# An Interval Type-2 Fuzzy Logic Approach for Dynamic Parameter Adaptation in a Whale Optimization Algorithm Applied to Mathematical Functions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Whale Optimization Algorithm

#### 3.1. Original WOA

Algorithm 1: Pseudo-Code of the WOA Algorithm |

Initialize the whale population Xi (i = 1, 2,……, n) Calculate the fitness of each search agent X* = the best search agent while (t < maximum number of iterations)for each search agent update a, A, C, l, and pif1 (p < 0.5)if2 ($\left|A\right|<1$)Update the position of the current agent by Equation (1) else if2 ($\left|A\right|\ge 1$) Select a random search agent (${X}_{rand}$) Update the position of the current agent by Equation (6) end if2else if1 (p $\ge $ $0.5$) Update the position of the current search by Equation (5) end if1end for Check if any search agent goes beyond the search space and amend it Calculate the fitness of each search agent Update X* if there is a better solution t = t + 1 end whilereturn X* |

#### 3.1.1. Surround Prey

#### 3.1.2. Bubble-Net Attacking Method

#### 3.1.3. Search for Prey

#### 3.2. Fuzzy WOA

#### 3.2.1. Type-1 Fuzzy Logic System

#### 3.2.2. Interval Type-2 Fuzzy System

## 4. Set of Benchmark Functions

## 5. Experimental Results

## 6. Analysis of Results

#### 6.1. Statistical Test

_{a}= µ

_{1}$<$ µ

_{2}, H

_{0}= µ

_{1}$\ge $ µ

_{2}, and critical value of −1.645. The description in each hypothesis is as follows. H

_{o}is the proposed FWOA-IT2FLS method which is greater or equal to the original WOA with random values and H

_{a}denotes that the results of the proposed FWOA-IT2FLS method are smaller (better) than the original WOA. According to the values used in Table 3, Table 4, Table 5, Table 6 and Table 7, a sample of 30 experiments was randomly chosen, which shows a rejection zone for values lower than −1.64. Equation (11) expresses the mathematical function of the z-test:

_{a}.

#### 6.2. Discussion of the Results

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**General representation of a bubble-net search mechanism in WOA (X* is the best solution obtained to date): (

**a**) shrinking encircling mechanism and (

**b**) spiral updating position.

**Figure 9.**Plot of the mathematical functions. (

**a**) Sphere, Griewangk, Rastringin, Shewefel, and Sum of Different Powers. (

**b**) Zakharov, Dixon and Price, Levy, Sum of Squares, and Rotated Hyper Ellipsoid.

${\mathit{f}}_{\mathit{x}}$ | Name | Search Domain | f Min | Mathematical Representation |
---|---|---|---|---|

F1 | Sphere | ${\left[-5.12,5.12\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}^{2}$ |

F2 | Griewangk | ${\left[-600,600\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}\frac{{\mathrm{x}}_{\mathrm{i}}^{2}}{400}-{\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{{\mathrm{x}}_{\mathrm{i}}}{\sqrt{\mathrm{i}}}\right)+1$ |

F3 | Rastringin | ${\left[-5.12,5.12\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)=10\mathrm{n}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}\left[{\mathrm{x}}_{\mathrm{i}}^{2}-10\mathrm{c}\mathrm{o}\mathrm{s}\left(2{\mathrm{x}}_{\mathrm{i}}\right)\right]$ |

F4 | Shewefel | ${\left[-500,500\right]}^{n}$ | −837.9658 | $\mathrm{f}\left(\mathrm{x}\right)=418.9829\mathrm{n}-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{d}\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}\mathrm{sin}\left(\sqrt{\left|{\mathrm{x}}_{\mathrm{i}}\right|}\right)$ |

F5 | Sum of Different Powers | ${\left[-1,1\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\left|{\mathrm{x}}_{\mathrm{i}}\right|}^{\mathrm{i}+1}$ |

