Original Whale Optimization Algorithm Research Articles

Application of spiral enhanced whale optimization algorithm in solving optimization problems

The Whale Optimization Algorithm (WOA) is regarded as a classic metaheuristic algorithm, yet it suffers from limited population diversity, imbalance between exploitation and exploration, and low solution accuracy. In this paper, we propose the Spiral-Enhanced Whale Optimization Algorithm (SEWOA), which incorporates a nonlinear time-varying self-adaptive perturbation strategy and an Archimedean spiral structure into the original WOA. The Archimedean spiral structure enhances the diversity of the solution space, aiding the algorithm in escaping local optima. The nonlinear time-varying self-adaptive optimization dynamic perturbation strategy improves the algorithm’s local search capability and enhances solution accuracy. The effectiveness of the proposed algorithm is validated from multiple perspectives using CEC2014 test functions, CEC2017 test functions, and 23 benchmark test functions. The experimental results demonstrate that the enhanced Whale Optimization Algorithm significantly improves population diversity, balances global and local search, and enhances solution accuracy. Additionally, SEWOA exhibits excellent performance in solving three engineering design problems, showcasing its value and wide range of potential applications.

A nonlinear randomly reuse-based mutated whale optimization algorithm and its application for solving engineering problems

The whale optimization algorithm (WOA) is a meta-heuristic optimization algorithm inspired by the social behavior of whale hunting. However, the WOA has the defects of easily falling into local optimum and slow convergence speed. Therefore, to overcome the shortcomings of the original WOA, a novel variant of WOA called nonlinear randomly reuse-based mutated whale optimization algorithm (NRRMWOA) is proposed in this study. In detail, the proposed NRRMWOA includes three novel strategies as compared with original WOA. Firstly, a nonlinear adaptive parameter strategy is introduced to achieve nonlinear adjustment of search pattern along with the iteration time. The second improvement is the random reuse strategy, which can fully utilize the current optimal whale to improve the solution accuracy. The thirdly proposed late disturbance mutation strategy has the function of increasing population diversity in the later stage of iteration. The three proposed strategies complement each other and significantly improve the convergence efficiency and global exploration capability of WOA. To verify the superiority of the proposed NRRMWOA, a series of typical benchmark functions are utilized to conduct comprehensive numerical experiments. Compared with three sets of comparison algorithms (five variants of WOA, five advanced algorithms, and five well-known algorithms), the ARVs (average ranking values) obtained by NRRMWOA are 1.4667, 1.9167, and 2.1333, respectively. The rankings are all the best in the corresponding group. The detailed experimental and statistics results indicate that the presented NRRMWOA has higher convergence speed and better solution quality compared with the three sets of existing algorithms. In addition, 15 real-world optimization problems are employed to further test the performance of NRRMWOA for solving engineering optimization problems. For the results of 15 real-world optimization problems, the proposed NRRMWOA achieves the best results including 8 best “Mean”, 10 best “Min.”, and 7 best “Max.”. As a conclusion, the proposed NRRMWOA is a promising and excellent algorithm.

ABWOA: adaptive boundary whale optimization algorithm for large-scale digital twin network construction

Digital twin network (DTN) as an emerging network paradigm, have garnered growing attention. For large-scale networks, a crucial problem is how to effectively map physical networks onto the infrastructure platform of DTN. To address this issue, we propose a heuristic method of the adaptive boundary whale optimization algorithm (ABWOA) to solve the digital twin network construction problem, improving the efficiency and reducing operational costs of DTN. Extensive comparison experiments are conducted between ABWOA and various algorithms such as genetic algorithm, particle swarm optimization, artificial bee colony, differential evolution algorithm, moth search algorithm and original whale optimization algorithm. The experimental results show that ABWOA is superior to other algorithms in terms of solution quality, convergence speed, and time cost. It can solve the digital twin network construction problem more effectively.

