant, gas, and hydropower, in a single-area power framework with two cases as a common single regulator over multiple controller’s behaviors. ii. To apply an evolutionary intelligence procedure, namely bio-inspired and socio-inspired techniques for LFC is considered for multi-source with multicontrollers in a single-area power network. iii. To study the profitability of all four cost functions for several nature-inspired algorithms such as teaching–learning-based optimization, genetic algorithm, cohort intelligence, and differential evolution-based PID controller.

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iv. To observe and investigate frequency irregularity (Δf) in the multi-controllerbased power system using bio-inspired and socio-inspired techniques to show the effectiveness and suitability of the presented methods.

2 Related Works A single-area multi-source power framework with three separate proportional integral derivative regulators in Fig. 1 and using a single regulator in Fig. 2 are designed for the investigation of nature-inspired algorithms based on various profitability indices. Simulations were organized on an Intel, windows 10 Pro CPU of 2.0 GHz and 4 GB RAM computer in the MATLAB (R2014a) environment. The Simulink design of several sources and multi-controllers of a single-area power network with utilization is given in Fig. 1 and the Simulink model of several sources along with a single controller is presented in Fig. 2. Most of the previous research works are related to tuning PID controller gain values for integrated power systems purely based on biologically inspired and natural-inspired algorithms separately. This article fully fills the research gap with the recently developed cohort intelligence algorithm to get steady-state precision with very lower computation time and reduced oscillation control in the dynamic power which is necessary for the considerations such as safety, economic reasons, or stable operation for an automatic generation control process. In this article, GA, TLBO, Dee, and CIO are implemented to minimize the cost function J1, J2, J3, and J4 in equations [1–4]. In LFC issues, for improved performance of the framework, the parameters of the J1, J2, J3, and J4 must be as minimum as achievable. The rate of profitability (cost function values) determines the realization of the dynamic varying system. To substantiate the suitability and power of the nature-inspired technique in the under-study system, the outcomes are correlated with all the J1, J2, J3, and J4 profitability indices.

Fig. 1 Simulink model of multi-source with multi-controllers strategy 1 (S1) in single-area power system

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Fig. 2 Simulink model of multi-source with single controller strategy 2 (S2) in single-area power system

(Tsim J1 = ITAE = t · |(ΔF)|dt

(1)

0

(Tsim J2 = IAE = |(ΔF)|dt

(2)

0

J3 = ISE =

(Tsim (ΔF)2 dt

(3)

0

(Tsim J4 = ITSE = t.(ΔF)2 dt

(4)

0

where frequency changes in multi-source with single/multi-controller in one-area power framework are represented by ΔF and Tsim is the optimization running time. In load frequency control issues, the control signals are generated based on Area Control Error (ACE) into the thermal plant (UT), Hydro Plant (UH), and Gas Plant (UG) by multi-PID Controller. The control signals are mathematically formed in Eqs. (5–7). Further, the boundary of PID regulator gain values, i.e., Kp, Ki, Kd are generally the constraints of the problem. Hence, taking this problem constraint, the design issues for the LFC regulator can be developed as an optimization issue and could be treated as reducing objective function values, i.e., profitability. Required to this, the chosen upper and lower decision variables of the PID regulator designed are 1 and 0 appropriately. The filter coefficient parameter (N) is taken as known as 100. The load frequency control issue is a minimization optimization issue and optimal gain values of Kp, Ki, and Kd are those parameters through which the cost function/objective function is minimum.

8 Comparison of Biologically Inspired Algorithm with Socio-inspired …

( UT = K p1 AEC + K i1

(AEC)dt + K d1

d AEC dt dt

(5)

(AEC)dt + K d2

d AEC dt dt

(6)

(AEC)dt + K d3

d AEC dt dt

(7)

( U H = K p2 AEC + K i2 ( UG = K p3 AEC + K i3

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3 Employed Optimization Techniques 3.1 Genetic Algorithm The genetic algorithm [17] is the adaptive heuristics search technique. Adaptive means handling different types and different sizes of data set effectively and it performs well for both large and small amount of data as well. Heuristics means using practical methods and rules to generate high-quality solutions for optimization problems. The GA evolution based which is using the following operators, namely reproduction, crossover, mutation, and selection. In the reproduction, the parents are selected from the initial population (μ) using the tournament operator (k) which reduces diverse of the initial solution. Further, crossover and mutation are applied to the parents’ chromosomes to get different offspring (λ) solutions. Finally, a selection of new solutions from the combined solution of μ and λ.The functional flow of the genetic algorithm is mentioned below as pseudocode.

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3.2 Differential Evolution Algorithm It is a difference vector-based evolutionary technique developed by Storn and Price [18, 19]. Each solution is known as a target/donor/trial vector. The choice of choosing good solutions is processed only after getting the all trial vectors. Greedy selection is performed between target and trial vectors. Target vectors are not involved in mutation. DE plays with two vectors; target vectors and newly obtained trial vectors. The size of the initial population can be varied by the user or programmer. The minimum requirement of Np should be greater than or equal to four. The mutant vector is mathematically represented in equation [8], and the generation of a trial vector is mathematically formulated in equation [9]. Mutation: Based on difference vectors, a new donor vector (V ) of chromosomes (Xi) is created as V = X ri + F (X r 1 − X r 2 )

(8)

F = Scaling factor (0–2), r 1 , r 2 , r 3 = Random solution r 1 , r 2 , r 3 e{1, 2, 3 … Np} and r 1 /= r 2 /= r 3 /= i. Recombination: Binomial (uniform crossover), Recombination is performed to improve the diversity of the solution. The creation of a trial vector mathematically

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expressed as ( u = j

v j ifr ≤ pc OR j = δ x j if r ≥ pc AND j /= δ

) (9)

where Pc = Crossover probability, r = random number (0–1) and δ = randomly chosen variables position, uj = trial vector, vj = donor vector, and x j = target vector. The probability for crossover Pc is generally high. High Pc results in more variables from donors and δ is providing a chance of solution from the donor vector, i.e., a minimum of one solution. Further, evaluate the objective function of the entire offspring. Finally, the current solution is updated using greedy selection. Various mutation techniques (DE/x/y/z) are employed normally. In this, DE: Differential Evolution, x: Target vector for mutation, y: difference vectors count, and z: nature of crossover scheme (can be either exponential crossover/binomial crossover). The algorithm flow is given below as a pseudocode of DE.

3.3 Socio-inspired Algorithm 3.3.1

Teaching and Learning Optimization Technique

This is a rather modernistic refined modern technique created by Rao et al. [20– 22] following the activities of the teacher and student interaction in a lecture room environment. Again it also comes under one of the nature-inspired algorithms, i.e., population-based to get the best solution from the initial population. In TLBO, the

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initial population is treated as design parameters. Here, the various design parameters will be related to distinct subjects rendered to learners and the result copy is corresponding to the cost function. The teacher is treated as a good solution until now. There are two general phases of learning treated in this method which are the teacher stage and followed by learner stage. The teacher stage indicates learning from teachers and the learner stage represents the learning among the students through communication as well as interaction. The function flow of TLBO is mentioned below as a pseudocode.

3.3.2

Cohort Intelligence Optimization Technique

The CI algorithm [23–26] is directed by the social bias of learning participants of a cohort. Each participant in the cohort iterates to reach a goal that is universal to all. For this, each participant applies the roulette wheel approach and chooses another participant to follow which may lead to the development of their own behaviour. As a result, each participant learns from the other and contributes to the further development behavior of the whole. A cohort’s behavior can be considered saturated if, after a significant number of learning trials attempted, i.e., there is no further improvement among the participants. User-defined dimension of cohort intelligence algorithm is given in Table 1 and the peculiarity of CI techniques is as follows:

8 Comparison of Biologically Inspired Algorithm with Socio-inspired … Table 1 Parameter of cohort intelligence algorithm

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Dimensions

Multi-controllers Single controller

Decision variables (N)

9

3

Cohorts (C)

5

5

Saturation rate (E)

0.001

0.001

Interval reduction values, 0.97 (r)

0.97

1. It illustrates the training mechanism of cohort participants. Every participant essentially shares a common goal of achieving better behavior by developing their qualification. The discussion and tournament are the two natural talents of each cohort participant. These are accomplished by choosing the roulette wheel choice and further sampling close proximity to the chosen participants. 2. Each participant analyzes himself/herself with other participants in the cohort to improve their individual performance and associated behavior. 3. In cohort, at the end of each learning experiment, all the participants autonomously in the search area. 4. The problem with various sizes and different applications can be effectively handled to get feasible solutions by CI algorithm.

