ABSTRACT Real coded genetic algorithms (RCGAs) are among the most versatile metaheuristic algorithms used to determine the global optimal solutions for nonlinear optimization problems, aimed at solving real‐life complex optimization problems. After the selection operator, the crossover and mutation operators are two crucial strategies upon which the performance of a genetic algorithm (GA) depends. At present, various types of crossover and mutation operators are available in the literature. Therefore, this study attempts to present a state‐of‐the‐art review on real coded crossover operators and real coded mutation operators. Each operator is explained with the help of examples. The objective of this state‐of‐the‐art paper is to serve as a foundation for researchers who wish to design new real coded crossover operators or mutation operators. Moreover, to evaluate the effectiveness of various variants of RCGA, 23 classical benchmark problems are utilised, offering insights into their performance across different optimization problems.

Real Coded Genetic Algorithms (RCGAs) are pop-ular, versatile, non-traditional optimization techniques and can be applicable to a numerous variety of optimization problems. The performance of RCGAs depends on the choice of crossover and mutation operators as well as the selection operator. In order to improve the performance of RCGAs, it is necessary to develop new real-coded crossover operators and new real-coded mutation operators. In this research paper, a new real-coded mutation operator is proposed named Beta Mutation. The CEC 2017 and CEC 2022 benchmark problem sets are used to evaluate its performance. The results are compared with existing and well-known RCGAs. For evaluating its performance, the proposed mutation operator is compared with similar variants of RCGAs. Based on mean, standard deviation, best, worst of the objective function values, and Friedman's mean rank test. It is concluded that the proposed Beta Mutation operator outperforms the other mutation operator under investigation.

High-dimensional optimization problems present a significant challenge in various scientific and engineering domains due to their complexity and the exponential increase in the search space. Traditional optimization algorithms often struggle to balance exploration and exploitation efficiently in such settings. To address this challenge, Reinforcement Learning (RL) is integrated with metaheuristic algorithms in this paper. The proposed framework dynamically selects among Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Artificial Bee Colony (ABC) algorithms based on their performance history. The RL agent is trained via a Q- Learning algorithm with dynamically allocated rewards to ensure a fair evaluation of the improvements in objective values. It determines the most suitable algorithm to apply in each iteration, adapting its strategy as the optimization progresses. QL-H(GDPA) is evaluated on five widely recognized high-dimensional benchmark functions using statistical analyses such as the Friedman test and the Nemenyi post hoc test. The experimental results demonstrate the superior performance of QL-H(GDPA) over individual algorithms, highlighting its effectiveness in high-dimensional optimization. The adaptive nature of the algorithm selection process allows for more effective navigation through complex solution spaces, particularly in high-dimensional contexts. The study underscores the potential of RL in improving optimization strategies and opens avenues for more intelligent and adaptable optimization frameworks in high-dimensional scenarios.

Real-world optimisation problems that are not endowed in mathematical characteristics like differentiability, convexity etc. require non-traditional optimisation approaches that explore the promising regions of the search space stochastically to achieve the optima of the problem. Grey Wolf Optimizer (GWO) is one of the efficient and recently developed approaches in the area of Swarm Intelligence to solve real-world optimisation problems over continuous space. However, in some cases, due to the insufficient diversity, GWO still suffers from the problem of stagnation in local optimums. Therefore, this article presents the novel algorithm OCS-GWO that enhances the performance of original GWO by introducing the opposition-based learning to approximate the closer search candidate solution to the global optima and chaotic local search for the exploitation of the search regions efficiently. In OCS-GWO, a chaotic local search is used for balancing the exploration and exploitation operators that are the underlying features of any stochastic search algorithm. The performance of the proposed algorithm OCS-GWO has been evaluated on a set of 23 standard benchmark test problems and on three engineering application problems – gear train, cantilever beam and speed reducer design problems. The experimental results on test problems and engineering applications confirm the efficiency and reliability of the proposed algorithm over original GWO.

Abstract Swarm intelligence is one of the most promising area of numerical optimization to solve real‐world optimization problems. Grey wolf optimizer (GWO), which is based on leadership hierarchy of grey wolves, is one of the relatively new algorithm in the field of swarm intelligence–based algorithms. In order to solve constrained real‐world optimization problems, in this paper, a constrained version of GWO has been proposed by incorporating a simple constraint handling technique in GWO, and then an attempt is made to improve the ability of the leaders in original GWO by proposing random walk GWO (RW‐GWO) by pointing out some drawbacks in their process of searching prey. (To the best of the knowledge of the authors, a constrained version of GWO has not been developed yet. The unconstrained version of RW‐GWO has been proposed in the authors' earlier work.) The efficiency of both these proposed algorithms have been tested on the Institute of Electrical and Electronics Engineers Congress on Evolutionary Computation 2006 benchmark problems and on 3 engineering application problems to observe their comparative performance. It is concluded from the results that the proposed improved version of GWO, namely, RW‐GWO, has better potential to solve these constraint problems compared to GWO very efficiently as a constrained optimizer.

