# A New Hybrid Whale Optimizer Algorithm with Mean Strategy of Grey Wolf Optimizer for Global Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Whale Optimizer Algorithm (WOA)

**Encircling prey:**Whale encircles the small fishes (prey) then modifies its position towards the global optimum solution over the course of increasing number of generation from start to a maximum number of generations.

- Shrinking encircling mechanism: this method is used by linearly decreasing the value of $\overrightarrow{a}~\left[0,2\right]$. Random value for a vector $\overrightarrow{a}~\left[-1,1\right]$.
- Spiral updating position: position update amid whale and small fishes (prey) that showed a helix-shaped movement is given as follows:$$M={\overrightarrow{d}}^{\prime}{e}^{bt}\mathrm{cos}(2\pi l)+{\overrightarrow{x}}_{t}^{*}$$

## 4. Mean Grey Wolf Optimizer (MGWO)

**Hunting:**In order to mathematically simulate hunting behavior, we suppose that the alpha, beta and delta have better knowledge about the potential location of the prey. The following equations are developed in this regard.

**Search for prey and attacking prey**: The $\overrightarrow{a}$ is random value in the gap $[-2a,2a]$. When random value $\left|\overrightarrow{a}\right|<1$ the wolves are forced to attack the prey. Searching for prey is the exploration ability and attacking the prey is the exploitation ability. The arbitrary values of $\overrightarrow{a}$ are utilized to force the search to move away from the prey.

## 5. Hybrid Algorithm

Pseudo Code of HAGWO |

Initialize the population |

Find the fitness of each search member |

$\overrightarrow{x}*$ is the best search member |

While ($t<$ max. number of generations) |

For every search member |

Update $a,\overrightarrow{a},\overrightarrow{c},l$ and $p$ |

if ($p<0.5$) |

if ($\left|\overrightarrow{a}\right|<1$) |

update the position of the current search member by the Equation (1) |

else if ($\left|\overrightarrow{a}\right|\ge 1$) |

select a random search member (${\overrightarrow{x}}_{rand}$) |

update the position of the current search member by the Equation (8) |

end if |

else if ($p\ge 0.5$) |

update the position of the present search member by using Equations (16) and (17) |

end if |

end for |

Find the fitness of all search members |

Update $\overrightarrow{x}*$,${\overrightarrow{d}}_{\alpha}$,${\overrightarrow{d}}_{\beta}$ and ${\overrightarrow{d}}_{\delta}$ |

$t=t+1$ |

end while |

return $\overrightarrow{x}*$ |

## 6. Parameter Setting

## 7. Test Problems

## 8. The Performance of the HAGWO Algorithm

## 9. Analysis

## 10. Experiments and Discussion on the Results

## 11. Bio-Medical Science Real Life Applications

## 12. Welded Beam Design

## 13. Pressure Vessel Design

## 14. Conclusions and Future Work

## Author Contributions

## Conflicts of Interest

## Appendix A

**Table A1.**Classification datasets (Mirjalili [48]).

Classification Datasets | Number of Attributes | Number of Training Samples | Number of Test Samples | Number of Classes |
---|---|---|---|---|

3-bits XOR | 3 | 8 | 8 as training samples | 2 |

Baloon | 4 | 16 | 16 as training samples | 2 |

Iris | 4 | 150 | 150 as training samples | 3 |

Breast Cancer | 9 | 599 | 100 | 2 |

Parameter | Value |
---|---|

$\overrightarrow{a}$ | Linearly decreased from 2 to 0 |

Search Agents | 200 |

Maximum number of iterations | 100–200 |

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**Figure 1.**Convergence graph of metaheuristics. (

**a**) Dim 20; (

**b**) Dim 40; (

**c**) Dim 60; (

**d**) Dim 80; (

**e**) Dim 100.

**Figure 2.**Convergence Curve of PSO, GWO, WOA, MGWO and HAGWO variants on Unimodal benchmark functions.

**Figure 3.**Convergence Curve of PSO, GWO, WOA, MGWO and HAGWO variants on Multimodal benchmark functions.

**Figure 4.**Convergence Curve of PSO, GWO, WOA, MGWO and HAGWO variants on Fixed-dimension multimodal benchmark functions.

