# Circle Search Algorithm: A Geometry-Based Metaheuristic Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

- Introducing a novel geometry-based optimization method, called CSA.
- Presenting a mathematical model for the proposed CSA, including the states of exploration and exploitation processes.
- Applying the proposed CSA and other comparative algorithms to determine the optimal solution of 23 well-known functions and three engineering design issues
- Applying the CSA to solve high-dimensional functions (100 and 1000 dimensions).
- Testing the superiority and significance of the CSA in comparison with other algorithms, performed by using a variety of statistical tests, including the mean, standard deviation, rank test, and p-values.

## 2. Circle Search Algorithm

#### 2.1. Background

_{c}). The radius (R) is the line that links any point on the circle to the center. The perimeter of a circle is equal to the length of the curve that surrounds it. As observed in Figure 1, the tangent line segment is a straight line that intersects the circle at a single point (x

_{t}) and is perpendicular to the radius intersecting this point. According to Pythagorean equations, the orthogonal function (Tan) of the right triangle is the ratio between the radius and the perpendicular tangent line segment. It is obvious that the radius is defined as the distance between x

_{t}and x

_{c}, whereas the tangent line segment is the distance between points x

_{t}and x

_{p}; then, the orthogonal function (Tan) is expressed as in the following equations:

#### 2.2. CSA Formulation

_{t}is considered the search agent of the CSA, and the center point X

_{c}is assumed to be the best position in the algorithm. As shown in Figure 2, the CSA updates the search agent in response to the movement of the touching point toward the center. Nevertheless, to prevent the CSA from being stuck in a local solution, the contact point will be updated randomly by changing the angle in a random manner. The main steps of the CSA optimizer are explained below:

**Step 1: Initialization**: This step is important in the CSA, where whole dimensions of each search agent should be equally randomized, as depicted in Algorithm 1. Most of the previous published code randomizes the dimensions unequally, which sometimes make the algorithms surprisingly obtain the best results. Then, the search agents are initialized between the upper limit values (UB) and lower limit values (LB) of the search space as in Equation (4):

**Step 2: Update search agent position**: the position of the search agents

**X**is updated according to the evaluated best position

_{t}**X**as shown in Equation (5):

_{c}**Case 1:**Iter > (c.Maxiter): this case means that the angle $\theta =w\times rand$ all the time, which can applied to improve the exploration process of the CSA and escape the local stagnation.**Case 2:**Iter < (c.Maxiter): this case makes the angle $\theta =w\times p$ all the time, which can be used to improve the exploitation process of the CSA.

Algorithm 1 Initialization of the CSA |

InputLBandUB.Do for all search agentsr = random number between [0, 1]. Use Equation (4) to initialize the search agent X._{t}End Do |

Algorithm 2 Pseudo-code of the CSA |

Initialize the search agentsXusing Algorithm 1_{t}Input the constant value c, Iter = 0, and Maxiter While Iter less than MaxiterUse Equation (8) to find the value of a Do for all search agentsUse Equation (7) to find the value of w Use Equation (9) to find the value of p Use Equation (6) to find the value of the angle θ Use Equation (5) to update the search agent X_{t}if the updated search agents are out of the boundaries then set search agents equal to the boundaries find the fitness function f( X)_{t}End Do Evaluate the f( X) with the stored best solution f(_{t}X)_{c}Update f( X) and _{c}X_{c}Iter = Iter + 1 End WhileOutput f( X) and _{c}X_{c} |

## 3. Computational Complexity of the CSA

## 4. Experimental Results and Discussion

#### 4.1. Standard Functions

#### 4.2. Comparative Algorithms

#### 4.3. Statistical Analysis

#### 4.4. High-Dimensional Functions

#### 4.5. Computational Time

#### 4.6. Convergence Speed

## 5. Real-World Engineering Problems

#### 5.1. Welded Beam

_{c}), deflection (δ), and other restrictions. The formulations of objective function and its constraints are stated in Equation (9). The design factors x = [x

_{1}, x

_{2}, x

_{3}, x

_{4}] = [h, l, t, b] refer to thickness of weld, length, height, and thickness of the bar, respectively. Table 12 lists the optimal design factors that were achieved by the CSA and other algorithms. It is obvious that the CSA obtained the lowest cost compared with PSO, SSA, SCA, and GWO algorithms. Moreover, the standard deviation of 30 runs’ results of the CSA was low, and thus the CSA was stable at all 30 runs. Therefore, the CSA was considered robust enough to be used in real-world engineering problems.

#### 5.2. Pressure Vessel

_{s}and T

_{h}) and the inner radius and length of the vessel (R and L) without bounce. Table 13 lists the optimal design factors of the pressure vessel that were obtained by the CSA and other algorithms, including PSO, SSA, SCA, and GWO algorithms. It was obvious that the CSA was competing with other algorithms to find the best minimum; however, the GWO algorithm obtained better mean and standard deviations for 30 independent runs.

#### 5.3. Tension Spring

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Algorithm | Name | Reference | Classification | Mimicking |
---|---|---|---|---|

Orca predation algorithm | OPA | [76] | Biology-based (hunters) | The orcas’ hunting habit. |

Komodo mlipir algorithm | KMA | [77] | Biology-based | Komodo dragons and miliper foraging and reproduction |

Integrated optimization algorithm | IOA | [78] | Evolutionarily based | Follower search, leader search, wanderer search, crossover search, and role learning are all terms used to construct IOA |

Reptile search algorithm | RSA | [79] | Biology-based (hunters) | Crocodiles’ hunting habit |

African vultures optimization | AVOA | [80] | Biology-based | African vultures’ feeding and navigational behaviors |

Elephant clan optimization | ECO | [81] | Biology-based | Elephants’ clan behavior. |

Cooperation search algorithm | CoSA | [82] | Human-learning-based | The behaviors of teamwork in contemporary business |

Group teaching optimization | GTOA | [83] | Human-learning-based | The relationship between the instructor and his or her pupils |

Mayfly algorithm | MA | [84] | Biology-based | Mayfly flying and mating behavior |

Search and rescue optimization algorithm | SAR | [85] | Human-learning-based | The study of human behavior during search and rescue missions |

Function | Expression | Dimension (d) | Solution Space | Best Solution |
---|---|---|---|---|

F1 | $f(x)={\displaystyle \sum _{i=1}^{d}{x}_{i}{}^{2}}$ | 30, 100, 1000 | [−100, 100]^{d} | 0 |

F2 | $f(x)={\displaystyle \sum _{i=1}^{d}\left|{x}_{i}\right|}+{\displaystyle \prod _{i=1}^{d}\left|{x}_{i}\right|}$ | 30, 100, 1000 | [−10, 10]^{d} | 0 |

F3 | $f(x)={\displaystyle \sum _{i=1}^{d}{({\displaystyle \sum _{j=1}^{i}{x}_{j}})}^{2}}$ | 30, 100, 1000 | [−100, 100]^{d} | 0 |

F4 | $f(x)={\mathrm{max}}_{i}\{\left|{x}_{i}\right|,1\le i\le d\}$ | 30, 100, 1000 | [−100, 100]^{d} | 0 |

F5 | $f(x)={\displaystyle \sum _{i=1}^{d}[100{({x}_{i+1}-{x}_{i}{}^{2})}^{2}+{({x}_{i}-1)}^{2}]}$ | 30, 100, 1000 | [−30, 30]^{d} | 0 |

F6 | $f(x)={\displaystyle \sum _{i=1}^{d}{({x}_{i}+0.5)}^{2}}$ | 30, 100, 1000 | [−100, 100]^{d} | 0 |

F7 | $f(x)={\displaystyle \sum _{i=1}^{d}i.{x}_{i}{}^{4}+random[0,1)}$ | 30, 100, 1000 | [−1.28, 1.28]^{d} | 0 |

