370
_
_
_
y
prod pos max
i
I
m
M
sp
t
s
S
p
t
=
∑ m
M
=
∑
(
*
)
t
* onnage
t
im
I m
I
∑
≥
1
1
i
s=
p
s
i=
∑
(
)
g
imp
im
p
i i=
∑ 1
prod p p
p
Pt
T
p
d
t
∀ =
p p
=
t
…
1 2
1 2
; ;
P P
, , ,
(6)
_
_
m in_
y
prod
g
i
I
m
M
sp
t
s
S
p
g
t
=
∑
∑
(
*
)
t
* onnage
t
im
I
m
M
m
I
=
∑
≥
1
1
i
s=
p
i
s
=
∑
(
)
g
imp
im
p
i i=
∑ 1
prod p p
p
Pt
T
p
d
t
∀ =
p p
=
t
…
1 2
1 2
;
P P
, , ,
(7)
y
i
I
m
M
sp
t
m
I
(
(
)
x
tonnage
imp
t
im
*
*
tonnage (
)
grade
min grade
im
e
ep
t
_
−
+
=
=
∑
∑
1
1
s s
S
es
ep
t
ep
t
avg grade min grade
dev grade neg
e
=
∑
−
+
ep
t
min grade
≥
∀ =
e
1
0
1
* (
_
es
g
))
_
g
, 2 2
1
2 2
2
12
, ,
,
1 , , ;
,
1
,
, ;
p
; ;
;
(8)
y
i
I
m
M
sp
t
m
I
(
(
)
x
tonnage
imp
t
im
*
*
tonnage (
)
grade
max grade
im
e
ep
t
−
+
=
=
∑
∑
1
1
*( * *
))
_
,
s
S
es
ep
t
ep
t
avg g
_ rade max g
_
x grade
dev g
_ rade pos
e
=
∑
−
))
− max grade
≤
∀ =
e
1
0
1 2
, 2
1
2 2
2
12
, ,
,
1 , , ;
,
1 , ,
,
, ;
p
; ;
;
T
(9)
e
i
I
m
M
m
I
,
(
)
x
tonnage
ims
t
im
*
*
tonnage (
)
grade
min grade
im
e
es
_
−
≥
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(10)
e
i
I
m
M
m
I
,
(
)
x
tonnage
ims
t
im
*
*
tonnage (
)
grade
max grade
im
e
es
−
≤
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(11)
Inv
tonnage
y Cap C C
s
ims
im
r
t
i
I
m
M
p
P t
sp
r
m
I
0
1
1
1
1 1
r
+
−
tonnage im
=
=
=
=1 r
∑
∑
∑
∑∑ y y
(
*
x ims
r
x ims
r
)
s s
s
St
T
∀ =
s
…
=
t
…
1 2
1 2
, , , ;
S S
, , ,
(12)
y Inv
I
s
St
sp
p
P
s
1
1
0
1 2
1
≤
∀
Inv I I s
0
…
=
=
∑
, , , ;
S
(13)
y
p
P
r
i
I
m
M
sp
r
=
r
p
=
i
m
( )
t−
∑
∑
y
Inv I
sp
t
p
P
s
i
I
m
M
m
I
p
=
i
m
Inv I I s
(
)
x
tonnage
ims
r
im +
1
1
1
1
r r
p
P
s
St
T
=
( )
t−
=
∑
∑
∀ =
s
=
t
…
…
1
1
1 2
2 3
, , , ;
S S
, , ,
(14)
e
i
I
m
M
m
I
,
(
)
x
tonnage
imw
t
im
*
*
tonnage (
)
grade
min grade
im
e
ew
_
−
≥
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(15)
e
i
I
m
M
m
I
,
(
)
x
tonnage
imw
t
im
*
*
tonnage (
)
grade
max grade
im
e
ew
−
≤
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(16)
The objective function (Equation (1)) aims to minimize total costs. This include total
transportation cost associated with hauling of materials from different mines to different
destinations, total rehandling cost associated with reclaiming of materials from stockpiles
to different plants and penalty costs associated with violations from target tonnage requirements, target grade requirements and maximum possible mining capacity of the mines. The
objective function has to be minimized subject to a number of constraints. The fact that
the blocks from each of the mines cannot mined more than once is ensured by constraint
(2). Constraints (3) and (4) ensure that the maximum possible production capacity for
each mine is met in each period with any deviations from the former being penalized in the
objective function. Constraint (5) ensures that the precedence relationships between the
blocks for each mine always remain satisfied. Constraint (6) and (7) ensure that the tonnage
_
_
_
y
