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This paper addresses optimization problems subject to an objective function's evaluation‐number constraint arbitrarily given in advance. These problems are practical for black‐box objective functions which cost much money and/or time per evaluation because money and/or time constraints are typically applicable in real‐world projects. This paper suggests a new solution strategy that can be used to adapt metaheuristics to evaluation‐number constraints arbitrarily given in advance. The proposed strategy updates the setting parameters of a target metaheuristic algorithm for every generation by solving a fixed small‐size optimization problem related to the evaluation‐number constraint. It has the advantage of being insensitive to the effects of dimensionality increases in principle. The effectiveness of the proposed strategy is confirmed by numerical experiments with two well‐known metaheuristics, i.e., the particle swarm optimization algorithm and the differential evolution algorithm. © 2024 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

The prospect of the energy transition is exciting and sure to benefit multiple aspects of daily life. However, various challenges, such as planning, business models, and energy access are still being tackled. Energy Communities have been gaining traction in the energy transition, as they promote increased integration of Renewable Energy Sources (RESs) and more active participation from the consumers. However, optimization becomes crucial to support decision making and the quality of service for the effective functioning of Energy Communities. Optimization in the context of Energy Communities has been explored in the literature, with increasing attention to metaheuristic approaches. This paper contributes to the ongoing body of work by presenting the results of a benchmark between three classical metaheuristic methods—Differential Evolution (DE), the Genetic Algorithm (GA), and Particle Swarm Optimization (PSO)—and three more recent approaches—the Mountain Gazelle Optimizer (MGO), the Dandelion Optimizer (DO), and the Hybrid Adaptive Differential Evolution with Decay Function (HyDE-DF). Our results show that newer methods, especially the Dandelion Optimizer (DO) and the Hybrid Adaptive Differential Evolution with Decay Function (HyDE-DF), tend to be more competitive in terms of minimizing the objective function. In particular, the Hybrid Adaptive Differential Evolution with Decay Function (HyDE-DF) demonstrated the capacity to obtain extremely competitive results, being on average 3% better than the second-best method while boasting between around 2× and 10× the speed of other methods. These insights become highly valuable in time-sensitive areas, where obtaining results in a shorter amount of time is crucial for maintaining system operational capabilities. [ABSTRACT FROM AUTHOR]

A multi-modal search space can be defined as having multiple attraction basins - each basin has a single local optimum which is reached from all points in that basin when greedy local search is used. Optimization in multi-modal search spaces can then be viewed as a two-phase process. The first phase is exploration in which the most promising attraction basin is identified. The second phase is exploitation in which the best solution (i.e. the local optimum) within the previously identified attraction basin is attained. The goal of thresheld convergence is to improve the performance of search techniques during the first phase of exploration. The effectiveness of thresheld convergence has been demonstrated through applications to existing metaheuristics such as particle swarm optimization and differential evolution, and through the development of novel metaheuristics such as minimum population search and leaders and followers. [ABSTRACT FROM AUTHOR]

To overcome the disadvantages of premature convergence and easy trapping into local optimum solutions, this paper proposes an improved particle swarm optimization algorithm (named NDWPSO algorithm) based on multiple hybrid strategies. Firstly, the elite opposition-based learning method is utilized to initialize the particle position matrix. Secondly, the dynamic inertial weight parameters are given to improve the global search speed in the early iterative phase. Thirdly, a new local optimal jump-out strategy is proposed to overcome the "premature" problem. Finally, the algorithm applies the spiral shrinkage search strategy from the whale optimization algorithm (WOA) and the Differential Evolution (DE) mutation strategy in the later iteration to accelerate the convergence speed. The NDWPSO is further compared with other 8 well-known nature-inspired algorithms (3 PSO variants and 5 other intelligent algorithms) on 23 benchmark test functions and three practical engineering problems. Simulation results prove that the NDWPSO algorithm obtains better results for all 49 sets of data than the other 3 PSO variants. Compared with 5 other intelligent algorithms, the NDWPSO obtains 69.2%, 84.6%, and 84.6% of the best results for the benchmark function ( f 1 - f 13 ) with 3 kinds of dimensional spaces (Dim = 30,50,100) and 80% of the best optimal solutions for 10 fixed-multimodal benchmark functions. Also, the best design solutions are obtained by NDWPSO for all 3 classical practical engineering problems. [ABSTRACT FROM AUTHOR]

To overcome the disadvantages of premature convergence and easy trapping into local optimum solutions, this paper proposes an improved particle swarm optimization algorithm (named NDWPSO algorithm) based on multiple hybrid strategies. Firstly, the elite opposition-based learning method is utilized to initialize the particle position matrix. Secondly, the dynamic inertial weight parameters are given to improve the global search speed in the early iterative phase. Thirdly, a new local optimal jump-out strategy is proposed to overcome the "premature" problem. Finally, the algorithm applies the spiral shrinkage search strategy from the whale optimization algorithm (WOA) and the Differential Evolution (DE) mutation strategy in the later iteration to accelerate the convergence speed. The NDWPSO is further compared with other 8 well-known nature-inspired algorithms (3 PSO variants and 5 other intelligent algorithms) on 23 benchmark test functions and three practical engineering problems. Simulation results prove that the NDWPSO algorithm obtains better results for all 49 sets of data than the other 3 PSO variants. Compared with 5 other intelligent algorithms, the NDWPSO obtains 69.2%, 84.6%, and 84.6% of the best results for the benchmark function ( f 1 - f 13 ) with 3 kinds of dimensional spaces (Dim = 30,50,100) and 80% of the best optimal solutions for 10 fixed-multimodal benchmark functions. Also, the best design solutions are obtained by NDWPSO for all 3 classical practical engineering problems. [ABSTRACT FROM AUTHOR]

