369
pen grade neg ep
t
_
g
: Penalty cost for negative deviations of element e from its target grade
requirements for plant p in period t (in Rs/tonne)
Inv s
0 :
Initial inventory of stockpile s
Inv s
t :
Inventory of stockpile s at the start of period t
Cap a s :
Capacity of stockpile s
min grade ep
t
_
:
grade ep
t
Minimum acceptable grade of element e at destination d in time period t
max g d ep
t
_
:
grade ep
t
Maximum acceptable grade of element e at destination d in time perd t
avg g d es
_
:
grade es
Average grade of element e reclaimed from stockpile s
dist ab :
Distance between element a and element b of the mineral value chain
haul
t
_
:
cost
Cost associated with hauling materials
rehandlin d d g c t sp
_
:
cost sp Cost associated with reclaiming materials from stock s to plant p
pen MC m
_
:
MC m
C
Penalty cost associated with violation of production from maximum
mining capacity for mine m
2.3 Decision variables
x imd
t
:
A binary decision variable which takes a value of 1 when block i of mine
m is sent to destination d in time period t and 0 otherwise
y sp
t :
A continuous decision variable representing the amount of material
reclaimed from stockpile s and sent to processing plant p in period t
dev prod pos p
t
_
p
: A continuous decision variable representing surplus production from
the upper limit of tonnage target of plant p in period t
dev prod neg p
g
t
_
p
: A continuous variable representing shortage in production from the
lower limit of tonnage target of plant p in period t
dev grade pos ep
t
_
g
: A continuous variable representing the tonnage of deviation from the
upper limit of grade requirement for element e in plant p in period t
dev grade neg ep
t
_
g
: A continuous variable representing the tonnage of deviation from lower
limit of grade requirement for element e in plant p in period t
dev MC m
t
_
:
MC m
C
t
A continuous variable representing negative deviation from maximum
possible mining capacity of mine m in period t
2.4 Objective function
Minimize
x
tonnage dist
t
y
imd
t
im
md
t
sp
t
s
S
p
P
t
T
i
*
*
tonnage
*
_
haul co
(
S
P
st
T
(
) )
=
=
=
=
∑ ∑
∑ ∑
∑
1
1
1
1
I I
m
M
d
D
t
T
sp
p
t
m
I
rehandlin d d g
t
dev prod pos p
∑
∑
∑
∑
=
=
=
1
1
1
*
)
(
+
_
p
*
_
pen _
_
_ _
_
*
_
)
(
prod pos dev p
_ rod neg
pen p
_ rod neg
dev
p
t
p
g
t
p
P
t
T
p
g
t
+
=
= ∑
∑
1
1
_
_
_ _
*
_
_
_
grade pos p n g
_ rade pos
dev g
_ rade n
ep
t
ep
t
e
E
p
P
t
T
=
=
=
∑
∑
∑
+
1
1
1
eg e e
p n grade
dev d d
pen MC
ep
t
ep
t
m
t
m
C
m
M
t
T
*
_
pen
_
)
neg ep
_
*
MC m
C C
t
_
(
)
=
= ∑
∑
1
1
(1)
2.5 Constraints
x
m
i
imd
t
d
D
t
T
=
= ∑
∑
≤ ∀ =
…
=
…
1
1
1
m
∀ = 2
1
M i
…
= 2
, , , ;
M M
, , , I m
I
(2)
(
*
)
,, , ;
, , ,
t
* onnage
M
,
t
T
,
imd
t
im
d
D
t
T
m
t
=
=
∑
∑
, ,
, ,
, ,
, ,
1
1
2
,
1 2
,
(3)
(
*
)
_
, , , ;
, ,
t
* onnage
MC
MC
m
M t
;
imd
t
im
d
D
t
T
m
C C
t
m
C C
t
=
=
∑
∑
MC C
∀ =
m
…
=
, ;t
;
1
1
1 2
,
1 2
, … …,T
(4)
x
m
M
d
D
r
t
jmd
t
t
T
=
∑ r
t
∑
x imd
r
d
D
=
∑
∀
x jmd
t
∑
…
1
1
d
t=
d
t
=
∑
imd
d d =
∑ 1
1 2
, , , ;
M (
)
i j
i j j
∈ {
}
I m
I
…
≠
∈
j j
j ∈ { …
; i j
≠ j
pr p p e im