4 Result and Discussion In this article, the design of multi-source with multi-controllers in a single-area framework is modeled in MATLAB (.mdl file). The design parameters of the power framework model are considered in the examination from [9] and presented in Figures. 1 and 2. GA, DE, TLBO, and cohort algorithm programs have been coded independently (in the.m file). The profitability is determined in the.m file and used in DE, GA, TLBO, and CIO methods to find the suitable parameters of the PID regulator for LFC. The size of the population is considered as 5 then the maximum iteration is fixed for GA, DE, and TLBO is 100. The rest of the user-defined parameters for optimization are given in the appendix. The results of bio-inspired code are correlated with the socio-inspired code to show the strength and suitability of cohort intelligence for LFC in both multi-source with multi-controllers of single-area power systems and multi-source with the single controller of single-area power network. The tuned gain values of the PID regulator for various cost functions and algorithms are mentioned in Tables 2 and 3. Time domain transient performance results are given in Tables 4 and 5. It is noticeable from Table 4 that the least area control error value (0.001314 Hz) and undershoot (−0.0272 Hz) of cost function ISE based is obtained with the TLBO algorithm-based PID regulator as correlated to all other techniques and cost functions. Furthermore, in overshoot (0 Hz) and settling time (14 s), cost ITAE is calculated

CIO (S1)

TLBO

GA

0.98467

0.95165

0.80582

0.90452

ITAE

ISE

ITSE

1

ITSE

IAE

1

0.99706

IAE

ISE

1

ITAE

0.99788

0.99996

0.94512

ISE

ITSE

0.9761

0.95244

0.78889

0.7102

0.92815

1

0.94289

0.9774

0.7923

0.97551

0.99997

0.22097

0.83145

IAE

1

ITSE

1

0.99998

0.64423

ISE

0.99993 0.94547

ITAE

0.86165

0.86613

0.16136

0.65621

0.29936

0.061489

0.5654

0.99963

0

0.099755

0.29901

0.69365

0.50096

0.0050951

0.2392

0.96883

0.12073

0

0.55162

0.58236

0.99506

0.96088

0.63675

0.37361

0.51739

0.99885

0.99999

0.93001

0.96316

0.99994

0.91011

0.52465

0.9812

0.59491

Kp2

ITAE

DE

Hydro Kd1

Kp1

Ki1

Thermal

IAE

Cost function

Algorithms

0.14418

0.70112

0.088224

0.0012103

0.00012998

0.0031767

3.2034e-05

0

0.056265

0.012607

0.24918

0.059054

0.56876

0.78472

0

0.023938

Ki2

Table 2 Optimized gain values for multi-source with multi-controllers in single-area power system

0.60768

0.91831

0.73483

0.81038

1

0.99953

0.87176

0.99628

0.96171

0.87358

0.99999

0.055084

0.37621

0.29522

0.80283

0.83031

Kd2

0.41242

0.70856

0.96289

0.98982

0.99766

0.90434

0.99989

1

1

1

1

1

0.95653

0.92879

0.91997

0.97823

Kp3

Gas

0.99839

0.85545

0.98313

0.99609

0.99999

1

0.99534

1

1

1

1

1

0.94317

1

0.58236

0.66235

Ki3

0.17836

0.11148

0.52722

0.3887

3.5982e-08

1

0.80394

0

0

0

0

0

0.56281

0.66441

0.015208

0

Kd3

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Table 3 Optimized gain values for multi-source with single controller in single-area power system Cost function

Algorithms

Gain parameters Kp

DE

GA

TLBO

CIO

Ki

Kd

ITAE

1

1

0.38163

IAE

1

0.5725

0.23238

ISE

1

0.99857

1

ITSE

1

0.6268

0.5685

ITAE

0.99998

0.99999

0.48134

IAE

0.99877

0.9941

0.47194

ISE

0.89042

0.99999

0.29375

ITSE

0.99956

0.91056

0.57984

ITAE

0.99994

0.99582

0.21619

IAE

1

1

0.99758

ISE

1

1

1

ITSE

1

0.79208

0.73814

ITAE

0.99391

0.97848

0.41518

IAE

0.98181

0.97258

0.77923

ISE

0.91561

0.84594

0.93668

ITSE

0.92417

0.68054

0.81764

Table 4 Time domain parameters of single-area multi-source power system with multi-controllers Algorithms

Objective function

Area control error

Undershoot

Overshoot

Settling time

GA

ITAE

0.186378

−0.0427

0

14

IAE

0.07825

−0.036

0.003

29

ISE

0.00143

−0.034

0.00275

24

ITSE

0.001949

−0.038

0.002

28

ITAE

0.2872

−0.042

0.0034

23

IAE

0.08535

−0.041

0.0029

22

ISE

0.0017817

−0.0313

0.0106

31

DE

TLBO

CIO* (S1)

ITSE

0.0023994

−0.036

0.006

32

ITAE

0.237

−0.04

0.0032

31

IAE

0.08028

−0.035

0.0003

33

ISE

0.001314

−0.0272

0.005

26

ITSE

0.001852

−0.0345

0.0027

25

ITAE

0.2539

−0.0382

0.0018

27

IAE

0.081320

−0.0350

0.0022

29

ISE

0.001835

−0.0350

0.0008

30

ITSE

0.002839

−0.040

0.0088

24

Bold fonts represent the best cost function for the respective computation techniques and * indicates the best of best performance

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Table 5 Time domain parameters of single-area multi-source power system with single controller Algorithms Objective Area function control error

Undershoot changes Overshoot changes in Settling in (Hz) (Hz) time in (s)

GA

DE

TLBO

CIO* (S2)

ITAE

0.350914

−0.034

0.0085

72

IAE

0.09002

−0.034

0.0084

74

ISE

0.003034

−0.0377

0.0125

31

ITSE

0.002319

−0.033

0.0065

75

ITAE

0.34992

−0.035

0.01

31

IAE

0.098449

−0.039

0.0025

51

ISE

0.001315

−0.028

0.0068

35

ITSE

0.0027161 −0.0336

0.003

42

ITAE

0.35278

−0.038

0.0125

31

IAE

0.087592

−0.028

0.00685

34

ISE

0.001314

−0.028

0.00683

34

ITSE

0.00241

−0.031

0.005

37

ITAE

0.3563

−0.035

0.0088

31

IAE

0.08934

−0.03

0.007

34

ISE

0.001504

−0.03

0.00633

39

ITSE

0.002845

−0.03

0.005

40

Bold fonts represent the best cost function for the respective computation techniques and * indicates the best of best performance

with a GA algorithm-based PID regulator that gives minimum values over other algorithms and objective functions. Further, It is noticeable from Table 5 that the least area control error value (0.001314 Hz) and undershoot (−0.028 Hz) of cost function ISE based is obtained with the TLBO algorithm-based PID regulator as correlated to all other techniques and cost functions. Furthermore, in overshoot (0.0125 Hz) and settling time (31 s), cost ISE is calculated with a GA algorithm-based PID regulator that gives minimum values over other algorithms and objective functions. The deviation of frequency in the understudy system with GA, DE, TLBO, and CIO-based PID controllers are shown in Figs. 3, 4, 5, and 6, respectively, for multiple controllers. Figure 3 clearly shows that the GA-ISE-based solid black color line gives best results than other cost functions in terms of reduced oscillation and settling time. Figure 4 shows that DE-IAE-based solid red color line gives the best result over other cost functions in terms of overshoot and settling time. Figure 5 clearly reveals that TLBO-ISE- and TLBO-ITSE-based PID controllers give more are less best performance than other cost functions in terms of reduced oscillations and settling time. Figure 6 shows CIO-ITAE-based solid black color line gives the best results over other cost functions in terms of reduced oscillation and settling time. The frequency deviation of selected best cost functions for the respective soft computing technique involving multi-controller is shown in Fig. 7. Figure 7 reveals

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Fig. 3 Frequency response (S1) of genetic algorithm for various cost functions

Fig. 4 Frequency response (S1) of differential evolution algorithm for various cost functions

that CIO-ITAE used PID regulator gives improved performance in terms of reduced oscillation and settling time. Figures 8, 9, 10, and 11 show the bar chart analysis of bio-inspired with socioinspired optimization techniques for various cost function in terms of time domain specifications and error values. Figure 8 clearly shows that ISE- and ITSE-based system gives very minimum errors over other performance indexes. Figure 9 shows that the IAE-based PID regulator in TLBO algorithms gives minimum undershoot

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Fig. 5 Frequency response (S1) of TLBO algorithm for various cost functions

Fig. 6 Frequency response (S1) of cohort intelligence algorithm for various cost functions

values than other computing techniques. Figure 10 shows that the ITAE-based PID regulator in GA algorithms gives minimum overshoot values than other computing techniques also observed CIO-ISE-based PID controller gives a reduced overshoot. Figure 11 shows that both bio-inspired and socio-inspired algorithm-based PID controllers have capable of handling steady state precision within 35 s effectively. Finally, This proves the superiority of CIO-based PID controllers over GA, DE, and TLBO-based PID regulators in terms of oscillation control with lesser computation time.

