strategic mine planning 1

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GEMCOM SOFTWARE INTERNATIONAL INC.

www.gemcomsoftware.com

Whittle

Strategic Mine Planning

Prepared by Norm HansonFor Witwatersrand University

School of Mining Engineering



Training Objectives

This course is design to introduce participants to pit optimization concept and allow them to become proficient at preparing design constrains, prepared suitable models and run pit designs using Whittle Programming’s Four-X Pit Optimization software.

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Today’s Program

Introduction to Pit Optimization

o Introduction to Optimization Concepts

o Quick Tour Of Four-X o Exporting the block model and

topography Validation of model export

o My “first” optimum pit design



What is Optimal?

MINERAL

AIR

WASTE

Any Feasible Outline has a Value

The Outline with the highest value is Optimal

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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. 150 tonnes of 0.5% Copper & 0.25 g/t Gold



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 * 84%* 101.27 ) - (50 * R90)]- (50 * R7.20)

[(8506.94) - (4500)]- (360)

R3646.94



CostsRevenue

100 tonnes of 1 g/t Gold

[(1*100 * 84%* 101.27) - (100 * R90)]- (100 * R7.20)

[(8506.94) - (9000)]- (720)

-1213.06

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But wait!

If we just call this truck load waste

• We only pay R720 to mine it.

• We would be R463.06better off



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*CostM

The 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 =



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

150 Tonnes of 0.5%

Copper & 0.25 g/t Gold

= [(0.25*150 * 50%* 101.27 + 0.5%*150 *75%*14767 )

- (150 * R48)]- (150 * R 7.20)

[(2012.74)+ (8306.33) - (7200) ]- (1080)

10319.07- 8280

2039.07

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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

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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



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

R 2.00Waste

R 1.00

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Size .vs. Value

R 0

R 500

R 1,000

R 1,500

R 2,000

R 2,500

R 3,000

0 2,000 4,000 6,000 8,000 10,000 12,000

Pit Tonnes

Pit Value



Design Sensitivity

R 0

R 500

R 1,000

R 1,500

R 2,000

R 2,500

R 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• 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



Arc Chaining

• All slopes are translated into a large number of block relationships

• It is wrong to assume we need an arc from each block to every block which is “above”it

• This is because arcs can chainA

B

C

If A is mined

so is C

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Chaining of Three

Arcs per Block



Let’s Do It

Demonstration using Four-X

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

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



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



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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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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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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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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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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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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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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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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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

strategic mine planning 1

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