# An Improved Whale Optimization Algorithm with Random Evolution and Special Reinforcement Dual-Operation Strategy Collaboration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Whale Optimization Algorithm (WOA)

#### 2.1. Encircling Prey

#### 2.2. Bubble Net Attacking Method

#### 2.3. Searching for Prey

## 3. Random Evolutionary Whale Optimization Algorithm (REWOA)

#### 3.1. Random Evolutionary

#### 3.2. Special Reinforcement

#### 3.3. Main Procedure of the REWOA

#### 3.4. Complexity Analysis

## 4. Experimental Results and Discussion

#### 4.1. Evaluation of Exploitation Capability

#### 4.2. Evaluation of Exploration Capability

#### 4.3. Analysis of Convergence Behavior

## 5. Hammerstein Model Identification Using REWOA

#### 5.1. Hammertein Model

- (a)
- Two-term Gaussian mixture distribution

- (b)
- The t-distribution

#### 5.2. The Identification Proceduce

**Step1:**Obtain the input sample data $u(k)$ and output sample data $y(k)$ of the system;**Step2:**Calculate the output $\widehat{y}\left(k\right)$ of the model according to the weight vector of the auxiliary model;**Step3:**Initialization of positions and parameters;**Step4:**Minimize the fitness value using REWOA to get the best solution in the current iteration;**Step5:**Check whether the identification result is satisfied or not. If satisfied, then stop the algorithm and get the best solutions; if not, go back to step 4 and set $k=k+1$.

#### 5.3. Simulation Study

- Experiment 1

- Experiment 2

- Experiment 3

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Random walk through of 100 consecutive steps, starting at the origin (0, 0): (

**a**) Levy flight step-size distribution; (

**b**) RW step-size distribution; (

**c**) Gaussian step-size distribution.

**Figure 3.**Flowchart for the REWOA. where $cr$ indicates the crossover probability using Equation (14), $pr$ indicates adaptive probability using Equation (23).

**Figure 4.**Comparison of convergence curves of REWOA and literature algorithms obtained for some of the benchmark problems.

**Figure 11.**The residual test results for the three experiments: (

**a**) experiment 1; (

**b**) experiment 2; (

**c**) experiment 3.

No. | Formula | Dim | Range | ${\mathit{f}}_{\mathbf{min}}$ |
---|---|---|---|---|

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

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

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

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

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

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

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

F8 | $f(x)={\displaystyle {\sum}_{i=1}^{n}-{x}_{i}}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})$ | 30 | [−500, 500] | −12,569.49 |

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

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

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

F12 | $\begin{array}{l}f(x)=\frac{\pi}{n}\{10\mathrm{sin}(\pi {y}_{1})+{\displaystyle {\sum}_{i=1}^{n-1}({y}_{i}}-1{)}^{2}[1+10{\mathrm{sin}}^{2}(\pi {y}_{i+1})]+{({y}_{n}-1)}^{2}\}+{\displaystyle {\sum}_{i=1}^{n}u({x}_{i},10,100,4)}\\ {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}\\ \\ \end{array}\begin{array}{c}{x}_{i}>a\\ -a<{x}_{i}<a\\ {x}_{i}<a\end{array}\end{array}$$\begin{array}{l}f(x)=0.1\{{\mathrm{sin}}^{2}(3\pi {x}_{1})+{\displaystyle {\sum}_{i=1}^{n}({x}_{i}}-1{)}^{2}[1+{\mathrm{sin}}^{2}(3\pi {x}_{i}+1)]\\ \begin{array}{cc}& \end{array}+{({x}_{n}-1)}^{2}[1+{\mathrm{sin}}^{2}(2\pi {x}_{n})\}+{\displaystyle {\sum}_{i=1}^{n}u({x}_{i},5,100,4)}\end{array}$ | 30 | [−50, 50] | 0 |

