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August 2007
Copyright 2007, Gemcom Software International Inc. 1
Copyright 2008, Imageo & Gemcom Software International Inc.
WhittleCut-off Optimization
1
Prepared by Norm Hanson
Pit Optimization Copyright 2008, Imageo & Gemcom Software International Inc.
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This Morning’s Objectives
The purpose of this session is to present the theory behind cut-off optimisation, in easy to understand manner and to provide participants with the knowledge required to apply an elevated cut-off strategy to their own deposit.
August 2007
Copyright 2007, Gemcom Software International Inc. 2
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This session
Introduction to Value Concepts
� Understanding Value� Marginal Condition� How cut-off can be calculated?
� Risk of not using the “best” cut-off� Subsidizing Waste� Sterilizing Resource/Reserves
� Time Value of Money
� Cut-off & Cut-Overs
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What is Value?
Which Truck is Worth the Most?
1. 50 tonnes of 2g/t Gold
2. 100 tonnes of 1 g/t Gold
3. 250 tonnes of 0.5 g/t Gold
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What is Value?
Dollar Value = Revenues – Costs
Revenues can be calculated from:– Ore tonnages– Grades– Recoveries– Product price
Costs can be calculated from:– Mining cost– Milling cost
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CostsRevenue
50 tonnes of 2g/t Gold
= [(2* 50 * 92.5%* $27.97) - (50 * $17.5)]- (50 * $2.00)
[(2587) - (875)]- (100)
$1612
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CostsRevenue
100 tonnes of 1g/t Gold
= [(1* 100 * 92.5%* $27.97.15) - (100 * $17.5)]- (100 * $2.00)
[(2587) - (1750)]- (200)
$637
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CostsRevenue
250 tonnes of 0.5g/t Gold
= [(0.5* 250 * 92%* $27.97) - (250 * $17.5)]- (250 * $2.00)
[(3234) - (4375)]- (500)
-$1641
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But wait!
If we just call this truck load waste
We only pay $500 to mine it.
We would be $1141 better off
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What is the marginal Condition?
Whenever the cost of processing is higher than the revenue, we should treat the truck load as waste
Value =
[ (Ore*Grade*Recovery* Price) - (Ore*CostP) ] - Rock*CostMThe Section in square Brackets must => 0
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CostRevenue
The Marginal Situation
by transformation this becomes
Ore* Grade* Recovery* Price Ore* CostP=
Price*Recovery*OreCostP*Ore
Grade Marginal =
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Marginal Cut-off
Price*Recovery
CostPGrade Marginal =
This marginal cut-off condition will change whenever, Processing costs, Recoveries or Prices change!
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Revenue from CopperRevenue from gold
Costs
250 Tonnes of 0.25% Copper & 0.5 g/t Gold (process SX EW)
= [(0.25*250 * 65%* 27.97 + 0.25%*250 *30%*7840)
- (250 * $7.5)]- (250 * $2.00)
[(2273)+ (1470) - (1875) ]- (500)
3743- 2375
$1368
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Revenue from CopperRevenue from gold
Costs
250 Tonnes of 0.25% Copper & 0.5 g/t Gold (process Fl oatation)
= [(0.25*250 * 25%* 27.97 + 0.25%*250 *75%*7840)
- (250 * $12.5)]- (250 * $2.00)
[(874)+ (3675) - (3125) ]- (500)
4559- 3625
$924
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Value
Dollar Value = Revenues – CostsRevenues can be calculated from:– Ore tonnages– Grades– Recoveries– Product price
Costs can be calculated from:– Mining cost– Milling cost– Selling Costs– Overheads
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What affects the optimal outline?
