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Harnessing Uncertainty for Orebody Modelling and Strategic Mine Planning Roussos Dimitrakopoulos Canada Research Chair in Sustainable Mineral Resource Development and Optimization under Uncertainty

Department of Mining, Metals and Materials Engineering

Keynote speech to AMEC Internal conference, Vancouver, November 2005

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

Traditional DCF View (now or never)

Accurate Uncertainty Assessment Needed

Probability

1

Unknown, true answer

Reserves

Single, often precise, wrong answer

Accurate uncertainty estimation

Reserves

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

Single estimated model

1

Unknown, true answer

Probability

Mining Project Valuation

Single, often precise, wrong answer

Traditional view Reserves

Mining Process or Transfer Function

Multiple probable models

Mine Design Production Scheduling Financial and Production Forecasts

Risk oriented view

1

Probability

Orebody Model

Accurate uncertainty estimation

Reserves

Quantitative Models of Geological Uncertainty:

Stochastic or geostatistical conditional simulations

Describing the Uncertainty about a Mineral Deposit Actual but unknown mineral deposit

Information about the deposit

Probable models of the deposit

Describing the Uncertainty about a Gold Deposit

Model characteristics: o Large number of blocks o Multiple domains o Resource classes with specific sample selection criteria

A gold load

Lode 1502 Simulation #1

Lode 1502 Simulation #2

Lode 1502 Simulation #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 Methodology Mine Design (Scenario) . . .

Mine Design (Scenario X)

Multiple simulations

Distribution of outcomes for a scenario

Open Pit Mine Design and Production Scheduling Intermediate pushbacks

Pit Limit

Limits of Traditional Modelling The expected project NPV has only 2 – 4% probability to be realised

NPV (m$, i = 8%)

25

Probability

20 15 10 Stochastic Orebodies

5

Conventional 0 0

5

10

15

20

25

Pit Shells

30

35

40

45

50

Limits of Traditional Modelling Discounted Cash Flow Cash Flow (m$ p.a.)

9

First 2 years of production

7 5

Final year likely to be negative cash flow

3 1 0 -1 -3 -5 0

2

4

6

8

Production Period (1/4 Year)

10

12

14

Probabilities on Pit Limits

100% probability of falling within the pit for a given metal price Pit limit determined conventionally

This is Not ...

Moving Forward ….. Step 1

Exploring existing technologies

Past Work – Open Pit Mine Design Upside Potential / Downside Risk

DCF

Upside Min acceptable return

Downside Value 1

Upside or D ow nside =

2

Pit design

A R ] * probability ) ∑ ([V alue − MAvg

Past Work – Open Pit Mine Design Upside Potential / Downside Risk Upside Potential (m$)

Pit Design

Downside Potential (m$)

CB-1

CB-2

CB-3

CB-1

2.3

2.41

1.8

0.0

1.3

2.1

1.6

2.4

2.43

2.9

2.40

CB-2

CB-3

-0.079

-0.20

-0.78

-0.15

-0.51

1.9

0.0

-0.022

-0.28

1.2

0.0

-0.16

2

4

6

12

-0.96

Moving Forward ….. Step 2

Re-writing optimizers

Models of Uncertainty in Optimization Integer Programming An objective function Maximise (c1x11+c2x21+…. ) … Subject to

Orebody model

c1

c2

c3

c4 c = constant X11 = binary variable

c1x11+c2x21+…. ≥ b1

Period 1

c1x1p+c2x2p+…. ≥ bp

Period p

Stochastic Integer Programming The objective function now …..

s11 s21 s31

Maximise (s11x11+s21x21+…. s12x11+s22x21+….) … Subject to s11x11+s21x21+…. ≥ b1

s11x1p+s21x2p+….

