Optimization of complex engineering problems using modified sine cosine algorithm

Abstract

In this article, a modified version of the Sine Cosine algorithm (MSCA) is proposed to solve the optimization problem. Based on the Sine Cosine algorithm (SCA), the position update formula of SCA is redefined to increase the convergence speed, then the Levy random walk mutation strategy is adopted to improve the population diversity. In order to verify the performance of MSCA, 24 well-known classical benchmark problems and IEEE CEC2017 test suites were introduced, and by comparing MSCA with several popular methods, it is demonstrated that MSCA has good convergence and robustness. Finally, MSCA is used to address six complex engineering design problems, demonstrating the engineering utility of the algorithm.

Introduction

Optimization is the process of finding the best solution to a problem by specific rules¹. Traditional optimization techniques need to satisfy the preconditions of continuity and differentiability of the objective function, which makes it impossible to apply anywhere in the real world². The heuristic algorithm treats the optimization problem as a black box, and does not need to consider the problem information, which brings a boon to the many complex optimization problems. In recent years, as an emerging optimization technology, heuristic algorithms have received widespread popularity in the field of engineering³. Heuristic algorithms are mainly inspired by natural experience and observations, which obtain beneficial information by simulating the cooperation or competition of individuals in the population to find possible better positions in the search space. In general, heuristic algorithms can be classified into three main types⁴: evolution-based, physics-based and swarm-based algorithms. Evolution-based algorithms are inspired by the laws of biological evolution, individuals exchange information to ensure that they can find a favorable position in the search space. The representative algorithms are Genetic Algorithms (GA)⁵, Differential Evolution (DE)⁶ and Evolutionary Strategies (ES)⁷. Physics-based algorithms inspired by the laws of physics in nature. The most popular methods are Gravitational Search Algorithm (GSA)⁸, Charged System Search (CSS)⁹ and Henry gas solubility optimization(HGSO). Swarm-based algorithm mimic social behaviors such as cooperation, competition, and foraging. The most typical algorithms are Ant Colony Optimization (ACO)¹⁰, Particle Swarm Optimization (PSO)¹¹ and Harris Hawks Optimization (HHO)¹².

Recently, many algorithms have been proposed and received more attention from many researchers. For example—Sadoun et al. ¹³ have applied the Dwarf Mongoose Optimization algorithm (DMO)¹⁴ to predict the effect of Al₂O₃ nanoparticle content. In¹⁵, Reptile search Algorithm (RSA)¹⁶ combines with ant colony optimization was developed for churn prediction. In¹⁷, Ebola Optimization Search Algorithm (EOSA)¹⁸ is used to evaluation the performance convolutional neural networks. In¹⁹, Aquila Optimizer (AO)²⁰ is used for weights allocated of forecasting model. In²¹, Discrete Equilibrium Optimizer and Simulated Annealing is hybridized to solved structural optimization and multi-level image segmentation problems. In²², Snake Optimizer (SO) is proposed to tackle real-world engineering problems. In²³, several modified versions of HHO were comprehensively reviewed for engineering application. In²⁴, an Enhanced Remora Optimization Algorithm (EROA) is developed for constrained engineering problems. In²⁵, a modified COOT algorithm is presented to the dimensionality reduction problem. In²⁶, a modified version of Aquila Optimizer (MAO) is developed to solve CEC2017 test suit and five different engineering problems. In²⁷, an Improved Wild Horse Optimizer (IWHO) is proposed to solve high-dimensional cases. In²⁸, a self‑adaptive Harris Hawks optimization algorithm with opposition‑based learning and chaotic local search strategy is used to solve constrained problems. In²⁹, Arithmetic Optimization Algorithm (AOA) is used for the image fusion process.

Sine Cosine Algorithm (SCA)is a swarm-based optimization algorithm that evolves with the help of mathematical models³⁰. The SCA algorithm via a set of random solutions as the beginning, which perfectly avoids the defect that other classical algorithms fall into local optimum in the early search process of optimization. During the search process, periodic sine and cosine trigonometric functions were adopted to adaptively modify the search range to ensure that individuals can fluctuate toward or away from the global optimum³¹. At the same time, the currently found global best position in the population is continuously updated to improve the ability of individual to move to the optimal position. In², SCA was tested on several different types of benchmark problems and successfully optimized the cross-section of an aircraft wing. In³², SCA is mixed with Salp Swarm Algorithm (SSA) for feature selection problems. In³³, an advanced sine cosine algorithm is employed for assign the weight of ensemble prediction model. In³⁴, a modified version of the SCA is proposed for training multilayer perceptron. In³⁵, a dynamic sine cosine algorithm is employed for solve large scale global optimization problems. In³⁶, a hybrid self-adaptive sine cosine algorithm is proposed to solve engineering application problems.

Although many experts employ different strategies to strive to improve its performance of original SCA, it can be seen from the optimization metrics obtained by the improved version on the test problem that in some cases, SCA exhibits low convergence and inability to jump out of local optima.

Therefore, the aim of this paper is to propose a modified version of the SCA named MSCA. The original formula of SCA is adjusted, which effectively avoids the defect of slow convergence, and improves the search range and optimization ability of the algorithm. After that, while retaining the excellent individuals, the personal best position and the Levy flight strategy in the cuckoo are introduced into the mutation operator to ensure the individual diversity.

To prove the superior performance of MSCA on optimization problems, the proposed method has undergone preliminary performance tests on the classic benchmark functions and the CEC2017 test set. Furthermore, to demonstrate the engineering practicality, the proposed method is experimented on complex engineering problems. Compared with the existing advanced algorithms, the proposed method can obtain more reliable results.

The rest of the article is organized as follows: Section “The modified sine cosine algorithm” briefly introduces the Sine Cosine Algorithm (SCA) and proposes a modified version of Sine Cosine Algorithm (MSCA). In Section “MSCA for numerical optimization problems”, 24 classical benchmark functions and CEC2017 test suites are adopted for numerical experiments to prove search performance of MSCA. Six classical engineering optimization problems are adopted to verify the engineering practicability of MSCA in Section “MSCA for engineering optimization problems”. Finally, Section “Conclusion” summarizes the work of this article.

The modified sine cosine algorithm

Brief introduction of SCA

SCA is a swarm-based optimization method proposed by Mirjalili in 2016². In the optimization process of SCA, all solutions find the best possible position in the search space according to the pattern of sine and cosine random cycles. Same as other algorithm optimization stages, the exploration phase with high disturbance and the development phase with low randomness constitute the optimization process of SCA. The diagram of SCA search mode as shown in Fig. 1, and the two-phases position update equation of SCA is shown as follows:

$$X_{ij}^{k+1}=\left\{{}_{X_{ij}^k+r_1\times cos\left(r_2\right)\times \left|r_3\times P_j^k-X_{ij}^k\right|,\begin{array}{c}{r}_4\ge 0.5\end{array}}^{X_{ij}^k+r_1\times sin\left(r_2\right)\times \left|r_3\times P_j^k-X_{ij}^k\right|,\begin{array}{c}{r}_4<0.5\end{array}}\right)$$ 1

where $X_{ij}^{k+1}$ denotes the position of ith individual in jth dimension at (k + 1)th iteration. $P_j^k$ denotes the global best position in jth dimension at kth iteration. The parameter $r_1$ decreases linearly with the iterative process, which is used to ensure the balance between exploration and exploitation, the $r_1$ is defined as follows:

$$r_1=a-k\frac{a}{\overline{k}}$$ 2

where $a$ is a constant number, and $\overline{k}$ is the maximum number of iterations. $r_2$ is the random number uniformly distributed in [0, 2π], $r_3$ is the random number uniformly distributed in [0,2], when $r_3>1$, The exchange of information between $P_j^k$ and $X_{ij}^k$ increases; when $r_3<1$, the influence between $P_j^k$ and $X_{ij}^k$ is reduced. $r_4$ is a random number in the interval [0,1], which is used to switch with equal probability between sine cosine trigonometric functions.

Figure 1.

Diagram of SCA search mode.

The MSCA algorithm

In this section, a modified version of SCA is proposed to affronts the drawbacks of SCA exhibits optimization stagnation and slow convergence.

Modified position updating of SCA

In the original SCA, only the global optimal position is multiplied by a random factor, which only ensures that the population performs a local search to the current individual position, and ignores the existence of better potential solutions near the global optimal position.

In order to improve the convergence rate and ensure the search balance between the global optimal position and the current individual position, the update formula of SCA is reconstructed. When $r_4$ is less than 0.5, a search mode containing a sine function is used to find a potential better solution near the global optimal position. Otherwise, the cosine function formula is adopted to find a better solution near the current individual position. The modified position update as follows:

$${\mathit{U}}_i^k=\left\{{}_{{\mathit{X}}_i^k+r_1\times cos\left(r_2\right)\times \left|r_3\times {\mathit{P}}_{}^k-{\mathit{X}}_i^k\right|,\begin{array}{c}\end{array}{r}_4\ge 0.5}^{{\mathit{X}}_i^k+r_1\times sin\left(r_2\right)\times \left|{\mathit{P}}_{}^k-r_3\times {\mathit{X}}_i^k\right|,\begin{array}{c}\end{array}{r}_4<0.5}\right)$$ 3

where ${\mathit{U}}_i^k$ represents the position vector of ith temporary individual at kth iteration. ${\mathit{X}}_i^k$ represents the position vector of ith individual at kth iteration. $r_2$, $r_3$ and $r_4$ are random numbers for each position vector of ith temporary individual. ${\mathit{P}}_{}^k$ represents the position vector of global best-known position at kth iteration.

Levy random walk mutation strategy

In SCA, only the global optimal individual is considered to guide the evolution direction of the population. When individual falls into a local optimal, it is difficult to jump out of the local unfavorable position only with the help of the global optimum position, which eventually leads to optimization stagnation.

Therefore, the personal best position¹¹ and the Levy random walk³⁷ is introduced into MSCA to make up for the shortcoming of cannot jump out of the local optimal. In order to take full advantage of the mutation operator, the global best position is used to form a difference vector with random individuals to act with the Levy random operator, and finally randomly superimposed on the current position or the personal best position to form a new individual. The Mantegna simulates Levy flight with different beta values as presented in Fig. 2. The expression of ${\mathit{X}}_i^{k+1}$ is defined as follows:

$${\mathit{X}}_i^{k+1}=\left\{\begin{array}{cc}{\mathit{U}}_{r_5}^k+\left({\mathit{P}}^k-{\mathit{U}}_{r_6}^k\right)·\mathit{\phi }(-1,1)·\frac{\overline{k}-k}{\overline{k}}·\mathit{Levy}\left(\beta \right),\hfill & r_7<0.5\\ {\mathit{pBest}}_i^k+\left({\mathit{P}}^k-{\mathit{U}}_{r_6}^k\right)·\mathit{\phi }(-1,1)·\frac{\overline{k}-k}{\overline{k}}·\mathit{Levy}\left(\beta \right),\hfill & r_7\ge 0.5\end{array}\right)$$ 4

where ${\mathit{X}}_i^{k+1}$ represents the position vector of ith individual at (k + 1)th iteration. ${\mathit{pBest}}_i^k$ represents the position vector of personal best position of ith individual at kth iteration. $r_5$ and $r_6$ represent the index of randomly selected individuals in the temporary population, $r_5\ne r_6$. $\mathit{\phi }(-1,1)$ represents the random vector in the interval [-1,1]. $\mathit{Levy}\left(\beta \right)$ represents random vector with Levy distribution, $\beta$ is the tuning parameter.