**Figure 5.**(

**a**) Convergence graph of Iris dataset problem; (

**b**) Convergence graph of XOR dataset problem; (

**c**) Convergence graph of Baloon dataset problem; (

**d**) Convergence graph of Breast cancer dataset problem.

Function | Dim | Range | ${\mathit{f}}_{\mathbf{min}}$ | Graph |
---|---|---|---|---|

${F}_{1}(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}$ | 30 | [−100, 100] | 0 | |

${F}_{2}(x)={\displaystyle \sum _{i=1}^{n}\left|{x}_{i}\right|}+{\displaystyle \prod _{i=1}^{n}\left|{x}_{i}\right|}$ | 30 | [−10, 10] | 0 | |

${F}_{3}(x)={\displaystyle \sum _{i=1}^{n}}{({\displaystyle \sum _{j-1}^{i}{x}_{j}})}^{2}$ | 30 | [−100, 100] | 0 | |

${F}_{4}(x)={\mathrm{max}}_{i}\left\{\left|{x}_{i}\right|,\text{}1\le i\le n\right\}$ | 30 | [−100, 100] | 0 | |

${F}_{5}(x)={\displaystyle \sum _{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]}$ | 30 | [−30, 30] | 0 | |

${F}_{6}(x)={\displaystyle \sum _{i=1}^{n}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}}$ | 30 | [−100, 100] | 0 | |

${F}_{7}(x)={\displaystyle \sum _{i=1}^{n}i{x}_{i}^{4}+rand[0,1)}$ | 30 | [−1.28, 1.28] | 0 |

Function | Dim | Range | ${\mathit{f}}_{\mathbf{min}}$ | Graph |
---|---|---|---|---|

${F}_{8}(x)={\displaystyle \sum _{i=1}^{n}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)}$ | 30 | [−500, 500] | −418.9829 × 5 | |

${F}_{9}(x)={\displaystyle \sum _{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}(2\pi {x}_{i})+10\right]}$ | 30 | [−5.12, 5.12] | 0 | |

$\begin{array}{cc}\hfill {F}_{10}(x)=& -20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}}\right)\hfill \\ & -\mathrm{exp}\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)}\right)+20+e\hfill \end{array}$ | 30 | [−32, 32] | 0 | |

${F}_{11}(x)=\frac{1}{4000}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1}}$ | 30 | [−600, 600] | 0 | |

$\begin{array}{c}\begin{array}{cc}\hfill {F}_{12}(x)=& \frac{\pi}{n}\left\{10\mathrm{sin}\left(\pi {y}_{i}\right)+{\displaystyle \sum _{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)+{\left({y}_{n-1}\right)}^{2}\right]}\right\}\hfill \\ & +{\displaystyle \sum _{i=1}^{n}u\left({x}_{i},10,100,4\right)}\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill {y}_{i}=& 1+\frac{{x}_{i}+1}{4}\hfill \end{array}\hfill \\ u\left({x}_{i},a,k,m\right)=\{\begin{array}{cc}k{\left({x}_{i}-a\right)}^{m}& {x}_{i}>a\\ 0& -a<{x}_{i}<a\\ k{\left(-{x}_{i}-a\right)}^{m}& {x}_{i}<-a\end{array}\hfill \end{array}$ | 30 | [−50, 50] | 0 | |

$\begin{array}{cc}\hfill {F}_{13}(x)=& 0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{i}\right)+{\displaystyle \sum _{i=1}^{n}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i}+1\right)\right]}+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}(2\pi {x}_{n})\right]\right\}\hfill \\ & +{\displaystyle \sum _{i=1}^{n}u\left({x}_{i},5,100,4\right)}\hfill \end{array}$ | 30 | [−50, 50] | 0 |

Function | Dim | Range | ${\mathit{f}}_{\mathbf{min}}$ | Graph |
---|---|---|---|---|