F6 | Zakharov | ${\left[-5,10\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}^{2}+({\displaystyle \sum _{\mathrm{i}+1}^{\mathrm{n}}}0.5\mathrm{i}{\mathrm{x}}_{\mathrm{i}}{)}^{2}+({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}0.5\mathrm{i}{\mathrm{x}}_{\mathrm{i}}{)}^{4}$ |

F7 | Dixon and Price | ${\left[-10,10\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\left({\mathrm{x}}_{1}-1\right)}^{2}+{\displaystyle \sum _{\mathrm{i}=2}^{\mathrm{n}}}\mathrm{i}{\left(2{\mathrm{x}}_{\mathrm{i}}^{2}-{\mathrm{x}}_{\mathrm{i}-1}\right)}^{2}$ |

F8 | Levy | ${\left[-10,10\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{s}\mathrm{i}\mathrm{n}}^{2}\left(\mathsf{\pi}{\mathsf{\omega}}_{1}\right)+{\sum}_{\mathrm{i}=1}^{\mathrm{n}-1}{({\mathsf{\omega}}_{1}-1)}^{2}[1+10{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}(\mathsf{\pi}{\mathsf{\omega}}_{1}+1)]+{\left({\mathsf{\omega}}_{\mathrm{n}}-1\right)}^{2}[1+{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}\left(2\mathsf{\pi}{\mathsf{\omega}}_{\mathrm{n}}\right)],\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{\mathsf{\omega}}_{\mathrm{i}}=1+\frac{{\mathrm{x}}_{\mathrm{i}}-1}{4}$, for all i = 1,……,n |

F9 | Sum of Squares | ${\left[-10,10\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{i}{\mathrm{x}}_{\mathrm{i}}^{2}$ |

F10 | Rotated Hyper Ellipsoid | ${\left[-65.536,65536\right]}^{n}$ | 0 | $\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{i}}}{\mathrm{x}}_{\mathrm{j}}^{2}$ |

**Table 2.**Benchmark functions of varying ${\overrightarrow{\mathit{r}}}_{1}$ and ${\overrightarrow{\mathit{r}}}_{2}$ values with 10 dimensions.

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Fixed | Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | FWOA-T1FLS | FWOA-IT2FLS | |

F1 | 6.92 × 10^{−51} | 2.49 × 10^{−1} | 3.41 × 10^{−1} | 6.29 × 10^{−53} |

F2 | 1.61 × 10^{−9} | 6.99 × 10^{+1} | 7.72 × 10^{+6} | 2.91 × 10^{−9} |

F3 | 8.79 × 10^{−10} | 2.15 × 10^{+1} | 1.71 × 10^{+13} | 3.74 × 10^{−10} |

F4 | 2.91 × 10^{−2} | 2.15 × 10^{+1} | 5.00 × 10^{+2} | 9.88 × 10^{+1} |

F5 | 6.35 × 10^{−35} | 8.21 × 10^{−2} | −1.69 × 10^{−15} | 1.86 × 10^{−28} |

F6 | 4.54 × 10^{−1} | 3.06 × 10^{−1} | 8.96 × 10^{+13} | 3.97 × 10^{−1} |

F7 | 1.20 × 10^{−2} | 2.90 × 10^{−2} | 2.90 × 10^{−2} | 1.26 × 10^{−2} |

F8 | 3.97 × 10^{−1} | 1.16 × 10^{0} | 3.84 × 10^{+14} | 6.20 × 10^{−1} |

F9 | 4.94 × 10^{−54} | 1.16 × 10^{−1} | 3.93 × 10^{−32} | 1.22 × 10^{−48} |

F10 | 2.21 × 10^{−52} | 1.01 × 10^{+1} | −6.55 × 10^{+13} | 7.48 × 10^{−44} |

**Table 3.**Benchmark functions of varying ${\overrightarrow{\mathit{r}}}_{1}$ and ${\overrightarrow{\mathit{r}}}_{2}$ values with 20 dimensions.