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Fig. 7 Comparative best frequency deviation for various algorithms and best cost functions

Fig. 8 Bar chart analysis of area control error for single area with multiple controllers

The deviation of frequency in the understudy system with GA, DE, TLBO, and CIO-based PID controllers are shown in Figs. 12, 13, 14, and 15, respectively, for a single controller. Figure 12 clearly shows that the GA-ISE-based solid black color line gives best results than other cost functions in terms of reduced settling time. Figures 13, 14, and 15 clearly reveal that DE-ISE, TLBO-ISE, and CIO-ISE-based PID controllers give the best performance over the other cost functions in terms of reduced oscillations and settling time. The frequency deviation plot of selected best cost functions for the respective soft computing technique is shown in Fig. 16. From Fig. 16 reveals that the CIO-ISEbased PID controller gives superior performance in terms of reduced oscillation and

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Fig. 9 Bar chart analysis of undershoot for single area with multiple controllers

Fig. 10 Bar chart analysis of overshoot for single area with multiple controllers

settling time. A system with single and multiple controllers analysis based on cohort intelligence is shown in Fig. 17. From Fig. 17 gives the system with a multi-controller reduced oscillation and settling time over the single controller. Figures 19, 20, 21, and 22 shows the bar chart analysis of bio-inspired with socioinspired optimization techniques for various cost function in terms of time domain specifications and error values. Figure 19 clearly show that ISE and ITSE-based system gives very minimum errors over other performance indexes. Figure 20 shows that the CIO-ITSE-based PID regulator gives minimum undershoot values than other computing techniques. Figure 21 shows that CIO based PID regulator gives superior performance over the other computing technique. Figure 22 shows that both bioinspired and socio-inspired algorithm-based PID controllers have capable of handling steady state precision within 80 s effectively. It is noticed that TLBO- and CIO-based

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Fig. 11 Bar chart analysis of settling time for single area with multiple controllers

Fig. 12 Frequency response (S2) of genetic algorithm for various cost functions

controllers achieve the settling time within 40 s which is closer to multi-controller techniques. Table 6 represents the computation time taken by the soft computing techniques. Figure 23 shows the bar chart analysis of bio-inspired with socio-inspired optimization techniques for various cost functions in terms of computation time. Figure 23 clearly shows that CIO and DE-based system takes very lesser computation time than the other techniques. The challenges and limitations of this article are not focused on the variant operator involved in genetic and cohort intelligence algorithms to know the better performance. The selection of new solutions is also depending on various types of selection operators employed in the optimization

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Fig. 13 Frequency response (S2) of differential evolution for various cost functions

Fig. 14 Frequency response (S2) of Teaching–learning-based optimization algorithm for various cost functions

process, namely Roulette wheel selection, Tournament selection, Uniform selection techniques, etc. This may be addressed in our forthcoming research work effectively.

8 Comparison of Biologically Inspired Algorithm with Socio-inspired …

Fig. 15 Frequency response (S2) of cohort intelligence algorithm for various cost functions

Fig. 16 Comparative best frequency deviation for various algorithms and ISE cost functions

203

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Fig. 17 Frequency response (S1 and S2) of cohort using two different control methods

Fig. 19 Bar chart analysis of area control error for single area with single controller

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Fig. 20 Bar chart analysis of undershoot for single area with single controller

Fig. 21 Bar chart analysis of overshoot for single area with single controller

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Fig. 22 Bar chart analysis of settling time for single area with single controller Table 6 Computation time of various optimization algorithms for different cost functions Cost function

GA

DE

TLBO

CIO

ITAE

17

7

13

7.53

IAE

26

8

17

5.3

ISE

23

7

11

2.28

ITSE

11

6

13

2.13

Fig. 23 Bar chart analysis of computation time for various algorithms

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207

5 Conclusion This paper presents the application of four different algorithms for LFC in both single and multi-controllers involved in a single area with a multi-source power system. The comparison is done among each cost function with each soft computing technique utilized PID controller to view the strength and superiority of nature-inspired technique. From the bar chart analysis, it could be brought down the curtain that cohort intelligence-based ITAE and ISE cost function gives better dynamic response as well as consumes lesser computation time over other optimization techniques. Further, it is observed that the choice of multiple controllers is the best choice for the penetration of more renewable energy sources with the base load.

Appendix A. Parameters of optimization technique Genetic Algorithm: Population (Np = 5), Iteration (T = 100 − 225) Differential Evolution: Population (Np = 5), Iteration (T = 100), Cross over Probability (Pcr = 0.6), Scaling factor (F = 0.2 − 0.8) TLBO: Population (Np = 5), Iteration (T = 100), Teaching Factor (TF = 1–2) Cohort Intelligence: Cohorts (C = 5), Iteration (T < 30), convergence parameter (E = 0.001), Reduction factor (r = 0.97). B. Design parameters of multisource with single controller/multi controllers in single area power system [9].

References 1. Kundur PS, Malik OP (2022) Power system stability and control. McGraw-Hill Education 2. Elgerd OI, Fosha CE (1970) Optimum megawatt-frequency control of multiarea electric energy systems. IEEE Trans Power Appar Syst 4:556–563 3. Bevrani H (2014) Robust power system frequency control, vol 4. Springer, New York 4. Dey N, Ashour AS, Bhattacharyya S (eds) Applied nature-inspired computing: algorithms and case studies. Springer Singapore 5. Murugesan D, Jagatheesan K, Kulkarni AJ, Shah P (2023) A socio inspired technique in nuclear power plant for load frequency control by using cohort intelligence optimization-based PID controller. In: Renewable energy optimization, planning and control: proceedings of ICRTE 2022. Springer Nature Singapore, Singapore, pp 1–12 6. Kumarakrishnan V, Vijayakumar G, Boopathi D, Jagatheesan K, Saravanan S, Anand B (2020) Optimized PSO technique based PID controller for load frequency control of single area power system. Solid State Technol 63(5):7979–7990

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7. Murugesan D, Jagatheesan K, Boopathi D (2021) Meta-heuristic strategy planned controller for frequency supervision of integrated thermal plant with renewable source. In: 2021 IEEE 3rd Ph.D. colloquium on ethically driven innovation and technology for society (PhD EDITS). IEEE, pp 1–2 8. Jagatheesan K, Anand B, Samanta S, Dey N, Santhi V, Ashour AS, Balas VE (2017) Application of flower pollination algorithm in load frequency control of multi-area interconnected power system with nonlinearity. Neural Comput Appl 28(1):475–488 9. Jagatheesan K, Anand B, Samanta S, Dey N, Ashour AS, Balas VE (2017) Design of a proportional-integral-derivative controller for an automatic generation control of multi-area power thermal systems using firefly algorithm. IEEE/CAA J Autom Sinica 6(2):503–515 10. Deb K (2014) Multi-objective optimization. In: Search methodologies. Springer, Boston, MA, pp 403–449 11. Mohanty B, Panda S, Hota PK (2014) Controller parameters tuning of differential evolution algorithm and its application to load frequency control of multi-source power system. Int J Electr Power Energy Syst 54:77–85 (DE) 12. Murugesan D, Shah P, Jagatheesan K, Sekhar R, Kulkarni AJ (2022) Cohort intelligence optimization based controller design of isolated and interconnected thermal power system for automatic generation control. In: 2022 second international conference on computer science, engineering and applications (ICCSEA). IEEE, pp 1–6 13. Paliwal N, Srivastava L, Pandit M (2020) Application of grey wolf optimization algorithm for load frequency control in multi-source single area power system. Evol Intell 1–22 14. Ramjug-Ballgobin R, Ramlukon C (2021) A hybrid metaheuristic optimisation technique for load frequency control. SN Appl Sci 3(5):1–14 15. Acharyulu BVS, Swamy SK, Mohanty B, Hota PK (2019) Performance comparison of FOA optimized several classical controllers for multi-area multi-sources power system. In: Soft computing in data analytics. Springer, Singapore, pp 417–426 16. Naidu K, Mokhlis H, Bakar AA (2014) Multiobjective optimization using weighted sum artificial bee colony algorithm for load frequency control. Int J Electr Power Energy Syst 55:657–667 17. Eshelman LJ (2000) Genetic algorithms. Evol Comput 1:64–80 18. Storn R, Price K (1997) Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359 (DE) 19. Das S, Suganthan PN (2010) Differential evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput 15(1):4–31 20. Rao RV (2016) Teaching-learning-based optimization algorithm. In: Teaching learning based optimization algorithm. Springer, Cham, pp 9–39 21. Gorripotu TS, Samalla H, Rao JM, Azar AT, Pelusi D (2019) TLBO algorithm optimized fractional-order PID controller for AGC of interconnected power system. In: Soft computing in data analytics. Springer, Singapore, pp 847–855 22. Rao RV, Savsani VJ, Vakharia DP (2012) Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problems. Inf Sci 183(1):1–15 23. Shah P, Agashe S, Kulkarni AJ (2018) Design of a fractional PIλDμ controller using the cohort intelligence method. Front Inf Technol Electron Eng 19(3):437–445 24. Sekhar R, Shah P (2020) Predictive modeling of a flexible robotic arm using cohort intelligence socio-inspired optimization. In: 2020 1st international conference on information technology, advanced mechanical and electrical engineering (ICITAMEE). IEEE, pp 193–198 25. Shah P, Kulkarni AJ (2019) Application of variations of cohort intelligence in designing fractional PID controller for various systems. In:Socio-cultural inspired metaheuristics. Springer, Singapore, pp 175–192 26. Murugesan D, Jagatheesan K, Shah P, Sekhar R (2023) Fractional order PIλDμ controller for microgrid power system using cohort intelligence optimization. Results Control Optim 11:100218

Chapter 9

Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing and Topology Design of Real-Size Tall Building Systems Salar Farahmand-Tabar

1 Introduction Nowadays, due to the lack of land for construction and the increasing population in major cities, tall buildings have found a special priority. On the other hand, the shape, form, and appearance of tall buildings are different from the past, and if these changes continue in the future, it won’t be long that humans dream of realizing the construction of buildings as high as a mile. Some people call a building tall whose height causes lateral forces (wind and earthquake) to have a significant effect on its design or a structure whose period is more than 0.7 s. Some consider the heightto-dimension ratio of the structure to be the criterion of such a classification. The height-to-dimension ratios of 1.5π, π, π/2, and π/3 are considered as very tall, tall, medium, and short structures, respectively. In addition to ensuring safety, efficiency, comfort, and aesthetics criteria, tall buildings should not have any problems in terms of design. Hence, the structural form is presented through architects and engineers, and then, the design is started according to that initial form. The importance of the effect of lateral forces rises sharply as the height of the building increases, such that the lateral stiffness and displacement control the design of a tall building. The stiffness depends on the type of structural system, while the efficiency of each building system is directly related to the amount of the utilized materials; Therefore, for optimization purposes, sufficient stiffness should be obtained with minimum weight. This leads to the creation of suitable systems for any given height. Basically, an important subject in the design of tall buildings is to change the form of buildings to more rigid ones that limit the displacement and increase the S. Farahmand-Tabar (B) Department of Civil Engineering Eng, Faculty of Engineering, University of Zanjan, Zanjan, Iran e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_9

