# Optimisation of Three-Dimensional Stope Layouts Using a Dual Interchange Algorithm for Improved Value Creation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Creation of a Synthetic Platreef Block Model

- Platinum at 43%.
- Palladium at 48%.
- Rhodium at 3%.
- Gold at 6%.

- Step 1 involved creating x, y, and z block dimensions along X, Y, and Z directions which were 5 m, 5 m, and 5 m, respectively. The block dimensions were selected because block sizes and non-cubic shapes affect the performance of the stope layout optimisation process but may enhance computational time. Smaller block sizes that are cubic in shape are desirable to minimise dilution since block model peripheries tend to be irregular. However, as the block size becomes smaller, more blocks must be evaluated, thus slowing down the optimisation process and increasing the computational time. The block size of 25 m
^{3}created an acceptable balance between block size and performance of the optimisation process. - Step 2 involved applying average density values in t/m
^{3}to the blocks. - Step 3 involved the generation of random 4E grade values between typical low values of about 0.01 g/t and high values of about 21.9 g/t since Thormann et al. [7] noted that the Platreef orebody has a highly variable grade distribution.

## 3. Construction of the Economic Block Model

_{xyz}[11]. Equations (3) and (4) were applied to the block model to convert it into an economic block model. For compatibility, it was necessary to convert the metal price in USD/oz and grade in g/t to their equivalent USD/t and percentage (%), respectively, assuming that 1t is equivalent to 32150.75 troy ounces [3]. Table 2 summarises the conversion factors for transforming the block model into an economic block model.

- $BE{V}_{r}\hspace{1em}$ block economic value (USD) for block r
- ${p}_{b}\hspace{1em}$ basket metal price (USD/oz)
- ${r}_{r}\hspace{1em}$ refining cost of block r (USD/t)
- ${g}_{r}\hspace{1em}$ grade of block r (g/t)
- ${y}_{r}\hspace{1em}$ recovery from block r (%)
- ${e}_{r}\hspace{1em}$ extraction or mining cost for block r (USD/t)
- ${c}_{r}\hspace{1em}$ concentrating or processing cost of block r (USD/t)
- ${t}_{r}\hspace{1em}$ tonnage of block r (t)
- ${d}_{r}\hspace{1em}$ dilution and mining losses incurred in block r (%)

## 4. Extraction of the Platreef Deposit

## 5. Mining Constraints

## 6. Dual Interchange Algorithm Anatomy

## 7. Comparison of MSO and DIA Performance

#### 7.1. Comparison of MSO and DIA for Scenario A

#### 7.2. Summary of Results from the Four Scenarios

## 8. DIA Performance Evaluation under Different Numbers of Iterations

#### 8.1. Optimisation Results for Scenario A under Different Numbers of Iterations

#### 8.2. Optimisation Results for Scenario B under Different Numbers of Iterations

#### 8.3. Optimisation Results for Scenario C under Different Numbers of Iterations

#### 8.4. Optimisation Results for Scenario D under Different Numbers of Iterations

## 9. Summary of the DIA Structure and Findings in Relation to Similar Studies

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Synthetic Platreef block model with the legend showing BEV in USD. Source: [3].

**Figure 3.**Pseudocode for the dual interchange algorithm. Source: [3].

**Figure 4.**Schematic illustration of the proposed dual interchange algorithm integrating the PSO and GA. Source: [3].

**Figure 5.**Scenario A: visualisations of the optimum stope layout solution using the MSO with the legend showing SEV in USD. Source: [3].

**Figure 6.**Scenario A: optimum solution convergence using the DIA. Source: [3].

**Figure 7.**Scenario A: visualisations of the optimum stope layout solution using the DIA with the legend showing SEV in USD. Source: [3].

