# Design of a Predictive Model of Rock Breakage by Blasting Using Artificial Neural Networks

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## Abstract

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## 1. Introduction

## 2. Theoretical Frame

- Controllable variables: Explosives, Geometric blasting design and Startup sequences.
- Uncontrollable variables: Geological and Geomechanical characteristics of the rock mass.

#### 2.1. Kuz–Ram Equation

- ${X}_{50}$ = Percentage of passing fragments less than 50%.A = Rock factor.${Q}_{e}$ = Explosive mass per drill.E = Relative weight Strength of explosive${V}_{o}/{Q}_{e}$ = Volume per kg of explosive.

#### 2.2. Use of the Artificial Neural Network (Ann)

#### 2.3. Design of a Feedforward Neural Network (Fnn) for the Case Study

#### 2.3.1. Number of Layers

#### 2.3.2. Number of Neurons in Each Layer

#### 2.3.3. Initialization of Weight

#### 2.3.4. Activation Function of Each Layer

- f: Sigmoid functionx: variable

#### 2.3.5. Training Algorithm

- ${W}_{t}$: Weights$\alpha $: Coefficient of friction$\gamma $: Learning rate$\nabla f$: Gradient of the function f

#### 2.4. Multiple Linear Regression (Mlr)

- ${\beta}_{0}$: is the ordinate in the origin, namely is the value of the dependent variable Y when all the predictors are zero.${\beta}_{i}$: is the average effect that the increase in one unit of the predictor variable ${X}_{i}$ has on the dependent variable Y, holding all else constant. This are known as partial regression coefficients.${e}_{i}$: is the residual or error, namely the difference between the observed value and the one estimated by the model.

## 3. Methodology for the Design of the Ann Computer Model

## 4. Collection of Field Data

## 5. Design and Experimentation of ANN

- $\sum _{i=0}^{n}{C}_{N}^{i}=\frac{N!}{i!(N-i)!}>K$n: number of input parameters = 8K: used dataset number = 47N: number of hidden neurons to be determinedResulting N > 8. Therefore: 9 ≤ N ≤ 17.

## 6. Experimental Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | linear dichroism |

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**Figure 1.**Environmental impacts vs. degree of Breakage [20].

**Figure 2.**Operating costs versus blasting costs [21].

**Figure 3.**Unit cost of operations vs. cost f blasting [22].

**Figure 10.**${P}_{80}$ Breakage vs. ANN model vs. the Multiple Linear Regression (MLR) model (training).

**Table 1.**Parameters used in ANN [29].

B(m) | S(m) | GU | Mineral Density (t/m^{3}) | Diameter (Inches) | Bench (m) | Overdrilling (m) | Stemming (m) | D.Expl (ton/m^{3}) | Kg.Expl | Powder Factor (kg/ton) | P_{80} | P_{50} | P_{20} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

9.3 | 10.7 | 4 | 2.45 | 10.625 | 15 | 2 | 6.5 | 1.3 | 709 | 194 | 83.4 | 47.2 | 12.3 |

9.3 | 10.7 | 4 | 2.45 | 10.625 | 15 | 2 | 6.5 | 1.3 | 709 | 194 | 91.8 | 51 | 12.81 |

9.3 | 10.7 | 3 | 2.35 | 10.625 | 15 | 2 | 6.5 | 1.3 | 709 | 202 | 86.3 | 47.9 | 9.34 |

9.3 | 10.7 | 3 | 2.35 | 10.625 | 15 | 2 | 6.5 | 1.3 | 709 | 202 | 88.3 | 47.9 | 9.76 |

9.3 | 10.7 | 4 | 2.45 | 10.625 | 15 | 2 | 6.5 | 1.3 | 709 | 194 | 80.5 | 44.3 | 7.9 |

8 | 8 | 5 | 2.48 | 10.625 | 14 | 0 | 7 | 1.2 | 436 | 196 | 74.1 | 39.9 | 4.21 |

8.8 | 10.2 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 225 | 70.3 | 38.6 | 7.61 |

8.8 | 10.2 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 225 | 75.6 | 40.7 | 8.04 |

9 | 12 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 187 | 63.4 | 37.8 | 6.7 |

