# An Improved Whale Optimization Algorithm Based on Different Searching Paths and Perceptual Disturbance

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## Abstract

**:**

## 1. Introduction

## 2. Whale Optimization Algorithm

#### 2.1. Inspiration

#### 2.2. Search for Prey (Exploration Phase)

#### 2.3. Encircling Prey

#### 2.4. Bubble-Net Attacking Method (Exploitation Phase)

- 1.
- Shrinking encircling mechanism

- 2.
- Spiral updating position method

#### 2.5. Idea of Improving Whale Optimization Algorithm

## 3. Complex Path-Perceptual Disturbance WOA

#### 3.1. Selection of Mathematical Model of Searching Path

#### 3.1.1. Logarithmic Spiral Curve (Lo)

#### 3.1.2. Archimedes Spiral Curve (Ar)

#### 3.1.3. Rose Spiral Curve (Ro)

#### 3.1.4. Epitrochoid-I (Ep-I)

#### 3.1.5. Hypotrochoid (Hy)

#### 3.1.6. Epitrochoid-II (Ep-II)

#### 3.1.7. Fermat Spiral Curve (Fe)

#### 3.1.8. Lituus Spiral Curve (Li)

#### 3.2. Introduction of Disturbance Factor

#### 3.3. Improved WOA with Perceptual Disturbances and Complex Paths

## 4. Simulation and Results Analysis

#### 4.1. Selection of Testing Functions

#### 4.2. Simulation Results and Analysis

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Function | Function Expression | Range | F_{min} |
---|---|---|---|

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

F2 | ${f}_{2}(x)={\displaystyle \sum _{i=1}^{d}\left|{x}_{i}\right|}+{\displaystyle {\prod}_{i=1}^{n}\left|{x}_{i}\right|}$ | [−10,10] | 0 |

F3 | ${f}_{3}(x)={{\displaystyle \sum _{i=1}^{d}\left({\displaystyle {\sum}_{j-1}^{i}{x}_{j}}\right)}}^{2}$ | [−100,100] | 0 |

F4 | ${f}_{4}\left(x\right)={\mathrm{max}}_{i}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | [−100,100] | 0 |

F5 | ${f}_{5}(x)={\displaystyle \sum _{i=1}^{d-1}(100({x}_{i+1}-{x}_{i}{}^{2})+{({x}_{i}-1)}^{2})}$ | [−30,30] | 0 |

F6 | ${f}_{6}\left(x\right)={\displaystyle {\sum}_{i=1}^{d}{\left([{x}_{i}+0.5]\right)}^{2}}$ | [−100,100] | 0 |

F7 | ${f}_{7}\left(x\right)={\displaystyle {\sum}_{i=1}^{d}i{x}_{i}^{4}+random[0,1)}$ | [−1.28,1.28] | 0 |

F8 | ${f}_{8}\left(x\right)={\displaystyle {\sum}_{i=1}^{n}-{x}_{i}}\mathrm{sin}\left(\sqrt{{x}_{i}}\right)$ | [−500,500] | −418.9 $\times $ 5 |

F9 | ${f}_{9}(x)={\displaystyle \sum _{i=1}^{d}({x}_{i}{}^{2}-10cos(2\pi {x}_{i})+10)}$ | [−5.12,5.12] | 0 |

F10 | $\begin{array}{l}{f}_{10}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}}\right)-\mathrm{exp}\left(\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)}\right)\\ +20+e\end{array}$ | [−32,32] | 0 |

F11 | ${f}_{11}(x)=\frac{1}{4000}\left({\displaystyle \sum _{i=1}^{n}({x}^{2}i)}\right)-\left({\displaystyle \prod _{i=1}^{n}\mathrm{cos}(\frac{xi}{\sqrt{i}})}\right)+1$ | [−600,600] | 0 |