prod pos max
i
I
m
M
sp
t
s
S
p
t
=
∑ m
M
=
∑
(
*
)
t
* onnage
t
im
I m
I
∑
≥
1
1
i
s=
p
s
i=
∑
(
)
g
imp
im
p
i i=
∑ 1
prod p p
p
Pt
T
p
d
t
∀ =
p p
=
t
…
1 2
1 2
; ;
P P
, , ,
(6)
_
_
m in_
y
prod
g
i
I
m
M
sp
t
s
S
p
g
t
=
∑
∑
(
*
)
t
* onnage
t
im
I
m
M
m
I
=
∑
≥
1
1
i
s=
p
i
s
=
∑
(
)
g
imp
im
p
i i=
∑ 1
prod p p
p
Pt
T
p
d
t
∀ =
p p
=
t
…
1 2
1 2
;
P P
, , ,
(7)
y
i
I
m
M
sp
t
m
I
(
(
)
x
tonnage
imp
t
im
*
*
tonnage (
)
grade
min grade
im
e
ep
t
_
−
+
=
=
∑
∑
1
1
s s
S
es
ep
t
ep
t
avg grade min grade
dev grade neg
e
=
∑
−
+
ep
t
min grade
≥
∀ =
e
1
0
1
* (
_
es
g
))
_
g
, 2 2
1
2 2
2
12
, ,
,
1 , , ;
,
1
,
, ;
p
; ;
;
(8)
y
i
I
m
M
sp
t
m
I
(
(
)
x
tonnage
imp
t
im
*
*
tonnage (
)
grade
max grade
im
e
ep
t
−
+
=
=
∑
∑
1
1
*( * *
))
_
,
s
S
es
ep
t
ep
t
avg g
_ rade max g
_
x grade
dev g
_ rade pos
e
=
∑
−
))
− max grade
≤
∀ =
e
1
0
1 2
, 2
1
2 2
2
12
, ,
,
1 , , ;
,
1 , ,
,
, ;
p
; ;
;
T
(9)
e
i
I
m
M
m
I
,
(
)
x
tonnage
ims
t
im
*
*
tonnage (
)
grade
min grade
im
e
es
_
−
≥
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(10)
e
i
I
m
M
m
I
,
(
)
x
tonnage
ims
t
im
*
*
tonnage (
)
grade
max grade
im
e
es
−
≤
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(11)
Inv
tonnage
y Cap C C
s
ims
im
r
t
i
I
m
M
p
P t
sp
r
m
I
0
1
1
1
1 1
r
+
−
tonnage im
=
=
=
=1 r
∑
∑
∑
∑∑ y y
(
*
x ims
r
x ims
r
)
s s
s
St
T
∀ =
s
…
=
t
…
1 2
1 2
, , , ;
S S
, , ,
(12)
y Inv
I
s
St
sp
p
P
s
1
1
0
1 2
1
≤
∀
Inv I I s
0
…
=
=
∑
, , , ;
S
(13)
y
p
P
r
i
I
m
M
sp
r
=
r
p
=
i
m
( )
t−
∑
∑
y
Inv I
sp
t
p
P
s
i
I
m
M
m
I
p
=
i
m
Inv I I s
(
)
x
tonnage
ims
r
im +
1
1
1
1
r r
p
P
s
St
T
=
( )
t−
=
∑
∑
∀ =
s
=
t
…
…
1
1
1 2
2 3
, , , ;
S S
, , ,
(14)
e
i
I
m
M
m
I
,
(
)
x
tonnage
imw
t
im
*
*
tonnage (
)
grade
min grade
im
e
ew
_
−
≥
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(15)
e
i
I
m
M
m
I
,
(
)
x
tonnage
imw
t
im
*
*
tonnage (
)
grade
max grade
im
e
ew
−
≤
∀ =
e
=
=
∑
∑
1
1
0
1 2
, 2
1
2 2
2
12
, , ;
,
1 , , ;
,
1 , ,
, ;
, ;
;
;
T
(16)
The objective function (Equation (1)) aims to minimize total costs. This include total
transportation cost associated with hauling of materials from different mines to different
destinations, total rehandling cost associated with reclaiming of materials from stockpiles
to different plants and penalty costs associated with violations from target tonnage requirements, target grade requirements and maximum possible mining capacity of the mines. The
objective function has to be minimized subject to a number of constraints. The fact that
the blocks from each of the mines cannot mined more than once is ensured by constraint
(2). Constraints (3) and (4) ensure that the maximum possible production capacity for
each mine is met in each period with any deviations from the former being penalized in the
objective function. Constraint (5) ensures that the precedence relationships between the
blocks for each mine always remain satisfied. Constraint (6) and (7) ensure that the tonnage