9 | 12 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 187 | 80.6 | 45.6 | 8.28 |

9 | 12 | 3 | 2.35 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 195 | 94.4 | 51.6 | 14.08 |

9 | 12 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 187 | 76.6 | 45.7 | 11.89 |

9 | 12 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 187 | 86 | 51 | 13.42 |

9 | 12 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 187 | 82.3 | 47.8 | 11.5 |

9 | 12 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 187 | 66.2 | 39.5 | 8.35 |

9 | 12 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 187 | 87.3 | 49.1 | 7.8 |

9 | 12 | 3 | 2.35 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 195 | 71.6 | 42.2 | 5.97 |

9 | 12 | 3 | 2.35 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 195 | 72.3 | 41.5 | 4.74 |

10 | 13 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 155 | 61.9 | 37.4 | 7.07 |

10 | 13 | 3 | 2.35 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 162 | 60.9 | 36.8 | 6.54 |

10 | 13 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 155 | 69.5 | 40.3 | 5.66 |

10 | 13 | 4 | 2.45 | 10.625 | 15 | 2 | 6 | 1.3 | 743 | 155 | 84 | 48.1 | 8.19 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 260.5 | 135.9 | 43.6 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 225.9 | 119.8 | 37.3 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 237.6 | 131.6 | 42.3 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 222.9 | 118.1 | 36.8 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 232.1 | 128.8 | 41.3 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 216.2 | 119 | 37.8 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 236.3 | 121.3 | 38.5 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 161 | 85.1 | 24.8 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 218.8 | 127.6 | 40.8 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 218.5 | 119.5 | 37.5 |

7 | 8 | 5 | 2.48 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 318 | 184.4 | 99.8 | 30.3 |

7 | 8 | 6 | 2.57 | 10.625 | 15 | 2 | 7.5 | 1.22 | 663 | 307 | 180 | 103.2 | 31.6 |

8.8 | 10.2 | 5 | 2.48 | 10.625 | 15 | 2 | 6 | 1.22 | 768 | 230 | 276.6 | 152.6 | 54.6 |

8.8 | 10.2 | 6 | 2.57 | 10.625 | 15 | 2 | 6 | 1.22 | 768 | 222 | 234.2 | 122.9 | 42.8 |

8.8 | 10.2 | 6 | 2.57 | 10.625 | 15 | 2 | 6 | 1.22 | 768 | 222 | 178.2 | 85.2 | 26.1 |

6 | 7 | 6 | 2.57 | 10.625 | 15 | 2 | 6 | 1.22 | 719 | 444 | 194.8 | 102.7 | 24.1 |

6 | 7 | 5 | 2.48 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 460 | 140.3 | 79.8 | 21.2 |

6 | 7 | 5 | 2.48 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 460 | 272.9 | 141.7 | 24.1 |

6 | 7 | 5 | 2.48 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 460 | 192.1 | 91.8 | 24.7 |

6 | 7 | 6 | 2.47 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 444 | 314.7 | 179.3 | 62.4 |

6 | 7 | 6 | 2.57 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 444 | 348 | 199 | 60.3 |

6 | 7 | 6 | 2.57 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 444 | 322.1 | 179.2 | 51.6 |

6 | 7 | 6 | 2.57 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 444 | 220.9 | 108.7 | 31.4 |

6 | 7 | 6 | 2.57 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 444 | 288.8 | 152.1 | 36.1 |

6 | 7 | 6 | 2.57 | 10.625 | 15 | 2 | 6.7 | 1.22 | 719 | 444 | 241.2 | 127.6 | 38.5 |

Mean Squared Error by Number of Simulations | |||||||||
---|---|---|---|---|---|---|---|---|---|

Number of Hidden Neurons | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Average |

10 | 0.013934 | 0.010181 | 0.010772 | 0.011604 | 0.013516 | 0.010188 | 0.009633 | 0.014887 | 0.011839 |

11 | 0.014779 | 0.010263 | 0.00894 | 0.0091 | 0.009728 | 0.010336 | 0.011956 | 0.0098 | 0.010613 |

12 | 0.011681 | 0.009246 | 0.009824 | 0.012855 | 0.010517 | 0.011888 | 0.008056 | 0.010981 | 0.010631 |