F12 | $\begin{array}{l}{f}_{12}\left(x\right)=\frac{\pi}{n}\left\{10\mathrm{sin}\left(\pi {y}_{1}\right)+{\displaystyle {\sum}_{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}(\pi {y}_{i+1}\right]+{({y}_{n}+1)}^{2}}\right\}\\ +{\displaystyle {\sum}_{i=1}^{n}u({x}_{i}},10,100,4)\end{array}$${y}_{i}=1+\frac{{x}_{i}+1}{4};u({x}_{i},a,k,m)=\{\begin{array}{c}k{({x}_{i}-a)}^{m}\\ 0\\ k{(-{x}_{i}-a)}^{m}\end{array}\begin{array}{c}{x}_{i}>a\\ -a<{x}_{i}<a\\ {x}_{i}<-a\end{array}$ | [−50,50] | 0 |

F13 | $\begin{array}{l}{f}_{13}\left(x\right)=0.1\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle {\sum}_{i=1}^{n}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}(3\pi {x}_{i+1})\right]}\\ +{({x}_{n}-1)}^{2}\left[1+{\mathrm{sin}}^{2}(3\pi {x}_{i+1})\right]\}+{\displaystyle {\sum}_{i=1}^{n}u({x}_{i}},5,100,4)\end{array}$ | [−50,50] | 0 |

F14 | ${f}_{14}={(\frac{1}{500}+{\displaystyle {\sum}_{j=1}^{25}\frac{1}{j+{\displaystyle {\sum}_{i=1}^{2}{({x}_{i}-{a}_{ij})}^{6}}}})}^{-1}$ | [−65,65] | 1 |

F15 | ${f}_{15}={\displaystyle {\sum}_{i=1}^{11}[{a}_{i}-\frac{{x}_{1}({b}_{i}^{2}+{b}_{1}{x}_{2})}{{b}_{i}^{2}+{b}_{1}{b}_{3}+{b}_{4}}}{]}^{2}$ | [−5,5] | 0.0003 |

F16 | ${f}_{16}=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}x{}_{1}{}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ | [−5,5] | −1.0316 |

F17 | ${f}_{17}={({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6)}^{2}+10(1-\frac{1}{8\pi})\mathrm{cos}{x}_{1}+10$ | [−5,5] | 0.398 |

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

F19 | ${f}_{19}=-{\displaystyle {\sum}_{i=1}^{4}{c}_{i}}\mathrm{exp}(-{{\displaystyle {\sum}_{j=1}^{3}{a}_{ij}({x}_{ij}-{p}_{ij})}}^{2})$ | [1,3] | −3.80 |

F20 | ${f}_{20}=-{\displaystyle {\sum}_{i=1}^{4}{c}_{i}}\mathrm{exp}(-{{\displaystyle {\sum}_{j=1}^{6}{a}_{ij}({x}_{ij}-{p}_{ij})}}^{2})$ | [0,1] | −3.32 |

F21 | ${f}_{21}=-{{\displaystyle {\sum}_{i=1}^{5}\left[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}\right]}}^{-1}$ | [0,10] | −10.1513 |

F22 | ${f}_{22}=-{{\displaystyle {\sum}_{i=1}^{7}\left[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}\right]}}^{-1}$ | [0,10] | −10.4028 |

F23 | ${f}_{23}=-{{\displaystyle {\sum}_{i=1}^{10}\left[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}\right]}}^{-1}$ | [0,10] | −10.5363 |

**Table 2.**Comparison of optimization results obtained for the unimodal, multimodal and fixed-dimension multimodal benchmark functions.

Function | L0 | Ar | Ro | Hy | Pe-I | Pe-II | Fe | Li | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AVE | STD | AVE | STD | AVE | STD | AVE | STD | AVE | STD | AVE | STD | AVE | STD | AVE | STD | |

F1 | 1.41 × 10^{−30} | 4.91 × 10^{−30} | 3.64 × 10^{−106} | 4992.4116 | 2.065 × 10^{−47} | 6816.3959 | 1.001 × 10^{−48} | 8014.5287 | 1.031 × 10^{−86} | 6343.7262 | 7.242 × 10^{−71} | 5094.0843 | 3.096 × 10^{−60} | 5251.4799 | 3.638 × 10^{−49} | 7360.7530 |

