Including stockpiles into mathematical programming models for mine planning
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Including stockpiles into mathematical programming models for mine planning
Felipe Ferreira, Ms. Sc. Student, Universidad Adolfo Ibañez, Santiago, Chile Eduardo Moreno, Associate Professor, Universidad Adolfo Ibañez, Santiago,
Chile.
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Introduction
• Mine Planning software. What do they optimize?
• Introduction of optimization models in the 60’s
– Ultimate Pit Limit, Lane, Lerchs-Grossman’s Algorithm
– Johnson 1968
– More details: Newman et al. (2010) and Osanloo et al. (2008)
• Many models and methods reach better solutions than commercial software.
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Literature Review • Stockpile has great importance in mining operations
• Few authors include the stockpile option in long-term optimization models
• Bley et al. (2012a)
– Difficulty in modelling stockpiles: mixing behavior
• Ramazan & Drimitrakopoulos (2013)
– Stochastic method with stockpile option
• Smith & Wicks (2014)
– Use an optimization model with stockpile option for medium-term planning.
• Geovia Whittle’s manual
– Stockpile withdrawals are considered to be at the average grade of material sent to it
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Literature Review
• Tabesh et al. (2015) – Shows a non-linear model with stockpile option
– Linear model: uses a lot of stockpiles with predefined metal grades
• Bley et al. (2012b) Introduces two non-linear models
– They consider instant mixing of the material send to the stockpile • Non-linear and non-convex restriction
• Models can’t be used in great size (let’s say real size) instances
• We present three linear models for including stockpiles in long-term mine planning
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Non-linear model: Blocks
• For each block 𝑏 ∈ 𝐵 – 𝑤𝑏: total tonnage
– 𝑚𝑏: metal tonnage
• Time period 𝑡 ∈ 𝑇
• If block 𝑏 is extracted in 𝑡: 𝑥𝑏,𝑡𝑒 ∈ {0,1}
– Fraction of block 𝑏 sent to processing plant: 𝑥𝑏,𝑡𝑝
– Fraction of block 𝑏 sent to stockpile: 𝑥𝑏,𝑡𝑠
• This lead us to some constraints:
– Block destination: 𝑥𝑏,𝑡𝑝+ 𝑥𝑏,𝑡𝑠 ≤ 𝑥𝑏,𝑡
𝑒
– Block must be extracted in one period only:
𝑥𝑏,𝑡𝑒
𝑡∈𝑇
≤ 1
Metal Grade: 𝑚𝑏
𝑤𝑏
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Non-linear model: Stockpile
• Assumption: Extracted Ore arrives to stockpile at the end of period 𝑡, and Ore is reclaimed from stockpile at the beginning of period 𝑡
• Variables:
– Ore, metal available in stock at the end of period 𝑡: ots, 𝑎𝑡𝑠
– Ore, metal sent to mill from stock at the beginning of period 𝑡: otp
, 𝑎𝑡𝑝
• Then:
– 𝑜𝑡𝑝≤ 𝑜𝑡−1𝑠
– 𝑎𝑡𝑝≤ 𝑎𝑡−1𝑠
• Finally the amount of Ore and Metal in stockpile a the end of period 𝑡 is:
𝑜𝑡𝑠 =
𝑤𝑏 ∗ 𝑥𝑏,0𝑠
𝑏∈𝐵𝑡
𝑡 = 0
𝑜𝑡−1𝑠 − 𝑜𝑡
𝑝+ 𝑤𝑏 ∗ 𝑥𝑏,𝑡
𝑠
𝑏∈𝐵𝑡
𝑡 > 0
𝑎𝑡𝑠 =
𝑚𝑏 ∗ 𝑥𝑏,0𝑠
𝑏∈𝐵𝑡
𝑡 = 0
𝑎𝑡−1𝑠 − 𝑎𝑡
𝑝+ 𝑚𝑏 ∗ 𝑥𝑏,𝑡
𝑠
𝑏∈𝐵𝑡
𝑡 > 0
t t+1
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Non-linear model: Instant Mixing
• Other assumptions (but important):
– Blocks sent to stockpile are instantly mixed reaching homogeneity
– Other processes are not considered (for example: Comminution Process)
• Instant mixing constraint: 𝑎𝑡𝑝
𝑜𝑡𝑝 ≤𝑎𝑡−1𝑠
𝑜𝑡−1𝑠 ∀𝑡 ∈ 𝑇
𝑚1 𝑚2 𝑚3 𝑚4 𝑚5
𝑤1 𝑤2 𝑤3 𝑤4 𝑤5
Average Metal Grade
Stockpile
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Non-linear model: Objective function and other Constraints
Incomes Expenses
Precedence constraint
Extraction Capacity
Processing Capacity
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Linear Models: Upper Bound
• Blocks are stockpiled (and reclaimed) independently from each other
• There’s no instant mixing
• Infeasible solution!
