Metaheuristic Solutions for Capacitated Open Pit Mining Problem

1. http://www.iro.umontreal.ca/~ferland/ http://www.researchandpractise.com/vrp/

2. Capacitated Open Pit Mining Problem Semya Elaoud, Sfax University (Tunesia) Jacques A. Ferland, University of Montreal Jonathan Bellemare, University of Montreal Jorge Amaya, University of Chile

4. RIOT Mining Problem web site: http://riot.ieor.berkeley.edu/riot/Applications/OPM/OPMInteractive.html the net value of extracting block i objective function Maximal Open Pit problem: to determine the maximal gain expected from the extraction

5. Maximal pit slope constraints to identify the set Bi of predecessor blocks that have to be removed before block i

6. Maximal pit slope constraints to identify the set Bi of predecessor blocks that have to be removed before block i

7. Maximal pit slope constraints to identify the set Bi of predecessor blocks that have to be removed before block i

9. Use the open pit graph G = (V, A) to specify the maximal pit slope constraints The maximal pit slope constraints:

10. (MOP) equivalent to determine the maximal closure of G = (V, A) • Equivalent to determine the minimum cut of the associated Picard’s graph where The maximal open pit is equal to N* = (S – {s})

12. Scheduling block extraction Account for operational constraints: Ct the maximal weight that can be extracted during period t and for the discount factor during the extracting horizon: discount rate per period

13. piweight of block i the net value of extracting block i N can be replaced by the maximal open pit N* = (S – {s})

14. Scheduling block extraction ↔ RCPSP • Open pit extraction ↔ project • Each block extraction ↔ activity • Precedence relationship derived from the open pit graph

15. Scheduling block extraction ↔ RCPSP

16. Scheduling block extraction ↔ RCPSP

17. Scheduling block extraction ↔ RCPSP • Open pit extraction ↔ project • Each block extraction ↔ activity • Precedence relationship derived from the open pit graph • Reward associated with activity (block) i depends of the extraction period t

18. Solution encoding and decoding

19. Decoding a block listinto a schedule Serial decoding • Initiate the first extraction period t = 1 • During any period t: - The next block to be extracted is the first block in the rest of the block list (including the blocks not extracted yet) having all their predecessors already extracted such that the capacity Ct is not exceeded by its extraction. Include this block in the newsol block list. - If no such block exists, then a new extraction period (t + 1) is initiated.

20. Metaheuristic solution approach

21. Outline of the solution approach

22. Initial solution

37. First neighborhood moving one ore block

41. Unique ore block process on cluster i

42. Second neighborhood moving multiple ore blocks

44. Multiple ore block process on cluster i

45. Implementation of a metaheuristic procedure

46. Numerical experimentation

47. Second encoding of the solutionandParticle Swarm Solution Approach

48. Genotype representation of solution Similar to Hartman’s priority value encoding for RCPSP priority of scheduling block i extraction

49. Decoding of a representation PR into a solution x • Serial decoding to schedule blocks sequentially one by one to be extracted • To initiate the first extraction period t = 1: remove the block among those having no predecessor (i.e., in the top layer) having the highest priority. • During any period t, at any stage of the decoding scheme: the next block to be removed is one of those with the highest priority among those having all their predecessors already extracted such that the capacity Ct is not exceeded by its extraction. If no such block exists, then a new extraction period (t + 1) is initiated.

50. Priority of a block • Consider its net value bi and impact on the extraction of other blocks in future periods • Block lookahead value (Tolwinski and Underwood) determined by referring to the spanning cone SCi of block i