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stability. In general, the behavior of buildings and their vulnerability under lateral loads depends on the shape, size, and overall form in plan and height, as well as the way of transferring lateral forces to the ground. The more symmetrical and simpler the configuration of the structure is, the better it will perform when subjected to different loads, and as a result, it will be less vulnerable. Therefore, due to the importance and connection of architectural features with the structural system in fulfilling the desired performance of the building, the selection of architectural design and compatible systems in high-rise buildings is particularly important. During the past centuries, many systems have been designed for tall structures and many tall buildings have been constructed based on them. At first, the structural systems in high-rise buildings only included columns, beams, and slabs. This category of structural systems in which the most load-bearing elements are located inside the building is called interior structures. However, a huge revolution was taking place in tall buildings over time and new systems such as tube systems, braced tubes, bundled tubes, diagrids, and exoskeleton systems were a transformation in tall buildings both in terms of architecture and structure. This category of structural systems is called exterior structures since the load-bearing elements are placed around the building. Considering the requirements of strength and safety while increasing the height of the structures besides the cost-related issues in terms of utilized material, the design process of tall buildings becomes complementary and multi-faceted. Different alternative solutions can exist that meet the design requirements. However, a significant issue is finding the best solution among the possible solutions. The economical design is related to the optimum material usage and minimum weight of the structure. The real-size tall buildings impose construction at an expensive cost. In contrast, by lowering the weight and the cost of the structure the design limitations are threatened and exceeded. Handling such a multidisciplinary objective demands a computer-aided procedure utilizing computer intelligence for economical design so as to reach an optimum solution. This optimum design in which the structure has the balance between reliable load-carrying capacity, safety, and minimum cost becomes a decision-making problem that cannot be achieved through the conventional design process. Here, structural optimization assists the designers in solving the complex mathematical relations of structural problems, especially tall buildings. The first category of optimization methods, entitled classical, is based on mathematical expressions resulting in deterministic solutions. However, they have some drawbacks and cannot be used in various problems including large-scale tall buildings. Thus, an alternative method inspired by nature, entitled metaheuristics, with high performance and flexibility for general purposes can be considered. Contrary to classical methods, metaheuristics use the historical information of iterative processes while searching. They start with a set of random solutions within the lower and upper limits followed by the steps of the algorithm to reach the optimum solution considering the defined fitness function. The final solution is achieved when the stopping criteria such as the maximum number of fitness evaluations or specified tolerance value are reached.

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Nature-inspired metaheuristic methods [1] are categorized into several groups including evolutionary, swarm intelligence, physics-related, and human behavior. Evolutionary algorithms are the basic group of this category, and Genetic Algorithm (GA) is the most famous representative of this group of metaheuristics [2–4]. Using the operators inspired by biology including mutation, crossover, and selection, the GA investigates the search area for a better solution. In each generation/iteration, a stochastic or score-based selection of parent individuals is done to generate the child population for the following generation. In this process, the parents can be combined, called the crossover, or they can be changed randomly, called the mutation. The fitness of a new population is evaluated and the process continues till reaching the termination criteria. In this chapter, the design optimization of tall buildings in terms of size and topology is exhibited. Considering the LRFD-AISC [5], the optimal design problem is formulated to be solved using GA and Fuzzy-GA. The accelerated version of Fuzzy-GA, considered here, utilizes the bi-linear membership function for fuzzification to cover the limitations of the GA in complex problems. The merit of the GA and accelerated Fuzzy-GA is investigated by considering two types of tall building systems, the outrigger-braced system with belt truss as an interior system, and the diagrid as an exterior system of tall buildings. The 40-story tall building with outrigger system and belt truss is optimized through GA, and the complex problem of the 60-story diagrid is optimized using accelerated Fuzzy-GA. Both threedimensional real-size structures are designed for optimum topology and minimum size/weight of the structure. Finding the optimum solutions of large-scale problems detected in real-world practice represents the robustness of the GA and its variants in the field of engineering/structural optimization.

2 Related Works on Real-Size Tall Building Systems Numerous tall buildings have been constructed over the past few decades thanks to advancements in creative construction methods. Tall buildings can be described as large-scale structural systems and can be used to describe buildings with many structural members. Additionally, the optimal design of such buildings has long been acknowledged as one of the most difficult topics of structural design engineering due to the complexity involved in the construction of real-size tall buildings. As a result, from the past to the present, comparatively a few researches on this topic have been investigated. Outrigger system with belt truss in tall buildings is an example of interior structures in which the interior part of the building serves as the location of the main lateral load-resisting system [6, 7]. Besides the size optimization, the important type of optimization regarding these structures is topology optimization [8, 9] in terms of locating the outrigger system with belt truss for better and optimum structural behavior. The optimal location of the outrigger system or belt truss has been investigated a lot [10–15]. Also, the number and location of the multi-outrigger system

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have been optimized simultaneously [16, 17]. Among various optimization methods, GA has been mostly used in tackling the optimization problem of these structures [18–21]. On the other hand, diagrids [22–27] are categorized as an exterior structure so that the lateral load-resisting system is placed at its perimeter. The pattern/arrangement of diagrid elements is the key factor in the design process of diagrids [28–31]. Therefore, the geometrical pattern of these structures is subjected to topology-based design optimization [32–37]. Similar to outrigger-braced tall buildings, GA has been applied to the optimization problems of diagrids [38, 39].

3 Mathematical Formation of Size and Topology Optimization of Real-Size Tall Building Systems The structural system must satisfy the structural constraints established by specifications to achieve the intended structural behavior in the minimum-weighted, or optimally designed, tall building. This section discusses the mathematical formulas required to obtain the desired minimal-weighted tall building with optimum topology. It is aimed to discover the values of the structural sizing parameters from a list of standard sections, as well as the topology-related parameters such as the outrigger’s location in the structures, and the geometric pattern of elements in diagrids. The following equation is the objective function for the minimum structural weight: W =

Ntot Σ

γi Ai L i

(1)

i=1

where W is the structural weight achieved by the unit weight of the material (γi ) considering the volume of the utilized members ( Ai L i ) with respect to their lengths (L i ) and the cross-sectional areas (Ai ). Classifying the members into separate groups with similar characteristics would be necessary to decrease the computational cost. Considering the problem, several constraints such as constructability, strength, and serviceability limits should be satisfied, and the violated ones should be penalized. For the strength constraint, the members subjected to compression and flexural loads should be evaluated through the following equation according to the design method of the AISC-LRFD: ] ] [ ] [ [ Mry 8 Mrx Pr Pr + ≤ 1.0 for ≥ 0.2 + ϕc Pn 9 ϕb Mnx ϕb Mny ϕc Pn ] [ ] [ Mry Pr Pr Mrx ≤ 1.0 for < 0.2 (2) + + 2ϕc Pn ϕb Mnx ϕb Mny ϕc Pn

9 Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing …

213

where the members’ demand-to-capacity ratios are achieved using the combination of applied axial load (Pr ), and flexural loads about the major and minor axis (Mrx , Mry ) with respect to their nominal factored values, ϕc Pn and ϕb Mn . In tall buildings, the limit states of serviceability have a determinative role in the process of design. The second constraint which should be considered in the design optimization is related to the comfort of occupants regarding the maximum allowable inter-story drift, written as follows: C Drift = f

δf −1≤0 δ limit f

(3)

( ) where the inter-story drift δ f is limited in each story by its permitted value which is 0.002 of the height of each story under lateral loads. Another form of this constraint is the total drift or the maximum displacement of the structure formulated as follows: Drift_tot Cind =

Δi −1≤0 Δilimit

(4)

where the total displacement measured from the roof is permitted up to 1/500 of the overall building height. Finally, a constraint is defined as representing the capability of members’ construction with proper and compatible sizes in adjacent stories. By neglecting this limitation, bigger sections may be selected and placed above the smaller ones, and that is not permitted. This constraint is formulated as follows: top.

i,Const CIel =

di −1≤0 dibot.

(5)

( ) top where the height of the vertical or inclined sections at the top floor di should be ( bot. ) equal to or smaller than those located at the bottom di .

4 Genetic Algorithm (GA) 4.1 Standard GA The efficient and reliable method of genetic algorithm for global optimization is inspired by the concept of survival of the fittest and natural selection. It is proposed to investigate the adaptive behavior by Holland [2]. Through the process of evolution starting from candidate solutions, GA is capable of providing near-optimum solutions. Therefore, it can be utilized in any kind of conceivable optimization problem.