Function | Expression | Dimension (d) | Solution Space | Best Solution |
---|---|---|---|---|

F8 | $f(x)={\displaystyle \sum _{i=1}^{d}{x}_{i}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})}$ | 30, 100, 1000 | [−500, 500]^{d} | −418.9829 × d |

F9 | $f(x)={\displaystyle \sum _{i=1}^{n}[{x}_{i}{}^{2}-10\mathrm{cos}(2\pi {x}_{i})}+10]$ | 30, 100, 1000 | [−5.12, 5.12]^{d} | 0 |

F10 | $f(x)=-20\mathrm{exp}(-0.2\sqrt{\frac{1}{d}{\displaystyle \sum _{j=1}^{d}{x}_{j}}})-\mathrm{exp}(\frac{1}{n}\mathrm{cos}(2\pi {x}_{j}))+20+e$ | 30, 100, 1000 | [−32, 32]^{d} | 0 |

F11 | $f(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{d}{x}_{i}{}^{2}}-{\displaystyle \prod _{i=1}^{d}\mathrm{cos}(\frac{{x}_{i}}{\sqrt{i}})}+1$ | 30, 100, 1000 | [−600, 600]^{d} | 0 |

F12 | $\begin{array}{l}f(x)=\frac{\pi}{d}\{10\mathrm{sin}(\pi {y}_{1})+{\displaystyle \sum _{i=1}^{d}{({y}_{i}-1)}^{2}[1+10{\mathrm{sin}}^{2}(\pi {y}_{i+1})]+{({y}_{d}-1)}^{2}\}}\\ \hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{d}u({x}_{i},10,100,4)}\\ {y}_{i}=1+\frac{{x}_{i}+1}{4}\\ u({x}_{i},a,k,m)=\{\begin{array}{ll}k{({x}_{i}-a)}^{m}& {x}_{i}>a\\ 0& -a<{x}_{i}<a\\ k{(-{x}_{i}-a)}^{m}& {x}_{i}<-a\end{array}\end{array}$ | 30, 100, 1000 | [−50, 50]^{d} | 0 |

F13 | $\begin{array}{ll}f(x)& =0.1\times \{10{\mathrm{sin}}^{2}(3\pi {x}_{1})+{\displaystyle \sum _{i=1}^{d}{({x}_{i}-1)}^{2}[1+{\mathrm{sin}}^{2}(3\pi {x}_{i}+1)]}\\ & +{({x}_{d}-1)}^{2}[1+{\mathrm{sin}}^{2}(2\pi {x}_{d})]\}+{\displaystyle \sum _{i=1}^{d}u({x}_{i},5,100,4)}\end{array}$ | 30, 100, 1000 | [−50, 50]^{d} | 0 |

Function | Expression | Dimension (d) | Solution Space | Best Solution |
---|---|---|---|---|

F14 | $f(x)=(\frac{1}{500}+{\displaystyle \sum _{i=1}^{25}\frac{1}{i+{\displaystyle \sum _{j=1}^{2}{({x}_{i}-{a}_{ij})}^{6}}}{)}^{-1}}$ | 2 | [−65, 65]^{d} | 1 |

F15 | $f(x)={\displaystyle \sum _{i=1}^{11}{[{a}_{i}-\frac{{x}_{i}({b}_{i}{}^{2}+{b}_{i}{x}_{2})}{{b}_{i}{}^{2}+{b}_{i}{x}_{3}+{x}_{4}}]}^{2}}$ | 4 | [−5, 5]^{d} | 0.00030 |

F16 | $f(x)=4{x}_{1}{}^{2}-2.1{x}_{1}{}^{4}+{\scriptscriptstyle \frac{1}{3}}{x}_{1}{}^{6}+{x}_{1}{x}_{2}-4{x}_{2}{}^{2}+4{x}_{2}{}^{4}$ | 2 | [−5, 5]^{d} | −1.0316 |

F17 | $f(x)={({x}_{2}-{\scriptscriptstyle \frac{5.1}{4{\pi}^{2}}}{x}_{1}{}^{2}+{\scriptscriptstyle \frac{5}{\pi}}{x}_{1}-6)}^{2}+10(1-{\scriptscriptstyle \frac{1}{8\pi}})\mathrm{cos}({x}_{1})+10$ | 2 | [−5, 5]^{d} | 0.398 |

F18 | $\begin{array}{ll}f(x)=& [1+{({x}_{1}+{x}_{2}+1)}^{2}(19-14{x}_{1}+3{x}_{1}{}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}{}^{2})]\times \\ & [30+{(2{x}_{1}-3{x}_{2})}^{2}\times (18-32{x}_{1}+12{x}_{1}{}^{2}\\ & +48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}{}^{2})]\end{array}$ | 2 | [−2, 2]^{d} | 3 |

F19 | $f(x)=-{\displaystyle \sum _{i=1}^{4}{c}_{i}\mathrm{exp}{(-{\displaystyle \sum _{j=1}^{3}{a}_{ij}({x}_{j}-{p}_{ij}})}^{2})}$ | 3 | [1, 3]^{d} | −3.86 |

F20 | $f(x)=-{\displaystyle \sum _{i=1}^{4}{c}_{i}\mathrm{exp}{(-{\displaystyle \sum _{j=1}^{6}{a}_{ij}({x}_{j}-{p}_{ij}})}^{2})}$ | 6 | [0, 1]^{d} | −3.32 |

F21 | $f(x)=-{\displaystyle \sum _{i=1}^{5}{[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}^{-1}}$ | 4 | [0, 10]^{d} | −10.1532 |

F22 | $f(x)=-{\displaystyle \sum _{i=1}^{7}{[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}^{-1}}$ | 4 | [0, 10]^{d} | −10.4028 |

F23 | $f(x)=-{\displaystyle \sum _{i=1}^{10}{[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}^{-1}}$ | 4 | [0, 10]^{d} | −10.5363 |

Algorithm | Parameters |
---|---|

Proposed CSA | w decreased from 1.5 to 0 and constant c = 0.75 for F1–F13 and c = 0.3 for F14–F23 |

PSO | Inertia weight w decreased from 0.5 to 0.3, c_{1} = 2, and c_{2} = 2 |

GWO | The parameter a changed from 2 to 0 |

SSA | Probability update was 0.5 |

SCA | Constant a = 2 and probability update was 0.5 |

WOA | The parameter a changed from 2 to 0, b = 1 |

HHO | The decreasing energy E1 changed from 2 to 0 |

CGO | α, β, and γ were random numbers |

TSO | The parameter a changed from 2 to 0 |

GWO | SCA | SSA | HHO | WOA | PSO | TSO | CGO | CSA | ||
---|---|---|---|---|---|---|---|---|---|---|

F1 | Avg. | 5.7838 × 10^{−38} | 6.5806 × 10^{−11} | 9.5464 × 10^{−08} | 1.4759 × 10^{−109} | 6.6351 × 10^{−69} | 4.6281 × 10^{−07} | 1.0826 × 10^{−03} | 1.1934 × 10^{−136} | 9.5326 × 10^{−219} |

STD. | 9.1734 × 10^{−38} | 2.7669 × 10^{−10} | 5.2552 × 10^{−08} | 8.0839 × 10^{−109} | 3.6342 × 10^{−68} | 1.0552 × 10^{−06} | 3.9713 × 10^{−03} | 4.7846 × 10^{−136} | 0.0000 × 10^{00} | |

Min | 1.1944 × 10^{−40} | 1.9641 × 10^{−15} | 3.3386 × 10^{−08} | 0.0000 × 10^{00} | 6.6909 × 10^{−86} | 2.3249 × 10^{−10} | 8.9536 × 10^{−08} | 0.0000 × 10^{00} | 2.9648 × 10^{−278} | |