Figure 2.

Mantegna simulates Levy flight with different beta values.

Execution steps of MSCA

The flowchart of the MSCA is shown in Fig. 3:

Figure 3.

Flowchart of the MSCA method.

MSCA for numerical optimization problems

Compared with the SCA, MSCA has the advantages of better convergence speed and avoiding premature convergence because of the improved position update strategy and the Levy random walk mutation strategy. To further demonstrate the performance of MSCA, 24 classical benchmark problems³⁸ and IEEE CEC2017 test suites³⁹ are adopted to verify the effectiveness of proposed strategy.

Benchmark problem set I: classical benchmark problems

The first set of classical benchmark problems can be divided into unimodal (F₁- F₆), multimodal (F₇-F₁₃) and fixed-dimension multimodal (F₁₄-F₂₄). The unimodal was used to validate the convergence performance of the algorithm, the multimodal was used to test the ability of the method to avoid local stagnation, and the fixed-dimension multimodal was designed to examination the balance levels of the algorithm between exploration and exploitation. The 2D shape of the classical problem are listed in Fig. 4. The problems are listed in detail in Table 1. It is worth pointing out that f_min, range and n represent the theoretical optimal value, the upper and lower bounds and the dimension of the problem, respectively.

Figure 4.

A 2D shape of classical benchmark problems.

Table 1.

Details of 24 classical benchmark problems.

Type	Equation	n	Range	fmin
Unimodal	$F_1(x)=\sum _{i=1}^n{x}_i^2$	30	[−100, 100]	0
Unimodal	$F_2\left(x\right)=\sum _{i=1}^n\left|x_i\right|+\prod _{i=1}^n\left|x_i\right|$	30	[−10, 10]	0
Unimodal	$F_3(x)=\sum _{i=1}^n{\left(\sum _{j=1}^i{x}_j\right)}^2$	30	[−100, 100]	0
Unimodal	$F_4\left(x\right)=max\left\{\left|x_i\right|,1\le i\le n\right\}$	30	[−100, 100]	0
Unimodal	$F_5(x)=\sum _{i=1}^{n-1}[100{(x_{i+1}^{}-x_i^2)}^2+{(x_i^{}-1)}^2]$	30	[−30, 30]	0
Unimodal	$F_6(x)=\sum _{i=1}^n{(\left|x_i+0.5\right|)}^2$	30	[−100, 100]	0
Unimodal	$F_7(x)=\sum _{i=1}^{n}i{x}_i^4+random[0,1)$	30	[−1.28, 1.28]	0
Multimodal	$F_8(x)=\sum _{i=1}^n-x_{i}sin(\sqrt{|x_i|})$	30	[−500, 500]	−12,567
Multimodal	$F_9(x)=\sum _{i=1}^n[x_i^2-10cos(2\pi x_i)+10]$	30	[−5.12, 5.12]	0
Multimodal	$F_{10}(x)=-20exp(-0.2\sqrt[]{\frac{1}{n}\sum _{i=1}^n{x}_i^2})-exp(\frac{1}{n}\sum _{i=1}^{n}cos(2\pi x_i))+20+e$	30	[−32, 32]	0
Multimodal	$F_{11}(x)=\frac{1}{4000}\sum _{i=1}^n{x}_i^2-\prod _{i=1}^{n}cos(\frac{x_i}{\sqrt{i}})+1$	30	[−600, 600]	0
Multimodal	$\begin{array}{c}{F}_{12}\left(x\right)=\frac{\pi }{n}\{10{sin}^2(\pi y_1)+\sum _{i=1}^{n-1}{(y_i-1)}^2[1+10{sin}^2(\pi y_{i+1})]\\ +{(y_n-1)}^2\}+\sum _{i=1}^{n}u(x_i,10,100,4)\\ y_i=1+\frac{x_i+1}{4},u(x_i,a,k,m)=\left\{\begin{array}{c}k{(x_i-a)}^m{x}_i>a\\ 0-a\le x_i<a\\ k{(-x_i-a)}^m{x}_i<-a\end{array}\right)\\ \end{array}$	30	[−50, 50]	0
Multimodal	$\begin{array}{c}{F}_{13}(x)=0.1\{{sin}^2(3\pi x_1)+\sum _{i=1}^n{(x_i-1)}^2[1+{sin}^2(3\pi x_{i+1}+1)]\\ +{(x_n-1)}^2[1+{sin}^2(2\pi x_n)]\}+\sum _{i=1}^{n}u(x_i,5,100,4)\\ \end{array}$	30	[−50, 50]	0
Fixed	$F_{14}(x)={[\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^2{(x_i-a_{\mathit{ij}}^{})}^6}]}^{-1}$	2	[−65.536,65.536]	1
Fixed	$F_{15}(x)=\sum _{i=1}^{11}{\left[a_i-\frac{x_1(b_i^2+b_i^{}{x}_2^{})}{b_i^2+b_i^{}{x}_3^{}+x_4^{}}\right]}^2$	4	[−5, 5]	0.0003075
Fixed	$F_{16}(x)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1^{}{x}_2^{}-4x_2^2+4x_2^4$	2	[−5, 5]	−1.0316285
Fixed	$F_{17}(x)={(x_2^{}-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6)}^2+10(1-\frac{1}{8\pi })cos{x}_1+10$	2	[−5,10] × [0,15]	0.398
Fixed	$\begin{array}{c}{F}_{18}(x)=[1+{(x_1^{}+x_2^{}+1)}^2(19-14x_1^{}+3x_1^2-14x_2^{}\\ +6x_1^{}{x}_2^{}+3x_1^2)]\times [30+{(2x_1^{}-3x_2^{})}^2(18-32x_1^{}\\ +12x_1^2+48x_2^{}-36x_1^{}{x}_2^{}+27x_2^2)]\\ \end{array}$	2	[−2, 2]	3
Fixed	$F_{19}(x)=-\sum _{i=1}^4{c}_{i}exp[-\sum _{j=1}^3{a}_{\mathit{ij}}{(x_j-p_{\mathit{ij}})}^2]$	3	[0,1]	−3.86
Fixed	$F_{20}(x)=-\sum _{i=1}^4{c}_{i}exp[-\sum _{j=1}^6{a}_{\mathit{ij}}{(x_j-p_{\mathit{ij}})}^2]$	6	[0,1]	−3.32
Fixed	$F_{22}(x)=-\sum _{i=1}^5{[(x-a_i){(x-a_i)}^T+c_i]}^{-1}$	4	[0,10]	−10.1532
Fixed	$F_{23}(x)=-\sum _{i=1}^7{[(x-a_i){(x-a_i)}^T+c_i]}^{-1}$	4	[0,10]	−10.4028
Fixed	$F_{23}(x)=-\sum _{i=1}^{10}{[(x-a_i){(x-a_i)}^T+c_i]}^{-1}$	4	[0,10]	−10.5363
Fixed	$F_{24}(x)=x_1^2+x_2^2+25\left({sin}^2(x_1)+{sin}^2(x_2)\right)$	2	[−5, 5]	0

Parameter settings

For fair comparison, seven existing persuasive algorithms are introduced to compare with MSCA, including GA⁴⁰, PSO¹¹, GSA⁸, JAYA⁴¹, ALO⁴², MVO⁴³, GWO⁴⁴, WOA⁴, SSA⁴⁵, HGSO¹², AOA⁴⁶ and SCA². the population size was set to 50, and the maximum iterations was set to 500 in the selected algorithm. For avoid the influence of randomness, each algorithm was independently repeated for 20 times. The default parameter value settings of all the algorithms are listed in Table 2.

Table 2.

The default parameters of selected algorithm.

Algorithm	Detailed parameter	Value
GA	Mutation probability	0.05
	Crossover probability	0.6
PSO	Cognitive coefficient(C1)	2.0
	Cognitive coefficient(C2)	2.0
	Weight (wmax)	0.8
	Weight (wmin)	0.3
GSA	Attenuation factor a	20
	Initial gravitational constant G0	100
MVO	Minimum of Wormhole Existence Probability	0.2
	Maximum of Wormhole Existence Probability	1.0
GWO	Constant a	2.0
WOA	Constant a	2.0
SSA	Coefficient c1	$2exp{\left[\left(-4k\right)/\overline{k}\right]}^2$
HGSO	Cluster number	5
AOA	Control parameter $\mu$	0.5
	Control parameter $\alpha$	5
SCA	Constant a	2.0
MSCA	Constant a	2.0
	Tuning parameter β	1.5

Experimental results analysis

To obey the univariate principle, all algorithms are run independently in the same environment. The comparison results of selected algorithms for 24 benchmark problems are given in Table 3, including the mean and standard deviation (STD). The results of the Wilcoxon ranksum and signed-rank test at 5% level significance level are reported in Tables 4 and 5, respectively. It should be pointed out that in Table 5 , if the result of MSCA is better than the competitive algorithms, that the MSCA is recorded as win, if equal, recorded as tie, otherwise recorded as lose.

Table 3.

Comparison results of selected algorithms for classical benchmark problem.