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Fixed | FWOA-T1FLS | FWOA-IT2FLS | |

F1 | 1.18 × 10^{−70} | 5.26 × 10^{−1} | 1.06 × 10^{−41} | 7.57 × 10^{−60} |

F2 | 2.57 × 10^{−9} | 8.75 × 10^{−1} | 3.33 × 10^{+6} | 1.75 × 10^{−10} |

F3 | 9.47 × 10^{−10} | 6.70 × 10^{−1} | 1.17 × 10^{+13} | 8.22 × 10^{−12} |

F4 | 4.06 × 10^{+2} | 1.33 × 10^{+1} | 1.40 × 10^{+13} | 9.20 × 10^{+1} |

F5 | 4.15 × 10^{−49} | 2.43 × 10^{−2} | 2.25 × 10^{−36} | 2.97 × 10^{−42} |

F6 | 2.17 × 10^{−1} | 3.29 × 10^{−1} | 1.22 × 10^{+13} | 3.88 × 10^{−1} |

F7 | 1.13 × 10^{−2} | 1.06 × 10^{0} | 1.06 × 10^{0} | 1.40 × 10^{−2} |

F8 | 8.45 × 10^{−1} | 1.16 × 10^{0} | 6.27 × 10^{−1} | 5.67 × 10^{−1} |

F9 | 3.17 × 10^{−72} | 4.03 × 10^{−1} | 1.21 × 10^{−52} | 5.25 × 10^{−55} |

F10 | 1.16 × 10^{−71} | 4.19 × 10^{0} | 4.29 × 10^{−53} | 1.25 × 10^{−57} |

**Table 4.**Benchmark functions of varying ${\overrightarrow{\mathit{r}}}_{1}$ and ${\overrightarrow{\mathit{r}}}_{2}$ values with 30 dimensions.

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Fixed | FWOA-T1FLS | FWOA-IT2FLS | |

F1 | 9.58 × 10^{−84} | 2.39 × 10^{−1} | 1.20 × 10^{−51} | 2.84 × 10^{−59} |

F2 | 2.57 × 10^{−9} | 6.35 × 10^{+1} | 5.60 × 10^{+5} | 7.22 × 10^{−9} |

F3 | 1.36 × 10^{−10} | 4.36 × 10^{+2} | 1.17 × 10^{+13} | 1.39 × 10^{−9} |

F4 | 3.76 × 10^{+2} | 4.39 × 10^{+2} | 1.40 × 10^{+13} | 2.23 × 10^{+2} |

F5 | 3.75 × 10^{−54} | 1.03 × 10^{−1} | 4.47 × 10^{−27} | 1.56 × 10^{−42} |

F6 | 5.65 × 10^{−1} | 3.73 × 10^{−1} | 5.65 × 10^{−1} | 1.24 × 10^{−1} |

F7 | 1.39 × 10^{−2} | 4.50 × 10^{−1} | 8.01 × 10^{+5} | 1.31 × 10^{−2} |

F8 | 9.39 × 10^{−1} | 9.35 × 10^{−1} | 4.77 × 10^{+14} | 4.32 × 10^{−1} |

F9 | 6.38 × 10^{−81} | 6.65 × 10^{+1} | −1.28 × 10^{−62} | 1.19 × 10^{−61} |

F10 | 2.14 × 10^{−78} | 4.60 × 10^{0} | −4.37 × 10^{0} | 7.51 × 10^{−66} |

**Table 5.**Benchmark functions of varying ${\overrightarrow{\mathit{r}}}_{1}$ and ${\overrightarrow{\mathit{r}}}_{2}$ values with 50 dimensions.