F2 | Avg. | 1.8357 × 10^{−22} | 3.1710 × 10^{−08} | 2.6003 × 10^{−01} | 3.4043 × 10^{−53} | 6.7368 × 10^{−50} | 5.3551 × 10^{−04} | 8.4374 × 10^{−06} | 1.7566 × 10^{−71} | 1.3380 × 10^{−92} |

STD. | 3.6643 × 10^{−22} | 4.5318 × 10^{−08} | 1.8497 × 10^{−01} | 1.2317 × 10^{−52} | 3.3959 × 10^{−49} | 8.9901 × 10^{−04} | 1.2036 × 10^{−05} | 7.9508 × 10^{−71} | 7.3287 × 10^{−92} | |

Min | 1.1603 × 10^{−23} | 1.1753 × 10^{−09} | 4.5012 × 10^{−03} | 9.7243 × 10^{−172} | 1.1799 × 10^{−57} | 2.8776 × 10^{−06} | 6.5020 × 10^{−07} | 0.0000 × 10^{00} | 3.4777 × 10^{−140} | |

F3 | Avg. | 1.2328 × 10^{−08} | 1.0265 × 10^{−07} | 3.0191 × 10^{02} | 5.3195 × 10^{−78} | 1.7475 × 10^{−01} | 1.0393 × 10^{03} | 3.9684 × 10^{02} | 1.0135 × 10^{−98} | 3.5072 × 10^{−192} |

STD. | 3.2665 × 10^{−08} | 3.1441 × 10^{−07} | 4.7355 × 10^{02} | 2.0975 × 10^{−77} | 4.9298 × 10^{−01} | 9.4985 × 10^{02} | 1.0181 × 10^{03} | 2.6061 × 10^{−98} | 0.0000 × 10^{00} | |

Min | 3.4277 × 10^{−12} | 9.8383 × 10^{−13} | 1.1474 × 10^{00} | 4.6827 × 10^{−103} | 7.1459 × 10^{−08} | 3.8895 × 10^{−02} | 4.5120 × 10^{−04} | 0.0000 × 10^{00} | 1.2646 × 10^{−274} | |

F4 | Avg. | 5.7254 × 10^{−01} | 1.0814 × 10^{00} | 1.0281 × 10^{00} | 1.5750 × 10^{−119} | 4.8675 × 10^{−02} | 3.0927 × 10^{00} | 7.2259 × 10^{−01} | 6.5523 × 10^{−59} | 1.2504 × 10^{−98} |

STD. | 1.0996 × 10^{00} | 2.5084 × 10^{00} | 8.5943 × 10^{−01} | 8.6266 × 10^{−119} | 1.3524 × 10^{−01} | 2.9911 × 10^{00} | 6.7465 × 10^{−01} | 1.2075 × 10^{−58} | 6.8486 × 10^{−98} | |

Min | 4.2053 × 10^{−08} | 9.9628 × 10^{−05} | 2.8062 × 10^{−02} | 8.0269 × 10^{−181} | 6.5725 × 10^{−06} | 2.8062 × 10^{−02} | 6.0508 × 10^{−03} | 0.0000 × 10^{00} | 3.0109 × 10^{−139} | |

F5 | Avg. | 2.5381 × 10^{01} | 7.0200 × 10^{01} | 2.5675 × 10^{01} | 1.8902 × 10^{−06} | 8.5474 × 10^{00} | 1.9573 × 10^{01} | 1.1420 × 10^{01} | 1.5351 × 10^{01} | 0.0000 × 10^{00} |

STD. | 1.5260 × 10^{01} | 1.3659 × 10^{02} | 2.4616 × 10^{01} | 1.0192 × 10^{−05} | 1.2854 × 10^{01} | 2.3932 × 10^{01} | 1.2831 × 10^{01} | 5.4322 × 10^{00} | 0.0000 × 10^{00} | |

Min | 7.5169 × 10^{−02} | 7.5169 × 10^{−02} | 9.3447 × 10^{−04} | 0.0000 × 10^{00} | 2.3611 × 10^{−07} | 4.2949 × 10^{−05} | 4.6622 × 10^{−03} | 3.2114 × 10^{−06} | 0.0000 × 10^{00} | |

F6 | Avg. | 5.4155 × 10^{−01} | 5.3173 × 10^{00} | 2.0211 × 10^{−07} | 2.8436 × 10^{−07} | 1.3848 × 10^{−02} | 4.7315 × 10^{−07} | 4.8451 × 10^{−01} | 1.8244 × 10^{−18} | 0.0000 × 10^{00} |

STD. | 3.5419 × 10^{−01} | 8.0594 × 10^{−01} | 3.5045 × 10^{−07} | 1.4406 × 10^{−06} | 2.4111 × 10^{−02} | 8.7532 × 10^{−07} | 4.2675 × 10^{−01} | 5.9259 × 10^{−18} | 0.0000 × 10^{00} | |

Min | 2.8333 × 10^{−04} | 1.2272 × 10^{00} | 2.4022 × 10^{−08} | 0.0000 × 10^{00} | 7.5679 × 10^{−08} | 5.0032 × 10^{−09} | 3.5827 × 10^{−02} | 6.2486 × 10^{−24} | 0.0000 × 10^{00} | |

F7 | Avg. | 1.3202 × 10^{−03} | 4.7389 × 10^{−04} | 4.0651 × 10^{−02} | 1.0839 × 10^{−04} | 1.2714 × 10^{−03} | 2.7046 × 10^{−02} | 2.5284 × 10^{−03} | 4.6345 × 10^{−04} | 3.8180 × 10^{−04} |

STD. | 6.4299 × 10^{−04} | 4.4640 × 10^{−04} | 4.0466 × 10^{−02} | 1.0664 × 10^{−04} | 2.4965 × 10^{−03} | 2.5014 × 10^{−02} | 2.8216 × 10^{−03} | 3.4197 × 10^{−04} | 6.6990 × 10^{−04} | |

Min | 2.8985 × 10^{−04} | 4.5957 × 10^{−06} | 2.1466 × 10^{−03} | 1.3803 × 10^{−05} | 7.2309 × 10^{−07} | 7.7340 × 10^{−04} | 1.1353 × 10^{−04} | 7.3420 × 10^{−05} | 2.6012 × 10^{−05} | |

F8 | Avg. | −1.1269 × 10^{04} | −1.1117 × 10^{04} | −1.2096 × 10^{04} | −1.2569 × 10^{04} | −1.2427 × 10^{04} | −1.2071 × 10^{04} | −1.2223 × 10^{04} | −1.2451 × 10^{04} | −1.2569 × 10^{04} |

STD. | 1.6040 × 10^{03} | 1.5893 × 10^{03} | 1.2285 × 10^{03} | 2.3541 × 10^{−09} | 6.5709 × 10^{02} | 1.2389 × 10^{03} | 9.5254 × 10^{02} | 6.4871 × 10^{02} | 1.9404 × 10^{−12} | |

Min | −1.2557 × 10^{04} | −1.2551 × 10^{04} | −1.2569 × 10^{04} | −1.2569 × 10^{04} | −1.2569 × 10^{04} | −1.2569 × 10^{04} | −1.2569 × 10^{04} | −1.2569 × 10^{04} | −1.2569 × 10^{04} | |

F9 | Avg. | 4.1807 × 10^{01} | 4.0559 × 10^{01} | 9.9496 × 10^{00} | 0.0000 × 10^{00} | 1.8948 × 10^{−15} | 3.0214 × 10^{01} | 2.2908 × 10^{00} | 0.0000 × 10^{00} | 0.0000 × 10^{00} |

STD. | 3.5269 × 10^{01} | 4.2016 × 10^{01} | 1.4311 × 10^{01} | 0.0000 × 10^{00} | 1.0378 × 10^{−14} | 3.3687 × 10^{01} | 1.2547 × 10^{01} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | |

Min | 0.0000 × 10^{00} | 0.0000 × 10^{00} | 2.3251 × 10^{−08} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | 2.7281 × 10^{−08} | 1.7390 × 10^{−08} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | |

F10 | Avg. | 1.1978 × 10^{−01} | 2.5187 × 10^{−01} | 9.4409 × 10^{−01} | 8.8818 × 10^{−16} | 5.0330 × 10^{−15} | 9.1797 × 10^{−01} | 1.3331 × 10^{−01} | 2.7830 × 10^{−15} | 8.8818 × 10^{−16} |

STD. | 6.5604 × 10^{−01} | 9.5974 × 10^{−01} | 1.2595 × 10^{00} | 0.0000 × 10^{00} | 2.9626 × 10^{−15} | 1.9652 × 10^{00} | 7.2493 × 10^{−01} | 1.8027 × 10^{−15} | 0.0000 × 10^{00} | |

Min | 7.9936 × 10^{−15} | 6.1332 × 10^{−09} | 4.1912 × 10^{−05} | 8.8818 × 10^{−16} | 8.8818 × 10^{−16} | 8.9230 × 10^{−06} | 7.4523 × 10^{−06} | 8.8818 × 10^{−16} | 8.8818 × 10^{−16} | |

F11 | Avg. | 3.7391 × 10^{−03} | 2.1526 × 10^{−07} | 1.8495 × 10^{−02} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | 1.1234 × 10^{−02} | 6.6144 × 10^{−02} | 0.0000 × 10^{00} | 0.0000 × 10^{00} |

STD. | 1.0062 × 10^{−02} | 1.1396 × 10^{−06} | 1.4792 × 10^{−02} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | 1.3511 × 10^{−02} | 1.5859 × 10^{−01} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | |

Min | 0.0000 × 10^{00} | 3.2196 × 10^{−15} | 4.8962 × 10^{−04} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | 2.9305 × 10^{−09} | 1.8765 × 10^{−08} | 0.0000 × 10^{00} | 0.0000 × 10^{00} | |

F12 | Avg. | 1.2064 × 10^{00} | 1.4563 × 10^{00} | 1.9961 × 10^{−02} | 1.0540 × 10^{−08} | 3.7369 × 10^{−04} | 7.9748 × 10^{−01} | 7.4001 × 10^{−03} | 1.3909 × 10^{−20} | 1.5705 × 10^{−32} |

STD. | 2.5890 × 10^{00} | 2.4947 × 10^{00} | 7.7080 × 10^{−02} | 3.8589 × 10^{−08} | 1.1984 × 10^{−03} | 1.7667 × 10^{00} | 1.2526 × 10^{−02} | 6.6394 × 10^{−20} | 5.5674 × 10^{−48} | |

Min | 1.6660 × 10^{−05} | 2.1551 × 10^{−04} | 4.8395 × 10^{−07} | 6.3387 × 10^{−21} | 2.1843 × 10^{−09} | 3.6269 × 10^{−13} | 1.3758 × 10^{−06} | 1.0149 × 10^{−25} | 1.5705 × 10^{−32} | |

F13 | Avg. | 2.3921 × 10^{−01} | 1.8630 × 10^{00} | 4.0527 × 10^{−01} | 7.6694 × 10^{−07} | 2.4754 × 10^{−03} | 2.3928 × 10^{−01} | 1.5123 × 10^{−01} | 2.2315 × 10^{−02} | 1.3498 × 10^{−32} |

STD. | 2.3561 × 10^{−01} | 9.1497 × 10^{−01} | 1.5566 × 10^{00} | 3.5053 × 10^{−06} | 6.0994 × 10^{−03} | 1.2677 × 10^{00} | 1.5863 × 10^{−01} | 5.2790 × 10^{−02} | 5.5674 × 10^{−48} | |

Min | 5.3035 × 10^{−04} | 3.4038 × 10^{−02} | 7.9766 × 10^{−06} | 1.3498 × 10^{−32} | 2.8468 × 10^{−09} | 1.1833 × 10^{−10} | 6.0890 × 10^{−05} | 8.1780 × 10^{−24} | 1.3498 × 10^{−32} | |

F14 | Avg. | 3.2156 × 10^{00} | 1.9345 × 10^{00} | 1.0643 × 10^{00} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 1.0311 × 10^{00} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} |

STD. | 3.8889 × 10^{00} | 9.9707 × 10^{−01} | 2.5219 × 10^{−01} | 1.5699 × 10^{−10} | 1.3046 × 10^{−08} | 1.8148 × 10^{−01} | 1.7835 × 10^{−07} | 0.0000 × 10^{00} | 1.7494 × 10^{−16} | |

Min | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | 9.9800 × 10^{−01} | |

F15 | Avg. | 4.0459 × 10^{−04} | 5.7433 × 10^{−04} | 6.4506 × 10^{−04} | 3.2725 × 10^{−04} | 3.2225 × 10^{−04} | 4.6375 × 10^{−04} | 3.7770 × 10^{−04} | 3.3801 × 10^{−04} | 3.0806 × 10^{−04} |

STD. | 1.1637 × 10^{−04} | 2.5944 × 10^{−04} | 3.3345 × 10^{−04} | 2.2421 × 10^{−05} | 2.5146 × 10^{−05} | 2.4872 × 10^{−04} | 1.7279 × 10^{−04} | 1.6718 × 10^{−04} | 7.8742 × 10^{−07} | |

Min | 3.0749 × 10^{−04} | 3.3338 × 10^{−04} | 3.0769 × 10^{−04} | 3.0751 × 10^{−04} | 3.0784 × 10^{−04} | 3.0749 × 10^{−04} | 3.0758 × 10^{−04} | 3.0749 × 10^{−04} | 3.0749 × 10^{−04} | |

F16 | Avg. | −1.0316 × 10^{00} | −1.0315 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} |

STD. | 7.0549 × 10^{−08} | 8.7695 × 10^{−05} | 4.0464 × 10^{−14} | 3.0122 × 10^{−09} | 3.6944 × 10^{−09} | 6.6486 × 10^{−16} | 1.5990 × 10^{−05} | 6.7752 × 10^{−16} | 4.6137 × 10^{−09} | |

Min | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | −1.0316 × 10^{00} | |

F17 | Avg. | 3.9789 × 10^{−01} | 4.0100 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} | 3.9790 × 10^{−01} | 3.9789 × 10^{−01} | 3.9791 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} |

STD. | 5.3714 × 10^{−06} | 3.2491 × 10^{−03} | 7.3008 × 10^{−15} | 5.2004 × 10^{−06} | 1.7830 × 10^{−05} | 0.0000 × 10^{00} | 2.2182 × 10^{−05} | 0.0000 × 10^{00} | 5.2398 × 10^{−08} | |

Min | 3.9789 × 10^{−01} | 3.9811 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} | 3.9789 × 10^{−01} | |

F18 | Avg. | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0001 × 10^{00} | 3.0000 × 10^{00} | 2.9715 × 10^{01} | 3.0000 × 10^{00} | 3.0000 × 10^{00} |

STD. | 4.1823 × 10^{−05} | 3.5195 × 10^{−05} | 1.8243 × 10^{−13} | 6.0229 × 10^{−07} | 4.6026 × 10^{−04} | 9.9301 × 10^{−16} | 5.0596 × 10^{00} | 9.3663 × 10^{−16} | 4.9599 × 10^{−06} | |

Min | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0000 × 10^{00} | 3.0032 × 10^{00} | 3.0000 × 10^{00} | 3.0000 × 10^{00} | |