The Whale Optimization Algorithm (WOA) is an efficient meta-heuristic algorithm inspired by the feeding behavior of whales. Although it has successfully solved many optimization problems, it still suffers from premature convergence and poor accuracy when solving complex problems. To address these issues, this paper proposes an improved whale optimization algorithm called EIWOA to improve the search efficiency and accuracy. First, we introduce a new global search mechanism and encircling prey strategy in EIWOA, which utilizes differential evolution, and sine-cosine search strategy to improve the global search efficiency and avoid premature convergence. Second, we use a lévy flight-based spiral update position strategy to improve the local search efficiency of the algorithm, thus improving the convergence speed and accuracy. Again, we introduce a balancing factor with fluctuating decay properties into EIWOA to better balance exploration and exploitation. Finally, we introduce a dynamic opposite learning-based whale-fall strategy in EIWOA to equip the algorithm with the ability to jump out of the local optimum. The qualitative analysis of the algorithm shows that the EIWOA algorithm converges quickly, is highly accurate, and has the ability to jump out of the local optimum. In order to validate the performance of the proposed EIWOA algorithm, we evaluated the algorithm on CEC2017 benchmark function and four real world engineering problems. We also conduct a comparative study of the EIWOA algorithm with EWOA, an excellent variant of WOA, as well as some excellent meta-heuristics developed recently. The results of numerical experiments demonstrate the superiority of the proposed EIWOA, which is further confirmed statistically by the results of Friedman's test and Wilcoxon signed rank test. The proposed EIWOA algorithm is significantly better than WOA and other competing algorithms in terms of convergence speed, accuracy and optimization ability in dealing with complex optimization problems. [ABSTRACT FROM AUTHOR]

With the rapid development of the geometric modeling industry and computer technology, the design and shape optimization of complex curve shapes have now become a very important research topic in CAGD. In this paper, the Hybrid Artificial Hummingbird Algorithm (HAHA) is used to optimize complex composite shape-adjustable generalized cubic Ball (CSGC–Ball, for short) curves. Firstly, the Artificial Hummingbird algorithm (AHA), as a newly proposed meta-heuristic algorithm, has the advantages of simple structure and easy implementation and can quickly find the global optimal solution. However, there are still limitations, such as low convergence accuracy and the tendency to fall into local optimization. Therefore, this paper proposes the HAHA based on the original AHA, combined with the elite opposition-based learning strategy, PSO, and Cauchy mutation, to increase the population diversity of the original algorithm, avoid falling into local optimization, and thus improve the accuracy and rate of convergence of the original AHA. Twenty-five benchmark test functions and the CEC 2022 test suite are used to evaluate the overall performance of HAHA, and the experimental results are statistically analyzed using Friedman and Wilkerson rank sum tests. The experimental results show that, compared with other advanced algorithms, HAHA has good competitiveness and practicality. Secondly, in order to better realize the modeling of complex curves in engineering, the CSGC–Ball curves with global and local shape parameters are constructed based on SGC–Ball basis functions. By changing the shape parameters, the whole or local shape of the curves can be adjusted more flexibly. Finally, in order to make the constructed curve have a more ideal shape, the CSGC–Ball curve-shape optimization model is established based on the minimum curve energy value, and the proposed HAHA is used to solve the established shape optimization model. Two representative numerical examples comprehensively verify the effectiveness and superiority of HAHA in solving CSGC–Ball curve-shape optimization problems. [ABSTRACT FROM AUTHOR]

The Internet of Things (IoT) edge is an emerging technology of sensors and devices that communicate real-time data to a network. IoT edge computing was introduced to handle the latency concerns related to cloud computing data management, as the data are processed closer to their point of origin. Clustering and scheduling tasks on IoT edge computing are considered a challenging problem due to the diverse nature of task and resource characteristics. Metaheuristics and optimization methods are widely used in IoT edge task clustering and scheduling. This paper introduced a new task clustering and scheduling mechanism using differential evolution optimization on IoT edge computing. The proposed mechanism aims to optimize task clustering and scheduling to find optimal execution times for submitted tasks. The proposed mechanism for task clustering is based on the degree of similarity of task characteristics. The proposed mechanisms use an evolutionary mechanism to distribute system tasks across suitable IoT edge resources. The clustering tasks process categorizes tasks with similar requirements and then maps them to appropriate resources. To evaluate the proposed differential evolution mechanism for IoT edge task clustering and scheduling, this study conducted several simulation experiments against two established mechanisms: the Firefly Algorithm (FA) and Particle Swarm Optimization (PSO). The simulation configuration was carefully created to mimic real-world IoT edge computing settings to ensure the proposed mechanism's applicability and the simulation results' relevance. In the heavyweight workload scenario, the proposed DE mechanism started with an execution time of 916.61 milliseconds, compared to FA's 1092 milliseconds and PSO's 1026.09 milliseconds. By the 50th iteration, the proposed DE mechanism had reduced its execution time significantly to around 821.27 milliseconds, whereas FA and PSO showed lesser improvements, with FA at approximately 1053.06 milliseconds and PSO stabilizing at 956.12 milliseconds. The simulation results revealed that the proposed differential evolution mechanism for edge task clustering and scheduling outperforms FA and PSO regarding system efficiency and stability, significantly reducing execution time and having minimal variation across simulation iterations. [ABSTRACT FROM AUTHOR]

Socio-cognitive computing is a paradigm developed for the last several years in our research group. It consists of introducing mechanisms inspired by inter-individual learning and cognition into metaheuristics. Different versions of the paradigm have been successfully applied in hybridizing Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Genetic Algorithms, Differential Evolution, and Evolutionary Multi-agent System (EMAS) metaheuristics. In this paper, we have followed our previous experiences in order to propose a novel mutation based on socio-cognitive mechanism and test it based on Evolution Strategy (ES). The newly constructed versions were applied to popular benchmarks and compared with their reference versions. [ABSTRACT FROM AUTHOR]

The sparse recovery (SR) space-time adaptive processing (STAP) method has excellent clutter suppression performance under the condition of limited observation samples. However, when the cluttering is nonlinear in a spatial-Doppler profile, it will cause an off-grid effect and reduce the sparse recovery performance. A meshless search using a meta-heuristic algorithm (MH) can completely eliminate the off-grid effect in theory. Therefore, genetic algorithm (GA), differential evolution (DE), particle swarm optimization (PSO), and grey wolf optimization (GWO) methods are applied to SR-STAP for selecting exact clutter atoms in this paper. The simulation results show that MH-STAP can estimate the clutter subspace more accurately than the traditional algorithm; PSO-STAP and GWO-STAP showed better clutter suppression performance in four MH-STAP methods. To search for more accurate clutter atoms, PSO and GWO are combined to improve the method's capacity for global optimization. Meanwhile, the fitness function is improved by using prior knowledge of the clutter distribution. The simulation results show that the improved PSO-GWO-STAP algorithm provides excellent clutter suppression performance, which solves the off-grid problem better than does single MH-STAP. [ABSTRACT FROM AUTHOR]

