Particle Swarm Procedure for the Capacitated Open Pit Mining Problem Jacques A. Ferland, University of Montreal Jorge Amaya, University of Chile Melody Suzy Djuimo, University of Montreal ICARA, December 2006

RIOT Mining Problem web site: Maximal Open Pit problem: to determine the maximal gain expected from the extraction the net value of extracting block i objective function

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

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

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

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

Scheduling block extraction ↔ RCPSP Open pit extraction ↔ project Each block extraction ↔ activity Precedence relationship derived from the maximal pit slope constraints

Scheduling block extraction ↔ RCPSP Open pit extraction ↔ project Each block extraction ↔ activity Precedence relationship derived from the maximal pit slope constraints Reward associated with activity (block) i depends of the extraction period t

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

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 C t is not exceeded by its extraction. If no such block exists, then a new extraction period (t + 1) is initiated.

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

Genotype priority vector generation Several different genotype priority vectors can be randomly generated with a GRASP procedure biased to give higher priorities to blocks i having larger lookahead values Several feasible solutions of (SBE) can be obtained by decoding different genotype vectors generated with the GRASP procedure.

Particle Swarm Procedure Evolutionary process evolving in the set of genotype vectors to converge to an improved feasible solution of (SBE). Initial population P of M genotype vectors (individuals) generated using GRASP Denote the best achievement of the individual k up to the current iteration the best overall genotype vector achieved up to the current iteration

Particle Swarm Procedure Denote the best achievement of the individual k up to the current iteration the best overall genotype vector achieved up to the current iteration Modification of the individual vector k at each iteration

Particle Swarm Procedure Denote the best achievement of the individual k up to the current iteration the best overall genotype vector achieved up to the current iteration Modification of the individual vector k at each iteration

Numerical Results 20 problems randomly generated over a two dimensions grid having 20 layers and being 60 blocks wide. The 10 problems having smaller optimal pit are used to analyse the impact of the parameters. We compare the results for 12 different set of parameter values l

For each set of parameter values l, each problem ρ is solved 5 times to determine va lρ : the average of the best values v(PRb) achieved vb lρ : the best values v(PRb) achieved % lρ : the average % of improvement it lρ : the last iteration where an improvement of PRb occurs. Then for each set of parameter l, we compute the average values va l, vb l, % l, and it l over the 10 problems.

Results

Impact of β in GRASP Comparing rows 1, 2, and 3, we observe that the values of va l and vb l decrease while the value of % l increases as the value of β increases. The same observations apply for rows 4, 5, and 6. Bias to increase priority of blocks with larger lookahead value as β decreases. ( β = 100 is equivalent to assign priority randomly). Individual genotype vectors in initial population tend to be better as β decreases.

Impact of population size M The value va l in row 1 is larger than in row 4. The same is true if we compare rows 2 and 5, and rows 3 and 6. This indicates that the values of the solutions generated are better when the size of the population is larger. This makes sense since we generate a larger number of different solutions.

Impact of particle swarm parameters Comparing the results in rows 2, 7, 8,and 9, and those in rows 5, 10, 11, and 12, there is no clear impact of modifying the values of the parameters w, c 1 and c 2. Note that the values w = 0.7, c1 = 1.4, and c2 = 1.4 were selected accordingly to the authors in [19] who shown that setting the values of the parameters close to w = and c1 = c2 = gives acceptable results.

Impact of particle swarm process Each of the 10 other larger problems are solved 5 times with parameter values in set 1. Average value Best value Worst value v greedy value of the solution generated by decoding the genotype vector where priority of blocks are proportional to their lookahead values the worst values vw 1ρ is better than v greedy for all problems. the percentage of improvement of va 1ρ over v greedy ranges from 2.32% to 52.24%

Future work Solve larger problems Include other operational constraints found in real world applications Compare with other evolutionary approaches (genetic algorithm)