F19 | Avg. | −3.8616 × 10^{00} | −3.8507 × 10^{00} | −3.8628 × 10^{00} | −3.8583 × 10^{00} | −3.8518 × 10^{00} | −3.8628 × 10^{00} | −3.8040 × 10^{00} | −3.8628 × 10^{00} | −3.8625 × 10^{00} |

STD. | 2.3429 × 10^{−03} | 7.7056 × 10^{−03} | 1.6209 × 10^{−11} | 6.6567 × 10^{−03} | 1.7466 × 10^{−02} | 2.6684 × 10^{−15} | 2.0896 × 10^{−01} | 2.7101 × 10^{−15} | 1.4192 × 10^{−03} | |

Min | −3.8628 × 10^{00} | −3.8605 × 10^{00} | −3.8628 × 10^{00} | −3.8628 × 10^{00} | −3.8628 × 10^{00} | −3.8628 × 10^{00} | −3.8628 × 10^{00} | −3.8628 × 10^{00} | −3.8628 × 10^{00} | |

F20 | Avg. | −3.3219 × 10^{00} | −2.6135 × 10^{00} | −3.2161 × 10^{00} | −3.0656 × 10^{00} | −3.3118 × 10^{00} | −3.2638 × 10^{00} | −3.3133 × 10^{00} | −3.2784 × 10^{00} | −3.3059 × 10^{00} |

STD. | 2.2933 × 10^{−05} | 3.1389 × 10^{−01} | 5.5503 × 10^{−02} | 1.0806 × 10^{−01} | 3.4871 × 10^{−02} | 7.1334 × 10^{−02} | 9.1886 × 10^{−03} | 5.8273 × 10^{−02} | 4.1813 × 10^{−02} | |

Min | −3.3220 × 10^{00} | −3.0564 × 10^{00} | −3.3220 × 10^{00} | −3.1993 × 10^{00} | −3.3217 × 10^{00} | −3.3220 × 10^{00} | −3.3211 × 10^{00} | −3.3220 × 10^{00} | −3.3220 × 10^{00} | |

F21 | Avg. | −8.3916 × 10^{00} | −6.8959 × 10^{00} | −1.0153 × 10^{01} | −9.8130 × 10^{00} | −9.8109 × 10^{00} | −8.9697 × 10^{00} | −1.0128 × 10^{01} | −1.0153 × 10^{01} | −1.0153 × 10^{01} |

STD. | 2.1174 × 10^{00} | 2.2821 × 10^{00} | 5.5169 × 10^{−11} | 1.2933 × 10^{00} | 1.2928 × 10^{00} | 2.1819 × 10^{00} | 2.7575 × 10^{−02} | 6.7923 × 10^{−15} | 1.4067 × 10^{−07} | |

Min | −1.0152 × 10^{01} | −1.0152 × 10^{01} | −1.0153 × 10^{01} | −1.0153 × 10^{01} | −1.0153 × 10^{01} | −1.0153 × 10^{01} | −1.0153 × 10^{01} | −1.0153 × 10^{01} | −1.0153 × 10^{01} | |

F22 | Avg. | −8.9990 × 10^{00} | −7.2308 × 10^{00} | −1.0227 × 10^{01} | −9.8713 × 10^{00} | −9.8701 × 10^{00} | −8.3816 × 10^{00} | −1.0391 × 10^{01} | −1.0403 × 10^{01} | −1.0403 × 10^{01} |

STD. | 2.2109 × 10^{00} | 2.4640 × 10^{00} | 9.6292 × 10^{−01} | 1.6218 × 10^{00} | 1.6214 × 10^{00} | 2.7339 × 10^{00} | 1.0467 × 10^{−02} | 1.3601 × 10^{−15} | 3.3477 × 10^{−05} | |

Min | −1.0402 × 10^{01} | −1.0394 × 10^{01} | −1.0403 × 10^{01} | −1.0403 × 10^{01} | −1.0403 × 10^{01} | −1.0403 × 10^{01} | −1.0403 × 10^{01} | −1.0403 × 10^{01} | −1.0403 × 10^{01} | |

F23 | Avg. | −8.6580 × 10^{00} | −7.2350 × 10^{00} | −9.8216 × 10^{00} | −1.0175 × 10^{01} | −1.0354 × 10^{01} | −8.7432 × 10^{00} | −1.0518 × 10^{01} | −1.0358 × 10^{01} | −1.0536 × 10^{01} |

STD. | 2.5993 × 10^{00} | 2.5826 × 10^{00} | 1.8535 × 10^{00} | 1.3719 × 10^{00} | 9.8705 × 10^{−01} | 2.5794 × 10^{00} | 2.0234 × 10^{−02} | 9.7874 × 10^{−01} | 2.7028 × 10^{−05} | |

Min | −1.0536 × 10^{01} | −1.0536 × 10^{01} | −1.0536 × 10^{01} | −1.0536 × 10^{01} | −1.0536 × 10^{01} | −1.0536 × 10^{01} | −1.0536 × 10^{01} | −1.0536 × 10^{01} | −1.0536 × 10^{01} |

GWO | SCA | SSA | HHO | WOA | PSO | TSO | CGO | |
---|---|---|---|---|---|---|---|---|

F1 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 6.0350 × 10^{−03} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.9209 × 10^{−06} |

F2 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 5.0383 × 10^{−01} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 2.3534 × 10^{−06} |

F3 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 3.1817 × 10^{−06} |

F4 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 3.7243 × 10^{−05} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 2.3534 × 10^{−06} |

F5 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.5625 × 10^{−02} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} |

F6 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.3183 × 10^{−04} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} |

F7 | 5.3070 × 10^{−05} | 6.5641 × 10^{−02} | 1.7344 × 10^{−06} | 1.7088 × 10^{−03} | 3.3269 × 10^{−02} | 1.7344 × 10^{−06} | 5.7924 × 10^{−05} | 1.3194 × 10^{−02} |

F8 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 6.2500 × 10^{−02} | 2.5631 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.2383 × 10^{−06} |

F9 | 2.5631 × 10^{−06} | 2.5596 × 10^{−06} | 1.7344 × 10^{−06} | 1.0000 × 10^{00} | 1.0000 × 10^{00} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.0000 × 10^{00} |

F10 | 1.4383 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.0000 × 10^{00} | 2.1912 × 10^{−05} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 6.3342 × 10^{−05} |

F11 | 1.2500 × 10^{−01} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.0000 × 10^{00} | 1.0000 × 10^{00} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.0000 × 10^{00} |

F12 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} |

F13 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 2.7016 × 10^{−05} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7300 × 10^{−06} |

F14 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.0231 × 10^{−05} | 1.8435 × 10^{−04} | 1.7344 × 10^{−06} | 1.0000 × 10^{00} | 1.7344 × 10^{−06} | 5.0000 × 10^{−01} |

F15 | 2.8434 × 10^{−05} | 1.7344 × 10^{−06} | 1.9209 × 10^{−06} | 6.3391 × 10^{−06} | 2.8786 × 10^{−06} | 3.3173 × 10^{−04} | 3.5152 × 10^{−06} | 3.1123 × 10^{−05} |

F16 | 3.8822 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 8.9443 × 10^{−04} | 4.1140 × 10^{−03} | 1.7344 × 10^{−06} | 2.1266 × 10^{−06} | 1.7344 × 10^{−06} |

F17 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 5.2165 × 10^{−06} | 1.9209 × 10^{−06} | 1.7344 × 10^{−06} | 1.9209 × 10^{−06} | 1.7344 × 10^{−06} |

F18 | 3.0650 × 10^{−04} | 1.2453 × 10^{−02} | 1.7344 × 10^{−06} | 2.1630 × 10^{−05} | 1.1499 × 10^{−04} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} |

F19 | 1.4936 × 10^{−05} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 9.3157 × 10^{−06} | 1.1265 × 10^{−05} | 1.7344 × 10^{−06} | 2.3534 × 10^{−06} | 1.7344 × 10^{−06} |

