# Stochastic Open-Pit Mine Production Scheduling: A Case Study of an Iron Deposit

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## Abstract

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_{2}, S, P and K, magnetic ratio and specific gravity). To assess the effect of integrating these two sources of uncertainty in mine planning decision, stochastic and deterministic production scheduling models are applied based on the simulated block models. The results show the capacity of the stochastic mine planning model to identify and minimize risks, obtaining valuable information in ore content or quality at early stages of the project, and improving decision-making with respect to the deterministic production scheduling. Numerically speaking, the stochastic mine planning model improves 6% expected cumulative discounted cash flow and generates 16% more iron ore than deterministic model.

## 1. Introduction

- geological block modeling: data are obtained from the geological logging and analytical assays of drill hole samples taken at different locations and depths in the deposit, and are used to delineate lithological and mineralogical domains and to interpolate the grades of elements of interest (products, by-products and contaminants) and other attributes, such as the specific gravity and the metal recovery. This step usually combines expert geological knowledge with geostatistical methods. Then, the deposit is modeled as a three-dimensional array of rectangular cuboids or blocks, each of which is assigned a prediction of its grade(s) and other attributes (rock type, specific gravity, metal recovery, etc.) [1,2,3,4,5,6];
- economic model: based on the block model information, alternative destinations, such as mills, waste dumps or stockpiles, and economic parameters, such as metal prices, mining and processing costs, a specific economic value is assigned to each block, indicating the amount of money that one obtains or loses by extracting/processing this block;
- ultimate pit limit determination: this step consists of delimiting the sub-region of the deposit in which extraction will be carried out. Among many different possible configurations of blocks for extraction that respect the overall slope angle of pit walls, the configuration that maximizes the profit, termed the ultimate pit limit (UPL), is often chosen [7];
- production scheduling: this step consists of deciding which blocks should be extracted, and when, and how extracted blocks should be treated. For this purpose, the UPL is divided into smaller pits called nested pits. A sequence of pushbacks or phases is then defined by considering operational spaces. Finally, a long-term open-pit mine production scheduling (LOMPS) is defined, which considers a number of operational constraints in each phase and maximizes the cumulative discounted cash flow (DCF) of the project [7].

## 2. Background

#### 2.1. Modeling Geological Uncertainty

#### 2.2. Production Scheduling under Geological Uncertainty

## 3. Methodology

- the layout of the geological domains (rock types) are simulated in the area of study, via the plurigaussian approach [12]. In this manner, it is possible to account for the uncertainty in the modeling of geological domains (rock type in this case);
- the quantitative variables of interest are jointly simulated within each simulated geological domain (obtained from the previous step) separately. Here, a multigaussian model together with a turning bands simulation algorithm [41] are used. By having multiple realizations, the uncertainty in the quantitative variables conditioned to the rock type model can be accounted for;
- a mixed stochastic integer programming (SIP) model is used to assess the impact of geological uncertainty. The model uses mine blocks along the planning horizon with the aim of maximizing the expected cumulative DCF and minimizing the total cost of uncertainty associated with deviations from production objectives (both in quantity and quality). The model is subject to several constraints such as precedence slope angles for safe pit walls, maximum and minimum operational resource consumption (mining and processing) associated with the quantity of materials, and maximum and minimum blending requirements, associated with the quality of the processed material;
- the approach for directly solving the SIP formulation is highly impractical at a large scale. To reduce the number of integer variables required by the SIP model, we use a temporal decomposition strategy to divide the original problem into sub-problems associated with fewer periods, together with a block pre-selection procedure based on a linear programming relaxation solution [42], but applied to a stochastic case.

## 4. Case Study

#### 4.1. Presentation

- host rock, composed mostly of intrusive or hypabyssal rocks such as andesites, diorites and trachytes. These lithologies are porphyritic in texture and are affected by hydrothermal alteration, including argilization, epidotization and chloritization. The iron grades are low, generally less than 14.5%;
- skarn, formed by contact metamorphism at high temperatures and ductile rheological environments. This lithology contains epidote, chlorite, pyroxenes and clay minerals, as well as contact mineralization (magnetite) associated with faults and irregular fracturing, with an iron grade that reaches 25%;
- mineralized breccias, resulting from the cooling and fracturing of the skarn in a posterior event with lower temperature and a fragile rheological environment. The main iron ore mineral observed is magnetite, while the main gangue minerals are gypsum, chalcedony, mica and quartz.

_{2}, S, P, K) and the magnetic ratio (MR) that quantifies the recovery of iron ore processing. The drill holes are mainly vertical and unevenly spaced (Figure 1).

