368
It aims to minimize deviations from quality and quantity targets of processing plants, total
transportation costs and total rehandling costs subject to all the practical and operational
constraints of production. The notation used in the mathematical model follows:
2.1 Indices
m: Mine index, m = 1, 2, …, M
p: Processing plant index, p = 1, 2, …, P
s: Stockpile index, s = 1, 2, …, S
w: Waste dump index, w = 1, 2, …, W
d: Destination index, d = 1, 2, …, D
t: Time period index, t = 1, 2, …, T
e: Element index, e = 1, 2, …, E
i m : block index for mine m, i = 1, 2, …, I m
2.2 Parameters
I m :
Number of blocks in mine m
Pre im :
set of predecessors of block i of mine m
MC m
C
t :
Mining Capacity of mine m in period t (in tonnes)
grade im
e :
Percentage of element e in block i of mine m
tonnage im :
Tonnage of block i of mine m (in tonnes)
tar p d p
t
_
:
prod p
d
t
Target production requirement of processing plant p in period t (in tonnes)
tol prod pos p
t
_
p
: Allowed positive tolerance from production target of plant p in period t (%)
tol prod neg p
g
t
_
p
: Allowed negative tolerance from production target of plant p in period t (%)
min prod p
t
_
:
prod p
d
t
Minimum production requirement for plant p in period t (in tonnes)
min prod
tar prod
t
prod
l prod neg
p
p
d
t
p
d
t
p
d
t
p
g
t
_ p p
( _
tar
* _
tol
_
,
neg
p
p
g
,
tar prod
= ( (
) ∀ =
p p
…
1 2
,
, ;
, ,
, , ,
P t
;
T
=
…
, ,
1 2
,
max p d p
t
_
:
prod p
d
t
Maximum production requirement for plant p in period t (in tonnes)
max prod
ta prod
prod
prod pos
p
p
d
t
p
d
t
p
d
t
p
t
_
p p
( _
tar
* _
tol
_
,
pos
p
p
,
+
tar prod p
d
t
( (
) ∀ =
p p
…
1 2
,
, ;
, ,
, , ,
P t
;
T
=
…
, ,
1 2
,
tar g d ep
t
_
:
grade ep
t
Target grade requirement of element e at plant p in time period t
tol grade pos ep
t
_
g
: Percentage of positive tolerance allowed for element e at plant p in period t
tol grade neg ep
t
_
g
: Percentage of negative tolerance allowed for element e at plant p in period t
min grade ep
t
_
:
grade ep
t
Minimum grade requirement of element e at plant p in period t
min grade
tar grade
tol grade neg
ep
t
ep
t
ep
t
_ g
ep
_
g
−
tar grade
= (
)
∀ =
…
=
…
e
E
=
… p
P
=
… t
T
=
…
1 2
1 2
1 2
, , , ;
E
;
P
, , ,
max g d ep
t
_
:
grade ep
t
Maximum grade requirement of element e at plant p in period t
max grade
ta grade
tol grade pos
ep
t
ep
t
ep
t
g
ep
_
g
+
tar grade ep
t
= (
)
∀ =
…
=
…
e
E
=
… p
P
=
… t
T
=
…
1 2
1 2
1 2
, , , ;
E
;
P
, , ,
pen prod pos p
t
_
p
: Penalty cost for positive deviations from target tonnage requirements for
plant p in period t (in Rs/tonne)
pen prod neg p
g
t
_
p
: Penalty cost for negative deviations from target tonnage requirements for
plant p in period t (in Rs/tonne)
pen grade pos ep
t
_
g
: Penalty cost for positive deviations of element e from its target grade
requirements for plant p in period t (in Rs/tonne)
It aims to minimize deviations from quality and quantity targets of processing plants, total
transportation costs and total rehandling costs subject to all the practical and operational
constraints of production. The notation used in the mathematical model follows:
2.1 Indices
m: Mine index, m = 1, 2, …, M
p: Processing plant index, p = 1, 2, …, P
s: Stockpile index, s = 1, 2, …, S
w: Waste dump index, w = 1, 2, …, W
d: Destination index, d = 1, 2, …, D
t: Time period index, t = 1, 2, …, T
e: Element index, e = 1, 2, …, E
i m : block index for mine m, i = 1, 2, …, I m
2.2 Parameters
I m :
Number of blocks in mine m
Pre im :
set of predecessors of block i of mine m
MC m
C
t :
Mining Capacity of mine m in period t (in tonnes)
grade im
e :
Percentage of element e in block i of mine m
tonnage im :
Tonnage of block i of mine m (in tonnes)
tar p d p
t
_
:
prod p
d
t
Target production requirement of processing plant p in period t (in tonnes)
tol prod pos p
t
_
p
: Allowed positive tolerance from production target of plant p in period t (%)
tol prod neg p
g
t
_
p
: Allowed negative tolerance from production target of plant p in period t (%)
min prod p
t
_
:
prod p
d
t
Minimum production requirement for plant p in period t (in tonnes)
min prod
tar prod
t
prod
l prod neg
p
p
d
t
p
d
t
p
d
t
p
g
t
_ p p
( _
tar
* _
tol
_
,
neg
p
p
g
,
tar prod
= ( (
) ∀ =
p p
…
1 2
,
, ;
, ,
, , ,
P t
;
T
=
…
, ,
1 2
,
max p d p
t
_
:
prod p
d
t
Maximum production requirement for plant p in period t (in tonnes)
max prod
ta prod
prod
prod pos
p
p
d
t
p
d
t
p
d
t
p
t
_
p p
( _
tar
* _
tol
_
,
pos
p
p
,
+
tar prod p
d
t
( (
) ∀ =
p p
…
1 2
,
, ;
, ,
, , ,
P t
;
T
=
…
, ,
1 2
,
tar g d ep
t
_
:
grade ep
t
Target grade requirement of element e at plant p in time period t
tol grade pos ep
t
_
g
: Percentage of positive tolerance allowed for element e at plant p in period t
tol grade neg ep
t
_
g
: Percentage of negative tolerance allowed for element e at plant p in period t
min grade ep
t
_
:
grade ep
t
Minimum grade requirement of element e at plant p in period t
min grade
tar grade
tol grade neg
ep
t
ep
t
ep
t
_ g
ep
_
g
−
tar grade
= (
)
∀ =
…
=
…
e
E
=
… p
P
=
… t
T
=
…
1 2
1 2
1 2
, , , ;
E
;
P
, , ,
max g d ep
t
_
:
grade ep
t
Maximum grade requirement of element e at plant p in period t
max grade
ta grade
tol grade pos
ep
t
ep
t
ep
t
g
ep
_
g
+
tar grade ep
t
= (
)
∀ =
…
=
…
e
E
=
… p
P
=
… t
T
=
…
1 2
1 2
1 2
, , , ;
E
;
P
, , ,
pen prod pos p
t
_
p
: Penalty cost for positive deviations from target tonnage requirements for
plant p in period t (in Rs/tonne)
pen prod neg p
g
t
_
p
: Penalty cost for negative deviations from target tonnage requirements for
plant p in period t (in Rs/tonne)
pen grade pos ep
t
_
g
: Penalty cost for positive deviations of element e from its target grade
requirements for plant p in period t (in Rs/tonne)