330
L x
c x
C
j
j
c x
i
j
x
i
C C
γ
γ
c x
γ
μ i i
( )
−
x
γ
x
= c c ( (
) ∀ ∈
x x
2
(3)
where x j
γ is the value of attribute γ of the jth block in the ith cluster, which is denoted by C i ,
with μ
γ
i i
μ being the value of attribute γ of its center. L j
( )
x j represents the loss in recovery of
attribute γ and c is a constant that magnifies the penalization.
Revenue and cost calculations are performed on each scenario and the one that maximizes
profit is deemed as optimal. Formulas for calculations of revenue and cost are shown in
Equations (4) and (5). Meanings of parameters are shown in Table 1.
Total revenue =
( )
( )
⎡
⎣
⎡ ⎡
⎤
⎦
⎤ ⎤
=
=
∑
∑
( )
j
N
j
j
N
1
1
j
x
∑
) =
V P
r L
−
x
j
∑
)
x
∑
)
k
i
i
⎣
⎡ ⎡
P P
r r
k
j
k
a
× ×
×
P i
P P
ρ
(4)
Total cost Total cost for processing streams construction
Total m
+
inin i i g cost Total processing cost
=
+
× ×
×
=
∑ ∑
m M
×
N m
×
y
V + ∑ ∑
c
j
N
i
m
ji
ρ
1
1
=
i
ρ ρ × V p
× i
(5)
3 CASE STUDIES
3.1 Overview
Two case studies were carried out using k-means clustering with spatial coordinates as
additional variables. The two datasets were both based on drill hole data, one with easting
and northing as x and y coordinates, the other with full x-y-z coordinates. Both datasets
contained the same attributes of interest, namely copper and gold grades. Grades of other
materials including MgO, CaO etc. were disregarded. 11 options for processing streams were
considered and listed in Table 2. List of detailed parameter values is shown in Table 3.
3.2 Case study of a copper and gold dataset with 2D coordinates
Easting and northing were used as x-y coordinates in this dataset. The dataset contained 60
aggregated blocks in total with aggregation having a tonnage of around 100 kilo-tons. The
attributes (copper and gold grades) and the geo-spatial coordinates were normalized prior to
k-means clustering, so that the scales of the variables would be the same. The TWSS versus
Table 1. List of parameters for revenue and cost calculations.
Parameter
Representation
Unit
a
Vector of attributes
ρ
Block bulk density
ton/m
3
V
Block volume
m
3
x j
k
j
th block grade of k
th attribute
%
N
Total number of blocks
P k
Price of k
th attribute
$
i
r r
k
Recovery from i
th processing stream of k
th attribute
%
L j
k
( )
x j
k
Loss of recovery from i
th processing stream of k
th attribute
%
y ji
Binary variable (1 if j
th block sent to i
th processing, 0 otherwise)
p i
Processing cost of i
th processing stream
$/ton
M
Cost of constructing a processing stream
$
m
Total number of clusters/processing streams
m c
Mining cost
$/ton
L x
c x
C
j
j
c x
i
j
x
i
C C
γ
γ
c x
γ
μ i i
( )
−
x
γ
x
= c c ( (
) ∀ ∈
x x
2
(3)
where x j
γ is the value of attribute γ of the jth block in the ith cluster, which is denoted by C i ,
with μ
γ
i i
μ being the value of attribute γ of its center. L j
( )
x j represents the loss in recovery of
attribute γ and c is a constant that magnifies the penalization.
Revenue and cost calculations are performed on each scenario and the one that maximizes
profit is deemed as optimal. Formulas for calculations of revenue and cost are shown in
Equations (4) and (5). Meanings of parameters are shown in Table 1.
Total revenue =
( )
( )
⎡
⎣
⎡ ⎡
⎤
⎦
⎤ ⎤
=
=
∑
∑
( )
j
N
j
j
N
1
1
j
x
∑
) =
V P
r L
−
x
j
∑
)
x
∑
)
k
i
i
⎣
⎡ ⎡
P P
r r
k
j
k
a
× ×
×
P i
P P
ρ
(4)
Total cost Total cost for processing streams construction
Total m
+
inin i i g cost Total processing cost
=
+
× ×
×
=
∑ ∑
m M
×
N m
×
y
V + ∑ ∑
c
j
N
i
m
ji
ρ
1
1
=
i
ρ ρ × V p
× i
(5)
3 CASE STUDIES
3.1 Overview
Two case studies were carried out using k-means clustering with spatial coordinates as
additional variables. The two datasets were both based on drill hole data, one with easting
and northing as x and y coordinates, the other with full x-y-z coordinates. Both datasets
contained the same attributes of interest, namely copper and gold grades. Grades of other
materials including MgO, CaO etc. were disregarded. 11 options for processing streams were
considered and listed in Table 2. List of detailed parameter values is shown in Table 3.
3.2 Case study of a copper and gold dataset with 2D coordinates
Easting and northing were used as x-y coordinates in this dataset. The dataset contained 60
aggregated blocks in total with aggregation having a tonnage of around 100 kilo-tons. The
attributes (copper and gold grades) and the geo-spatial coordinates were normalized prior to
k-means clustering, so that the scales of the variables would be the same. The TWSS versus
Table 1. List of parameters for revenue and cost calculations.
Parameter
Representation
Unit
a
Vector of attributes
ρ
Block bulk density
ton/m
3
V
Block volume
m
3
x j
k
j
th block grade of k
th attribute
%
N
Total number of blocks
P k
Price of k
th attribute
$
i
r r
k
Recovery from i
th processing stream of k
th attribute
%
L j
k
( )
x j
k
Loss of recovery from i
th processing stream of k
th attribute
%
y ji
Binary variable (1 if j
th block sent to i
th processing, 0 otherwise)
p i
Processing cost of i
th processing stream
$/ton
M
Cost of constructing a processing stream
$
m
Total number of clusters/processing streams
m c
Mining cost
$/ton