An Interval Type-2 Fuzzy Logic Approach for Dynamic Parameter Adaptation in a Whale Optimization Algorithm Applied to Mathematical Functions

In this paper, an improved whale optimization algorithm (WOA) based on the utilization of an interval type-2 fuzzy logic system (IT2FLS) is presented. The main idea is to present a proposal for adjusting the values of the r→1 and r→2 parameters in the WOA using an IT2FLS to achieve excellent results in the execution of the WOA. The original WOA has already proven itself as an algorithm with excellent results; therefore, a wide variety of improvements have been made to it. Herein, the main purpose is to provide a hybridization of the WOA algorithm employing fuzzy logic to find the appropriate values of the r→1 and r→2 parameters that can optimize the mathematical functions used in this study, thereby providing an improvement to the original WOA algorithm. The performance of the fuzzy WOA using IT2FLS (FWOA-IT2FLS) shows good results in the case study of the benchmark function optimization. An important comparative with other metaheuristics is also presented. A statistical test and the comparative with other bio-inspired algorithms, namely, the original WOA with type-1 FLS (FWOA-T1FLS) are analyzed. The performance index used is the average of the minimum errors in each proposed method.

Stable Feature Selection using Improved Whale Optimization Algorithm for Microarray Datasets

A microarray is a collection of DNA sequences that reflect an organism’s whole gene set and are organized in a grid pattern for use in genetic testing. Microarray datasets are extremely high-dimensional and have a very small sample size, posing the challenges of insufficient data and high computational complexity. Identification of true biomarkers that are the most significant features (a very small subset of the complete feature set) is desired to solve these issues. This reduces over-fitting, and time complexity, and improves model generalization. Various feature selection algorithms are used for this biomarker identification. This research proposed a modification to the whale optimization algorithm (WOAm) for biomarker discovery, in which the fitness of each search agent is evaluated using the hinge loss function during the hunting for prey phase to determine the optimal search agent. Also compared the results of the proposed modified algorithm with the original whale optimization algorithm and also with contemporary algorithms like the marine predator algorithm and grey wolf optimization. All these algorithms are evaluated on six different high-dimensional microarray datasets. It has been observed that the proposed modification for the whale optimization algorithm has significantly improved the results of feature selection across all the datasets. Domain experts trust the resultant biomarker/ associated genes by the stability of the results obtained. The chosen feature set’s stability was also evaluated during the research work. According to the findings, our proposed WOAm has superior stability compared to other algorithms for the CNS, colon, Leukemia, and OSCC. datasets.

An improved hybrid whale optimization algorithm for global optimization and engineering design problems

The whale optimization algorithm (WOA) is a widely used metaheuristic optimization approach with applications in various scientific and industrial domains. However, WOA has a limitation of relying solely on the best solution to guide the population in subsequent iterations, overlooking the valuable information embedded in other candidate solutions. To address this limitation, we propose a novel and improved variant called Pbest-guided differential WOA (PDWOA). PDWOA combines the strengths of WOA, particle swarm optimizer (PSO), and differential evolution (DE) algorithms to overcome these shortcomings. In this study, we conduct a comprehensive evaluation of the proposed PDWOA algorithm on both benchmark and real-world optimization problems. The benchmark tests comprise 30-dimensional functions from CEC 2014 Test Functions, while the real-world problems include pressure vessel optimal design, tension/compression spring optimal design, and welded beam optimal design. We present the simulation results, including the outcomes of non-parametric statistical tests including the Wilcoxon signed-rank test and the Friedman test, which validate the performance improvements achieved by PDWOA over other algorithms. The results of our evaluation demonstrate the superiority of PDWOA compared to recent methods, including the original WOA. These findings provide valuable insights into the effectiveness of the proposed hybrid WOA algorithm. Furthermore, we offer recommendations for future research to further enhance its performance and open new avenues for exploration in the field of optimization algorithms. The MATLAB Codes of FISA are publicly available at https://github.com/ebrahimakbary/PDWOA.

A fusion algorithm based on whale and grey wolf optimization algorithm for solving real-world optimization problems

