# Estimating Flyrock Distance Induced Due to Mine Blasting by Extreme Learning Machine Coupled with an Equilibrium Optimizer

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## Abstract

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## 1. Introduction

## 2. Models for the Prediction of Flyrock

#### 2.1. Empirical Models for the Prediction of Flyrock

**Figure 1.**Schematic diagram of a mechanism of flyrock [56].

#### 2.2. Mathematical Models for the Prediction of Flyrock

#### 2.3. Semi-Empirical Trajectory Physics-Based Models for the Prediction of Flyrock

#### 2.4. Artificial Intelligence Techniques

## 3. Background of Model

#### 3.1. Extreme Learning Machine (ELM)

#### Essence of ELM

- Randomly assign the hidden node’s parameter
- Calculate the output matrix of the hidden layer
- Compute the output weights.

#### 3.2. Artificial Neural Network

- Transfer Function
- Network Architecture
- Learning Law.

#### 3.3. Equilibrium Optimization

#### 3.4. Particle Swarm Optimization (PSO)

#### 3.5. Case Study and Data Collection

## 4. Model Development

#### 4.1. Hybridization of PSO-ANN

- Personal experiences of individuals that give their best results.
- Experiences of other individuals that give the best results of the entire swarms.

#### 4.2. Hybridization of PSO-ELM

#### 4.3. Hybridization of EO-ELM

Algorithm 1: The algorithm of EO-ELM. Exponential term (F),
λ is a turnover rate and defined as a random vector in between 0 and 1, a_{2} is used to control the exploitation task. a_{1} is used to control the exploration task,
$\mathit{s}\mathit{i}\mathit{g}\mathit{n}\mathbf{(}\overrightarrow{\mathit{r}}\mathbf{-}\mathbf{0.5}\mathbf{)}$ component consequences the direction of intensification and diversification of particles, r is defined as a random vector in between 0 and 1, generation rate (G), r_{1,} and r_{2} denote the random values between 0 and 1. GCP is called generation rate control parameter |

1. Select training and testing dataset 2. Begin ELM training 3. Set hidden units of ELM 4. Obtain the number of input weights and hidden biases 5. Initialize the populations (P) 6. Initialize the fitness of four equilibrium candidates 7. Assignment of EO parameters value (a _{1} = 2, a_{2} = 1, GP = 0.5)8. for it = 1 to maximum iteration number do 9. for i = 1 to P do 10. Estimate the fitness of the i ^{th} particle11. if fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[1\right]}$) 12. Replace fitness (${\overrightarrow{P}}_{eq\left[1\right]}$) with fitness (${\overrightarrow{P}}_{i}$) and ${\overrightarrow{P}}_{eq\left[1\right]}$ with ${\overrightarrow{P}}_{i}$ 13. elseif fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[1\right]}$) & fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[2\right]}$) 14. Replace fitness (${\overrightarrow{P}}_{eq\left[2\right]}$) with fitness (${\overrightarrow{P}}_{i}$) and ${\overrightarrow{P}}_{eq\left[2\right]}$ with ${\overrightarrow{P}}_{i}$ 15. elseif fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[1\right]}$) & fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[2\right]}$) & fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[3\right]}$) 16. Replace fitness (${\overrightarrow{P}}_{eq\left[3\right]}$) with fitness (${\overrightarrow{P}}_{i}$) and ${\overrightarrow{P}}_{eq\left[3\right]}$ with ${\overrightarrow{P}}_{i}$ 17. elseif fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[1\right]}$) & fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[2\right]}$) & fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[3\right]}$) & fitness (${\overrightarrow{P}}_{i}$) < fitness (${\overrightarrow{P}}_{eq\left[4\right]}$) 18. Replace fitness (${\overrightarrow{P}}_{eq\left[4\right]}$) with fitness (${\overrightarrow{P}}_{i}$) and ${\overrightarrow{P}}_{eq\left[4\right]}$ with ${\overrightarrow{P}}_{i}$ 19. end if 20. end for 21. ${\overrightarrow{P}}_{mean}$ = $({\overrightarrow{P}}_{eq\left[1\right]}+{\overrightarrow{P}}_{eq\left[2\right]}+{\overrightarrow{P}}_{eq\left[3\right]}+{\overrightarrow{P}}_{eq\left[4\right]})/4$ 22. ${\overrightarrow{P}}_{eq,pool}=\{{\overrightarrow{P}}_{eq\left[1\right]}+{\overrightarrow{P}}_{eq\left[2\right]}+{\overrightarrow{P}}_{eq\left[3\right]}+{\overrightarrow{P}}_{eq\left[4\right]}+{\overrightarrow{P}}_{eq\left[mean\right]}\}$ (Equilibrium pool) 23. Allocate $t={\left(1-\frac{Iteration}{Ma{x}_{iteration}}\right)}^{\left({a}_{2}\times \frac{Iteration}{Ma{x}_{iteration}}\right)}$ 24. for i = 1 to P do 25. Random generation of vectors $\overrightarrow{\lambda}$ and $\overrightarrow{r}$ 26. Random selection of equilibrium candidate from equilibrium pool 27. Evaluate $\overrightarrow{F}={a}_{1}\times sign(\overrightarrow{r}-0.5)[{e}^{-\overrightarrow{\lambda}t}-1]$ 28. Evaluate $\overrightarrow{GCP}=\{\begin{array}{c}0.5\ast {r}_{1}{r}_{2}\ge GP\\ 0{r}_{2}<GP\end{array}$ 29. Evaluate ${\overrightarrow{G}}_{0}=\overrightarrow{GCP}\ast ({\overrightarrow{P}}_{eq}-\overrightarrow{\lambda}\times \overrightarrow{P})$ 30. Evaluate $\overrightarrow{G}={\overrightarrow{G}}_{0}\times \overrightarrow{F}$ 31. $\overrightarrow{P}={\overrightarrow{P}}_{eq}+(\overrightarrow{P}-{\overrightarrow{P}}_{eq})\xb7\overrightarrow{F}+\frac{\overrightarrow{G}}{\overrightarrow{\lambda}V}\ast (1-\overrightarrow{F})$ (Concentration update) 32. end for 33. end for 34. Set ELM optimal input weights and hidden biases using ${\overrightarrow{P}}_{eq\left[1\right]}$ 35. Obtain output weights 36. ELM testing |