#### 4.2. Rock Type Modeling

#### 4.3. Modeling of Quantitative Variables

_{2}, S, P, K, MR) were normal-score transformed. The experimental direct and cross-variograms of the normal scores data were then calculated. The fitting of theoretical variogram models was realized as per the following strategy:

- for the host rock, a linear coregionalization model was fitted for all the quantitative variables of interest using a semi-automated least-square algorithm [43];
- for the skarn domain, a linear coregionalization model was fitted for Fe, SiO
_{2}, K and MR, and other two models were fitted for S and P, insofar as none of these variables present significant correlations with the remaining ones in this domain; - for the mineralized breccia, a linear coregionalization model was fitted for all the variables of interest except S, which was modeled separately.

#### 4.4. Ultimate Pit Limit

#### 4.5. Production Scheduling

_{2}< 15%, S < 0.2%, P < 0.6%, K < 1.2%. The reason for this decision is that no penalties or bonuses were applied for low impurity content in iron ore. Penalties for shortage and/or surplus deviations of production targets were computed according to Mai’s approach [38], which guarantees that the total costs of producing shortage and/or surplus deviations from production targets (iron ore production and grades of (un)desirable elements) are higher than the benefits gained. A discount rate of 10% was applied on revenues and costs.

_{2}), sulfur (S), phosphorus (P) and potassium (K) and their risk profiles. Silica (Figure 8c), one of the most important impurities to be controlled, shows deviations from period 8 to 12, with a high probability of non-compliance of the silica production target. The predominant rock type in the ore reserve is breccia, where Fe and SiO

_{2}correlate negatively. It follows that low (high) SiO

_{2}grades are present in periods with high (low) Fe grades. Figure 8d shows sulfur content and the most likely deviations in periods 5, 6, 7 and 11. Similarly, Figure 8e reports phosphorus content in the iron ore, with slight deviations reported in periods 2 and 3, and the most likely deviations in periods 5 to 7 also being observed. Finally, Figure 8f presents potassium content per period, showing the most likely deviations in periods 1, 6, 7, 8 and 11. These deviations reduce the selling price of the iron ore because additional processes are required to reduce impurities.

## 5. Conclusions and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Economic Block Valuation

_{2}, $e=6$: S grade, $e=7$: P grade, $e=8$: K grade.

## Appendix B. Stochastic Integer Programming (SIP) Model

## Appendix C. Deterministic Integer Programming (DIP) Model

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**Figure 1.**Isometric views of drill hole data, colored according to (

**a**) total iron grade, (

**b**) rock type.

**Figure 2.**(

**a**) Histogram of iron grade and (

**b**) scatter plot of iron vs. silica grades. Colors represent rock types.

**Figure 3.**Contact analysis between rock types. Left: mean iron grade as a function of the signed distance to the rock type boundary; right: cross-correlation between iron grades measured in different rock types, as a function of the distance between sampling data. Red line: original data; blue, green and black lines: simulated iron grades.

**Figure 4.**(

**a**,

**b**) Two plurigaussian realizations of rock types conditioned to drill hole data. (

**c**) Most probable rock type, i.e., the most frequent rock type over the realizations. White areas correspond to grid nodes above the topographic surface or outside the sampled region.

**Figure 5.**Probability maps of the different rock types, calculated on the basis of 100 plurigaussian realizations.

**Figure 6.**One realization of iron (top) and silica (bottom) grades at (

**a**,

**c**) a quasi-point support (before upscaling) and (

**b**,

**d**) a block support (after upscaling).

**Figure 8.**Production plan obtained from the stochastic integer-programming (SIP) and deterministic integer programming (DIP) models: (

**a**) ore and rock tonnes, and grades of (

**b**) iron, (

**c**) silica, (

**d**) sulfur, (

**e**) phosphorus, and (

**f**) potassium. The error bars represent the P5 and P95 values obtained on the set of simulated block models; the continuous lines represent the average (expected) values.

**Figure 9.**Expected discounted cash flow (DCF) from the DIP model compared to that of the SIP model as a reference case (100%). The error bars represent the P5 and P95 values obtained on the set of simulated block models; the continuous lines represent the average (expected) values.

Variable | Statistics | Global | Host Rock | Skarn | Breccia |
---|---|---|---|---|---|