In general:– If the price increases, the pit gets bigger– If the costs increase, the pit gets smaller– If the slopes are steeper, the pit gets deeper
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Finding the Optimal
MINERAL
AIR
WASTE
• Once price, costs and slope are fixed• The optimal outline is fixed
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A Simple Example
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Pit Tonnages and Value
Pit 1 2 3 4 5 6 7 8
Ore 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000Waste 100 400 900 1,600 2,500 3,600 4,900 6,400
Total 600 1,400 2,400 3,600 5,000 6,600 8,400 10,400
Tonnages
Pit 1 2 3 4 5 6 7 8
Value 900 1,600 2,100 2,400 2,500 2,400 2,100 1,600
Values
Ore is Worth$ 2.00Waste$ 1.00
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Size .vs. Value
$0
$500
$1,000
$1,500
$2,000
$2,500
$3,000
0 2,000 4,000 6,000 8,000 10,000 12,000
Pit Tonnes
Pit Value
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Design Sensitivity
$0
$500
$1,000
$1,500
$2,000
$2,500
$3,000
0 2,000 4,000 6,000 8,000 10,000 12,000
Pit Tonnes
Pit Value
A
B
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Guarantee One Optimal Solution
Finding the Outline
Four-X
Heuristics (searches)
Trial & Error
Floating Cone
Lerchs-Grossman
Johnson’s Network Flow
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How does 3-D Lerchs-Grossman Algorithm Work?
Works with block values
Works with block mining precedences (arcs)
Guarantees to find the three-dimensional outline with the highest possible value
Completely Searches the model
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Arc Relationships
If A is to be mined, B must be mined to expose A
The reverse is not true
If B is to be mined, A may or may not be mined
A
B
Arc from
A to B
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Arc ChainingAll slopes are translated into a large number of block relationshipsIt is wrong to assume we need an arc from each block to every block which is “above” itThis is because arcs can chain
A
B
C
If A is mined
so is C
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Chaining of Three Arcs per Block
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Let’s Do It
Demonstration using Gemcom Whittle
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Block Value - Rule 1
The value must be calculated on the assumption that the block has already been uncovered.
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Block Value - Rule 2
The value must be calculated on the assumption that the block will be mined.
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Block Value - Rule 3
Any expenditure that would stop if mining stopped must be included in the cost of mining, processing or selling.
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Minimum Arcs per Block
Desired Slope
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Demonstration of L-G Algorithm
A simple example
45 degree slopes
2-dimensions
Blocks are cubic
Principles are the same for 3-dimensions but harder to show.
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Three Arcs per Block
2-Dimensions & 45° slopes = 3 arcs per block
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Start
Starting with a 2-dimensional cross sectional model.
Only 3 blocks contain ore & have values as shown. All other
blocks are waste and have a value of –1.0
23.96.923.9
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Step 1
23.96.923.9
The first arc from a block containing value that we
find is to a block which is not flagged for mining
� � � �
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23.96.923.9
Step 2
We link the two blocks together. The total value of the two-block
branch is 22.9, therefore both blocks are now flagged to be mined.
22.9
����
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23.96.923.9
Step 3
We deal with the other two arcs from this block in the same way.
The total value of the four-block branch is 20.9
20.9
����
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23.96.923.9
Step 4
We can continue the same process to the end of the first bench
20.9 20.93.9
����� ��������� � � � � � � � �
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23.96.923.9
Step 5
We then moved along the next bench, and find a block which has
no value itself, but is part of a branch with value
17.9 20.93.9
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� � � ����
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23.96.923.9
Step 6
The next flagged block has an arc to a block which is also
flagged. We do not create a link for this arc or for the vertical one
from the same block, because nothing new has to be resolved.
17.9 20.93.9
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����
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23.96.923.9
Step 7
The next arc from a flagged to another flagged block is between two
branches. The procedure is unchanged – we do not insert a link
15.9 20.93.9
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23.96.923.9
Step 8
We continue adding links. The dotted link when added will change
the value of the branch to –0.1. All blocks in this branch have their
flags turned off.
15.9 20.90.9
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�
� � �
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23.96.923.9
Step 9
The Lerchs-Grossman includes a procedure for combining the two
linked branches into one branch, with only one total value. Note
that there is no requirement to always branch upwards from the
root.
15.9 20.8
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����
����
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Step 10
23.96.923.9
At the end of the second bench we have now have only two branches
15.9 16.8
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����
���� ���� ����
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Step 11
23.96.923.9
Lerchs-Grossman detects that the extra waste will remove the ability
of the centre branch to co-operate with the right hand branch in
paying for the mining of the circled block.
