robust planning for an open-pit mining problem under ore ......robust planning for an open-pit...

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




Chuquicamata, Chile




Block model. Color ∼ amount of mineral (ore grade)




Which blocks to extract?




Between extracted ones, which to process?




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)




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+




Minimization of VaRMinimization of CVaRMCH & MCH-� models




4 Conclusions





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





−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 %





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.





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





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%







4 Conclusions




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




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




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




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




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




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







4 Conclusions




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


PresentaciónOpen-pit production planning problemStochastic programming modelsMinimization of VaRMinimization of CVaRMCH & MCH- models


robust planning for an open-pit mining problem under ore ......robust planning for an open-pit...

Documents