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Abstract

Mining is the art of extraction of material from the earth surface for profit. The main objective of mining companies is to maximize net present value (NPV) while considering health, safety and environmental issues. Revolution in the mining industry has seen mathematical modelling being the key component in improving the mining process. Mining models has been proven as the next frontier that needs all the focus it can get. There are three mining stages namely: mining or extraction, processing and refining involved in this exercise. The major optimisation problems found in surface mining operations are ultimate pit limit problem, cut-off grade optimisation and open pit production scheduling. In order to maximise the output from the three mining stages, there is need to apply mathematical equations and models to optimise the whole process. There are extensive research studies since 1960s on development of mathematical models optimising mining operations. However, the mining industry has not fully appreciated and applied all these optimisation models. This paper highlights the strides made in development of mathematical models in the mining industry and their application. It discusses ways in which the mining industry can increase the application of these models to improve the output generated from the mining projects.

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References

  • Asad, M.W.A., Dimitrakopoulos, R.: A heuristic approach to stochastic cut-off grade optimisation for open pit mining complexes with multiple processing streams. Resour. Policy 38, 591–597 (2013)

    Article  Google Scholar 

  • Asad, M.W.A.: Development of generalized cut-off grade optimisation algorithm for open pit mining operations. J. Eng. Appl. Sci. 21(2), 119–127 (2002)

    Google Scholar 

  • Ataee-pour, M.: A linear model for determination of block economic values. In: The 19th International Mining Congress and Fair of Turkey, Ä°zmir, pp. 289–294 (2005)

    Google Scholar 

  • Carlson, T.R., Erickson, J.D., O’Brain, D.T., Pana, M.T.: Computer techniques in mine planning. Min. Eng. 18(5), 53–56 (1966)

    Google Scholar 

  • Cetin, E., Dowd, P.A.: Multiple cut-off grade optimization by genetic algorithms and comparison with grid search method and dynamic programming. J. South Afr. Inst. Min. Metall. 116(7), 681–688 (2016)

    Article  Google Scholar 

  • Dagdelen, K., Kawahata, K.: Value creation through strategic mine planning and cut-off grade optimization. Min. Eng. 60(1), 39–45 (2008)

    Google Scholar 

  • David, M., Dowd, P., Korobov, S.: Forecasting departure from planning in open pit design and grade control. In: Proceedings of the 12th Symposium on the Application of Computers and Operations Research in the Mineral Industries (APCOM), vol. 2, pp. F131–F142. Colorado School of Mines, Golden (1974)

    Google Scholar 

  • Dehghani, H., Ataee-pour, M.: The role of economic uncertainty on the block economic value – a new valuation approach. Arch. Min. Sci. 57(4), 991–1014 (2012)

    Google Scholar 

  • Dimitrakopoulos, R.G., Abdel Sabour, S.A.: Evaluating mine plans under uncertainty: can the real options make a difference? Resour. Policy 32(3), 116–125 (2007)

    Article  Google Scholar 

  • Dowd, P.A., Xu, C.: MINVEST User’s Manual, p. 272. Computer Aided Mine Design Centre, University of Leeds (2000)

    Google Scholar 

  • Dowd, P.A.: Application of dynamic and stochastic programming to optimise cut-off grades and production rates. Trans. Inst. Min. Metall. Sect. A. Min. Technol. 85(1), 22–31 (1976)

    Google Scholar 

  • Elahizeyni, E., Kakaie, R., Yousefi, A.: A new algorithm for optimum open pit design: floating cone method III. J. Min. Environ. 2(2), 118–125 (2011)

    Google Scholar 

  • Gholamnejad, J.: Incorporation of rehabilitation cost into the optimum cut-off grade determination. J. South Afr. Inst. Min. Metall. 108(2), 89–94 (2009)

    Google Scholar 

  • Githiria, J., Musingwini, C.: Comparison of cut-off grade models in mine planning for improved value creation based on NPV. In: Proceedings of the 6th Regional Conference of the Society of Mining Professors (SOMP), Johannesburg, pp. 347–362 (2018)

    Google Scholar 

  • Githiria, J., Musingwini, C.: A stochastic cut-off grade optimisation model to incorporate uncertainty for improved project value. J. South Afr. Inst. Min. Metall. 119(3), 1–12 (2019)

    Article  Google Scholar 

  • Githiria, J., Muriuki, J., Musingwini, C.: Development of a computer-aided application using Lane’s algorithm to optimise cut-off grade. J. South Afr. Inst. Min. Metall. 116(11), 1027–1035 (2016)

    Article  Google Scholar 

  • Goldfarb, D., Chen, W.: On strongly polynomial dual simplex algorithms for the maximum flow problem. Spec. Issue Math. Program. B 78(2), 159–168 (1997)

    Article  Google Scholar 

  • Henning, U.: Calculation of cut-off grade. Can. Min. J. 84(3), 54–57 (1963)

