Tuna Swarm Optimization: A Novel Swarm-Based Metaheuristic Algorithm for Global Optimization

Abstract

In this paper, a novel swarm-based metaheuristic algorithm is proposed, which is called tuna swarm optimization (TSO). The main inspiration for TSO is based on the cooperative foraging behavior of tuna swarm. The work mimics two foraging behaviors of tuna swarm, including spiral foraging and parabolic foraging, for developing an effective metaheuristic algorithm. The performance of TSO is evaluated by comparison with other metaheuristics on a set of benchmark functions and several real engineering problems. Sensitivity, scalability, robustness, and convergence analyses were used and combined with the Wilcoxon rank-sum test and Friedman test. The simulation results show that TSO performs better compared to other comparative algorithms.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Real-world optimization problems have become more challenging, which requires more efficient solution methods. Different scholars have studied various approaches to solve these complex and difficult problems from the real world. A part of researchers solve these optimization problems using traditional methods such as quasi-Newton, conjugate gradient, and sequential quadratic programming methods. However, owing to the nonlinear, nonproductivity characteristics of most real-world optimization problems and the involvement of multiple decision variables and complex constraints, these traditional algorithms are difficult to be solved effectively [1, 2]. The metaheuristic algorithm has the advantages of not relying on the problem model, not requiring gradient information, having strong search capability and wide applicability, and can achieve a good balance between solution quality and computational cost [3]. Therefore, the metaheuristic algorithms have been proposed to solve real-world optimization problems, such as image segmentation [4, 5], feature selection [6, 7], mission planning [8, 9], parameter optimization [10, 11], job shop scheduling [12, 13], etc.

Metaheuristic algorithms are usually classified into three categories [14]: evolution-based algorithms, physical-based algorithms, and swarm-based algorithms. The evolution-based algorithm is inspired by the laws of evolution in nature. Genetic algorithm (GA) [15], inspired by Darwin's theory of superiority and inferiority, is a well-known evolution-based algorithm. With the popularity of GA, several other widely used evolution-based algorithms have been proposed, including differential evolution (DE) [16], genetic programming (GP) [17], evolutionary strategies (ES) [18], and evolutionary programming (EP) [19]. In addition, several new evolution-based algorithms have been proposed, such as artificial algae algorithm (AAA) [20], biogeography-based optimization (BBO) [21], and monkey king evolutionary (MKE) [22]. The physical-based algorithms are inspired by various laws of physics. One of the most famous algorithms of this category is simulated annealing (SA) [23]. SA is inspired by the law of thermodynamics in which a material is heated up and then cooled slowly. There are other physical-based algorithms proposed, including gravitational search algorithm (GSA) [24], nuclear reaction optimization (NRO) [25], water cycle algorithm (WCA) [26], and sine cosine algorithm (SCA) [27]. The swarm-based algorithms are inspired by the social behavior of different species in natural groups. Particle swarm optimization (PSO) [28] and ant colony optimization (ACO) [29] are two typical swarm-based algorithms. PSO and ACO mimic the aggregation behavior of bird colonies and the foraging behavior of ant colonies, respectively. Some other algorithms of this category include: grey wolf optimizer (GWO) [30], monarch butterfly optimization (MBO) [31], elephant herding optimization (EHO) [32], moth search algorithm (MSA) [33], manta ray foraging optimization (MRFO) [34],earthworm optimization algorithm (EOA) [35], etc. With the development of metaheuristics, a type of human-based metaheuristic algorithm is also emerging. These algorithms are inspired by the characteristics of human activity. Teaching-learning-based optimization (TLBO) [36], inspired by traditional teaching methods, is a typical example of this category among metaheuristic algorithms. Other human-based metaheuristics include: social evolution and learning optimization (SELO) [37], group teaching optimization algorithm (GTOA) [38], heap-based optimizer (HBO) [39], political optimizer (PO) [40], etc.

There is a common feature of all these metaheuristic algorithms that rely on exploration and exploitation in the search space to find the optimal solution [41, 42]. Exploration means that the algorithm searches for promising regions in a wide search space and exploitation is a further search for the best solution in the promising regions. The balance of the two search behaviors affects the quality of the solution. When exploration dominates, exploitation declines, and vice versa. Therefore, it is a big challenge to balance exploration and exploitation for metaheuristics. Although there are constantly new algorithms being developed, the no free lunch (NFL) [43] theory states that no particular algorithm can solve all optimization problems perfectly. The NFL has motivated researchers to develop effective metaheuristic algorithms to solve various fields of optimization problems.

In this paper, a novel swarm-based metaheuristic is presented called tuna swarm optimization (TSO). It is inspired by two types of swarm foraging behavior of tunas. The TSO is evaluated in 23 benchmark functions and 3 engineering design problems. Test results reveal that the method proposed in this paper significantly outperforms those popular and recent metaheuristics. This paper is structured as follows: Section 2 describes the inspiration for TSO and builds the corresponding mathematical model. A benchmark function set and three engineering design problems are employed to check the performance of TSO in Sections 3 and 4, respectively. Section 5 concludes the overall work and provides an outlook for the future.

