Skip to main content
SpringerLink
Log in

A New Approach to Optimize Ultimate Geometry of Open Pit Mines with Variable Overall Slope Angles

  • Original Paper
  • Published:
Natural Resources Research Aims and scope Submit manuscript

Abstract

The optimization method for determining open pit limit is important, as it is crucial to achieve the most profitable outcome from the limited amount of reserves available. However, importance must be also given to geotechnical aspects to ensure that safety requirements are met. Thus, overall slope angle (OSAs) must be incorporated in the optimization process. In the conventional methods, there were problems in incorporating various OSAs, or they were included in the pit design after the completion of the optimization procedure. To include variable OSAs in the optimization, cone-based method is still considered as one of the most suitable approaches. To apply this method, a new mathematical model incorporating various OSAs into the ultimate pit problem through mixed integer programming (MIP) and simulated annealing is proposed. Four different cases of change in OSAs in the pit were included in the algorithm. The proposed method was verified by applying it to five different cases, and the OSA was achieved with a considerably low difference of 0°–2° while optimizing the ultimate pit. The comparison between the introduced algorithm and the Lerchs–Grossman algorithm indicated that an improvement within a range of 8–20% can be achieved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10

Similar content being viewed by others

References

  • Aarts, E. H. L., & Van Laarhoven, R. J. (1987). Simulated annealing: Theory and applications. Reidel.

    Google Scholar 

  • Achireko, P. K. (1998). Application of modified conditional simulation and artificial neural networks to open pit optimization. Technical University of Nova Scotia.

    Google Scholar 

  • Albor Consuegra, F. R., & Dimitrakopoulos, R. (2009). Stochastic mine design optimisation based on simulated annealing: Pit limits, production schedules, multiple orebody scenarios and sensitivity analysis. Mining Technology, 118(2), 79–90.

    Google Scholar 

  • Askari-Nasab, H., Pourrahimian, Y., Ben-Awuah, E., & Kalantari, S. (2011). Mixed integer linear programming formulations for open pit production scheduling. Journal of Mining Science, 47(3), 338–359.

    Google Scholar 

  • Bai, V. X., Turczynski, G., Baxter, N., Place, D., Sinclair-Ross, H., & Ready, S. (2017). Pseudoflow method for pit optimization. Whitepaper, Geovia-Whittle, Dassault Systems.

  • Barnes, R. J., & Johnson, T. B. (1982). Bounding techniques for the ultimate pit limit problem. In Proc. 17th APCOM (pp. 263–273). AIME.

  • Blom, M., Pearce, A. R., & Stuckey, P. J. (2017). Short-term scheduling of an open-pit mine with multiple objectives. Engineering Optimization, 49(5), 777–795.

    Google Scholar 

  • Blom, M., Pearce, A. R., & Stuckey, P. J. (2018). Multi-objective short-term production scheduling for open-pit mines: A hierarchical decomposition-based algorithm. Engineering Optimization, 50(12), 2143–2160.

    Google Scholar 

  • Campos, P. H. A., Arroyo, C. E., & Morales, N. (2018). Application of optimized models through direct block scheduling in traditional mine planning. The Journal of the Southern African Institute of Mining and Metallurgy, 118, 381–386.

    Google Scholar 

  • Cerny, V. A. (1985). Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications, 45, 41–51.

    Google Scholar 

  • Chaves, L. S., Carvalho, L. A., Souza, F. R., Nader, B., Ortiz, C. E. A., Torres, V. F. N., Câmara, T. R., Napa-García, G. F., & Valadão, G. E. S. (2020). Analysis of the impacts of slope angle variation on slope stability and NPV via two different final pit definition techniques. REM - International Engineering Journal, 73(1), 119–126.

    Google Scholar 

  • Chen, T. (1976). 3D pit design with variable wall slope capabilities. In Proc. 14th APCOM (pp. 615–625). AIME.

  • CIM. (2019). CIM Estimation of Mineral Resources and Mineral Reserves Best Practice Guidelines. Retrieved April 21, 2021, from https://www.bcsc.bc.ca/-/media/PWS/Resources/For_Companies/Mining/CIM-Best-Practices-Guidelines-November-29-2019.pdf

  • Dagasan, Y., Renard, P., Straubhaar, J., Erten, O., & Topal, E. (2019). Pilot point optimization of mining boundaries for lateritic metal deposits: Finding the trade-off between dilution and ore loss. Natural Resources Research, 28, 153–171.

    Google Scholar 

  • Dekkers, A., & Aarts, E. (1991). Global optimization and simulated annealing. Mathematical Programming, 50, 367–393.

    Google Scholar 

  • Deutsch, M., González, E., & Williams, M. (2015). Using simulation to quantify uncertainty in ultimate-pit limits and inform infrastructure placement. Mining Engineering, 67(12), 49–55.

    Google Scholar 

  • Dowd, P., & Onur, A. H. (1993). Open pit optimization—part 1: Optimal open pit design. Trans. Instn Min Metall (Sect. A: Min. industry), 102, A95–A104.