#### 4.4. Model Verification and Evaluation

## 5. Results and Discussion

#### 5.1. Average Performance of Models

#### 5.2. Anderson–Darling (A–D) Test

#### 5.3. Sensitivity Analysis

_{i}is a vector of length m in array Z, that as Equation (9):

_{ij}) between the data set Z

_{i}and Z

_{j}as Equation (10):

## 6. Conclusions

^{2}, RMSE, MAPE, NSE, MAE, VAF, and A20) are used for comparing the efficacy of the developed model. The developed EO-ELM model performed better compared to PSO-the ELM and PSO-ANN in predicting flyrock (Table 4). Further, all models were run 10-times and average results are shown in Table 5. It was observed that the EO-ELM model outperformed the PSO-ELM and PSO-ANN in average results. Furthermore, the 10-times run of the EO-ELM model showed better convergence capability (Figure 17) than the other two. Further, the A–D test showed that the EO-ELM model has better performance efficiency compared to the PSO-ELM and PSO-ANN. A sensitivity analysis was done introducing a new parameter, WI. The PF and BI showed the highest sensitivity with 0.98 each (Figure 18).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Standard flow chart of PSO [85].

**Figure 4.**Location map of Thailand quarry [86].

Parameters | Hole Diameter | Burden | Stemming Length | Rock Density | Charge per M | Powder Factor | Blastability Index | Weathering Index | Flyrock |
---|---|---|---|---|---|---|---|---|---|

Symbol | D | B | ST | ρ | CPM | PF | BI | WI | FR |

Unit | mm | m | m | Cum.t | kg/m | kg/cum | % | Ratio | m |

Minimum | 76 | 2.5 | 1.2 | 1.8 | 4.54 | 0.08 | 18.5 | 0.13 | 27 |

Quartile1 | 76 | 2.7 | 2 | 1.8 | 4.99 | 0.19 | 28.6 | 0.25 | 37 |

Average | 90 | 3 | 2 | 2 | 7 | 0.30 | 43 | 0.76 | 81 |

Quartile3 | 102 | 3.6 | 2.95 | 2.5 | 8.99 | 0.40 | 54.6 | 0.88 | 82 |

Maximum | 102 | 4.6 | 4 | 2.5 | 9.4 | 0.50 | 80.8 | 0.99 | 436 |

Model | Parameters | Value |
---|---|---|

EO-ELM | Maximum Iteration | 500 |

Size of Population | 25 | |

a1 | 2.5 | |

a2 | 2.5 | |

GP | 0.6 | |

PSO-ELM | Maximum Iteration | 500 |

Size of Population | 25 | |

C1 | 1 | |

C2 | 2 | |

W (inertia weight) | 0.9 | |

PSO-ANN | Maximum Iteration | 500 |

Size of Population | 25 | |

C1 | 1 | |

C2 | 2 | |

W | 0.98 |

**Table 3.**Linear equations for predicted and measured values for the EO-ELM, PSO-ANN, and PSO-ELM for training and testing data.

