Cut Off Optimization

August 2007

1

Whittle Cut-off Cut-o ff Optimi Optimizatio zation n Prepared by Norm Hanson

Copyright 2008, Imageo & Gemcom Software International Inc.

2

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.

Pit Optimization

Copyright 2007, Gemcom Software International Inc.


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August 2007

3

This session Introduction to Value Concepts 

Understanding Value  



Marginal Condition How cut-off can be calculated?

Risk of not using using the “best” “best” cut-off cut-off  

Subsidizing Waste Sterilizing Resource/Reserves



Time Value of Money



Cut-off & Cut-Overs

Pit Optimization


4

What is Value? Which Truck is Worth the Most? 1. 2. 3.

50 tonnes of 2g/t Gold 100 tonnes of 1 g/t Gold 250 tonnes of 0 .5 .5 g/t Gold

Pit Optimization



2

August 2007

3

This session Introduction to Value Concepts 

Understanding Value  



Marginal Condition How cut-off can be calculated?

Risk of not using using the “best” “best” cut-off cut-off  

Subsidizing Waste Sterilizing Resource/Reserves



Time Value of Money



Cut-off & Cut-Overs

Pit Optimization


4

What is Value? Which Truck is Worth the Most? 1. 2. 3.

50 tonnes of 2g/t Gold 100 tonnes of 1 g/t Gold 250 tonnes of 0 .5 .5 g/t Gold

Pit Optimization



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August 2007

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What is Value? Dollar Value = Revenues Revenues – – Costs Revenues can be calculated from: – – – –

Ore tonnages Grades Recoveries Product price

Costs can be calculated from: – Mining cost – Milling cost

Pit Optimization


6

50 tonnes tonnes of 2g/t 2g/t Gold

= [(2* 50 * 92.5%* $27.97 ) - (50 * $17.5)]$17.5)]- (50 * $2.00) $2.00)

Revenue

Costs

[(2587) - (87 (875)] 5)]-- (10 (100) 0)

$1612

Pit Optimization



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August 2007

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100 tonnes of 1g/t Gold

= [(1* 100 * 92.5%* $27.97.15 ) - (100 * $17.5)]- (100 * $2.00)

Revenue

Costs

[(2587) - (1750)]- (200)

$637

Pit Optimization


8

250 tonnes of 0.5g/t Gold

= [(0.5* 250 * 92%* $27.97 ) - (250 * $17.5)]- (250 * $2.00)

Revenue

Costs

[(3234) - (4375)]- (500)

-$1641

Pit Optimization



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August 2007

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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

Pit Optimization


10

What is the marginal Condition?

Whenever the cost of processing is higher than the revenue, we should treat the truck load as waste Value = The Section in square Brackets must => 0

[ (Ore*Grade*Recovery* Price) - (Ore*CostP) ] - Rock*CostM

Pit Optimization



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August 2007

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The Marginal Situation

Ore *Grade *Recovery* Price = Ore * CostP

Revenue

Cost

by transformation this becomes

Marginal Grade

Pit Optimization

=

Ore * CostP Ore * Recovery * Price


12

Marginal Cut-off

Marginal Grade =

CostP Recovery* Price

This marginal cut-off condition will change whenever, Processing costs, Recoveries or Prices change! Pit Optimization



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August 2007

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250 Tonnes of 0.25% Copper & 0.5 g/t Gold (process SXEW)

Revenue from gold

Revenue from Copper

= [(0.25*250 * 65%*

0.25%*250 *30%* 7840 )

27.97 +

- (250 * $7.5)]- (250 * $2.00)

Costs [(2273)+ (1470) - (1875) ]- (500) 3743- 2375

$1368 Pit Optimization


14

250 Tonnes of 0.25% Copper & 0.5 g/t Gold (process Floatation)

Revenue from gold

Revenue from Copper

= [(0.25*250 * 25%* 27.97 + 0.25%*250 *75%*7840 ) - (250 * $12.5)]- (250 * $2.00)

Costs [(874)+ (3675) - (3125) ]- (500) 4559- 3625

$924 Pit Optimization



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August 2007

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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 Selling Costs Overheads 15

Pit Optimization


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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

Pit Optimization



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August 2007

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Finding the Optimal

AIR

WASTE

MINERAL

• Once price, costs and slope are fixed • The optimal outline is fixed Pit Optimization


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A Simple Example

Pit Optimization



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August 2007

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Pit Tonnages and Value

Tonnages Pit

1

Ore Waste

3

4

5

6

7

8

Ore is Worth 500 1,000 1,500 $Waste 2.00 100 400 900 $ 1.00

2,000 1,600

2,500 2,500

3,000 3,600

3,500 4,900

4,000 6,400

3,600

5,000

6,600

8,400 10,400

Total

600

2

1,400

2,400

Values Pit Value

Pit Optimization

1

2

3

4

5

6

7

8

900

1,600

2,100

2,400

2,500

2 ,4 00

2 ,1 00

1 ,6 00


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Size .vs. Value

$3,000 $2,500 e $2,000 u l a V $1,500 t i P $1,000

$500 $0 0

2,000

4,000

6,000

8,000

10,000

12,000

Pit Tonnes

Pit Optimization



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August 2007

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Design Sensitivity

$3,000 $2,500 B

e $2,000 u l a V $1,500 t i P $1,000

A

$500 $0 0

2,000

4,000

6,000

8,000

10,000

12,000

Pit Tonnes

Pit Optimization


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Finding the Outline

Heuristics (searches) Trial & Error Floating Cone Lerchs-Grossman

Four-X

Johnson’s Network Flow

Guarantee One Optimal Solution

Pit Optimization



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August 2007

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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