2. Tuna Swarm Optimization

2.1. Inspiration

Tuna, scientifically named Thunnini, is a marine carnivorous fish. There are many species of tuna, and the size varies greatly. Tuna are top marine predators, feeding on a variety of midwater and surface fish. Tunas are continuous swimmers, and they have a unique and efficient way of swimming (called fishtail shape) in which the body stays rigid while the long, thin tail swings rapidly. Although the single tuna swims very fast, it is still not as fast as the nimble small fish response. Therefore, the tuna will use the “ group travel “ method for predation. They use their intelligence to find and attack their prey. These creatures have evolved a variety of effective and intelligent foraging strategies.

The first strategy is spiral foraging. When tuna are feeding, they swim by forming a spiral formation to drive their prey into shallow water where they can be attacked more easily.

The second strategy is parabolic foraging. Each tuna swims after the previous individual, forming a parabolic shape to enclose its prey.

Tuna successfully forage by the above two methods. In this paper, a new swarm-based metaheuristic optimization algorithm, namely, tuna swarm optimization, is proposed based on modeling these natural foraging behaviors.

2.2. Mathematical Model

In this section, the mathematical model of the proposed algorithm is described in detail.

2.2.1. Initialization

Similar to most swarm-based metaheuristics, TSO starts the process of optimization by generating initial populations at random uniformly in the search space,$$\begin{array}{}\text{(1)}& {\mathbf{X}}_i^{\mathrm{int}}=\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{d}\cdot \mathbf{u}\mathbf{b}-\mathbf{l}\mathbf{b}+\mathbf{l}\mathbf{b},\hspace{1em}i=\mathrm{1,2},\mathrm{...},NP,\end{array}$$where ${\mathbf{X}}_i^{\mathrm{int}}$ is the $i^{th}$ initial individual, $\mathbf{u}\mathbf{b}$ and $\mathbf{l}\mathbf{b}$ are the upper and lower boundaries of the search space, $NP$ is the number of tuna populations, and $\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{d}$ is a uniformly distributed random vector ranging from 0 to 1.

2.2.2. Spiral Foraging

When sardines, herring, and other small schooling fish encounter predators, the entire school of fish forms a dense formation constantly changing the swimming direction, making it difficult for predators to lock a target. At this time, the tuna group chase the prey by forming a tight spiral formation. Although most of the fish in the school have little sense of direction, when a small group of fish swim firmly in a certain direction, the nearby fish will adjust their direction one after another and finally form a large group with the same goal and start to hunt. In addition to spiraling after their prey, schools of tuna also exchange information with each other. Each tuna follows the previous fish, thus enabling information sharing among neighboring tuna. Based on the above principles, the mathematical formula for the spiral foraging strategy is as follows:$$\begin{array}{}\text{(2)}& {\mathbf{X}}_i^{t+1}=\begin{array}{c}{\alpha }_1\cdot X_{best}^t+\beta \cdot X_{best}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_i^t,\hspace{1em}i=1,\\ {\alpha }_1\cdot X_{best}^t+\beta \cdot X_{best}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_{i-1}^t,\hspace{1em}i=\mathrm{2,3},\mathrm{...},NP,\end{array}\text{(3)}& {\alpha }_1=a+1-a\cdot \frac{t}{t_{\max }},\text{(4)}& {\alpha }_2=1-a-1-a\cdot \frac{t}{t_{\max }},\text{(5)}& \beta =e^{bl}\cdot \cos 2\pi b,\text{(6)}& l=e^{3\cos t_{\max }+1/t-1\pi },\end{array}$$where ${\mathbf{X}}_i^{t+1}$ is the $i^{th}$ individual of the $t+1$ iteration, ${\mathbf{X}}_{best}^t$ is the current optimal individual (food), ${\alpha }_1$ and ${\alpha }_2$ are weight coefficients that control the tendency of individuals to move towards the optimal individual and the previous individual, $a$ is a constant used to determine the extent to which the tuna follow the optimal individual and the previous individual in the initial phase, $t$ denotes the number of current iteration, $t_{\max }$ is the maximum iterations, and $b$ is a random number uniformly distributed between 0 and 1.

When all tuna forage spirally around the food, they have good exploitation ability for the search space around the food. However, when the optimal individual fails to find food, blindly following the optimal individual to forage is not conducive to group foraging. Therefore, we consider generating a random coordinate in the search space as a reference point for spiral search. This facilitates each individual to search a wider space and makes TSO with global exploration ability. The specific mathematical model is described as follows:$$\begin{array}{}\text{(7)}& {\mathbf{X}}_i^{t+1}=\begin{array}{c}{\alpha }_1\cdot X_{rand}^t+\beta \cdot X_{rand}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_i^t,\hspace{1em}i=1,\\ {\alpha }_1\cdot X_{rand}^t+\beta \cdot X_{rand}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_{i-1}^t,\hspace{1em}i=\mathrm{2,3},\mathrm{...},NP,\end{array}\end{array}$$where ${\mathbf{X}}_{rand}^t$ is a randomly generated reference point in the search space.