≥ b1

Period 1 Simulated model 1 Simulated model 2 Simulated model r

s12x1p+s22x2p+…. ≥ b1 s1rx1p+s2rx2p+…. ≥ b1

1 1 1 s s1 4 s2 s31 s11 s1 21 s31 s1ns4s21n s3n s4 s41

Period p

“Uncertainty Will Create Opportunities and Value”

Higher NPV for less risk

Base Case: Geological Risk Assessment of Ore Production Mt

70%

Mt

60%

Mt

50%

Mt Mt 18.2%

Mt 13.9%

13.5%

12.7%

12.4%

12.5%

8.7%

Mt

40%

Avrg. Deviation Target Ore Production Maximum Ore Expected Ore Minimum Ore 10.8%

11.4% 12.4%

10.4%

30%

14.5%

12.3%

11.9%

20% 12.3%

12.9%

9.0%

10%

Mt

0% 1

2

3

4

5

6

7

8

9

10

Period

11

12

13

14

15

16

17

Average Deviation (%)

Ore Production

Uncertainty in Ore Production - Base Case Schedule

Risk-based: Assessment in Ore Production Mt

70%

Mt

60%

Mt

50%

Mt

40% Avrg. Deviation Target Ore Production Maximum Ore Expected Ore Minimum Ore

Mt Mt Mt 0.5%

3.0%

1.2% 0.7%

3.5%

30% 20% 10%

1.9%

0.2%

2.7% 1.6%

0.0%

0.6%

0.4%

0.1% 0.0%

0.7%

0%

Mt 1

2

3

4

5

6

7

8

Period

9

10

11

12

13

14

15

Average Deviation (%)

Ore Production

Uncertainty in Ore Production - Risk-based Schedule

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

NPV

Difference 28%

Traditional and Risk Traditional “Expected” 2001

2003

2005

2007

2009

Year

2011

2013

2015

2017

Uncertainty is Good: Discounting Geological Risk

The discounting goes along with production sequencing

SIP - Production Scheduling Model Objective function P

N

Max ∑ [∑ E{(NPV) } b t =1

t i

i =1

U

t i

Part 1

Mill & dump

- ∑ E{(NPV) + MC } s t i

i =1 M

+ ∑ (SV) (P) q t s

s =1

M

t t i * i

Stockpile input

t s

ty - ∑ (c d su + clty d slty)] s =1

ty u

Part 2

Part 3

Stockpile output Part 4

Risk management

Stochastic Integer Programming - SIP Deviation 1 Orebody Model 1 A production schedule

3

2 4

Orebody Model 2

Ore Grade 2 Metal …

……

1

Ore Grade 1 Metal …

Orebody Model R

- TARGET [ ] Deviation 2 - TARGET [ ]

Deviation R Ore Grade R Metal …

- TARGET [ ]

Cross-Sectional Views of the Schedules SIP

Periods 1 2 3 4 5 6

Whittle Four-X

Managing Risk Between Periods Deviations from metal production target Metal quantity (1000 Kg)

33 2.5

22 1.5

11 0.5

00 0

11

22

3 3

4

Periods Ct=Ct-1 * RDFt-1 RDF – risk discounting factor

RDFt=1/(1+r)t r – orebody risk discount rate

Case Study on a Large Gold Mine The SIP specific information

Orebody risk discounting rate

20 %

Cost of shortage in ore production

10,000 /t

Cost of excess ore production

1,000 /t

Cost of shortage in metal production

20 /gr

Cost of excess metal production

20 /gr

Number of simulated orebody models

15

Deviations from Production Targets Metal Production

Metal quantity (1000 Kg)

- 20 - 16 - 12

SIP model WFX

-8 -4 0 1

2

3

4

Periods

5

6

Stockpile’s Profile Tonnes (million)

6

Available ore at the end of each period

4 2 0 1

2

3

4

5

SIP model

6

WFX

Tonnes (million)

5 4 3

Ore taken out from the stockpile

2 1 0 1

2

3

4

5

6

Periods

Uncertainty is Good: Traditional vs Risk-Based Stochastic Integer Programming

$723 M Risk Based

1000 800

$609 M Traditional

$ (million)

600 400

Difference = 17%

200 0 1

Cumulative NPV values SIP model

WFX

2

3

4

5

6 Periods

Average NPV values SIP model

WFX

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

And

•

It is all about good people: Education and training in a long term sense

Please join us!

