# Improved Chaotic Particle Swarm Optimization Algorithm with More Symmetric Distribution for Numerical Function Optimization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of Previous Work

#### Particle Swarm Optimizer

## 3. Chaotic Particle Swarm Optimization- Arctangent Acceleration Coefficient Algorithm

#### 3.1. Chaotic Particle Swarm Optimization (CPSO)

- (1)
- A random sequence Z between chaotic [0, 1] is generated by an iteration of an initial value between [0, 1] through the iteration of the logistic equation: ${a}_{0},{a}_{1},{a}_{2}$… Through linear mapping using Equation (6), the chaos is extended to the value range of the optimization variable $X\left[a,b\right]$ to achieve traversal of the range of values of the optimized variables.$$Z\to X:X=a+(b-a)\times \mathrm{cos}\left(Z\right)$$
- (2)
- Generate a chaotic random sequence Z between [0, 1] using the logistic equation, and then pass the carrier map in Equation (7), introducing chaos into $gbest$, a nearby area, to achieve local chaotic search:$$Z\to Y:X=gbest+R\times \mathrm{cos}\left(Z\right)$$

#### 3.2. Arc Tangent Acceleration Coefficients (AT)

#### 3.3. Cosine Map Inertia Weight

Algorithm 1. Pseudo-code of the Chaotic Particle Swarm Optimization—Arctangent Acceleration algorithm. |

1 Initialize the parameters (PS, D, ${V}_{\mathrm{min}}$, ${V}_{\mathrm{max}}$, ${M}_{\mathrm{max}}$, ${c}_{1}$, ${c}_{2}$, ${X}_{\mathrm{min}}$, ${X}_{\mathrm{max}}$)2 The particle swarm positions ${X}_{i}(i=1,2,\cdots ,PS)$ are initialized by chaos theory by Equation (6)3 Randomly generate N initial velocities within the maximum range4 PSO algorithm is used to search for individual extremum and global optimal solutions5 Local search:6 While Iter < M_{max} do7 Update the inertia weight ω using Equation (10)8 Using Equations (8) and (9) to update the cognitive component c_{1} and social component c_{2}.9 for i = 1:PS (population size) do10 The speed ${V}_{i}$ of the particle ${X}_{i}$ is updated using Equation (1)11 Use Equation (2) to update the position of particle ${X}_{i}$12 Calculate the fitness values ${f}_{i}$ of the new particle ${X}_{i}$13 if ${X}_{i}$ is superior to $pbes{t}_{i}$14 Set ${X}_{i}$ to be $pbes{t}_{i}$15 End if16 if ${X}_{i}$ is superior to $gbest$17 Set ${X}_{i}$ to be $gbest$18 End if19 Global search:20 Chaotic search K times near $gbest$21 Use the following formula $Z\to Y:X=gbest+R\times \mathrm{cos}\left(Z\right)$ (Equation (7))22 K chaotic search points near $pbes{t}_{i}$ are obtained23 for j=1:K do24 if $f(X)$ < $f(gbest)$ 25 Set $f(X)$ to be $gbest$26 End if27 End for28 Iter = Iter +129 End While |

## 4. Simulation Experiments, Settings and Strategies

## 5. Experimental Results and Discussion

**1**(bold), the CPSO-AT optimization was superior to the other algorithms and the effect was more obvious. In addition, the best solutions are shown in bold in Table 3, Table 4, Table 6 and Table 7.

#### 5.1. Comparison of CPSO-AT with Classical PSO, Basic PSO, and CPSO

#### 5.2. Comparison of Chaotic Particle Swarm Optimization-Arctangent Acceleration with Moth-Flame Optimization, Krill Herd and Biogeography-Based Optimization

## 6. Conclusions and Future Work

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Comparison of the convergence curves of four algorithms for different test functions (Dim = 30): (

**a**) ${f}_{1}$; (

**b**) ${f}_{5}$; (

**c**) ${f}_{6}$; (

**d**) ${f}_{7}$; (

**e**) ${f}_{8}$; (

**f**) ${f}_{10}$.

**Figure 6.**Comparison of the convergence curves of four algorithms for different test functions (Dim = 50): (

**a**) ${f}_{1}$; (

**b**) ${f}_{2}$; (

**c**) ${f}_{4}$; (

**d**) ${f}_{5}$; (

**e**) ${f}_{8}$; (

**f**) ${f}_{10}$.

**Figure 7.**Comparison of the convergence curves of four algorithms for different test functions (Dim = 30): (

**a**) ${f}_{1}$; (

**b**) ${f}_{4}$; (

**c**) ${f}_{6}$; (

**d**) ${f}_{7}$; (

**e**) ${f}_{8}$; (

**f**) ${f}_{10}$.

**Figure 8.**Comparison of the convergence curves of four algorithms for different test functions (Dim = 50): (

**a**) ${f}_{1}$; (

**b**) ${f}_{2}$; (

**c**) ${f}_{5}$; (

**d**) ${f}_{7}$; (

**e**) ${f}_{8}$; (

**f**) ${f}_{10}$.

**Table 1.**Parameter settings for Chaotic Particle Swarm Optimization- Arctangent Acceleration (CPSO-AT) and other Particle Swarm Optimization (PSO) variants.