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

${F}_{15}(x)={{\displaystyle \sum _{i=1}^{11}\left[{a}_{i}-\frac{{x}_{1}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{x}_{i}+{x}_{4}}\right]}}^{2}$ | 4 | [−5, 5] | 0.00030 | |

$\begin{array}{cc}\hfill {F}_{16}(x)=& 4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}\hfill \\ & +{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}\hfill \end{array}$ | 2 | [−5, 5] | −1.0316 | |

$\begin{array}{cc}\hfill {F}_{17}(x)=& {\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}\hfill \\ & +10\left(1-\frac{1}{8\pi}\right)\mathrm{cos}{x}_{1}+10\hfill \end{array}$ | 2 | [−5, 5] | 0.398 | |

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

${F}_{19}(x)=-{\displaystyle \sum _{i=1}^{4}{c}_{i}\mathrm{exp}\left(-{\displaystyle \sum _{j=1}^{3}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2}}\right)}$ | 3 | [1, 3] | −3.86 | |

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

${F}_{21}(x)=-{{\displaystyle \sum _{i=1}^{5}\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}}^{-1}$ | 4 | [0,10] | −10.1532 | |

${F}_{22}(x)=-{{\displaystyle \sum _{i=1}^{7}\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}}^{-1}$ | 4 | [0, 10] | −10.4028 | |

${F}_{23}(x)=-{{\displaystyle \sum _{i=1}^{10}\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}}^{-1}$ | 4 | [0, 10] | −10.5363 |

Problem | PSO | GWO | WOA | MGWO | HAGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\downarrow}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ |

1 | 7.8901 × 10^{−28} | 6.2609 × 10^{4} | 1.2292 × 10^{−293} | 5.7592 × 10^{4} | 0 | 7.2834 × 10^{4} | 7.4502 × 10^{−267} | 7.2336 × 10^{4} | 0 | 7.5361 × 10^{4} |

2 | 5.9781 × 10^{−12} | 1.1398 × 10^{12} | 1.2084 × 10^{−153} | 8.2863 × 10^{11} | 0 | 7.6425 × 10^{13} | 9.3431 × 10^{−154} | 6.0482 × 10^{13} | 0 | 8.8066 × 10^{13} |

3 | 27.2552 | 1.4283 × 10^{5} | 8.2741 × 10^{−12} | 1.9326 × 10^{5} | 6.6018 × 10^{4} | 1.3360 × 10^{5} | 6.0565 × 10^{−13} | 1.6858 × 10^{5} | 5.9829 × 10^{−17} | 2.4735 × 10^{5} |

4 | 1.9922 | 89.3164 | 3.3355 × 10^{−64} | 84.5135 | 13.6245 | 89.1133 | 1.8414 × 10^{−65} | 82.5356 | 1.5274 × 10^{−86} | 89.9799 |

5 | 130.8650 | 3.0054 × 10^{8} | 27.9593 | 3.5339 × 10^{8} | 28.7815 | 3.3241 × 10^{8} | 27.1707 | 2.9514 × 10^{8} | 27.1630 | 3.7213 × 10^{8} |

6 | 7.6619 × 10^{−5} | 6.4409 × 10^{4} | 1.9983 | 6.7418 × 10^{4} | 1.0113 | 7.5845 × 10^{4} | 1.7551 | 7.3870 × 10^{4} | 1.7255 | 7.5449 × 10^{4} |

7 | 0.0691 | 82.0814 | 0.0012 | 81.8383 | 4.2362 × 10^{−4} | 137.7861 | 3.1935 × 10^{−4} | 140.1725 | 0.0012 | 141.0199 |

Problem | PSO | GWO | WOA | MGWO | HAGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\downarrow}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ |

1 | 99.1220 | 1.7580 × 10^{3} | 90.2939 | 1.8259 × 10^{3} | 71.4189 | 1.7312 × 10^{3} | 102.3734 | 2.0195 × 10^{3} | 77.6527 | 1.9623 × 10^{3} |