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Linear Models: Lower Bound
• Material reclaimed from stockpile has a fixed metal grade 𝐿
– We replace the instant mixing constraint with: 𝑎𝑡𝑝= 𝐿 ∗ 𝑜𝑡
𝑝
L-Bound model
• Blocks sent to stockpile must have a metal grade above 𝐿
L-Average model
• The cumulative average metal grade of the blocks sent to the stockpile must be at least 𝐿
xb,ts = 0 ∀b ∈ B t. q. :
mbwb< L
𝑚𝑏 ∗ xb,𝑡′s ≥ 𝐿 ∗ 𝑤𝑏 ∗ xb,𝑡′
s
𝑏∈𝐵𝑡′≤𝑡𝑏∈𝐵 𝑡′≤𝑡
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Results
• Instance 1: Marvin (MineLib) – Solved with a fixed extraction sequence (so we can solve the non-
linear model with SCIP)
• Variation of processing capacities to observe economical impact of stockpiles
Cap. UB Non-Linear L-Average L-Bound No stock. 60% 2.1% $ 742,292,000 -0.3% -4.8% -11.8%
70% 1.3% $ 820,693,000 -0.1% -3.8% -8.2%
80% 0.6% $ 882,863,000 0.0% -2.5% -5.1%
90% 0.3% $ 928,833,000 0.0% -1.4% -2.9%
100% 0.1% $ 961,253,000 0.0% -0.7% -1.3%
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Solution Analysis
Mill Waste dump Stockpile
Extracted material destination Material sent to mill
Non-linear
L-Average
L-Bound
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Results
• Instance 2: Tampakan – Great Size Instance, we couldn’t use the non-linear model
– High presence of arsenic contaminant capacity constraint for this element
– Stockpile used for lowering arsenic average level in material sent to mill
Model NPV % difference
Upper Bound $ 4,848,040,000 -
L-Average $ 4,677,720,000 -3.51%
L-Bound $ 4,451,700,000 -8.18%
No Stockpile $ 4,296,550,000 -11.38%
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Solution Analysis
-
20
40
60
80
100
120
140
160
-
10.000.000
20.000.000
30.000.000
40.000.000
50.000.000
60.000.000
70.000.000
80.000.000
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Ars
en
ic G
rad
e
Ton
ns
of
Mat
eri
al
Period
Incoming material to mill, and arsenic grade in it (L-Average Solution)
0
500
1000
1500
2000
2500
3000
3500
4000
0 0,5 1 1,5 2 2,5 3
Ars
en
ic G
rad
e
Metal Grade
L-Average
Planta
Stockpile
Desecho
Destination of blocks extracted on first period
Cut-off grade: 0.21%
Mill
Waste Dump
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• Instance: Marvin (same than before)
• Our model reaches a higher NPV, even without using a stockpile, than commercial software Whittle.
Considering extraction decision
Solution NPV Variation
Whittle $ 847,035,400
Whittle + L-average $ 855,442,430 +0.99%
Optimal schedule (no stock)
$ 877,732,900 +3.62%
Optimal schedule (with L-average stockpile)
$ 911,356,530 +7.59%
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Solution Analysis
-
10.000.000
20.000.000
30.000.000
40.000.000
50.000.000
60.000.000
70.000.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ton
ns
of
Mat
eri
al
Period
Waste Dump
Sento to Stock
Sent to Mill
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
-
10.000.000
20.000.000
30.000.000
40.000.000
50.000.000
60.000.000
70.000.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Me
tal G
rad
e
Ton
ns
of
Mat
eri
al
Period
Stock to Mill
Sent to Mill
Metal Grade in Mill
Destination of Extracted Material
Material sent to mill, and metal grade in it
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Conclusions
• Stockpile use increases NPV of mine operations
• Linear model with stockpile option
– Practical way, can be used in large instances
– Behaves similar than non-linear models
• Optimization models defy classical ways to perform mine planning
• Stockpile use affects extraction sequence and block destination, but this is not considered by Com. Softwares