Determining an optimum long term production schedule is an important part of the planning process of any open pit mine; however, the associated optimization problem is demanding and hard to deal with, as it involves large datasets and multiple hard and soft constraints which makes it a large combinatorial optimization problem. In this paper a procedure has been proposed to apply a relatively new and computationally less expensive metaheuristic technique known as particle swarm optimization (PSO) algorithm to this computationally challenging problem of the open pit mines. The performance of different variants of the PSO algorithm has been studied and the results are presented. 1. Introduction Production scheduling of the open pit mines is a difficult and complex optimization problem. It aims to define the most profitable extraction sequence of the mineralized material from the ground that produces maximum possible discounted profit while satisfying a set of physical and operational constraints. Block model representation of the ore body commonly plays the role of the starting point for the planning and production scheduling of the open pit mines. The block model divides the mineralized body and the surrounded rock into a three-dimensional array of regular size blocks. A set of attributes, for example, grade, tonnage, density, and so forth, are then assigned to each one of these blocks estimated using some form spatial interpolation technique, for example, kriging, inverse distance weighting method, and so forth, and the exploratory drill hole sample data. The blocks are then divided into two categories, that is, ore and waste blocks. The blocks whose prospective profit exceeds their processing cost are categorized as an ore block to be sent for processing once mined while the rest are the waste blocks. An economic value is then assigned to each block by taking into account its group; that is, it is categorized as ore or waste, the commodity price, mining, processing, and marketing costs. The next step is to define the final contour of the pit, that is, the limit to which it is economically feasible to mine by solving the ultimate pit limit problem (UPL). The mathematical formulation of the UPL problem can be defined as follows: where represents the economic value of block , represents the total number of blocks in the block model, represents a binary variable corresponding to block which takes the value 1 if block is inside the ultimate pit limits and 0 otherwise, and represents the predecessor group of block . The solution to this problem identifies the
References
[1]
H. Lerchs and I. F. Grossmann, “Optimum design of open pit mines,” Canadian Institute of Mining Transactions, vol. 68, pp. 17–24, 1965.
[2]
T. B. Johnson, Optimum Open Pit Mine Production Scheduling, University of California, Berkeley, Calif, USA, 1968.
[3]
K. Dagdelen and T. B. Johnson, “Optimum open pit mine production scheduling by Lagrangian parameterization,” in Proceedings of the 19th International Symposium Application of Computers and Operations Research in the Mineral Industry (APCOM '86), pp. 127–142, Littleton, Colo, USA, 1986.
[4]
L. Caccetta, P. Kelsey, and L. Giannini, “Open pit mine production scheduling,” in Proceedings of 3rd Regional APCOM International Symposium, Series 5/98, The Australasian Institute of Mining and Metallurgy, Kalgoorlie, Australia, 1998.
[5]
J. Whittle, Four-X User Manual, Whittle Programming, Melbourne, Australia, 1998.
[6]
M. E. Gershon, “Mine scheduling optimization with mixed integer programming,” Mining Engineering, vol. 35, no. 4, pp. 351–354, 1983.
[7]
L. Caccetta and S. P. Hill, “An application of branch and cut to open pit mine scheduling,” Journal of Global Optimization, vol. 27, no. 2-3, pp. 349–365, 2003.
[8]
A. H. Onur and P. A. Dowd, “Open-pit optimization. Part 2. Production scheduling and inclusion of roadways,” Transactions of the Institution of Mining and Metallurgy Section A: Mining Technology, vol. 102, pp. 105–113, 1993.
[9]
Q. Wang, “Long-term open-pit production scheduling through dynamic phase-bench sequencing,” Transactions of the Institute of Mining and Metallurgy Section A, vol. 105, pp. A99–A104, 1996.
[10]
S. Ramazan, “The new fundamental tree algorithm for production scheduling of open pit mines,” European Journal of Operational Research, vol. 177, no. 2, pp. 1153–1166, 2006.
[11]
B. Denby and D. Schofield, “Open-pit design and scheduling by use of genetic algorithms,” Transactions of the Institution of Mining and Metallurgy, Section A: Mining Technology, vol. 103, pp. A21–A26, 1994.
[12]
B. Denby and D. Schofield, “Genetic algorithms for open pit scheduling-extension into 3-dimentions,” in Proceedings of the Mine Planning and Equipment Selection Conference (MPES '96), pp. 177–185, Sao Paulo, Brazil, 1996.
[13]
B. Denby, D. Schofield, and T. Surme, “Genetic algorithms for flexible scheduling of open pit operations,” in Proceedings of the 27th Application of Computer in Mineral Industry Symposium (APCOM '98), pp. 473–483, London, UK, 1998.
[14]
M. Kumral and P. A. Dowd, “Short-term mine production scheduling for industrial minerals using multi-objective simulated annealing,” in Proceedings of the 30th Application of Computer in Mineral Industry Symposium (APCOM '02), pp. 731–742, Fairbanks, Alaska, USA, 2002.
[15]
M. Kumral and P. A. Dowd, “A simulated annealing approach to mine production scheduling,” Journal of the Operational Research Society, vol. 56, no. 8, pp. 922–930, 2005.
[16]
J. Sattarvand and C. Niemann-Delius, Long Term Open Pit Planning by Ant Colony Optimization, Institut für Bergbaukunde III, RWTH Aachen University, 2009.
[17]
J. Sattarvand and C. Niemann-Delius, “A new metaheuristic algorithm for long-term open-pit production planning / Nowy meta-heurystyczny algorytm wspomagaj？cy d？ugoterminowe planowanie produkcji w kopalni odkrywkowej,” Archives of Mining Sciences, vol. 58, no. 1, pp. 107–118, 2013.
[18]
J. A. Ferland, J. Amaya, and M. S. Djuimo, “Application of a particle swarm algorithm to the capacitated open pit mining problem,” in Autonomous Robots and Agents, S. Mukhopadhyay and G. Sen Gupta, Eds., pp. 127–133, Springer, Berlin, Germany, 2007.
[19]
M. P. Gaupp, Methods for Improving the Tractability of the Block Sequencing Problem for Open Pit Mining, Business and Economics Division, Colorado School of Mines, 2008.
[20]
R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the 6th International Symposium on Micro Machine and Human Science, pp. 39–43, October 1995.
[21]
J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948, December 1995.
[22]
F. van den Bergh, An Analysis of Particle Swarm Optimizers, Faculty of Natural and Agricultural Science, University of Pretoria, 2001.
[23]
F. van den Bergh, An Analysis of Particle Swarm Optimizers, Department of Computer Science, University of Pretoria, 2002.
[24]
P. Angeline, “Evolutionary optimization versus particle swarm optimization: philosophy and performance differences,” in Evolutionary Programming VII, V. W. Porto, N. Saravanan, D. Waagen, and A. E. Eiben, Eds., vol. 1447 of Lecture Notes in Computer Science, pp. 601–610, Springer, Berlin, Germany, 1998.
[25]
Z. Michalewicz and M. Schoenauer, “Evolutionary algorithms for constrained parameter optimization problems,” Evolutionary Computation, vol. 4, no. 1, pp. 1–32, 1996.
[26]
M. Clerc and J. Kennedy, “The particle swarm-explosion, stability, and convergence in a multidimensional complex space,” IEEE Transactions on Evolutionary Computation, vol. 6, no. 1, pp. 58–73, 2002.