Algorithms | Population Size | Dimension | Parameter Settings | Iteration |
---|---|---|---|---|

Basic PSO | 100 | 30,50 | ${c}_{1}={c}_{2}=2.0,\omega =1,{V}_{\mathrm{max}}=0.2\times Bound$ | 2000 |

Classical PSO | 100 | 30,50 | ${c}_{1}={c}_{2}=2.0,\omega =0.9~0.4,{V}_{\mathrm{max}}=0.2\times Bound$ | 2000 |

CPSO | 100 | 30,50 | ${c}_{1}={c}_{2}=2.0,\omega =0.9~0.4,{V}_{\mathrm{max}}=0.2\times Bound$ | 2000 |

CPSO-AT | 100 | 30,50 | $\begin{array}{c}{c}_{1}:2.5~0.5,{c}_{2}=0.5~2.5,\omega =0.9333~0.2667,\\ {V}_{\mathrm{max}}=0.2\times Bound\end{array}$ | 2000 |

Name | Test function | Dim | S | ${\mathit{f}}_{\mathbf{min}}$ | Group |
---|---|---|---|---|---|

sphere | ${f}_{1}\left(\overrightarrow{x}\right)={\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}$ | 30 50 | ${\left[-100,100\right]}^{n}$ | 0 | Unimodal |

Schwefel’s 1.2 | ${f}_{2}\left(\overrightarrow{x}\right)={\displaystyle \sum _{i=1}^{n}}{\left({\displaystyle \sum _{j=1}^{i}}{x}_{j}\right)}^{2}$ | 30 50 | ${\left[-100,100\right]}^{n}$ | 0 | Unimodal |

Rosenbrock | ${f}_{3}\left(\overrightarrow{x}\right)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]$ | 30 50 | ${\left[-30,30\right]}^{n}$ | 0 | Unimodal |

Dixon & Price | ${f}_{4}\left(\overrightarrow{x}\right)={\left({x}_{1}-1\right)}^{2}+{\displaystyle \sum _{i=2}^{n}}i{\left(2{x}_{i}^{2}-{x}_{i-1}\right)}^{2}$ | 30 50 | ${\left[-10,10\right]}^{n}$ | 0 | Unimodal |

Sum Squares | ${f}_{5}\left(\overrightarrow{x}\right)={\displaystyle \sum _{i=1}^{n}}i{x}_{i}^{2}$ | 30 50 | ${\left[-10,10\right]}^{n}$ | 0 | Unimodal |

Griewank | ${f}_{6}\left(\overrightarrow{x}\right)={\displaystyle \sum _{i=1}^{n}}\frac{{x}_{i}^{2}}{4000}-{\displaystyle \prod _{i=1}^{n}}\mathrm{cos}\left({x}_{i}/\sqrt{i}\right)-1$ | 30 50 | ${\left[-600,600\right]}^{n}$ | 0 | Multimodal |

Ackley | $\begin{array}{ll}{f}_{7}\left(\overrightarrow{x}\right)=& -20\text{}\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}}\right)\\ & -\mathrm{exp}\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)\\ & +20+\mathrm{e}\end{array}$ | 30 50 | ${\left[-32,32\right]}^{n}$ | 0 | Multimodal |

Rastrigin | ${f}_{8}\left(\overrightarrow{x}\right)={\displaystyle \sum _{i=1}^{n}}({x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10)$ | 30 50 | ${\left[-5.12,5.12\right]}^{n}$ | 0 | Multimodal |

Levy | $\begin{array}{ll}{f}_{9}\left(\overrightarrow{x}\right)=& {\mathrm{sin}}^{2}\pi {\omega}_{i}\\ & {\displaystyle \sum _{i=1}^{n-1}}{({\omega}_{i}-1)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {\omega}_{i}+1\right)\right]\\ & +{\left({\omega}_{d}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(2\pi {\omega}_{d}\right)\right]\end{array}$ $\mathrm{where}\text{}{\omega}_{i}=1+\frac{{x}_{i}-1}{4},\mathrm{for}\text{}\mathrm{all}\text{}\mathrm{i}=1,\cdots ,\mathrm{n}$ | 30 50 | [−10, 10]^{n} | 0 | Multimodal |

Zakharov | ${f}_{10}\left(\overrightarrow{x}\right)={\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}+{\left({\displaystyle \sum _{i=1}^{n}}0.5i{x}_{i}\right)}^{2}+{\left({\displaystyle \sum _{i=1}^{n}}0.5i{x}_{i}\right)}^{4}$ | 30 50 | [−5, 10]^{n} | 0 | Multimodal |

**Table 3.**Experimental results obtained by basic particle swarm optimization (PSO), classical PSO, CPSO, and Chaotic Particle Swarm Optimization-Arctangent Acceleration (CPSO-AT) with 30 dimensions.