F13 | 30 | [−50, 50] | 0 | |

F14 | $f(x)={(\frac{1}{500}+{\displaystyle {\sum}_{j=1}^{25}\frac{1}{j+{\displaystyle {\sum}_{i=1}^{2}{({x}_{i}-{a}_{ij})}^{6}}}})}^{-1}$ $f(x)={\displaystyle {\sum}_{i=1}^{11}[{a}_{i}}-\frac{{x}_{1}({b}_{i}^{2}+{b}_{i}{x}_{2})}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}{]}^{2}$ $f(x)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ $f(x)={({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6)}^{2}+10(1-\frac{1}{8\pi})\mathrm{cos}{x}_{1}+10$ | 2 | [−65, 65] | 1 |

F15 | 4 | [−5, 5] | 0.0003 | |

F16 | 2 | [−5, 5] | −1.0316 | |

F17 | 2 | [−5, 5] | 0.398 | |

F18 | $\begin{array}{l}f(x)=[1+{({x}_{1}+{x}_{2}+1)}^{2}(19-14{x}_{1}+3{x}_{1}^{2}-14{x}^{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2})]\\ \begin{array}{cc}& \times \end{array}[30+{(2{x}_{1}-3{x}_{2})}^{2}\times (18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2})]\end{array}$ | 2 | [−2, 2] | 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] | −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] | −3.32 |

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

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

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

Method | Control Parameter | Value |
---|---|---|

REWOA | Convergence factor $(a)$ | [1, 2] |

Mutation probability $(mr)$ | 0.2 | |

Crossover probability $(cr)$ | [0, 0.5] | |

Adaptive weight $(\omega )$ | [0, 1] | |

WOA [21] | Convergence factor $(a)$ | [0, 2] |

Probability coefficient $(p)$ | 0.5 | |

WOABAT [43] | Convergence factor $(a)$ | [0, 2] |

Probability coefficient $(p)$ | 0.5 | |

Pulse rate $(r)$ | 0.5 | |

Sound loudness $({A}_{i})$ | 0.5 | |

SSA [22] | Probability coefficient $(p)$ | 0.5 |

WOASAT [51] | Reduction rate $(\alpha )$ | 0.99 |

Initial temp $(t0)$ | 0.1 | |

Maximum Number of Iterations | 30 | |

Convergence factor $(a)$ | [0, 2] | |

Probability coefficient $(p)$ | 0.5 | |

DE [10] | Mutation operator $(F)$ | 0.5 |

Crossover probability $(cr)$ | 0.3 |

**Table 3.**Comparison of REWOA with other algorithms for use on unimodal problems. Note that the bold letters in the table indicate the respective best results.

Function | Metric | REWOA | WOA | WOABAT | SSA | WOASAT | DE |
---|---|---|---|---|---|---|---|

F1 | avg | 4.2 × 10^{−322} | 3.67 × 10^{−73} | 1.57 × 10^{−06} | 1.60 × 10^{−07} | 0 | 3.44 × 10^{−15} |

std | 0 | 1.12 × 10^{−72} | 7.11 × 10^{−07} | 2.24 × 10^{−07} | 0 | 1.04 × 10^{−14} | |

F2 | avg | 2.60 × 10^{−213} | 5.57 × 10^{−52} | 7.33 × 10^{−03} | 2.34 | 0 | 2.10 × 10^{−09} |

std | 0 | 1.82 × 10^{−51} | 1.39 × 10^{−03} | 1.49 | 0 | 3.82 × 10^{−09} | |

F3 | avg | 0 | 4.21 × 10^{4} | 9.56 × 10^{−06} | 1.53 × 10^{3} | 7.93 × 10^{−01} | 4.07 × 10^{3} |

std | 0 | 1.48 × 10^{4} | 2.17 × 10^{−06} | 7.48 × 10^{2} | 5.00 × 10^{−01} | 1.99 × 10^{3} | |

F4 | avg | 5.65 × 10^{−151} | 49.6 | 1.01 × 10^{−03} | 11.4 | 8.04 × 10^{−01} | 8.30 |

std | 2.79 × 10^{−150} | 29.4 | 8.71 × 10^{−05} | 3.93 | 3.92 × 10^{−01} | 3.90 | |