  • Elahi, E., Kakaie, R., & Yousefi, A. (2011). A new algorithm for optimum open pit design: Floating cone method III. Journal of Mining & Environment, 2(2), 118–125. https://doi.org/10.22044/JME.2012.63

  • Farmer, I., & Dimitrakopoulos, R. (2018). Schedule-based pushback design within the stochastic optimisation framework. International Journal of Mining, Reclamation and Environment, 32(5), 327–340.

    Google Scholar 

  • Fu, Z., Asad, M. W. A., & Topal, E. (2019). A new model for open-pit production and waste-dump scheduling. Engineering Optimization, 51(4), 718–732.

    Google Scholar 

  • Giannini, L. M. (1990). Optimum design of open pit mines. Curtin University of Technology.

    Google Scholar 

  • Gilani, S. O., & Sattarvand, J. (2015). A new heuristic non-linear approach for modeling the variable slope angles in open pit mine planning algorithms. Acta Montanistica Slovaca, 20(4), 251–259.

    Google Scholar 

  • Hochbaum, D. S., & Chen, A. (2000). Performance analysis and best implementations of old and new algorithms for the open pit mining problem. Operation Research, 48(6), 894–914.

    Google Scholar 

  • Hochbaum, D. S. (2008). The pseudoflow algorithm: A new algorithm for the maximum-flow problem. Operations Research, 56(4), 992–1009. https://doi.org/10.1287/opre.1080.0524

    Article  Google Scholar 

  • Huttagosol, P. (1988). Modified tree graph algorithm for ultimate pit limit analysis. MSc diss., Colorado School of Mines.

  • Jalali, S. E., Ataee-Pour, M., & Shahriar, K. (2006). Pit limit optimization using stochastic process. CIM Bulletin, 99(1024), 1–11.

    Google Scholar 

  • Javadzadeh, S., Ataee-pour, M., & Hosseinpour, V. (2019). Modeling optimum mining limits with imperialist competitive algorithm. In Proceedings of the 27th international symposium on mine planning and equipment selection-MPES 2018 (pp. 197–211). Springer.

  • Johnson, T. B. (1968). Optimum open pit mine production scheduling. California University, Berkeley, Operations Research Center.

    Google Scholar 

  • Johnson, T. B., & Sharp, W. R. (1971). A three-dimensional dynamic programming method for optimal ultimate open pit design. USBM, 7553(25).

  • Khalokakaie, R. (1999). Computer-aided optimal open pit design with variable slope angles. University of Leeds.

    Google Scholar 

  • Khalokakaie, R. (2006). Optimum open pit design with modified moving cone II methods. Journal of Engineering in Tehran University, 41(3), 297–307. (In Persian).

    Google Scholar 

  • Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671–680.

    Google Scholar 

  • Koenigsberg, E. (1982). The optimum contours of an open pit mine: an application of dynamic programming. In Proc. 17th APCOM (pp. 274–287). AIME.

  • Kumral, M., & Dowd, P. A. (2004). A simulated annealing approach to mine production scheduling. Journal of the Operational Research Society, 56(8), 922–930.

    Google Scholar 

  • Kumral, M. (2012). Production planning of mines: Optimisation of block sequencing and destination. International Journal of Mining, Reclamation and Environment, 26(2), 93–103. https://doi.org/10.1080/17480930.2011.644474

    Article  Google Scholar 

  • Kumral, M. (2013). Optimizing ore–waste discrimination and block sequencing through simulated annealing. Applied Soft Computing, 13(8), 3737–3744. https://doi.org/10.1016/j.asoc.2013.03.005

    Article  Google Scholar 

  • Lerchs, H., & Grossmann, I. F. (1965). Optimum design of open pit mines. CIM Bulletin, 58, 47–54.

    Google Scholar 

  • L’Heureux, G., Gamache, M., & Soumis, F. (2013). Mixed integer programming model for short term planning in open-pit mines. Mining Technology, 122(2), 101–109.

    Google Scholar 

  • Li, S., Sari, Y. A., & Kumral, M. (2020). Optimization of Mining-Mineral Processing Integration Using Unsupervised Machine Learning Algorithms. Natural Resources Research, 29, 3035–3046. https://doi.org/10.1007/s11053-020-09628-0

    Article  Google Scholar 

  • Lipkewich, M. P., & Borgman, L. (1969). Two- and three-dimensional pit design optimization techniques (pp. 505–523). A Decade of Digital Computing in the Mineral Industry. AIME.

    Google Scholar 

  • Locatelli, M. (2000). Simulated annealing algorithms for continuous global optimization: Convergence conditions. Journal of Optimization Theory and Applications, 104, 121–133.

    Google Scholar 

  • Madowe, A. (2016). Design and implementation of steeper slope angles on a kimberlite open pit diamond operation—a practical approach. The Journal of the Southern African Institute of Mining and Metallurgy, 116, 723–730.

    Google Scholar 

  • Malli, T., Pamukcu, C., & Kose, H. (2015). Determination of optimum production capacity and mine life considering net present value in open pit mining at different overall slope angles. Acta Montanistica Slovaca, 20(1), 62–70.

    Google Scholar 

  • Meyer, M. (1969). Applying linear programming to the design of ultimate pit limits. Management Science, 16(2), B121–B135.