This paper represents the execution of a reliable, steady and wideband fractional order digital integrator (FODI) in respect of infinite impulse response (IIR) filter using a recent nature inspired meta-heuristic optimization technique known as mayfly optimization algorithm (MOA). MOA significantly functions stronger against some of the popular algorithms like real-coded genetic algorithm (RGA), particle swarm optimization (PSO), differential evolution (DE), improved particle swarm optimization (IPSO), whale optimization algorithm (WOA) and chaotic oppositional whale optimization algorithm (COWOA) with reference to different magnitude error performances, explanation character accuracy, convergence profile and computational time required to find the optimal solution. In this paper, the authors thoroughly analyze the convergence behavior and magnitude error metrices of the different order FODI's using WOA, COWOA and MOA. MATLAB simulation results also powerfully uphold the magnitude response efficiency and stability of the proposed different order FODI's over some recently published research papers. [ABSTRACT FROM AUTHOR]

This paper presents an overview of the history and recent efforts in combining chaos theory and evolutionary computation techniques. Various algorithms from the evolutionary computation domain, also known as metaheuristic algorithms, have been successfully enhanced with chaotic components in the past. Numerous ways to incorporate chaos have been examined, and many impressive results have been reported. Implementations of discrete chaotic maps such as Lozi, Hénon, and logistic map as generators of chaotic pseudo-random sequences for controlling evolution operators in metaheuristics have achieved significant popularity. In this survey, we focus on the research field itself and perform a bibliographical analysis to show how broad and active is nowadays the research field of chaos-enhanced metaheuristics and what are some of the most recent works published. [ABSTRACT FROM AUTHOR]

In this paper, three metaheuristic optimization algorithms: genetic algorithm (GA), particle swarm optimization (PSO), and differential evolution (DE) are compared in terms of minimizing the total owning cost (TOC) of the active part of a three-phase shell-type distribution transformer. The three methods use six inputs: power rating, primary voltage, secondary voltage, primary and secondary winding connections, and frequency. The TOC of the transformer, which includes the cost of the basic materials of the transformer plus the cost of losses, is minimized under the imposed constraints (excitation current, impedance, no-load losses, load losses, and efficiency) usually specified in the standards. As a case study, the three algorithms are applied to optimize the design of a three-phase shell-type distribution transformer of 750 kVA. All applied metaheuristic algorithms provide good results, while DE avoids local optima leading to better TOC reduction. The results of the optimization algorithms used are superior to those of the manufacturer, showing a 6% TOC reduction. Optimization of the design of a power transformer may have important implications for reducing greenhouse gas emissions and extending the lifetime of the equipment. [ABSTRACT FROM AUTHOR]

Grasshopper optimization algorithm (GOA) is a meta-heuristic algorithm for solving optimization problems by modeling the biological habit and social behavior of grasshopper swarms in nature. Compared with other optimization algorithms, GOA still has room to improve its performance on solving complex problems. Therefore, this paper proposes an improved grasshopper optimization algorithm (EMGOA) based on dynamic dual elite learning and sinusoidal mutation. First of all, dynamic elite learning strategy is adopted to improve the influence of elites on the update process, enabling the algorithm to have a faster convergence speed. Then, sinusoidal function is utilized to guide the mutation of the current global optimal individual during each iteration to avoid the algorithm falling into the local optimum and improve the convergence accuracy of the algorithm. In order to investigate the performance of the proposed EMGOA algorithm, experiments are conducted on 26 benchmark functions and CEC2019 in this paper. The experimental results show that the optimization performance of EMGOA is obviously better than GOA, and EMGOA is competitive with six state-of-the-art meta-heuristic optimization algorithms. [ABSTRACT FROM AUTHOR]

Identifying parameters in photovoltaic (PV) cell and module models is one of the primary challenges of the simulation and design of photovoltaic systems. Metaheuristic algorithms can find near-optimal solutions within a reasonable time for such challenging real-world optimization problems. Control parameters must be adjusted with many existing algorithms, making them difficult to use. In real-world problems, many of these algorithms must be combined or hybridized, which results in more complex and time-consuming algorithms. This paper presents a new artificial parameter-less optimization algorithm (APLO) for parameter estimation of PV models. New mutation operators are designed in the proposed algorithm. APLO's exploitation phase is enhanced by each individual searching for the best solution in this updating operator. Moreover, the current best, the old best, and the individual's current position are utilized in the differential term of the mutation operator to assist the exploration phase and control the convergence speed. The algorithm uses a random step length based on a normal distribution to ensure population diversity. We present the results of a comparative study using APLO and well-known existing parameter-less meta-heuristic algorithms such as grey wolf optimization, the salp swarm algorithm, JAYA, teaching-learning based optimization, colliding body optimization, as well as three major parameter-based algorithms such as differential evolution, genetic algorithm, and particle swarm optimization to estimate the parameters of PV the modules. The results revealed that the proposed algorithm could provide excellent exploration–exploitation balance and consistency during the iterations. Furthermore, the APLO algorithm shows high reliability and accuracy in identifying the parameters of PV cell models. [ABSTRACT FROM AUTHOR]

A biomedical application of a novel metaheuristic optimizer is proposed in this paper by constructing an enhanced arithmetic optimization algorithm (AOA). The latter algorithm was constructed using the logarithmic spiral (Ls) search mechanism from the whale optimization algorithm and the greedy selection scheme from the differential evolution algorithm. The proposed algorithm (Ls-AOA) was tested against unimodal and multimodal benchmark functions and demonstrated better capability comparatively using other efficient metaheuristic algorithms reported in the literature. The constructed Ls-AOA algorithm was then proposed to design a proportional-integral-derivative (PID) controller employed in a functional electrical stimulation (FES) system for the first time. The initial statistical and convergence profile assessment showed better performance of the proposed algorithm. The comparative analyses for transient and frequency responses were performed for the PID-controlled FES system using the original AOA, sine–cosine and particle swarm optimization algorithms and the traditional Ziegler-Nichols tuning scheme. Similarly, the FES system tuned with the latter methods was also assessed for disturbance rejection and noise elimination. All the comparative analyses demonstrated that the proposed Ls-AOA has the greater capability for the challenging biomedical FES system. [ABSTRACT FROM AUTHOR]