Function	Item	GA	PSO	GSA	JAYA	ALO	MVO	GWO	WOA	SSA	HGSO	AOA	SCA	MSCA
F1	Mean	1.45E + 03	8.84E-04	4.85E-12	3.98E + 01	1.43E-04	8.62E-01	1.93E-19	1.91E-69	2.85E-05	3.02E-17	3.51E-23	4.72E + 00	1.13E-75
	STD	2.51E + 02	9.82E-04	6.97E-12	9.40E + 00	7.88E-05	2.07E-01	2.63E-19	7.12E-69	5.78E-05	1.14E-16	1.57E-22	8.71E + 00	6.21E-75
F2	Mean	1.28E + 01	5.00E + 00	7.37E-02	6.14E + 00	4.80E + 01	1.12E + 01	4.98E-12	8.59E-52	3.25E + 00	1.58E-15	3.26E-115	1.38E-02	6.76E-47
	STD	1.71E + 00	6.82E + 00	2.64E-01	2.54E + 00	4.86E + 01	3.22E + 01	2.95E-12	3.72E-51	1.91E + 00	5.42E-15	1.01E-114	2.05E-02	1.94E-46
F3	Mean	2.07E + 04	5.90E + 03	7.04E + 02	3.85E + 04	2.35E + 03	7.84E + 01	5.07E-03	5.96E + 04	2.25E + 03	2.04E + 02	5.01E-03	6.92E + 03	8.51E + 00
	STD	4.07E + 03	4.76E + 03	2.22E + 02	5.66E + 03	1.13E + 03	3.74E + 01	1.38E-02	1.91E + 04	7.83E + 02	3.76E + 02	1.12E-02	4.67E + 03	1.48E + 01
F4	Mean	2.26E + 01	7.77E + 00	4.03E + 00	2.74E + 01	1.75E + 01	1.47E + 00	1.92E-04	4.51E + 01	1.54E + 01	9.38E-02	3.51E-02	2.46E + 01	2.18E-32
	STD	9.68E-01	1.77E + 00	1.14E + 00	5.56E + 00	5.31E + 00	6.18E-01	1.64E-04	2.87E + 01	5.75E + 00	1.94E-01	1.31E-02	9.57E + 00	1.19E-31
F5	Mean	2.56E + 05	3.85E + 03	8.91E + 01	3.88E + 03	2.53E + 02	3.62E + 02	2.79E + 01	2.85E + 01	4.50E + 02	2.82E + 01	2.85E + 01	1.61E + 04	2.55E + 01
	STD	6.47E + 04	1.63E + 04	6.94E + 01	2.18E + 03	3.73E + 02	5.68E + 02	7.92E-01	3.41E-01	7.22E + 02	2.86E-01	2.40E-01	4.24E + 04	1.66E-01
F6	Mean	1.41E + 03	1.03E-03	6.09E-12	6.79E + 01	1.04E-04	8.26E-01	1.95E + 00	1.09E + 00	1.74E-05	3.40E + 00	3.84E + 00	8.48E + 00	8.46E-04
	STD	2.46E + 02	1.10E-03	1.05E-11	2.22E + 01	5.34E-05	2.07E-01	6.30E-01	3.43E-01	2.26E-05	3.39E-01	2.60E-01	5.47E + 00	1.99E-03
F7	Mean	6.54E-01	1.04E + 00	5.55E-01	7.09E-01	5.72E-01	5.25E-01	3.67E-01	5.05E-01	6.95E-01	5.13E-01	4.54E-01	6.49E-01	4.52E-01
	STD	3.14E-01	2.13E + 00	2.95E-01	2.85E-01	2.67E-01	2.74E-01	3.19E-01	2.21E-01	3.47E-01	2.84E-01	2.33E-01	2.57E-01	2.98E-01
F8	Mean	-1.13E + 04	-9.78E + 03	-2.87E + 03	-5.19E + 03	-5.78E + 03	-8.13E + 03	-5.45E + 03	-9.38E + 03	-7.23E + 03	-4.07E + 03	-3.03E + 38	-3.90E + 03	-1.24E + 04
	STD	2.02E + 02	6.03E + 02	4.19E + 02	5.98E + 02	8.90E + 02	7.61E + 02	8.05E + 02	1.77E + 03	8.03E + 02	5.81E + 02	1.35E + 39	2.99E + 02	6.26E + 02
F9	Mean	5.55E + 01	5.72E + 01	1.87E + 01	2.56E + 02	8.34E + 01	1.12E + 02	1.74E + 01	0.00E + 00	5.16E + 01	8.27E + 00	0.00E + 00	3.90E + 01	0.00E + 00
	STD	6.38E + 00	1.81E + 01	4.11E + 00	2.00E + 01	2.03E + 01	3.14E + 01	5.25E + 00	0.00E + 00	1.43E + 01	2.69E + 01	0.00E + 00	3.87E + 01	0.00E + 00
F10	Mean	8.93E + 00	2.44E-01	9.00E-07	7.70E + 00	2.38E + 00	1.51E + 00	9.54E-11	4.09E-15	3.57E + 00	5.34E-10	0.00E + 00	1.38E + 01	0.00E + 00
	STD	6.41E-01	4.66E-01	4.97E-07	6.11E + 00	8.25E-01	6.08E-01	6.30E-11	2.89E-15	1.18E + 00	1.82E-09	0.00E + 00	9.08E + 00	0.00E + 00
F11	Mean	1.40E + 01	1.72E-02	1.70E + 01	1.39E + 00	2.28E-02	7.58E-01	8.04E-03	0.00E + 00	4.53E-02	4.53E-12	3.47E-01	8.24E-01	0.00E + 00
	STD	2.11E + 00	1.56E-02	3.87E + 00	1.11E-01	1.27E-02	9.93E-02	1.47E-02	0.00E + 00	2.88E-02	1.96E-11	1.82E-01	3.17E-01	0.00E + 00
F12	Mean	9.83E + 01	2.84E-01	1.18E + 00	1.93E + 01	1.15E + 01	1.35E + 00	1.55E-01	7.90E-02	8.39E + 00	5.33E-01	6.18E-01	4.07E + 03	2.42E-05
	STD	1.85E + 02	3.95E-01	7.00E-01	9.80E + 00	4.75E + 00	1.10E + 00	1.32E-01	3.32E-02	3.15E + 00	1.50E-01	6.88E-02	2.21E + 04	5.04E-06
F13	Mean	7.12E + 04	5.61E-02	8.22E + 00	4.26E + 01	1.17E + 01	1.21E-01	1.40E + 00	1.28E + 00	2.86E + 01	2.48E + 00	2.67E + 00	3.81E + 04	3.78E-03
	STD	3.87E + 04	7.19E-02	6.16E + 00	1.01E + 02	1.47E + 01	4.92E-02	2.78E-01	3.61E-01	1.40E + 01	1.77E-01	1.24E-01	1.91E + 05	1.87E-02
F14	Mean	9.98E-01	9.98E-01	4.03E + 00	9.98E-01	1.63E + 00	9.98E-01	5.93E + 00	3.20E + 00	1.25E + 00	1.38E + 00	8.37E + 00	1.40E + 00	9.98E-01
	STD	9.96E-09	4.12E-17	2.98E + 00	2.73E-04	8.82E-01	5.00E-11	4.81E + 00	3.66E + 00	4.42E-01	4.71E-01	4.53E + 00	8.07E-01	1.43E-15
F15	Mean	1.05E-03	2.77E-03	6.28E-03	6.32E-04	3.50E-03	5.42E-03	4.41E-03	6.49E-04	4.81E-03	5.46E-04	1.88E-02	9.99E-04	6.13E-04
	STD	4.01E-04	5.97E-03	3.33E-03	4.37E-04	6.73E-03	8.39E-03	8.19E-03	3.83E-04	8.04E-03	1.30E-04	3.39E-02	3.36E-04	4.11E-04
F16	Mean	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00
	STD	2.27E-05	6.71E-16	1.00E-15	3.05E-05	7.51E-14	5.14E-07	1.26E-08	6.85E-09	6.21E-14	1.80E-05	1.89E-07	3.75E-05	1.58E-13
F17	Mean	3.98E-01	3.98E-01	3.98E-01	3.99E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.99E-01	4.18E-01	4.00E-01	3.98E-01
	STD	8.57E-05	0.00E + 00	8.97E-15	1.66E-03	3.89E-14	7.72E-07	2.13E-06	3.99E-04	6.10E-13	1.37E-03	1.27E-02	1.67E-03	7.78E-11
F18	Mean	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	2.17E + 01	3.00E + 00	3.00E + 00
	STD	4.14E-04	1.24E-15	1.10E-13	7.97E-04	3.65E-13	4.09E-06	1.53E-04	3.99E-03	8.44E-13	3.75E-04	3.20E + 01	2.47E-05	7.71E-13
F19	Mean	-3.86E + 00	-3.86E + 00	-3.70E + 00	-3.86E + 00	-3.86E + 00	-3.86E + 00	-3.86E + 00	-3.83E + 00	-3.86E + 00	-3.86E + 00	-3.85E + 00	-3.86E + 00	-3.86E + 00
	STD	7.35E-06	2.65E-15	2.46E-01	2.71E-15	1.00E-13	2.19E-06	2.09E-03	5.36E-02	2.39E-02	3.36E-03	4.64E-03	3.40E-03	9.53E-10
F20	Mean	-3.28E + 00	-3.28E + 00	-2.29E + 00	-3.24E + 00	-3.26E + 00	-3.25E + 00	-3.27E + 00	-3.08E + 00	-3.28E + 00	-3.27E + 00	-3.00E + 00	-3.03E + 00	-3.29E + 00
	STD	5.69E-02	6.57E-02	7.37E-01	6.01E-02	6.06E-02	5.89E-02	8.36E-02	2.68E-01	8.77E-02	6.66E-02	1.06E-01	1.37E-01	5.11E-02
F21	Mean	-6.95E + 00	-8.48E + 00	-4.98E + 00	-7.22E + 00	-5.94E + 00	-6.87E + 00	-9.53E + 00	-7.52E + 00	-9.40E + 00	-8.09E + 00	-4.40E + 00	-2.56E + 00	-9.30E + 00
	STD	3.54E + 00	2.90E + 00	4.22E-01	2.46E + 00	2.75E + 00	3.04E + 00	1.97E + 00	2.54E + 00	1.85E + 00	1.99E + 00	1.74E + 00	1.94E + 00	1.93E + 00
F22	Mean	-7.45E + 00	-8.28E + 00	-8.10E + 00	-8.99E + 00	-7.25E + 00	-9.80E + 00	-1.01E + 01	-7.80E + 00	-8.96E + 00	-7.49E + 00	-3.62E + 00	-4.34E + 00	-1.02E + 01
	STD	3.49E + 00	3.12E + 00	2.68E + 00	2.15E + 00	3.08E + 00	1.88E + 00	1.19E + 00	2.81E + 00	2.60E + 00	2.41E + 00	1.66E + 00	1.62E + 00	9.70E-01
F23	Mean	-7.39E + 00	-9.61E + 00	-9.11E + 00	-8.96E + 00	-8.08E + 00	-9.73E + 00	-9.72E + 00	-7.09E + 00	-1.05E + 01	-8.09E + 00	-3.85E + 00	-4.70E + 00	-9.82E + 00
	STD	3.64E + 00	2.44E + 00	2.70E + 00	2.27E + 00	3.36E + 00	2.14E + 00	2.50E + 00	2.83E + 00	3.26E-10	2.29E + 00	1.91E + 00	1.23E + 00	1.87E + 00
F24	Mean	9.09E-05	8.77E-73	3.38E-16	6.39E-47	2.41E-13	1.90E-06	2.75E-148	8.74E-94	1.15E-13	1.07E-118	0.00E + 00	8.00E-78	2.31E-229
	STD	1.13E-04	1.77E-72	6.02E-16	2.08E-46	2.21E-13	1.67E-06	1.21E-147	3.91E-93	1.12E-13	3.92E-118	0.00E + 00	2.99E-77	0.00E + 00

Significant values are in bold.

Table 4.

Wilcoxon ranksum test on classical benchmark functions at 5% level of significance.