2 | 2.2796 × 10^{10} | 1.6119 × 10^{10} | 1.9355 × 10^{8} | 1.1882 × 10^{10} | 1.5336 × 10^{10} | 1.0808 × 10^{12} | 1.2096 × 10^{10} | 8.5535 × 10^{11} | 1.7613 × 10^{10} | 1.2454 × 10^{12} |

3 | 2.3823 × 10^{3} | 1.3443 × 10^{4} | 1.9391 × 10^{3} | 1.2191 × 10^{4} | 5.3442 × 10^{4} | 2.9219 × 10^{4} | 1.9383 × 10^{3} | 1.1327 × 10^{4} | 1.8372 × 10^{3} | 1.4026 × 10^{4} |

4 | 1.7351 | 3.6600 | 0.5260 | 5.2238 | 22.8982 | 11.5619 | 0.4520 | 5.0290 | 1.9694 | 12.1502 |

5 | 1.2183 × 10^{6} | 1.5789 × 10^{7} | 2.7240 × 10^{6} | 2.5709 × 10^{7} | 3.5593 × 10^{6} | 2.6816 × 10^{7} | 2.4950 × 10^{6} | 2.1397 × 10^{7} | 2.0059 × 10^{6} | 2.0732 × 10^{7} |

6 | 1.0927 × 10^{3} | 6.6662 × 10^{3} | 796.3752 | 5.1671 × 10^{3} | 887.1449 | 6.1512 × 10^{3} | 1.0961 × 10^{3} | 6.8357 × 10^{3} | 691.9697 | 5.6904 × 10^{3} |

7 | 23.5823 | 32.2722 | 0.1980 | 2.9834 | 0.3679 | 5.8363 | 0.3163 | 5.5002 | 0.2879 | 5.5672 |

Problem | PSO | GWO | WOA | MGWO | HAGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\downarrow}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ |

8 | −6.5676 × 10^{3} | −2.5481 × 10^{3} | −6.1169 × 10^{3} | −864.3118 | −8.7219 × 10^{3} | −3.0712 × 10^{3} | −5.0428 × 10^{3} | −1.5535 × 10^{3} | −9.0083 × 10^{3} | −1.4592 × 10^{3} |

9 | 46.7630 | 432.1257 | 0 | 457.9386 | 0 | 466.5042 | 0 | 425.4633 | 0 | 477.4335 |

10 | 5.5924 × 10^{−6} | 20.9199 | 7.9936 × 10^{−15} | 20.84.27 | 4.4409 × 10^{−15} | 209134 | 7.9936 × 10^{−15} | 20.8774 | 4.4409 × 10^{−15} | 20.9385 |

11 | 0.0123 | 690.0217 | 0 | 666.3022 | 0 | 623.4277 | 0.0083 | 705.5196 | 0 | 710.0255 |

12 | 2.7804 | 5.7234 × 10^{8} | 4.3888 | 6.7091 × 10^{8} | 0.5737 | 5.4619 × 10^{8} | 5.5598 | 6.1162 × 10^{8} | 0.1551 | 7.1939 × 10^{8} |

13 | 3.0253 | 1.1558 × 10^{9} | 2.0670 | 9.2722 × 10^{8} | 1.5688 | 1.6628 × 10^{9} | 3.0729 | 1.1356 × 10^{9} | 1.2865 | 9.3192 × 10^{8} |

Problem | PSO | GWO | WOA | MGWO | HAGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\downarrow}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathbf{\downarrow}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ |

8 | −6.2997 × 10^{3} | 857.6904 | −4.1179 × 10^{3} | 1.0767 × 10^{3} | −8.4186 × 10^{3} | 378.1677 | −3.6282 × 10^{3} | 761.7072 | −8.4892 × 10^{3} | 622.2040 |

9 | 151.9952 | 129.9151 | 12.6542 | 45.1218 | 5.1226 | 37.8648 | 6.0697 | 34.7850 | 2.9663 | 29.5347 |

10 | 2.5215 | 3.2684 | 0.2513 | 1.8198 | 0.2607 | 1.7464 | 0.2415 | 1.7742 | 0.1296 | 1.3559 |