214

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Similar to all metaheuristics, GA is appealing for two primary reasons: (1) it has an inherent capability for parallel computing; (2) it relies on “payoff” data, i.e., not derivative data, which is crucial for combinatorial or extremely non-linear problems. However, it should be highlighted that GA’s success in identifying the true local or global optimum is constrained by the “anytime” behavior; the best member of the population evolves quickly at first, then the pace is decreased until evolution essentially ceases [3]. Natural selection is modeled by the algorithm. Using methods inspired by concept of the evolution in nature, such as selection, mutation, and crossover, a population of potential solutions is developed and evolved through generations. A population of random individuals is the starting point for evolution. Fitness evaluation is carried out for each member of the population in generations. The quality of the solutions is measured by the fitness evaluation considering the values obtained from the objective function. Then, several individuals are randomly chosen from the population (based on fitness values) to generate the parent population. To create offspring and establish a new population, their genetic material is recombined and perhaps stochastically mutated. The algorithm’s subsequent iteration uses the new population. The algorithm typically comes to an end when either it reaches the maximum number of generations, or the elite member of the population is attained a sufficient fitness level. The step-by-step process given below can be used to represent the standard genetic algorithm: (1) Set the initial random chromosomes as individuals’ population. (2) Determine the fitness values of individuals within the population. (3) Choose individuals to include in a new population based on their fitness level (Selection). (4) Create a new population by doing crossover and mutation. 5) Replicate the elite individual in the new population. (6) Continue performing steps 2 through 5 until the stopping criteria are met. Keep in mind that there isn’t a “standard” variation of the GA that is approved by everyone. For instance, step (5) is included here since it significantly improves the GA’s performance in almost all problems. There are two types of standard GA: realvalued GA and binary-encoded GA. For the binary representation of GA, each design variable is encoded in a series of bits of the proper length. This design variable is uniquely mapped to a particular value in the real problem by the string of 0s and 1s known as a gene. All genes are put together to form a chromosome, which is a potential answer to the underlying issue. According to the mentioned pseudo-code and the flowchart presented in Fig. 1, a detailed procedure of GA is: Initialization: Solutions are first created at random to generate an initial population. The size of the population is specified by the nature of the issue.

9 Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing …

215

Fig. 1 Flowchart of a typical genetic algorithm (GA)

Fitness evaluation: The value of each chromosome’s associated objective function determines its fitness. This objective function is selected by the user. Because it frequently returns the cost of the suggested solution, it is known as the “cost function.” Selection: Each subsequent generation chooses a portion of the present population to breed a new generation. An approach based on fitness is utilized to choose individual solutions, in which fitter solutions often have a higher chance of being chosen. Some selection techniques evaluate the rate of each solution’s fitness and give preference to the best ones. There are common techniques for selection: (1) Roulette wheel method, (2) Tournament selection, and (3) Stochastic remainder selection.

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Reproduction (Crossover and mutation): Utilizing the genetic operators such as crossover (recombination) and mutation, the next stage is to create a new population of solutions. A pair of “parent” solutions are selected for breeding for every new solution that will be developed. There are common techniques for crossover: (1) One-site crossover, (2) Two-site crossover, and (3) Trade of Uniform Crossover. The GA can avoid being trapped in local optima and explore new areas of the search space by stochastically applying mutation to the offspring. For every new child, new parents are selected, and this process iterates until a new population of solutions with the proper size is generated. Elitism: Elitism is a tactic employed to guarantee that the best member of the present generation is maintained for the following one. The best individual is often conveyed unaltered to the following generation. Termination: Until the stopping criteria are met, this generational process is repeated. The following are examples of typical stopping criteria: • A solution that meets a few requirements is discovered. • The allotted computation budget is used up. • The fitness of the top-ranked solution is approaching, or it has already reached a point where further iterations do not provide better outcomes. • The aforementioned in combination.

4.2 Fuzzy-Assisted GA with Bi-Linear Membership Function Numerous optimization applications have made extensive use of the GA and its fuzzy variant (Fuzzy-GA) [40–45]. The performance of the fuzzy-GA can be influenced by the membership function used through the fuzzification process. Here, the bi-linear membership function is utilized in Fuzzy-GA for both constraints and objective functions to accelerate its performance. According to crisp logic, a phenomenon can either be true or false. The application of crisp logic will complicate the circumstances for structural optimal design since the problem is affected by a significant level of ambiguity and imprecision. Fuzzy logic takes into account the transition between exclusion and inclusion using a membership function, μY , which converts each member of set X into a fuzzy set Y by doing the following: (

μY (x) ∈ [0, 1] x∈X

(6)

9 Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing …

217

The value of membership in crisp logic for structural element i is equal to 1 if σi < σ u and equal to zero if the constraint is not fulfilled. However, the membership value (α) is altered from 0 to 1 for fuzzy logic, where α = 1 denotes that the constraint is fulfilled and α = 0 denotes that the constraint has been violated relative to the specified tolerance. Consequently, 0 < α < 1 denotes that the constraint has been met to the specified amount. When using fuzzy optimization with the membership function, both inequality constraints and the objective function are fuzzified. ) μ D¯ (x) = μ F (x) ∩

m ⋂

( μgi (x)

(7)

i=1

where μgi (x) and μ F (x) denote the membership functions for the ith inequality constraint and the objective function, respectively. The xopt is achieved from the region as (

μ D xopt

)

( ) = max μ D (x), μ D (x) = min μ F (x), min μgi (x) i=1,...,m

(8)

The membership functions related to the objective function and constraints (Fig. 2) are intersected to provide a fuzzy domain. The optimization problem can be expressed using a general satisfaction parameter (λ) as follows: maximize λ subject to λ ≤ μ F (x) λ ≤ μgi (x); i = 1, . . . , m 0≤λ≤1

Fig. 2 Intersection of membership functions

(9)

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An optimization approach is suggested using a formulation that combines fuzzy logic with enhanced Lagrangian penalty functions considering the membership parameters of α and β for the objective function and γ and ω for the constraints. To minimize ) [ ) noe [ ]2 ( ]2 ( λ 1 1 Σ λ −1+β γi a − 1 + ωi ϕ(x, y) = − S f λ + + α 2 μF 2 i=1 μσ i ⎫ ⎧ ) [ ]2 ⎬ [ ]2 ( nod nm 1 ⎨Σ λ λ 1 Σ + γj − 1 + ωj γk − 1 + ωk + (10) ⎭ 2 2 ⎩ j=1 μδ j μgk k=1 It is intended to maximize the penalized λ (S f λ) while taking into account the constraints. In structural optimization problems, stress (σ ), displacement (δ), and constructability (g) are typically the kinds of constraints expressed as follows using the membership function (μ): λ ≤ μub σi (x); i = 1, . . . , total number of members (nm) λ ≤ μlb σi (x) λ ≤ μδ j (x); j = 1, . . . , number of constrained DOFs (nd) λ ≤ μgk (x); k = 1, . . . , total number of diagrid units (nu)

(11)

All constraints can be expressed as ( min μ F (x),

) min

i=1,...,nm

μaσi ,

min μδ j ,

j=1,...,nd

min μgk

k=1,...,nu

to form an unconstrained problem as maximize ϕ(x) = λ

(12)

First, each chromosome is given a design variable (x) via GA. Then, using structural analysis, membership values for the objective functions and constraints are determined. Using Eq. (9), the fitness value of each chromosome is determined. It’s crucial to pick the right membership function for the objective function to reach a solution. If the objective function’s lower and upper bounds for minimization , ,, problems are assumed to be F and F , then ( μF =

,

1; if F ≤ F ,, 0; if F > F

(13)

9 Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing …

(a)

219

(b)

Fig. 3 Bi-linear membership functions [39] ,

,,

The membership value (F < F < F ) can be used to implement the necessary ,, membership function. The F may be calculated using basic GA iterations or by ,, using design experience. Even if the violated constraints emerge, F can be lowered to a minimal value or a specific portion of the objective function to reach the value , of the lower bound (F ). There might be several issues with the objective function’s linear membership function. The possibility of an expedited convergence to a local optimum solution is raised if the fuzzy-GA technique is employed after the process of standard GA [29]. ,, However, in the initial iterations of the fuzzy-GA, we frequently have F > F if ,, F is identical to the best solution, and the membership value (μ F ) for the majority of chromosomes becomes zero, making it impossible to compare them with one another. Therefore, it is possible to accelerate the convergence by using a bi-linear membership function (Fig. 3b). If membership function No. 2 is taken into account in Fig. 3a, Fu ≤ F, the slope of the line decreases relative to line No. 1, indicating that there is less diversity in the fitness values of the chromosomes. As a result, the possibility of the algorithm to be trapped in a locally optimal ,, solution will rise, which will lead to a lower convergence rate. The value of F in , Fig. 3a can either be the average of F and Fu or the best value from earlier studies. The allowable values (ga ) of the constraints can be used to choose the membership function for them. The applied membership function for the constraints in the current study is displayed in Fig. 3b, where gu is equal to n × ga (n > 1) and Δg is the amount of the relaxation for the constraint.

