---2 4 3 Lerchs-Grossman Algorithm

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Lerchs-Grossman Algorithm 
An ultimate pit limit method 
 
 Block model. 
 
 The deposit can then be represented by a set of blocks. 
With geostatistical modelling, it is possible to know the grade of each 
block. Using known parameters such as ore price, cut-off grade, mining 
cost, block mass etc., the economic value of a block can be calculated. 
 
Feasible Pit. 
• There is a slope angle constraint in a surface mining pit. We assume that the 
slope angle is 45◦. 
• In a 2-dimensional pit optimisation problem, to extract an underlying block, 
you have to make sure that the blocks extracted above satisfy the slope angle 
constraint. 
• Example: To extract block A, you have to extract at least 3 blocks (B, C, D) in 
the row above to satisfy the angle constraint. For blocks B, C and D, the same 
principle applies, and so on until reaching the surface. You cannot extract 
block J since only extracting the two blocks above (E and F) won’t satisfy the 
constraint (90 degrees slope on the left!). 
E F G H I 
J B C D 
A A 
Optimal pit contour. 
 
 The purpose of block modeling is to find the optimal pit contour and the 
sequence of extraction. 
 The optimal pit contour is composed of a set of blocks forming a feasible 
pit, such that the selected parameter (usually the profit) is maximized. 
This is called pit limit optimization. 
 For instance, when adding the weight of the blocks forming the optimal 
pit, the sum obtained should be the highest of all feasible pits. 
 
E F G H I 
B C D 
A 
Example: In this block model, EFGB, FGHC, 
GHID, EFGHBC, or EFGHIBCDA etc. are all 
feasible pits. However, one of them has 
the highest sum of weights and this one 
would be the optimal pit contour. 
Graphic representation. 
 After obtaining the block model, the first thing to do is to eliminate 
the blocks on the side that can’t form any feasible pit (in blue). 
 
 
 
 
 
 
 The blocks can then be represented by a set of nodes. 
 
 
E F G H I 
B C D 
A 
A 
C B D 
G H I F E 
Graphic Representation. 
 Edges are straight lines that connect nodes to the nearest overlying 
neighbors. 
 Arcs, as opposed to edges have a direction (arrowhead). An arc denoted 
by (A,B) means that the flow goes from node A to B. The arc are always 
directed to the surface. 
 A set of nodes connected by arcs is called a directed graph G=(X,A). 
 
 
 
 
 
 A 
C B D 
G H I F E 
Definitions 
 Path: Sequence of arcs such that the terminal node of each arcs is the initial 
node of the following one 
 Circuit: Path in which the initial node is the same as the terminal node 
 Chain: Sequence of edges in which each edge has one node in common with 
the following one 
 Cycle: Chain in which the initial and final node coincide 
 Tree: Connected and directed graph with no cycles. A tree is denoted Tn. A 
rooted tree is a tree with a special node, called the root 
 Root: Node selected from a tree, usually denoted as X0. A tree can only have 
one root. 
 Dummy arc: Arc connected to the root node. 
 Branch: When a tree is cut into two part, the part of not containing the root 
is called a branch. 
 
Solving the graph. 
Step 1 : Assign weight to the nodes and build the initial Tree T0 by 
connecting all nodes to the root node Xo with arcs. 
 
 
 
 
 
 
I H G F E 
C D B 
A 
-6 -2 -3 -1 2 
-11 1 8 
10 
X0 
Step 2 : Label nodes using the following information's : 
 
 
 
 
Initially, all nodes in the initial tree are “plus” as they are all directed away 
from the root. 
 
Direction Cumulative weight Label 
plus + Strong 
plus - or 0 Weak 
minus + Weak 
minus - or 0 Strong 
Plus: directed away from the root. 
Minus: directed towards the root. 
-6 
W+ 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
S+ 
8 
S+ 
10 
S+ 
X0 
Distinguish two groups in this rooted initial 
tree : The group TA that contains all strong-
plus (S+) nodes connected to root and the 
group TB that contains all weak-plus (W+) 
nodes connected to root. 
Important note 
 The cumulative weight of an arc is the sum of all the nodes supported by 
this arc. 
 Example: 
 
 
 
 
 
 
 The cumulative weight of arc A1 is -3+5+4=6. 
 The cumulative weight of arc A2 is 4. 
 