13 | 0.012474 | 0.011985 | 0.008593 | 0.008027 | 0.008891 | 0.008518 | 0.00879 | 0.00918 | 0.009557 |

14 | 0.009478 | 0.00777 | 0.012028 | 0.008874 | 0.012441 | 0.010617 | 0.010151 | 0.01007 | 0.010179 |

15 | 0.007212 | 0.008788 | 0.009467 | 0.012912 | 0.011154 | 0.00952 | 0.011585 | 0.010223 | 0.010108 |

**Table 3.**Comparison table real ${P}_{80}$, ANN ${P}_{80}$ and MLR ${P}_{80}$ (Training and Testing).

Training | Testing | |||||
---|---|---|---|---|---|---|

Statistical parameters | ${P}_{80}$ | ${P}_{80}$ | ${P}_{80}$ | ${P}_{80}$ | ${P}_{80}$ | ${P}_{80}$ |

(Real) | (ANN) | (MLR) | (Real) | (ANN) | (MLR) | |

Correlation coefficient ${R}^{2}$ | 0.87 | 0.85 | 0.81 | 0.78 | ||

Mean (mm) | 148.1 | 150.2 | 148.11 | 204.61 | 195.36 | 193.65 |

Standard deviation | 85.82 | 80.8 | 78.98 | 68.86 | 61.99 | 63.27 |

Coefficient of variation | 0.58 | 0.54 | 0.53 | 0.33 | 0.32 | 0.32 |

Training | Testing | |||||
---|---|---|---|---|---|---|

Statistical parameters | ${P}_{50}$ | ${P}_{50}$ | ${P}_{50}$ | ${P}_{50}$ | ${P}_{50}$ | ${P}_{50}$ |

(Real) | (ANN) | (MLR) | (Real) | (ANN) | (MLR) | |

Correlation coefficient ${R}^{2}$ | 0.83 | 0.79 | 0.79 | 0.79 | ||

Mean (mm) | 81.36 | 85.03 | 81.36 | 109.37 | 107.47 | 105.06 |

Standard deviation | 46.53 | 45.34 | 41.59 | 35.87 | 36.56 | 33.91 |

Coefficient of variation | 0.57 | 0.53 | 0.51 | 0.32 | 0.34 | 0.32 |

Training | Testing | |||||
---|---|---|---|---|---|---|

Statistical parameters | ${P}_{20}$ | ${P}_{20}$ | ${P}_{20}$ | ${P}_{20}$ | ${P}_{20}$ | ${P}_{20}$ |

(Real) | (ANN) | (MLR) | (Real) | (ANN) | (MLR) | |

Correlation coefficient ${R}^{2}$ | 0.82 | 0.78 | 0.78 | 0.78 | ||

Mean (mm) | 22.38 | 22.93 | 22.38 | 32.45 | 32.49 | 32.01 |

Standard deviation | 17.09 | 15.27 | 15.15 | 12.92 | 12.13 | 12.65 |

Coefficient of variation | 0.76 | 0.66 | 0.67 | 0.39 | 0.37 | 0.39 |

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## Share and Cite

**MDPI and ACS Style**

Rosales-Huamani, J.A.; Perez-Alvarado, R.S.; Rojas-Villanueva, U.; Castillo-Sequera, J.L. Design of a Predictive Model of Rock Breakage by Blasting Using Artificial Neural Networks. *Symmetry* **2020**, *12*, 1405.
https://doi.org/10.3390/sym12091405

**AMA Style**

Rosales-Huamani JA, Perez-Alvarado RS, Rojas-Villanueva U, Castillo-Sequera JL. Design of a Predictive Model of Rock Breakage by Blasting Using Artificial Neural Networks. *Symmetry*. 2020; 12(9):1405.
https://doi.org/10.3390/sym12091405

**Chicago/Turabian Style**

Rosales-Huamani, Jimmy Aurelio, Roberth Saenz Perez-Alvarado, Uwe Rojas-Villanueva, and Jose Luis Castillo-Sequera. 2020. "Design of a Predictive Model of Rock Breakage by Blasting Using Artificial Neural Networks" *Symmetry* 12, no. 9: 1405.
https://doi.org/10.3390/sym12091405