In order to better understand and analyze population-based meta-heuristic optimization algorithms, this paper proposed a new hybrid algorithm combined Lévy flight with modified Whale Optimization Algorithm (WOA) and Grey Wolf Optimizer (GWO) , which is called LMWOAGWO to discard the dross and select the essence. Firstly, the population is initialized by using the uniform distribution space combined with the pseudo-reverse learning strategy, which lays the foundation for global search. Then, modifications were made to both WOA and GWO. For WOA algorithm, random adjustment control parameters strategy and different chaotic maps are used to adjust the main parameters of WOA to avoid the algorithm falling into local optimum in the later stage. For GWO algorithm, a new optimal solution is added to the grey wolf population to increase the optimal update position of the algorithm. On this basis, the dynamic weighting strategy is introduced to improve the convergence accuracy and convergence speed of the algorithm. Subsequently, new conditions were added during the WOA exploitation phase to formulate LMWOAGWO and the greedy strategy is used to retain better iteration update locations. Finally, Lévy flight is used to improve the global search ability of the algorithm. Extensive numerical experiments were conducted using 23 standard test benchmark functions, 25 CEC2005 functions, 15 popular benchmark functions and 10 CEC2019 functions to test the performance of LMWOAGWO compared with other well-known swarm optimization algorithms. Experimental and statistical results show that the performance of LMWOAGWO algorithm is better than many state-of-the-art algorithms. Then, 22 real-world optimization problems were used to further study the effectiveness of LMWOAGWO. Winners of CEC2020 Real World Single Objective Constraint Optimization Competition, such as iLSHADEϵ algorithm, sCMAgES algorithm, COLSHADE algorithm and EnMODE algorithm are selected as four comparison algorithms in real world optimization problems. Experimental results show that the proposed LMWOAGWO has the capability to solve real-world optimization problems. Finally, the application efficiency of LMWOAGWO in solving two basic optimization problems in wireless networks is briefly introduced, and compared with the original WOA and GWO. Simulation results show that the performance of the LMWOAGWO is better than WOA and GWO.

Inversed Control Parameter in Whale Optimization Algorithm and Grey Wolf Optimizer for Wrapper-based Feature Selection: A comparative study

Whale Optimization Algorithm (WOA) and Grey Wolf Optimizer (GWO) are well-perform metaheuristic algorithms used by various researchers in solving feature selection problems. Yet, the slow convergence speed issue in Whale Optimization Algorithm and Grey Wolf Optimizer could demote the performance of feature selection and classification accuracy. Therefore, to overcome this issue, a modified WOA (mWOA) and modified GWO (mGWO) for wrapper-based feature selection were proposed in this study. The proposed mWOA and mGWO were given a new inversed control parameter which was expected to enable more search area for the search agents in the early phase of the algorithms and resulted in a faster convergence speed. The objective of this comparative study is to investigate and compare the effectiveness of the inversed control parameter in the proposed methods against the original algorithms in terms of the number of selected features and the classification accuracy. The proposed methods were implemented in MATLAB where 12 datasets with different dimensionality from the UCI repository were used. kNN was chosen as the classifier to evaluate the classification accuracy of the selected features. Based on the experimental results, mGWO did not show significant improvements in feature reduction and maintained similar accuracy as the original GWO. On the contrary, mWOA outperformed the original WOA in terms of the two criteria mentioned even on high-dimensional datasets. Evaluating the execution time of the proposed methods, utilizing different classifiers, and hybridizing proposed methods with other metaheuristic algorithms to solve feature selection problems would be future works worth exploring.

Improved Whale Optimization Algorithm for Solving Microgrid Operations Planning Problems

Microgrid operations planning is one of the keys to ensuring the safe and efficient outputs of distributed energy resources (DERs) and the stable operation of a power system in a microgrid (MG). In this study, for the symmetry in renewable energy and microgrid systems, and coordinated control based on a storage battery system, an MG dispatching model with DER conditions and integrated costs is established in grid-connected mode, on the basis of MG operation costs, interaction costs, and pollutant emissions costs. Moreover, an optimization objective for minimizing integrated costs is established. Therefore, based on the original whale optimization algorithm (WOA), an improved whale optimization algorithm (IWOA) with adaptive weight strategy and Levy flight trajectory is proposed in this paper, to solve the optimal operations planning problem of MGs. Finally, in computing comparisons with methods such as the genetic algorithm (GA), particle swarm algorithm (PSO), WOA, wild horse optimizer (WHO), and enhanced whale optimization algorithm (EWOA), the results show that the IWOA computation had lower integrated costs and higher operational efficiency. Moreover, it is verified that the IWOA performed better in solving the MG operations planning problem.