F20 | 1.5658 × 10^{−02} | 1.7344 × 10^{−06} | 5.7924 × 10^{−05} | 1.7344 × 10^{−06} | 1.1748 × 10^{−02} | 3.0861 × 10^{−01} | 1.4795 × 10^{−02} | 7.3433 × 10^{−01} |

F21 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.3591 × 10^{−01} | 2.6033 × 10^{−06} | 6.6858 × 10^{−01} | 1.7344 × 10^{−06} | 6.0496 × 10^{−07} |

F22 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 3.1123 × 10^{−05} | 1.7988 × 10^{−05} | 9.3157 × 10^{−06} | 3.8202 × 10^{−01} | 2.1266 × 10^{−06} | 1.7300 × 10^{−06} |

F23 | 1.7344 × 10^{−06} | 1.7344 × 10^{−06} | 1.4795 × 10^{−02} | 1.9729 × 10^{−05} | 3.1123 × 10^{−05} | 6.4352 × 10^{−01} | 1.7344 × 10^{−06} | 3.1123 × 10^{−05} |

Function | Test | CSA | PSO | SSA | SCA | GWO |
---|---|---|---|---|---|---|

F1 | Best | 4.05207 × 10^{−66} | 1.40909 × 10^{−02} | 8.41001 × 10^{−02} | 7.65220 × 10^{−05} | 1.15671 × 10^{−16} |

Mean | 9.49793 × 10^{−37} | 3.23392 × 10^{01} | 1.98479 × 10^{01} | 4.22900 × 10^{−01} | 2.72480 × 10^{−14} | |

Std | 5.20184 × 10^{−36} | 4.83756 × 10^{01} | 2.80619 × 10^{01} | 9.12128 × 10^{−01} | 3.45607 × 10^{−14} | |

F2 | Best | 1.79982 × 10^{−33} | 1.51649 × 10^{−01} | 3.28652 × 10^{−01} | 9.66793 × 10^{−05} | 4.82345 × 10^{−10} |

Mean | 2.92106 × 10^{−22} | 3.89166 × 10^{00} | 4.75249 × 10^{00} | 3.31250 × 10^{−03} | 6.43809 × 10^{−09} | |

Std | 1.53426 × 10^{−21} | 4.88194 × 10^{00} | 3.71882 × 10^{00} | 3.35433 × 10^{−03} | 4.85863 × 10^{−09} | |

F3 | Best | 6.77086 × 10^{−66} | 2.43120 × 10^{01} | 2.58477 × 10^{02} | 3.09196 × 10^{−04} | 5.83050 × 10^{−01} |

Mean | 1.18525 × 10^{−40} | 8.05289 × 10^{04} | 2.33034 × 10^{04} | 4.24039 × 10^{00} | 1.82227 × 10^{02} | |

Std | 6.39505 × 10^{−40} | 3.84529 × 10^{04} | 3.16829 × 10^{04} | 6.62382 × 10^{00} | 2.88485 × 10^{02} | |

F4 | Best | 5.64112 × 10^{−36} | 1.97453 × 10^{−01} | 1.96642 × 10^{−01} | 1.97453 × 10^{−01} | 1.97453 × 10^{−01} |

Mean | 3.81723 × 10^{−22} | 2.62001 × 10^{00} | 9.88436 × 10^{−01} | 2.62001 × 10^{00} | 2.59439 × 10^{00} | |

Std | 1.83964 × 10^{−21} | 2.72100 × 10^{00} | 7.22522 × 10^{−01} | 2.72100 × 10^{00} | 2.71903 × 10^{00} | |

F5 | Best | 0.00000 × 10^{00} | 8.09021 × 10^{−01} | 1.15830 × 10^{00} | 1.80961 × 10^{00} | 1.80961 × 10^{00} |

Mean | 0.00000 × 10^{00} | 3.38351 × 10^{03} | 4.75416 × 10^{02} | 4.89317 × 10^{03} | 1.26269 × 10^{02} | |

Std | 0.00000 × 10^{00} | 1.14975 × 10^{04} | 9.22158 × 10^{02} | 1.39892 × 10^{04} | 1.65257 × 10^{02} | |

F6 | Best | 0.00000 × 10^{00} | 9.17634 × 10^{−05} | 3.55921 × 10^{−04} | 1.76154 × 10^{−03} | 8.62047 × 10^{−04} |

Mean | 0.00000 × 10^{00} | 3.30019 × 10^{01} | 1.32821 × 10^{01} | 2.93302 × 10^{01} | 7.17494 × 10^{00} | |

Std | 0.00000 × 10^{00} | 5.13931 × 10^{01} | 1.86246 × 10^{01} | 2.88694 × 10^{01} | 3.74740 × 10^{00} | |

F7 | Best | 4.23203 × 10^{−06} | 1.54553 × 10^{−02} | 4.83797 × 10^{−03} | 3.57837 × 10^{−04} | 3.66995 × 10^{−03} |

Mean | 4.00859 × 10^{−04} | 1.86225 × 10^{−01} | 1.41949 × 10^{−01} | 1.01286 × 10^{−01} | 8.42239 × 10^{−03} | |

Std | 4.78167 × 10^{−04} | 1.75616 × 10^{−01} | 1.17454 × 10^{−01} | 2.46964 × 10^{−01} | 4.68636 × 10^{−03} | |

F8 | Best | −4.18983 × 10^{04} | −4.18952 × 10^{04} | −4.18914 × 10^{04} | −4.18464 × 10^{04} | −4.18464 × 10^{04} |

Mean | −4.18983 × 10^{04} | −3.96372 × 10^{04} | −4.03440 × 10^{04} | −3.75470 × 10^{04} | −3.82785 × 10^{04} | |

Std | 2.96014 × 10^{−11} | 4.49162 × 10^{03} | 3.63037 × 10^{03} | 5.82693 × 10^{03} | 5.27419 × 10^{03} | |

F9 | Best | 0.00000 × 10^{00} | 3.27575 × 10^{−03} | 8.23333 × 10^{−03} | 3.73030 × 10^{−05} | 3.25244 × 10^{−06} |

Mean | 0.00000 × 10^{00} | 8.27521 × 10^{01} | 4.28287 × 10^{01} | 1.24652 × 10^{02} | 1.51656 × 10^{02} | |

Std | 0.00000 × 10^{00} | 7.32871 × 10^{01} | 4.87230 × 10^{01} | 1.19664 × 10^{02} | 1.11244 × 10^{02} | |

F10 | Best | 8.88178 × 10^{−16} | 7.84248 × 10^{−02} | 2.24259 × 10^{−02} | 8.35089 × 10^{−04} | 1.59905 × 10^{−09} |

Mean | 8.88178 × 10^{−16} | 2.88503 × 10^{00} | 1.76283 × 10^{00} | 1.31387 × 10^{00} | 9.57690 × 10^{−01} | |

Std | 0.00000 × 10^{00} | 2.62189 × 10^{00} | 1.12465 × 10^{00} | 2.18567 × 10^{00} | 2.49990 × 10^{00} | |

F11 | Best | 0.00000 × 10^{00} | 4.25764 × 10^{−03} | 5.71780 × 10^{−02} | 3.41024 × 10^{−06} | 0.00000 × 10^{00} |

Mean | 0.00000 × 10^{00} | 1.52847 × 10^{00} | 9.20741 × 10^{−01} | 2.27770 × 10^{−01} | 1.55201 × 10^{−03} | |

Std | 0.00000 × 10^{00} | 1.66071 × 10^{00} | 5.63135 × 10^{−01} | 2.79365 × 10^{−01} | 5.96126 × 10^{−03} | |

F12 | Best | 4.71163 × 10^{−33} | 1.97180 × 10^{−04} | 1.12509 × 10^{−03} | 5.24202 × 10^{−03} | 1.23122 × 10^{−04} |