Mobile robots are relevant dynamic systems in recent applications. Path planning is an essential task for these robots since it allows them to move from one location to another safely and at an affordable cost. Path planning has been studied extensively for static scenarios. However, when the scenarios are dynamic, research is limited due to the complexity and high cost of continuously re-planning the robot's movements to ensure its safety. This paper proposes a new, simple, reliable, and affordable method to plan safe and optimized paths for differential mobile robots in dynamic scenarios. The method is based on the online re-optimization of the static parameters in the state-of-the-art deterministic path planner Bug0. Due to the complexity of the dynamic path planning problem, a metaheuristic optimization approach is adopted. This approach utilizes metaheuristics from evolutionary computation and swarm intelligence to find the Bug0 parameters when the mobile robot is approaching an obstacle. The proposal is tested in simulation, and well-known metaheuristic methods are compared, including Differential Evolution (DE), the Genetic Algorithm (GA), and Particle Swarm Optimization (PSO). The dynamic planner based on PSO generates paths with the best performances. In addition, the results of the PSO-based planner are compared with different Bug0 configurations, and the former is shown to be significantly better. [ABSTRACT FROM AUTHOR]

The quality of operation of neural networks in solving application problems is determined by the success of the stage of their training. The task of learning neural networks is a complex optimization task. Traditional learning algorithms have a number of disadvantages, such as «sticking» in local minimums and a low convergence rate. Modern approaches are based on solving the problems of adjusting the weights of neural networks using metaheuristic algorithms. Therefore, the problem of selecting the optimal set of values of algorithm parameters is important for solving application problems with symmetry properties. This paper studies the application of a new metaheuristic optimization algorithm for weights adjustment—the algorithm of the spiders-cycle, developed by the authors of this article. The approbation of the proposed approach is carried out to adjust the weights of recurrent neural networks used to solve the time series forecasting problem on the example of three different datasets. The results are compared with the results of neural networks trained by the algorithm of the reverse propagation of the error, as well as three other metaheuristic algorithms: particle swarm optimization, bats, and differential evolution. As performance criteria for the comparison of algorithms of global optimization, in this work, descriptive statistics for metrics of the estimation of quality of predictive models, as well as the number of calculations of the target function, are used. The values of the MSE and MAE metrics on the studied datasets were obtained by adjusting the weights of the neural networks using the cycling spider algorithm at 1.32, 25.48, 8.34 and 0.38, 2.18, 1.36, respectively. Compared to the inverse error propagation algorithm, the cycling spider algorithm reduced the value of the error metrics. According to the results of the study, it is concluded that the developed algorithm showed high results and, in the assessment of performance, was not inferior to the existing algorithm. [ABSTRACT FROM AUTHOR]

With the rapid development of the Internet of Things (IoT), the curse of dimensionality becomes increasingly common. Feature selection (FS) is to eliminate irrelevant and redundant features in the datasets. Particle swarm optimization (PSO) is an efficient metaheuristic algorithm that has been successfully applied to obtain the optimal feature subset with essential information in an acceptable time. However, it is easy to fall into the local optima when dealing with high-dimensional datasets due to constant parameter values and insufficient population diversity. In the paper, an FS method is proposed by utilizing adaptive PSO with leadership learning (APSOLL). An adaptive updating strategy for parameters is used to replace the constant parameters, and the leadership learning strategy is utilized to provide valid population diversity. Experimental results on 10 UCI datasets show that APSOLL has better exploration and exploitation capabilities through comparison with PSO, grey wolf optimizer (GWO), Harris hawks optimization (HHO), flower pollination algorithm (FPA), salp swarm algorithm (SSA), linear PSO (LPSO), and hybrid PSO and differential evolution (HPSO-DE). Moreover, less than 8% of features in the original datasets are selected on average, and the feature subsets are more effective in most cases compared to those generated by 6 traditional FS methods (analysis of variance (ANOVA), Chi-Squared (CHI2), Pearson, Spearman, Kendall, and Mutual Information (MI)). [ABSTRACT FROM AUTHOR]

Artificial neural networks have achieved a great success in simulating the information processing mechanism and process of neuron supervised learning, such as classification. However, traditional artificial neurons still have many problems such as slow and difficult training. This paper proposes a new dendrite neuron model (DNM), which combines metaheuristic algorithm and dendrite neuron model effectively. Eight learning algorithms including traditional backpropagation, classic evolutionary algorithms such as biogeography-based optimization, particle swarm optimization, genetic algorithm, population-based incremental learning, competitive swarm optimization, differential evolution, and state-of-the-art jSO algorithm are used for training of dendritic neuron model. The optimal combination of user-defined parameters of model has been systemically investigated, and four different datasets involving classification problem are investigated using proposed DNM. Compared with common machine learning methods such as decision tree, support vector machine, k-nearest neighbor, and artificial neural networks, dendritic neuron model trained by biogeography-based optimization has significant advantages. It has the characteristics of simple structure and low cost and can be used as a neuron model to solve practical problems with a high precision. [ABSTRACT FROM AUTHOR]

This paper evaluates the performance and suitability of four different metaheuristic algorithms for optimal sizing of standalone microgrids in remote area. The studied metaheuristic algorithms are particle swarm optimization, differential evolution, water cycle algorithm and grey wolf optimization. These algorithms are applied to optimize the capacity of diesel generator, fuel tank, solar photovoltaic, wind turbine, and battery energy storage in four different AC-coupled standalone microgrids for a remote area community in South Australia. The objective function is selected as the net present value of electricity over a 20-year lifetime. The optimisation study is conducted based on the real data of annual load consumption, ambient temperature, solar insolation, and wind speed of the site. Capital, replacement, and maintenance costs of components in Australian market are incorporated for the economic analysis. An operating power reserve is maintained based on the static and dynamic reserve concepts. Uncertainty analysis based on 10-year real data of renewable energies and load consumption is conducted. Sensitivity analysis is provided for variations of the battery price and capacity. The performance of the applied algorithms is evaluated by comparing the economic and operational results, as well as the computational time and optimization convergence. It is found that differential evolution algorithm is unreliable for optimal sizing problem of the studied standalone microgrids.. [ABSTRACT FROM AUTHOR]