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5 Discussion on the Optimum Design Examples of Real-Size Tall Building Systems The performance of the standard GA and accelerated Fuzzy-GA is evaluated using the optimum design of two real-size tall buildings with different structural systems, that are employed as design examples in this chapter. For the first design example, a 40-story, 4240-member outrigger-braced space frame containing 49 groups of members is selected. The second example is the design of a sixty-story, diagrid structure including varying number of diagrid units in which beams and inclined

9 Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing … Table 1 Parameters of calculating wind load (ASCE 10–16)

221

Parameter

Values

Direction angle

0

Pressure coefficient, C p

Windward Leeward

0.8 0.5

Wind speed (V)

110 mph

Exposure type (K z )

Category B

Importance factor (Iw )

1

Topology factor (K zt )

1

Gust factor (G)

0.85

Directionality factor (K d )

0.85

members classified in a separate group. According to ASCE-LRFD, the following load combination is considered: Q = 1.2D + 1.0L ∓ 1.0W

(14)

The wind load is computed using the analytical approach (ACSE7-16) with a wind speed of 47 m/s (∼ =105 mph) in a considered enclosed structure. The wind load is computed, based on the specified parameters of Table 1. The following formula is used to get the velocity pressure at height z: qz = K z K zt K d V 2

(15)

The velocity pressure (qh ) at the average height of the building (h) is calculated considering the K z at the height (h) using the same formula as before. For partially enclosed or enclosed structures of any height, the wind load pressure for lateral load-resisting systems is calculated as follows: P = qz GC p−windward − qh GC p−leeward

(16)

The GA uses a set of parameters that must be preset at the beginning. Additionally, the performance of the GA parameter set varies for each design problem and is closely correlated with problem size. The GA parameter settings for the following design examples are chosen following suggested ranges and values from earlier research as well as in-depth analyses carried out in this work (Table 2). Each design example was solved many times using various parameter values, and the best designs were then given here.

222 Table 2 Selected parameters for GA and fuzzy-GA

S. Farahmand-Tabar

Values

Parameters

Outrigger-braced (Standard GA)

Diagrid (Fuzzy-GA)

Population

60

30

Function of selection

Roulette wheel

Roulette wheel

Mutation

Percent

0.5

0.5

Rate

0.07

0.08

Crossover ratio

0.75

0.08

Termination

1000 iterations

350 iterations

5.1 Outrigger-Braced Tall Building with Inclined Truss Belt Utilizing a core system for reducing the inter-story drifts is one of the advantageous and efficient lateral load-resisting solutions for structures with a specific number of floors. For taller structures, however, the core system’s rigidity is insufficient to constrain the structure to the permitted drift limits subjected to the wind load. For such buildings, a method that limits lateral load-related drift without using more structural material is necessary. The outrigger system is one of the technologies that helps tall structures have enough rigidity. Outrigger systems with belt trusses are a frequent choice for tall moment-resisting frames considering easy and inexpensive construction compared to other structural systems. This design includes a core system that is made of shear walls or braced frames, with the outriggers which are trusses linking the outside columns to the core. Stiff arms on outriggers allow them to interact with surrounding columns. Under strong lateral loads, the rotation of the core at the outriggers’ level creates a tensioncompression couple in the peripheral columns through belt trusses, reducing the core deflection and distributing the force to exterior columns. Figure 4 shows the schematic outrigger system with the belt truss. In the design example, the two-way slab is used as the system of flooring. Beams are subjected to distributed gravity loads for both dead and live load scenarios of 500 and 250 kg/m for the outer columns, respectively, and 1000 and 500 kg/m for the inner beams. The material properties include A913-Grade 50 steel with a yield stress of 35,153,481 kg/m2 and a modulus of elasticity 2.039 × 1010 kg/m2 . The specified material unit weight is 7849 kg/m3 . A list of 267 standard W-sections is provided for the design problem. The position of the outriggers (topology) was simultaneously optimized with weight optimization (member sizing). Convergence history, roof displacement, interstory drifts, optimal cross sections, and the proper position of the outriggers are all examples of the obtained results of the design optimization. The element groups in the 40-story structure with six spans are specified in Fig. 5. To obtain reliable results, the optimization process was repeated 15 times and the statistical findings

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223

Fig. 4 The outrigger system with truss belt in tall buildings

are presented in Table 3. The optimum cross sections, structural weight, and outrigger position are listed in Table 3. The ratio of the outrigger position to the structure’s height (X/H = 0.52) is nearly identical to the one suggested by Taranath [10]. This means that when the outrigger system is placed on story 21 in the 40-story building of the considered model, the optimum structural weight (2387.30 tons) considering the constraints of design and displacement can be achieved. The convergence history of an outrigger-braced structure is illustrated in Fig. 6a. Also, Fig. 6b shows the inter-story drift at various stories. Drift values are within the permissible range as stated by ASCE 7–16. Optimizing the design of tall buildings is a complex task that involves multiple design variables and objectives, which makes it a challenging problem to solve. Genetic algorithm has certain limitations and challenges that can hinder its effectiveness in optimizing large-scale problems such as tall buildings. A large number of design variables with uncertainties and related constraint handling issues in largescale complex problems are the challenges resulting in a time-consuming process and premature convergence in a local minimum, requiring a robust version of the algorithm such as accelerating fuzzy-GA.

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(a)

(b)

Fig. 5 Specifying separate groups for the members of the model

Table 3 The obtained w-shaped cross sections of the 40-story outrigger-braced tall building Story (n)

Group

C(2n + 1)

C(2n)

C(2n −1)

Br(n)

B(2n + 1)

B(2n)

1:5

1:6

24 × 84

8 × 67

21 × 83

18 × 35

8 × 67

10 × 49

5:10

7:12

10 × 60

18 × 71

10 × 60

18 × 35

8 × 67

10 × 39

10:15

13:18

12 × 58

8 × 58

8 × 67

18 × 35

18 × 40

8 × 31

15:20

19:24

8 × 67

18 × 65

8 × 58

10 × 45

18 × 35

12 × 30

20:25

25:30

18 × 71

18 × 55

8 × 58

10 × 45

18 × 35

12 × 40

25:30

31:36

8 × 48

18 × 50

21 × 62

14 × 26

10 × 45

12 × 40

30:35

37:42

8 × 48

8 × 31

10 × 45

10 × 45

18 × 35

12 × 35

35:40

43:48

16 × 31

12 × 35

10 × 39

14 × 26

10 × 45

12 × 35

Outrigger

49

Outrigger level

50

Outrigger level/total height

24 × 84 21 0.525

Best weight

2387.30 (ton)

Mean weight

2563.28 (ton)

Standard deviation

42.659

5.2 Diagrid Structure To construct the tall building using a diagrid system, the optimization approach for this example uses a GA with modified fuzzification. The parameters and objective function for simultaneous size and topology optimization are determined by the structure’s geometry (diagrid pattern/angle). The range of angles in the building

9 Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing …

(a)

225

(b)

Fig. 6 The achieved convergence and optimum inter-story drift of the outrigger-braced tall building

varies between 30 and 85° depending on the number of units that may be piled along the structure’s overall height. It should be mentioned that the diagrid system’s joints should be installed on the floors. Because of this, the range of angles becomes discrete depending on the size of the structure. A 60-story structure with dimensions of 21 m has been optimized. The structure has a symmetrical plan and an average story height of 3.5 m. Steel material with a yield stress of 2.8e05, a modulus of elasticity equals to 2.3e08 kN/m2 , and a weight per volume of 76.98 kN/m3 has been utilized. The structure is subjected to the 7 kN/m2 of dead load and 4 kN/m2 of live load. A list of steel box sections from the standard AISC sections is taken into account for the design case. A group is assigned to inclined members and beams of each unit of the diagrid system. A typical model is shown in three dimensions in Fig. 7. Due to the nature of the problem and possibility of various topologies for this structure, the length of chromosome will be varied in different cases. Considering the number of diagrid units covering the building height and the corresponding beams Fig. 7 Typical diagrid system in a tall building

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Fig. 8 Typical chromosome for a diagrid system

and inclined members, the length of the chromosome can be achieved as follows: L chromosome = log 2n p +

) ( n total−stories × log 2n diagrid + log 2n beam n covered−stories

(17)

where the number of all possible cases for covering the total height with diagrid elements (using different angles) is represented by n p . The members in each unit are grouped into beams and inclined elements. Thus, the parent chromosome includes the specified number of covering diagrid units (n covered−stories ), followed by the same number of blocks related to both beams and inclined members of each diagrid unit (Fig. 8). To generate children, the n covered−stories is determined by GA operators initially. Considering the achieved value, the gene blocks for each unit are generated which may result in chromosomes with different lengths in comparison with previous ones. For equalizing the length, the crossover is utilized for shorter child chromosome, and the repetition of blocks is used when parent chromosome becomes shorter than the children. According to algorithm 1, the optimization is carried out. The obtained optimum results considering the size and topology of the structure are illustrated in Table 4. The achieved values of cross-sectional areas for beams and inclined diagrid members, the average demand-to-capacity ratio for the elements, base reaction forces, maximum displacement, and inter-story drift are listed in the table providing the final minimum structural weight and optimal angle of the diagrid elements. The achieved optimum angle of the diagrid is in accordance with the suggested range presented within the literature [32]. The convergence curve of the structure is provided in Fig. 9(a) in comparison to convergence histories attained by basic GA to demonstrate the effectiveness of the accelerating fuzzification in the GA. In this figure, the premature convergence of the standard GA is obvious. The optimum inter-story drift is depicted in Fig. 9b which represents that the algorithm is successful in satisfying the constraints while providing an optimum solution.

9 Genetic Algorithm and Accelerating Fuzzification for Optimum Sizing … Table 4 Optimum results of the 60-story diagrid structures

(a)

227

Diagrid Unit

Diagrids

Beams

1

80 × 6.5

30 × 1

2

70 × 6.5

30 × 1

3

70 × 6.5

30 × 1

4

70 × 6.5

30 × 1

5

70 × 6

30 × 1

6

70 × 4

30 × 1

7

70 × 4

30 × 1

8

70 × 3

30 × 1

9

70 × 2.5

30 × 1

10

35 × 2.5

30 × 1

Avg. D/C ratio

0.370785

0.81950

Base shear (kN)

13242

Base Moment (kN m)

13687.3

Max. displacement (m)

0.27780

Max. inter-story drift

0.001862

Opt. Weight (ton)

4828.10

Opt. Diagrid angle

71.6°

Conv. Iter.