In particular, metaheuristic algorithms usually perform extensive global exploration in the early stage and then gradually transition to precise local exploitation. Therefore, TSO changes the reference points of spiral foraging from random individuals to optimal individuals as the iteration increases. In summary, the final mathematical model of the spiral foraging strategy is as follows:$$\begin{array}{}\text{(8)}& {\mathbf{X}}_i^{t+1}=\begin{array}{c}\begin{array}{c}{\alpha }_1\cdot X_{rand}^t+\beta \cdot X_{rand}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_i^t,\hspace{1em}i=1,\\ {\alpha }_1\cdot X_{rand}^t+\beta \cdot X_{rand}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_{i-1}^t,\hspace{1em}i=\mathrm{2,3},\mathrm{...},NP,\end{array},\text{if\hspace{0.17em}rand}<\frac{t}{t_{\max }},\\ \begin{array}{c}{\alpha }_1\cdot {\mathbf{X}}_{best}^t+\beta \cdot {\mathbf{X}}_{best}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_i^t,\hspace{1em}i=1,\\ {\alpha }_1\cdot {\mathbf{X}}_{best}^t+\beta \cdot {\mathbf{X}}_{best}^t-{\mathbf{X}}_i^t+{\alpha }_2\cdot {\mathbf{X}}_{i-1}^t,\hspace{1em}i=\mathrm{2,3},\mathrm{...},NP,\end{array},\text{if\hspace{0.17em}rand}\ge \frac{t}{t_{\max }}.\end{array}\end{array}$$

2.2.3. Parabolic Foraging

In addition to feeding by forming a spiral formation, tunas also form a parabolic cooperative feeding. Tuna forms a parabolic formation with food as a reference point. In addition, tuna hunt for food by searching around themselves. These two approaches are performed simultaneously, with the assumption that the selection probability is 50% for both. The specific mathematical model is described as follows:$$\begin{array}{}\text{(9)}& {\mathbf{X}}_i^{t+1}=\begin{array}{c}{\mathbf{X}}_{best}^t+\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{d}\cdot {\mathbf{X}}_{best}^t-{\mathbf{X}}_i^t+TF\cdot p^2\cdot {\mathbf{X}}_{best}^t-{\mathbf{X}}_i^t,\hspace{1em}\text{if\hspace{0.17em}rand\hspace{0.17em}}<0.5\text{,}\\ TF\cdot p^2\cdot {\mathbf{X}}_i^t,\hspace{1em}\text{if\hspace{0.17em}rand}\ge \text{0.}5\text{,}\end{array}\text{(10)}& p={1-\frac{t}{t_{\max }}}^{t/t_{\max }},\end{array}$$where $TF$ is a random number with a value of 1 or −1.

Tuna hunt cooperatively through two foraging strategies and then find their prey. For the optimization process of TSO, the population is first randomly generated in the search space. In each iteration, each individual randomly chooses one of the two foraging strategies to execute, or chooses to regenerate the position in the search space according to probability $z$. The value of parameter $z$ will be discussed in the parameter setting simulation experiments. During the entire optimization process, all individuals of TSO are continuously updated and calculated until the end condition is met, and then the optimal individual and the corresponding fitness value are returned. The TSO pseudocode is shown in Algorithm 1. The detailed process of TSO is shown in Figure 1.

[figure omitted; refer to PDF]

Subject to$$\begin{array}{c}\text{(12)}& g_{1}X=-x_1+0.0193x_3\le 0,g_{2}X=-x_2+0.00954x_3\le 0,\\ g_{3}X=-\pi x_3^2{x}_4-\frac{4}{3}\pi x_3^2+\mathrm{1,296,000}\le 0,\\ g_{4}X=x_4-240\le 0,\\ Variableranges:1\times 0.0625\le x_1,\\ x_2\le 99\times 0.0625,10\le x_3,x_4\le 200.\end{array}$$

The results of TSO for solving this problem are compared with other algorithms such as DDSCA, ISCA, MBA, CPSO, TEO, hHHO-SCA, HPSO, MVO, and AFA, and the comparison is shown in Table 19. The results show that the TSO solution is superior to the solutions provided by the comparison algorithms with optimal solutions for each parameter [0.7782, 0.3846, 40.3196, and 199.9999], corresponding to a minimum cost of 5885.3327.

Table 19

Comparisons of the best solutions offered by reported optimizers for pressure vessel design.