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Fixed | FWOA-T1FLS | FWOA-IT2FLS | |

F1 | 1.69 × 10^{−93} | −3.57 × 10^{−1} | 1.00 × 10^{−56} | 2.35 × 10^{−60} |

F2 | 5.28 × 10^{−9} | −6.03 × 10^{0} | 1.42 × 10^{6} | 2.15 × 10^{−9} |

F3 | 8.64 × 10^{−10} | 7.55 × 10^{−2} | 2.16 × 10^{5} | 1.96 × 10^{−10} |

F4 | 4.21 × 10^{2} | 1.27 × 10^{2} | 5.00 × 10^{+2} | 3.03 × 10^{+2} |

F5 | 1.55 × 10^{−64} | 1.00 × 10^{−1} | 3.91 × 10^{−38} | 3.50 × 10^{−42} |

F6 | 5.65 × 10^{−1} | 1.71 × 10^{0} | 3.75 × 10^{13} | 4.35 × 10^{−1} |

F7 | 1.12 × 10^{−2} | 5.63 × 10^{−1} | 2.28 × 10^{14} | 1.30 × 10^{−2} |

F8 | 9.93 × 10^{−1} | 7.21 × 10^{−1} | 3.97 × 10^{14} | 5.81 × 10^{−1} |

F9 | 1.14 × 10^{−91} | 6.25 × 10^{−1} | −1.33 × 10^{0} | 4.88 × 10^{−59} |

F10 | 1.64 × 10^{−89} | −2.57 × 10^{0} | −4.37 × 10^{0} | 2.80 × 10^{−59} |

**Table 6.**Benchmark functions of varying ${\overrightarrow{\mathit{r}}}_{1}$ and ${\overrightarrow{\mathit{r}}}_{2}$ values with 100 dimensions.

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Fixed | FWOA-T1FLS | FWOA-IT2FLS | |

F1 | 2.34 × 10^{−101} | 2.68 × 10^{−1} | 5.17 × 10^{−63} | 2.63 × 10^{−59} |

F2 | 4.00 × 10^{−9} | 7.57 × 10^{+1} | 3.44 × 10^{+5} | 4.59 × 10^{−10} |

F3 | 8.19 × 10^{−10} | 4.53 × 10^{−1} | 1.17 × 10^{+13} | 8.94 × 10^{−10} |

F4 | 4.21 × 10^{2} | 5.00 × 10^{2} | 1.40 × 10^{+13} | 4.87 × 10^{+2} |

F5 | 2.40 × 10^{−64} | 7.40 × 10^{−2} | 7.65 × 10^{−79} | 2.67 × 10^{−43} |

F6 | 5.00 × 10^{0} | 1.61 × 10^{0} | 1.62 × 10^{+14} | 1.29 × 10^{0} |

F7 | 1.12 × 10^{−2} | 1.33 × 10^{0} | 1.33 × 10^{0} | 1.20 × 10^{−2} |

F8 | 1.00 × 10^{0} | 6.78 × 10^{−1} | 7.05 × 10^{+14} | 4.38 × 10^{−1} |

F9 | 2.52 × 10^{−102} | 2.91 × 10^{0} | 3.33 × 10^{−1} | 7.47 × 10^{−59} |

F10 | 1.03 × 10^{−98} | −2.17 × 10^{0} | 2.53 × 10^{−80} | 2.01 × 10^{−60} |

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

FWOA-IT2FLS | Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | Z-Value | Evidence | |

F1 | −2.63 × 10^{−59} | 2.34 × 10^{−101} | −1.265 | NS |

F2 | 4.59 × 10^{−10} | 4.00 × 10^{−9} | −1.004 | NS |

F3 | 8.94 × 10^{−10} | 8.19 × 10^{−10} | −3.531 | S |

F4 | 4.87 × 10^{+2} | 4.21 × 10^{+2} | −97.494 | S |

F5 | 2.67 × 10^{−43} | 2.40 × 10^{−64} | 1.281 | NS |

F6 | 1.29 × 10^{0} | 5.00 × 10^{0} | −0.016 | NS |

F7 | 1.20 × 10^{−2} | 1.12 × 10^{−2} | 0.0543 | NS |

F8 | 4.38 × 10^{−1} | 1.00 × 10^{0} | 77.992 | NS |

F9 | 7.47 × 10^{−59} | −2.52 × 10^{−102} | −0.852 | NS |

F10 | 2.01 × 10^{−60} | −1.03 × 10^{−98} | −1.0054 | NS |

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

FWOA-T1FLS | Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | Z-Value | Evidence | |