210

(b)

Fig. 9 The achieved convergence history and optimum inter-story drift of the diagrid structure

6 Conclusions In this chapter, the standard GA and its accelerated fuzzy version were introduced to achieve the optimum size and topology of tall buildings. GA is a well-known representative of Evolutionary Algorithms (EA) that is inspired by biology concepts for populating through generations. Without the need for gradient-based information,

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GA is easy to implement considering the three operators of mutation, crossover, and selection. Using these operators in standard GA, the parent population is selected, and the child population is generated by crossover and mutation. Two real-size tall buildings with interior and exterior structural systems were chosen from the literature to be optimized by the algorithm and to indicate the algorithm’s performance and capacity in determining optimum solutions. The outrigger and diagrid systems were examples of tall building systems with 40 and 60 stories, respectively. Starting from the optimum design of an outrigger-braced building, GA was presented to be reliable and effective to obtain the optimum solution for this challenging problem of structural engineering. The optimal location of the outrigger system with the belt truss (topology) achieved by the standard GA was reliable compared to those presented in the literature. Especially, for the second example (diagrid structure) with more complexity, variables, and design domain, the accelerated Fuzzy-GA with bi-linear membership function succeeded to reach more optimal weight than those achieved by the standard GA. Also, the obtained diagrid angles, as topology optimization, proved the better performance of accelerated fuzzy-GA. These results represent the merit and capability of the genetic algorithm in the design optimization of real-size tall building systems.

References 1. Applied nature-inspired computing: algorithms and case studies. Springer Singapore (2020); Advancements in applied metaheuristic computing. IGI global (2017) 2. Holland JH (1975) Adaptation in natural and artificial systems. The Uni-versity of Michigan Press, Ann Arbor 3. Eiben AE, Smith JE (2003) Introduction to evolutionary computing. Springer, New York 4. Katoch S, Chauhan SS, Kumar VA (2021) review on genetic algorithm: past, present, and future. Multimed Tools Appl 80:8091–8126 5. AISC (2010) Specification for structural steel buildings, American Institute of Steel Construction. Chicago, Illinois, USA 6. Taranath BS (1998) Steel, concrete, and composite design of tall buildings, 2nd edn. McGrawHill, New York, USA 7. Smith BS, Coull A (1991) Tall building structures. Wiley, New York, USA 8. Lee DK, Kim JH, Starossek U, Shin SM (2012) Evaluation of structural outrigger belt truss layouts for tall buildings by using topology optimization. Struct Eng Mech 43(6):711–724 9. Lee S, Tovar A (2014) Outrigger placement in tall buildings using topology optimization. Eng Struct 74:122–129 10. Taranath BS (1975) Optimum belt truss locations for high rise structures. Struct Eng 53(8):345– 348 11. Zeidabadi NA, Mirtalae K, Mobasher B (2004) Optimized use of the outrigger system to stiffen the couple shear walls in tall buildings. Struct Des Tall Spec 13(1):9–27 12. Zhou Y, Zhang C, Lu X (2015) An inter-story drift-based parameter analysis of the optimal location of outriggers in tall buildings. Struct Design Tall Spec Build 25(5):215–231 13. Ding JM (1991) Optimum belt truss location for high-rise structures and toplevel drift coefficient. J Build Struct 4:10–13

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Chapter 10

Evaluation of Underwater Images Using Genetic Algorithm-Monitored Preprocessing and Morphological Segmentation Venkatesan Rajinikanth and Arulmozhi Rama

1 Introduction A variety of domains are adopting computerized automatic image examination procedures due to rapid advances in science and technology [1, 2]. According to the existing literature, image examination procedures analyze the vital information contained in greyscale/RGB scale images of a chosen dimension, and this analysis is performed using a computer algorithm. The implemented computer algorithms will examine digital image frames using selected pre- and post-processing methodologies, and the results of this assessment will be considered in order to make the appropriate decision. Segmentation involves extracting a region of interest (RoI) from an image and then analyzing it [3]. It is one of the most common image examination procedures. It is typically necessary to compare the RoI extracted from the image frame with the Ground Truth (GT) in order to verify the validity of the proposed scheme. The segmentation of RoI is commonly implemented in various domains and the necessary information regarding this can be accessed from the literature [4–6], Underwater Imaging (UI) is one of the domains related with the deep-sea exploration and these images are captured using the methods, like Autonomous Underwater Vehicle (AUV), Remotely Operated Vehicles (ROV), Human-Occupied Vehicles (HOV), and autonomous vehicles, and the information captured from these devices are then transferred to the ground processing station for further processing [7]. The UI captured in a deep-sea environment is more complex compared to the traditional images and

V. Rajinikanth (B) · A. Rama Department of Computer Science and Engineering, Division of Research and Innovation, Saveetha School of Engineering, SIMATS, Chennai 602105, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dey (ed.), Applied Genetic Algorithm and Its Variants, Springer Tracts in Nature-Inspired Computing, https://doi.org/10.1007/978-981-99-3428-7_10

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these images are associated with an uneven background, poor illumination, noisy image, and unclear/distorted RoI. Several methods of user interface processing have been proposed and implemented by researchers in the literature, each of which aims to extract the ROI more accurately for further investigation. Using a pre-processing and post-processing methodology integrated to extract and evaluate RoI with greater accuracy, the proposed research aims to implement an integrated pre-processing and post-processing methodology. A number of phases are involved in the proposed scheme, including (i) Image collection and resizing, (ii) Pre-processing using Kapur’s tri-level thresholding and MA, (iii) Morphology-based segmentation, and (iv) Comparison of the segmented section with GT. The pre-processing implemented in this approach involves entropy-supported thresholding, which helps in enhancing the RoI by grouping the similar pixels based on the chosen threshold level (Th). After the enhancement, the RoI in the image frame is then mined using the morphology-supported scheme. In this work, the morphological segmentation is executed using (i) Watershed algorithm (WA) and (ii) Markov random field (MRF), and based on the achieved result, the merit of the implemented scheme is confirmed. This work considers the MA to identify the optimal threshold by maximizing Kapur’s Entropy (KE) and this algorithm is a hybrid version of the Particle Swarm Optimization (PSO), Firefly Algorithm (FA), and Genetic Algorithm (GA). The final performance of the MA depends mainly on the GA, which helps to update the whole process with a newer offspring. The implemented MA helps in getting the better-pre-processed RoI, which can be easily mined using the chosen segmentation process. The segmented RoI and the GT is compared and the necessary metrics, such as Jaccard (JA), Dice (DI), accuracy (AC), precision (PR), sensitivity (SE), and specificity (SP) are computed, and based on these values, the eminence of the implemented morphological segmentation scheme is confirmed. This research obtained the images from the UFO-120 benchmark database and the proposed experimental work is implemented using Matlab2022, and this scheme helped to get enhanced JA (>90%), DI (>95%), and AC (>98%) with the implemented approach. The contribution of this scheme includes: . Implementation of KE and MA-based tri-level thresholding to pre-process the UI and . Morphological operation assisted segmentation to effectively extract the RoI from the thresholded image. Other sections of this work are as follows: Sect. 2 presents the literature review, Sect. 3 demonstrates the methodology, and Sects. 4 and 5 show the experimental outcome and the conclusion of the proposed study, respectively.

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2 Literature Review Automatic extraction and evaluation of the RoI are necessary to verify the information available in the chosen digital image. The previous research in the literature authorizes that segmentation is one of the widely accepted procedures for evaluating the necessary information in the image and it can be achieved using manual approach, conventional segmentation, and deep-learning schemes. Examination of the UI is necessary to mine and evaluate the information available from the collected image and this process is quite complex due to the background section, poor illumination, and lesser visibility of the RoI due to the water density and other reasons. The researchers proposed various computer algorithm-based methods to extract the RoI from the UI and the summary of a few chosen procedures which considered the UFO-120 dataset is accessible in Table 1. Table 1 Summary of chosen UI examination methods Reference

Procedure

Fulton et al. [8]

Developed a methodology to process the UI and to monitor the movement of the AUV based on the achieved results.

Islam et al. [9]

This research developed a model based on the image saliency to guide the AUV during deep-sea exploration

Fulton et al. [10]

Implementation of an automatic underwater robot and human communication is achieved based on the image-guided scheme

Mo [11]

A detailed evaluation of UI for simultaneous localization and mapping is presented

Islam et al. [12]

This work implemented super-resolution-based enhancement of UI for enhanced visual perception

Islam et al. [13]

Semantic segmentation execution for the UI using the chosen benchmark database is presented

Islam et al. [14]

Improvement and analysis of the UI based on deep residual multipliers are presented

Huang et al. [15]

Improvement of the visibility of UI based on multiscale cascade transformer with adaptive group attention is presented

Sharma et al. [16]

Enhancement and evaluation of the UI based on deep neural network (DNN) is presented

Haroutunian Development of virtual UI database for the automatic deep-sea inspection is et al. [17] presented Sabbagh and A detailed survey of the UI enhancement procedure is presented. Kelly [18] Vats and Detailed examination of the UI enhancement and segmentation procedures are Patnaik [19] presented and discussed

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3 Methodology The achievement of the automatic image investigation procedure depends on the methodology constructed to examine the chosen digital image. In this work, the RGB-scale UI is considered for the analysis and every image and the GT is resized into 512 × 512 × 3 pixels. The complete procedure implemented in this research work can be found in Fig. 1. Initially, the necessary image and the GT are collected from the UFO-120 database (100 numbers) and then every image is resized to the required level. The RoI in these images is then enhanced using the KE- and MA-based tri-level thresholding (Th = 3) and the thresholded image is then considered for the RoI mining process based on the chosen morphological segmentation. This work implements WS- and MRF-based approaches to extract the RoI from the chosen UI and the mined piece is then matched with the GT to compute the necessary metrics. Based on the computed metrics, the merit of the proposed scheme is verified.