Algorithm	Optimal values for variables				Optimal cost
	$x_1$	$x_2$	$x_3$	$x_4$
DDSCA [50]	0.7782	0.3855	40.3198	176.6389	5888.3366
ISCA [51]	0.8125	0.4375	42.0982	176.6389	6059.7410
MBA [52]	0.7802	0.3856	40.4292	198.4964	5889.3216
CPSO [53]	0.8125	0.4375	42.0912	176.7465	6061.0777
TEO [54]	0.7791	0.3852	40.3698	199.3018	5887.5110
hHHO-SCA [55]	0.9459	0.4471	46.8513	125.468	6393.0927
HPSO [56]	0.8125	0.4375	42.0984	176.6366	6059.7143
MVO [57]	0.8215	0.4375	42.0907	176.7386	6060.8066
AFA [58]	0.8125	0.4375	42.0984	176.6366	6059.7143
TSO	0.7782	0.3846	40.3196	199.9999	5885.3327

4.2. Tension/Compression Spring Design

The tension/compression spring design problem is a mechanical engineering design optimization problem. As shown in Figure 7, the goal of this problem is to reduce the weight of the spring. It includes four nonlinear inequalities and three continuous variables: wire diameter w($x_1$), average coil diameter d($x_2$), and coil length or number L($x_3$). This problem can be described by the following equation:$$\begin{array}{}\text{(13)}& \min f{x}_1,x_2,x_3=x_3+2x_1^2{x}_2.\end{array}$$

[figure omitted; refer to PDF]

Subject to$$\begin{array}{c}\text{(14)}& g_{1}X=1-\frac{x_2^3{x}_3}{71785x_1^4}\le 0,g_{2}X=\frac{x_{2}4x_2-x_1}{12566x_1^3{x}_2-x_1}+\frac{1}{5108x_1^2}-1\le 0,\\ g_{3}X=1-\frac{140.45x_1}{x_2^2{x}_3}\le 0,\\ g_{4}X=\frac{2x_1+x_2}{3}-1\le 0,\\ Variablerange:0.05\le x_1\le 2,0.25\le x_2\le 1.3,\\ 2.0\le x_3\le \mathrm{15.0.}\end{array}$$

The solution of TSO is compared with other methods given in the literature, including GA3, CPSO, CDE, DDSCA, GSA, hHHO-SCA, AEO, and MVO. Table 20 shows the parameters and costs corresponding to the optimal solution of each algorithm. As can be seen from Table 10, TSO is the best algorithm for solving the problem. The optimal solution for each parameter corresponding to the lowest cost of 1.724852 is [0.205729, 3.470488, 9.036623, 0.205729].

Table 20

Comparisons of best solutions offered by reported optimizers for tension/compression spring design.

Algorithm	Optimal values for variables			Optimal cost
	$x_1$	$x_2$	$x_3$
GA3 [59]	0.051989	0.363965	10.890522	0.0126810
CPSO [53]	0.051728	0.357644	11.244543	0.0126747
CDE [60]	0.051609	0.354714	11.410831	0.0126702
DDSCA [50]	0.052669	0.380673	10.0153	0.012688
GSA [24]	0.050276	0.323680	13.525410	0.0127022
hHHO-SCA [55]	0.054693	0.433378	7.891402	0.0128229
AEO [61]	0.051897	0.361751	10.879842	0.0126662
MVO [57]	0.05251	0.3762	10.33513	0.012970
TSO	0.051642	0.355609	11.354247	0.0126652

4.3. Welded Beam Design

The welded beam design problem is the classical structural optimization problem. As shown in Figure 8, the objective of this design problem is to minimize the fabrication cost of the welded beam. The optimization variables include welding thickness h($x_1$), joint beam length l($x_2$), beam height t($x_3$), and beam thickness b($x_4$). The mathematical model is as follows:$$\begin{array}{}\text{(15)}& \min f{x}_1,x_2,x_3,x_4=1.10471x_1^2{x}_2+0.04811x_3{x}_{4}14.0+x_2.\end{array}$$

[figure omitted; refer to PDF]

Subject to$$\begin{array}{c}\text{(16)}& g_{1}X={\tau }_d-\tau X\ge 0,g_{2}X={\sigma }_d-\sigma X\ge 0,\\ g_{3}X=x_4-x_1\ge 0,\\ g_{4}X=P_{c}X-P\ge 0,\\ g_{5}X={\delta }_d-\delta X\ge 0,\end{array}$$where$$\begin{array}{c}\text{(17)}& \tau X=\sqrt{\frac{{{\tau }^{\prime }X}^2+{{\tau }^{″}X}^2+x_2{\tau }^{\prime }X{\tau }^{″}X}{\sqrt{0.25x_2^2+{x_1+x_3}^2}}},\sigma X=\frac{50400}{x_3^2{x}_4},\\ P_{c}X=64746.0021-0.0282346x_3{x}_3{x}_4^3,\\ \delta X=\frac{2.1952}{x_3^2{x}_4},\\ {\tau }^{\prime }X=\frac{6000}{\sqrt{2}{x}_1{x}_2},\\ {\tau }^{″}X=\frac{600014+0.5x_2\sqrt{0.25x_2^2+{x_1+x_3}^2}}{20.707x_1{x}_2{x}_2^2/12+0.25{x_1+x_3}^2}.\end{array}$$

This problem has been solved by different algorithms such as DDSCA, HGA, MGWO-III, IAPSO, TEO, hHHO-SCA, HPSO, CPSO, and WCA. Table 21 summarizes the results of the above algorithms and compares them with the best results of TSO. The results show that TSO can provide a parameter design plan with lower cost compared to other algorithms. TSO generates the best solution at design variables of 0.205729, 3.470490, 9.036626, and 0.205729 with a minimum cost of 1.724854.