Function	GA vs MSCA	PSO vs MSCA	GSA vs MSCA	JAYA vs MSCA	ALO vs MSCA	MVO vs MSCA	GWO vs MSCA	WOA vs MSCA	SSA vs MSCA	HGSO vs MSCA	AOA vs MSCA	SCA vs MSCA
F1	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	1.92E-07	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F2	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	3.94E-07	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F3	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	1.06E-07	3.29E-05	6.80E-08	6.80E-08	4.90E-01	2.69E-06	6.80E-08
F4	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F5	6.80E-08	1.20E-06	6.80E-08	6.80E-08	1.20E-06	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F6	6.80E-08	4.57E-01	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F7	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	1.16E-04	6.80E-08	6.80E-08	6.80E-08	3.94E-07	2.56E-07
F8	1.20E-06	6.92E-07	6.80E-08	6.80E-08	6.46E-08	1.23E-07	6.80E-08	1.80E-06	6.80E-08	6.80E-08	1.60E-05	6.80E-08
F9	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	N/A	8.01E-09	6.68E-05	N/A	8.01E-09
F10	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	2.71E-06	8.01E-09	7.86E-09	N/A	8.01E-09
F11	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	8.01E-09	3.37E-07	N/A	8.01E-09	1.98E-02	8.01E-09	8.01E-09
F12	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F13	6.80E-08	6.92E-07	7.90E-08	6.80E-08	9.17E-08	2.56E-07	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F14	4.67E-08	4.91E-09	8.85E-07	4.67E-08	4.41E-01	4.67E-08	4.61E-08	4.67E-08	5.88E-01	4.67E-08	4.67E-08	4.67E-08
F15	3.15E-02	9.07E-02	6.80E-08	1.64E-01	3.15E-02	6.04E-03	9.25E-01	3.51E-01	9.79E-03	3.94E-01	9.21E-04	9.79E-03
F16	6.66E-08	7.80E-09	7.80E-09	6.66E-08	1.14E-02	6.66E-08	6.66E-08	6.66E-08	2.48E-03	6.66E-08	6.66E-08	6.66E-08
F17	6.80E-08	8.01E-09	4.63E-08	1.71E-01	8.50E-07	6.80E-08	6.80E-08	6.80E-08	1.41E-05	6.80E-08	6.80E-08	6.80E-08
F18	6.76E-08	7.95E-09	2.26E-04	6.76E-08	4.02E-01	6.76E-08	6.76E-08	6.76E-08	8.29E-01	6.76E-08	8.47E-07	6.76E-08
F19	6.80E-08	8.01E-09	6.80E-08	8.01E-09	6.46E-08	6.80E-08	6.80E-08	6.80E-08	9.13E-07	6.80E-08	6.80E-08	6.80E-08
F20	6.61E-05	2.34E-01	1.61E-04	8.37E-01	9.25E-01	5.87E-06	1.16E-04	2.06E-06	7.11E-03	3.75E-04	6.80E-08	2.22E-07
F21	2.60E-05	2.81E-03	7.94E-04	1.78E-03	9.79E-03	2.60E-05	8.36E-04	1.60E-05	4.17E-05	2.47E-04	5.87E-06	1.23E-07
F22	6.92E-07	2.25E-02	4.39E-01	6.74E-01	4.90E-01	1.05E-06	1.05E-06	3.42E-07	5.12E-03	4.54E-07	1.23E-07	1.92E-07
F23	1.58E-06	6.80E-05	8.00E-05	3.09E-01	7.64E-02	7.58E-06	9.75E-06	6.92E-07	6.80E-08	4.54E-06	2.22E-07	1.66E-07
F24	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	8.01E-09	6.80E-08

Significant values are in bold.

Table 5.

Wilcoxon signed-rank test on classical benchmark functions at 5% level of significance.

Item	Win	Tie	Lose	R+	R-	p-value	Decision
GA vs MSCA	24	0	0	300	0	1.82E-05	+
PSO vs MSCA	19	0	5	280	20	2.04E-04	+
GSA vs MSCA	20	0	4	283	17	1.45E-04	+
JAYA vs MSCA	23	0	1	298	2	2.35E-05	+
ALO vs MSCA	19	0	5	281	19	1.82E-04	+
MVO vs MSCA	24	0	0	300	0	1.82E-05	+
GWO vs MSCA	21	0	3	248	52	5.11E-03	+
WOA vs MSCA	21	2	1	293.5	6.5	6.08E-05	+
SSA vs MSCA	19	0	5	263	37	1.24E-03	+
AOA vs MSCA	18	2	4	245.5	54.5	8.15E-03	+
SCA vs MSCA	24	0	0	300	0	1.82E-05	+

From Table 3, MSCA outperforms GA, MVO and SCA on the mean of all benchmark problems, and the standard deviation of MSCA is smaller than SCA. Furthermore, MSCA wins most problems compared to PSO, GSA, JAYA and ALO.

In Table 4, The p-value of the vast majority of rank sum tests are less than 5%, which illustrates the variability in the overall distribution of target values for the proposed and control methods. At the same time, MSCA achieved p-values of less than 5% in all sign-rank comparisons in Table 5, indicating that the results obtained by MSCA are statistically significant. Overall, MSCA can balance the process between exploration and exploitation.

Robustness and convergence analysis

For validate the robustness and convergence performance of MSCA in classical benchmark problems, the box-plot of all results and convergence curve of best result for JAYA, ALO, MVO, SCA and MSCA are plotted in Figs. 5 and 6.respectively.

Figure 5.

The box-plot of different algorithms on classical benchmark problems.

Figure 6.

The convergence curve on classical benchmark problems.

Box-plot revealed the distribution of results obtained by each method. It can be seen from Fig. 5 that JAYA, ALO, MVO and SCA show different degrees of dispersion in different benchmark problems, while MCSA get more stable and better distribution. And it can be seen from the convergence curve in Fig. 6 that MSCA has a faster convergence rate than other algorithms. Hence, the modified SCA position update formula combined with Levy random walk mutation strategy can get more accurate results.

Benchmark problem set II: CEC2017 test suites

In this section, IEEE CEC2017 test suites are adopted to prove the performance of MSCA, which are mainly divided into four groups: (a) Unimodal, (b) Multimodal, (c) Hybrid and (d) Composition. The definition of the CEC2017 benchmark problems is shown in Table 6. Unlike classical benchmark function problems, this problem involves matrix rotation and global optimum shifted operations.

Table 6.

The definition of CEC2017 test suites.

Type	Function	Name	n	Range	fmin
Unimodal	F1:	Shifted and Rotated Bent Cigar Function	10	[−100, 100]	100
Unimodal	F2:	Shifted and Rotated Sum of Different Power Function	10	[−100, 100]	200
Unimodal	F3:	Shifted and Rotated Zakharov Function	10	[−100, 100]	300
Multimodal	F4:	Shifted and Rotated Rosenbrock’s Function	10	[−100, 100]	400
Multimodal	F5:	Shifted and Rotated Rastrigin’s Function	10	[−100, 100]	500
Multimodal	F6:	Shifted and Rotated Expanded Scaffer’s F6 Function	10	[−100, 100]	600
Multimodal	F7:	Shifted and Rotated Lunacek Bi_Rastrigin Function	10	[−100, 100]	700
Multimodal	F8:	Shifted and Rotated Non−Continuous Rastrigin’s Function	10	[−100, 100]	800
Multimodal	F9:	Shifted and Rotated Levy Function	10	[−100, 100]	900
Multimodal	F10:	Shifted and Rotated Schwefel’s Function	10	[−100, 100]	1000
Hybrid	F11:	Hybrid Function 1 (N = 3)	10	[−100, 100]	1100
Hybrid	F12:	Hybrid Function 2 (N = 3)	10	[−100, 100]	1200
Hybrid	F13:	Hybrid Function 3 (N = 3)	10	[−100, 100]	1300
Hybrid	F14:	Hybrid Function 4 (N = 4)	10	[−100, 100]	1400
Hybrid	F15:	Hybrid Function 5 (N = 4)	10	[−100, 100]	1500
Hybrid	F16:	Hybrid Function 6 (N = 4)	10	[−100, 100]	1600
Hybrid	F17:	Hybrid Function 6 (N = 5)	10	[−100, 100]	1700
Hybrid	F18:	Hybrid Function 6 (N = 5)	10	[−100, 100]	1800
Hybrid	F19:	Hybrid Function 6 (N = 5)	10	[−100, 100]	1900
Hybrid	F20:	Hybrid Function 6 (N = 6)	10	[−100, 100]	2000
Composition	F21:	Composition Function 1 (N = 3)	10	[−100, 100]	2100
Composition	F22:	Composition Function 2 (N = 3)	10	[−100, 100]	2200
Composition	F23:	Composition Function 3 (N = 4)	10	[−100, 100]	2300
Composition	F24:	Composition Function 4 (N = 4)	10	[−100, 100]	2400
Composition	F25:	Composition Function 5 (N = 5)	10	[−100, 100]	2500
Composition	F26:	Composition Function 6 (N = 5)	10	[−100, 100]	2600
Composition	F27:	Composition Function 7 (N = 6)	10	[−100, 100]	2700
Composition	F28:	Composition Function 8 (N = 6)	10	[−100, 100]	2800
Composition	F29:	Composition Function 9 (N = 3)	10	[−100, 100]	2900
Composition	F30:	Composition Function 10 (N = 3)	10	[−100, 100]	3000

Experimental results analysis

The statistical results for CEC2017 are illustrated in Table 7. From the table, it can be concluded that MSCA outperforms the other popular methods in most of the CEC2017 benchmark problems. In addition, from the results of the Wilcoxon ranksum and signed-rank test in Tables 8 and 9, it can be concluded that the proposed method is effective in improving the search efficiency of SCA and provides better results than other methods. In conclusion, MSCA is a quite competitive algorithm.

Table 7.

Statistic result of the algorithms on CEC2017 test suit with 10 dimensions.