One of the most promising technologies to be employed in beyond 5G networks is based on the concept of cell-free (CF) massive MIMO, in which a predetermined set of access points (APs) jointly cooperate in the data transmission and reception to/from the user equipment (UE). In order to efficiently manage radio resources, the CF central processing unit implement uplink power control policies. These policies aim to optimize a given network utility function. In this paper, we investigate the max-min fairness optimization problem, in which the spectral efficiency performance of the UE with the worst channel conditions is prioritized, taking into account a per-UE power constraint and assuming linear maximum ratio combining at APs. Existing solutions have typically relied on second-order cone programming with convex approximations, which exhibit high computational complexity and scalability issues. Therefore, meta-heuristics (MHs) are explored as alternative optimization schemes, capable of providing (near)-optimal solutions with reasonable computational effort. Three MH approaches with different operation principles are compared. Numerical results show that the differential evolution algorithm exhibits the best trade-off between solution quality and run time, being able to reach (near)-optimal solutions faster than the bisection approach while coping with the scalability issues of the geometric programming-based algorithm. [ABSTRACT FROM AUTHOR]

Biomedical image analysis based on metaheuristic algorithms is one of the most important research areas encountered in recent years. Due to the low contrast differences between the diseased areas and the image background in high-contrast biomedical images, effective methods are required to diagnose diseases with high accuracy. To overcome the difficulties encountered in this field, metaheuristic approaches may offer effective solutions due to their advantages such as the ability of converging to the global optimum, higher convergence rate, and having few control parameters. In this work, Jellyfish Search (JS), Marine Predators (MPA), Tunicate Swarm (TSA), Mayfly Optimization (MA), Chimp Optimization (ChOA), Slime Mould Optimization (SMA), Archimedes Optimization (AOA), and Equilibrium Optimizer (EO) algorithms, which are the most recently proposed metaheuristic algorithms in the literature, have been improved as clustering based in order to achieve vessel segmentation with high precision. Also, a detailed performance comparison of these algorithms has been realized for the rate of convergences, error values reached, CPU time, standard deviation, sensitivity, specificity, accuracy, F-score, and Wilcoxon rank sum-test. In order to present the compatibility of the results obtained with the literature, the performances of these novel algorithms have also been compared to that of Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), and Differential Evolution (DE) algorithms. The simulation results represent that each algorithm produces similar convergence and error performance. Also, it can be emphasized from the statistical analyses that the stability and robustness of each metaheuristic approach are quite adequate in separating the vessel pixels and the background pixels of a retinal image. In general, this paper proves that although having fewer number of control parameters, the JS, MPA, TSA, MA, ChOA, SMA, AOA, and EO algorithms produce similar but a bit better results in terms of image segmentation when compared to PSO, GWO, and DE algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for optimizing a planar elliptical dipole antenna with elliptical slots, for ultrawideband electromagnetic compatibility applications. The antenna is required to achieve minimum return loss and boresight realized gain with an adequate gain flatness across the frequency range of 1–5 GHz. Such an antenna is a powerful tool for electromagnetic measurements, due to its very compact size and its wide operating bandwidth. The main optimization method used in this paper is the invasive weed optimization (IWO), which is a nature-inspired metaheuristic evolutionary algorithm. The conventional and a modified version of IWO are compared to other prior-art evolutionary algorithms, such as the particle swarm optimization and differential evolution. The comparison is performed by applying all the methods on a set of mathematical test functions and also on the specific antenna design problem presented in this paper. The comparative results demonstrate the superiority of the modified IWO over the other optimization methods. [ABSTRACT FROM AUTHOR]

• New heuristic to solve the supplier selection and order quantity allocation problem. • Best solution compared to others reported in the literature for a known problem. • Better performance compared to Particle Swarm Optimization and Differential Evolution. This paper addresses the so-called supplier-selection and order-quantity allocation problem. Because of the complexity of this of problem (e.g., non-linear, discontinuous cost function), a new heuristic method is proposed and tested. This heuristic method explores the search space in a short period of time to find cheaper solutions. In order to test the efficiency of the proposed heuristic, two metaheuristic algorithms are applied: particle swarm optimization (PSO) and differential evolution (DE). Additionally, two numerical examples were solved. In the first one, it is shown that the proposed heuristic performed best compared to other solutions previously published in the literature, both in terms of computational time and total cost. The complexity analysis found that in the worst case, the proposed heuristic executes on average 99.9% less numerical operations than PSO and DE. In the second numerical example, larger instances were studied. Our findings show that the proposed heuristic was able to find a feasible solution in 15 out 15 instances, while the PSO and the DE algorithms were unable to find a solution in 9 out of 15 instances. Therefore, the proposed heuristic does not just lead to lower total cost solutions, but it also performs a more exhaustive search in shorter computational times for larger instances of the problem. Finally, Wilcoxon and Kruskal-Wallis statistical tests demonstrate significant difference between the proposed heuristic and the PSO and DE; the proposed heuristic presented a lower median in most cases. [ABSTRACT FROM AUTHOR]

In this study, an energy efficiency study has been carried out through the moment values of a quadrotor applied in the position control with the differential evolution algorithm (DE), which is under the evolutionary algorithms among meta-heuristic algorithms particle swarm optimization (PSO) which is among swarm-based algorithms and gravity search algorithm (GSA) among physics-based algorithms and charged system search (CSS) algorithm, which are all well-known in the literature. The efficient use of energy in electrically powered flying mechanisms is important in terms of completing its task. The aim of this study is to obtain proportional–integral–derivative (PID) coefficients that minimize the energy consumption caused by quadrotor moments. First of all, 12 PID coefficients calculated with traditional Ziegler Nicholes, and trying-error methods for the control of the quadrotor have been trained with DE, PSO, GSA and CSS algorithms and their cost values have been found for this problem. In the study, optimum PID coefficients have been found separately with each algorithm and torque data giving energy efficiency have been obtained by applying them to the state equations. Although the cost minimization convergence value obtained from DE does not look good compared to other algorithms, it has been observed that the moment values of the obtained PID coefficients in the control program are lower. In the study, it has been shown that the PID coefficients obtained from DE reach the target with approximately less energy than the maximum-moment value of the PSO, which provides the closest moment value. It can be thought that this value contributes to energy consumption in each orbital movement of the quadrotor. The simulation results show that the coefficients obtained from DE provide better energy efficiency and are superior to other algorithms compared in terms of energy efficiency. • PID coefficients for quadrotor control are optimized with DE,PSO,CSS,GSA algorithms • The torque values to be applied to each motor were adjusted with these coefficients • The total force applied with DEPID in the z direction is smaller than others • DEPID consumes less energy than the other algorithms in the paper [ABSTRACT FROM AUTHOR]