Function | Algorithm | k | The best | The worst | Mean | S.D. |
---|---|---|---|---|---|---|

f_{1} | Basic PSO | 1 | 2.6248 × 10^{01} | 7.7947 × 10^{01} | 4.6323 × 10^{01} | 6.7444 × 10^{01} |

Classical PSO | 2.0757 × 10^{−02} | 6.6373 × 10^{−01} | 1.9719 × 10^{−01} | 7.8747 × 10^{−01} | ||

CPSO | 5.2121 × 10^{−03} | 2.1287 × 10^{−02} | 1.1753 × 10^{−02} | 8.2205 × 10^{−04} | ||

CPSO-AT | 6.3925 × 10^{−10} | 3.4988 × 10^{−08} | 4.7194 × 10^{−09} | 3.4958 × 10^{−08} | ||

f_{2} | Basic PSO | 1 | 2.2111 × 10^{03} | 9.9471 × 10^{03} | 4.9502 × 10^{03} | 1.0172 × 10^{04} |

Classical PSO | 1.1954 × 10^{01} | 1.6882 × 10^{02} | 5.3816 × 10^{01} | 1.6148 × 10^{02} | ||

CPSO | 1.1748 × 10^{00} | 5.6324 × 10^{00} | 3.0123 × 10^{00} | 3.0028 × 10^{−01} | ||

CPSO-AT | 2.4951 × 10^{−05} | 7.2287 × 10^{−03} | 1.2410 × 10^{−03} | 8.8665 × 10^{−03} | ||

f_{3} | Basic PSO | 1 | 2.6521 × 10^{05} | 1.6800 × 10^{06} | 9.3596 × 10^{05} | 1.0188 × 10^{05} |

Classical PSO | 3.6194 × 10^{02} | 1.9558 × 10^{06} | 1.1596 × 10^{06} | 1.4155 × 10^{05} | ||

CPSO | 2.7934 × 10^{01} | 1.4180 × 10^{02} | 5.5210 × 10^{01} | 2.7520 × 10^{01} | ||

CPSO-AT | 2.0086 × 10^{01} | 2.7560 × 10^{01} | 2.4461 × 10^{01} | 2.4490 × 10^{−01} | ||

f_{4} | Basic PSO | 1 | 2.6551 × 10^{03} | 1.1167 × 10^{05} | 2.3649 × 10^{04} | 6.2701 × 10^{03} |

Classical PSO | 7.7642 × 10^{00} | 7.8947 × 10^{04} | 3.3909 × 10^{04} | 6.8874 × 10^{03} | ||

CPSO | 1.2618 × 10^{00} | 5.2299 × 10^{00} | 2.9283 × 10^{00} | 5.3780 × 10^{−01} | ||

CPSO-AT | 6.6667 × 10^{−01} | 6.6939 × 10^{−01} | 6.6699 × 10^{−01} | 7.5939 × 10^{−04} | ||

f_{5} | Basic PSO | 1 | 1.0021 × 10^{03} | 2.0249 × 10^{03} | 1.5684 × 10^{03} | 6.4703 × 10^{01} |

Classical PSO | 1.7245 × 10^{01} | 2.2872 × 10^{03} | 1.3307 × 10^{03} | 1.8101 × 10^{02} | ||

CPSO | 1.2397 × 10^{−01} | 4.4059 × 10^{−01} | 2.4581 × 10^{−01} | 3.8646 × 10^{−02} | ||

CPSO-AT | 4.4910 × 10^{−06} | 4.4064 × 10^{−04} | 1.1348 × 10^{−04} | 2.6671 × 10^{−05} | ||

f_{6} | Basic PSO | 1 | 3.2239 × 10^{01} | 5.5316 × 10^{01} | 4.0062 × 10^{01} | 9.2496 × 10^{−01} |

Classical PSO | 1.0188 × 10^{00} | 5.1722 × 10^{01} | 2.7296 × 10^{01} | 2.6616 × 10^{00} | ||

CPSO | 8.6921 × 10^{−04} | 1.1118 × 10^{−02} | 3.1795 × 10^{−03} | 8.4473 × 10^{−04} | ||

CPSO-AT | 1.1528 × 10^{−08} | 5.5167 × 10^{−04} | 6.0028 × 10^{−05} | 2.0000 × 10^{−05} | ||

f_{7} | Basic PSO | 1 | 8.8528 × 10^{00} | 1.6029 × 10^{01} | 1.1915 × 10^{01} | 5.3159 × 10^{−01} |

Classical PSO | 7.4734 × 10^{00} | 1.2232 × 10^{01} | 1.1736 × 10^{01} | 2.1728 × 10^{−01} | ||

CPSO | 9.3623 × 10^{−02} | 2.0819 × 10^{00} | 5.1825 × 10^{−01} | 2.2687 × 10^{−01} | ||

CPSO-AT | 5.2116 × 10^{−05} | 1.8759 × 10^{−04} | 1.0254 × 10^{−04} | 1.6110 × 10^{−05} | ||

f_{8} | Basic PSO | 0 | 2.4279 × 10^{02} | 3.2104 × 10^{02} | 2.6438 × 10^{02} | 3.4627 × 10^{00} |

Classical PSO | 6.8569 × 10^{01} | 3.1272 × 10^{02} | 2.9253 × 10^{02} | 4.5919 × 10^{00} | ||

