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Robust Planning for an Open-Pit MiningProblem under Ore-Grade Uncertainty
Daniel Espinoza1 Guido Lagos2 Eduardo Moreno3
Juan Pablo Vielma4
1Department of Industrial Engineering, Universidad de Chile
2H. Milton Stewart School of Industrial and Systems Engineering, Georgia Tech
3School of Engineering, Universidad Adolfo Ibáñez
4Sloan School of Management, MIT
INFORMS 2012; Phoenix, AZ
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
1 Open-pit production planning problem
2 Stochastic programming modelsMinimization of VaRMinimization of CVaRMCH & MCH-� models
3 Computational results
4 Conclusions
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Chuquicamata, Chile
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
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Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Block model. Color ∼ amount of mineral (ore grade)
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
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Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Which blocks to extract?
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
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Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Between extracted ones, which to process?
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
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Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Problem without uncertainty
Decisions: for each block b ∈ B,
extraction decision xeb ∈ {0,1}, processing decision xpb ∈ {0,1}
Objective: minimize loss
L((xe, xp), ρ
)= (ve)Txe︸ ︷︷ ︸
ext. cost
+ (vp)Txp︸ ︷︷ ︸proc. cost
− (W pρ)Txp︸ ︷︷ ︸proc. profit
Constraints:Precedence constraintsExtraction & processing capacity
⇒ Precedence-constrainedKnapsack problem (NP-hard)
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
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Open-pit production planning problemStochastic programming models
Computational resultsConclusions
UNCERTAINTY IN ORE GRADE ρ
Why? High costs & operation is done only once =⇒ HIGHLYRISKY
Production plan (xe, xp) =⇒ random loss L ((xe, xp), ρ)
Objective of our work:
Assess different risk-averse approaches
Basic hypothesis: we can obtain an iid sample, as largeas we want, of the ore grades vector ρ ∈ RB+
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Minimization of VaRMinimization of CVaRMCH & MCH-� models
1 Open-pit production planning problem
2 Stochastic programming modelsMinimization of VaRMinimization of CVaRMCH & MCH-� models
3 Computational results
4 Conclusions
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Minimization of VaRMinimization of CVaRMCH & MCH-� models
Minimization of Value-at-Risk (VaR)
Given � ∈ (0,1), for L loss r.v.:
VaR�(L) : “1−� percentile of losses”
Is a non-convex risk measure.Model: given risk level �,
min(xe,xp)∈X
VaR�[L((xe, xp), ρ
)]SAA approximation: take iid sample ρ1, . . . , ρN =⇒approximate P with PN := 1N
∑i 1{ρ=ρi}
→ Consistency: a.s. convergence in objective value andoptimal solution set under mild assumptions
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Minimization of VaRMinimization of CVaRMCH & MCH-� models
−12 −8.4 −4.8 −1.2 2.4 60
140
280
420
560
700
Losses
His
tog
ram
−12 −10 −8 −6 −4 −2 0 2 4 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Losses
Cu
mu
lati
ve d
istr
ibu
tio
n f
un
cti
on
ε = 10 %
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Minimization of VaRMinimization of CVaRMCH & MCH-� models
Minimization of Conditional Value-at-Risk (CVaR)
Given � ∈ (0,1], for L atom-less loss r.v.:
CVaR�(L) : “mean of � worst losses”
Is distortion risk measure: coherent, law-invariant &co-monotonic.Model: given risk level �,
min(xe,xp)∈X
CVaR�[L((xe, xp), ρ
)]SAA approximation: approximate E using iid sampleρ1, . . . , ρN . Consistency under mild hypothesis.