Pit Optimization


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Arc Relationships If A is to be mined, B must be mined to expose A The reverse is not true

B

If B is to be mined, A may or may not be mined

Arc from A to B A

Pit Optimization



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August 2007

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Arc Chaining C

All slopes are translated into a large number of block relationships It is wrong to assume we need a n arc from each block to every block which is “above” it This is because arcs can chain

B If A is mined so is C A Pit Optimization


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Chaining of Three Arcs per Block

Pit Optimization



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August 2007

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Let’s Do It

Demonstration using Gemcom Whittle

Pit Optimization


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Block Value - Rule 1

The value must be calculated on the assumption that the block has already been uncovered.

Pit Optimization



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August 2007

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The value must be calculated on the assumption that the block will be mined.

Pit Optimization


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Any expenditure that would stop if mining stopped must be included in the cost of mining, processing or selling.

Pit Optimization



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August 2007

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Minimum Arcs per Block

Desired Slope

Pit Optimization


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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.

Pit Optimization



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August 2007

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Three Arcs per Block

°

2-Dimensions & 45 slopes = 3 arcs per block Copyright 2008, Imageo & Gemcom Software International Inc.

Pit Optimization

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Start

23.9

6.9

23.9

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 Pit Optimization



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August 2007

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Step 1

   

23.9

6.9

23.9

The first arc from a block containing value that we find is to a block which is not flagged for mining

Pit Optimization


36

Step 2



23.9

6.9

23.9

22.9 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. Pit Optimization



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August 2007

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Step 3



23.9

6.9

23.9

20.9 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 Pit Optimization


38

Step 4

6.9              23.9

23.9

20.9

3.9

20.9

We can continue the same process to the end of the first bench Pit Optimization



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August 2007

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Step 5

    

23.9

17.9

6.9 

3.9

 23.9

20.9

We then moved along the next bench, and find a block which has no value itself, but is part of a branch with value Pit Optimization


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Step 6

 

23.9

17.9

6.9 

3.9

 23.9

20.9

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. Pit Optimization



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August 2007

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Step 7

   

23.9

15.9

6.9 

3.9

 23.9

20.9

The next arc from a flagged to another flagged block is between two branches. The procedure is unchanged – we do not insert a link Pit Optimization


42

Step 8

 

  





15.9

0.9

23.9

6.9

 23.9

20.9

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. Pit Optimization



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August 2007

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Step 9

  

23.9

   

6.9

  23.9

20.8

15.9

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. Pit Optimization


44

Step 10

  

23.9

  

  



23.9

6.9

15.9



16.8

At the end of the second bench we have now have only two branches Pit Optimization



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August 2007

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Step 11

        

23.9

  

  



23.9

6.9



16.8

8.9

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. Pit Optimization


46

Step 12

           

23.9

   

6.9

8.9

    23.9

15.9

Lerchs-Grossman includes a procedure for breaking the single branch into two branches by removing a link Pit Optimization



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August 2007

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Step 13

      

        

23.9

   



6.9

23.9

8.9

8.9

At the end of this third bench we have drop the central sub branch above the low grade block Pit Optimization


48

Step 14

           

    

  

  



23.9



6.9

-0.1

 23.9

8.9

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. Pit Optimization



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August 2007

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Step 15

       

           

    

  

  



23.9





6.9

23.9

0.8 At the fourth bench we have just one branch and the combined value is now only 0.8 Pit Optimization


50

Step 16

                            

      

  

  



23.9

     



6.9

23.9

0.8 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. Pit Optimization



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August 2007

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Optimal Pit

23.9

6.9

23.9

0.8 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. Pit Optimization


52

Let’s Do It

Demonstration using Gemcom Whittle

Pit Optimization



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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 Pit Optimization


54

DCF Analysis

Actual

1

2

3

4

1.00

1.00

1.00

1.00

0.91

0.83

0.75

0.91

0.83

0.75

5

6

7

8

9

10

1.00

1.00

1.00

1.00

1.00

1.00

0.68

0.62

0.56

0.51

0.47

0.42

0.39

0.68

0.62

0.56

0.51

0.47

0.42

0.39

11

Total

10.0

Cash Flow

Discount Factor

DCF

6.14

NPV •"Financial" NPV Factor 10%(1/(1+D/100))

Pit Optimization



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August 2007

55

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

Pit Optimization


56

The Time Value of Money

A dollar we receive today is more valuable than a dollar we may receive in the future CashFlow NPV = ∑ 1 (1 + Disc ) m

n

n−

This is important when we wish to make a decision about a some Long Term Investments.

