Stochastic Open-Pit Mine Production Scheduling: A Case Study of an Iron Deposit
- 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 ;
- 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 .
2.1. Modeling Geological Uncertainty
2.2. Production Scheduling under Geological Uncertainty
- the layout of the geological domains (rock types) are simulated in the area of study, via the plurigaussian approach . 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  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 , but applied to a stochastic case.
4. Case Study
- 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.
4.2. Rock Type Modeling
4.3. Modeling of Quantitative Variables
- for the host rock, a linear coregionalization model was fitted for all the quantitative variables of interest using a semi-automated least-square algorithm ;
- for the skarn domain, a linear coregionalization model was fitted for Fe, SiO2, 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
5. Conclusions and Perspectives
Conflicts of Interest
Appendix A. Economic Block Valuation
Appendix B. Stochastic Integer Programming (SIP) Model
Appendix C. Deterministic Integer Programming (DIP) Model
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|Statistics||Rock Ton (Mton)||Ore Ton (Mton)||Waste Ton (Mton)||Stripping Ratio|
|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 , 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 ).
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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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
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/min10070585Chicago/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