CPSO | 2.9624 × 10^{01} | 4.2594 × 10^{01} | 3.6522 × 10^{01} | 1.9274 × 10^{00} | ||

CPSO-AT | 9.2920 × 10^{01} | 1.7194 × 10^{02} | 1.3559 × 10^{02} | 5.0030 × 10^{00} | ||

f_{9} | Basic PSO | 1 | 1.0570 × 10^{01} | 5.6268 × 10^{01} | 2.9851 × 10^{01} | 2.6466 × 10^{00} |

Classical PSO | 5.2905 × 10^{00} | 4.9856 × 10^{01} | 2.1421 × 10^{01} | 2.8645 × 10^{00} | ||

CPSO | 9.4416 × 10^{−03} | 2.7481 × 10^{00} | 8.9063 × 10^{−01} | 1.4012 × 10^{−01} | ||

CPSO-AT | 3.5811 × 10^{−01} | 8.0576 × 10^{−01} | 5.5508 × 10^{−01} | 4.5298 × 10^{−01} | ||

f_{10} | Basic PSO | 1 | 8.5199 × 10^{01} | 2.4830 × 10^{02} | 1.4751 × 10^{02} | 2.0541 × 10^{01} |

Classical PSO | 5.3003 × 10^{−01} | 1.1802 × 10^{02} | 8.4578 × 10^{01} | 6.7666 × 10^{00} | ||

CPSO | 5.7915 × 10^{−01} | 1.1559 × 10^{00} | 8.3102 × 10^{−01} | 5.2794 × 10^{−02} | ||

CPSO-AT | 1.3389 × 10^{−04} | 3.6616 × 10^{−03} | 1.2061 × 10^{−03} | 2.1842 × 10^{−04} |

**Table 4.**Experimental results obtained by basic particle swarm optimization (PSO), classical PSO, CPSO, and Chaotic Particle Swarm Optimization- Arctangent Acceleration (CPSO-AT) with 50 dimensions.

Function | Algorithm | k | The best | The worst | Mean | S.D. |
---|---|---|---|---|---|---|

f_{1} | Basic PSO | 1 | 6.2992 × 10^{01} | 1.4072 × 10^{02} | 9.6568 × 10^{01} | 1.1705 × 10^{02} |

Classical PSO | 1.9810 × 10^{00} | 9.4058 × 10^{00} | 4.7912 × 10^{00} | 1.0715 × 10^{01} | ||

CPSO | 7.9989 × 10^{−02} | 1.9665 × 10^{−01} | 1.3500 × 10^{−01} | 4.0335 × 10^{−03} | ||

CPSO-AT | 4.3797 × 10^{−07} | 1.3892 × 10^{−05} | 2.6618 × 10^{−06} | 1.3012 × 10^{−05} | ||

f_{2} | Basic PSO | 1 | 1.7735 × 10^{04} | 4.4191 × 10^{04} | 3.3163 × 10^{04} | 4.0007 × 10^{04} |

Classical PSO | 7.0766 × 10^{02} | 2.4271 × 10^{03} | 1.3298 × 10^{03} | 2.2228 × 10^{03} | ||

CPSO | 3.4384 × 10^{01} | 1.9495 × 10^{02} | 7.9785 × 10^{01} | 5.0406 × 10^{00} | ||

CPSO-AT | 1.3885 × 10^{−01} | 2.7291 × 10^{00} | 7.4383 × 10^{−01} | 3.4377 × 10^{00} | ||

f_{3} | Basic PSO | 1 | 2.9077 × 10^{06} | 1.4292 × 10^{07} | 7.9705 × 10^{06} | 6.3951 × 10^{05} |

Classical PSO | 6.9551 × 10^{03} | 1.6232 × 10^{07} | 6.8072 × 10^{06} | 1.1339 × 10^{06} | ||

CPSO | 1.0277 × 10^{02} | 3.5589 × 10^{02} | 2.3726 × 10^{02} | 4.4241 × 10^{01} | ||

CPSO-AT | 4.5713 × 10^{01} | 4.8066 × 10^{01} | 4.6730 × 10^{01} | 1.3927 × 10^{−01} | ||

f_{4} | Basic PSO | 1 | 8.5353 × 10^{04} | 6.5400 × 10^{05} | 2.3039 × 10^{05} | 4.4620 × 10^{04} |

Classical PSO | 2.1823 × 10^{02} | 2.5940 × 10^{05} | 1.0611 × 10^{05} | 1.7552 × 10^{04} | ||

CPSO | 8.5470 × 10^{00} | 3.0465 × 10^{01} | 2.0967 × 10^{01} | 3.9523 × 10^{00} | ||

CPSO-AT | 6.6783 × 10^{−01} | 2.0588 × 10^{00} | 8.4647 × 10^{−01} | 5.6966 × 10^{−02} | ||

f_{5} | Basic PSO | 1 | 2.9846 × 10^{03} | 7.5971 × 10^{03} | 5.4303 × 10^{03} | 3.7474 × 10^{02} |

