# A Novel Large-Scale Stochastic Pushback Design Merged with a Minimum Cut Algorithm for Open Pit Mine Production Scheduling

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## Abstract

**:**

## 1. Introduction

## 2. Literature Survey

#### Stochastic Mine Planning under Uncertainties

## 3. Proposed Methodology

#### 3.1. Stochastic Pushback Design Formulation

#### 3.1.1. Stochastic Graph Closure with Multiple Scenarios

- 1.
- Set a parameter $\lambda $ such that 0 < $\lambda $ < 1.
- 2.
- Start $\lambda $ = 0.
- 3.
- Assign best production
^{k}_{s}= 0 at λ = 0 for all resource limitations $k$ and scenario $s$. - 4.
- Increase λ by a small value ∆λ, i.e., $\lambda =\lambda +\Delta \lambda $
- 5.
- Solve $\Phi \left(\lambda \right)$.
- 6.
- If ($bestproductio{n}^{k}{}_{s}\le {b}^{k},\forall s,\forall k$) from a solution of $\Phi \left(\lambda \right)$,Update λ value by $\Delta \lambda i.e.,\lambda =\lambda +\Delta \lambda $;best production
^{k}_{s}$={\sum}_{i=1}^{N}{a}^{k}{}_{i,s}{x}_{i}$. - 7.
- Go to step 5.
- 8.
- End.

^{th}resource constraints for grade scenario s and ε is a small constant number. Since the values of b

^{k}are constant and the values of best production

^{k}

_{s}are increased after each iteration, Equation (10) ensures that $\nabla \lambda $ value will have a monotonically decreasing function. When the algorithm of the pseudo-code is close to violating the resource constraints, the $\nabla \lambda $ value will be incremented by a small magnitude.

## 4. Results and Discussion

## 5. Conclusions and Future Scope

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Graph constructions after merging three block economic models generated using multiple scenarios.

Statistics | Sample No. | Mean (%) | Variance (% ^{2}) | Coefficient of Variation | Skewness | Kurtosis |
---|---|---|---|---|---|---|

Iron | 1456 | 63.85 | 64.52 | 12.58 | −2.74 | 8.39 |

Description | Value |
---|---|

Slope angle (degree) | 45° |

Block dimensions (m × m × m) | 20 × 20 × 10 |

Recovery (%) | 0.9 |

Cutoff grade of iron (%) | 55.26 |

Discount cash flow (%) | 0.10 |

Iron selling price (USD/ton) | 40 |

Iron ore selling cost (USD/ton) | 3.6 |

Iron ore processing cost (USD/ton) | 12 |

Iron ore mining cost (USD/ton) | 3 |

Year (T) | No. of Blocks (N) | No. of Blocks Extracted per Year | Solution Time in Seconds (t) | Gap (%) | No. of Scenarios (s) |
---|---|---|---|---|---|

