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A Stochastic Optimisation Formulation for the Transition from Open Pit to Underground Mining
James MacNeil
November, 2014
COSMO – Stochastic Mine Planning LaboratoryDepartment of Mining and Materials Engineering 1
Outline
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1. Introduction2. Method and Formulation3. Application at a Gold Deposit4. Conclusions and Future Work
Consideration for Transitioning Between Methods
312/3/2014
• As open‐pit mining deepens:• Stripping ratio (waste:ore) increases• Cost of haulage to surface increases
• Underground mining can provide:• Increased mine life by allowing access to deep ore• Make use of processing facilities already in‐place
Problem Setting
412/3/2014
• Multidisciplinary decision which requires:
Geotechnical Analysis
Detailed Cost/Revenue Structure
Underground Mine Design Received as inputs
Accurate Valuation Proceduresfor both Portions of the Mine
Focus of this presentation
Method and Formulation
512/3/2014
Simplifying the Problem
612/3/2014
• This problem is limited to a finite number of transition opportunities based on:• Viable crown pillar locations• Discretization of space into selective mining units
• Need to define opportunities in 3‐dimensions through:• Ultimate pit for open‐pit resource• Extent of underground orebody• Crown pillar size and location
Method Procedure
712/3/2014
• In order to accurately value the action of transitioning at a certain opportunity, we must create an optimized schedule which considers uncertainty
Open‐Pit Production Year 1
Underground Production Year 1
. . . Open‐Pit
Production Year N
Transition Year
Life of Mine Schematic:
. . .
Underground Production Year M
Transition year is fixed, but yearly cash‐flows change across transition opportunities
Broad Scope Schematic of Method
812/3/2014
… Crown Pillar
PotentialOpen‐pit Resource
PotentialUnderground Resource
Determine extent of each corresponding above and below ground orebody
…
Optimize production schedule above and below ground to assess value at each transition opportunity
UG Production Schedule
OP Production Schedule
…
Transition Opportunity 1
Transition Opportunity 2
Transition Opportunity n
Optimization Procedure
912/3/2014
Open‐pit Underground
20 geological simulations Input:
Two‐stage stochastic optimization which aims to maximize value while limiting deviations from metal quantity and processing
tonnage targetsFormulation:
Parallel Implementation of Tabu Search Metaheuristic
IBM ILOG CPLEX 12.51 C++ Concert TechnologySolution Method:
Portion of Mine:
Different Optimization Frameworks
1012/3/2014
Stochastic Optimization
Several orebody simulations
…Stochastic Scheduler
Risk‐based Production Schedule
21
3
Deterministic Scheduler
Traditional Production Schedule
21
3Single orebody model
Deterministic Optimization
Formulation Details
11
• Open‐pit Objective Function:
, , , , , , , ,
• Underground Objective Function:
, , , , , , , ,
Economic value of stopes mined in a given period
Deviations from ore and metal targets
Economic value of blocks mined in a given period
Deviations from ore and waste targets
Evaluation of Optimal Transition Opportunity
1212/3/2014
• Optimal extraction sequence and resulting cash‐flows under uncertainty provide accurate valuation of mining complex
NPV of OP Operation
NPV of UG Operation
NPV of Mining Complex
Transition opportunity specific values
Risk Analysis Procedure
1312/3/2014
90% chance of being below this value
Value
Period
P90
P10
10% chance of being below this value
Range Describes variation
Several simulations
…Schedule Produced
by Optimizer
Simulations
Case Study
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Application at a Gold Deposit
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• Fictitious gold deposit containing an open‐pit and an underground stoping operation
• Four transition opportunities were selected with varying crown pillar locations
TransitionOpportunity 1
TransitionOpportunity 2
TransitionOpportunity 3
TransitionOpportunity 4
Number of OP Blocks
66,000 63,000 59,000 55,000
Number of UG Stopes
235 315 395 450
Crown Pillar Locations
16
Location 2 – Crown Pillar Depth: 820ft
Location 3 – Crown Pillar Depth: 760ft Location 4 – Crown Pillar Depth: 700ft
• Crown pillar size is assumed to be constant at each locationHeight = 60ft, Width = 740ft, Length = 1260ft
Location 1 – Crown Pillar Depth: 880ft
Crown Pillar