View more...
Department of Mining, Metals and Materials Engineering

Keynote speech to AMEC Internal conference, Vancouver, November 2005

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

Traditional DCF View (now or never)

Accurate Uncertainty Assessment Needed

Probability

1

Unknown, true answer

Reserves

Single, often precise, wrong answer

Accurate uncertainty estimation

Reserves

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

Single estimated model

1

Unknown, true answer

Probability

Mining Project Valuation

Single, often precise, wrong answer

Traditional view Reserves

Mining Process or Transfer Function

Multiple probable models

Mine Design Production Scheduling Financial and Production Forecasts

Risk oriented view

1

Probability

Orebody Model

Accurate uncertainty estimation

Reserves

Quantitative Models of Geological Uncertainty:

Stochastic or geostatistical conditional simulations

Describing the Uncertainty about a Mineral Deposit Actual but unknown mineral deposit

Information about the deposit

Probable models of the deposit

Describing the Uncertainty about a Gold Deposit

Model characteristics: o Large number of blocks o Multiple domains o Resource classes with specific sample selection criteria

A gold load

Lode 1502 Simulation #1

Lode 1502 Simulation #2

Lode 1502 Simulation #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 Methodology Mine Design (Scenario) . . .

Mine Design (Scenario X)

Multiple simulations

Distribution of outcomes for a scenario

Open Pit Mine Design and Production Scheduling Intermediate pushbacks

Pit Limit

Limits of Traditional Modelling The expected project NPV has only 2 – 4% probability to be realised

NPV (m$, i = 8%)

25

Probability

20 15 10 Stochastic Orebodies

5

Conventional 0 0

5

10

15

20

25

Pit Shells

30

35

40

45

50

Limits of Traditional Modelling Discounted Cash Flow Cash Flow (m$ p.a.)

9

First 2 years of production

7 5

Final year likely to be negative cash flow

3 1 0 -1 -3 -5 0

2

4

6

8

Production Period (1/4 Year)

10

12

14

Probabilities on Pit Limits

100% probability of falling within the pit for a given metal price Pit limit determined conventionally

This is Not ...

Moving Forward ….. Step 1

Exploring existing technologies

Past Work – Open Pit Mine Design Upside Potential / Downside Risk

DCF

Upside Min acceptable return

Downside Value 1

Upside or D ow nside =

2

Pit design

A R ] * probability ) ∑ ([V alue − MAvg

Past Work – Open Pit Mine Design Upside Potential / Downside Risk Upside Potential (m$)

Pit Design

Downside Potential (m$)

CB-1

CB-2

CB-3

CB-1

2.3

2.41

1.8

0.0

1.3

2.1

1.6

2.4

2.43

2.9

2.40

CB-2

CB-3

-0.079

-0.20

-0.78

-0.15

-0.51

1.9

0.0

-0.022

-0.28

1.2

0.0

-0.16

2

4

6

12

-0.96

Moving Forward ….. Step 2

Re-writing optimizers

Models of Uncertainty in Optimization Integer Programming An objective function Maximise (c1x11+c2x21+…. ) … Subject to

Orebody model

c1

c2

c3

c4 c = constant X11 = binary variable

c1x11+c2x21+…. ≥ b1

Period 1

c1x1p+c2x2p+…. ≥ bp

Period p

Stochastic Integer Programming The objective function now …..

s11 s21 s31

Maximise (s11x11+s21x21+…. s12x11+s22x21+….) … Subject to s11x11+s21x21+…. ≥ b1

s11x1p+s21x2p+….