    Google Scholar 

  • Henry, E., Marcotte, D., Samis, M.: Valuing a mine as a portfolio of European call options: the effect of geological uncertainty and implications for strategic planning. Geostat. Banff. Quant. Geol. Geostat. 14, 501–510 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Johnson, T.B.: Optimum open-pit mine production scheduling. In: Weiss A. (ed.) A Decade of Digital Computing in the Mineral Industry, pp. 539–562. AIME, New York (1969)

    Google Scholar 

  • Khalokakaie, R.: Optimum open pit design with modified moving cone II methods. J. Eng. Tehran Univ. 4(3), 297–307 (2006)

    Google Scholar 

  • Koenigsberg, E.: The optimum contours of an open pit mine: an application of dynamic programming. In: Proceedings of the 17th Symposium on the Application of Computers and Operations Research in the Mineral Industries (APCOM), pp. 247–287. AIME, New York (1982)

    Google Scholar 

  • Lane, K.F.: Choosing the optimum cut-off grade. Colorado Sch. Min. Q. 59(4), 811–829 (1964)

    Google Scholar 

  • Lane, K.F.: The Economic Definition of Ore: Cut-off Grade in Theory and Practice, p. 149. Mining Journal Books, London (1988)

    Google Scholar 

  • Lerchs, H., Grossmann, I.F.: Optimum design of open pit mines. Trans. CIM 68, 17–24 (1965). Joint CORS and ORSA Conference, Montreal, May

    Google Scholar 

  • Li, S., Yang, C., Lu, C.: Cut-off grade optimization using stochastic programming in open-pit mining. In: IEEE, pp. 66–69 (2012). https://doi.org/10.1109/icicse.2012.24

  • Muir, D.C.W.: Pseudoflow, new life for Lerchs-Grossmann pit optimisation. Orebody Model. Strateg. Mine Plan. Spectr. Ser. 14, 97–104 (2004)

    Google Scholar 

  • Musingwini, C.: Presidential address: optimization in underground mine planning – developments and opportunities. J. South Afr. Inst. Min. Metall. 116(9), 1–12 (2016)

    Article  Google Scholar 

  • Myburgh, C.A., Deb, K., Craig, S.: Applying modern heuristics to maximising net present value through cut-off grade optimisation. In: Proceedings of Orebody Modelling and Strategic Mine Planning Symposium, Melbourne, pp. 155–164 (2014)

    Google Scholar 

  • Osanloo, M., Gholamnejad, J., Karimi, B.: Long-term open pit mine production planning: a review of models and algorithms. Int. J. Min. Reclam. Environ. 22(1), 3–35 (2008). https://doi.org/10.1080/17480930601118947

    Article  Google Scholar 

  • Roman, R.J.: The role of time value of money in determining an open pit mining sequence and pit limits. In: Proceedings of the 12th International Symposium on Application of Computers and Operation Research in the Minerals Industries (APCOM), pp. 72–85 (1974)

    Google Scholar 

  • Samavati, M., Essam, D., Nehring, M., Sarker, R.: Open-pit mine production planning and scheduling: a research agenda. In: Data and Decision Sciences in Action, pp. 221–226 (2018)

    Google Scholar 

  • Whittle, J.: A decade of open pit mine planning and optimisation – the craft of turning algorithms into packages. In: Proceedings APCOM 1999 Computer Applications in the Minerals Industries 28th International Symposium, Colorado School of Mines, Golden, pp. 15–24 (1999)

    Google Scholar 

  • Whittle, J., Wharton, C.: Optimising cut-offs over time. In: Proceedings of the 25th International Symposium on the Application of Computers and Mathematics in the Mineral Industries, Brisbane, pp. 261–265 (1995)

    Google Scholar 

  • Wright, E.A.: MOVING CONE II - a simple algorithm for optimum pit limits design. In: Proceedings of the 28th Symposium on the Application of Computers and Operations Research in the Mineral Industries (APCOM), pp. 367–374 (1999)

    Google Scholar 

  • Yamatomi, J., Mogi, G., Akaike, A., Yamaguchi, U.: Selective extraction dynamic cone algorithm for three-dimensional open pit designs. In: Proceedings of the 25th Symposium on the application of computers and operations research in the mineral industries (APCOM), Brisbane, pp. 267–274 (1995)

    Google Scholar 

  • Yasrebi, A.B., Wetherelt, A., Foster, P., Kennedy, G., Ahangaran, D.K., Afzal, P., Asadi, A.: Determination of optimised cut-off grade utilising non-linear programming. Arab. J. Geosci. 8(10), 8963–8967 (2015)

    Article  Google Scholar 

  • Zhao, H., Kim, Y.C.: A new optimum pit limit design algorithm. In: Proceedings of the 23rd International Symposium on the Application of Computers and Operations Research in the Mineral Industries, pp. 423–434 (1992)

    Google Scholar 

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Githiria, J. (2020). Review of Mathematical Models Applied in Open-Pit Mining. In: Topal, E. (eds) Proceedings of the 28th International Symposium on Mine Planning and Equipment Selection - MPES 2019. MPES 2019. Springer Series in Geomechanics and Geoengineering. Springer, Cham. https://doi.org/10.1007/978-3-030-33954-8_10

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  • DOI: https://doi.org/10.1007/978-3-030-33954-8_10

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