11 | 12.6332 | 70.7797 | 1.9879 | 25.2536 | 1.9234 | 27.8438 | 1.6198 | 23.8614 | 1.6759 | 26.8415 |

12 | 2.5010 × 10^{7} | 1.0165 × 10^{8} | 2.5134 × 10^{7} | 1.0722 × 10^{8} | 4.2413 × 10^{7} | 1.1477 × 10^{8} | 5.7916 × 10^{7} | 1.5932 × 10^{8} | 4.6413 × 10^{7} | 1.3041 × 10^{8} |

13 | 2.6298 × 10^{7} | 1.4156 × 10^{8} | 3.2950 × 10^{7} | 1.5177 × 10^{8} | 8.8731 × 10^{7} | 2.9685 × 10^{8} | 3.8630 × 10^{7} | 1.6992 × 10^{8} | 3.1642 × 10^{7} | 1.3828 × 10^{8} |

Problem | PSO | GWO | WOA | MGWO | HAGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\downarrow}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}$ |

14 | 7.8740 | 412.9323 | 13.6186 | 22.0408 | 2.9822 | 210.6945 | 9.6391 | 18.7819 | 2.9821 | 493.6227 |

15 | 8.6440 × 10^{−4} | 0.4044 | 4.6612 × 10^{−4} | 0.0760 | 4.2691 × 10^{−4} | 0.1871 | 4.4429 × 10^{−4} | 0.0702 | 3.1020 × 10^{−4} | 0.4770 |

16 | −1.0316 | 0.5530 | −1.0316 | −0.4506 | −1.0316 | 0.2993 | −1.0316 | 0.4158 | −1.0316 | 3.1477 |

17 | 0.3982 | 1.1732 | 0.4004 | 1.1432 | 0.4145 | 1.4559 | 0.3985 | 4.0587 | 1.4624 | 0.4635 |

18 | 3.0009 | 110.4515 | 3.0046 | 187.0153 | 3.0011 | 241.6666 | 3.0009 | 207.6478 | 3.0009 | 415.2426 |

19 | −3.8624 | −2.6790 | −3.8617 | −2.6833 | −3.7304 | −3.3467 | −3.8624 | −2.7356 | −3.8624 | −2.5019 |

20 | −3.2031 | −2.2384 | −3.3025 | −0.7864 | −2.9526 | −2.9526 | −2.8402 | −1.6544 | −3.3201 | −0.7698 |

21 | −2.6305 | −0.6444 | −10.1517 | −0.3554 | −5.0551 | −0.3800 | −101521 | −0.6119 | −10.1532 | −0.2680 |

22 | −10.4016 | −1.1161 | −10.4028 | −1.0080 | −5.0876 | −0.8341 | −10.4022 | −0.8415 | −10.4028 | −0.8049 |

23 | −5.1756 | −1.0843 | −10.5361 | −0.8808 | −9.9864 | −0.5751 | −10.5353 | −0.5825 | −10.5361 | −0.4436 |

Problem | PSO | GWO | WOA | MGWO | HAGWO | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\downarrow}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ |

8 | 20.1530 | 56.9403 | 13.9616 | 1.6660 | 7.8693 | 29.3410 | 9.4639 | 2.5297 | 13.4478 | 69.3887 |

9 | 0.0035 | 0.0373 | 0.0013 | 0.0054 | 0.0030 | 0.0142 | 0.0012 | 0.0047 | 0.0015 | 0.0214 |

10 | −0.9955 | 0.2106 | −1.0194 | 0.0817 | −1.0038 | 0.1513 | −0.9955 | 0.2106 | −0.9679 | 0.4297 |

11 | 0.5563 | 0.2268 | 0.5161 | 0.1831 | 0.7692 | 0.9017 | 0.6865 | 0.9173 | 0.4635 | 0.2174 |

12 | 30.5241 | 37.9585 | 7.6988 | 26.3454 | 8.8125 | 34.1104 | 7.8123 | 28.9425 | 12.1049 | 58.4134 |

