GEMCOM SOFTWARE INTERNATIONAL INC.
Whittle
Strategic Mine Planning
Norm Hanson
University
School of Mining Engineering
Prepared by
For Witwatersrand
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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
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What is Optimal?
AIR
WASTE
MINERAL
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.
2.
3.
50 tonnes of 2g/t Gold
100 tonnes of 1 g/t Gold
150 tonnes of 0.5% Copper & 0.25 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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50 tonnes of 2g/t Gold
= [(2* 50 * 84%* 101.27 ) - (50 * R90)]- (50 * R7.20)
Revenue
Costs
[(8506.94) - (4500)]- (360)
R3646.94
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100 tonnes of 1 g/t Gold
[(1*100 * 84%* 101.27) - (100 * R90)]- (100 * R7.20)
Revenue
Costs
[(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 R463.06
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 =
The Section in square
Brackets must => 0
[ (Ore*Grade*Recovery* Price) - (Ore*CostP) ]
- Rock*CostM
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The Marginal Situation
Ore *Grade *Recovery*Price = Ore *CostP
Revenue
Cost
by transformation this becomes
Marginal Grade =
Ore * CostP
Ore * Recovery * Price
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Marginal Cut-off
Marginal Grade =
CostP
Recovery* Price
This marginal cut-off condition will change
whenever, Processing costs, Recoveries or Prices
change!
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150 Tonnes of 0.5%
Copper & 0.25 g/t Gold
Revenue from gold
Revenue from Copper
= [(0.25*150 * 50%* 101.27 + 0.5%*150 *75%*14767 )
- (150 * R48)]- (150 * R 7.20)
Costs
[(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
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Finding the Optimal
AIR
WASTE
MINERAL
• 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
Tonnages
Pit
Ore
Waste
Total
3
4
5
6
7
8
Ore is Worth
500
1,000 1,500
Waste
R
2.00
100
400
900
R 1.00
1
2
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
600
1,400
2,400
1
2
3
4
5
6
7
8
900
1,600
2,100
2,400
2,500
2,400
2,100
1,600
Values
Pit
Value
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Size .vs. Value
R 3,000
Pit Value
R 2,500
R 2,000
R 1,500
R 1,000
R 500
R0
0
2,000
4,000
6,000
8,000
10,000 12,000
Pit Tonnes
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Design Sensitivity
R 3,000
Pit Value
R 2,500
B
R 2,000
A
R 1,500
R 1,000
R 500
R0
0
2,000
4,000
6,000
8,000
10,000 12,000
Pit Tonnes
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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
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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
B
Arc from
A to B
A
• 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
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Arc Chaining
C
B
If A is mined
so is C
A
• 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 chain
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Chaining of Three
Arcs per Block
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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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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 LG Algorithm
•
•
•
•
•
A simple example
45 degree slopes
2-dimensions
Blocks are cubic
Principles are the same for 3dimensions 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
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
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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
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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.
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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
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GEMCOM SOFTWARE INTERNATIONAL INC.
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
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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
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Step 6
6.9
23.9
3.9
17.9
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.
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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
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Step 8
23.9
6.9
15.9
0.9
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.
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Step 9
23.9
15.9
6.9
23.9
20.8
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.
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Step 10
23.9
6.9
15.9
23.9
16.8
At the end of the second bench we have now have only two branches
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Step 11
23.9
6.9
8.9
23.9
16.8
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.
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Step 12
23.9
6.9
23.9
15.9
8.9
Lerchs-Grossman includes a procedure for breaking the single branch
into two branches by removing a link
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Step 13
23.9
8.9
6.9
23.9
8.9
At the end of this third bench we have drop the central sub branch
above the low grade block
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Step 14
23.9
6.9
23.9
8.9
-0.1
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.
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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
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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.
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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.
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