Table 21

Comparisons of best solutions offered by reported optimizers for welded beam design problem.

Algorithm	Optimal values for variables				Optimal cost
	$x_1$	$x_2$	$x_3$	$x_4$
DDSCA [50]	0.20516	3.4759	9.0797	0.20552	1.7305
HGA [62]	0.205712	3.470391	9.039693	0.205716	1.725236
MGWO-III [63]	0.205667	3.471899	9.036679	0.205733	1.724984
IAPSO [64]	0.205729	3.470886	9.036623	0.205729	1.724852
TEO [54]	0.205681	3.472305	9.035133	0.205796	1.725284
hHHO-SCA [55]	0.190086	3.696496	9.386343	0.204157	1.779032
HPSO [56]	0.20573	3.470489	9.036624	0.20573	1.724852
CPSO [53]	0.202369	3.544214	9.048210	0.205723	1.728024
WCA [26]	0.205728	3.470522	9.036620	0.205729	1.724856
TSO	0.205729	3.470490	9.036626	0.205729	1.724854

5. Conclusions

This work presents a novel swarm-based metaheuristic algorithm: tuna swarm optimization. The algorithm is inspired by the cooperative foraging mechanisms of tuna, including spiral foraging and parabolic foraging. The method has few adjustable parameters and can be implemented easily. TSO was comprehensively evaluated using a set of benchmark functions in different dimensions and was compared with other state-of-the-art algorithms. The results show that TSO is superior to the comparative algorithms. In addition, the pressure vessel design problem, the tension/compression spring design problem, and the welded beam design problem are investigated. The statistical results show that TSO has a high potential for solving real-world optimization problems compared to the reported methods. A major factor in TSO's success is the balance of exploitation and exploration achieved through the two foraging strategies. Meanwhile, fewer iterative steps bring less time costs, which is one of the strengths of TSO. However, while TSO performs excellently in most functions, there is still potential for enhancement regarding the small percentage of functions. This can be done by further enhancing TSO's ability to get rid of local optimum, using methods such as hybridisation of algorithms, adaptive parameters, etc.

For future work, binary and multiobjective versions of TSO can be developed for discrete problems and multiobjective optimization problems. Moreover, TSO will be applied to solve UAV mission planning problems such as trajectory planning problems, target allocation problems, etc. A further interesting direction would be to investigate the performance of different constraint handling methods in solving constrained optimization problems.

Authors’ Contributions

Andi Tang and Lei Xie made a major contribution to this work by presenting the conception and code.

Acknowledgments

The authors acknowledge funding received from the following science foundations: National Natural Science Foundation of China (No. 62101590) and The Science Foundation of the Shanxi Province, China (2020JQ-481 and 2021JM-224).

References

[1] G. Wu, "Across neighborhood search for numerical optimization," Information Sciences, vol. 329, pp. 597-618, DOI: 10.1016/j.ins.2015.09.051, 2016.

[2] G. Wu, W. Pedrycz, P. N. Suganthan, R. Mallipeddi, "A variable reduction strategy for evolutionary algorithms handling equality constraints," Applied Soft Computing, vol. 37, pp. 774-786, DOI: 10.1016/j.asoc.2015.09.007, 2015.

[3] A.-D. Tang, T. Han, H. Zhou, L. Xie, "An improved equilibrium optimizer with application in unmanned aerial vehicle path planning," Sensors, vol. 21 no. 5,DOI: 10.3390/s21051814, 2021.

[4] E. H. Houssein, M. M. Emam, A. A. Ali, "An efficient multilevel thresholding segmentation method for thermography breast cancer imaging based on improved chimp optimization algorithm," Expert Systems with Applications, vol. 185,DOI: 10.1016/j.eswa.2021.115651, 2021.

[5] M. Abd Elaziz, D. Yousri, M. A. A. Al-qaness, A. M. AbdelAty, A. G. Radwan, A. A. Ewees, "A Grunwald-Letnikov based Manta ray foraging optimizer for global optimization and image segmentation," Engineering Applications of Artificial Intelligence, vol. 98,DOI: 10.1016/j.engappai.2020.104105, 2021.

[6] Y. Xu, H. Huang, A. Asghar Heidari, W. Gui, X. Ye, Y. Chen, H. Chen, Z. Pan MFeature, "MFeature: towards high performance evolutionary tools for feature selection," Expert Systems with Applications, vol. 8,DOI: 10.1016/j.eswa.2021.115655, 2021.

[7] M. Alweshah, S. A. Khalaileh, B. B. Gupta, A. Almomani, A. I. Hammouri, M. A. Al-Betar, "The monarch butterfly optimization algorithm for solving feature selection problems," Neural Computing & Applications, vol. 23,DOI: 10.1007/s00521-020-05210-0, 2020.