13 | −3.7394 | 0.2649 | −3.8041 | 0.1681 | −3.6656 | 0.1125 | −3.7793 | 0.1772 | −3.7514 | 0.2717 |

14 | −2.9378 | 0.3013 | −3.0727 | 0.3977 | −2.8403 | 0.2221 | −2.7525 | 0.1956 | −2.8669 | 0.4184 |

15 | −2.5199 | 0.2835 | −7.0626 | 2.6720 | −5.0296 | 0.2538 | −7.6430 | 1.6995 | −8.5355 | 2.5679 |

16 | −9.5802 | 2.0466 | −7.8806 | 2.1526 | −5.0617 | 0.2709 | −9.0605 | 1.5746 | −9.9991 | 0.9721 |

17 | −4.8679 | 0.7136 | −7.6961 | 2.4874 | −7.7094 | 2.5030 | −8.7497 | 2.7833 | −8.7497 | 2.7833 |

(i) | Iris Dataset Problem | |||||

Algorithm | Best Min Value | Best Max Value | Average | S.D. | Classification Rate | |

PSO | 0.6667 | 0.8418 | 0.6895 | 0.0336 | 37.22% | |

WOA | 0.7029 | 0.8572 | 0.7263 | 0.0374 | 89.31% | |

GWO | 0.6667 | 0.8756 | 0.6714 | 0.0249 | 91.333% | |

MGWO | 0.6667 | 0.8133 | 0.6789 | 0.0154 | 91.334% | |

HAGWO | 0.6668 | 0.8807 | 0.6728 | 0.0281 | 93.00% | |

(ii) | XOR Dataset Problem | |||||

Algorithm | Best Min Value | Best Max Value | Average | S.D. | Classification Rate | |

PSO | 2.0621 × 10^{−23} | 0.1771 | 0.0162 | 0.0481 | 37.50% | |

WOA | 0.0705 | 0.1523 | 0.0943 | 6.9504 | 98% | |

GWO | 8.2721 × 10^{−6} | 0.1327 | 0.0156 | 0.0375 | 100% | |

MGWO | 5.0578 × 10^{−5} | 0.2159 | 0.0348 | 0.0634 | 100% | |

HAGWO | 0.0427 | 0.2300 | 0.0029 | 0.0469 | 100% | |

(iii) | Baloon Dataset Problem | |||||

Algorithm | Best Min Value | Best Max Value | Average | S.D. | Classification Rate | |

PSO | 5.6029 × 10^{−}^{28} | 0.1596 | 0.0161 | 0.0409 | 100% | |

WOA | 7.7005 × 10^{−}^{4} | 0.0313 | 0.0076 | 0.0080 | 100% | |

GWO | 3.5126 × 10^{−17} | 0.1168 | 0.0064 | 0.0261 | 100% | |

MGWO | 2.2483 × 10^{−}^{15} | 0.0556 | 0.0071 | 0.0173 | 100% | |

HAGWO | 1.6372 × 10^{−5} | 0.1798 | 0.0143 | 0.0438 | 100% | |

(iv) | Breast Cancer Dataset Problem | |||||

Algorithm | Best Min Value | Best Max Value | Average | S.D. | Classification Rate | |

PSO | 0.0054 | 0.0441 | 0.0130 | 0.0070 | 14.00% | |

WOA | 0.0018 | 0.0416 | 0.0033 | 0.0043 | 97.21% | |

GWO | 0.0014 | 0.0464 | 0.0065 | 0.0093 | 99.00% | |

MGWO | 0.0017 | 0.0387 | 0.0066 | 0.0096 | 99.11% | |

HAGWO | 0.0013 | 0.0464 | 0.0026 | 0.0042 | 100% |

Algorithm | Optimum Variables | $\mathit{f}\mathbf{\left(}\mathit{Y}\mathbf{\right)}$ | |||
---|---|---|---|---|---|