3.1 Database The success of the automatic image examination scheme relies mainly on the database considered for the assessment and in this work, the UFO-120 benchmark UI database is considered. This database is recorded in RGB-scale format and it is available with the related GT. Prior works on this database can be accessed from [20], and in this work, only 100 images are considered and every image is resized to 512 × 512 × 3 pixels before the processing. Figure 2 presents the sample test images and the related GT considered in this study. These images are associated with complex background and hence the extraction of the RoI is complex and needs integrated pre-processing and post-processing methods.

Fig. 1 Proposed scheme for UI examination

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Fig. 2 Sample images of UFO-120 database

3.2 Image Enhancement The earlier works on the image thresholding procedures using approaches, such as Otsu, Kapur, Tsallis, and Shannon, can be found in [21–24]. These methods implemented a tri-level thresholding (Th = 3) to separate the given gray/RGB-scaled digital image frame into RoI, normal region, and the background, and the RoI from the enhanced image is then mined using a chosen segmentation process. The literature confirms that the integration of pre-processing (thresholding) and post-processing (segmentation) helps to achieve a better result. Further, the performance of the segmentation scheme mainly depends on the thresholding outcome; hence, appropriate care must be taken during the pre-processing task [25, 26]. In this work, the pre-processing is achieved using the KE and to find the optimal threshold, it considered the MA, a recent heuristic algorithm developed by integrating the PSO, FA and GA. The necessary information about the KE and MA is already discussed in the literature and hence, this section presents only the overview of the thresholding depicted in Fig. 3. The MA is a recently developed heuristic algorithm by Zervoudakis and Tsafarakis [27] based on the mimicking of the behavior of mayflies [27]. Due to its merit, it is widely adopted to solve a variety of optimization tasks and also for image processing applications. The complete information regarding this algorithm can be found in [28–30]. In this work, the following algorithm values are assigned; number of agents = 25, dimension of search = 3, maximum iterations (Iter max ) = 2000, and (stopping criteria = Iter max ). In this work, the initial population of the GA is obtained from PSO + FA and the GA computes the fitness function to identify the best agent. After the identification, the selection of an agent is achieved using a roulette-wheel scheme [31, 32] and the

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Fig. 3 Flow chart of thresholding process

selected agents are then considered for reproduction using a two-point crossover. Finally, the flip-bit mutation is then applied and the offspring are ranked based on its value and a few offspring are discarded to maintain the actual population of the agents. The necessary information regarding this approach can be found in [33, 34].

3.3 Segmentation The segmentation process is widely adopted to mine the necessary section from the chosen image using manual, conventional, and deep learning schemes. The ultimate goal of the segmentation is to mine the RoI from the image frame with better accuracy. The manual segmentation is commonly not preferred and the operator needs some expertise regarding the software employed to extract the section. The deep learning scheme needs appropriate training, validation, and testing and it can provide a better result when the database size is large. Hence, conventional segmentation scheme is considered in this work to extract the RoI from all the 100 test images of the UFO-120 database. The conventional segmentation procedure helps to implement semi-automatic and automatic extraction methods and the automatic approaches are widely preferred

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to mine the RoI from the UI with better accuracy, This work implemented the morphology-supported segmentation scheme to mine the necessary section from the pre-processed UI. The morphological approach is efficient in identifying the RoI from the image and also helps to effectively extract this section. The procedures, such as WS and MRF are the two schemes, which work based on the concept of morphological segmentation. The necessary information regarding WS and MRF can be accessed from the literature and both approaches help to mine the RoI automatically with better accuracy. The WS working is as follows; initial watershed generation, morphological dilation, and morphology-based enhancement of the RoI and segmentation of the RoI. This scheme needs edge detection combined with segmentation. The Canny edge detection is considered and is then integrated with the morphological operation to mine the section [35, 36]. The MRF approach will separate the chosen image into various sections and presents the outcome as the background, normal section, and the RoI. Both these procedures work well on the gray scale image and hence, after the pre-processing, each image needs to be converted into gray using RGB to gray conversion. Other related information regarding MRF can be found in [37, 38].

3.4 Performance Evaluation and Validation After implementing a suitable computer algorithm to assess the digital images, it is necessary to compute few recommended metrics to verify the eminence of the method. In this work, the comparison of the extracted RoI (binary image) and the GT (binary image) helps to get the necessary parameters, such as True Positive (TP), True Negative (TN), False Positive (FP), and False Negative (FN). In the chosen image, the actual RoI pixels are considered as binary 1 (TP) and the background image pixels are considered as binary 0 (TN). During the comparison, the wrongly identified binary 0/1 helps to get the FN and FP values and these parameters are then considered to identify the necessary quality metrics sown in Eqs. (1)–(6) and based on these values, the merit of the proposed technique is verified [39, 40]. Jaccard = JA = Dice = DI = Accuracy = AC =

TP TP + FP + FN

(1)

2TP 2TP + FP + FN

(2)

TP + TN TP + TN + FP + FN

(3)

TP TP + FP

(4)

Precision = PR =

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Sensitivity = SE =

TP TP + FN

(5)

Specificit y = SP =

TN TN + FP

(6)

4 Result and Discussions This part of the chapter shows results achieved using MATLAB simulation and its discussions. The investigational process is implemented using 100 images of the UFP-120 database images and the average outcome achieved is considered to verify the merit of the scheme. This work initially implements the KE + MA supported thresholding (Th = 3), which separates the image into three regions. In this work, the RGB-scaled image is considered for the analysis and the result of the sample image is depicted in Fig. 4. Figure 4a shows the original and the pre-processed image and Fig. 4b presents the related histogram values. The original image histogram consists of a major pixel distribution compared to the image treated with Th = 3. The thresholded image is then considered for extracting the RoI. A similar procedure is then implemented for all other considered images and the outcome is then considered to mine the RoI using the morphological segmentation scheme. The result of this pre-processing confirms that the implemented KE + MA thresholding completely improves the image section by eliminating the background pixel. This image is then considered for the segmentation process using morphological segmentation. The proposed research implements the WS- and MRF-based methods to extract the RoI and these two methods work with the help of the morphological enhancement scheme. The WS comes under the automatic segmentation process in which the necessary RoI is mined based on the chosen marker value, which must be assigned by the operator during the initiation. The outcome of the WS is shown in Fig. 5 in which, Fig. 5a presents the initial watershed, Fig. 5b, c presents the morphological enhancements and Fig. 5d shows the segmented outcome. The binary form of image Fig. 5d is chosen as the final RoI and it is then compared and verified against the GT. An MRF-based segmentation method is implemented in this study to extract the segment needed from preprocessed images, which are then analyzed further. Figure 6 presents the outcome of the proposed scheme where Fig. 6a illustrates the energy calculated during Expectation Maximization (EM), Fig. 6b illustrates the morphological enhancement, and Fig. 6c, d illustrate the outcomes. To calculate the essential image metrics, we compare the final ROI with the GT after morphological closing is implemented (Fig. 6d).

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Fig. 4 Results achieved with the KE + MA thresholding process

To justify the merit of the proposed technique, a pixel-wise comparison of the GT and RoI is implemented in order to compute the essential image metrics. As shown in Fig. 7a, the GT is shown in Fig. 7a–c are the ROIs. Based on these images, Table 2 presents the metrics computed.

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Fig. 5 Results of WS scheme

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Fig. 6 Results of MRF scheme

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Fig. 7 Sample results obtained in this research

Table 2 Performance metrics computed Scheme WS

MRF

Image

JA

DI

AC

PR

SE

SP

1

93.5027

96.6422

98.8426

94.4738

98.9126

98.8285

2

92.4832

96.0948

98.6637

96.9541

95.2506

99.3757

3

94.1489

96.9863

98.4447

96.1892

97.7967

98.6676

Average

93.0364

95.3964

98.0352

95.8363

96.0893

98.1073

1

93.9356

96.8730

98.9166

95.1856

98.6213

98.9772

2

94.1149

96.9683

98.9784

96.3422

97.6025

99.2550

3

94.3831

97.1104

98.5088

96.3094

97.9247

98.7097

Average

92.9465

95.0735

98.0183

95.3864

95.8934

98.0835

5 Conclusion The proposed work developed an integrated pre- and post-processing methodologies to extract and evaluate the RoI from the UI with better accuracy. Initially, a tri-level thresholding is implemented using KE + MA to group the pixels and then this image

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is considered for morphological-based segmentation. In this work, the performance of WS and the MRF is verified and the achieved result of this scheme is compared with the GT to confirm the merit of the proposed scheme. The result of this study confirms that both these methods help to get a better outcome and, hence, it is confirmed that the morphology-supported segmentation offers better result when the complex UI is considered. In future, the performance of the proposed scheme can be verified using the Tsallis entropy and Shannon’s technique.

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