Function	Item	GA	PSO	GSA	JAYA	ALO	MVO	GWO	WOA	SSA	HGSO	AOA	SCA	MSCA
F1	Mean	1.12E + 07	9.35E + 06	3.28E + 02	3.81E + 08	1.84E + 03	1.69E + 04	1.78E + 08	4.47E + 08	2.13E + 03	4.40E + 08	1.02E + 10	8.37E + 08	1.55E + 05
	STD	5.63E + 06	2.42E + 07	4.24E + 02	1.49E + 08	1.89E + 03	5.93E + 03	2.06E + 08	2.87E + 08	2.22E + 03	4.52E + 08	4.29E + 09	2.45E + 08	8.44E + 04
F2	Mean	6.44E + 06	1.04E + 09	4.60E + 10	4.31E + 07	1.43E + 03	7.26E + 02	5.45E + 09	5.69E + 09	2.04E + 05	6.10E + 08	8.30E + 14	1.19E + 09	1.74E + 05
	STD	1.22E + 07	5.58E + 09	7.19E + 10	4.05E + 07	2.69E + 03	1.03E + 03	9.60E + 09	1.58E + 10	3.55E + 05	1.86E + 09	1.39E + 15	2.48E + 09	3.18E + 05
F3	Mean	3.79E + 03	3.00E + 02	1.58E + 04	8.46E + 03	3.00E + 02	3.00E + 02	5.17E + 03	1.24E + 04	3.45E + 02	3.28E + 03	1.50E + 04	2.14E + 03	3.79E + 02
	STD	1.47E + 03	3.60E-03	3.32E + 03	2.05E + 03	1.68E-01	7.99E-02	2.85E + 03	8.10E + 03	1.36E + 02	1.30E + 03	4.11E + 03	1.35E + 03	5.77E + 01
F4	Mean	4.07E + 02	4.24E + 02	4.25E + 02	4.10E + 02	4.11E + 02	4.05E + 02	4.29E + 02	4.80E + 02	4.06E + 02	4.35E + 02	1.13E + 03	4.54E + 02	4.07E + 02
	STD	9.05E-01	3.94E + 01	1.79E + 01	6.11E-01	1.76E + 01	1.47E + 00	2.82E + 01	5.93E + 01	2.20E + 00	1.82E + 01	4.70E + 02	2.06E + 01	5.94E-01
F5	Mean	5.68E + 02	1.08E + 03	6.13E + 02	1.13E + 03	5.24E + 02	5.29E + 02	5.25E + 02	5.69E + 02	5.30E + 02	5.44E + 02	5.71E + 02	1.95E + 03	5.34E + 02
	STD	1.42E + 01	8.77E + 02	1.73E + 02	2.05E + 02	8.60E + 00	1.19E + 01	1.37E + 01	2.17E + 01	1.32E + 01	8.66E + 00	1.89E + 01	4.93E + 02	6.38E + 00
F6	Mean	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.05E + 02	6.45E + 02	6.21E + 02	6.14E + 02	6.42E + 02	6.00E + 02	6.00E + 02
	STD	3.47E-03	4.87E-03	3.24E-02	1.10E-02	2.79E-02	1.13E-02	3.49E + 00	1.33E + 01	1.05E + 01	7.71E + 00	5.27E + 00	1.58E-02	1.51E-04
F7	Mean	7.37E + 02	7.24E + 02	7.24E + 02	7.63E + 02	7.41E + 02	7.29E + 02	7.41E + 02	7.94E + 02	7.39E + 02	7.65E + 02	7.93E + 02	7.81E + 02	7.29E + 02
	STD	5.81E + 00	5.27E + 00	5.67E + 00	8.03E + 00	1.41E + 01	9.68E + 00	1.57E + 01	2.01E + 01	1.16E + 01	1.17E + 01	1.27E + 01	1.01E + 01	4.03E + 00
F8	Mean	8.16E + 02	8.13E + 02	8.23E + 02	8.47E + 02	8.21E + 02	8.19E + 02	8.20E + 02	8.47E + 02	8.30E + 02	8.40E + 02	8.45E + 02	8.45E + 02	8.09E + 02
	STD	3.26E + 00	5.14E + 00	6.23E + 00	6.18E + 00	1.03E + 01	9.59E + 00	7.64E + 00	1.84E + 01	9.80E + 00	7.50E + 00	1.25E + 01	7.41E + 00	2.69E + 00
F9	Mean	9.00E + 02	9.00E + 02	9.00E + 02	9.00E + 02	9.00E + 02	9.00E + 02	9.46E + 02	1.81E + 03	1.03E + 03	9.79E + 02	1.39E + 03	9.00E + 02	9.00E + 02
	STD	1.93E-02	1.45E-01	5.83E-01	1.00E-01	3.14E-01	6.88E-01	6.89E + 01	6.45E + 02	1.50E + 02	5.27E + 01	2.13E + 02	1.44E-01	5.26E-05
F10	Mean	1.49E + 03	1.52E + 03	2.90E + 03	2.25E + 03	2.00E + 03	1.70E + 03	1.88E + 03	2.32E + 03	1.96E + 03	2.59E + 03	2.43E + 03	2.40E + 03	1.29E + 03
	STD	1.51E + 02	2.15E + 02	2.40E + 02	2.26E + 02	3.94E + 02	2.66E + 02	4.13E + 02	3.71E + 02	3.27E + 02	2.07E + 02	2.66E + 02	2.01E + 02	1.36E + 02
F11	Mean	1.27E + 03	1.17E + 03	1.17E + 04	2.61E + 03	1.17E + 03	1.14E + 03	1.18E + 03	1.36E + 03	1.17E + 03	1.18E + 03	5.60E + 03	2.68E + 03	1.13E + 03
	STD	8.15E + 01	3.13E + 02	2.65E + 03	4.85E + 02	6.92E + 01	7.33E + 01	9.02E + 01	2.60E + 02	4.89E + 01	2.79E + 01	6.71E + 03	1.14E + 03	2.14E + 01
F12	Mean	6.64E + 05	1.11E + 06	1.50E + 06	9.96E + 06	1.97E + 06	1.27E + 06	8.08E + 05	4.53E + 06	3.27E + 06	7.92E + 06	3.71E + 08	1.72E + 07	1.84E + 05
	STD	5.07E + 05	2.83E + 06	8.49E + 05	9.25E + 06	2.11E + 06	1.13E + 06	8.92E + 05	5.44E + 06	4.58E + 06	6.79E + 06	3.07E + 08	1.68E + 07	3.41E + 05
F13	Mean	1.22E + 04	8.39E + 03	1.13E + 04	1.85E + 04	1.51E + 04	1.30E + 04	1.35E + 04	1.77E + 04	1.66E + 04	3.16E + 04	2.51E + 07	4.33E + 04	3.97E + 03
	STD	8.41E + 03	8.48E + 03	2.18E + 03	1.63E + 04	1.30E + 04	1.07E + 04	9.55E + 03	1.91E + 04	1.10E + 04	2.71E + 04	6.38E + 07	2.47E + 04	4.03E + 03
F14	Mean	2.34E + 03	2.08E + 03	1.42E + 04	4.40E + 03	1.24E + 04	1.98E + 03	5.24E + 03	2.97E + 03	4.13E + 03	2.77E + 03	7.80E + 03	6.49E + 03	1.46E + 03
	STD	1.03E + 03	1.03E + 03	2.86E + 03	1.70E + 03	6.85E + 03	8.71E + 02	2.40E + 03	1.75E + 03	4.15E + 03	1.35E + 03	7.29E + 03	2.24E + 03	1.46E + 01
F15	Mean	2.95E + 03	4.63E + 03	2.87E + 04	1.02E + 04	3.08E + 04	3.01E + 03	9.88E + 03	1.00E + 04	9.93E + 03	4.50E + 03	1.61E + 04	8.43E + 03	1.59E + 03
	STD	1.37E + 03	6.30E + 03	5.87E + 03	6.34E + 03	1.65E + 04	2.27E + 03	6.75E + 03	6.62E + 03	7.74E + 03	2.42E + 03	7.72E + 03	3.37E + 03	1.32E + 02
F16	Mean	1.66E + 03	1.74E + 03	2.25E + 03	1.73E + 03	1.90E + 03	1.75E + 03	1.79E + 03	2.05E + 03	1.82E + 03	1.80E + 03	2.13E + 03	1.80E + 03	1.62E + 03
	STD	6.33E + 01	1.34E + 02	1.05E + 02	4.75E + 01	1.85E + 02	1.25E + 02	1.29E + 02	2.08E + 02	1.62E + 02	8.73E + 01	1.81E + 02	9.61E + 01	3.22E + 01
F17	Mean	1.74E + 03	1.78E + 03	1.88E + 03	1.93E + 03	1.83E + 03	1.82E + 03	1.78E + 03	1.84E + 03	1.78E + 03	1.78E + 03	1.89E + 03	1.90E + 03	1.73E + 03
	STD	1.93E + 01	6.23E + 01	1.51E + 02	5.63E + 01	1.03E + 02	7.46E + 01	3.27E + 01	6.59E + 01	2.39E + 01	1.25E + 01	8.68E + 01	7.55E + 01	8.08E + 00
F18	Mean	1.40E + 04	3.36E + 04	2.44E + 04	1.01E + 05	4.40E + 04	2.07E + 04	2.14E + 04	1.90E + 04	2.13E + 04	2.44E + 05	2.54E + 08	2.29E + 05	8.86E + 03
	STD	1.03E + 04	2.16E + 04	6.49E + 03	5.95E + 04	2.99E + 04	1.44E + 04	1.73E + 04	1.32E + 04	1.44E + 04	3.60E + 05	4.88E + 08	2.43E + 05	6.47E + 03
F19	Mean	5.01E + 03	1.36E + 04	6.44E + 05	1.10E + 04	2.58E + 04	2.47E + 03	2.94E + 04	4.73E + 05	7.15E + 03	6.58E + 03	4.99E + 06	1.24E + 04	1.97E + 03
	STD	3.61E + 03	1.65E + 04	5.57E + 05	1.01E + 04	1.77E + 04	8.24E + 02	6.97E + 04	8.96E + 05	6.74E + 03	4.53E + 03	2.15E + 07	8.02E + 03	1.06E + 02
F20	Mean	2.03E + 03	2.08E + 03	2.39E + 03	2.27E + 03	2.20E + 03	2.16E + 03	2.14E + 03	2.24E + 03	2.13E + 03	2.11E + 03	2.16E + 03	2.29E + 03	2.02E + 03
	STD	8.67E + 00	8.39E + 01	1.34E + 02	6.44E + 01	1.02E + 02	8.87E + 01	7.87E + 01	8.93E + 01	6.71E + 01	3.51E + 01	6.19E + 01	6.56E + 01	2.21E + 01
F21	Mean	2.27E + 03	2.56E + 03	2.32E + 03	2.67E + 03	2.21E + 03	2.25E + 03	2.32E + 03	2.34E + 03	2.28E + 03	2.24E + 03	2.36E + 03	2.62E + 03	2.20E + 03
	STD	5.06E + 01	2.47E + 02	9.78E + 01	6.89E + 01	3.19E + 01	6.49E + 01	9.00E + 00	4.81E + 01	5.97E + 01	2.01E + 01	3.66E + 01	2.15E + 02	5.03E-01
F22	Mean	2.31E + 03	2.32E + 03	2.39E + 03	2.33E + 03	2.30E + 03	2.37E + 03	2.37E + 03	2.63E + 03	2.30E + 03	2.34E + 03	3.16E + 03	2.39E + 03	2.31E + 03
	STD	1.60E + 00	2.20E + 01	3.40E + 02	4.46E + 00	2.72E + 00	2.45E + 02	1.72E + 02	5.83E + 02	1.69E + 01	3.46E + 01	3.91E + 02	2.86E + 01	8.96E-01
F23	Mean	2.53E + 03	3.00E + 03	4.76E + 03	2.92E + 03	2.42E + 03	2.41E + 03	2.63E + 03	2.68E + 03	2.62E + 03	2.66E + 03	2.75E + 03	3.30E + 03	2.41E + 03
	STD	3.16E + 01	3.93E + 02	9.91E + 02	2.28E + 02	1.09E + 02	1.12E + 00	1.33E + 01	3.35E + 01	8.74E + 00	2.07E + 01	3.86E + 01	1.51E + 02	2.38E + 00
F24	Mean	2.73E + 03	3.24E + 03	2.54E + 03	4.02E + 03	2.61E + 03	2.61E + 03	2.76E + 03	2.78E + 03	2.75E + 03	2.64E + 03	2.90E + 03	3.85E + 03	2.61E + 03
	STD	3.93E + 01	9.10E + 02	6.33E + 01	3.34E + 02	4.87E + 01	5.81E + 01	1.57E + 01	5.69E + 01	1.19E + 01	6.55E + 01	1.05E + 02	2.28E + 02	4.12E + 01
F25	Mean	2.94E + 03	2.95E + 03	2.95E + 03	2.97E + 03	2.93E + 03	2.93E + 03	2.95E + 03	3.06E + 03	2.93E + 03	2.96E + 03	3.42E + 03	2.97E + 03	2.94E + 03
	STD	1.59E + 01	3.17E + 01	3.43E + 00	9.57E + 00	2.18E + 01	2.89E + 01	3.27E + 01	1.59E + 02	2.45E + 01	1.57E + 01	2.31E + 02	2.96E + 01	1.90E + 01
F26	Mean	4.70E + 03	7.29E + 03	5.65E + 03	7.20E + 03	4.60E + 03	3.73E + 03	3.32E + 03	4.03E + 03	2.94E + 03	3.13E + 03	4.20E + 03	8.11E + 03	4.38E + 03
	STD	1.35E + 03	2.36E + 03	1.91E + 02	3.80E + 02	1.54E + 03	1.36E + 03	3.55E + 02	5.70E + 02	2.38E + 02	1.37E + 02	3.91E + 02	6.80E + 02	1.54E + 03
F27	Mean	3.10E + 03	3.13E + 03	3.64E + 03	3.11E + 03	3.12E + 03	3.10E + 03	3.13E + 03	3.18E + 03	3.10E + 03	3.12E + 03	3.61E + 03	3.11E + 03	3.10E + 03
	STD	3.45E + 00	3.67E + 01	2.55E + 02	2.29E + 01	2.54E + 01	1.78E + 01	4.32E + 01	5.20E + 01	3.77E + 00	1.71E + 01	2.86E + 02	3.19E + 00	1.29E + 01
F28	Mean	3.22E + 03	3.35E + 03	3.55E + 03	3.42E + 03	3.33E + 03	3.33E + 03	3.40E + 03	3.54E + 03	3.35E + 03	3.31E + 03	4.00E + 03	3.31E + 03	3.35E + 03
	STD	8.40E + 01	8.50E + 01	5.74E + 01	1.38E + 02	1.28E + 02	1.18E + 02	8.62E + 01	1.74E + 02	1.69E + 02	8.05E + 01	2.50E + 02	8.27E + 01	1.05E + 02
F29	Mean	3.22E + 03	3.31E + 03	3.95E + 03	3.38E + 03	3.34E + 03	3.21E + 03	3.26E + 03	3.46E + 03	3.22E + 03	3.27E + 03	3.45E + 03	3.54E + 03	3.19E + 03
	STD	5.22E + 01	9.69E + 01	3.83E + 02	7.49E + 01	1.14E + 02	4.16E + 01	6.39E + 01	1.52E + 02	6.25E + 01	3.59E + 01	1.48E + 02	1.08E + 02	2.38E + 01
F30	Mean	3.13E + 05	5.32E + 05	3.36E + 06	4.45E + 05	4.56E + 05	4.65E + 05	1.12E + 06	2.53E + 06	1.07E + 06	1.29E + 06	2.31E + 07	1.72E + 06	3.53E + 04
	STD	4.61E + 05	5.15E + 05	1.22E + 06	3.77E + 05	7.43E + 05	5.64E + 05	9.98E + 05	2.93E + 06	1.54E + 06	9.50E + 05	4.82E + 07	1.20E + 06	2.51E + 04