Classical PSO | 6.5151 × 10^{01} | 7.6068 × 10^{03} | 4.5427 × 10^{03} | 6.5260 × 10^{02} | ||

CPSO | 3.1804 × 10^{00} | 5.6937 × 10^{00} | 4.2359 × 10^{00} | 1.9161 × 10^{−01} | ||

CPSO-AT | 4.7685 × 10^{−03} | 6.7831 × 10^{−02} | 2.0803 × 10^{−02} | 5.0860 × 10^{−03} | ||

f_{6} | Basic PSO | 1 | 4.3759 × 10^{01} | 1.0136 × 10^{02} | 7.8097 × 10^{01} | 4.6646 × 10^{00} |

Classical PSO | 6.1000 × 10^{00} | 1.0849 × 10^{02} | 8.7209 × 10^{01} | 4.9779 × 10^{00} | ||

CPSO | 6.9568 × 10^{−03} | 1.4624 × 10^{−02} | 9.6156 × 10^{−03} | 7.3385 × 10^{−04} | ||

CPSO-AT | 4.4032 × 10^{−06} | 5.4671 × 10^{−04} | 8.9087 × 10^{−05} | 2.3055 × 10^{−05} | ||

f_{7} | Basic PSO | 1 | 1.3003 × 10^{01} | 1.5286 × 10^{01} | 1.3916 × 10^{01} | 1.7718 × 10^{−01} |

Classical PSO | 8.0444 × 10^{00} | 1.5888 × 10^{01} | 1.4502 × 10^{01} | 4.5279 × 10^{−01} | ||

CPSO | 4.6800 × 10^{−01} | 1.9761 × 10^{00} | 1.5023 × 10^{00} | 4.2769 × 10^{−02} | ||

CPSO-AT | 2.0332 × 10^{−03} | 8.7914 × 10^{−01} | 9.1370 × 10^{−02} | 2.9368 × 10^{−02} | ||

f_{8} | Basic PSO | 0 | 5.1852 × 10^{02} | 6.0608 × 10^{02} | 5.7156 × 10^{02} | 1.6262 × 10^{00} |

Classical PSO | 1.1434 × 10^{02} | 5.7794 × 10^{02} | 4.5988 × 10^{02} | 8.5234 × 10^{00} | ||

CPSO | 7.4647 × 10^{01} | 1.4715 × 10^{02} | 1.1398 × 10^{02} | 7.3222 × 10^{00} | ||

CPSO-AT | 2.0437 × 10^{02} | 3.2078 × 10^{02} | 2.6771 × 10^{02} | 8.6923 × 10^{00} | ||

f_{9} | Basic PSO | 1 | 3.2829 × 10^{01} | 7.5323 × 10^{01} | 5.1270 × 10^{01} | 3.5370 × 10^{00} |

Classical PSO | 3.8076 × 10^{00} | 4.8866 × 10^{01} | 3.6627 × 10^{01} | 2.9257 × 10^{00} | ||

CPSO | 1.3024 × 10^{00} | 5.1136 × 10^{00} | 2.8866 × 10^{00} | 4.1186 × 10^{−01} | ||

CPSO-AT | 6.2676 × 10^{−01} | 1.4325 × 10^{00} | 1.0566 × 10^{00} | 8.0876 × 10^{−01} | ||

f_{10} | Basic PSO | 1 | 4.3138 × 10^{03} | 9.8686 × 10^{06} | 1.2329 × 10^{06} | 4.0793 × 10^{05} |

Classical PSO | 7.0378 × 10^{01} | 2.9135 × 10^{05} | 6.9641 × 10^{04} | 2.1887 × 10^{04} | ||

CPSO | 6.6807 × 10^{00} | 1.3292 × 10^{01} | 1.0134 × 10^{01} | 8.8523 × 10^{−01} | ||

CPSO-AT | 2.2299 × 10^{−01} | 1.3807 × 10^{00} | 6.4995 × 10^{−01} | 3.0979 × 10^{−02} |

Algorithm | Population | Maximum Iterations | Dim | Other |
---|---|---|---|---|

MFO | 50 | 2000 | 30,50 | $t$ is random number in the range [−2,1] |

KH | 50 | 2000 | 30,50 | ${N}^{max}=0.01,{V}_{f}=0.02,{D}^{max}=0.005$ |

BBO | 50 | 2000 | 30,50 | $\mathrm{Mu}=0.005,\mathsf{\mu}=0.8$c |

CPSO-AT | 50 | 2000 | 30,50 | $\omega =\phi \times \mathrm{cos}\left(\left(\frac{{M}_{j}}{{M}_{max}}\right)\times \mathsf{\pi}\right)+\tau $ |

${c}_{1}=\partial \times \mathrm{arctan}\left(\left(\frac{{M}_{J}}{{M}_{max}}\right)\times \sigma \right)+{\delta}_{1}$ | ||||

${c}_{2}=-\partial \times \mathrm{arctan}\left(\left(\frac{{M}_{j}}{{M}_{max}}\right)\times \sigma \right)+{\delta}_{2}$ |

**Table 6.**Experimental results produced by the Moth-Flame Optimization Algorithm (MFO), Krill Herd Algorithm (KH) and Biogeography-Based Optimization Algorithm (BBO) and Chaotic Particle Swarm Optimization-Arctangent Acceleration (CPSO-AT) with 30 dimensions.