Potential Open‐pit Resource
Transition Opportunities (TO)
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TO 1 TO 2
TO 3 TO 4
135
140
145
150
155
Cumulative Discoun
ted Ca
sh‐Flow
(Millions)
Net Present Value at Each Transition Opportunity
TO 1
TO 2
TO 3
TO 4
Deterministic Study
1812/3/2014
0
20
40
60
80
100
120
140
160
1 2 3 4 5 6 7 8 9 10 11 12 13
Cumulative Discoun
ted
Cash‐Flow
(Millions)
Period
Cumulative Cash‐Flow Generated throughout LOM
TO 1
TO 2
TO 3
TO 4
Optimal to transition at Opportunity 4
Transition Year
Risk Analysis on Deterministic Result
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0123456789
1 2 3 4 5Metal To
nnes Produ
ced
Underground Production Period
0
50
100
150
200
250
1 2 3 4 5
Cumulative Discoun
ted
Cash‐Flow
(Millions)
Underground Production Period
Deterministic
p50
p10
p90
Likelihood of meeting yearly target:25% 45% 8% 0% 29%
Target Simulations
Stochastic Result
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020406080100120140160180
1 2 3 4 5 6 7 8 9 10 11 12 13
Cumulative Discou
nted
Cash‐Flow
(Millions)
Period
TO1TO2TO3TO4
130
135
140
145
150
155
160
Cumulative Discou
nted
Cash‐Flow
(Millions)
Net Present Value at End of Life of Mine Plan
TO 1TO 2TO 3TO 4
Optimal to transition at Opportunity 3
Transition Year
Risk Analysis on Stochastic Result
2112/3/2014
012345678
1 2 3 4 5
Tonn
es of M
etal
Prod
uced
Period
Likelihood of meeting yearly target:89% 71% 37% 74% 11%
0
100
200
300
400
500
600
1 2 3 4 5
Tonn
es Sen
t to Mill
(Tho
usands)
Period
Deterministic Target Simulations
Visual Comparison
22
Stochastic ResultDeterministic Result
Ope
n‐pit
Resource
Und
ergrou
nd
Resource
Plan View
700ft 760ft
Comparison of Two Studies
23
Transition Opportunity 3
• 59,000 Open‐Pit Blocks• 395 Underground Stopes
• Crown Pillar at Depth of 760 ft
Stochastic Result
Transition Opportunity 4
• 55,000 Open‐Pit Blocks• 450 Underground Stopes
• Crown Pillar at Depth of 700 ft
Deterministic Result
• Different schedules and design for both portions of the deposit for stochastic versus deterministic results
Comparison in Result Variation
24
• Able to measure variation based on difference between P90 and P10• Apparent that there is higher certainty in cash‐flows predicted by
stochastic scheduler
0
20
40
60
80
100
120
1 2 3 4 5
Cash‐Flow Variatio
n(M
illions)
Underground Production Period
Deterministic
Stochastic
Comparison in Result Value
2512/3/2014
151
152
153
154
155
156
157
Cumulative Discou
nted
Cash‐Flow
(Millions)
Net Present Value of Optimal Transition Opportunity
Stochastic
Deterministic
• 3% increase in NPV for stochastic decision to transition at opportunity 3
• Deterministic optimizer makes a risky and less profitable decision to transition at opportunity 4
Conclusions and Future Work
2612/3/2014
Conclusions
2712/3/2014
• Made an initial attempt at solving a large pertinent problem
• Stochastic transition depth is different than deterministic
• Benefits of incorporating geological uncertainty:• Increased ability to meet targets• Higher NPV
Future Work
2812/3/2014
• Attempt a more thorough evaluation of the solution space
• Incorporate into context of a mining complex
• Observe the impact of price uncertainty
• Improve ability to accurately model and value underground mines
Widescreen Test Pattern (16:9)
Aspect Ratio Test
(Should appear circular)
16x9
4x3
4x3
4x3
4x3
4x3
4x3
4x34x3
4x34x3
4x34x3
4x34x3
16x9
16x9
16x9
16x9
16x9
16x916x9
16x916x9
16x916x9
Relevant Parameters
3112/3/2014
Economic Parameters
Metal Price $750
Crown Pillar Height 60ft
Economic Discount Rate 10%
OP Mining cost/ton $1.5
UG Mining cost/ton $135
Processing cost/ton $31.5
Open‐Pit Stochastic Optimization Formulation
32
• Objective function::
, , , , , , , ,
• Subject to: 1,2, … , ; 1,2, … ,
1 1,2, … , 0 1,2, … , ; 1,2, … , ; ∈
Economic value of blocks mined in a given period
Deviations from ore and waste targets
Ore target in each period Waste target in each period
Reserve Constraint Slope Constraint
Methods Used to Solve Scheduling Problems
33
Open-Pit MethodTabu Search Heuristic
• Smart algorithm that efficiently searches the solution space for optimal solution
• Uses a diversification strategy to overcome local optima
• Implemented in parallel to reduce solving times
• No optimality guarantee
CPLEX
• Commercially available mathematical programming solver
• Implements branch and bound algorithm to
• Implemented in parallel to reduce solving times
Underground Method