Significant values are in bold.

Table 8.

Wilcoxon ranksum test on CEC17 test suit at 5% level of significance.

Function	GA vs MSCA	PSO vs MSCA	GSA vs MSCA	JAYA vs MSCA	ALO vs MSCA	MVO vs MSCA	GWO vs MSCA	WOA vs MSCA	SSA vs MSCA	HGSO vs MSCA	AOA vs MSCA	SCA vs MSCA
F1	6.80E-08	1.56E-04	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.01E-02	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F2	2.22E-07	4.25E-01	6.80E-08	1.92E-07	1.63E-03	1.61E-04	2.56E-07	6.80E-08	1.72E-01	6.80E-08	6.80E-08	6.80E-08
F3	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	2.92E-05	6.80E-08	6.80E-08	6.80E-08
F4	7.64E-02	1.49E-05	1.10E-05	6.80E-08	1.23E-02	9.13E-07	2.00E-04	1.20E-06	4.70E-03	6.80E-08	6.80E-08	6.80E-08
F5	9.17E-08	3.15E-02	4.90E-01	6.80E-08	6.22E-04	9.05E-03	1.78E-03	2.56E-07	1.99E-01	3.75E-04	6.92E-07	6.80E-08
F6	6.80E-08	5.98E-01	6.80E-08	6.80E-08	6.80E-08	6.01E-07	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F7	6.61E-05	3.06E-03	3.64E-03	6.80E-08	4.32E-03	5.65E-02	2.75E-02	6.80E-08	6.22E-04	6.80E-08	6.80E-08	6.80E-08
F8	3.42E-07	2.80E-03	9.07E-08	6.80E-08	4.17E-05	3.05E-04	3.07E-06	6.80E-08	7.90E-08	6.80E-08	6.80E-08	6.80E-08
F9	6.80E-08	6.03E-06	9.75E-06	6.80E-08	3.15E-02	5.08E-01	6.80E-08	6.80E-08	1.60E-05	6.80E-08	6.80E-08	6.80E-08
F10	1.41E-05	7.58E-04	6.80E-08	6.80E-08	6.80E-08	3.94E-07	2.96E-07	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F11	2.96E-07	5.63E-04	6.80E-08	6.80E-08	1.08E-01	7.35E-01	9.79E-03	3.42E-07	4.60E-04	7.58E-06	6.80E-08	6.80E-08
F12	1.25E-05	7.41E-05	5.23E-07	9.17E-08	2.14E-03	8.60E-06	8.35E-03	3.50E-06	6.61E-05	9.17E-08	6.80E-08	6.80E-08
F13	2.75E-04	1.79E-02	6.61E-05	8.36E-04	1.95E-03	2.14E-03	5.09E-04	2.75E-04	2.04E-05	7.95E-07	2.92E-05	9.13E-07
F14	6.80E-08	4.32E-03	6.80E-08	6.80E-08	6.80E-08	5.25E-05	1.23E-07	6.80E-08	6.80E-08	6.80E-08	7.90E-08	6.80E-08
F15	4.54E-07	1.81E-05	6.80E-08	6.80E-08	6.80E-08	2.36E-06	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F16	7.71E-03	6.22E-04	6.80E-08	3.94E-07	1.66E-07	9.75E-06	5.23E-07	1.43E-07	6.01E-07	2.56E-07	7.90E-08	1.66E-07
F17	1.79E-02	2.34E-03	6.01E-07	6.80E-08	4.68E-05	7.58E-06	9.17E-08	6.80E-08	1.66E-07	6.80E-08	6.80E-08	6.80E-08
F18	9.09E-02	3.05E-04	9.75E-06	6.80E-08	2.04E-05	1.55E-02	4.11E-02	1.79E-02	1.14E-02	3.94E-07	1.01E-03	6.80E-08
F19	1.23E-07	6.92E-07	6.80E-08	7.90E-08	6.80E-08	4.32E-03	1.20E-06	6.80E-08	1.06E-07	7.90E-08	7.90E-08	6.80E-08
F20	6.61E-05	1.95E-03	6.80E-08	6.80E-08	1.66E-07	2.22E-07	2.22E-07	9.17E-08	2.56E-07	6.01E-07	1.66E-07	6.80E-08
F21	6.80E-08	1.60E-05	1.08E-01	6.80E-08	6.80E-08	3.10E-01	6.80E-08	6.80E-08	4.11E-02	6.80E-08	6.80E-08	6.80E-08
F22	6.80E-08	5.98E-01	2.23E-02	6.80E-08	1.12E-03	8.36E-04	1.14E-02	1.60E-05	5.25E-05	6.80E-08	6.80E-08	6.80E-08
F23	6.80E-08	7.11E-03	6.80E-08	6.80E-08	1.20E-06	7.90E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08	6.80E-08
F24	3.42E-07	4.57E-01	4.54E-06	6.80E-08	7.58E-04	1.23E-02	1.06E-07	1.05E-06	1.66E-07	3.10E-01	9.17E-08	6.80E-08
F25	2.07E-02	5.07E-01	8.36E-04	6.92E-07	3.79E-01	6.56E-03	3.79E-01	6.92E-07	5.61E-01	8.29E-05	6.80E-08	5.87E-06
F26	6.56E-03	9.75E-06	5.98E-01	6.80E-08	2.73E-01	5.25E-05	8.60E-01	5.98E-01	1.58E-06	8.82E-01	5.98E-01	6.80E-08
F27	1.23E-03	1.25E-05	6.80E-08	4.17E-05	5.90E-05	4.57E-01	7.11E-03	5.23E-07	3.51E-01	1.10E-05	6.80E-08	5.25E-05
F28	9.28E-05	1.25E-01	6.80E-08	4.11E-02	4.57E-01	3.38E-04	3.85E-02	8.29E-05	8.39E-01	1.29E-04	9.75E-06	6.39E-02
F29	1.67E-02	4.70E-03	6.80E-08	7.90E-08	1.79E-04	8.82E-01	3.06E-03	6.80E-08	2.18E-01	9.13E-07	3.42E-07	6.80E-08
F30	1.44E-04	3.96E-03	6.80E-08	1.80E-06	3.64E-03	4.70E-03	1.80E-06	2.92E-05	2.14E-03	6.80E-08	6.80E-08	7.90E-08

Significant values are in bold.

Table 9.

Wilcoxon signed-rank test on CEC17 test suit at 5% level of significance.

Item	Win	Tie	Lose	R+	R-	p-value	Decision
GA vs MSCA	29	0	1	448	17	9.32E-06	+
PSO vs MSCA	27	0	3	443	22	1.49E-05	+
GSA vs MSCA	27	0	3	429	36	5.31E-05	+
JAYA vs MSCA	30	0	0	465	0	1.73E-06	+
ALO vs MSCA	23	0	7	364	101	6.84E-03	+
MVO vs MSCA	19	0	11	322	143	6.56E-02	≈
GWO vs MSCA	28	0	2	443	22	1.49E-05	+
WOA vs MSCA	29	0	1	448	17	9.32E-06	+
SSA vs MSCA	21	0	9	384	81	1.83E-03	+
HGSO vs MSCA	28	0	2	435	30	3.11E-05	+
AOA vs MSCA	29	0	1	457	8	3.88E-06	+
SCA vs MSCA	29	0	1	459	6	3.18E-06	+

Robustness and convergence analysis

The box-plot for MSCA and SCA for CEC2017 are illustrated in Figs. 7 and 8, respectively. From Fig. 7, the overall distribution of the MSCA optimal solution is more concentrated and better. Notably, in Fig. 8, the overall convergence speed of MSCA is faster than that of SCA. On the whole, the proposed algorithm can improve the convergence rate and solution accuracy, which is an effective tool for solving numerical problems.

Figure 7.

The box-plot of SCA and MSCA on CEC2017.

Figure 8.

The convergence curve of SCA and MSCA on CEC2017.

MSCA for engineering optimization problems

In this subsection, the proposed MSCA tries to handle six famous constrained engineering problems. The parameters of the MCSA are set the same as in Section “MSCA for numerical optimization problems”. In addition, the static penalty function is employed to handle the equation and inequality constraints involved in the problem. The static penalty function is expressed as follows:

$$fit(\mathit{x})=f(\mathit{x})+\alpha [{\sum }_{i=1}^n{g}_i^2(\mathit{x})+{\sum }_{j=1}^p\left|h_j^{}(\mathit{x})\right|]$$ 5

where $fit(\mathit{x})$ and $f(\mathit{x})$ are the fitness value and objective value, respectively. $\alpha$ is the penalty coefficient. $n$ is the number of inequality constraints. $g_i^{}(\mathit{x})$ is the degree of violation of the ith inequality constraint. $p$ is the number of equality constraints. $h_j^{}(\mathit{x})$ is the degree of violation of the jth equality constraint.

Case I: Three-bar truss design problem

Trusses belong to one of the prefabricated concrete structures. This problem requires finding the minimum volume of the truss according to the decision variables of cross-sectional areas $\mathit{x}$ = $\left(x_1,x_2\right)$ = $\left(A_1,A_2\right)$⁴⁷, as plotted in Fig. 9, the problem are described as follows:

$$\begin{array}{c}\text{min}f(\mathit{x})=\left(2\sqrt{2}{x}_1+x_2\right)\times l\\ s.t.g_1=\frac{\sqrt{2}{x}_1+x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0\\ g_2=\frac{x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0\\ g_3=\frac{1}{x_1^{}+\sqrt{2}{x}_2}P-\sigma \le 0\\ 0\le x_1,x_2\le 1\\ \end{array}$$ 6

Figure 9.

The three-bar truss design.