Model | Training Data | Testing Data |
---|---|---|

EO-ELM | 67.10x + 73.93 | 96.66x + 100.59 |

PSO-ANN | 58.06x + 74.58 | 88.9x + 92.43 |

PSO-ELM | 64.58x + 73.92 | 99.10x + 98.85 |

Training Data Sets | |||||||

R^{2} | RMSE | MAE | MAPE | NSE | VAF | A20 | |

EO-ELM | 0.942 | 17.02 | 11.26 | 21.20 | 0.946 | 94.62 | 0.53 |

PSO-ANN | 0.827 | 29.5 | 21.07 | 27.04 | 0.821 | 82.19 | 0.43 |

PSO-ELM | 0.907 | 21.56 | 15.64 | 24.27 | 0.900 | 90.08 | 0.45 |

Testing Data Sets (Continued) | |||||||

R^{2} | RMSE | MAE | MAPE | NSE | VAF | A20 | |

EO-ELM | 0.973 | 34.82 | 20.3 | 17.60 | 0.978 | 97.88 | 0.65 |

PSO-ANN | 0.924 | 48.12 | 31.68 | 24.25 | 0.93 | 92.89 | 0.35 |

PSO-ELM | 0.959 | 35.7 | 23.53 | 21.84 | 0.96 | 95.79 | 0.56 |

Training Data Sets | |||||||

R^{2} | RMSE | MAE | MAPE | NSE | VAF | A20 | |

EO-ELM | 0.95 | 16.66 | 12.13 | 19.87 | 0.95 | 94.46 | 0.60 |

PSO-ANN | 0.83 | 29.68 | 19.33 | 29.30 | 0.82 | 82.06 | 0.41 |

PSO-ELM | 0.88 | 23.68 | 16.76 | 26.96 | 0.88 | 88.29 | 0.47 |

Testing Data Sets (Continued) | |||||||

R^{2} | RMSE | MAE | MAPE | NSE | VAF | A20 | |

EO-ELM | 0.97 | 32.14 | 19.78 | 20.37 | 0.93 | 93.97 | 0.57 |

PSO-ANN | 0.87 | 64.44 | 36.02 | 29.96 | 0.72 | 74.72 | 0.33 |

PSO-ELM | 0.88 | 48.55 | 26.97 | 26.71 | 0.84 | 84.84 | 0.51 |

Count | Mean | Median | SD | AD | p-Value | |
---|---|---|---|---|---|---|

Actual | 114 | 81.307 | 50.5 | 85.927 | 0 | 1 |

PSO-ELM | 114 | 78.951 | 54.632 | 75.877 | 2.843 | 0.03244 |

PSO-ANN | 114 | 78.183 | 55.035 | 70.484 | 2.308 | 0.00619 |

EO-ELM | 114 | 79.311 | 53.492 | 76.092 | 0.8886 | 0.004215 |

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## Share and Cite

**MDPI and ACS Style**

Bhatawdekar, R.M.; Kumar, R.; Sabri Sabri, M.M.; Roy, B.; Mohamad, E.T.; Kumar, D.; Kwon, S.
Estimating Flyrock Distance Induced Due to Mine Blasting by Extreme Learning Machine Coupled with an Equilibrium Optimizer. *Sustainability* **2023**, *15*, 3265.
https://doi.org/10.3390/su15043265

**AMA Style**

Bhatawdekar RM, Kumar R, Sabri Sabri MM, Roy B, Mohamad ET, Kumar D, Kwon S.
Estimating Flyrock Distance Induced Due to Mine Blasting by Extreme Learning Machine Coupled with an Equilibrium Optimizer. *Sustainability*. 2023; 15(4):3265.
https://doi.org/10.3390/su15043265

**Chicago/Turabian Style**

Bhatawdekar, Ramesh Murlidhar, Radhikesh Kumar, Mohanad Muayad Sabri Sabri, Bishwajit Roy, Edy Tonnizam Mohamad, Deepak Kumar, and Sangki Kwon.
2023. "Estimating Flyrock Distance Induced Due to Mine Blasting by Extreme Learning Machine Coupled with an Equilibrium Optimizer" *Sustainability* 15, no. 4: 3265.
https://doi.org/10.3390/su15043265