Mean | 4.71163 × 10^{−33} | 1.01808 × 10^{00} | 4.60484 × 10^{−01} | 1.64094 × 10^{00} | 1.44908 × 10^{00} | |

Std | 1.39185 × 10^{−48} | 1.49538 × 10^{00} | 1.11625 × 10^{00} | 1.78266 × 10^{00} | 2.82336 × 10^{00} | |

F13 | Best | 1.34978 × 10^{−32} | 3.52700 × 10^{−05} | 2.58636 × 10^{−04} | 7.25952 × 10^{−04} | 4.19563 × 10^{−04} |

Mean | 1.34978 × 10^{−32} | 1.45863 × 10^{01} | 2.90656 × 10^{00} | 2.47516 × 10^{01} | 2.72461 × 10^{00} | |

Std | 5.56740 × 10^{−48} | 2.21891 × 10^{01} | 5.27444 × 10^{00} | 4.78364 × 10^{01} | 2.82642 × 10^{00} |

Function | Test | CSA | PSO | SSA | SCA | GWO |
---|---|---|---|---|---|---|

F1 | Best | 1.74114 × 10^{−68} | 7.86330 × 10^{−01} | 8.69625 × 10^{−01} | 9.85563 × 10^{−01} | 1.28824 × 10^{−03} |

Mean | 6.75805 × 10^{−43} | 2.57658 × 10^{04} | 1.87823 × 10^{03} | 3.13124 × 10^{04} | 6.86880 × 10^{01} | |

Std | 3.69123 × 10^{−42} | 5.49528 × 10^{04} | 4.45580 × 10^{03} | 6.67314 × 10^{04} | 1.47024 × 10^{02} | |

F2 | Best | 4.94485 × 10^{−35} | 1.70351 × 10^{00} | 1.86340 × 10^{00} | 2.11466 × 10^{00} | 7.00593 × 10^{−04} |

Mean | 2.44060 × 10^{−21} | 2.85661 × 10^{02} | 9.02285 × 10^{01} | 4.69947 × 10^{01} | 4.62960 × 10^{−02} | |

Std | 1.33304 × 10^{−20} | 2.42764 × 10^{02} | 7.96943 × 10^{01} | 2.46857 × 10^{01} | 3.27695 × 10^{−02} | |

F3 | Best | 7.28940 × 10^{−62} | 1.44335 × 10^{06} | 7.08473 × 10^{03} | 3.52592 × 10^{04} | 3.55999 × 10^{05} |

Mean | 4.45675 × 10^{−39} | 9.83821 × 10^{06} | 5.17443 × 10^{06} | 8.89991 × 10^{05} | 1.07487 × 10^{06} | |

Std | 2.13852 × 10^{−38} | 3.95888 × 10^{06} | 4.61038 × 10^{06} | 6.68057 × 10^{05} | 3.87543 × 10^{05} | |

F4 | Best | 9.39244 × 10^{−41} | 2.82903 × 10^{−02} | 2.82903 × 10^{−02} | 2.82903 × 10^{−02} | 2.82903 × 10^{−02} |

Mean | 4.97072 × 10^{−23} | 3.13296 × 10^{00} | 1.47575 × 10^{00} | 3.13296 × 10^{00} | 3.01621 × 10^{00} | |

Std | 2.35097 × 10^{−22} | 2.36133 × 10^{00} | 1.34765 × 10^{00} | 2.36133 × 10^{00} | 2.28900 × 10^{00} | |

F5 | Best | 0.00000 × 10^{00} | 1.11144 × 10^{02} | 1.10805 × 10^{02} | 1.13580 × 10^{02} | 1.13580 × 10^{02} |

Mean | 0.00000 × 10^{00} | 3.96596 × 10^{06} | 1.47830 × 10^{05} | 4.84064 × 10^{06} | 1.08327 × 10^{06} | |

Std | 0.00000 × 10^{00} | 1.44863 × 10^{07} | 7.61203 × 10^{05} | 1.75361 × 10^{07} | 4.23453 × 10^{06} | |

F6 | Best | 0.00000 × 10^{00} | 1.11552 × 10^{00} | 1.18127 × 10^{00} | 1.37640 × 10^{00} | 1.16202 × 10^{00} |

Mean | 0.00000 × 10^{00} | 9.06200 × 10^{03} | 1.26919 × 10^{03} | 1.13358 × 10^{04} | 1.43531 × 10^{02} | |

Std | 0.00000 × 10^{00} | 1.17632 × 10^{04} | 2.22354 × 10^{03} | 1.46871 × 10^{04} | 1.19839 × 10^{02} | |

F7 | Best | 1.34248 × 10^{−05} | 2.95314 × 10^{−02} | 3.36464 × 10^{−03} | 8.28122 × 10^{−03} | 5.86957 × 10^{−03} |

Mean | 2.40186 × 10^{−04} | 3.51020 × 10^{01} | 7.03426 × 10^{−01} | 3.98208 × 10^{01} | 8.03855 × 10^{00} | |

Std | 2.09618 × 10^{−04} | 1.65495 × 10^{02} | 1.69178 × 10^{00} | 1.89489 × 10^{02} | 2.83190 × 10^{01} | |

F8 | Best | −4.18983 × 10^{05} | −4.18978 × 10^{05} | −4.18978 × 10^{05} | −4.18977 × 10^{05} | −4.18977 × 10^{05} |

Mean | −4.18983 × 10^{05} | −3.87824 × 10^{05} | −3.88408 × 10^{05} | −3.84621 × 10^{05} | −3.84898 × 10^{05} | |

Std | 1.18405 × 10^{−10} | 4.42563 × 10^{04} | 4.34900 × 10^{04} | 4.69810 × 10^{04} | 4.68726 × 10^{04} | |

F9 | Best | 0.00000 × 10^{00} | 4.15359 × 10^{−01} | 4.65994 × 10^{−01} | 5.27444 × 10^{−01} | 5.27444 × 10^{−01} |

Mean | 0.00000 × 10^{00} | 1.34989 × 10^{03} | 4.44170 × 10^{02} | 1.47652 × 10^{03} | 1.47580 × 10^{03} | |

Std | 0.00000 × 10^{00} | 1.02977 × 10^{03} | 4.18444 × 10^{02} | 1.10934 × 10^{03} | 1.10745 × 10^{03} | |

F10 | Best | 8.88178 × 10^{−16} | 1.17690 × 10^{−01} | 1.21822 × 10^{−01} | 1.30832 × 10^{−01} | 9.76292 × 10^{−04} |

Mean | 8.88178 × 10^{−16} | 4.07750 × 10^{00} | 2.10393 × 10^{00} | 4.24648 × 10^{00} | 2.82129 × 10^{00} | |

Std | 0.00000 × 10^{00} | 2.78347 × 10^{00} | 1.56515 × 10^{00} | 2.84110 × 10^{00} | 3.32683 × 10^{00} | |

F11 | Best | 0.00000 × 10^{00} | 1.54060 × 10^{00} | 5.76886 × 10^{−01} | 1.65931 × 10^{00} | 4.45692 × 10^{−03} |

Mean | 0.00000 × 10^{00} | 1.40518 × 10^{02} | 1.45067 × 10^{01} | 1.74547 × 10^{02} | 7.56728 × 10^{−01} | |

Std | 0.00000 × 10^{00} | 1.93782 × 10^{02} | 3.49974 × 10^{01} | 2.45555 × 10^{02} | 6.70865 × 10^{−01} | |

F12 | Best | 4.71163 × 10^{−34} | 3.49707 × 10^{−05} | 2.95159 × 10^{−05} | 3.53593 × 10^{−05} | 3.22334 × 10^{−05} |

