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.

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.

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

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

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

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

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

Non-linear model: Objective function and other Constraints

Incomes Expenses

Precedence constraint

Extraction Capacity

Processing Capacity

Linear Models: Upper Bound

• Blocks are stockpiled (and reclaimed) independently from each other

• There’s no instant mixing

• Infeasible solution!

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

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%

Solution Analysis

Mill Waste dump Stockpile

Extracted material destination Material sent to mill

Non-linear

L-Average

L-Bound

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%

Solution Analysis

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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

• 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%

Solution Analysis

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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

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