F1 | 5.71 × 10^{−63} | 2.34 × 10^{−101} | 0.560 | NS |

F2 | 3.44 × 10^{+5} | 4.00 × 10^{−9} | 3.085 | NS |

F3 | 1.17 × 10^{−13} | 8.19 × 10^{−10} | −2.902 | S |

F4 | 1.40 × 10^{+13} | 4.21 × 10^{+2} | 591.500 | NS |

F5 | 7.65 × 10^{−79} | 2.40 × 10^{−64} | −1.600 | S |

F6 | 1.62 × 10^{+14} | 5.00 × 10^{0} | −0.183 | NS |

F7 | 1.33 × 10^{0} | 1.12 × 10^{−2} | −1.002 | NS |

F8 | 7.05 × 10^{+14} | 1.00 × 10^{0} | −1.686 | S |

F9 | 3.33 × 10^{−1} | −2.52 × 10^{−102} | −1.000 | NS |

F10 | 2.53 × 10^{−80} | −1.03 × 10^{−98} | 3.122 | NS |

${\mathit{f}}_{\mathit{x}}$ | Methods | |||
---|---|---|---|---|

FWOA-IT2FLS | FWOA-T1FLS | Z-Value | Evidence | |

F1 | 2.63 × 10^{−59} | 9.04 × 10^{−62} | −1.265 | NS |

F2 | −4.59 × 10^{−10} | 3.44 × 10^{+5} | −3.09 | S |

F3 | −8.94 × 10^{−10} | −2.02 × 10^{+5} | 2.902 | NS |

F4 | −4.87 × 10^{+2} | −5.02 × 10^{+2} | 1.4388 | NS |

F5 | 2.67 × 10^{−43} | 7.65 × 10^{−79} | 1.281 | NS |

F6 | 1.29 × 10^{0} | 5.00 × 10^{0} | 0.166 | NS |

F7 | 1.20 × 10^{−2} | 1.22 × 10^{−2} | 1.807 | NS |

F8 | 4.38 × 10^{−1} | 9.69 × 10^{−1} | −8.242 | S |

F9 | 7.47 × 10^{−59} | 3.33 × 10^{−1} | 1.000 | NS |

F10 | 2.01 × 10^{−60} | 2.53 × 10^{−80} | −1.005 | NS |

${\mathit{f}}_{\mathit{x}}$ | Methods | |||||||
---|---|---|---|---|---|---|---|---|

Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | Fuzzy WOA-T1FLS | Fuzzy WOA-IT2FLS | FBCO-IT2FLS [61] | |||||

Best | Worst | Best | Worst | Best | Worst | Best | Worst | |

F1 | 1.60 × 10^{−97} | 1.98 × 10^{−92} | 5. 43 × 10^{−55} | 2. 23 × 10^{−55} | 1.74 × 10^{−59} | 2.94 × 10^{−60} | 7.74 × 10^{−8} | 6.94 × 10^{−7} |

F2 | 3.57 × 10^{−8} | 1.32 × 10^{−8} | 1.04 × 10^{−8} | 2.34 × 10^{7} | 3.73 × 10^{−8} | 4.02 × 10^{−8} | 8.40 × 10^{−5} | 3.26 × 10^{−2} |

F3 | 4.81 × 10^{−9} | 9.58 × 10^{−9} | 4.67 × 10^{+4} | 8.69 × 10^{+5} | 4.78 × 10^{−9} | 8.90 × 10^{−9} | 7.52 × 10^{0} | 1.51 × 10^{+1} |

F4 | 4.20 × 10^{2} | 4.22 × 10^{+2} | 5.00 × 10^{2} | 5.00 × 10^{+2} | 3.03 × 10^{+2} | 3.03 × 10^{+2} | 1.01 × 10^{3} | 2.47 × 10^{+3} |

F5 | 3.92 × 10^{−63} | 3.24 × 10^{−94} | 5.00 × 10^{2} | 1.35 × 10^{−36} | 1.77 × 10^{−86} | 7.93 × 10^{−41} | 1.87 × 10^{−10} | 1.08 × 10^{−6} |