F5 | avg | 2.64 | 27.9 | 6.58 | 3.40 × 10^{2} | 29.6 | 62.1 |

std | 7.88 | 4.88 × 10^{−01} | 11.9 | 4.59 × 10^{2} | 19.0 | 51.9 | |

F6 | avg | 2.01 × 10^{−08} | 3.64 × 10^{−01} | 1.71 × 10^{−06} | 2.72 × 10^{−07} | 1.28 × 10^{−03} | 6.90 × 10^{−14} |

std | 3.94 × 10^{−08} | 2.20 × 10^{−01} | 8.14 × 10^{−07} | 3.02 × 10^{−07} | 7.99 × 10^{−04} | 1.56 × 10^{−13} | |

F7 | avg | 1.47 × 10^{−03} | 2.92 × 10^{−03} | 4.85 × 10^{−04} | 1.74 × 10^{−01} | 5.99 × 10^{−02} | 2.76 × 10^{−01} |

std | 1.26 × 10^{−03} | 3.34 × 10^{−03} | 8.45 × 10^{−04} | 5.91 × 10^{−02} | 3.59 × 10^{−02} | 2.61 × 10^{−01} | |

$best$ $sd-best$ | 3 | 0 | 1 | 0 | 2 | 1 | |

4 | 0 | 0 | 0 | 0 | 0 |

**Table 4.**Comparison of REWOA with other algorithms for use on multimodal problems. Note that the bold letters in the table indicate the respective best results.

Function | Metric | REWOA | WOA | WOABAT | SSA | WOASAT | DE |
---|---|---|---|---|---|---|---|

F8 | avg | −1.25 × 10^{4} | −1.03 × 10^{4} | −1.22 × 10^{4} | −7.60 × 10^{3} | −9.97 × 10^{3} | −1.03 × 10^{4} |

std | 58.7 | 2.05 × 10^{3} | 1.07 × 10^{3} | 8.92 × 10^{2} | 1.65 × 10^{3} | 6.42 × 10^{2} | |

F9 | avg | 0 | 5.68 × 10^{−15} | 5.97 | 53.4 | 0 | 32.0 |

std | 0 | 2.25 × 10^{−14} | 11.9 | 18.8 | 0 | 12.6 | |

F10 | avg | 8.88 × 10^{−16} | 4.44 × 10^{−15} | 9.36 × 10^{−04} | 2.55 | 8.88 × 10^{−16} | 2.27 |

std | 0 | 2.59 × 10^{−15} | 2.11 × 10^{−04} | 5.52 × 10^{−01} | 0.00 × 10^{+00} | 1.96 | |

F11 | avg | 0 | 5.87 × 10^{−03} | 8.55 × 10^{−08} | 1.87 × 10^{−02} | 0.00 × 10^{+00} | 2.43 × 10^{−02} |

std | 0 | 3.16 × 10^{−02} | 3.79 × 10^{−08} | 1.69 × 10^{−02} | 0.00 × 10^{+00} | 2.39 × 10^{−02} | |

F12 | avg | 3.18 × 10^{−09} | 2.61 × 10^{−02} | 1.32 × 10^{−08} | 6.55 × 10^{+00} | 1.81 × 10^{−04} | 7.60 × 10^{−01} |

std | 1.40 × 10^{−08} | 2.93 × 10^{−02} | 5.60 × 10^{−09} | 3.38 | 1.76 × 10^{−04} | 1.41 | |

F13 | avg | 2.17 × 10^{−03} | 4.97 × 10^{−01} | 2.22 × 10^{−07} | 18.5 | 1.35 × 10^{−32} | 5.15 × 10^{−01} |

std | 5.11 × 10^{−03} | 2.42 × 10^{−01} | 1.03 × 10^{−07} | 14.8 | 5.47 × 10^{−48} | 8.97 × 10^{−01} | |

$best$ $sd-best$ | 5 | 0 | 0 | 0 | 4 | 0 | |

0 | 2 | 2 | 0 | 0 | 0 |

**Table 5.**Comparison of REWOA with other algorithms for use on fixed-dimension multimodal problems. Note that the bold letters in the table indicate the respective best results.

