TRANSACTION PAPER

 

Investigating continuous time open pit dynamics

 

 

H. Askari-Nasab^I; S. Frimpong^II; J. Szymanski^I

^ISchool of Mining and Petroleum Engineering, University of Alberta, Edmonton, Alberta, Canada
^IIDepartment of Mining and Nuclear Engineering, University of Missouri-Rolla, Rolla, USA

 

 

---

SYNOPSIS

Current mine production planning, scheduling, and allocation of resources are based on mathematical programming models. In practice, the optimized solution cannot be attained without examining all possible combinations and permutations of the extraction sequence. Operations research methods have limited applications in large-scale surface mining operations because the number of variables becomes too large. The primary objective of this study is to develop and implement a hybrid simulation framework for the open pit scheduling problem. The paper investigates the dynamics of open pit geometry and the subsequent material movement as a continuous system described by time-dependent differential equations. The continuous open pit simulator (COPS) implemented in MATLAB, based on modified elliptical frustum is used to model the evolution of open pit geometry in time and space. Discrete open pit simulator (DOPS) mimics the periodic expansion of the open pit layouts. Function approximation of the discrete simulated push-backs provides the means to convert the set of partial differential equations (PDEs), capturing the dynamics of open pit layouts, to a system of ordinary differential equations (ODEs). Numerical integration with the Runge-Kutta scheme yields the trajectory of the pit geometry over time with the respective volume of materials and the net present value (NPV) of the mining operation. A case study of an iron ore mine with 114 000 blocks was carried out to verify and validate the model. The optimized pit limit was designed using Lerchs-Grossman's algorithm. The best-case annual schedule, generated by the shells node in Whittle Four-X yielded an NPV of $449 million over a 21-year mine life at a discount rate of 10% per annum. DOPS best scenario out of 2 500 simulation iterations resulted in an NPV of $443 million and COPS yielded an NPV of $440 million over the same time span. The hybrid simulation model is the basis for future research using reinforcement learning based on goal-directed intelligent agents.

---

 

 

“Full text available only in PDF format”

 

 

References

1. CHANDA, E.K. and WILKE, F.L. An EPD model of open pit short term production scheduling optimization for stratiform orebodies. Proceedings of 23rd APCOM Symposium, SME (ed.), 1992. pp. 759-768.         [ Links ]

2. ELVELI, B. Open pit mine design and extraction sequencing by use OR and AI concepts. International Journal of Surface Mining. Reclamation and Environment. 1995. vol. 9, pp. 149-153.         [ Links ]

3. ERARSLAN, K. AND CELEBI, N. A simulative model for optimum open pit design. The Canadian Mining and Metallurgical Bulletin. 2001. vol. 94, pp. 59-68.         [ Links ]

4. HALATCHEV, R.A. A model of discounted profit variation of open pit production sequencing optimization. Proceedings of Application of Computers and Operations Research in the Mineral Industry, Tucson, Arizona. Taylor & Francis Group(ed.), London, 2005. pp. 315-323.         [ Links ]

5. ONUR, A.H. AND DOWD, P.A. Open pit optimization-part 2: production scheduling and inclusion of roadways. Transactions of the Institution of Mining and Metallurgy. 1993. vol. 102, pp. A105-A113.         [ Links ]

6. TOLWINSKI, B. and UNDERWOOD, R. An algorithm to estimate the optimal evolution of an open pit mine. Proceedings of 23rd APCOM Symposium, University of Arizona. SME (ed.), Littleton, Colorado, 1992. pp. 399-409.         [ Links ]

7. DOWD, P.A. and ELVAN, L. Dynamic programming applied to grade control in sub-level open stopping. Trans. IMM. 1987. vol. 96, pp. A171-A178.         [ Links ]

8. CHANDA, E.K. and DAGDELEN, K. Optimal blending of mine production using goal programming and interactive graphics system. International Journal of Surface Mining Reclamation and Environment. 1995. vol. 9, pp. 203-208.         [ Links ]

9. YOUDI, Z., QINGZIANG, C., and LIXIN, W. Combined approach for surface mine short-term planning optimization. Proceedings of 23rd APCOM Symposium, SME (ed.), Colorado, 1992. pp. 499-506.         [ Links ]

10. MANN, C. and WILKE, F.L. Open pit short term mine planning for grade control-a combination of CAD techniques and linear programming. Proceedings of 23rd APCOM Symposium, SME (ed.), Colorado, 1992. pp. 487-497.         [ Links ]

11. FRIMPONG, S., ASA, E., and SZYMANSKI, J. MULSOPS: multivariate optimized pit shells simulator for tactical mine planning. International Journal of Surface Mining, Reclamation & Environment. 1998. vol. 12, pp. 163-169.         [ Links ]

12. FRIMPONG, S., ASA, E., and SUGLO, R.S. Numerical simulation of surface mine production system using pit shell simulator. Mineral Resources Engineering. 2001. vol. 10, pp. 185-203.         [ Links ]

13. ASKARI-NASAB, H. and SZYMANKSI, J. Modelling open pit dynamics using Monte Carlo simulation. Proceedings of Computer Applications in the Minerals Industry (CAMI), Banff, Alberta, Canada. On CD-ROM, The Reading Matrix Inc., CA, USA, 2005. pp. 21-32.         [ Links ]

14. ASKARI-NASAB, H., AWUAH-OFFEI, K., and FRIMPONG, S. Stochastic simulation of open pit pushbacks with a production simulator. Proceedings of CIM Mining Industry Conference and Exhibition, Edmonton, Alberta, Canada. 2004. pp. on CD-ROM        [ Links ]

15. CARTWRIGHT, J.H.E. and PIRO, O. The dynamics of Runge-Kutta methods. Int. J. Bifurcations Chaos. 1992. vol. 2, pp. 427-449        [ Links ]

16. WHITTLE PROGRAMMING PTY, LTD. Whittle strategic mine planning software, Gemcom Software International Inc., 1998-2004.         [ Links ]

17. ABRAMOWITZ, M. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Stegun, I.A.E. (eds). New York: Dover, 1972. 880.         [ Links ]

18. LERCHS, H. and GROSSMANN, I.F. Optimum design of open-pit mines. The Canadian Mining and Metallurgical Bulletin, Transactions. 1965. vol. LXVIII, pp. 17-24.         [ Links ]

19. DEUTSCH, C.V. and JOURNEL, A.G. (eds). GSLIB geostatistical software library and user's guide. Applied geostatistics series. New York, Oxford University Press, 1998.         [ Links ]

20. KRIGE, D.G. A statistical approach to some basic mine valuation and allied problems at the Witwatersrand, Masters thesis, University of Witwatersrand, South Africa, 1951.         [ Links ]

21. MARQUARDT, D. An algorithm for least squares estimation of nonlinear parameters. SIAM J. Appl. Math. 1963. vol. 11, pp. 431-441.         [ Links ]

 

 

Paper received Apr. 2007
Revised paper received Apr. 2007