**Figure 8.**Comparison of the performance of the DIA and MSO. Source: [3].

**Table 1.**Synthetic Platreef block model specifications. Source [3].

Parameter | Details |
---|---|

Spatial extent | X: −7,642,592.5 to −7,641,952.5 |

Y: 742,722.5 to 742,897.5 | |

Z: −1097.5 to −1042.5 | |

Number of blocks | 50,000 |

Block size (x, y, z) | Fixed size: 5 × 5 × 5 m |

Density | 3.1 t/m^{3} |

4E PGE grade (g/t) | Variation: 0.01 to 21.9 |

Average: 1.28 | |

Blocks with positive BEV | 43,060 |

Blocks with negative BEV | 6940 |

BEV (USD) | Lowest: −15,531.000 |

Highest: 18,625,482.800 | |

Total positive BEV (USD) | 53,914,225,979.920 |

Total BEV (USD) | 53,819,387,407.039 |

**Table 2.**Conversion factors and input values for economic block model construction. Source: Adapted from [3].

Parameter | Conversion Factor | ||

Price | |||

US Dollar per tonne (USD/t) | 1 | ||

US Dollar per ounce (USD/oz) | 32,150.75 | ||

Grade | |||

Percent (%) | 1 | ||

Grade (g/t) | 0.0001 | ||

Parameter | Unit | Value | |

Price | Platinum (Pt) | USD/oz | 1250 |

Palladium (Pd) | USD/oz | 825 | |

Rhodium (Rh) | USD/oz | 1000 | |

Gold (Au) | USD/oz | 1300 | |

Refining cost | USD/t | 39.77 | |

Extraction cost | USD/t | 34.27 | |

Concentrating cost | USD/t | 15.83 | |

Recovery | % | 82 |

**Table 3.**Adaptation of the UPL optimisation principles to stope layout optimisation. Source: [3].

LG Algorithm | DIA |
---|---|

Open pit mining environment in 2D | Underground mining environment in 3D |

BEV (B_{ij}) | BEV (B_{ijk}) |

Block column value (M_{ij}) | Stope economic value (SEV_{ijk}) |

Ultimate pit economic value (P_{ij}) | Stope layout economic value (SLEV_{ijk}) |

**Table 4.**Input parameters considered for the optimisation problem Source: [3].

Scenario | Length (m) on x-axis | Width (m) on y-axis | Height (m) on z-axis |
---|---|---|---|

A | 20 | 40 | 25 |

B | 10–40 | ||

C | 50 | ||

D | 10–50 |

**Table 5.**Comparison of the optimum solutions generated by the MSO and DIA. Source: Adapted from [3].

Scenario | Parameter | MSO | DIA |
---|---|---|---|

A | Maximum stope layout economic value (USD) | 35,866,231,792.854 | 35,972,381,189.708 |

Number of stopes | 217 | 213 | |

Solution time (hh:mm:ss) | 00:00:47 | 05:39:34 | |

B | Maximum stope layout economic value (USD) | 35,730,081,740.688 | 37,004,613,547.534 |

Number of stopes | 229 | 647 | |

Solution time (hh:mm:ss) | 00:02:58 | 17:15:24 | |

C | Maximum stope layout economic value (USD) | 38,805,614,807.051 | 35,042,470,315.314 |

Number of stopes | 188 | 159 | |

Solution time (hh:mm:ss) | 00:00:39 | 09:25:10 | |

D | Maximum stope layout economic value (USD) | 34,044,114,429.081 | 37,108,222,577.360 |

Number of stopes | 181 | 677 | |

Solution time (hh:mm:ss) | 00:02:19 | 11:18:06 |

Scenario/Iteration | 50 | 100 | 150 | 200 | |
---|---|---|---|---|---|

Scenario A | SLEV (USD billion) | 35.95 | 32.13 | 35.77 | 35.97 |

Number of stopes | 192 | 190 | 191 | 213 | |

Solution time (hh:mm:ss) | 01:05:18 | 02:25:16 | 04:21:56 | 05:59:34 | |

Scenario B | SLEV (USD billion) | 36.6 | 37.07 | 37.12 | 37.23 |

Number of stopes | 660 | 665 | 644 | 667 | |

Solution time (hh:mm:ss) | 03:15:38 | 08:57:28 | 14:45:15 | 20:40:43 | |

Scenario C | SLEV (USD billion) | 35.29 | 30.28 | 35.34 | 36.50 |

Number of stopes | 211 | 123 | 175 | 181 | |

Solution time (hh:mm:ss) | 01:01:08 | 02:15:08 | 04:04:33 | 05:08:14 | |

Scenario D | SLEV (USD billion) | 36.46 | 36.74 | 36.88 | 37.83 |

Number of stopes | 647 | 655 | 672 | 669 | |

Solution time (hh:mm:ss) | 03:35:19 | 09:37:42 | 14:30:31 | 23:13:07 |

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**MDPI and ACS Style**

Nhleko, A.S.; Musingwini, C.
Optimisation of Three-Dimensional Stope Layouts Using a Dual Interchange Algorithm for Improved Value Creation. *Minerals* **2022**, *12*, 501.
https://doi.org/10.3390/min12050501

**AMA Style**

Nhleko AS, Musingwini C.
Optimisation of Three-Dimensional Stope Layouts Using a Dual Interchange Algorithm for Improved Value Creation. *Minerals*. 2022; 12(5):501.
https://doi.org/10.3390/min12050501

**Chicago/Turabian Style**

Nhleko, Adeodatus S., and Cuthbert Musingwini.
2022. "Optimisation of Three-Dimensional Stope Layouts Using a Dual Interchange Algorithm for Improved Value Creation" *Minerals* 12, no. 5: 501.
https://doi.org/10.3390/min12050501