F2 | 1.06 × 10^{−21} | 2.39 × 10^{−21} | 1.26 × 10^{−105} | 1012501 | 4.91 × 10^{−65} | 42530040813 | 8.33 × 10^{−43} | 19429812933 | 3.22 × 10^{−52} | 97498186729 | 3.22 × 10^{−52} | 5463750779 | 2.66 × 10^{−33} | 493771528062 | 4.66 × 10^{−36} | 14453630723 |

F3 | 21,533.06 | 15,903.34 | 17,308.09 | 61,432.76 | 49,342.12 | 22,646.55 | 37,706.46 | 22,712.248 | 55,440.039 | 19,449.57 | 50,199.04 | 10,850.31 | 69,415.32 | 25,663.15 | 30,810.03 | 36,337.14 |

F4 | 0.072581 | 0.39747 | 0.0240 | 10.750 | 8.1417 | 35.1489 | 7.8327 | 19.833 | 0.7614 | 5.7955 | 1.1439 | 15.942 | 1.0324 | 15.568 | 3.8082 | 24.589 |

F5 | 27.86558 | 0.763626 | 0.445 | 21966964 | 28.789 | 17463450 | 27.779 | 196730 | 10.637 | 197611 | 1.651 | 209162 | 8.598 | 258619 | 5.436 | 226382 |

F6 | 3.116266 | 0.532429 | 0.01752 | 8622.984 | 0.0001208 | 5866.599 | 0.010163 | 5806.449 | 0.01166 | 7645.330 | 0.00738 | 6855.553 | 0.01820 | 6264.7876 | 0.054570 | 4673.773 |

F7 | 0.001425 | 0.001149 | 0.000456 | 6.0642 | 0.00617 | 8.95037 | 0.001953 | 10.1904 | 0.00352 | 10.1742 | 0.00317 | 7.3495 | 0.00564 | 7.1875 | 0.000123 | 9.778 |

F8 | −5080.76 | 695.7968 | −12,569.062 | 882.164 | −8103.505 | 784.832 | −7574.253 | 385.518 | −7747.811 | 421.263 | −9937.307 | 625.755 | −8163.108 | 395.22 | −11,050.707 | 1125.405 |

F9 | 0 | 0 | 0 | 49.058 | 0 | 75.368 | 0 | 69.589 | 0 | 69.735 | 0 | 87.804 | 0 | 65.9588 | 0 | 60.5413 |

F10 | 7.4043 | 9.89757 | 8.88 × 10^{−16} | 2.9449 | 4.44 × 10^{−15} | 2.9965 | 4.44 × 10^{−15} | 3.5481 | 8.88 × 10^{−16} | 3.3239 | 8.88 × 10^{−16} | 3.0169 | 7.99 × 10^{−16} | 2.79940 | 1.39 × 10^{−13} | 3.05938 |

F11 | 0.00028 | 0.00158 | 9.9767 × 10^{−6} | 45.548 | 0 | 50.372 | 1.5259 × 10^{−10} | 62.525 | 7.414 × 10^{−10} | 54.537 | 1.1872 × 10^{−13} | 55.4545 | 0.00076571 | 66.1365 | 3.4649 × 10^{−5} | 50.2502 |

F12 | 0.33967 | 0.21486 | 0.00103 | 43593546 | 0.0335 | 48335777 | 0.5537 | 24074879 | 0.0631 | 46483837 | 0.1670 | 35041377 | 0.0273 | 39717531 | 0.0370 | 51384616 |

F13 | 1.88901 | 0.26608 | 0.0052 | 57915015 | 1.6642 | 73268462 | 0.1307 | 89315333 | 0.6859 | 96698854 | 0.2089 | 96675467 | 1.0066 | 68800044 | 1.3849 | 78754257 |

F14 | 2.11197 | 2.49859 | 0.99880 | 3.5538 | 2.9821 | 5.2941 | 0.99801 | 4.3043 | 0.9980 | 22.0250 | 0.9980 | 0.81930 | 2.9821 | 8.1928 | 2.9821 | 0.3263 |