 
A2 is directed away from 
the root. 
A1 is directed away from 
the root. 
Important Note. 
 The goal of the Leachs-Grossman algorithm is to find all possible 
connections between group TA (S+ nodes) and group TB (W+ nodes) 
 All strong arcs not directly connected to the root should be treated: 
• All strong plus (S+) arcs should be connected directly with the root x0 . 
• A strong minus (S-) arc should be connected directly with the root x0. 
 
Step 3 : Iterations. 
 Always start with the second row from surface. Take a S+ node and connect it with an 
upper W+ node (Iteration #1) 
 RULE # 1: If the cumulative weight is positive after adding the W+ node for the same 
S+ node, keep the S+ node arc connected to the root, and remove the W+ node arc 
connected to the root (Iteration #1). 
 
 
 Iteration #1: 
-6 
W+ 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
S+ 
8 
S+ 
10 
S+ 
X0 
-6 
W+ 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
S+ 
8 
S+ 
10 
S+ 
X0 
Positive Cumulative weight = 8+(-1)= 7 
 In Iteration #1, the cumulative weight was positive. We can connect the same 
S+ node with another upper W+ node (Iteration #2). 
 
 The cumulative weight after adding the second W+ node is still positive. (See 
Rule #1). 
 
 Iteration #2: 
-6 
W+ 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
S+ 
8 
S+ 
10 
S+ 
X0 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
S+ 
8 
S+ 
10 
S+ 
X0 
-6 
W+ 
Positive Cumulative weight = 8+(-1)+(-
3)= 4 
 In iteration #2, the same S+ node doesn’t have any possible connection left 
with a W+ node. We proceed at the next S+ nodes of the same row. 
 RULE #2 : If the cumulative weight is negative or 0 after adding a W+ node for 
the same S+ node, keep the W+ node arc connected to the root and remove 
the S+ node arc connected to the root. The S+ node becomes a W- node. 
Therefore, no further connection is possible for this node. (Iteration #3). 
 
 Iteration #3: 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
S+ 
8 
S+ 
10 
S+ 
X0 
-6 
W+ 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
W- 
8 
S+ 
10 
S+ 
X0 
-6 
W+ 
Negative Cumulative weight =1+(-2)=-1 
 There is no more S+ nodes in the second row. We now proceed in the third row ( 
row below). 
 We follow the same steps/rules as the two last iterations. 
 Reminder : We have to make the most possible connection between the TA group 
(S+ nodes) and the TB (W+ nodes). 
 
 Iteration #4: 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
W- 
8 
S+ 
10 
S+ 
X0 
-6 
W+ 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
W- 
8 
S+ 
10 
W- 
X0 
-6 
W+ 
Negative Cumulative weight =10+(-11)=-1 
Step 4: Ultimate pit. 
After finishing the third row, we notice that all possible connections 
between TA group (S+ nodes) and the TB (W+ nodes) have been made. 
The ultimate pit will be all the nodes connected to the S+ nodes tree. 
The final pit has a weight of 8+2-1-3=6. 
-2 
W+ 
-3 
W+ 
-1 
W+ 
2 
S+ 
-11 
W+ 
1 
W- 
8 
S+ 
10 
W- 
X0 
-6W+ 
Assignment 
• Find the ultimate pit limit and the corresponding closure for the 
following block models. 
1. First block model: 
2 -1 3 -2 -1 2 1 -1 -1 
4 2 -3 3 1 -1 -2 
1 -1 -3 1 5 
5 1 -2 
2. The grades in g per ton of each block of a gold deposit is given for the 
block model below. The weight of a block is 1000 tons. The mining and 
processing cost are respectively 6$/t and 27 $/t. The recovery is 85% . 
• Determine the ultimate pit limit for three iterations. In first iteration, use: 
50 $/g (parametrization factor of 0.80; 50*0.8= $40 and 40*0.8=$32) 
0.12 0.68 0.12 0.65 0.62 0.70 1.15 1.17 
0.63 1.2 1.29 0.64 3 0.68 1.13 1.57 
1.03 1.01 0.66 1.4 0.67 0.65 0.3 0.4 
0.2 0.4 1 0.71 0.64 1.3 0.4 1.16 
 
𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑝𝑟𝑖𝑐𝑒 × 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑦 × 𝑡𝑜𝑛𝑛𝑎𝑔𝑒 × 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 × 𝑔𝑟𝑎𝑑𝑒
− 𝑚𝑖𝑛𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 + 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 ∗ 𝑡𝑜𝑛𝑛𝑎𝑔𝑒