8.9 16.8
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23.96.923.9
Step 12
Lerchs-Grossman includes a procedure for breaking the single
branch into two branches by removing a link
8.9 15.9
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���� ���� �������� ���� ����
���� ���� ���� ���� ����
� ��
� � �
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23.96.923.9
Step 13
At the end of this third bench we have drop the central sub branch
above the low grade block
8.9 8.9
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��������
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23.96.923.9
Step 14
Continue adding links and eventually the total value of the left-hand
branch becomes negative. The next arc after this is again between
a positive and negative branch.
-0.1 8.9
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23.96.923.9
Step 15
At the fourth bench we have just one branch and the combined value
is now only 0.8
0.8
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23.96.923.9
Step 16
The L-G program scans for arcs from blocks which are flagged to
blocks which are not flagged. We can see The search has reach
the top of the model and not more block have to be removed.
0.8
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23.96.923.9
Optimal Pit
The flagged blocks constitute the optimal pit. The ‘W’-shaped pit
is worth 0.8. The centre branch has a negative value so none of
its blocks are flagged and none are mined.
0.8
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Let’s Do It
Demonstration using Gemcom Whittle
August 2007
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Which would you choose?
”Now you must decide. Do you want $6 million now in your hand or $1 million a year for 10 years?”
Let’s review Worksheet 3
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DCF
Discount Factor
ActualCash Flow
1 2 3 4 5 6 7 8 9 10 11 Total
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 10.0
0.91 0.83 0.75 0.68 0.62 0.56 0.51 0.47 0.42 0.39
0.91 0.83 0.75 0.68 0.62 0.56 0.51 0.47 0.42 0.39 6.14
DCF Analysis
•"Financial" NPV Factor 10%(1/(1+D/100))
NPV
August 2007
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DCF Analysis
The Discounted Casflow Method allows us to compare investment in Today’s Dollar Terms.
Expected future cash flows are discounted by a percentage each year
– Allow for cost of capital– Allow for risk
Sum of discounted cash flows is called NPV Net Present Value
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The Time Value of Money
A dollar we receive today is more valuable than a dollar we may receive in the future
This is important when we wish to make a decision about a some Long Term Investments.
∑− +
=m
nn
1 )Disc(1
CashFlowNPV
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Cut-Off Grade
Cut-off grade has been sacrosanct over long time periods
The formal method to define what might be considered economically viable to mine (ie what is the Ore Reserve)
Price*Recovery
CostPGrade Marginal =
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Cut-Off Grade - example
Cut-Off Grade = COSTP / (PRICE* REC) = $15 / ( $12.70* 92.5%)
Cut-Off Grade = 15 / 12.06
= 1.24
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Revenue vs Grade
Grade
Revenueper tonne
0
0
Cost of“processing”
Cut-off
Gradient = R
ecove
ry x P
rice
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Non-Linear Recovery
The percentage recovered in the mill depends on the head grade
Usually increases with increasing head grade
Some mills have a constant tailings grade
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Non-Linear Recovery
Grade
Revenueper tonne
0
0
Cost of"processing"
Cut-off
Thresholdgrade
Gradient =
Reco
very
x Pric
e
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Revenue vs Grade
Grade
Revenueper tonne
0
0
Cost of“processing”
Cut-off
Gradient = R
ecove
ry x P
rice
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Multiple Processing Methods
At any particular grade, we usually choose the processing method that produces the highest value (cash flow per tonne)
The “cut-over” is where there is a profit cross over.– A common mistake is to calculate the two cut-offs independently
Heap Leach
CIP
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Multiple Processing Methods