(5)
pen grade neg ep
t
_
g
: Penalty cost for negative deviations of element e from its target grade
requirements for plant p in period t (in Rs/tonne)
Inv s
0 :
Initial inventory of stockpile s
Inv s
t :
Inventory of stockpile s at the start of period t
Cap a s :
Capacity of stockpile s
min grade ep
t
_
:
grade ep
t
Minimum acceptable grade of element e at destination d in time period t
max g d ep
t
_
:
grade ep
t
Maximum acceptable grade of element e at destination d in time perd t
avg g d es
_
:
grade es
Average grade of element e reclaimed from stockpile s
dist ab :
Distance between element a and element b of the mineral value chain
haul
t
_
:
cost
Cost associated with hauling materials
rehandlin d d g c t sp
_
:
cost sp Cost associated with reclaiming materials from stock s to plant p
pen MC m
_
:
MC m
C
Penalty cost associated with violation of production from maximum
mining capacity for mine m
2.3 Decision variables
x imd
t
:
A binary decision variable which takes a value of 1 when block i of mine
m is sent to destination d in time period t and 0 otherwise
y sp
t :
A continuous decision variable representing the amount of material
reclaimed from stockpile s and sent to processing plant p in period t
dev prod pos p
t
_
p
: A continuous decision variable representing surplus production from
the upper limit of tonnage target of plant p in period t
dev prod neg p
g
t
_
p
: A continuous variable representing shortage in production from the
lower limit of tonnage target of plant p in period t
dev grade pos ep
t
_
g
: A continuous variable representing the tonnage of deviation from the
upper limit of grade requirement for element e in plant p in period t
dev grade neg ep
t
_
g
: A continuous variable representing the tonnage of deviation from lower
limit of grade requirement for element e in plant p in period t
dev MC m
t
_
:
MC m
C
t
A continuous variable representing negative deviation from maximum
possible mining capacity of mine m in period t
2.4 Objective function
Minimize
x
tonnage dist
t
y
imd
t
im
md
t
sp
t
s
S
p
P
t
T
i
*
*
tonnage
*
_
haul co
(
S
P
st
T
(
) )
=
=
=
=
∑ ∑
∑ ∑
∑
1
1
1
1
I I
m
M
d
D
t
T
sp
p
t
m
I
rehandlin d d g
t
dev prod pos p
∑
∑
∑
∑
=
=
=
1
1
1
*
)
(
+
_
p
*
_
pen _
_
_ _
_
*
_
)
(
prod pos dev p
_ rod neg
pen p
_ rod neg
dev
p
t
p
g
t
p
P
t
T
p
g
t
+
=
= ∑
∑
1
1
_
_
_ _
*
_
_
_
grade pos p n g
_ rade pos
dev g
_ rade n
ep
t
ep
t
e
E
p
P
t
T
=
=
=
∑
∑
∑
+
1
1
1
eg e e
p n grade
dev d d
pen MC
ep
t
ep
t
m
t
m
C
m
M
t
T
*
_
pen
_
)
neg ep
_
*
MC m
C C
t
_
(
)
=
= ∑
∑
1
1
(1)
2.5 Constraints
x
m
i
imd
t
d
D
t
T
=
= ∑
∑
≤ ∀ =
…
=
…
1
1
1
m
∀ = 2
1
M i
…
= 2
, , , ;
M M
, , , I m
I
(2)
(
*
)
,, , ;
, , ,
t
* onnage
M
,
t
T
,
imd
t
im
d
D
t
T
m
t
=
=
∑
∑
, ,
, ,
, ,
, ,
1
1
2
,
1 2
,
(3)
(
*
)
_
, , , ;
, ,
t
* onnage
MC
MC
m
M t
;
imd
t
im
d
D
t
T
m
C C
t
m
C C
t
=
=
∑
∑
MC C
∀ =
m
…
=
, ;t
;
1
1
1 2
,
1 2
, … …,T
(4)
x
m
M
d
D
r
t
jmd
t
t
T
=
∑ r
t
∑
x imd
r
d
D
=
∑
∀
x jmd
t
∑
…
1
1
d
t=
d
t
=
∑
imd
d d =
∑ 1
1 2
, , , ;
M (
)
i j
i j j
∈ {
}
I m
I
…
≠
∈
j j
j ∈ { …
; i j
≠ j
pr p p e im
(5)