[8] Y. Li, T. Han, H. Zhao, H. Gao, "An adaptive whale optimization algorithm using Gaussian distribution strategies and its application in heterogeneous ucavs task allocation," IEEE Access, vol. 7, pp. 110138-110158, DOI: 10.1109/ACCESS.2019.2933661, 2019.

[9] X. Wang, H. Zhao, T. Han, H. Zhou, C. Li, "A grey wolf optimizer using Gaussian estimation of distribution and its application in the multi-UAV multi-target urban tracking problem," Applied Soft Computing, vol. 78, pp. 240-260, DOI: 10.1016/j.asoc.2019.02.037, 2019.

[10] M. Abdel-Basset, R. Mohamed, R. K. Chakrabortty, K. Sallam, M. J. Ryan, "An efficient teaching-learning-based optimization algorithm for parameters identification of photovoltaic models: analysis and validations," Energy Conversion and Management, vol. 227,DOI: 10.1016/j.enconman.2020.113614, 2021.

[11] Q. Hao, Z. Zhou, Z. Wei, G. Chen, "Parameters identification of photovoltaic models using a multi-strategy success-history-based adaptive differential evolution," IEEE Access, vol. 8, pp. 35979-35994, DOI: 10.1109/ACCESS.2020.2975078, 2020.

[12] S.-W. Lin, C.-Y. Cheng, P. Pourhejazy, K.-C. Ying, C.-H. Lee, "New benchmark algorithm for hybrid flowshop scheduling with identical machines," Expert Systems with Applications, vol. 183,DOI: 10.1016/j.eswa.2021.115422, 2021.

[13] W. Liu, M. Dridi, H. Fei, A. H. El Hassani, "Hybrid metaheuristics for solving a home health care routing and scheduling problem with time windows, synchronized visits and lunch breaks," Expert Systems with Applications, vol. 183,DOI: 10.1016/j.eswa.2021.115307, 2021.

[14] W. Hare, J. Nutini, S. Tesfamariam, "A survey of non-gradient optimization methods in structural engineering," Advances in Engineering Software, vol. 59, pp. 19-28, DOI: 10.1016/j.advengsoft.2013.03.001, 2013.

[15] H. John Holland, Adaptation in Natural and Artificial Systems, 1992.

[16] R. A. Sarker, S. M. Elsayed, T. Ray, "Differential evolution with dynamic parameters selection for optimization problems," IEEE Transactions on Evolutionary Computation, vol. 18 no. 5, pp. 689-707, DOI: 10.1109/TEVC.2013.2281528, 2014.

[17] J. R. Koza, J. P. Rice, "Automatic programming of robots using genetic programming," Proceedings of the Tenth National Conference on Artificial Intelligence, .

[18] H.-G. Beyer, H.-P. Schwefel, "Evolution strategies – a comprehensive introduction," Natural Computing, vol. 1 no. 1,DOI: 10.1023/A:1015059928466, 2002.

[19] X. Xin Yao, Y. Yong Liu, G. Guangming Lin, "Evolutionary programming made faster," IEEE Transactions on Evolutionary Computation, vol. 3 no. 2, pp. 82-102, DOI: 10.1109/4235.771163, 1999.

[20] S. A. Uymaz, G. Tezel, E. Yel, "Artificial algae algorithm (AAA) for nonlinear global optimization," Applied Soft Computing, vol. 31, pp. 153-171, DOI: 10.1016/j.asoc.2015.03.003, 2015.

[21] D. Simon, "Biogeography-based optimization," IEEE Transactions on Evolutionary Computation, vol. 12 no. 6, pp. 702-713, DOI: 10.1109/TEVC.2008.919004, 2008.

[22] Z. Meng, J.-S. Pan, "Monkey King Evolution: a new memetic evolutionary algorithm and its application in vehicle fuel consumption optimization," Knowledge-Based Systems, vol. 97, pp. 144-157, DOI: 10.1016/j.knosys.2016.01.009, 2016.

[23] S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, "Optimization by simulated annealing," Science, vol. 220 no. 4598, pp. 671-680, DOI: 10.1126/science.220.4598.671, 1983.

[24] E. Rashedi, H. Nezamabadi-pour, S. Saryazdi, "GSA: a gravitational search algorithm," Information Sciences, vol. 179 no. 13, pp. 2232-2248, DOI: 10.1016/j.ins.2009.03.004, 2009.

[25] Z. Wei, C. Huang, X. Wang, T. Han, Y. Li, "Nuclear reaction optimization: a novel and powerful physics-based algorithm for global optimization," IEEE Access, vol. 7, pp. 66084-66109, DOI: 10.1109/ACCESS.2019.2918406, 2019.

[26] H. Eskandar, A. Sadollah, A. Bahreininejad, M. Hamdi, "Water cycle algorithm - a novel metaheuristic optimization method for solving constrained engineering optimization problems," Computers & Structures, vol. 110-111, pp. 151-166, DOI: 10.1016/j.compstruc.2012.07.010, 2012.