Grey Wolf Optimiser (GWO) is a recently developed optimisation approach to solve complex non-linear optimisation problems. It is relatively simple and leadership-hierarchy based approach in the class of Swarm Intelligence based algorithms. For solving complex real-world non-linear optimisation problems, the search equation provided in GWO is not of sufficient explorative behaviour. Therefore, in the present paper, an attempt has been made to increase the exploration capability along with the exploitation of a search space by proposing an improved version of classical GWO. The proposed algorithm is named as Cauchy-GWO. In Cauchy-GWO Cauchy operator has been integrated in which first two new wolves are generated with the help of Cauchy distributed random numbers and then another new wolf is generated by taking the convex combination of these new wolves. The performance of Cauchy-GWO is exhibited on standard IEEE CEC 2014 benchmark problem set. Statistical analysis of the results on CEC 2014 benchmark set and popular evaluation criteria, Performance Index (PI) proves that Cauchy-GWO outperforms GWO in terms of error values defined in IEEE CEC 2014 benchmarks collection. Later on in the paper, GWO and Cauchy-GWO algorithms have been used to solve three well-known engineering application problems and two problems of reliability. From the analysis conducted in the present paper, it can be concluded that the proposed algorithm, Cauchy-GWO is reliable and efficient algorithm to solve continuous benchmark test problems, as well as real-life applications problems.

Abstract Harmony search (HS) algorithm is inspired by the music improvisation process in which a musician searches for the best harmony and continues to polish the harmony to improve its aesthetics. The efficiency of evolutionary algorithms depends on the extent of balance between diversification and intensification during the course of the search. An ideal evolutionary algorithm must have efficient exploration in the beginning and enhanced exploitation toward the end. In this paper, a two‐phase harmony search (TPHS) algorithm is proposed that attempts to strike a balance between exploration and exploitation by concentrating on diversification in the first phase using catastrophic mutation and then switches to intensification using local search in the second phase. The performance of TPHS is analyzed and compared with 4 state‐of‐the‐art HS variants on all the 30 IEEE CEC 2014 benchmark functions. The numerical results demonstrate the superiority of the proposed TPHS algorithm in terms of accuracy, particularly on multimodal functions when compared with other state‐of‐the‐art HS variants; further comparison with state‐of‐the‐art evolutionary algorithms reveals excellent performance of TPHS on composition functions. Composition functions are combined, rotated, shifted, and biased version of other unimodal and multimodal test functions and mimic the difficulties of real search spaces by providing a massive number of local optima and different shapes for different regions of the search space. The performance of the TPHS algorithm is also evaluated on a real‐life problem from the field of computer vision called camera calibration problem, ie, a 12‐dimensional highly nonlinear optimization problem with several local optima.

Spider monkey optimization (SMO) algorithm, which simulates the food searching behavior of a swarm of spider monkeys, is a new addition to the class of swarm intelligent techniques for solving unconstrained optimization problems. The purpose of this article is to study the performance of SMO after incorporating quadratic approximation (QA) operator in it. The proposed version is named as QA‐based spider monkey optimization (QASMO). An experimental study has been carried out to check the validity and applicability of QASMO. For validation purpose, the performance of QASMO is tested over a benchmark set of 46 scalable and nonscalable problems, and results are compared with the original SMO algorithm. In order to test the applicability of the proposed algorithm in solving real‐life optimization problems, one of the most challenging optimization problems, namely, Lennard–Jones (LJ) problem is considered. LJ clusters containing atoms from three to ten have been taken into consideration, and results are presented. To the best of our knowledge, this is the first attempt to apply SMO and its proposed variant on a real‐life problem. The results demonstrate that incorporation of QA in SMO has positive effects on its performance in terms of reliability, efficiency, and accuracy.

Many real-world and engineering design problems can be formulated as constrained optimisation problems (COPs). Swarm intelligence techniques are a good approach to solve COPs. In this paper an efficient shrinking hypersphere-based particle swarm optimisation (SHPSO) algorithm is proposed for constrained optimisation. The proposed SHPSO is designed in such a way that the movement of the particle is set to move under the influence of shrinking hyperspheres. A parameter-free approach is used to handle the constraints. The performance of the SHPSO is compared against the state-of-the-art algorithms for a set of 24 benchmark problems. An exhaustive comparison of the results is provided statistically as well as graphically. Moreover three engineering design problems namely welded beam design, compressed string design and pressure vessel design problems are solved using SHPSO and the results are compared with the state-of-the-art algorithms.

