Particle Swarm Procedure for theCapacitated 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

Maximal Open Pit problem: to determine the maximal gain expected from the extraction

the net value of extracting block i

ib

otherwise. 0

extracted is block if 1 ixi

objective function .iNi

i xb

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

rspredecesso block iBi

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

(2) . or (1) , Subject to

Max (MOP)

NixNiBjxx

xb

i

iij

iNi

i

100

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 1

1

the net value of extracting block i

ib

pi weight of block i

otherwise. period during extracted is block if

01 tix t

i

N can be replaced by the maximal open pit N* = (S – {s})

(7) . or

(6)

(5)

4 1 Subject to

(3) Max (SBE)

t

i

i

1

TtNix

TtCxp

TtNiBjxx

Nix

xb

t

t

iN

i

i

t

i

t

l

l

j

T

t

t

i

t

iNi

t

iT

t

,,1,10

,,1

,,1,,0

)1(

1

1

1

Scheduling block extraction ↔ RCPSP

• Open pit extraction ↔ project

• Each block extraction ↔ activity

• Precedence relationship derived from the maximal pit slope constraints

rspredecesso block iBP ii

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

1)1( t

ib

rspredecesso block iBP ii

Genotype representation of solution

Similar to Hartman’s priority value encoding for RCPSP

priority of scheduling block i extraction

],,[ 1 NprprPR

]1,0[ipr

11

N

iipr

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 (Tolwinski and Underwood) determined by referring to the spanning cone SCi of block i

ib

iSCj

ji bb

.:* ijiNjSCi before extracted bemust

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.

ib

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

.,,1 MPRPRP

k

PR

PRb

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

k

PR

PRb

k

i

k

i

k

i

k

ii

k

i

k

i

k

ik

i

k

i

k

i

prvcppr

prprbrcprprrcwvcvc

:

)()(: 2211

define weand

populationnext in sol.current

sol.current

velocitycurrent new

locitycurrent vesol.best

of achiev.best sol.current

k

k

kk

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

k

PR

PRb

k

i

k

i

k

i

k

ii

k

i

k

i

k

ik

i

k

i

k

i

prvcppr

prprbrcprprrcwvcvc

:

)()(: 2211

define weand

.1

*,,10

1

N*

i

k

i

k

i

kk

pr

NiprPPRPR

have tognormalisinby and

thatso tingby transla obtained is

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

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.

100)(

)(%

iter.first at

iter.first at -

PRbv

PRbvvall

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 accordinglyto 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

Each of the 10 other larger problemsare solved 5 times with parametervalues in set 1.

Average value Best valueWorst value

vgreedy value of the solutiongenerated by decoding the genotype vector where priority of blocks are proportional to their lookahead values

the worst values vw1ρ is better than vgreedy for all problems.

the percentage of improvement of va1ρ over vgreedy 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)