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Minimization of VaRMinimization of CVaRMCH & MCH-� models
Modulated Convex-Hull (MCH) & MCH-� models
Robust model: given risk level � ∈ [0,1] and iid sampleρ1, . . . , ρN ,
min(xe,xp)∈X
maxρ∈U�
L((xe, xp), ρ
)In (ΩN ,P,PN), equivalent to minimizing the risk measure
� E(·) + (1− �) Worst-Case(·)︸ ︷︷ ︸=CVaR1/N(·)
of losses L ((xe, xp), ρ)MCH-� model: minimize risk measure
� E(·) + (1− �) CVaR�(·)
of losses, which allows to perform a convergence analysis
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Minimization of VaRMinimization of CVaRMCH & MCH-� models
Example: U� for N = 8 samples of ρ in mine with 2 blocks
0 2 4 6 8 10 12 140
2
4
6
8
10
12
14
ρ1
ρ2
Samples
ε = 0%
ε = 25%
ε = 50%
ε = 75%
ε = 100%
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
1 Open-pit production planning problem
2 Stochastic programming modelsMinimization of VaRMinimization of CVaRMCH & MCH-� models
3 Computational results
4 Conclusions
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Vein-type mine with ≈ 20K blocks, 20K scenarios (TBsimalgorithm)Solve SAA approximation of each model (VaR, CVaR,MCH, MCH-�)
taking N = 50, 100, 200, 400, . . . samplesfor several risk levels �
! Also solve minimization of worst loss and minimization ofexpected lossSAA approximation =⇒ repetitions algorithm:
Find statistically optimal solutionEstimate optimality gap to “true” problem
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
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Open-pit production planning problemStochastic programming models
Computational resultsConclusions
VaR, N = 100, in-sample
−2.5 −2 −1.5 −1 −0.5 00
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
* *
His
tog
ram
(fr
eq
uen
cy)
Loss
min. Exp. Loss
min. VaR ε=5%
min. VaR ε=1%
min. Worst Loss
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
CVaR, N = 200, in-sample
−2.5 −2 −1.5 −1 −0.5 00
0.5
1
1.5
2
2.5
3
3.5
His
tog
ram
(fr
eq
uen
cy)
Loss
min. Exp. Loss
min. CVaR ε=30%
min. CVaR ε=10%
min. Worst Loss
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
MCH, N = 200, in-sample
−2.5 −2 −1.5 −1 −0.5 00
0.5
1
1.5
2
2.5
His
tog
ram
(fr
eq
uen
cy)
Loss
min. Exp. Loss
min. MCH ε=30%
min. MCH ε=10%
min. Worst Loss
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
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Open-pit production planning problemStochastic programming models
Computational resultsConclusions
MCH-�, N = 200, in-sample
−2.5 −2 −1.5 −1 −0.5 00
0.5
1
1.5
2
2.5
3
3.5
His
tog
ram
(fr
eq
uen
cy)
Loss
min. Exp. Loss
min. MCH−eps ε=30%
min. MCH−eps ε=10%
min. Worst Loss
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
CVaR, N = 200, in-sample vs. out-of-sample
(a) In-Sample histogram
−2.5 −2 −1.5 −1 −0.5 00
0.5
1
1.5
2
2.5
3
3.5
His
tog
ram
(fr
eq
uen
cy)
Loss
min. Exp. Loss
min. CVaR ε=30%
min. CVaR ε=10%
min. Worst Loss
(b) Out-of-Sample histogram
−2.5 −2 −1.5 −1 −0.5 00
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
** ** **
His
tog
ram
(fr
eq
uen
cy)
Loss
min. Exp. Loss
min. CVaR ε=30%
min. CVaR ε=10%
min. Worst Loss
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
1 Open-pit production planning problem
2 Stochastic programming modelsMinimization of VaRMinimization of CVaRMCH & MCH-� models
3 Computational results
4 Conclusions
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
-
Open-pit production planning problemStochastic programming models
Computational resultsConclusions
Conclusions
VaR model: low risk aversion =⇒ inadequate for ourproblemCVaR, MCH & MCH-� models: risk averse performance,controllable with parameter �However behavior is not as clear when testing planout-of-sampleSlow statistical convergence of VaR, CVaR & MCH-�models: HIGH DIMENSIONALITY!Two-stage variant (extract→ see ρ→ process): nolosses / fast convergence / not much differencebetween models
Espinoza, Lagos, Moreno, Vielma Robust Planning for an Open-Pit Mining Problem
PresentaciónOpen-pit production planning problemStochastic programming modelsMinimization of VaRMinimization of CVaRMCH & MCH- models
Computational resultsConclusions
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