In this study, an energy efficiency study has been carried out through the moment values of a quadrotor applied in the position control with the differential evolution algorithm (DE), which is under the evolutionary algorithms among meta-heuristic algorithms particle swarm optimization (PSO) which is among swarm-based algorithms and gravity search algorithm (GSA) among physics-based algorithms and charged system search (CSS) algorithm, which are all well-known in the literature. The efficient use of energy in electrically powered flying mechanisms is important in terms of completing its task. The aim of this study is to obtain proportional–integral–derivative (PID) coefficients that minimize the energy consumption caused by quadrotor moments. First of all, 12 PID coefficients calculated with traditional Ziegler Nicholes, and trying-error methods for the control of the quadrotor have been trained with DE, PSO, GSA and CSS algorithms and their cost values have been found for this problem. In the study, optimum PID coefficients have been found separately with each algorithm and torque data giving energy efficiency have been obtained by applying them to the state equations. Although the cost minimization convergence value obtained from DE does not look good compared to other algorithms, it has been observed that the moment values of the obtained PID coefficients in the control program are lower. In the study, it has been shown that the PID coefficients obtained from DE reach the target with approximately less energy than the maximum-moment value of the PSO, which provides the closest moment value. It can be thought that this value contributes to energy consumption in each orbital movement of the quadrotor. The simulation results show that the coefficients obtained from DE provide better energy efficiency and are superior to other algorithms compared in terms of energy efficiency. • PID coefficients for quadrotor control are optimized with DE,PSO,CSS,GSA algorithms • The torque values to be applied to each motor were adjusted with these coefficients • The total force applied with DEPID in the z direction is smaller than others • DEPID consumes less energy than the other algorithms in the paper [ABSTRACT FROM AUTHOR]

This paper objective presents static synchronous series compensation FACTS device with biogeography based optimization (BBO) to deal for obtaining worthwhile power flow control. The biogeography based optimization method is utilized to find the optimal fitted child sets by surviving parents with the help of migration and mutation Jaipur process. The present BBO technique from the evolutionary strategy with static synchronous series compensator provides improved outcomes in comparison to other optimization methods. The simulation outcomes illustrate that the proposed BBO algorithm is efficacious, secure and correct to search the optimized values with SSSC based FACTS devices. The proposed method is considering the solution quality an optimistic substitute method for extricating the OPF problems. The simplification and effectiveness of this method are validated on the IEEE 57 bus and 75 bus Systems. In this paper, from the outcome results, it is clearly shown that the proposed technique execution can be properly and effectively applied to the optimal position of multiple OPF problems. i.e. BBO based algorithm with SSSC FACTS device is found better results when compared to without SSSC device in all aspects. [ABSTRACT FROM AUTHOR]

In this paper, we consider the contention aware task scheduling problem on a grid topology of processors. By contention awareness, we mean that simultaneous communication on a link has to be serialized. To solve this problem, we propose several nature inspired metaheuristic algorithms: Simulated Annealing (SA), Genetic Algorithm (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), Bat Algorithm (BA), Cuckoo Search (CS), and Firefly Algorithm (FA). We perform benchmark evaluation of these algorithms for the Normalized Schedule Length (NSL) parameter. The benchmark task graphs that we consider are: random task graphs, peer set task graphs, systolic array task graphs, Gaussian elimination task graphs, divide and conquer task graphs, and fast Fourier transform task graphs. [ABSTRACT FROM AUTHOR]

Water Wave Optimisation algorithm (WWO) is a new swarm-based metaheuristic inspired by shallow wave models for global optimisation. In this paper, an enhanced WWO, which combines with multiple assistant strategies (EWWO), is proposed. First, the random opposition-based learning (ROBL) mechanism is introduced to generate the initial population with high quality. Second, a new modified operation is designed and embedded into propagation operation to balance the global exploration and the local exploitation. Third, the covariance matrix self-adaptation evolution strategy (CMA-ES) is employed by the refraction operation to further strengthen the local exploitation. Furthermore, the diversity of the population is maintained in the evolution process by using a crossover operator. The experiment results based on CEC 2017 benchmarks indicate that the EWWO outperforms the state-of-the-art variant algorithms of the WWO and the standard WWO. [ABSTRACT FROM AUTHOR]

Nature-inspired algorithms take inspiration from living things and imitate their behaviours to accomplish robust systems in engineering and computer science discipline. Symbiotic organisms search (SOS) algorithm is a recent metaheuristic algorithm inspired by symbiotic interaction between organisms in an ecosystem. Organisms develop symbiotic relationships such as mutualism, commensalism, and parasitism for their survival in ecosystem. SOS was introduced to solve continuous benchmark and engineering problems. The SOS has been shown to be robust and has faster convergence speed when compared with genetic algorithm, particle swarm optimization, differential evolution, and artificial bee colony which are the traditional metaheuristic algorithms. The interests of researchers in using SOS for handling optimization problems are increasing day by day, due to its successful application in solving optimization problems in science and engineering fields. Therefore, this paper presents a comprehensive survey of SOS advances and its applications, and this will be of benefit to the researchers engaged in the study of SOS algorithm. [ABSTRACT FROM AUTHOR]

In this paper, six metaheuristic algorithms, in the form of artificial bee colony optimization, ant colony optimization, particle swarm optimization, differential evolution, firefly algorithm and teaching-learning-based optimization techniques are applied for parametric optimization of a multi-pass face milling process. Using those algorithms, the optimal values of cutting speed, feed rate and depth of cut for both roughing and finishing operations are determined for having minimum total production time and total production cost. It is observed that the teaching-learning-based optimization algorithm outperforms the others with respect to accuracy and consistency of the derived solutions as well as computational speed. Two statistical tests, i.e. paired t-test and Wilcoxson signed rank test also confirm its superiority over the remaining algorithms. Finally, these metaheuristics are employed for multi-objective optimization of the considered multi-pass milling process while concurrently minimizing both the objectives. [ABSTRACT FROM AUTHOR]