    Google Scholar 

  • Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., & Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1092.

    Google Scholar 

  • Milani, G. A. (2016). Genetic algorithm with zooming for the determination of the optimal open pit mines layout. The Open Civil Engineering Journal, 10, 301–322.

    Google Scholar 

  • Mousavi, A., Kozan, E., & Liu, S. Q. (2016). Open-pit block sequencing optimization: A mathematical model and solution technique. Engineering Optimization, 48(11), 1932–1950.

    Google Scholar 

  • Osanloo, M., Gholamnejad, J., & Karimi, B. (2008). Long-term open pit mine production planning: A review of models and algorithms. International Journal of Mining, Reclamation and Environment, 22(1), 3–35.

    Google Scholar 

  • Ozdamar, L., & Demirhan, M. (2000). Experiments with new stochastic global optimization search techniques. Computers and Operations Research, 27(9), 841–865.

    Google Scholar 

  • Pana, M. T. (1965). The simulation approach to open pit design. In Proc. 5th APCOM (pp. 1–24). University of Arizona, Tucson, Arizona, USA.

  • Parra, A., Morales, N., Vallejos, J., & Nguyen, P. M. V. (2018). Open pit mine planning considering geomechanical constraints. International Journal of Mining, Reclamation and Environment, 32(4), 221–238.

    Google Scholar 

  • Picard, J. C. (1976). Maximum closure of a graph and applications to combinatorial problems. Management Science, 22(11), 1268–1272.

    Google Scholar 

  • Poniewierski, J. (2018). Pseudoflow explained. Deswik, A discussion of Deswik pseudoflow pit optimization in comparison to Whittle LG pit optimization.

  • Ramazan, S., & Dimitrakopoulos, R. (2004). Recent applications of operations research and efficient MIP formulations in open pit mining. Mining, Metallurgy, and Exploration Transactions, 316, 73–78.

    Google Scholar 

  • Rychkun, E., & Chen, T. (1979). Open pit mine feasibility method and application at placer development. In A. Weiss (Ed.), Computer methods for the 80’s in the Mineral Industry (pp. 304–309). AIME.

    Google Scholar 

  • Salomon, D. (2006). Curves and surfaces for computer graphics. Springer Science & Business Media. https://doi.org/10.1007/0-387-28452-4

    Article  Google Scholar 

  • Sattarvand, J., & Shisvan, M. S. (2012). Modelling of accurate variable slope angles in open-pit mine design using spline interpolation. Archives of Mining Sciences, 57(4), 921–932.

    Google Scholar 

  • Sayadi, A. R., Fathianpour, N., & Mousavi, A. (2011). Open pit optimization in 3D using a new artificial neural network. Archives of Mining Sciences, 56(3), 389–403.

    Google Scholar 

  • Shenggui, Z., & Starfield, A. M. (1985). Dynamic programming with colour graphics smoothing for open pit design on a personal computer. Geotechnical and Geological Engineering, 3(1), 27–34. https://doi.org/10.1007/BF00881339

    Article  Google Scholar 

  • Soltani Khaboushan, A., & Osanloo, M. (2020). A set of classified integer programming (IP) models for optimum transition from open pit to underground mining methods. Natural Resources Research, 29, 1543–1559.

    Google Scholar 

  • Souza, F. R., Burgarelli, H. R., Nader, A. S., Ortiz, C. E. A., Chaves, L. S., Carvalho, L. A., Torres, V. F. N., Camara, T. R., & Galery, R. (2018). Direct block scheduling technology: Analysis of avidity. REM - International Engineering Journal, 71(1), 97–104.

    Google Scholar 

  • Stuart, N. J. (1992). Pit optimisation using solid modelling and the Lerchs Grossman algorithm. International Journal of Surface Mining and Reclamation, 6(1), 19–29.

    Google Scholar 

  • Talbi, E. (2009). Metaheuristics: From design to implementation. John Wiley & Sons Inc.

    Google Scholar 

  • Wilke, F. L., & Wright, E. A. (1984). Determining the optimal ultimate pit design for hard rock open pit mines using dynamic programming. Erzmetall, 37, 139–144.

    Google Scholar 

  • Wright, E. A. (1999). Moving Cone II—a simple algorithm for optimum pit limits design. In Proc. 28th APCOM, Colorado (pp. 367–374).

  • Zhao, Y., & Kim, Y. C. (1992). A new optimum pit limit design algorithm. In Proc. 23th APCOM (pp. 423–434).

Download references

Author information

Authors and Affiliations

  1. Mining Engineering Department, Middle East Technical University, Ankara, Turkey

    Firdevs Kübra Altuntov & Mustafa Erkayaoğlu

Authors
  1. Firdevs Kübra Altuntov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mustafa Erkayaoğlu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mustafa Erkayaoğlu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Altuntov, F.K., Erkayaoğlu, M. A New Approach to Optimize Ultimate Geometry of Open Pit Mines with Variable Overall Slope Angles. Nat Resour Res 30, 4047–4062 (2021). https://doi.org/10.1007/s11053-021-09911-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11053-021-09911-8

Keywords

Search

Navigation