The comparison of best solution result for this problem is displayed in Table 10. As can be observed, the results given by Tsai⁴⁸ are not feasible while MSCA is significantly better than the results reported in other competitive algorithms. From the comparative results, it is possible to conclude that that MSCA is work well over other methods.

Table 10.

The best solution for three-bar truss design problem.

Algorithm	x1	x2	g1	g2	g3	f(x)
Hernández49	0.788	0.408	NA	NA	NA	263.9
Ray and Saini50	0.795	0.395	−0.00169	−0.26124	−0.74045	264.3
Raj et al.51	0.78976441	0.40517605	−7.084 × 10–9	−1.4675992	−0.53240078	263.89671
Tsai48	0.788	0.408	0.00082	−0.2674	−0.73178	263.68a
Gandomi et al.37	0.78867	0.40902	−0.00029	−0.26853	−0.73176	263.9716
MSCA	0.788690415	0.408205144	−5.2854 × 10–8	−1.464150692	−0.53584936	263.89585052

a means not infeasible; NA means not available. Significant values are in bold.

Case II: Vertical deflection I beam problem

The I-beam structure is one of the design problems for prefabricated buildings with an H-shaped beam in cross-sectional form, as illustrated in Fig. 10. The objective of this problem is to minimize the vertical deflection of an I-beam while satisfying the cross-sectional area and stress constraints for a given load⁵². Consider the variable x = (x₁, x₂, x₃, x₄) = (h, l, t, b), the mathematical formulation of the problem is defined as follows:

$$\begin{array}{c}\text{min}f(\mathit{x})=\frac{5000}{\frac{t_w{\left(h-2t_f\right)}^3}{12}+\frac{b{t}_f^3}{6}+2b{t}_f{\left(\frac{h-t_f}{2}\right)}^2}\\ s.t.g_1\left(\mathit{x}\right)=2b{t}_f+t_w\left(h-2t_f\right)\le 300\\ g_2\left(\mathit{x}\right)=\frac{18h\times {10}^4}{t_w{\left(h-2t_f\right)}^3+2b{t}_f\left(4t_f^2+3h\left(h-2t_f\right)\right)}+\frac{15b\times {10}^4}{\left(h-2t_f\right)t_w^3+2t_f{b}^3}\le 6\\ 10\le h\le 80;10\le b\le 50;0.9\le t_w\le 5;0.9\le t_f\ne 5\\ \end{array}$$ 7

Figure 10.

The vertical deflection of an I-beam.

The best solution and statistical results of this problem are listed in Tables 11 and 12, respectively. From Table 11, it can be seen that MSCA obtain the same objective function value as CSA and get better results than ARSM, Improved ARSM and CS methods. It is possible to conclude in Table 12 that again MSCA has noteworthy performance in all metrics.

Table 11.

The best solution for vertical deflection of an I-beam problem.

Algorithm	h	b	tw	tf	f(x)
ARSM53	80	37.05	1.71	2.31	0.0157
Improved ARSM53	79.99	48.42	0.9	2.4	0.131
CS37	80	50	0.9	2.3216715	0.0130747
CSA54	80	49.99999999	0.9	2.3217923	0.01307412
MSCA	80	50	0.900000012	2.32179198	0.01307412

Significant values are in bold.

Table 12.

Statistical result for vertical deflection of an I-beam problem.

Algorithm	Best	Median	Mean	Worst	STD
CS37	0.0130747	N/A	0.0132165	0.01353646	1.345 × 10–4
CSA54	0.01307412	0.013091263	0.0131397	0.01339323	9.3686 × 10–5
MSCA	0.01307412	0.013074137	0.01307415	0.01307421	2.59 × 10–8

Significant values are in bold.

Case III: Tension string design problem.

The main goal of this problem is to minimizes the weight of the tension string with variable include wire diameter(d), mean coil diameter(D) and the number of active coils(n), as shown in Fig. 11. the mathematical model for this problem is given as follows:

$$\begin{array}{c}\text{min}f(\mathit{x})=\left(x_3+2\right)x_2{x}_1^2,\mathit{x}=\left(x_1,x_2,x_3\right)=\left(d,D,n\right)\\ s.t.g_1\left(\mathit{x}\right)=1-\frac{x_2^2{x}_3}{71785x_1^4}\le 0\\ g_2\left(\mathit{x}\right)=\frac{4x_2^2-x_1{x}_2}{12566\left(x_2{x}_1^3-x_1^4\right)}+\frac{1}{5108x_1^2}-1\le 0\\ g_3\left(\mathit{x}\right)=1-\frac{140.45x_1}{x_2^2{x}_3}\le 0\\ g_4\left(\mathit{x}\right)=\frac{x_1+x_2}{1.5}-1\le 0\\ 0.05\le x_1\le 2;0.25\le x_2\le 1.3;2\le x_3\le 15\\ \end{array}$$ 8

Figure 11.

Tension string design.

The comparison results with respect to the best solution between MSCA and existing algorithms are presented in Table 13, and the related statistical information is listed in Table 14. Based on both tables, MSCA exhibits more competitive results than other methods, indicating that the proposed MSCA algorithm has strong engineering practical value.

Table 13.

The best solution for tension string design problem.

Algorithm	x1	x2	x3	f(x)
Belegundu55	0.05	0.3159	14.25	0.0128334
Coello56	0.05148	0.351661	11.632201	0.01270478
Ray and Saini50	0.050417	0.321532	13.979915	0.01306
Coello and Montes57	0.051989	0.363965	10.890522	0.012681
Ray and Liew 58	0.052160217	0.368158695	10.64844226	0.012669249
Raj et al.51	0.053862	0.41128365	8.6843798	0.0127484
Mahdavi et al.59	0.05115438	0.34987116	12.0764321	0.0126706
He and Wang60	0.051728	0.357644	11.244543	0.0126747
Montes and Coello61	0.051643	0.35536	11.397926	0.012698
Kaveh and Talatahari62	0.051865	0.3615	11	0.0126432a
Coelho63	0.051515	0.352529	11.538862	0.012665a
Akay and Karaboga64	0.051749	0.358179	11.203763	0.012665a
MSCA	0.051781993	0.358944836	11.16078852	0.012666807

a means infeasible. Significant values are in bold.

Table 14.

Statistical result for tension string design problem.

Algorithm	Best	Median	Mean	Worst	STD
Belegundu55	0.0128334	NA	NA	NA	NA
Coello56	0.01270478	0.01275576	0.0127692	0.01282208	3.9390 × 10–5
Ray and Saini50	0.01306	NA	0.015526	0.018992	NA
Coello and Montes57	0.012681	NA	0.012742	0.012973	5.9000 × 10–5
Ray and Liew58	0.01266925	0.01292267	0.01292267	0.01671727	5.92 × 10–4
He and Wang60	0.0126747	NA	0.01273	0.012924	5.1985 × 10–5
Montes and Coello61	0.012698	NA	0.013461	0.16485	9.6600 × 10–4
Kaveh and Talatahari62	0.0126432a	NA	0.01272a	0.012884a	3.4888 × 10–5
Coelho63	0.012665a	0.012957a	0.013524a	0.017759a	0.001268
Akay and Karaboga64	0.012665a	NA	0.012709a	NA	0.012813
MSCA	0.01266681	0.01273110	0.01281752	0.01334238	1.90 × 10–4

NA means not available; a means infeasible. Significant values are in bold.

Case IV: Welded beam design problem

Welded beam design is an important part of the engineering design problem, as shown in Fig. 12, the objective is to minimize the costs of welded beam⁵¹ by selecting four variables: the thickness of weld h, length of welded joint l, width of the beam t and thickness of the beam b. The optimization model with variable $\mathit{x}=\left(x_1,x_2,x_3,x_4\right)=\left(h,l,t,b\right)$ is expressed as follows:

Figure 12.

Welded beam design.

Version I:

$$\begin{array}{c}\text{min}f(\mathit{x})=1.10471x_1^2{x}_2+0.04811x_3{x}_4(14+x_2)\\ s.t.g_1\left(\mathit{x}\right)=\tau \left(\mathit{x}\right)-{\tau }_{max}\le 0\\ g_2\left(\mathit{x}\right)=\sigma \left(\mathit{x}\right)-{\sigma }_{max}\le 0\\ g_3\left(\mathit{x}\right)=x_1-x_4\le 0;\\ g_4\left(\mathit{x}\right)=0\text{.125 -}{x}_1\le 0\\ g_5\left(\mathit{x}\right)=\delta \left(\mathit{x}\right)-0.25\le 0\\ g_6\left(\mathit{x}\right)=P-P_c\le 0\\ 0.1\le x_1,x_4\le 2;0.1\le x_2,x_3\le 10\\ \end{array}$$ 9

where the shear stress $\tau$ is defined as follows:

$$\tau =\sqrt{{\tau }_1^2+2{\tau }_1{\tau }_2\left(\frac{x_2}{2R}\right)+{\tau }_2^2};{\tau }_1=\frac{P}{x_1{x}_2\sqrt{2}};{\tau }_2=\frac{\mathit{MR}}{J}$$ 10

$$M=P\left(L+\frac{x_2}{2}\right);J=2\left\{\frac{x_1{x}_2}{\sqrt{2}}\left[\frac{x_2^2}{12}+{\left(\frac{x_1+x_3}{2}\right)}^2\right]\right\}$$ 11

$$R=\sqrt{\frac{x_2^2}{4}+{\left(\frac{x_1+x_3}{2}\right)}^2};\sigma =\frac{6PL}{x_1{x}_3^2};\delta =\frac{4P{L}^3}{E{x}_3^3{x}_4}$$ 12

$$P_c=\frac{4.013\sqrt{\frac{EG{x}_3^2{x}_4^6}{36}}}{L^2}\left(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}}\right)$$ 13

$$G=12\times {10}^6\text{psi},E=30\times {10}^6\text{psi},P=6000\text{lb},L=14\text{in}$$ 14

The MSCA is compared with four optimization methods, and the best solution and statistical results are reported in Tables 15 and 16. the results indicate that MSCA obtained the solution more accurately and more competitive than other methods.

Table 15.

The best solution for welded beam design problem on version I.

Algorithm	x1	x2	x3	x4	f(x)
Ragsdell and Phillips65	0.2455	6.196	8.273	0.2455	2.385937
Rao66	0.2455	6.196	8.273	0.2455	2.386
Ray and Liew58	0.244438276	6.237967234	8.288576143	0.244566182	2.3854347
Hwang and He67	0.2231	1.5815	12.8468	0.2245	2.25a
MSCA	0.244249519	6.206365305	8.312174308	0.24432385	2.383286722

a means infeasible. Significant values are in bold.

Table 16.

Statistical result for welded beam design problem on version I.

Algorithm	Best	Median	Mean	Worst	STD
Ragsdell and Phillips65	2.385937	NA	NA	NA	NA
Rao66	2.386	NA	NA	NA	NA
Ray and Liew58	2.3854347	3.0025883	3.2551371	6.3996785	0.959078
Hwang and He67	2.25a	NA	2.26a	2.28a	NA
MSCA	2.383286722	2.387845945	2.388104097	2.393825358	0.002758472

NA means not available; a means infeasible. Significant values are in bold.

