cut off optimization

39
August 2007 Copyright 2007, Gemcom Software International Inc. 1 Copyright 2008, Imageo & Gemcom Software International Inc. Whittle Cut-off Optimization 1 Prepared by Norm Hanson Pit Optimization 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.

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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.

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.

August 2007

Copyright 2007, Gemcom Software International Inc. 2

Pit Optimization Copyright 2008, Imageo & Gemcom Software International Inc.

3

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

Pit Optimization Copyright 2008, Imageo & Gemcom Software International Inc.

4

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

August 2007

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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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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 =

[ (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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15

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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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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���� ���� ���� �

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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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����

���� ���� �������� ���� ����

���� ���� ���� ���� ����

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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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��������

Pit Optimization Copyright 2008, Imageo & Gemcom Software International Inc.

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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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���� ���� ���� ���� ���� ���� ����

���� ���� ����

������������

���� ���� ���� ���� ���� ���� ���� ��

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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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���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� �

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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

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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

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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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60

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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61

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.

70Likely Cut-offs for Goldwith a US$325/oz Gold Price

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

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���� 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