Stochastic Mine Planning Concepts, Applications and Contributions:

From past developments to production scheduling with ‘future data’

Roussos Dimitrakopoulos

COSMO – Stochastic Mine Planning LaboratoryDepartment of Mining and Materials Engineering

Overview

• The economic side of uncertainty

• Models of geological uncertainly

• Limits of traditional mine design optimization

• Shifting the paradigm: Stochastic mine planning

• Using uncertainty to improve project performance

• Uncertainty is great!

Risk in Mining: A World Bank Survey

• 60% of mines had an average rate of production LESS THAN 70% of planned rate

• In the first year after start up, 70% of mills or concentrators had an average rate of production LESS THAN 70% of design capacity

• Key contributor to mining risk felt in all downstream phases: Geology and reserves

Mining Project Valuation

Orebody Model Mine DesignProduction Scheduling

Financial and Production Forecasts

Traditional view Unknown,trueanswer

Single,oftenprecise,wronganswer

Reserves

Prob

abili

ty

1Single estimated model

Risk oriented view

Multiple probable models

Mining Process or Transfer Function

Accurateuncertaintyestimation

Reserves

Prob

abili

ty

1 Accurateuncertaintyestimation

Reserves

Prob

abili

ty

1

Quantitative Models of Geological Uncertainty:

Stochastic or geostatisticalconditional simulations

Model characteristics:

o Large number of blockso Multiple domainso Resource classes with specific sample selection criteria A gold load

Describing the Uncertainty about a Gold Deposit

Lode 1502Simulation #1

Describing the Uncertainty about a Gold Deposit

Many managers believe that uncertainty is a problem and should be avoided…..

… you can take advantage of uncertainty. Yourstrategic investments will be sheltered from its adverse effects while remaining exposed to its upside potential.Uncertainty will create opportunities and value.

Once your way of thinking explicitly includes uncertainty, the whole decision-making framework changes. Martha Amram and Nalin Kulatilaka

in “Real Options”

Uncertainty is not a “Bad Thing”

Moving Forward in Optimization

Limits of traditional mine design

Using models of uncertainty

Intermediate pushbacks

Pit Limit

Open Pit Mine Design and Production Scheduling

Pit Shells

NPV

(m

$, i

= 8

%)

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

Stochastic OrebodiesConventional

Probability

0

Limits of Traditional Modelling

The expected project NPV has only 2 – 4% probability to be realised

Risk Analysis in a Mine Design

Moving Forward ….. Step 1

Exploring existing technologies

Upside Potential (m$) Downside Potential (m$)

CB-1 CB-2 CB-3 CB-1 CB-2 CB-3

2.3

1.3

2.4

2.9

Pit Design

2.41

2.1

2.43

2.40

0.0

-0.78

0.0

0.0

-0.079

-0.15

-0.022

-0.1612

6

4

21.8

1.6

1.9

1.2

-0.20

-0.51

-0.28

-0.96

Past Work – Open Pit Mine Design

Moving Forward ….. Step 2

Re-writing optimizers

Integer Programming

An objective function

Maximise (c1x11+c2x2

1+…. ) …

Subject to

c1x11+c2x2

1+…. ≥ b1

c1x1p+c2x2

p+…. ≥ bp

c4

c1 c2 c3

Period 1

Period p

Orebody model

c = constantX1

1 = binary variable

Models of Uncertainty in Optimization

The objective function now …..

Maximise (s11x11+s21x2

1+…. s12x11+s22x2

1+….) …

Subject to

s11x11+s21x2

1+…. ≥ b1

s11x1p+s21x2

p+…. ≥ b1

s12x1p+s22x2

p+…. ≥ b1

s1rx1p+s2rx2

p+…. ≥ b1

Stochastic Integer Programming

Simulated model 1Simulated model 2Simulated model r

Period 1

Period p

s41

s11 s2

1 s31

s41

s11 s2

1 s31

s41

s11 s2

1 s31

s41

s1n s2

n s3n

“Uncertainty will create opportunities and value”

Higher NPV for Less Risk

Difference 28%

Risk-Based

Traditional and Risk

Traditional “Expected”

NPV

Year

Uncertainty is Good: “Base case” vs “Risk-based”