Success-History-Based Parameter Adaptation Multi-Objective Differential Evolution (SHAMODE) has shown excellent performance recently compared to other multi-objective evolutionary algorithms (MOEAs) and therefore caught the interest of researchers lately. The SHAMODE framework employs historical memory of control parameters, which stores set values of control parameters that have performed well in the past generation and produces new set values of the parameter by directly sampling the parameter space close to one of these stored pairs. Since the algorithm's performance depends on the problem to be optimized, this paper proposes an enhanced version of SHAMODE that is compatible with optimizing the multi-objective wave energy converter problem. The improved version of SHAMODE is achieved by incorporating competing machine learning algorithms, Particle Swarm Optimization, and a modified bubble-net attacking method formula of the Whale optimization algorithm from its original form. Simulations and comparisons are used to verify the proposed method's performance based on some famous multi-objective problem (MOP) benchmark functions. The results reveal that the proposed algorithm performs better than other algorithms that have been recently developed. In addition, the suggested technique is used to optimize the multi-objective wave energy converter model in order to maximize the annual production of electric energy while minimizing the per-unit energy cost. • Machine Learning and PSO based Success-History Based Parameter Adaptation Multi-Objective Differential Evolution algorithm was proposed. • The Improved version of SHAMODE optimized the WEC system to maximize the annual power generated at a minimum Levelized cost of energy. • The Improved version of SHAMODE was applied to optimize the multi-objective wave energy converter model to maximize the annual generated power and minimize the per unit energy cost. [ABSTRACT FROM AUTHOR]

In joint replenishment problem with random number of imperfect items (IJRP), it is a great challenge for managers to correctly ascertain the number of imperfect items and take the right response strategy to make rectifications simultaneously. In this paper, the effectiveness of four common response strategies, i.e., scrap, all-unit quantity discount, rework & repair, and buy replacements for coping with imperfect items are investigated based on IJRP for the first time, respectively. Considering the NP -hard nature of the classical JRP and its extensions, as well as enhancing the explorable performance in crossing scale-levels of IJRP models and exploitable performance in diversifying the solution population of bare-bones differential evolutionary (BBDE), a meta-heuristic algorithm named as the adaptive BBDE (ABBDE) is designed by introducing an problem scale related adaptive recombination probability and a generalized opposition-based learning (OBL). Through numerical experiments, it has been found that (1) The adaptive improvements of ABBDE have been verified through obtaining the lower best-found total cost (TC), the lowest mean value, and the smallest standard error comparing to that of genetic evolution algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), bare-bones PSO (BBPSO), BBDE. (2) Parameter sensitivity analyses indicate that the scale of yearly demand, lead-time, and significance of uncertainty have significant impacts on system cost, whilst the parameters pertinent to specific response strategy are main factors causing system cost disturbance. (3) Management insights and limitations on adoption of a right response strategy and the practical usage of ABBDE are obtained. • Four response strategy influences for coping with defective items are examined. • Both random imperfect items and stochastic demands are considered. • An adaptive mutation probability and opposition-based learning are considered. • Numerical experiments are designed and extensively performed. • Insights are obtained through experiments on different scale cases. [ABSTRACT FROM AUTHOR]

Electromagnetic field optimization (EFO) is a relatively new physics-inspired population-based metaheuristic algorithm, which simulates the behavior of electromagnets with different polarities and takes advantage of a nature-inspired ratio, known as the golden ratio. In EFO, the population consists of electromagnetic particles made of electromagnets corresponding to variables of an optimization problem and is divided into three fields: positive, negative, and neutral. In each iteration, a new electromagnetic particle is generated based on the attraction-repulsion forces among these electromagnetic fields, where the repulsion force helps particle to avoid the local optimal point, and the attraction force leads to find global optimal. This paper introduces an improved version of the EFO called improved electromagnetic field optimization (iEFO). Distinct from the EFO, the iEFO has two novel modifications: new solution generation function for the electromagnets and adaptive control of algorithmic parameters. In addition to these major improvements, the boundary control and randomization procedures for the newly generated electromagnets are modified. In the computational studies, the performance of the proposed iEFO is tested against original EFO, existing physics-inspired algorithms, and state-of-the-art meta-heuristic algorithms as artificial bee colony algorithm, particle swarm optimization, and differential evolution. Obtained results are verified with statistical testing, and results reveal that proposed iEFO outperforms the EFO and other considered competitor algorithms by providing better results. [ABSTRACT FROM AUTHOR]

In the mid 1990s two landmark metaheuristics have been proposed: Particle Swarm Optimization and Differential Evolution. Their initial versions were very simple, but rapidly attracted wide attention. During the last quarter century hundreds of variants of both optimization algorithms have been proposed and applied in almost any field of science or engineering. However, no broader comparison of performance between both families of methods has been presented so far. In the present paper ten Particle Swarm Optimization and ten Differential Evolution variants, from historical ones from the 1990s up to the most recent ones from 2022, are compared on numerous single-objective numerical benchmarks and 22 real-world problems. On average Differential Evolution algorithms clearly outperform Particle Swarm Optimization ones. Such advantage of Differential Evolution over Particle Swarm Optimization is in contradiction with popularity: In the literature Particle Swarm Optimization algorithms are two–three times more frequently used than Differential Evolution ones. Problems for which Particle Swarm Optimization performs better than Differential Evolution do exist but are relatively few. Although this result may be an effect of the choice of specific variants, experimental settings or problems used for comparison, some re-consideration of algorithmic philosophy may be needed for Particle Swarm Optimization variants to make them more competitive. [ABSTRACT FROM AUTHOR]

Clustering is a powerful technique in data-mining, which involves identifing homogeneous groups of objects based on the values of attributes. Meta-heuristic algorithms such as particle swarm optimization, artificial bee colony, genetic algorithm and differential evolution are now becoming powerful methods for clustering. In this paper, we propose a new meta-heuristic clustering method, the Whale Clustering Optimization Algorithm, based on the swarm foraging behavior of humpback whales. After a detailed formulation and explanation of its implementation, we will then compare the proposed algorithm with other existing well-known algorithms in clustering, including PSO, ABC, GA, DE and k-means. Proposed algorithm was tested using one artificial and seven real benchmark data sets from the UCI machine learning repository. Simulations show that the proposed algorithm can successfully be used for data clustering. [ABSTRACT FROM AUTHOR]

This paper presents an efficient approach to design stable, wideband, and infinite impulse response digital integrators (DIs) and digital differentiators (DDs) of first, second, third, and fourth order using an evolutionary optimization algorithm called harmony search (HS). In recent years, although wideband DIs and DDs have been designed using metaheuristic optimization techniques such as simulated annealing, genetic algorithm, and particle swarm optimization (PSO), these algorithms lead to sub-optimal solutions because of stagnation and premature convergence. HS algorithm, however, promises an enhanced frequency response for DIs and DDs because of the better exploration and exploitation of the search space. Simulation results demonstrate the superiority of HS-based designs as compared with three well-known benchmark evolutionary optimization algorithms, namely real coded genetic algorithm (RGA), PSO, and differential evolution (DE) based designs by yielding the least values of different magnitude response error metrics. Parametric and non-parametric statistical hypothesis tests are also conducted to compare the consistency in the performance of HS-based DIs and DDs with those of the designs based on RGA, PSO, and DE. The proposed HS-based designs also outperform those of the designs based on both classical and evolutionary optimization approaches reported in recent literature in terms of the maximum absolute magnitude error metric. [ABSTRACT FROM AUTHOR]

