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: 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
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
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
.
.
.
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
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
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
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
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