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Multiple Products
Have to handle different proportions of products (metals)
Value of products may also vary of time
Common Approaches– Use equivalent metal– Use Value based cut-offs– Use CashFlow “grades”
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Equivalent Metal
GRADE1*REC1*PRICE1 + GRADE2*REC2*PRICE2 => PRCOST
Equiv GRADE1 = GRADE1 + K2*GRADE2where K2=(REC2*PR2)/(REC1*PR1)
Equiv GRADE2 = GRADE2 + K1*GRADE1where K1 = (REC1*PR1)/(REC2*PR2)
– Only applies if PRCOST is independent of grade, or varies linearly with grade
– Only applies if recovery is independent of grade
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Equivalent Metal
Alternatively calculate each cut-off separately and then use a cut-off of 1.0 with:
(GRADE1/CUTOFF1) + (GRADE2/CUTOFF2)
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Equivalent Metal (Graphically)
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Gold Price is the same but Project Size Varies
0.1 (0.1 (0.1 ( g/tg/tg/t )))
1 (1 (1 ( g/tg/tg/t )))
10 (10 (10 ( g/tg/tg/t )))
100 (100 (100 (g/tg/tg/t )))
Sm
all
Sm
all
Sm
all
U/G
roun
dU
/Gro
und
U/G
roun
d
& G
ravi
ty&
Gra
vity
& G
ravi
ty
Larg
eLa
rge
Larg
e
Dec
line
&D
eclin
e &
Dec
line
&
CIPCIP
CIP
Sm
all
Sm
all
Sm
all
Ope
npit
Ope
npit
Ope
npit
&C
IP&
CIP
&C
IP
Larg
eLa
rge
Larg
e
Ope
ns P
itO
pens
Pit
Ope
ns P
it
& C
IP&
CIP
& C
IP
Sm
all
Sm
all
Sm
all
Ope
n P
itO
pen
Pit
Ope
n P
it
& L
each
& L
each
& L
each
Larg
eLa
rge
Larg
e
Ope
n P
itO
pen
Pit
Ope
n P
it
& L
each
& L
each
& L
each
Mining/Milling CombinationsMining/Milling CombinationsMining/Milling Combinations
CutCut
Cut
-- - off
Gra
deof
f Gra
deof
f Gra
de
Underground Pit & CIP Pit & Leach
Pit Optimization Copyright 2008, Imageo & Gemcom Software International Inc.
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Type of Mining/Milling Likely Cut-Off
Underground Mines 5.02.5
Open Pit with CIP 2.01.51.0
Open Pit with Heap Leach 0.75
Large Scale Open Pit with Leach0.500.25
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Grade/Tonnage Relationship
Cut-off Tonnes Grade Contained Gold(Ounces)
0.25 70,291,800 1.66 3,739,7270.50 54,429,300 2.03 3,552,4840.75 43,173,000 2.40 3,332,8001.0 35,316,000 2.74 3,115,1921.5 26,152,200 3.28 2,757,8782.0 19,750,500 3.78 2,400,6482.5 15,106,500 4.26 2,067,2535.0 3,385,800 6.20 674,896
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Traditional Grade/Tonnage Curves
000
101010
202020
303030
404040
505050
606060
707070
808080
0 1 2 3 4 5
Cut-off Grade (g/t)
Ton
neT
onne
Ton
ne(m
)(m
)(m
)
000
111
222
333
444
555
666
777
Gra
de (
Gra
de (
Gra
de (
g/t
g/t
g/t )) )
Tonnes
Grade
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Alternate Grade/Tonnage Curve
0.250.50
0.751.0
2.02.5
5.0
1.5
0.000.000.00
1.001.001.00
2.002.002.00
3.003.003.00
4.004.004.00
5.005.005.00
6.006.006.00
7.007.007.00
000 202020 404040 606060 808080 100100100
TonnesTonnesTonnes (m)(m)(m)
Gra
de (
Gra
de (
Gra
de (
g/t
g/t
g/t )) )
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Mineral Resources
The shape, quantity and qualityof a resource varies with the concentration of mineral product
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The Economic Reality
The reality of mining today is that prices and economic constraints can significantly vary over time
Therefore the cut-off grade and the shape of a resource may also change
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Inappropriate Cut-off
If the Cut-off is too low.– Subsidizing processing of uneconomic mineralization.– Loose Value
If the cut if is too high– Lower resource can be exploited– [Potentially] Loose Value
August 2007
Copyright 2007, Gemcom Software International Inc. 39
Pit Optimization Copyright 2008, Imageo & Gemcom Software International Inc.
77
���� What you have Learnt
The value of any unit of material mined can calculated
It is sometimes better to throw “mineralization” away as waste, egcost of processing is higher than revenue generated. CUTOFF
A dollar we receive today is worth more than a dollar which may be received in the future.
When there are more than one possible processing stream, send ore to the one that generates the highest value, CUTOVER