The Social Spider Optimization algorithm (SSO) is a novel metaheuristic optimization algorithm. To enhance the convergence speed and computational accuracy of the algorithm, in this paper, an elite opposition-based Social Spider Optimization algorithm (EOSSO) is proposed; we use an elite opposition-based learning strategy to enhance the convergence speed and computational accuracy of the SSO algorithm. The 23 benchmark functions are tested, and the results show that the proposed elite opposition-based Social Spider Optimization algorithm is able to obtain an accurate solution, and it also has a fast convergence speed and a high degree of stability. [ABSTRACT FROM AUTHOR]

Comprehensive learning particle swarm optimization (CLPSO) is a powerful state-of-the-art single-objective metaheuristic. Extending from CLPSO, this paper proposes multiswarm CLPSO (MSCLPSO) for multiobjective optimization. MSCLPSO involves multiple swarms, with each swarm associated with a separate original objective. Each particle’s personal best position is determined just according to the corresponding single objective. Elitists are stored externally. MSCLPSO differs from existing multiobjective particle swarm optimizers in three aspects. First, each swarm focuses on optimizing the associated objective using CLPSO, without learning from the elitists or any other swarm. Second, mutation is applied to the elitists and the mutation strategy appropriately exploits the personal best positions and elitists. Third, a modified differential evolution (DE) strategy is applied to some extreme and least crowded elitists. The DE strategy updates an elitist based on the differences of the elitists. The personal best positions carry useful information about the Pareto set, and the mutation and DE strategies help MSCLPSO discover the true Pareto front. Experiments conducted on various benchmark problems demonstrate that MSCLPSO can find nondominated solutions distributed reasonably over the true Pareto front in a single run. [ABSTRACT FROM AUTHOR]

The growing environmental concerns, governmental regulations, and significant cost savings are the primary motivations for companies to consider the reuse and recovery of products in their inventory system. The previous research ignored several realistic features of reusable items inventory systems, such as the presence of multiple products and operational constraints. For the first time, this paper presents a new multiproduct economic order quantity inventory model for an inventory system of reusable products. The goal of the model is to determine the optimal replenishment quantity and reuse quantity of each item so that the system's total cost is minimized. Several operational constraints are considered to provide a more realistic framework, such as the total available budget, warehouse space, and holding cost. Due to the nonlinearity of the presented model, differential evolution (DE) and particle swarm optimization (PSO) algorithms are utilized as two solution approaches. However, these algorithms' performance is highly dependent on their control parameters. Therefore, for the first time, two new variants of these algorithms, called DEQL and PSOQL, are presented in this study, where the control parameters of algorithms are not pre-determined. A powerful reinforcement learning algorithm, Q-learning, adapts these values intelligently. In other words, as a research contribution, this research aims at introducing a new variant of hybrid the DE and PSO algorithms in which a machine learning algorithm controls the value of metaheuristic parameters. The other parameters of the proposed algorithms are tunned employing the Taguchi method. Extensive numerical examples are established in different sizes, and the outputs are discussed in terms of several criteria. Statistical analysis of the results is performed, demonstrating that the proposed reinforcement learning-based parameter adaption has significantly improved algorithms' performance. [ABSTRACT FROM AUTHOR]

Grey Wolf Optimizer (GWO) is a recently developed meta-heuristic search algorithm inspired by grey wolves (Canis lupus), which simulate the social stratum and hunting mechanism of grey wolves in nature and based on three main steps of hunting: searching for prey, encircling prey and attacking prey. This paper presents the application of GWO algorithm for the solution of non-convex and dynamic economic load dispatch problem (ELDP) of electric power system. The performance of GWO is tested for ELDP of small-, medium- and large-scale power systems, and the results are verified by a comparative study with lambda iteration method, Particle Swarm Optimization algorithm, Genetic Algorithm, Biogeography-Based Optimization, Differential Evolution algorithm, pattern search algorithm, NN-EPSO, FEP, CEP, IFEP and MFEP. Comparative results show that the GWO algorithm is able to provide very competitive results compared to other well-known conventional, heuristics and meta-heuristics search algorithms. [ABSTRACT FROM AUTHOR]

Storage equipment, such as batteries and thermal energy storage (TES), has become increasingly important recently for peak-load shifting in energy systems. Mathematical programming methods, used frequently in previous studies to optimize operating schedules, can always be used to derive a theoretically optimal solution, but are computationally time consuming. Consequently, we use metaheuristics, such as genetic algorithms (GAs), particle swarm optimization (PSO), and cuckoo search (CS), to optimize operating schedules of energy systems that include a battery, TES, and an air-source heat pump. In this paper, we used a GA, differential evolution (DE), our own proposed mutation-PSO (m-PSO), CS, and the self-adaptive learning bat algorithm (SLBA), of which m-PSO was the fastest, and CS was the most accurate. CS obtained the semi-optimal solution 135 times as fast as dynamic programming (DP), a mathematical programming method with 0.22% tolerance. Thus, we showed that metaheuristics, especially m-PSO and CS, have advantages over DP for optimization of the operating schedules of energy systems that include a battery and TES. [ABSTRACT FROM AUTHOR]

The execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step h. This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest h in either one-step or multi-step methods. Four chaotic oscillators are used as a case study, and the optimization of their Kaplan-Yorke dimension ( D K Y ) is performed by applying three metaheuristics, namely: particle swarm optimization (PSO), many optimizing liaison (MOL), and differential evolution (DE) algorithms. Three representative one-step and three multi-step methods are used to solve the four chaotic oscillators, for which the estimation of the highest h is obtained from their stability analysis. The optimization results show the effectiveness of using a high h value for the six numerical methods in reducing execution time while maximizing the positive Lyapunov exponent ( L E + ) and D K Y of the chaotic oscillators by applying PSO, MOL, and DE algorithms. [ABSTRACT FROM AUTHOR]