1 | 16,532 | 876 | 157.12 | 0.01 | 1 |

2 | 15,656 | 1815 | 37.19 | 0.00 | 1 |

3 | 13,841 | 1620 | 151.04 | 0.00 | 1 |

4 | 12,221 | 1919 | 263.66 | 0.01 | 1 |

5 | 10,302 | 2172 | 129.85 | 0.00 | 1 |

6 | 8130 | 2142 | 62.79 | 0.01 | 1 |

7 | 5988 | 1867 | 8.94 | 0.00 | 1 |

8 | 4121 | 1586 | 9.28 | 0.00 | 1 |

9 | 2535 | 1268 | 2.79 | 0.00 | 1 |

10 | 1267 | 1267 | 0.36 | 0.00 | 1 |

Total | 823.02 s |

Year | Ore (Mt) | Waste (Mt) | Metal (Kt) | NPV (USD 1 Million) |

1 | 7,170,000 | 2,850,000 | 50,000 | 101,000,000 |

2 | 8,000,000 | 12,800,000 | 40,400 | 56,900,000 |

3 | 8,000,000 | 10,500,000 | 36,700 | 43,900,000 |

4 | 8,000,000 | 14,000,000 | 36,900 | 38,800,000 |

5 | 8,000,000 | 16,900,000 | 38,800 | 38,100,000 |

6 | 8,000,000 | 16,500,000 | 39,600 | 36,200,000 |

7 | 8,000,000 | 13,400,000 | 38,300 | 31,800,000 |

8 | 8,000,000 | 10,100,000 | 35,900 | 26,100,000 |

9 | 8,000,000 | 6,510,000 | 32,200 | 19,600,000 |

10 | 7,700,000 | 6,800,000 | 27,500 | 12,700,000 |

Total | 7.88 × 10^{7} | 1.10 × 10^{8} | 3.76 × 10^{5} | 4.05 × 10^{8} |

Year (T) | No. of Blocks (N) | No. of Blocks Extracted per Year | Solution Time in Seconds (t) | Gap (%) | No. of the Scenarios (s) |
---|---|---|---|---|---|

1 | 16,532 | 917 | 351.49 | 0.23 | 20 |

2 | 15,615 | 1925 | 577.38 | 0.58 | 20 |

3 | 13,690 | 1616 | 233.55 | 1.46 | 20 |

4 | 12,074 | 1867 | 1309.13 | 2.79 | 20 |

5 | 10,207 | 2112 | 589.09 | 2.76 | 20 |

6 | 8095 | 2185 | 111.98 | 2.43 | 20 |

7 | 5910 | 2167 | 22.36 | 0.05 | 20 |

8 | 3743 | 1750 | 15.37 | 1.77 | 20 |

9 | 1993 | 1464 | 3.46 | 0.88 | 20 |

10 | 529 | Left | No solution or infeasible sol. | ||

Total | 3213.81s |

Year | Ore (Mt) | Waste (Mt) | Metal (Kt) | NPV (USD 1 Million) |
---|---|---|---|---|

1 | 7,114,500 | 33,755,00 | 49,160 | 98,380,000 |

2 | 7,738,000 | 14,280,000 | 41,705 | 61,000,000 |

3 | 7,816,000 | 10,670,000 | 38,840 | 49,795,000 |

4 | 7,709,000 | 13,650,000 | 37,660 | 41,975,000 |

5 | 7,755,500 | 16,425,000 | 39,345 | 40,275,000 |

6 | 7,643,000 | 17,355,000 | 40,355 | 38,535,000 |

7 | 7,776,500 | 17,020,000 | 41,575 | 36,690,000 |

8 | 7,717,500 | 12,305,000 | 37,295 | 28,465,000 |

9 | 7,629,000 | 9,117,500 | 32,530 | 20,465,000 |

Sum | 6.89 × 10^{7} | 1.14 × 10^{8} | 3.58 × 10^{5} | 4.16 × 10^{8} |

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**MDPI and ACS Style**

Joshi, D.; Chithaluru, P.; Singh, A.; Yadav, A.; Elkamchouchi, D.H.; Pérez-Oleaga, C.M.; Anand, D.
A Novel Large-Scale Stochastic Pushback Design Merged with a Minimum Cut Algorithm for Open Pit Mine Production Scheduling. *Systems* **2022**, *10*, 159.
https://doi.org/10.3390/systems10050159

**AMA Style**

Joshi D, Chithaluru P, Singh A, Yadav A, Elkamchouchi DH, Pérez-Oleaga CM, Anand D.
A Novel Large-Scale Stochastic Pushback Design Merged with a Minimum Cut Algorithm for Open Pit Mine Production Scheduling. *Systems*. 2022; 10(5):159.
https://doi.org/10.3390/systems10050159

**Chicago/Turabian Style**

Joshi, Devendra, Premkumar Chithaluru, Aman Singh, Arvind Yadav, Dalia H. Elkamchouchi, Cristina Mazas Pérez-Oleaga, and Divya Anand.
2022. "A Novel Large-Scale Stochastic Pushback Design Merged with a Minimum Cut Algorithm for Open Pit Mine Production Scheduling" *Systems* 10, no. 5: 159.
https://doi.org/10.3390/systems10050159