F6 | 4.92 × 10^{0} | 9.24 × 10^{0} | 8.52 × 10^{+14} | 7.69 × 10^{+14} | 4.99 × 10^{0} | 8.53 × 10^{0} | 5.46 × 10^{2} | 7.20 × 10^{+2} |

F7 | 2.18 × 10^{−6} | 3.33 × 10^{−1} | 1.13 × 10^{+8} | 1.00 × 10^{+1} | 4.14 × 10^{−4} | 3.18 × 10^{−1} | 1.76 × 10^{0} | 9.06 × 10^{0} |

F8 | 7.72 × 10^{−1} | 1.03 × 10^{0} | 1.01 × 10^{+14} | 9.98 × 10^{+14} | 9.63 × 10^{−2} | 1.22 × 10^{0} | 6.37 × 10^{−7} | 1.04 × 10^{−5} |

F9 | 2.96 × 10^{−90} | 2.56 × 10^{−90} | 1.00 × 10^{+1} | 1.63 × 10^{−7} | 1.70 × 10^{−57} | 7.68 × 10^{−58} | 5.10 × 10^{6} | 1.29 × 10^{−4} |

F10 | 2.56 × 10^{−89} | 2.88 × 10^{−88} | 6.55 × 10^{+1} | 5.68 × 10^{−7} | 6.29 × 10^{−59} | 5.96 × 10^{−58} | 1.35 × 10^{−4} | 2.44 × 10^{−3} |

${\mathit{f}}_{\mathit{x}}$ | Methods | ||
---|---|---|---|

Original WOA ${\overrightarrow{\mathit{r}}}_{1}$$\mathbf{and}{\overrightarrow{\mathit{r}}}_{2}$ Random | FWOA-IT2FLS | FBCO-IT2FLS [61] | |

F1 | 2.34 × 10^{−101} | 2.63 × 10^{−59} | 1.74 × 10^{−7} |

F2 | 4.00 × 10^{−9} | 1.36 × 10^{−8} | 6.12 × 10^{−3} |

F3 | 8.19 × 10^{−10} | 8.14 × 10^{−10} | 1.98 × 10^{0} |

F4 | 4.21 × 10^{2} | 3.37 × 10^{−12} | 3.59 × 10^{+2} |

F5 | 2.40 × 10^{−64} | 2.67 × 10^{−43} | 2.46 × 10^{−7} |

F6 | 5.00 × 10^{0} | 1.29 × 10^{0} | 3.89 × 10^{+1} |

F7 | 1.12 × 10^{−2} | 1.20 × 10^{−2} | 1.76 × 10^{0} |

F8 | 1.00 × 10^{0} | 4.38 × 10^{−1} | 1.91 × 10^{−6} |

F9 | 2.52 × 10^{−102} | 7.47 × 10^{−59} | 2.50 × 10^{−5} |

F10 | 1.03 × 10^{−98} | 2.01 × 10^{−60} | 6.83 × 10^{−4} |

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**MDPI and ACS Style**

Amador-Angulo, L.; Castillo, O.
An Interval Type-2 Fuzzy Logic Approach for Dynamic Parameter Adaptation in a Whale Optimization Algorithm Applied to Mathematical Functions. *Axioms* **2024**, *13*, 33.
https://doi.org/10.3390/axioms13010033

**AMA Style**

Amador-Angulo L, Castillo O.
An Interval Type-2 Fuzzy Logic Approach for Dynamic Parameter Adaptation in a Whale Optimization Algorithm Applied to Mathematical Functions. *Axioms*. 2024; 13(1):33.
https://doi.org/10.3390/axioms13010033

**Chicago/Turabian Style**

Amador-Angulo, Leticia, and Oscar Castillo.
2024. "An Interval Type-2 Fuzzy Logic Approach for Dynamic Parameter Adaptation in a Whale Optimization Algorithm Applied to Mathematical Functions" *Axioms* 13, no. 1: 33.
https://doi.org/10.3390/axioms13010033