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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