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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: http://riot.ieor.berkeley.edu/riot/Applications/OPM/OPMInteractive.html bi the net value of extracting block i 1 if block i is extracted xi 0 otherwise. objective function bi xi . iN Maximal Open Pit problem: to determine the maximal gain expected from the extraction Maximal pit slope constraints to identify the set Bi of predecessor blocks that have to be removed before block i Bi block i predecessors Maximal pit slope constraints to identify the set Bi of predecessor blocks that have to be removed before block i (MOP) Max bi xi iN Subject to x j xi 0 j Bi , i N (1) xi 0 or 1 i N. (2) 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: 1 discount rate per period 1 pi weight of block i bi the net value of extracting block i xit 1 if block i is extractedduring period t 0 otherwise. T bi (SBE) Max xt (3) iN (1 ) t 1 i t 1 N can be replaced by 4 T Subject to xi 1 iN t t t 1 the maximal open pit x j xi 0 j Bi , i N , t 1, , T (5) N* = (S – {s}) l t l 1 pi xi Ct t 1, , T (6) t iN xit 0 or 1 i N , t 1,, T . (7) Scheduling block extraction ↔ RCPSP • Open pit extraction ↔ project • Each block extraction ↔ activity • Precedence relationship derived from the maximal pit slope constraints Pi Bi block i predecessors Scheduling block extraction ↔ RCPSP • Open pit extraction ↔ project • Each block extraction ↔ activity • Precedence relationship derived from the maximal pit slope constraints Pi Bi block i predecessors • Reward associated with activity (block) i depends of the extraction period t bi (1 )t 1 Genotype representation of solution Similar to Hartman’s priority value encoding for RCPSP PR [ pr1 ,, prN ] pri [0,1] priority of scheduling block i extraction N pr 1 i 1 i 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. Priority of a block • Consider its net value bi and impact on the extraction of other blocks in future periods • Block lookahead value bi (Tolwinski and Underwood) determined by referring to the spanning cone SCi of block i SCi j N * : i must be extracted before j . i bi b jSCi j 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 bi • 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 P PR1 , , PR M . GRASP • Denote k PR the best achievement of the individual k up to the current iteration PRb the best overall genotype vector achieved up to the current iteration Particle Swarm Procedure • Denote k PR the best achievement of the individual k up to the current iteration PRb the best overall genotype vector achieved up to the current iteration Modification of the individual vector k at each iteration k vc : wvc c r ( pr i pri k ) c2 r2ki ( prbi pri k ) i k i k k 1 1i and wedefine ppri k : vcik pri k currentsol. k best achiev. of k new current velocity currentsol. k best sol. in next population current velocity currentsol. k Particle Swarm Procedure • Denote k PR the best achievement of the individual k up to the current iteration PRb the best overall genotype vector achieved up to the current iteration Modification of the individual vector k at each iteration k vc : wvc c r ( pr i pri k ) c2 r2ki ( prbi pri k ) i k i k k 1 1i and wedefine ppri k : vcik pri k PRk is obtained by transla ting PPRk so that pri k 0 i 1,, N * and by normalising to have N* pri 1. k i 1 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 valρ : the average of the best values v(PRb) achieved vblρ : the best values v(PRb) achieved %lρ : the average % of improvement val - v( PRb) at first iter. %l 100 v( PRb) at first iter. itlρ : the last iteration where an improvement of PRb occurs. • Then for each set of parameter l, we compute the average values val, vbl, %l , and itl over the 10 problems. Results Impact of β in GRASP Comparing rows 1, 2, and 3 , we observe that the values of val and vbl 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 val 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, c1 and c2. 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 = 0.7298 and c1 = c2 = 1.49618 gives acceptable results. Impact of particle swarm process the percentage of improvement of va1ρ over vgreedy better the worst values vw1ρ isranges than from problems. vgreedy for all2.32% to 52.24% Average Each of the 10 other larger problems value are solved 5 times with parameter Best vgreedy value of the solution values in set 1. value generated by decoding the Worst genotype vector where value priority of blocks are proportional to their lookahead values Future work • Solve larger problems • Include other operational constraints found in real world applications • Compare with other evolutionary approaches (genetic algorithm)

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open pit, scheduling problem, particle swarm, genetic algorithm, genotype representation, open pit mining, parameter values, united states, matthias ehrgott, beijing jiaotong university, university of applied sciences, harbin institute of technology, jan van vuuren, numerical results, autonomous robots

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posted: | 9/8/2010 |

language: | English |

pages: | 27 |

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