Function | Algorithm | k | The Best | The Worst | Mean | S.D. |
---|---|---|---|---|---|---|

f_{1} | MFO | 1 | 6.3466 × 10^{−15} | 1.0000 × 10^{04} | 2.0000 × 10^{03} | 1.2649 × 10^{04} |

KH | 9.6793 × 10^{−03} | 4.4021 × 10^{−02} | 2.0048 × 10^{−02} | 1.3669 × 10^{−02} | ||

BBO | 2.7007 × 10^{00} | 9.8717 × 10^{00} | 4.2946 × 10^{00} | 6.1697 × 10^{00} | ||

CPSO-AT | 3.1695 × 10^{−07} | 2.8893 × 10^{−06} | 8.6404 × 10^{−07} | 1.1386 × 10^{−07} | ||

f_{2} | MFO | 1 | 3.7805 × 10^{−13} | 1.1700 × 10^{06} | 4.7500 × 10^{05} | 1.4342 × 10^{06} |

KH | 7.4251 × 10^{00} | 7.7804 × 10^{02} | 3.1717 × 10^{02} | 3.4905 × 10^{02} | ||

BBO | 2.4456 × 10^{02} | 1.0627 × 10^{03} | 6.9743 × 10^{02} | 7.0483 × 10^{02} | ||

CPSO-AT | 1.2328 × 10^{−03} | 1.4670 × 10^{−01} | 3.7457 × 10^{−02} | 9.9016 × 10^{−03} | ||

f_{3} | MFO | 1 | 1.6385 × 10^{03} | 1.6385 × 10^{03} | 1.6385 × 10^{03} | 1.7316 × 10^{−12} |

KH | 2.9288 × 10^{01} | 1.2114 × 10^{02} | 5.5490 × 10^{01} | 4.0249 × 10^{01} | ||

BBO | 1.2701 × 10^{02} | 6.5835 × 10^{02} | 3.2701 × 10^{02} | 5.1255 × 10^{02} | ||

CPSO-AT | 2.3345 × 10^{01} | 2.9404 × 10^{01} | 2.4992 × 10^{01} | 9.1291 × 10^{−02} | ||

f_{4} | MFO | 1 | 6.6667 × 10^{−01} | 2.7370 × 10^{05} | 3.4679 × 10^{04} | 2.6107 × 10^{05} |

KH | 6.7509 × 10^{−01} | 1.2386 × 10^{00} | 8.4426 × 10^{−01} | 2.3154 × 10^{−01} | ||

BBO | 4.4060 × 10^{00} | 9.1277 × 10^{00} | 6.9919 × 10^{00} | 4.7954 × 10^{00} | ||

CPSO-AT | 6.6664 × 10^{−01} | 6.6762 × 10^{−01} | 6.6692 × 10^{−01} | 2.1520 × 10^{−04} | ||

f_{5} | MFO | 1 | 1.1144 × 10^{−14} | 2.9000 × 10^{03} | 6.9000 × 10^{02} | 2.6738 × 10^{03} |

KH | 8.0391 × 10^{−03} | 2.6790 × 10^{−01} | 7.1299 × 10^{−02} | 1.1182 × 10^{−01} | ||

BBO | 3.3696 × 10^{−01} | 6.0732 × 10^{−01} | 4.6529 × 10^{−01} | 2.5268 × 10^{−01} | ||

CPSO-AT | 1.9718 × 10^{−05} | 1.8382 × 10^{−04} | 8.3644 × 10^{−05} | 2.0107 × 10^{−05} | ||

f_{6} | MFO | 1 | 9.3259 × 10^{−15} | 9.0535 × 10^{01} | 2.7080 × 10^{01} | 1.3078 × 10^{02} |

KH | 2.0406 × 10^{−03} | 2.3335 × 10^{−02} | 1.0695 × 10^{−02} | 8.9323 × 10^{−03} | ||

BBO | 1.0110 × 10^{00} | 1.0698 × 10^{00} | 1.0388 × 10^{00} | 4.8482 × 10^{−02} | ||

CPSO-AT | 8.4554 × 10^{−08} | 1.5809 × 10^{−06} | 3.6554 × 10^{−07} | 5.6011 × 10^{−08} | ||

f_{7} | MFO | 1 | 7.0957 × 10^{−09} | 1.9963 × 10^{01} | 9.4662 × 10^{00} | 2.9308 × 10^{01} |

KH | 2.0726 × 10^{−03} | 1.6499 × 10^{00} | 7.0606 × 10^{−01} | 7.0425 × 10^{−01} | ||

BBO | 7.0705 × 10^{−01} | 1.3181 × 10^{00} | 9.6088 × 10^{−01} | 4.6179 × 10^{−01} | ||

CPSO-AT | 2.9201 × 10^{−04} | 1.2453 × 10^{−03} | 5.9240 × 10^{−04} | 2.0650 × 10^{−04} | ||

f_{8} | MFO | 0 | 7.6612 × 10^{01} | 1.9331 × 10^{02} | 1.3427 × 10^{02} | 9.8256 × 10^{01} |