Information sharing search boosted whale optimizer with Nelder-Mead simplex for parameter estimation of photovoltaic models

With the recent emphasis on new energy sources, solar photovoltaic cells have received widespread attention from scholars as a highly efficient and clean new energy source. Researchers model solar photovoltaic cells and analyze the values of various unknown parameters to improve their efficiency in converting solar energy into electric energy. To extract unknown parameters more accurately and efficiently, we propose a whale optimization algorithm (WOA) driven by the information sharing search mechanism and the Nelder-Mead simplex (NMs) mechanism, which ISNMWOA. The WOA can perform a global search of the parameters of the photovoltaic model to obtain a number of initial vectors of model parameters that satisfy the parameter range. The information sharing search strategy performs a further coarse local search on a number of feasible solutions generated. In contrast, the NMs can perform a further refined local search in the vicinity of the optimal parameters to obtain the final vector of optimally performing model parameters. First, we tested ISNMWOA against six advanced algorithms in CEC 2014. We then used the ISNMWOA to evaluate the parameters of the solar cells and PV modules. The RMSE of ISNMWOA on single diode model (SDM), double diode model (DDM), three diode model (TDM), and photovoltaic module model (PV) are 9.8602E-04, 9.8248E-04, 9.8248E-04 and 2.4251E-03 respectively. Compared to the original WOA, the RMSE of ISNMWOA on SDM, DDM, TDM, and PV decreased by 30.01%, 24.24%, 50.98%, and 6.94%, respectively. Moreover, compared to the high-performing GOFPANM, ISNMWOA runs 298 times faster than GOFPANM with almost identical optimization results. We evaluated the performance of three commercial PV modules under different irradiance and temperature conditions using ISNMWOA to verify the reliability of the experiment. The experimental results show that the ISNMWOA outperforms similar algorithms currently available.

Rapid determination of Panax notoginseng origin by terahertz spectroscopy combined with the machine learning method

Panax notoginseng is a valuable herb with geographical indication, and the quality and price of P. notoginseng from different origins are very different. Therefore, this paper proposes a rapid and accurate method for identifying the origins of P. notoginseng by collecting the roots of P. notoginseng. This paper improves the whale optimization algorithm in terms of global convergence and convergence speed, introduces the Levy flight strategy and reconstructed whale synergy factor A, and applies it to the parameter optimization of support vector machines, to obtain a high-performance classification model. The improved whale optimization algorithm model identifies the origin of P. notoginseng by discriminating their terahertz spectra. Compared with the commonly used genetic algorithm and the original whale optimization algorithm, improvement in the whale optimization algorithm was able to avoid falling into local optimum solutions more effectively while having a high convergence rate. Accordingly, the improved whale optimization algorithm optimized support vector machine model obtained an overall accuracy of 98.44%, which was significantly higher than the 95.31% overall accuracy of the genetic algorithm optimized support vector machine model and the 96.88% overall accuracy of the whale optimization algorithm optimized support vector machine model. It was concluded that terahertz spectroscopy together with machine learning would be a promising technique for identifying the origins of P. notoginseng.

Individual disturbance and neighborhood mutation search enhanced whale optimization: performance design for engineering problems

AbstractThe whale optimizer is a popular metaheuristic algorithm, which has the problems of weak global exploration, easy falling into local optimum, and low optimization accuracy when searching for the optimal solution. To solve these problems, this paper proposes an enhanced whale optimization algorithm (WOA) based on the worst individual disturbance (WD) and neighborhood mutation search (NM), named WDNMWOA, which employed WD to enhance the ability to jump out of local optimum and global exploration, adopted NM to enhance the possibility of individuals approaching the optimal solution. The superiority of WDNMWOA is demonstrated by representative IEEE CEC2014, CEC2017, CEC2019, and CEC2020 benchmark functions and four engineering examples. The experimental results show that thes WDNMWOA has better convergence accuracy and strong optimization ability than the original WOA.