Mean | 4.71163 × 10^{−34} | 4.61458 × 10^{00} | 1.15674 × 10^{00} | 4.71509 × 10^{00} | 2.37655 × 10^{00} | |

Std | 8.69906 × 10^{−50} | 7.79023 × 10^{00} | 2.17639 × 10^{00} | 8.25036 × 10^{00} | 3.71613 × 10^{00} | |

F13 | Best | 1.34978 × 10^{−32} | 1.24017 × 10^{00} | 1.07560 × 10^{00} | 1.37886 × 10^{00} | 1.05795 × 10^{00} |

Mean | 1.34978 × 10^{−32} | 2.66755 × 10^{02} | 2.86286 × 10^{01} | 2.83343 × 10^{02} | 2.02468 × 10^{02} | |

Std | 5.56740 × 10^{−48} | 4.21627 × 10^{02} | 2.54245 × 10^{01} | 4.39341 × 10^{02} | 3.96828 × 10^{02} |

CSA | PSO | SSA | SCA | GWO | |
---|---|---|---|---|---|

F1 | 0.178432 | 0.190542 | 0.302927 | 0.20064 | 0.282661 |

F2 | 0.12315 | 0.159538 | 0.228658 | 0.195575 | 0.239618 |

F3 | 0.568033 | 0.611411 | 0.721536 | 0.730838 | 0.698834 |

F4 | 0.12295 | 0.188939 | 0.239374 | 0.196382 | 0.251754 |

F5 | 0.156947 | 0.170969 | 0.235264 | 0.209225 | 0.246743 |

F6 | 0.115594 | 0.169406 | 0.215389 | 0.193195 | 0.227015 |

F7 | 0.275001 | 0.305304 | 0.393652 | 0.364141 | 0.403618 |

F8 | 0.14028 | 0.223961 | 0.268633 | 0.246461 | 0.294057 |

F9 | 0.122207 | 0.160593 | 0.262984 | 0.209103 | 0.27208 |

F10 | 0.133938 | 0.175045 | 0.255505 | 0.214336 | 0.264772 |

F11 | 0.13927 | 0.173938 | 0.255895 | 0.230421 | 0.256377 |

F12 | 0.526553 | 0.582185 | 0.667134 | 0.6323 | 0.678474 |

F13 | 0.512411 | 0.559926 | 0.652665 | 0.616762 | 0.688092 |

Sum | 3.114766 | 3.671757 | 4.699616 | 4.239379 | 4.804095 |

Rank | (1) | (2) | (4) | (3) | (5) |

CSA | PSO | SSA | SCA | GWO | |
---|---|---|---|---|---|

F1 | 0.864606 | 0.920696 | 1.411428 | 1.526312 | 1.954178 |

F2 | 0.8211 | 0.927257 | 1.691217 | 1.690701 | 2.304578 |

F3 | 9.746041 | 9.914311 | 9.757791 | 9.416562 | 9.929068 |

F4 | 0.712832 | 0.857485 | 1.391238 | 1.644103 | 2.219137 |

F5 | 0.784051 | 0.858395 | 1.423493 | 1.550448 | 2.002662 |

F6 | 0.764518 | 0.883734 | 1.460421 | 1.610766 | 1.972805 |

F7 | 1.450708 | 1.453512 | 2.155236 | 2.387385 | 2.742038 |

F8 | 0.970483 | 1.17946 | 1.778078 | 2.079184 | 2.718625 |

F9 | 0.904283 | 1.135896 | 1.676209 | 1.895193 | 2.421654 |

F10 | 0.918339 | 1.132918 | 1.680022 | 1.861007 | 2.290501 |

F11 | 1.024792 | 1.171338 | 1.809926 | 2.044052 | 2.410413 |

F12 | 2.495781 | 2.802061 | 3.292436 | 3.421013 | 4.016006 |

F13 | 2.471004 | 2.700387 | 3.175498 | 3.381912 | 4.266181 |

Sum | 23.92854 | 25.93745 | 32.70299 | 34.50864 | 41.24785 |

CSA | PSO | SSA | SCA | GWO | |
---|---|---|---|---|---|

h | 0.205729639786 | 0.205729639786 | 0.205723211955 | 0.205811043402 | 0.205724311092 |

l | 3.470488665628 | 7.092414276557 | 7.092727008708 | 7.380674589109 | 7.092527090825 |

t | 9.036623910358 | 9.036623910358 | 9.036624222889 | 8.972002667140 | 9.036803373437 |

b | 0.205729639786 | 0.205729639786 | 0.205729638416 | 0.208874244972 | 0.205728938896 |

Minimum cost | 1.724852308597 | 2.218150861764 | 2.218172785211 | 2.273030395508 | 2.218180086668 |

Average cost | 1.724853828957 | 2.218150861764 | 2.244245629001 | 2.291353653772 | 2.218198907121 |

Std. | 0.000004807757 | 0.000000000000 | 0.052784172626 | 0.010496471904 | 0.000013807448 |

CSA | PSO | SSA | SCA | GWO | |
---|---|---|---|---|---|

T_{s} | 0.77816864138 | 0.7781686414 | 0.79357920102 | 0.78922547613 | 0.007781787 |

T_{h} | 0.38464916263 | 0.3846491626 | 0.39226677976 | 0.40639993523 | 0.003846530 |

R | 40.3196187241 | 40.3196187241 | 41.1180752663 | 40.6926147876 | 40.319922618 |

L | 200.000000000 | 200.000000000 | 189.175939539 | 196.033601579 | 200.000000000 |

Minimum cost | 5885.33277362 | 5885.33277362 | 5912.20652171 | 6004.52071673 | 5885.46666087 |

Average cost | 6011.55334154 | 6013.40373404 | 6191.42556961 | 6198.38074830 | 5974.52840595 |

Std | 175.417988776 | 179.129462647 | 307.601967961 | 116.552624682 | 79.439547307 |

CSA | PSO | SSA | SCA | GWO | |
---|---|---|---|---|---|

D | 0.0517190259 | 0.0516975399 | 0.0500000000 | 0.0500000000 | 0.0517410542 |

d | 0.3574390430 | 0.3569217527 | 0.3174254133 | 0.3155229746 | 0.3579696634 |

N | 11.2468029380 | 11.2770151263 | 14.0277750624 | 14.4243340035 | 11.2159545387 |

Min. weight | 0.0126652492 | 0.0126652341 | 0.0127190578 | 0.0129556368 | 0.0126652949 |

Avg. weight | 0.0126789335 | 0.0133988758 | 0.0127190585 | 0.0131845009 | 0.0126662267 |

Std | 0.0000327544 | 0.0015508356 | 0.0000000011 | 0.0001295549 | 0.0000011609 |

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## Share and Cite

**MDPI and ACS Style**

Qais, M.H.; Hasanien, H.M.; Turky, R.A.; Alghuwainem, S.; Tostado-Véliz, M.; Jurado, F.
Circle Search Algorithm: A Geometry-Based Metaheuristic Optimization Algorithm. *Mathematics* **2022**, *10*, 1626.
https://doi.org/10.3390/math10101626

**AMA Style**

Qais MH, Hasanien HM, Turky RA, Alghuwainem S, Tostado-Véliz M, Jurado F.
Circle Search Algorithm: A Geometry-Based Metaheuristic Optimization Algorithm. *Mathematics*. 2022; 10(10):1626.
https://doi.org/10.3390/math10101626

**Chicago/Turabian Style**

Qais, Mohammed H., Hany M. Hasanien, Rania A. Turky, Saad Alghuwainem, Marcos Tostado-Véliz, and Francisco Jurado.
2022. "Circle Search Algorithm: A Geometry-Based Metaheuristic Optimization Algorithm" *Mathematics* 10, no. 10: 1626.
https://doi.org/10.3390/math10101626