KH | 5.9930 × 10^{00} | 1.7947 × 10^{01} | 1.2761 × 10^{01} | 4.3635 × 10^{00} | ||

BBO | 1.0071 × 10^{00} | 2.2005 × 10^{00} | 1.7721 × 10^{00} | 1.2180 × 10^{00} | ||

CPSO-AT | 9.9508 × 10^{00} | 3.7812 × 10^{01} | 2.0800 × 10^{01} | 2.5423 × 10^{00} | ||

f_{9} | MFO | 0 | 2.5135 × 10^{01} | 3.7618 × 10^{01} | 3.0205 × 10^{01} | 9.4966 × 10^{00} |

KH | 2.2932 × 10^{−04} | 8.1506 × 10^{−01} | 2.0720 × 10^{−01} | 2.2620 × 10^{−01} | ||

BBO | 8.6060 × 10^{−03} | 2.8077 × 10^{−02} | 2.1078 × 10^{−02} | 2.0072 × 10^{−02} | ||

CPSO-AT | 2.6859 × 10^{−01} | 8.0576 × 10^{−01} | 5.3717 × 10^{−01} | 6.5790 × 10^{−01} | ||

f_{10} | MFO | 1 | 1.1036 × 10^{−02} | 4.7619 × 10^{02} | 2.6749 × 10^{02} | 4.3400 × 10^{02} |

KH | 6.6967 × 10^{01} | 1.3059 × 10^{02} | 9.7460 × 10^{01} | 2.5847 × 10^{01} | ||

BBO | 2.3055 × 10^{01} | 4.4664 × 10^{01} | 3.3702 × 10^{01} | 2.0264 × 10^{01} | ||

CPSO-AT | 9.4240 × 10^{−05} | 2.1734 × 10^{−04} | 1.7080 × 10^{−04} | 4.8481 × 10^{−06} |

**Table 7.**Experimental results of the Moth-Flame Optimization Algorithm (MFO), Krill Herd Algorithm (KH) and Biogeography-Based Optimization Algorithm (BBO), and Chaotic Particle Swarm Optimization-Arctangent Acceleration (CPSO-AT) with 50 dimensions.

Function | Algorithm | k | The Best | The Worst | Mean | S.D. |
---|---|---|---|---|---|---|

f_{1} | MFO | 1 | 9.0418 × 10^{−05} | 2.0000 × 10^{04} | 1.0000 × 10^{04} | 2.0000 × 10^{04} |

KH | 9.1848 × 10^{−02} | 3.8079 × 10^{−01} | 2.2432 × 10^{−01} | 1.0846 × 10^{−01} | ||

BBO | 6.8687 × 10^{01} | 1.4266 × 10^{02} | 9.9904 × 10^{01} | 6.6770 × 10^{01} | ||

CPSO-AT | 1.7020 × 10^{−05} | 5.5462 × 10^{−05} | 4.3225 × 10^{−05} | 8.3417 × 10^{−07} | ||

f_{2} | MFO | 1 | 1.0000 × 10^{04} | 5.0200 × 10^{06} | 1.8190 × 10^{06} | 4.6469 × 10^{06} |

KH | 1.0638 × 10^{03} | 8.3849 × 10^{03} | 3.1824 × 10^{03} | 3.0333 × 10^{03} | ||

BBO | 1.6172 × 10^{04} | 5.0529 × 10^{04} | 2.7620 × 10^{04} | 2.7312 × 10^{04} | ||

CPSO-AT | 1.0216 × 10^{00} | 4.4131 × 10^{00} | 2.3541 × 10^{00} | 2.6353 × 10^{−01} | ||

f_{3} | MFO | 1 | 2.7684 × 10^{03} | 2.7686 × 10^{03} | 2.7684 × 10^{03} | 1.3105 × 10^{−01} |

KH | 6.0262 × 10^{01} | 1.8692 × 10^{02} | 1.3323 × 10^{02} | 4.8077 × 10^{01} | ||

BBO | 1.4617 × 10^{03} | 4.8529 × 10^{03} | 2.5030 × 10^{03} | 2.7857 × 10^{03} | ||

CPSO-AT | 4.5596 × 10^{01} | 1.0267 × 10^{02} | 5.2237 × 10^{01} | 2.1782 × 10^{00} | ||

f_{4} | MFO | 1 | 6.2704 × 10^{00} | 4.6889 × 10^{05} | 1.8365 × 10^{05} | 4.7211 × 10^{05} |

KH | 2.3253 × 10^{00} | 5.6925 × 10^{00} | 3.2332 × 10^{00} | 1.4030 × 10^{00} | ||

BBO | 4.7422 × 10^{01} | 1.0958 × 10^{02} | 7.4611 × 10^{01} | 4.8684 × 10^{01} | ||

CPSO-AT | 6.6895 × 10^{−01} | 8.9973 × 10^{−01} | 7.4677 × 10^{−01} | 4.9214 × 10^{−02} | ||

f_{5} | MFO | 1 | 1.8984 × 10^{−05} | 9.7000 × 10^{03} | 2.2300 × 10^{03} | 8.7864 × 10^{03} |