Version II

In version II, the researchers modified the definitions of some items on the basis of version I and added a new constraint $g_7$, which are defined as follows:

$$g_7\left(\mathit{x}\right)=0.10471x_1^2+0.04811x_3{x}_4(14+x_2)-5\le 0$$ 15

$$J=2\left\{\sqrt{2}{x}_1{x}_2\left[\frac{x_2^2}{4}+{\left(\frac{x_1+x_3}{2}\right)}^2\right]\right\};\sigma =\frac{6P{L}^3}{x_1{x}_3^3{x}_4}$$ 16

$$P_c=\frac{4.013E\sqrt{\frac{x_3^2{x}_4^6}{36}}}{L^2}\left(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}}\right)$$ 17

Tables 17 and 18 summarize the best solution and statistical results for version II, respectively. It can be seen from the simulation results that MSCA can still obtain convincing results with the addition of constraint of $g_7$, indicating that the proposed MSCA performs better than existing studies.

Table 17.

The best solution for welded beam design problem on version II.

Algorithm	x1	x2	x3	x4	f(x)
Coello56	0.2088	3.4205	8.9975	0.21	1.748309
Coello and Montes57	0.205986	3.471328	9.020224	0.20648	1.728226
He and Wang60	0.202369	3.544214	9.04821	0.205723	1.728024
Dimopoulos68	0.2015	3.562	9.041398	0.205706	1.731186
Mahdavi et al.59	0.20573	3.47049	9.03662	0.20573	1.7248
Montes et al.69	0.20573	3.470489	9.036624	0.20573	1.724852
Montes and Coello61	0.199742	3.61206	9.0375	0.206082	1.7373
Cagnina et al.70	0.205729	3.470488	9.036624	0.205729	1.724852
Kaveh and Talatahari71	0.205729	3.469875	9.036805	0.205765	1.724849
Kaveh and Talatahari62	0.2057	3.471131	9.036683	0.205731	1.724918
Gandomi et al.72	0.2015	3.562	9.0414	0.2057	1.73121
Mehta and Dasgupta73	0.20572885	3.47050567	9.03662392	0.20572964	1.724855
Akay and Karaboga64	0.20573	3.470489	9.036624	0.20573	1.724852
MSCA	0.205187143	3.266067065	9.03380051	0.205913062	1.69710013

Significant values are in bold.

Table 18.

Statistical result for welded beam design problem on version II.

Algorithm	Best	Median	Mean	Worst	STD
Coello56	1.748309	NA	1.771973	1.785835	0.01122
Coello and Montes57	1.728226	NA	1.792654	1.993408	0.07471
He and Wang60	1.728024	NA	1.748831	1.782143	0.012926
Dimopoulos68	1.731186	NA	NA	NA	NA
Montes et al.69	1.724852	NA	1.725	NA	1.0 × 10–15
Montes and Coello61	1.7373	NA	1.81329	1.994651	0.0705
Cagnina et al.70	1.724852	NA	2.0574	NA	0.2154
Kaveh and Talatahari71	1.724849	NA	1.727564	1.759522	0.008254
Kaveh and Talatahari62	1.724918	NA	1.729752	1.775961	0.0092
Gandomi et al.72	1.7312065	NA	1.878656	2.3455793	0.2677989
Mehta and Dasgupta73	1.724855	1.724861	1.724865	1.72489	NA
Akay and Karaboga64	1.724852	NA	1.741913	NA	0.031
MSCA	1.69710013	1.700828362	1.70209677	1.722162002	0.005815737

NA means not available. Significant values are in bold.

Case V: Gear train design problem

The gear train design problem was first proposed by Sandgren⁷⁴ as an unconstrained optimization problem, presented in Fig. 13. The task of the problem is to find the best variable vector $\mathit{x}=\left(x_1,x_2,x_3,x_4\right)=\left(T_d,T_b,T_a,T_f\right)$ to minimize cost of gear ratio. The optimization formula is given as follows:

$$\begin{array}{c}\text{min}f(\mathit{x})={\left(\frac{1}{6.931}-\frac{x_1{x}_2}{x_3{x}_4}\right)}^2\\ s.t.12\le x_i\le 60,i=1,2,3,4\\ x_i\in Z^{+},i=1,2,3,4\\ \end{array}$$ 18

where gear ratio = $\frac{x_1{x}_2}{x_3{x}_4}$.

Figure 13.

Gear train design.

To address this issue, MSCA was compared with several classical methods, and the best solutions recorded for MSCA and other literature results are listed in Table 19. On the other hand, the corresponding statistical results are represented in Table 20. The both tables show that MSCA provides more stable and accurate solutions than other methods.

Table 19.

The best solution for gear train design problem.

Algorithm	Td(x1)	Tb(x2)	Ta(x3)	Tf(x4)	Gear ratio	f(x)
Sandgren75	18	22	45	60	0.146667	5.712 × 10–6
Kannan and Kramer76	13	15	33	41	0.144124	2.146 × 10–8
Deb and Goyal77	19	16	49	43	0.144281	2.701 × 10–12
Gandomi et al.37	19	16	43	49	0.144281	2.701 × 10–12
Garg78	19	16	43	49	0.14428096	2.7008571 × 10–12
MSCA	19	16	43	49	0.14428096	2.7008571 × 10–12

Significant values are in bold.

Table 20.

Statistical result for gear train design problem.

Algorithm	Best	Median	Mean	Worst	STD
Gandomi et al.78	2.7009 × 10–12	NA	1.9841 × 10–9	2.3576 × 10–9	3.5546 × 10–9
Garg37	2.7008571 × 10–12	9.9215795 × 10–10	1.2149276 × 10–9	3.2999231 × 10–9	8.77 × 10–10
MSCA	2.7008571 × 10–12	2.3078157 × 10–11	4.8092363 × 10–11	3.0675559 × 10–10	7.4270038 × 10–11

NA means not available. Significant values are in bold.

Case VI: Pressure vessel design problem

IN this problem, shown in Fig. 14, the task is to minimize the total cost of the vessel with decision vector $\mathit{x}$ = $\left(x_1,x_2,x_3,x_4\right)$ = $\left(T_s,T_h,R,L\right)$⁷⁹. The optimization model is constructed as follows:

$$\begin{array}{c}\text{min}f(\mathit{x})=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_3\\ s.t.g_1(\mathit{x})=-x_1+0.0193x_3\le 0\\ g_2(\mathit{x})=-x_2+0.00954x_3\le 0\\ g_3(\mathit{x})=-\pi x_3^2{x}_4+\frac{4}{3}\pi x_3^3+1,296,000\le 0\\ g_4(\mathit{x})=x_4-240\le 0\\ \end{array}$$ 19

Figure 14.

Pressure vessel design.

Version I:

$$0.0625\le x_1,x_2\le 99\times 0.0625;10\le x_3,x_4\le 200$$ 20

The best results and statistical results for version I are reported in Tables 21 and 22, respectively. As can be seen from Table 21, MCSA obtain more objective results than other literature, and from Table 22, MCSA further obtains a more concentrated solution distribution, demonstrating the superior search performance of the proposed method.

Table 21.

The best solution for pressure vessel design problem on version I.

Algorithm	x1	x2	x3	x4	f(x)
Sandgren74	1.125	0.625	47.7	117.701	8129.1036
Kannan and Kramer80	1.125	0.625	58.291	43.69	7198.0428
Coello and Montes57	0.8125	0.4375	42.097398	176.65405	6059.946
He and Wang60	0.8125	0.4375	42.091266	176.7465	6061.0777
Montes and Coello61	0.8125	0.4375	42.098087	176.640518	6059.7456
Coelho63	0.8125	0.4375	42.0984	176.6372	6059.7208
He et al.81	0.8125	0.4375	42.098445	176.636595	6059.7143
Montes et al.69	0.8125	0.4375	42.098446	176.636047	6059.70166
Gandomi et al.37	0.8125	0.4375	42.0984456	176.6365958	6059.714335
Akay and Karaboga64	0.8125	0.4375	42.098446	176.636596	6059.714339
MSCA	0.780583407	0.3917558	40.4190779	198.964126	5917.509793

Significant values are in bold.

Table 22.

Statistical result for pressure vessel design problem on version I.

Algorithm	Best	Median	Mean	Worst	STD
Sandgren74	8129.1036	NA	N/A	N/A	N/A
Kannan and Kramer80	7198.0428	NA	N/A	N/A	N/A
Coello and Montes57	6059.9463	NA	6177.2533	6469.322	130.9297
Montes and Coello61	6059.7456	NA	6850.0049	7332.8798	426
Gandomi et al.37	6059.714	NA	6447.736	6495.347	502.693
Coelho63	6059.7208	6257.5943	6440.3786	7544.4925	448.4711
He et al.81	6059.7143	NA	6289.92881	NA	305.78
Akay and Karaboga64	6059.714339	NA	6245.30814	NA	205
MSCA	5917.509793	5994.4224	6029.24374	6396.551211	113.2470944

NA means not available. Significant values are in bold.

Version II:

$$0.0625\le x_1,x_2\le 99\times 0.0625;10\le x_3\le 200;10\le x_4\le 240$$ 21

In version II, the upper bound of the decision variable x₄ is set as 240, the best results and statistical results summarized by MSCA and control methods for version II in Tables 23 and 24, respectively. Analysis from both tables, MSCA reveals better performance than the existing studies. The comparison results further prove that MCSA has better feasibility in engineering than the other approaches.

Table 23.

The best solution for pressure vessel design problem on version II.

Algorithm	x1	x2	x3	x4	f(x)
Dimopoulos68	0.75	0.375	38.8601	221.36549	5850.38306
Mahdavi et al.59	0.75	0.375	38.8601	221.36553	5849.76169
Gandomi et al.72	0.75	0.375	38.8601	221.36547	5850.38306
MSCA	0.73822151	0.36818507	38.16830296	232.73616127	5849.52062

Significant values are in bold.

Table 24.

Statistical result for pressure vessel design problem on version II.

Algorithm	Best	Median	Mean	Worst	STD
Dimopoulos68	5850.383060	NA	NA	NA	NA
Mahdavi et al.59	5849.761700	NA	NA	NA	NA
Gandomi et al.72	5850.383060	NA	5937.337900	6258.968250	164.547470
MSCA	5849.520622	5919.655417	5944.332084	6217.813607	87.356362

NA means not available. Significant values are in bold.

Conclusion

In this article, the position update adjustment strategy and Levy random walk mutation mechanism were adopted into the original SCA to create the proposed MCSA. The position update adjustment strategy considers the swarm search for potentially better solutions around the current individual and the global optimal individual, effectively expanding the search and improving the convergence speed of the original SCA. And the Levy random walk mutation mechanism effectively increases the original SCA diversity to ensure that the SCAs jump out of the local optimal position and improve the search accuracy.

For qualitative analysis of proposed method (MSCA), it has been passed two levels of benchmark test suites—the classic and the IEEE CEC2017 function. box plots and convergence curves were employed to verify the performance of robustness and convergence. In addition, Wilcoxon signed-rank test and ranksum test were adopted to verify statistical significance. The experimental results show that MSCA is able to balance the exploration and exploitation process. Finally, MSCA is used to solve six complex real-world engineering design problems, and the comparison with the results of several other state-of-the-art methods proves that the proposed method can achieve competitive results. Based on the convincing results obtained from existing research, in the future we will try to apply the proposed method to image recognition and time series forecasting problems.

Author contributions

Conceptualization, C.S.; writing—review and editing, C.S.; validation, T.Z.; data curation, T.Z.; writing—original draft preparation, T.Z. methodology, S.L.; software, S.L.; All authors read and agreed to the published version of the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon request. There are no restrictions on data availability.

Competing interests

The authors declare no competing interests.