Harnessing Uncertainty for Orebody Modelling and …cosmo.mcgill.ca/wp-content/uploads/2015/08/AMEC_Vanc_Nov_05.pdfOptimization under Uncertainty ... Harnessing Uncertainty for Orebody

Roussos Dimitrakopoulos

Canada Research Chair inSustainable Mineral Resource Development and

Optimization under Uncertainty

Department of Mining, Metals and Materials Engineering

Keynote speech to AMEC Internal conference, Vancouver, November 2005

Harnessing Uncertainty for

Orebody Modelling and Strategic Mine Planning

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

• Conclusions - Uncertainty is great!

Uncertainty Matters:The Economic Side of Uncertainty

Changing the way we do things

Uncertainty Matters: Return on Investment is Uncertain, therefore Risky

• Possibility of not making a return on capital (NPV<0)

NPV, $MM-100 0 600

• Alternative development plans may have different risk profiles and expected values. Example:

NPV, $MM-100 0 600

Design - can’t capture high reserves

NPV, $MM-100 0 600

Design … can capture…

Reserve 0

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

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

… you can take advantage of uncertainty. Your strategic 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 Kulatilakain “Real Options”

Uncertainty is not a “Bad Thing”

Real Options vs DCF View of ValueC

urre

nt A

sset

Val

ue

FutureGold Price

$0

$-

$+ Real Options View:Current Value ofOption to Produce

Traditional DCF View(now or never)

No productionNPV = 0

ProductionNPV > 0

Contingent Decision Payoff Function

(future price known)

Accurate Uncertainty Assessment Needed

Unknown,trueanswer

Reserves

Accurateuncertaintyestimation

Single,oftenprecise,wronganswer

Reserves

Pro

babi

lity

1

“The goal of technical evaluation should be to strive for an accurate assessment of uncertainty, not a single precise answer”

Mining Project Valuation

Orebody Model Mine DesignProduction Scheduling

Financial and Production Forecasts

Traditional view

Unknown,trueanswer

Single,oftenprecise,wronganswer

Reserves

Prob

abili

ty

1

Single estimated model

Risk oriented view

Accurateuncertaintyestimation

Reserves

Prob

abili

ty

1 Accurateuncertaintyestimation

Reserves

Prob

abili

ty

1Multiple probable models

Mining Process or Transfer Function

Quantitative Models of Geological Uncertainty:

Stochastic or geostatisticalconditional simulations

Information about the deposit

Actual but unknown mineral deposit Probable models of

the deposit

Describing the Uncertainty about a Mineral Deposit

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

Lode 1502Simulation #2

Lode 1502Simulation #3

Moving Forward in Optimization: Limits of Traditional Mine Design

Using Models of Uncertainty

Risk Analysis in a Mine Design

Objective Quantify the impact of grade uncertainty to tonnage, grades, metal and net present value - net present vnalue vs risk exposure

Mine Design(Scenario)

Multiple simulations

Distribution of outcomes

for a scenario

Mine Design (Scenario X)

Methodology

.

.

.

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 ModellingThe expected project NPV has only

2 – 4% probability to be realised

First 2 years of production Final year likely to be

negative cash flow

Limits of Traditional Modelling Discounted Cash Flow

-5-3

-1

1

3

5

7

9

0 2 4 6 8 10 12 14

Production Period (1/4 Year)

Cas

h Fl

ow (

m$

p.a.

)

0

Probabilities on Pit Limits

Pit limit determined conventionally

100% probability of falling within the pit for a given metal price

This is Not ...

Moving Forward ….. Step 1

Exploring existing technologies

Min acceptable return

Upside

Downside

D C

F

Pit design1 2

Value

Past Work – Open Pit Mine DesignUpside Potential / Downside Risk

Upside or Avg[ ]( )*Downside Value MAR probability= −∑

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 DesignUpside Potential / Downside Risk

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

Base Case: Geological Risk Assessment of Ore Production

Uncertainty in Ore Production - Base Case Schedule

13.9% 13.5%

8.7%

18.2%

12.7% 12.4% 12.5%10.8% 11.4% 12.4%

10.4%12.3%

9.0%11.9%

14.5%12.3% 12.9%

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Period

Ore

Pro

duct

ion

0%

10%

20%

30%

40%

50%

60%

70%

Ave

rage

Dev

iatio

n(%

)

Avrg. DeviationTarget Ore ProductionMaximum OreExpected OreMinimum Ore

Risk-based: Assessment in Ore Production

Uncertainty in Ore Production - Risk-based Schedule

0.5%3.0% 1.2% 0.7%

3.5% 1.9% 0.2%2.7% 1.6% 0.0% 0.6% 0.4% 0.1% 0.0% 0.7%

Mt

Mt

Mt

Mt

Mt

Mt

Mt

Mt

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Period

Ore

Pro

duct

ion

0%

10%

20%

30%

40%

50%

60%

70%

Ave

rage

Dev

iatio

n(%

)

Avrg. DeviationTarget Ore ProductionMaximum OreExpected OreMinimum Ore

2001 2003 2005 2007 2009 2011 2013 2015 2017

Difference 28%

Risk-Based

Traditional and Risk

Traditional “Expected”

NPV

Year

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

Uncertainty is Good: Discounting Geological Risk

The discounting goes along with production sequencing

Objective function

SIP - Production Scheduling Model

Part 1

Part 2

Part 3

Part 4

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 1

A production schedule

Orebody Model 2

Orebody Model R

OreGrade 2Metal…

OreGrade RMetal…

- TARGET [ ]

- TARGET [ ]

- TARGET [ ]

Deviation 1

Deviation 2

Deviation R

1 234

Cross-Sectional Views of the Schedules

SIP Whittle Four-X

123456

Periods

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 PeriodsDeviations from metal production target

RDF – risk discounting factor

Orebody risk discounting rate 20 %Cost of shortage in ore production 10,000 /tCost of excess ore production 1,000 /tCost of shortage in metal production 20 /grCost of excess metal production 20 /grNumber of simulated orebody models 15

The SIP specific information

Case Study on a Large Gold Mine

Deviations from Production Targets

1 2 3 4 5 6

Periods

0

- 4

Met

al q

uant

ity

(100

0 K

g)

- 8

- 12

- 16

- 20

Metal Production

SIP model

WFX

0Tonn

es(m

illio

n)

1 2 3 4 5 6

2

4

6Stockpile’s Profile

Available ore at the end of each period

1 2 3 4 5 6 Periods

0

Tonn

es(m

illio

n)

1

2

3 Ore taken out from the stockpile

4

5

SIP model

WFX

1 2 3 4 5 6 Periods0

200$ (m

illion

)

400

600

800

1000

SIP model WFX

Cumulative NPV values

SIP model WFX

Average NPV values

Uncertainty is Good: Traditional vs Risk-BasedStochastic Integer Programming

$723 M Risk Based

$609 M Traditional

Difference = 17%

Geological Risk Discounting= 20%

Some conclusions

• “…. uncertainty is (not) a problem and should be avoided ?”

• “… you can take advantage of uncertainty….”

• “….uncertainty will create opportunities and value.”

• “ …once your way of thinking explicitly includes uncertainty,the whole decision-making framework changes.”

• We need:

Stochastic mine planning and NEW mathematical models

• It is all about good people:

Education and training in a long term sense

And

Please join us!