Particle Swarm Optimization (PSO), classified as a swarm intelligence technique, mimics the well-informed swarming behavior of social species. A simple and effective searching strategy declares PSO as a potential member for solving various optimization problems. The present study embeds the concept of chaos at different stages of PSO, intending to enhance the convergence speed while trying to avoid stagnation and maintaining the solution quality. The proposed PSO variant is termed as “Totally disturbed PSO (TDPSO)”. The algorithm starts with a disturbed (chaotic) population, generated by considered chaotic system. Thereafter, when a certain number of iterations have elapsed and the searching process approaches equilibrium state, a relative velocity index is calculated for each particle to evaluate its present state and to decide whether or not the particle needs perturbation. The efficacy of proposed algorithm is tested against a set of benchmark problems and results are compared with existing Chaotic PSO and a standard PSO variant. Numerical results manifest that TDPSO works better over considered existing variants by effectively enhancing the searching capability and precision as well.

This paper proposes a new binary PSO for solving multi-task selection problem concerning various issues such as Part mix, Tool allocation and Process plans in CNC machining centers. The mathematical formulation of considered selection problem emerges as highly constrained and 0-1, combinatorial optimization, which further belongs to the category of NP-hard problems. The proposed Binary PSO variant embeds a new sigmoid function namely “Gompertz function” as a binary number generator with an additional benefit of controlling its parameters so as to induce the combined effect of sigmoid as well as linear function. The corresponding variant is termed as “Gompertz Binary Particle Swarm Optimization (GBPSO)”. Before applying GBPSO for considered selection problem, the efficacy of proposed GBPSO is tested on a set of 0-1 Multi-dimensional knapsack problems and results are compared with standard binary PSO. Thereafter two test cases for considered optimal selection problem are solved and analyzed using GBPSO. The simulation results manifest the superiority of proposed variant over standard BPSO for solving benchmark problems and practical application as well.

Particle Swarm Optimization (PSO) is a relatively recent swarm intelligence algorithm inspired from social learning of animals. Successful implementation of PSO depends on many parameters. Inertia weight is one of them. The selection of an appropriate strategy for varying inertia weight w is one of the most effective ways of improving the performance of PSO. Most of the works done till date for investigating inertia weight have considered small values of w, generally in the range [0,1]. This paper presents some experiments with widely varying values of w which adapts itself according to improvement in fitness at each iteration. The same strategy has been implemented in two different ways giving rise to two inertia weight variants of PSO namely Globally Adaptive Inertia Weight (GAIW) PSO, and Locally Adaptive Inertia Weight (LAIW) PSO. The performance of the proposed variants has been compared with three existing inertia weight variants of PSO employing a test suite of 6 benchmark global optimization problems. The experiments show that the results obtained by the proposed variants are comparable with those obtained by the existing ones but with better convergence speed and less computational effort.

In the present work, we propose a new variant of basic DE algorithm called CMDE-G which uses Cauchy mutation (CM) operator. In this algorithm, at the end of every generation, CM is applied as a local search mechanism to explore the neighborhood of the best individual in the population. The performance of CMDE-G algorithm is analyzed on a set of 10 standard benchmark problems and four nontraditional composite functions. Simulation results show that the proposed algorithm helps in improving the solution quality besides maintaining a good convergence rate.

Particle swarm optimization (PSO) has been extensively used in recent years for the optimization of nonlinear optimization problems. Two of the most popular variants of PSO are PSO-W (PSO with inertia weight) and PSO-C (PSO with constriction factor). Typically particles in swarm use information from global best performing particle, gbest and their own personal best, pbest. Recently, studies have focused on incorporating influences of other particles other than gbest. In this paper, we develop a methodology to share information between two particles using a Laplacian operator designed from Laplace probability density function. The properties of this operator are analyzed. Two particles share their positional information in the search space and a new particle is formed. The particle, called as Laplacian particle, replaces the worst performing particle in the swarm. Using this new operator, this paper introduces two algorithms namely Laplace Crossover PSO with inertia weight (LXPSO-W) and Laplace Crossover PSO with constriction factor (LXPSO-C). The performance of the newly designed algorithms is evaluated with respect to PSO-W and PSO-C using 15 benchmark test problems. The empirical results show that the new approach improves performance measured in terms of efficiency, reliability and robustness.

An important problem in electrical engineering is to determine the optimal directional overcurrent relay times. The problem is modeled as a constrained nonlinear optimization problem in which the decision variables are the devices that control the act of isolation of faulty lines from the system without disturbing the healthy lines. Two models are considered namely IEEE-3 bus system and IEEE-4 bus system. The problem is solved using different versions of particle swarm optimization (PSO). The results obtained by PSO are compared with the results available in the literature. It is shown that PSO is able to provide superior results in terms of optimality and reliability in comparison to other methods.

This paper presents a new Self Organizing Migrating Genetic Algorithm (SOMGA) for function optimization, which is inspired by the features of Self Organizing Migrating Algorithm (SOMA). The uniqueness of this algorithm is that it is hybridization of binary coded GA and real coded SOMA. We compare its performance to Simple Genetic Algorithm (GA) and SOMA on 25 test functions. This algorithm is shown to be far more robust than GA and SOMA, providing fast convergence across a broad range of parameter settings.