KH | 5.1658 × 10^{−02} | 4.7215 × 10^{−01} | 2.2164 × 10^{−01} | 1.6813 × 10^{−01} | ||

BBO | 2.1286 × 10^{01} | 3.1796 × 10^{01} | 2.6284 × 10^{01} | 9.5861 × 10^{00} | ||

CPSO-AT | 3.1182 × 10^{−03} | 3.4762 × 10^{−02} | 1.5271 × 10^{−02} | 2.2791 × 10^{−03} | ||

f_{6} | MFO | 1 | 2.6725 × 10^{−05} | 9.0793 × 10^{01} | 2.7162 × 10^{01} | 1.3102 × 10^{02} |

KH | 8.7800 × 10^{−03} | 5.1455 × 10^{−02} | 2.3202 × 10^{−02} | 1.7232 × 10^{−02} | ||

BBO | 1.8720 × 10^{00} | 2.5471 × 10^{00} | 2.2098 × 10^{00} | 5.5828 × 10^{−01} | ||

CPSO-AT | 2.7021 × 10^{−06} | 7.4758 × 10^{−03} | 7.5390 × 10^{−04} | 2.4893 × 10^{−04} | ||

f_{7} | MFO | 1 | 3.1461 × 10^{00} | 1.9963 × 10^{01} | 1.7381 × 10^{01} | 1.5640 × 10^{01} |

KH | 1.1614 × 10^{00} | 2.9876 × 10^{00} | 1.8086 × 10^{00} | 6.9580 × 10^{−01} | ||

BBO | 2.9456 × 10^{00} | 3.5964 × 10^{00} | 3.2432 × 10^{00} | 5.4301 × 10^{−01} | ||

CPSO-AT | 2.4651 × 10^{−03} | 5.1956 × 10^{−03} | 3.7252 × 10^{−03} | 4.6693 × 10^{−04} | ||

f_{8} | MFO | 0 | 2.0610 × 10^{02} | 4.0942 × 10^{02} | 2.7372 × 10^{02} | 1.7473 × 10^{02} |

KH | 1.6014 × 10^{01} | 3.4316 × 10^{01} | 2.4666 × 10^{01} | 6.5266 × 10^{00} | ||

BBO | 1.6638 × 10^{01} | 2.8529 × 10^{01} | 2.4282 × 10^{01} | 9.1892 × 10^{00} | ||

CPSO-AT | 2.4890 × 10^{01} | 4.4884 × 10^{01} | 3.4785 × 10^{01} | 9.8221 × 10^{−01} | ||

f_{9} | MFO | 0 | 1.2546 × 10^{01} | 8.8610 × 10^{01} | 3.2530 × 10^{01} | 6.3687 × 10^{01} |

KH | 4.4904 × 10^{−01} | 2.7557 × 10^{00} | 1.0771 × 10^{00} | 8.7791 × 10^{−01} | ||

BBO | 3.0271 × 10^{−01} | 6.0636 × 10^{−01} | 4.4337 × 10^{−01} | 2.8994 × 10^{−01} | ||

CPSO-AT | 5.3723 × 10^{−01} | 1.3432 × 10^{00} | 9.6706 × 10^{−01} | 7.5773 × 10^{−01} | ||

f_{10} | MFO | 1 | 6.2671 × 10^{02} | 9.7623 × 10^{02} | 7.9242 × 10^{02} | 4.5047 × 10^{02} |

KH | 3.0410 × 10^{02} | 3.9719 × 10^{02} | 3.4173 × 10^{02} | 3.5971 × 10^{01} | ||

BBO | 7.6162 × 10^{01} | 1.3712 × 10^{02} | 1.0980 × 10^{02} | 4.7729 × 10^{01} | ||

CPSO-AT | 9.2969 × 10^{−03} | 6.3677 × 10^{−02} | 2.5705 × 10^{−02} | 1.2072 × 10^{−02} |

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**MDPI and ACS Style**

Ma, Z.; Yuan, X.; Han, S.; Sun, D.; Ma, Y.
Improved Chaotic Particle Swarm Optimization Algorithm with More Symmetric Distribution for Numerical Function Optimization. *Symmetry* **2019**, *11*, 876.
https://doi.org/10.3390/sym11070876

**AMA Style**

Ma Z, Yuan X, Han S, Sun D, Ma Y.
Improved Chaotic Particle Swarm Optimization Algorithm with More Symmetric Distribution for Numerical Function Optimization. *Symmetry*. 2019; 11(7):876.
https://doi.org/10.3390/sym11070876

**Chicago/Turabian Style**

Ma, Zhiteng, Xianfeng Yuan, Sen Han, Deyu Sun, and Yan Ma.
2019. "Improved Chaotic Particle Swarm Optimization Algorithm with More Symmetric Distribution for Numerical Function Optimization" *Symmetry* 11, no. 7: 876.
https://doi.org/10.3390/sym11070876