446
shovel must be allocated in that phase f (Equation 6). To assign operative time in a bench b
of a phase f in a shovel p, it must be allocated in that phase f (Equation 7). Bench b of phase
f is not finished in period t until all the material are extracted (Equation 8). If a bench b of
a phase f is finished in a period t, it remains finished in period t + 1. Constraints 10 controls
the precedence between benches of the same phase. Thus, to a bench b of a certain phase f
can be active, the upper bench must be finished. To assign operative time, the phase-bench
must be active (Equation 11). Constraints 12 and 13 controls the precedence between benches
of consecutive phases. Constraints 14 to 15 control the area available in the phase-bench in
order to allocate shovels. Constraints 16 to 18 models the shovel movements between phases.
At most just one route is performed by shovel p in period t (Equation 16). Constraint 17 links
the binary variable w p f t
,
f
with the variable υ p,r,t . For each shovel p, the constraint 18 ensures
that the route r of period t and the route s of period t + 1 are coherent, that is, the last phase
f of the r route and the first phase of the s route is the same. The set FR(t) represents the set
of all the routes consistent with the r route. That is, the set of routes whose first phase is the
same as the first phase of the r route. Constraints 19 and 20 model the deviation between the
ore sent to the ore processing plant and the ore processing capacity per period. Constraints
21 to 23 model the deviation between the metal fines sent to ore processing capacity and the
expected fines by the ore processing plant per period.
p P f F b B
p f b
p f b t
f b
RM p
x
T
p f b t
T
p
f b
F b B f
P f
∑
⋅ x
∀
TM
p
p
f b
T
∈ F
,
,
f F
f
,
f
, f ,
,
t
f
f
, b B
∈
b
)
(1)
p P f F b B
p f b
p f b t
RM p
x
T
p f b t
T
C
t T
P f
∑
⋅ x
≤
∀
PC
∈
,
,
f F
f
,
f
, f ,
(2)
f F b B
r R
p r
p r t
p t
TI v
p r
T T
p
p P t T
F b ( )
f
∑
∑
p f b t
x p f b t
TI p r
T T
≤
∀
p t
AV p
p P t
,
f
p
∑
t
,
r
,
,
(3)
x
x
p P f F b B
t T
p f b t
p f b t
,
f ,t
p , ,
b
,
,
f F
≤
∀
x
p
p f b t
P f
, f
( )
f
∈
(4)
x
z
p P f F b B
t T
p f b t
f b t
,
f ,t
f ,
,
,
f F
+
≤
z f b t
∀ ∈
p p
F b
,
F
( )
f
∈
−1
1
(5)
w
t T f F b B
p f t
p
x f b t
,
f
, f ,
,
,
f F
≥
∀
x p
x f b t
T f
, f
( )
f
(6)
w
p P f F b B
p f t
p
x f b t
,
f
, f ,
,
,
f F
≥
∀
x
p
p
x f b t
P f
, f
( )
f
(7)
z
R M
x
f F b B
t T
f b t
p P
p f b
p
x f b
,
b
,
,
f
, f ,
,b B
RM M
⋅
∀
TT
f f F b
,b
( )
f
( )
f f
∈
∈
P, { }
t
,..,
∑
τ∈ ∈
τ
(8)
z
f F
f b t
fb t
,
b
, ,
b
≥
∀
z
f f
f
z b t−1
(9)
z
f F b B
t T
f b t
f
t
,
b
, ,
b
,
,
b B
f f ∈
∈
F b
b
( )
f
( )
f f
(10)
x
z
p P f F b B
t T
p f b t
f b t
,
f ,t
f ,
,
,
f F
≤
∀
z
p
f b t
P f
, f
( )
f
∈
(11)
z
FC b B
b B
t T
f b t
f
t
,
b
,
,
FC
,b B
(
)
f f
, f
∈
FC b
FC
( )
f
( )
f
( )
f
( )
f f
∈
b b
FC b B
f f
∈
FC b
)
f
f f
(
)
b ML f f
L
b ML f f
L
−
f f
(12)
z
b B
b B
t T
f b t
f
t
,
b
,
,
C
,
b B
≤
∀
z
(
)
f f
f ∈ FC b
,
FC
( )
f
( )
f
∈ ( )
f
( )
f f
′
b b
FC b B
)
f f ∈ FC b
( f
f f
′
f f (
)
f f
(
)
b mL f f
L
b mL f f
L
′
b b −
f f
(13)
p P
p
pf b t
f b
AS p
AD f
f F b B
t
∑ ⋅
≤
p f b t
x
∀f f
( )
f
( )
f
=
, f ,
,
t
f
,b B( f
1
(14)
p P
p P
p f b
pf b
RM p
x
T
p f b
T
P
p
∈{
}
t
∑
∑
p
pf b t
f b
AS p x
A
p f b t
D f
x p f b t
⋅ x
f t
f
, ∈{
,
f
, f ,
τ∈ ∈ ∈
τ
SG S S BH
f F b B
t T
⋅
∀ ∀f f
( )
f
( )
f
∈
{ }
,b B( f
(15)
r R
p r t
v
p
p r t
P t T
∑ ≤
∀p ∈
∈
P t
,
r r
,
(16)
w
v
p P t T f F
p f t
r R
p r t
,
f
,
r
,
,
t T
≤
∀
v
p
p r t
P t
,t
( )
f
∑
(17)