F15 | 0.00057 | 0.00032 | 0.00030 | 0.00055 | 0.00031 | 0.01003 | 0.00033 | 0.01561 | 0.00032 | 0.00195 | 0.00033 | 0.00550 | 0.00071 | 0.00363 | 0.00037 | 0.01032 |

F16 | −1.0316 | 4.2 × 10^{−7} | −1.0316 | 4.2 × 10^{−7} | -1.0316 | 4.2 × 10^{−7} | −1.0316 | 4.2 × 10^{−7} | −1.0316 | 4.2 × 10^{−7} | −1.0316 | 4.2 × 10^{−7} | −1.0316 | 4.2 × 10^{−7} | −1.0316 | 4.2 × 10^{−7} |

F17 | 0.39791 | 2.7 × 10^{−5} | 0.0817 | 2.7 × 10^{−5} | 0.39791 | 2.7 × 10^{−5} | 0.39791 | 2.7 × 10^{−5} | 0.39791 | 2.7 × 10^{−5} | 0.39791 | 2.7 × 10^{−5} | 0.39791 | 2.7 × 10^{−5} | 0.39791 | 2.7 × 10^{−5} |

F18 | 3 | 4.22 × 10^{−15} | 0.3098 | 7.65 × 10^{−18} | 3 | 6.51 × 10^{−15} | 3 | 4.36 × 10^{−15} | 3 | 3.12 × 10^{−15} | 3 | 2.13 × 10^{−15} | 3 | 5.63 × 10^{−15} | 3 | 4.22 × 10^{−15} |

F19 | −3.85616 | 0.002706 | −3.8621 | 0.0495 | −3.8625 | 0.1135 | −3.8627 | 0.0110 | −3.8622 | 0.00087 | −3.8599 | 0.0030 | −3.8486 | 0.0024 | −3.8612 | 0.0165 |

F20 | −2.98105 | 0.376653 | −3.32165 | 0.1012356 | −3.31256 | 0.118835 | −3.31936 | 0.191023 | −3.31844 | 0.750346 | −3.04178 | 0.055936 | −2.83532 | 0.078376 | −3.32126 | 0.066325 |

F21 | −7.04918 | 3.629551 | −10.152 | 0.8923 | −5.055 | 0.5159 | −5.054 | 0.4053 | −2.627 | 0.1699 | −5.0546 | 0.3044 | −5.05583 | 0.5427 | −9.629 | 1.25043 |

F22 | −8.18178 | 3.829202 | −10.4023 | 1.2051 | −3.7242 | 0.19741 | −10.4020 | 1.4238 | −10.4014 | 2.3170 | −2.7658 | 0.2601 | −5.0876 | 0.5905 | −10.4024 | 2.6105 |

F23 | −9.34238 | 2.414737 | −10.5361 | 1.1602 | −5.1185 | 1.3741 | −5.1241 | 1.1746 | −3.8351 | 0.3774 | −3.8354 | 0.4215 | −5.0740 | 1.16607 | −5.1166 | 1.2442 |

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**MDPI and ACS Style**

Sun, W.-z.; Wang, J.-s.; Wei, X.
An Improved Whale Optimization Algorithm Based on Different Searching Paths and Perceptual Disturbance. *Symmetry* **2018**, *10*, 210.
https://doi.org/10.3390/sym10060210

**AMA Style**

Sun W-z, Wang J-s, Wei X.
An Improved Whale Optimization Algorithm Based on Different Searching Paths and Perceptual Disturbance. *Symmetry*. 2018; 10(6):210.
https://doi.org/10.3390/sym10060210

**Chicago/Turabian Style**

Sun, Wei-zhen, Jie-sheng Wang, and Xian Wei.
2018. "An Improved Whale Optimization Algorithm Based on Different Searching Paths and Perceptual Disturbance" *Symmetry* 10, no. 6: 210.
https://doi.org/10.3390/sym10060210