Function | Metric | REWOA | WOA | WOABAT | SSA | WOASAT | DE |
---|---|---|---|---|---|---|---|

F14 | avg | 1.32 | 3.22 | 1.78 | 1.16 | 8.14 | 2.51 |

std | 1.75 | 3.47 | 2.50 | 4.50 × 10^{−01} | 5.09 | 2.26 | |

F15 | avg | 5.37 × 10^{−04} | 8.27 × 10^{−04} | 4.03 × 10^{−04} | 2.96 × 10^{−03} | 5.51 × 10^{−04} | 3.86 × 10^{−03} |

std | 1.67 × 10^{−04} | 5.57 × 10^{−04} | 3.56 × 10^{−04} | 1.11 × 10^{−02} | 3.59 × 10^{−04} | 7.39 × 10^{−03} | |

F16 | avg | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 |

std | 6.08 × 10^{−16} | 1.42 × 10^{−09} | 5.53 × 10^{−16} | 1.86 × 10^{−14} | 1.15 × 10^{−10} | 6.21 × 10^{−16} | |

F17 | avg | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} |

std | 0 | 8.08 × 10^{−06} | 2.12 × 10^{−15} | 1.11 × 10^{−14} | 8.79 × 10^{−09} | 0 | |

F18 | avg | 3.90 | 3.00 | 12.0 | 3.00 | 3.00 | 5.70 |

std | 4.85 | 4.82 × 10^{−04} | 12.7 | 1.79 × 10^{−13} | 2.21 × 10^{−08} | 8.10 | |

F19 | avg | −3.86 | −3.85 | −3.86 | −3.86 | −3.81 | −3.86 |

std | 2.55 × 10^{−15} | 1.29 × 10^{−02} | 1.97 × 10^{−03} | 9.36 × 10^{−11} | 1.93 × 10^{−01} | 2.65 × 10^{−15} | |

F20 | avg | −3.26 | −3.23 | −3.29 | −3.21 | −3.26 | −3.27 |

std | 5.94 × 10^{−02} | 9.53 × 10^{−02} | 5.06 × 10^{−02} | 5.70 × 10^{−02} | 5.93 × 10^{−02} | 5.93 × 10^{−02} | |

F21 | avg | −7.38 | −8.52 | −9.48 | −7.96 | −5.40 | −5.56 |

std | 3.07 | 2.48 | 1.72 | 2.74 | 1.27 | 3.17 | |

F22 | avg | −8.55 | −8.05 | −9.17 | −8.38 | −5.09 | −5.29 |

std | 2.86 | 3.20 | 2.24 | 3.15 | 2.26 × 10^{−07} | 3.17 | |

F23 | avg | −8.66 | −7.11 | −10.0 | −8.42 | −5.31 | −6.34 |

std | 3.19 | 3.51 | 1.61 | 3.30 | 9.71 × 10^{−01} | 3.52 | |

$best$ $sd-best$ | 4 | 2 | 6 | 4 | 2 | 4 | |

4 | 1 | 1 | 0 | 1 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Jin, Q.; Xu, Z.; Cai, W.
An Improved Whale Optimization Algorithm with Random Evolution and Special Reinforcement Dual-Operation Strategy Collaboration. *Symmetry* **2021**, *13*, 238.
https://doi.org/10.3390/sym13020238

**AMA Style**

Jin Q, Xu Z, Cai W.
An Improved Whale Optimization Algorithm with Random Evolution and Special Reinforcement Dual-Operation Strategy Collaboration. *Symmetry*. 2021; 13(2):238.
https://doi.org/10.3390/sym13020238

**Chicago/Turabian Style**

Jin, Qibing, Zhonghua Xu, and Wu Cai.
2021. "An Improved Whale Optimization Algorithm with Random Evolution and Special Reinforcement Dual-Operation Strategy Collaboration" *Symmetry* 13, no. 2: 238.
https://doi.org/10.3390/sym13020238