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Integration of bat algorithm and salp swarm intelligence with stochastic difference variants for global optimization

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Abstract

This paper introduces a novel hybrid optimization algorithm named bat-salp swarm algorithm (BASSA). BASSA integrates the local exploitation capability of bat algorithm (BA) and the global exploration capability of salp swarm algorithm (SSA). Firstly, by introducing the echolocation of BA, the follower updating strategy of SSA is improved. Secondly, the algorithm selects between BA and SSA based on specific conditions. Finally, individuals undergo random differential mutation to increase population diversity, thereby avoiding local optima. To verify the effectiveness of the algorithm, we carry out experiments BASSA on 23 benchmark functions with different dimensions and compare it with 7 optimization algorithms, including BA, SSA, and 7 enhanced versions of SSA. Simulation results indicate that BASSA outperforms standard BA, SSA, and other enhanced algorithms in terms of mean and standard deviation. This suggests a significant improvement in optimization performance, with higher solution accuracy and faster convergence speed. Additionally, through performance evaluation on three real engineering problems, the results indicate that BASSA possesses strong optimization capabilities.

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Data availability

The data and materials supporting the results of this study have been clearly described in the manuscript. The key algorithms used in this study have been presented as pseudo-code in this paper and are also available upon request. We are committed to ensuring the transparency and accessibility of relevant data and resources to facilitate the reproducibility and validation of research results.

Abbreviations

\({N}_{BA},{N}_{SSA}\) :

The number of individuals in each algorithm

\({T}_{BA},{T}_{SSA}\) :

The number of iterations of the BA and SSA

\({r}_{i}^{t}\) :

The pulse emission rate

\({A}_{i}^{t}\) :

The acoustic loudness

\({c}_{2}\), \({c}_{3},\varepsilon ,\beta ,{{r}_{1},r}_{2}\) :

The random numbers

\(v{\left(i\right)}^{t},v{\left(i\right)}^{t+1}\) :

The velocity of the agent

\({x\left(i\right)}^{t},{x\left(i\right)}^{t+1}\) :

The position of the agent

α,:

The Acoustic Loudness Attenuation factor

γ,:

The Pulse frequency enhancement factor

T :

The maximum number of iterations

\({I}_{c}\left(t\right)\) :

The dispersion of population of each iteration

\(N\) :

The total number of samples

\({r}_{i}^{0}\) :

The maximum value of the pulse emission rate

NFE:

The number function evaluation

D:

The dimension of the problem solution space

\({x}_{ij}^{{\text{t}}+1}\) :

The new position of the updated particle

\({F}_{j}\) :

The position of the food

\({c}_{1}\) :

The balancing factor

\({f}_{max},{f}_{min}\) :

The upper and lower limits of the frequency

\({ub}_{j}\),\({lb}_{j}\) :

The upper and lower bounds of the escape space

\({x}_{best}^{t}\) :

The best agent in the population

\({x}_{j}^{1}\) :

The position of the leader salp swarm individual

t :

The current number of iterations

\({C}_{d}\left(t\right)\) :

The mass center

\(n\) :

The total number of dimensions

\(Di{v}_{max}\) :

The maximum diversity of the whole iteration

BASSA:

Integration Of Bat Algorithm And Salp Swarm Intelligence With Stochastic Difference Variants

BA:

Bat Algorithm

PSO:

Particle Swarm Optimizer

GP:

Genetic Programming

EP:

Evolutionary Programming

DE:

Differential Evolution

WWO:

Water Wave Optimization

EA:

Evolutionary Algorithms

CBO:

Collision Body Optimizer

KOA:

Kepler Optimization Algorithm

AOA:

Archimedean Optimization Algorithm

NOA:

Nuthatch Optimization Algorithm

FOX:

Fox Optimizer

SWO:

Spider Wasp Optimizer

GRO:

Gold Rush Optimization

IBQANA:

Improved Binary Quantum-Based Avian Navigation Optimizer Algorithm

QRBL:

Quasi-Reflection Learning

GOA:

Grasshopper Optimization Algorithm

OBL:

Opposition-Based Learning

IABC-EO:

Improved Artificial Bee Colony-Extremal Optimization

MTDE:

Multi-Trial Vector-Based Differential Evolution

QSSA:

Quantized Salp Swarm Algorithm

IWOA:

Improved Whale Optimization Algorithm

TLBO:

Teaching–Learning-Based Optimization

SFR:

Stagnation Finding And Replacing

RIME:

Rime Optimizer

HOA:

Horse Herd algorithm

GWCA:

Great Wall Construction Algorithm

WHO:

Wild Horse Optimizer

iSSA:

Salp Swarm Algorithm With Integrated Random Inertia Weight And Differential Mutation

RDSSA:

Improved Salp Swarm Algorithm With Decay Factor And Dynamic Learning

FKNN:

Fuzzy K-Nearest Neighbor

ISSA:

Improved Salp Swarm Algorithm

MBA:

Modified BA

AD:

Alzheimer’s Disease

ERIME:

Enhanced Rime Algorithm

ECSA:

Evolutionary Crow Search Algorithm

AO:

Aquila Optimizer

VBAO:

V-Shaped Binary Aquila Optimizer

CMS:

Communication Strategy

CMS:

Cuckoo Mutation Strategy

EBA:

Enhanced Bat Algorithm

DMBA:

Dynamic Membrane-Driven Bat Algorithm

MBADE:

Modified Bat Algorithm Hybridizing By Differential Evolution

RE:

Regularity Evolution

SI:

Swarm Intelligence

ANN:

Artificial Neural Network

SSA:

Salp Swarm Algorithm

GA:

Genetic Algorithms

ES:

Evolutionary Strategies

TSO:

Transient Search Optimizer

ABC:

Artificial Bee Colony

CSS:

Charging System Search

LSO:

Light Spectrum Optimizer

EVO:

Energy Valley Optimizer

WEO:

Water Evaporation Optimizer

RO:

Ray Optimizer

BO:

Bonobo Optimizer

SFO:

Sailfish Optimizer

COA:

Coati Optimization Algorithm

SOA:

Skill Optimization Algorithm

SGABA:

Simulated Annealing Gaussian Bat Algorithm

GWO:

Grey Wolf Optimization

WOA:

Whale Optimization Algorithm

EO:

Extremal Optimization

DMSSA:

Dual-Mutation Salp Swarm Algorithm

ADMS:

Adaptive differential evolution mutation strategy

GMLSSA:

Gravity-Based And Multi-Leader Search Strategy Salp Swarm Algoritm

TLSSA:

Teaching–learning Guided Salp Swarm Algorithm

MFO:

Moth-Flame Optimization

MFO-SFR:

Moth-Flame Optimization Stagnation Finding And Replacing

RUN:

Runge Kutta Optimizer

GKSO:

Genghis Khan Shark Optimizer

RFO:

Red Fox Optimization

CASSA:

Crazy Adaptive Salp Swarm Algorithm

LECUSSA:

Salp Swarm Algorithm With Levy Flight Strategy And Conditional Update

ALSSA:

Adaptive leader Salp Swarm Algorithm For Global Search

MSSA:

Improved Salp Swarm Algorithm

SSA:

Simulated Annealing Algorithm

MCI:

Mild Cognitive Impairment

NC:

Normal Controls

BSMO:

Binary Starling Murmuration Optimizer

SWO:

Spider Wasp Optimizer

SBAO:

S-Shaped Binary Aquila optimizer

MA:

Migration Algorithm

SRS:

Selective Replacement Strategy

DI-GWOCD:

Discrete Version Of The Improved Grey Wolf Optimizer For Effectively Detecting Communities Of Different Networks

TVSSA:

Time-varying Salp Swarm Algorithm

QSSALEO:

Quadratic Interpolation Salp Swarm-Based Local Escape Operator

ESSA:

Emended Salp Swarm Algorithm

MOEAS:

Multi Objective Evolutionary Algorithms

MAS:

Metaheuristic Algorithms

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Funding

This work was supported by the research projects: the National Natural Science Foundation of China under Grant Nos. 62202378, 62002289, 62176146, Natural Science Foundation of Shaanxi Province nos. 2021JQ-711, Shaanxi Provincial Department of Education Fund nos. 20JK0910.

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Authors and Affiliations

  1. School of Computer Science and Technology, Xi’an University of Posts and Telecommunications, Xi’an, 710121, Shaanxi, China

    Hongye Li, Jianan Wang & Yanjie Zhu

  2. Shaanxi Key Laboratory of Network Data Analysis and Intelligent Processing, Xi’an University of Posts and Telecommunications, Xi’an, 710121, Shaanxi, China

    Hongye Li, Jianan Wang & Yanjie Zhu

  3. Xi’an Key Laboratory of Big Data and Intelligent Computing, Xi’an University of Posts and Telecommunications, Xi’an, 710121, Shaanxi, China

    Hongye Li, Jianan Wang & Yanjie Zhu

Authors
  1. Hongye Li
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  2. Jianan Wang
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Hongye Li: Algorithms, Writing, Revision. Jianan Wang: Preprocessing, Coding, Algorithms, Writing, Editing. Yabing Li: Methods, Writing, Editing, Revision.

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Correspondence to Jianan Wang.

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Appendices

Appendix A. Test functions

See Tables 23 and 24.

Table 23 The characteristics of the 23 classical benchmark functions
Full size table
Table 24 Summary of the 29 CEC 2017 benchmark problems
Full size table

Appendix B. Evaluation Metrics

Mean value

The mean (\(\overline{x }\)) reflects the average of all observations in the sample. It is calculated using Eq. (31). It is calculated by adding all the results and dividing by the total number of results \(N\), where \({x}_{i}\) represents each observation in the sample.

$$\overline{x }=\frac{1}{N}\sum_{i=1}^{N}{x}_{i}$$
(31)

Standard deviation

Standard deviation (STD) is a measure of variation or dispersion in a set of data. A larger standard deviation indicates that the values are more different from the mean, while a smaller standard deviation indicates that these values are closer to the mean. Usually, a smaller standard deviation is considered better because it indicates relatively more stable data. The formula is shown in Eq. (32) where \(N\) denotes the total number of outcomes in the sample, \({x}_{i}\) is one of the samples, and \(\overline{x }\) is the sample mean.

$$STD=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}{\left({x}_{i}-\overline{x }\right)}^{2}}$$
(32)

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Li, H., Wang, J. & Zhu, Y. Integration of bat algorithm and salp swarm intelligence with stochastic difference variants for global optimization. Cluster Comput (2024). https://doi.org/10.1007/s10586-024-04447-x

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