[27] S. Mirjalili, "SCA: a Sine Cosine Algorithm for solving optimization problems," Knowledge-Based Systems, vol. 96, pp. 120-133, DOI: 10.1016/j.knosys.2015.12.022, 2016.

[28] J. Kennedy, R. Eberhart, "Particle swarm optimization," Proceedings of the IEEE International Conference on Neural Networks - Conference Proceedings, .

[29] M. Dorigo, G. Di Caro, "Ant colony optimization: a new meta-heuristic," Proceedings of the 1999 Congress on Evolutionary Computation, .

[30] S. Mirjalili, S. M. Mirjalili, A. Lewis Optimizer, "Grey wolf optimizer," Advances in Engineering Software, vol. 69, pp. 46-61, DOI: 10.1016/j.advengsoft.2013.12.007, 2014.

[31] G.-G. Wang, S. Deb, Z. Cui, "Monarch butterfly optimization," Neural Computing & Applications, vol. 31 no. 7, pp. 1995-2014, DOI: 10.1007/s00521-015-1923-y, 2019.

[32] G. G. Wang, S. Deb, L. D. S. Coelho, "Elephant herding optimization," Proceedings of the 2015 3rd International Symposium on Computational and Business Intelligence,DOI: 10.1109/iscbi.2015.8, .

[33] G.-G. Wang, "Moth search algorithm: a bio-inspired metaheuristic algorithm for global optimization problems," Memetic Computing, vol. 10 no. 2, pp. 151-164, DOI: 10.1007/s12293-016-0212-3, 2018.

[34] W. Zhao, Z. Zhang, L. Wang, "Manta ray foraging optimization: an effective bio-inspired optimizer for engineering applications," Engineering Applications of Artificial Intelligence, vol. 87,DOI: 10.1016/j.engappai.2019.103300, 2020.

[35] G. G. Wang, S. Deb, L. D. S. Coelho, "Earthworm optimisation algorithm: a bio-inspired metaheuristic algorithm for global optimisation problems," International Journal of Bio-Inspired Computation, vol. 12 no. 1,DOI: 10.1504/ijbic.2018.093328, 2018.

[36] R. V. Rao, V. J. Savsani, D. P. Vakharia, "Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems," Computer-Aided Design, vol. 43 no. 3, pp. 303-315, DOI: 10.1016/j.cad.2010.12.015, 2011.

[37] M. Kumar, A. J. Kulkarni, S. C. Satapathy, "Socio evolution & learning optimization algorithm: a socio-inspired optimization methodology," Future Generation Computer Systems, vol. 81, pp. 252-272, DOI: 10.1016/j.future.2017.10.052, 2018.

[38] Y. Zhang, Z. Jin, "Group teaching optimization algorithm: a novel metaheuristic method for solving global optimization problems," Expert Systems with Applications, vol. 148,DOI: 10.1016/j.eswa.2020.113246, 2020.

[39] Q. Askari, M. Saeed, I. Younas, "Heap-based optimizer inspired by corporate rank hierarchy for global optimization," Expert Systems with Applications, vol. 161,DOI: 10.1016/j.eswa.2020.113702, 2020.

[40] Q. Askari, I. Younas, M. Saeed Optimizer, "Political Optimizer: a novel socio-inspired meta-heuristic for global optimization," Knowledge-Based Systems, vol. 195,DOI: 10.1016/j.knosys.2020.105709, 2020.

[41] E. Alba, B. Dorronsoro, "The exploration/exploitation tradeoff in dynamic cellular genetic algorithms," IEEE Transactions on Evolutionary Computation, vol. 9 no. 2, pp. 126-142, DOI: 10.1109/TEVC.2005.843751, 2005.

[42] L. Lin, M. Gen, "Auto-tuning strategy for evolutionary algorithms: balancing between exploration and exploitation," Proceedings of the Soft Computing, vol. 1,DOI: 10.1007/s00500-008-0303-2, 2009.

[43] D. H. Wolpert, W. G. Macready, "No free lunch theorems for optimization," IEEE Transactions on Evolutionary Computation, vol. 1 no. 1, pp. 67-82, DOI: 10.1109/4235.585893, 1997.

[44] S. Mirjalili, A. Lewis, "The whale optimization algorithm," Advances in Engineering Software, vol. 95, pp. 51-67, DOI: 10.1016/j.advengsoft.2016.01.008, 2016.

[45] S. Mirjalili, A. H. Gandomi, S. Z. Mirjalili, S. Saremi, H. Faris, S. M. Mirjalili, "Salp Swarm Algorithm: a bio-inspired optimizer for engineering design problems," Advances in Engineering Software, vol. 114, pp. 163-191, DOI: 10.1016/j.advengsoft.2017.07.002, 2017.