2001 2003 2005 2007 2009 2011 2013 2015 2017

Discounting Geological Risk

The discounting goes along with production sequencing

Objective function

SIP - Production Scheduling Model

Part 1

Part 3

Part 4

Part 2

U t t t*i ii

i 1- E{(NPV) }MC s

=+∑

M tts s

s 1+ (SV) (P) q

=∑

P N t tii

t 1 i 1Max [ E{(NPV) } b

= =∑ ∑

M ty ty tytysu l slu

s 1- )](c d c d

=+∑

Mill & dump

Stockpile input

Stockpile output

Risk management

Stochastic Integer Programming - SIP

……

OreGrade 1Metal…

Orebody Model 1A production

schedule

Orebody Model 2

Orebody Model R

OreGrade 2Metal…

OreGrade RMetal…

- TARGET [ ]

- TARGET [ ]

- TARGET [ ]

Deviation 1

Deviation 2

Deviation R

123

4

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4

01 2 3

1

2

3

Met

al q

uant

ity

(100

0 K

g)

PeriodsCt=Ct-1 * RDFt-1 RDFt=1/(1+r)t

r – orebody risk discount rate

Managing Risk Between Periods

Deviations from metal production target

RDF – risk discounting factor

Uncertainty is Good: Traditional vs Risk-Based

1 2 3 4 5 6 Periods0

200$ (m

illion

)

400

600

800

1000

SIP model WFXCumulative NPV values

SIP model WFX

Average NPV values

$723 M Risk Based

$609 M Traditional

Difference = 17%

Geological Risk Discounting= 20%

SIP

Whittle Four-X

Future Drilling Data

Production sequencing withsimulated grade control drilling

‘Future’ Grade Control Data

Exploration dataGrade control data

Bench/Section of pit already mined out

Define relationship

Exploration dataSimulate grade control data

Bench/Section of pit NOT yet mined out

Simulation of orebody models from exploration data

Derive production scheduleusing SIP formulation

Step 1

Updating of the existing orebodymodels with the future data

Step 2

Schedules:

• SIP schedule derived from simulations based on exploration data• SIP schedule derived from simulations based on simulated grade control information (updated models)• Risk analysis of mine’s schedule with the updated models

Step 3

Simulation of high density future grade control data

SIP formulation

Derive production scheduleusing SIP formulation

SIP formulation

Scheduling and Simulated Future Data

Average of the simulations SimulationsSimulationsSimulations

2.52.72.93.13.33.53.73.94.14.34.54.7

2004 2005 2006 2007 2008 2009

Mill

ions

Tonn

age

2010Period (years)

Period (years)

02468

1012141618

2004 2005 2006 2007 2008 2009 2010

Mill

ion

gram

sM

etal

con

tent

0.0

50.0

100.0

150.0

200.0

2004 2005 2006 2007 2008 2009 2010

Mill

ions

AU

D

Period (years)

NPV

2.52.72.93.13.33.53.73.94.14.3

2004 2005 2006 2007 2008 2009

Mill

ions

Period (years)

Tonn

age

02468

10121416182022

2004 2005 2006 2007 2008 2009

Mil

gr

Period (years)

Met

al c

onte

nt

0.0

50.0

100.0

150.0

200.0

250.0

2004 2005 2006 2007 2008 2009

Mill

ions

AU

D

Period (years)

NPV

Mill target ( ore production)

Scheduling and Simulated Future Data

SIP and Simulated Orebody SIP and Simulated Future Data

2000

2200

Y=99824

P1

P2

P3

P4

2000

2200

Y=99824

P1

P2

P3

P4

2000

2200

2000

2200

Y=99824

P1

P2

Y=99824

P1

P2

P1

P2

P3

P5

P1

P2

P3

P5

Scheduling and Simulated Future Data

Mine’s Schedule SIP & Simulated Orebody SIP & Future data

Period (years)

20052006200720082009

1

2

3

4

5

Period (years)

20052006200720082009

Period (years)

20052006200720082009

1

2

3

4

5

330560552NPV ($ Mil. )

385552Metal Tonnes (Mt)

101814Ore Tonnes (Mt)

Mine’s schedule

(future data)

Updated simulations (future data)

Simulations (exploration

data)

Y=99824

P1

P2P3

P4Y=99824

P1

P2P3

P4

P1

P2P3

P5

P1

P2P3

P5

Y=99824

P1

P2Y=99824

P1

P2

Scheduling and Simulated Future Data

Uncertainty is Great

And we will eventually find out