Pit Optimization



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August 2007

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Cut-Off Grade

Cut-off grade has been sacrosanct over long time periods

Marginal Grade =

CostP Recovery* Price

The formal method to define what might be considered economically viable to mine (ie what is the Ore Reserve)

Pit Optimization


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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 =

Pit Optimization


1.24


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August 2007

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Revenue vs Grade Revenue per tonne

i c e P r x e r y o v c R e t = n i e a d G r

0

Grade Cut-off

Cost of “processing”

0

Pit Optimization


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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

60 Pit Optimization



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August 2007

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Non-Linear Recovery Revenue per tonne


0

Grade Cut-off

Cost of "processing"

0

Threshold grade

61 Pit Optimization


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Revenue vs Grade

Revenue per tonne


0

Grade Cut-off

Cost of “processing”

0 Pit Optimization



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August 2007

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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) Heap Leach

CIP

The “cut-over” is where there is a profit cross over. – A common mistake is to calculate the two cut-offs independently

63 Pit Optimization


64

Multiple Processing Methods

64 Pit Optimization



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August 2007

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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”

Pit Optimization


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Equivalent Metal

GRADE1*REC1*PRICE1 + GRADE2*REC2*PRICE2

=>

PRCOST

Equiv GRADE1 = GRADE1 + K2*GRADE2 where K2=(REC2*PR2)/(REC1*PR1) Equiv GRADE2 = GRADE2 + K1*GRADE1 where 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

Pit Optimization



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August 2007

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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)

Pit Optimization


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Equivalent Metal (Graphically)

Pit Optimization



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August 2007

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Gold Price is the same but Project Size Varies 100 100 g/t))) 100 ((g/t (g/t

10 10 g/t 10 (((g/t g/t)))

e e e d d d a a a r r r G G G f f f f f f o o o t t t u u u C C C

111 (((g/t g/t g/t)))

0.1 0.1 ((g/t g/t g/t)))

d y d y d t y l t l t l l n n l i n i i l a v u a v a u u v a o a o a o m r m r r m r r r S G S G S / G G G G / / U & U & U &

t t t i l i l l l i P l p P l p P p I I a I a a n n n C C m e m e C m e p & p & S p & S S O O O

& & & e e e e e e P P g P n g n I g l i n I I r i r i r l C a a c C a l c c C L L L e e e D D D

t t i t i i P P P P I P P e I e I e s s s C g g g C r n r n C r n e a e a e & a & L p L p & L p O O O

t t i t i i h l l h l l P l P l c P h c c a a a n a n a n a e m e m e e m e e L S p S L p L S p O O & O & &

t t i t i i h h P c e P h c e c e P g a g n a g n n a r r e e r e e e e L a a a p p L p L L L L O O & O & &

Mining/Milling Combinations Mining/MillingCombinations Combinations

Underground

Pit Optimization

Pit & CIP

Pit & Leach


70

Likely Cut-offs for Gold with a US$325/oz Gold Price

Type of Mining/Milling

Underground Mines Open Pit with CIP

Open Pit with Heap Leach Large Scale Open Pit with Leach

Pit Optimization


Likely Cut-Off

5.0 2.5 2.0 1.5 1.0 0.75 0.50 0.25


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August 2007

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Grade/Tonnage Relationship

Cut-off

Grade Contained Gold

Tonnes

(Ounces)

0.25 0.50 0.75 1.0 1.5 2.0 2.5 5.0 Pit Optimization

70,291,800 54,429,300 43,173,000 35,316,000 26,152,200 19,750,500 15,106,500 3,385,800

1.66 2.03 2.40 2.74 3.28 3.78 4.26 6.20

3,739,727 3,552,484 3,332,800 3,115,192 2,757,878 2,400,648 2,067,253 674,896


72

Traditional Grade/Tonnage Curves

80 80

77

70 70

66

60 60 ) ) m m ( ( e e n n n n o o T T

55

50 50

44

40 40 30 30

Tonnes

20 20

Grade

33 22

10 10

11

00

00 0

1

2

3

4

) ) t t / / g g ( ( e e d d a a r r G G

5

Cut-off Grade (g/t)

Pit Optimization



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August 2007

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Alternate Grade/Tonnage Curve

7.00 7.00 5.0

6.00 6.00 5.00 5.00 ) ) t t / / g g ( ( e e d d a a r r G G

2.5 2.0 1.5

4.00 4.00 3.00 3.00

1.0

2.00 2.00

0.75

0.50

0.25

1.00 1.00 0.00 0.00 00

20 20

40 40

60 60

80 80

100 100

Tonnes Tonnes (m) (m)

Pit Optimization


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Mineral Resources The shape, quantity and quality of a resource varies with the concentration of mineral product

Pit Optimization



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August 2007

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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

Pit Optimization


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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

Pit Optimization



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Cut Off Optimization

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