# Mine Productivity Upper Bounds and Truck Dispatch Rules

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## Abstract

**:**

## 1. Introduction

## 2. Mine Model

#### 2.1. Optimization Problem

#### Greedy Search

Algorithm 1 Greedy search for mine productivity. |

#### 2.2. Simulation

#### Dispatch Rule

Algorithm 2 Dispatch rule to a loading site. |

## 3. Results

#### 3.1. Simulated Productivity Convergence and Its Upper Bound

#### 3.2. Fleet Sizing

#### 3.3. A Tight Upper Bound

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Productive convergence using a discrete event system simulation for the mine, which must be bellow the upper bound given by the linear optimization problem (5)–(9).

**Figure 4.**Mine productivity as a function of number of trucks available in the mine. Models with uncertainty $p\in \{\mathrm{20\%},\mathrm{50\%}\}$ consider the average productivity of 30 runs using symmetric triangular distributions with the mean at $\mu $, whose minimum and maximum values are at $(1-p)\mu $ and $(1+p)\mu $, respectively.

**Figure 5.**The mine’s realized productivity gap to the upper bound as a function of the number of trucks available in the mine. Models with uncertainty $p\in \{\mathrm{20\%},\mathrm{50\%}\}$ consider the average productivity of 30 runs using symmetric triangular distributions with the mean at $\mu $, whose minimum and maximum values are at $(1-p)\mu $ and $(1+p)\mu $ respectively.

**Figure 6.**Average mine realized productivity gap to the upper bound for 10 different random instances with 160 trucks as a function of the number of loaders available in the mine. Models with uncertainty $p\in \{\mathrm{20\%},\mathrm{50\%}\}$ consider the average productivity of 30 runs using symmetric triangular distributions with the mean at $\mu $, whose minimum and maximum values are at $(1-p)\mu $ and $(1+p)\mu $ respectively.

**Table 1.**Dumping time ${t}_{u}$ for each truck model (i.e., they are the same for every dumping site) following a triangular distribution in the interval $[a,b]$ with the mode at c.

Truck Model ${\mathit{i}}_{\mathit{m}}$ | Minimum a (s) | Maximum b (s) | Mode c (s) |
---|---|---|---|

1 | 23 | 47 | 35 |

2 | 30 | 54 | 42 |

**Table 2.**Loading time ${t}_{\mathcal{l}}$ for each truck model (i.e., they are the same for every loading site) following a triangular distribution in the interval $[a,b]$ with the mode at c.

Truck Model ${\mathit{i}}_{\mathit{m}}$ | Minimum a (s) | Maximum b (s) | Mode c (s) |
---|---|---|---|

1 | 146 | 298 | 222 |

2 | 185 | 349 | 267 |

**Table 3.**Truck speed v for each truck model following a triangular distribution in the interval $[a,b]$ with the mode at c.

Truck Model ${\mathit{i}}_{\mathit{m}}$ | Minimum a (km/h) | Maximum b (km/h) | Mode c (km/h) |
---|---|---|---|

1 | 15 | 31 | 23 |

2 | 17 | 33 | 25 |

**Table 4.**Truck load L for each truck model following a triangular distribution in the interval $[a,b]$ with the mode at c.

Truck Model ${\mathit{i}}_{\mathit{m}}$ | Minimum a (t) | Maximum b (t) | Mode c (t) |
---|---|---|---|

1 | 138 | 148 | 143 |

2 | 189 | 201 | 195 |

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**MDPI and ACS Style**

Lisboa, A.C.; Castro, F.L.B.; de Venâncio, P.V.A.B.
Mine Productivity Upper Bounds and Truck Dispatch Rules. *Mining* **2023**, *3*, 786-797.
https://doi.org/10.3390/mining3040043

**AMA Style**

Lisboa AC, Castro FLB, de Venâncio PVAB.
Mine Productivity Upper Bounds and Truck Dispatch Rules. *Mining*. 2023; 3(4):786-797.
https://doi.org/10.3390/mining3040043

**Chicago/Turabian Style**

Lisboa, Adriano Chaves, Felipe Luz Barbosa Castro, and Pedro Vinícius Almeida Borges de Venâncio.
2023. "Mine Productivity Upper Bounds and Truck Dispatch Rules" *Mining* 3, no. 4: 786-797.
https://doi.org/10.3390/mining3040043