≥ b1

Period 1 Simulated model 1 Simulated model 2 Simulated model r

s12x1p+s22x2p+…. ≥ b1 s1rx1p+s2rx2p+…. ≥ b1

1 1 1 s s1 4 s2 s31 s11 s1 21 s31 s1ns4s21n s3n s4 s41

Period p

“Uncertainty Will Create Opportunities and Value”

Higher NPV for less risk

Base Case: Geological Risk Assessment of Ore Production Mt

70%

Mt

60%

Mt

50%

Mt Mt 18.2%

Mt 13.9%

13.5%

12.7%

12.4%

12.5%

8.7%

Mt

40%

Avrg. Deviation Target Ore Production Maximum Ore Expected Ore Minimum Ore 10.8%

11.4% 12.4%

10.4%

30%

14.5%

12.3%

11.9%

20% 12.3%

12.9%

9.0%

10%

Mt

0% 1

2

3

4

5

6

7

8

9

10

Period

11

12

13

14

15

16

17

Average Deviation (%)

Ore Production

Uncertainty in Ore Production - Base Case Schedule

Risk-based: Assessment in Ore Production Mt

70%

Mt

60%

Mt

50%

Mt

40% Avrg. Deviation Target Ore Production Maximum Ore Expected Ore Minimum Ore

Mt Mt Mt 0.5%

3.0%

1.2% 0.7%

3.5%

30% 20% 10%

1.9%

0.2%

2.7% 1.6%

0.0%

0.6%

0.4%

0.1% 0.0%

0.7%

0%

Mt 1

2

3

4

5

6

7

8

Period

9

10

11

12

13

14

15

Average Deviation (%)

Ore Production

Uncertainty in Ore Production - Risk-based Schedule

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

NPV

Difference 28%

Traditional and Risk Traditional “Expected” 2001

2003

2005

2007

2009

Year

2011

2013

2015

2017

Uncertainty is Good: Discounting Geological Risk

The discounting goes along with production sequencing

SIP - Production Scheduling Model Objective function P

N

Max ∑ [∑ E{(NPV) } b t =1

t i

i =1

U

t i

Part 1

Mill & dump

- ∑ E{(NPV) + MC } s t i

i =1 M

+ ∑ (SV) (P) q t s

s =1

M

t t i * i

Stockpile input

t s

ty - ∑ (c d su + clty d slty)] s =1

ty u

Part 2

Part 3

Stockpile output Part 4

Risk management

Stochastic Integer Programming - SIP Deviation 1 Orebody Model 1 A production schedule

3

2 4

Orebody Model 2

Ore Grade 2 Metal …

……

1

Ore Grade 1 Metal …

Orebody Model R

- TARGET [ ] Deviation 2 - TARGET [ ]

Deviation R Ore Grade R Metal …

- TARGET [ ]

Cross-Sectional Views of the Schedules SIP

Periods 1 2 3 4 5 6

Whittle Four-X

Managing Risk Between Periods Deviations from metal production target Metal quantity (1000 Kg)

33 2.5

22 1.5

11 0.5

00 0

11

22

3 3

4

Periods Ct=Ct-1 * RDFt-1 RDF – risk discounting factor

RDFt=1/(1+r)t r – orebody risk discount rate

Case Study on a Large Gold Mine The SIP specific information

Orebody risk discounting rate

20 %

Cost of shortage in ore production

10,000 /t

Cost of excess ore production

1,000 /t

Cost of shortage in metal production

20 /gr

Cost of excess metal production

20 /gr

Number of simulated orebody models

15

Deviations from Production Targets Metal Production

Metal quantity (1000 Kg)

- 20 - 16 - 12

SIP model WFX

-8 -4 0 1

2

3

4

Periods

5

6

Stockpile’s Profile Tonnes (million)

6

Available ore at the end of each period

4 2 0 1

2

3

4

5

SIP model

6

WFX

Tonnes (million)

5 4 3

Ore taken out from the stockpile

2 1 0 1

2

3

4

5

6

Periods

Uncertainty is Good: Traditional vs Risk-Based Stochastic Integer Programming

$723 M Risk Based

1000 800

$609 M Traditional

$ (million)

600 400

Difference = 17%

200 0 1

Cumulative NPV values SIP model

WFX

2

3

4

5

6 Periods

Average NPV values SIP model

WFX

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

And

•

It is all about good people: Education and training in a long term sense

Please join us!

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