An Efficient and Robust Improved Whale Optimization Algorithm for Large Scale Global Optimization Problems

As an efficient meta-heuristic algorithm, the whale optimization algorithm (WOA) has been extensively applied to practical problems. However, WOA still has the drawbacks of converging slowly, and jumping out from extreme points especially for large scale optimization problems. To overcome these defects, a modified whale optimization algorithm integrated with a crisscross optimization algorithm (MWOA-CS) is proposed. In MWOA-CS, each dimension of the optimization problem updates its position by randomly performing improved WOA or crisscross optimization algorithm during the entire iterative process. The improved WOA adopts the new nonlinear convergence factor and nonlinear inertia weight to tune the ability of exploitation and exploration. To analyze the performance of MWOA-CS, a series of numerical experiments were performed on 30 test benchmark functions with dimension ranging from 300 to 1000. The experimental results revealed that the presented MWOA-CS provided better convergence speed and accuracy, and meanwhile, displayed a significantly more effective and robust performance than the original WOA and other state of the art meta-heuristic algorithms for solving large scale global optimization problems.

A bilevel whale optimization algorithm for risk management scheduling of information technology projects considering outsourcing

Information Technology (IT) outsourcing can help enterprises to improve their core competitiveness and save costs. But the schedule risk in IT outsourcing process may bring huge losses. Based on the theory of distributed decision making (DDM), a bi-level multi-objective schedule risk management model is established with considering the project schedule risk and risk management cost. In addition, a bi-level Whale Optimization Algorithm (BiWOA) is designed to solve the problem and compared with the original whale optimization algorithm (WOA) and the moth–flame​ optimization (MFO). Three numerical examples are designed to test the effectiveness of the model. And the analysis of results show that it can effectively control the schedule risks of IT outsourcing projects. The influence of different preference degree of the client on the decision result is also analyzed. The effectiveness of bi-level whale optimization algorithm is proved by solving and analyzing the numerical examples. It demonstrated that bi-level whale optimization algorithm is more competitive with higher accuracy and the ability to escape from local optima and can solve the problem more effectively.

A Hybrid Approach Based on Grey Wolf and Whale Optimization Algorithms for Solving Cloud Task Scheduling Problem

In the context of cloud computing, one problem that is frequently encountered is task scheduling. This problem has two primary implications, which are the planning of tasks on virtual machines and the attenuation of performance. In order to address the problem of task scheduling in cloud computing, requisite nontraditional optimization attitudes to attain the optima of the problem, the present paper puts forth a hybrid multiple-objective approach called hybrid grey wolf and whale optimization (HGWWO) algorithms, that integrates two algorithms, namely, the grey wolf optimizer (GWO) and the whale optimization algorithm (WOA), with the purpose of conjoining the advantages of each algorithm for minimizing costs, energy consumption, and total execution time needed for task implementation, beside that improving the use of resources. Assessment of the aims of the proposed approach is carried out with the help of the tool known as CloudSim. As pointed out by the results of the experimental work undertaken, the proposed approach has the capability of performing at a superior level by comparison to the original algorithms GWO and WOA on their own with regard to costs, energy consumption, makespan, use of resources, and degree of imbalance.

An application of the whale optimization algorithm with Levy flight strategy for clustering of medical datasets

Clustering, which is handled by many researchers, is separating data into clusters without supervision. In clustering, the data are grouped using similarities or differences between them. Many traditional and heuristic algorithms are used in clustering problems and new techniques continue to be developed today. In this study, a new and effective clustering algorithm was developed by using the Whale Optimization Algorithm (WOA) and Levy flight (LF) strategy that imitates the hunting behavior of whales. With the developed WOA-LF algorithm, clustering was performed using ten medical datasets taken from the UCI Machine Learning Repository database. The clustering performance of the WOA-LF was compared with the performance of k-means, k-medoids, fuzzy c-means and the original WOA clustering algorithms. Application results showed that WOA-LF has more successful clustering performance in general and can be used as an alternative algorithm in clustering problems.

Boosted Whale Optimization Algorithm With Natural Selection Operators for Software Fault Prediction

Software fault prediction (SFP) is a challenging process that any successful software should go through it to make sure that all software components are free of faults. In general, soft computing and machine learning methods are useful in tackling this problem. The size of fault data is usually huge since it is obtained from mining software historical repositories. This data consists of a large number of features (metrics). Determining the most valuable features (i.e., Feature Selection (FS) is an excellent solution to reduce data dimensionality. In this paper, we proposed an enhanced version of the Whale Optimization Algorithm (WOA) by combining it with a single point crossover method. The proposed enhancement helps the WOA to escape from local optima by enhancing the exploration process. Five different selection methods are employed: Tournament, Roulette wheel, Linear rank, Stochastic universal sampling, and random-based. To evaluate the performance of the proposed enhancement, 17 available SFP datasets are adopted from the PROMISE repository. The deep analysis shows that the proposed approach outperformed the original WOA and the other six state-of-the-art methods, as well as enhanced the overall performance of the machine learning classifier.