- | $\mathit{h}$ | $\mathit{l}$ | $\mathit{t}$ | $\mathit{b}$ | - |

GA | 0.208800 | 3.420500 | 8.997500 | 0.210000 | 1.748309 |

GA | 0.205986 | 3.471328 | 9.020224 | 0.206480 | 1.728226 |

GA | 0.2489 | 6.1730 | 8.1789 | 0.2533 | 2.4328 |

UPSO | 0.2407 | 6.4851 | 8.2399 | 0.2497 | 2.4426 |

ABC | 0.205730 | 3.470489 | 9.036624 | 0.205730 | 1.7248852 |

CDE | 0.203137 | 3.542998 | 9.033498 | 0.206179 | 1.733462 |

CPSO | 0.202369 | 3.544214 | 9.048210 | 0.205723 | 1.728024 |

HIS | 0.20573 | 3.47049 | 9.03662 | 0.20573 | 1.7248 |

MFO | 0.2057 | 3.4703 | 9.0364 | 0.2057 | 1.72452 |

AFA | 0.205730 | 3.40489 | 9.036624 | 0.205730 | 1.724852 |

CSS | 0.2058 | 3.4681 | 9.0380 | 0.2057 | 1.7249 |

LSA-SM | 0.2057296 | 3.253120 | 9.036624 | 0.2057296 | 1.695247 |

HAGWO | 0.2055235 | 3.201258 | 9.033258 | 0.2052125 | 1.661258 |

Algorithm | Optimum Variables | $\mathit{f}\mathbf{(}\mathit{Y}\mathbf{)}$ | |||
---|---|---|---|---|---|

- | ${\mathit{x}}_{\mathbf{1}}$ | ${\mathit{x}}_{\mathbf{2}}$ | ${\mathit{x}}_{\mathbf{3}}$ | ${\mathit{x}}_{\mathbf{4}}$ | - |

GA | 0.812500 | 0.437500 | 40.323900 | 200.000000 | 6288.7445 |

CPSO | 0.812500 | 0.437500 | 42.091266 | 176.746500 | 6061.0777 |

GA | 0.812500 | 0.437500 | 42.097398 | 176.654050 | 6059.9463 |

HIS | 0.75 | 0.375 | 38.86010 | 221.36553 | 5849.76169 |

CDE | 0.812500 | 0.437500 | 42.098411 | 176.746500 | 6061.0777 |

BA | 0.8125 | 0.4375 | 42.0984456 | 176.6365958 | 6059.7143348 |

ABC | 0.812500 | 0.437500 | 42.098446 | 176.636596 | 6059.714339 |

AFA | 0.8125 | 0.4375 | 42.09844611 | 176.6365894 | 6059.7142719 |

CS | 0.8125 | 0.4375 | 42.0984456 | 176.6365958 | 6059.7143348 |

ES | 0.8125 | 0.4375 | 42.098087 | 176.640518 | 6059.7456 |

ACO | 0.8125 | 0.4375 | 42.103624 | 176.572656 | 6059.0888 |

TLBO | NA | NA | NA | NA | 6059.714335 |

MFO | 0.8125 | 0.4375 | 42.098445 | 176.636596 | 6059.7143 |

LSA-SM | 0.8103764 | 0.4005695 | 41.98842 | 178.0048 | 5942.6966 |

HAGWO | 0.8102456 | 0.4003526 | 41.78451 | 178.0012 | 5924.2536 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Singh, N.; Hachimi, H. A New Hybrid Whale Optimizer Algorithm with Mean Strategy of Grey Wolf Optimizer for Global Optimization. *Math. Comput. Appl.* **2018**, *23*, 14.
https://doi.org/10.3390/mca23010014

**AMA Style**

Singh N, Hachimi H. A New Hybrid Whale Optimizer Algorithm with Mean Strategy of Grey Wolf Optimizer for Global Optimization. *Mathematical and Computational Applications*. 2018; 23(1):14.
https://doi.org/10.3390/mca23010014

**Chicago/Turabian Style**

Singh, Narinder, and Hanaa Hachimi. 2018. "A New Hybrid Whale Optimizer Algorithm with Mean Strategy of Grey Wolf Optimizer for Global Optimization" *Mathematical and Computational Applications* 23, no. 1: 14.
https://doi.org/10.3390/mca23010014