[46] A. A. Heidari, S. Mirjalili, H. Faris, I. Aljarah, M. Mafarja, H. Chen, "Harris hawks optimization: algorithm and applications," Future Generation Computer Systems, vol. 97, pp. 849-872, DOI: 10.1016/j.future.2019.02.028, 2019.

[47] A. Faramarzi, M. Heidarinejad, B. Stephens, S. Mirjalili, "Equilibrium optimizer: a novel optimization algorithm," Knowledge-Based Systems, vol. 191,DOI: 10.1016/j.knosys.2019.105190, 2020.

[48] S. Kaur, L. K. Awasthi, A. L. Sangal, G. Dhiman, "Tunicate Swarm Algorithm: a new bio-inspired based metaheuristic paradigm for global optimization," Engineering Applications of Artificial Intelligence, vol. 90,DOI: 10.1016/j.engappai.2020.103541, 2020.

[49] A. Tang, H. Zhou, T. Han, L. Xie, "A modified manta ray foraging optimization for global optimization problems," IEEE Access, vol. 9, pp. 128702-128721, DOI: 10.1109/ACCESS.2021.3113323, 2021.

[50] Y. Li, Y. Zhao, J. Liu, "Dimension by dimension dynamic sine cosine algorithm for global optimization problems," Applied Soft Computing, vol. 98,DOI: 10.1016/j.asoc.2020.106933, 2021.

[51] S. Gupta, K. Deep, "Improved sine cosine algorithm with crossover scheme for global optimization," Knowledge-Based Systems, vol. 165, pp. 374-406, DOI: 10.1016/j.knosys.2018.12.008, 2019.

[52] A. Sadollah, A. Bahreininejad, H. Eskandar, M. Hamdi, "Mine blast algorithm: a new population based algorithm for solving constrained engineering optimization problems," Applied Soft Computing, vol. 13 no. 5, pp. 2592-2612, DOI: 10.1016/j.asoc.2012.11.026, 2013.

[53] Q. He, L. Wang, "An effective co-evolutionary particle swarm optimization for constrained engineering design problems," Engineering Applications of Artificial Intelligence, vol. 20 no. 1, pp. 89-99, DOI: 10.1016/j.engappai.2006.03.003, 2007.

[54] A. Kaveh, A. Dadras, "A novel meta-heuristic optimization algorithm: thermal exchange optimization," Advances in Engineering Software, vol. 110, pp. 69-84, DOI: 10.1016/j.advengsoft.2017.03.014, 2017.

[55] V. K. Kamboj, A. Nandi, A. Bhadoria, S. Sehgal, "An intensify Harris Hawks optimizer for numerical and engineering optimization problems," Applied Soft Computing, vol. 89,DOI: 10.1016/j.asoc.2019.106018, 2020.

[56] Q. He, L. Wang, "A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization," Applied Mathematics and Computation, vol. 186 no. 2, pp. 1407-1422, DOI: 10.1016/j.amc.2006.07.134, 2007.

[57] S. Mirjalili, S. M. Mirjalili, A. Hatamlou, "Multi-Verse Optimizer: a nature-inspired algorithm for global optimization," Neural Computing & Applications, vol. 27 no. 2, pp. 495-513, DOI: 10.1007/s00521-015-1870-7, 2016.

[58] A. Baykasoğlu, F. B. Ozsoydan, "Adaptive firefly algorithm with chaos for mechanical design optimization problems," Applied Soft Computing, vol. 36, pp. 152-164, DOI: 10.1016/j.asoc.2015.06.056, 2015.

[59] C. A. Coello Coello, E. M. Montes, "Constraint-handling in genetic algorithms through the use of dominance-based tournament selection," Advanced Engineering Informatics, vol. 16,DOI: 10.1016/S1474-0346(02)00011-3, 2002.

[60] F.-z. Huang, L. Wang, Q. He, "An effective co-evolutionary differential evolution for constrained optimization," Applied Mathematics and Computation, vol. 186 no. 1, pp. 340-356, DOI: 10.1016/j.amc.2006.07.105, 2007.

[61] W. Zhao, L. Wang, Z. Zhang, "Artificial ecosystem-based optimization: a novel nature-inspired meta-heuristic algorithm," Neural Computing & Applications, vol. 32 no. 13, pp. 9383-9425, DOI: 10.1007/s00521-019-04452-x, 2020.

[62] X. Yan, H. Liu, Z. Zhu, Q. Wu, "Hybrid genetic algorithm for engineering design problems," Cluster Computing, vol. 20 no. 1, pp. 263-275, DOI: 10.1007/s10586-016-0680-8, 2017.

[63] V. Kumar, D. Kumar, "An astrophysics-inspired Grey wolf algorithm for numerical optimization and its application to engineering design problems," Advances in Engineering Software, vol. 112, pp. 231-254, DOI: 10.1016/j.advengsoft.2017.05.008, 2017.

[64] N. Ben Guedria, "Improved accelerated PSO algorithm for mechanical engineering optimization problems," Applied Soft Computing, vol. 40, pp. 455-467, DOI: 10.1016/j.asoc.2015.10.048, 2016.