Count | 5614 | 1177 | 2022 | 2415 | |

Fe | Minimum | 1.40 | 1.56 | 1.40 | 15.02 |

Maximum | 70.45 | 14.98 | 24.99 | 70.45 | |

Mean | 21.82 | 7.17 | 12.62 | 36.82 | |

Variance | 257.52 | 11.99 | 35.40 | 162.89 | |

Median | 17.45 | 6.72 | 12.63 | 34.49 | |

Upper Quartile | 31.82 | 9.57 | 17.24 | 45.15 | |

Lower Quartile | 8.56 | 4.13 | 7.72 | 26.97 | |

SiO_{2} | Minimum | 0.7 | 3.46 | 5.29 | 0.7 |

Maximum | 83.11 | 83.11 | 79.31 | 60.1 | |

Mean | 32.23 | 38.0 | 41.55 | 21.62 | |

Variance | 226.07 | 142.57 | 160.26 | 121.10 | |

Median | 31.13 | 34.18 | 41.42 | 20.84 | |

Upper Quartile | 41.56 | 40.59 | 50.86 | 29.05 | |

Lower Quartile | 22.107 | 30.75 | 30.55 | 13.37 | |

S | Minimum | 0.007 | 0.02 | 0.4 | 0.007 |

Maximum | 5.53 | 3.68 | 5.53 | 3.63 | |

Mean | 0.594 | 0.45 | 0.62 | 0.64 | |

Variance | 0.50 | 0.32 | 0.50 | 0.58 | |

Median | 0.12 | 0.08 | 0.11 | 0.13 | |

Upper Quartile | 0.21 | 0.20 | 0.21 | 0.22 | |

Lower Quartile | 0.049 | 0.023 | 0.039 | 0.08 | |

P | Minimum | 0.0 | 0.0 | 0.0 | 0.0 |

Maximum | 11.1 | 8.5 | 11.1 | 9.45 | |

Mean | 0.29 | 0.24 | 0.27 | 0.33 | |

Variance | 0.22 | 0.18 | 0.24 | 0.21 | |

Median | 0.46 | 0.29 | 0.48 | 0.52 | |

Upper Quartile | 0.78 | 0.59 | 0.79 | 0.84 | |

Lower Quartile | 0.26 | 0.15 | 0.30 | 0.32 | |

K | Minimum | 0.0 | 0.35 | 0.018 | 0.0 |

Maximum | 7.78 | 7.27 | 7.78 | 5.7 | |

Mean | 2.05 | 2.70 | 2.33 | 1.5 | |

Variance | 1.280 | 1.14 | 1.26 | 0.79 | |

Median | 1.92 | 2.55 | 2.12 | 1.42 | |

Upper Quartile | 2.7 | 3.31 | 2.88 | 2.07 | |

Lower Quartile | 1.27 | 1.96 | 1.52 | 0.85 | |

MR | Minimum | 0.0 | 0.0 | 0.0 | 0.0 |

Maximum | 0.98 | 0.40 | 0.69 | 0.98 | |

Mean | 0.22 | 0.05 | 0.11 | 0.41 | |

Variance | 0.048 | 0.002 | 0.009 | 0.044 | |

Median | 0.144 | 0.04 | 0.094 | 0.38 | |

Upper Quartile | 0.34 | 0.06 | 0.15 | 0.55 | |

Lower Quartile | 0.04 | 0.02 | 0.03 | 0.25 |

**Table 2.**Summary statistics (average, 5th and 95th percentiles over 100 realizations of grades, specific gravity and magnetic ratio) of the ultimate pit included in the 95% pit limit probability.

Statistics | Rock Ton (Mton) | Ore Ton (Mton) | Waste Ton (Mton) | Stripping Ratio |
---|---|---|---|---|

P5 | 401.85 | 105.23 | 295.62 | 2.71 |

Average | 404.88 | 107.72 | 297.16 | 2.74 |

P95 | 409.26 | 109.06 | 298.47 | 2.78 |

Approach | Advantages | Drawbacks |
---|---|---|

Stochastic integer programming (SIP) model | Geological and geometallurgical uncertainties are explicitly taken into account in the model. The model maximizes the expected cumulative discounted cash flow and, simultaneously, minimizes the impact of losses due to deviations from production targets. The solution under this approach has a better risk control and a higher probability of meeting production targets | Real-world size problems are computationally demanding (we applied a hybrid heuristic algorithm [42], allowing to reduce the required computer resources and running times). The choice of unit costs of under- and over-production strongly impacts the objective function (we followed the approach presented in [38]). Other sources of uncertainty such as commodity prices and costs are not considered. We propose this point as a future work. |

Deterministic integer programming (DIP) model | This model lacks the possibility to deal with uncertainties in geological and geometallurgical variables: all the parameters used in the model are considered known in advance. The geological information is modeled through a single, smoothed block model (kriging prediction, or average of simulated realizations). Real-world size problems are computationally demanding. |

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## Share and Cite

**MDPI and ACS Style**

Maleki, M.; Jélvez, E.; Emery, X.; Morales, N.
Stochastic Open-Pit Mine Production Scheduling: A Case Study of an Iron Deposit. *Minerals* **2020**, *10*, 585.
https://doi.org/10.3390/min10070585

**AMA Style**

Maleki M, Jélvez E, Emery X, Morales N.
Stochastic Open-Pit Mine Production Scheduling: A Case Study of an Iron Deposit. *Minerals*. 2020; 10(7):585.
https://doi.org/10.3390/min10070585

**Chicago/Turabian Style**

Maleki, Mohammad, Enrique Jélvez, Xavier Emery, and Nelson Morales.
2020. "Stochastic Open-Pit Mine Production Scheduling: A Case Study of an Iron Deposit" *Minerals* 10, no. 7: 585.
https://doi.org/10.3390/min10070585