Binary Whale Optimization Algorithm for Dimensionality Reduction

Feature selection (FS) was regarded as a global combinatorial optimization problem. FS is used to simplify and enhance the quality of high-dimensional datasets by selecting prominent features and removing irrelevant and redundant data to provide good classification results. FS aims to reduce the dimensionality and improve the classification accuracy that is generally utilized with great importance in different fields such as pattern classification, data analysis, and data mining applications. The main problem is to find the best subset that contains the representative information of all the data. In order to overcome this problem, two binary variants of the whale optimization algorithm (WOA) are proposed, called bWOA-S and bWOA-V. They are used to decrease the complexity and increase the performance of a system by selecting significant features for classification purposes. The first bWOA-S version uses the Sigmoid transfer function to convert WOA values to binary ones, whereas the second bWOA-V version uses a hyperbolic tangent transfer function. Furthermore, the two binary variants introduced here were compared with three famous and well-known optimization algorithms in this domain, such as Particle Swarm Optimizer (PSO), three variants of binary ant lion (bALO1, bALO2, and bALO3), binary Dragonfly Algorithm (bDA) as well as the original WOA, over 24 benchmark datasets from the UCI repository. Eventually, a non-parametric test called Wilcoxon’s rank-sum was carried out at 5% significance to prove the powerfulness and effectiveness of the two proposed algorithms when compared with other algorithms statistically. The qualitative and quantitative results showed that the two introduced variants in the FS domain are able to minimize the selected feature number as well as maximize the accuracy of the classification within an appropriate time.

3D mesh simplification with feature preservation based on Whale Optimization Algorithm and Differential Evolution

Large-scale 3D models consume large computing and storage resources. To address this challenging problem, this paper proposes a new method to obtain the optimal simplified 3D mesh models with the minimum approximation error. First, we propose a feature-preservation edge collapse operation to maintain the feature edges, in which the collapsing cost is calculated in a novel way by combining Gauss curvature and Quadratic Error Metrics (QEM). Second, we introduce the edge splitting operation into the mesh simplification process and propose a hybrid ‘undo/redo’ mechanism that combines the edge splitting and edge collapse operation to reduce the number of long and narrow triangles. Third, the proposed ‘undo/redo’ mechanism can also reduce the approximation error; however, it is impossible to manually choose the best operation sequence combination that can result in the minimum approximation error. To solve this problem, we formulate the proposed mesh simplification process as an optimization model, in which the solution space is composed of the possible combinations of operation sequences, and the optimization objective is the minimum of the approximation error. Finally, we propose a novel optimization algorithm, WOA-DE, by replacing the exploration phase of the original Whale Optimization Algorithm (WOA) with the mutate and crossover operations of Differential Evolution (DE) to compute the optimal simplified mesh model more efficiently. We conduct numerous experiments to test the capabilities of the proposed method, and the experimental results show that our method outperforms the previous methods in terms of the geometric feature preservation, triangle quality, and approximation error.

Whale optimization algorithm based on nonlinear convergence factor and chaotic inertial weight

SummaryThe whale optimization algorithm (WOA) is a new biological meta‐heuristic algorithm based on the social hunting behaviors of humpback whales. However, it can easily fall into a local optimum when solving complex problems and exhibits slow convergence speed and poor exploration. This study proposed three improved versions of the WOA based on the concepts of chaos initialization, nonlinear convergence factor, and chaotic inertial weight to enhance its exploration abilities. These properties were employed to improve the population diversity and maintain the balance between exploration and exploitation. The performance of the best version was compared with those of moth‐flame optimization, firefly algorithm, particle swarm optimization, gray wolf optimizer, flower pollination algorithm, original WOA, and two recently proposed hybrid WOA through 19 benchmark functions. Experimental results indicated that the proposed algorithms exhibit better performance in terms of complexity and convergence speed.