147
v p
ug t
,
the underground (ug) discounted revenue generated by selling the final product within
mining-stope p in period t minus the discounted extra cost of extracting mining-stope
p as ore and processing it.
q j
op t
,
the open pit (op) discounted cost of mining all the material in mining-cut j in period t
as waste.
q p
ug t
,
the underground (ug) discounted cost of mining all the material in mining-stope p in
period t as waste.
g j
average grade of element in ore portion of mining-cut j.
g p
g
average grade of element in ore portion of mining-stope p.
o j
ore tonnage in mining-cut j.
o p
o
ore tonnage in mining-stope p.
r
processing recovery; the proportion of mineral commodity recovered.
sp
selling price of mineral commodity in present value terms.
sc
selling cost of mineral commodity in present value terms.
pc extra cost in present value terms per tonne of ore for mining and processing in period t.
cm l
t
cost per bench or level l in present value terms of mining a tonne of rock material by
open pit mining in period t.
cm
t
cost in present value terms of mining a tonne of rock material by underground mining
in period t.
4 THE INTEGRATED MILP MODEL
An integrated MILP model is formulated to determine the time and sequence of extraction
of ore and waste blocks over the mine life for open pit and/or underground mining. The
proposed MILP model interrogates the orebody and determines the best mining option that
produces an optimal extraction sequence to maximize the Net Present Value (NPV) of the
mining project. The mining options could either be open pit mining or underground mining
or a combination of both open pit and underground mining. The model further determines
the capital and operational development schedules required to extract the orebody by underground mining. The NPV of the extraction strategy is maximized in the presence of technical, geotechnical, geological, and economic constraints to enforce the mining sequence, grade
blending requirements, capital and operational developments, and mining and processing
capacities.
4.1 Modeling the economic block value
The economic block values are defined based on the SMUs, thus, mining-cuts for open pit
mining and mining-stopes for underground mining. The value of the block is a function
of the recovered quantity of mineral present in the block (processing recovery), the discounted revenue from selling the commodity, and the discounted mining, processing and
selling costs.
The discounted revenues generated by selling the final product within block k being
extracted in period t by open pit mining v j
op t
, and underground mining v p
ug t
, are respectively
given in Equations (1) and (2). Similarly, the discounted costs of mining all the material
within block k being extracted in period t by open pit mining q j
op t
, and underground mining
q p
ug t
, are respectively given in Equations (3) and (4).
v
o g r
o pc
j
op t
j
j
g
j
j
J
j
J
,
(
)
sp sc
=
×
o j
× ×
r
−
)
sc
×
⎛
⎝
⎜
⎛ ⎛
⎜
⎝ ⎝
⎜ ⎜
⎞
⎠
⎟
⎞ ⎞
⎟
⎠ ⎠
⎟ ⎟
=
=
∑
⎛ ⎛
⎝ ⎝
∑
1
1
(1)
v
o g r
O pc
p
ug t
p
p
o g
p
O
p
P
p
p
,
(
)
sp sc
=
×
o
× ×
r
−
)
sc
⎛
⎝
⎜
⎛ ⎛
⎜
⎝ ⎝
⎜ ⎜
⎞
⎠
⎟
⎞ ⎞
⎟
⎠ ⎠
⎟ ⎟
=
=
∑
⎛ ⎛
⎝ ⎝
∑
1
1
(2)
v p
ug t
,
the underground (ug) discounted revenue generated by selling the final product within
mining-stope p in period t minus the discounted extra cost of extracting mining-stope
p as ore and processing it.
q j
op t
,
the open pit (op) discounted cost of mining all the material in mining-cut j in period t
as waste.
q p
ug t
,
the underground (ug) discounted cost of mining all the material in mining-stope p in
period t as waste.
g j
average grade of element in ore portion of mining-cut j.
g p
g
average grade of element in ore portion of mining-stope p.
o j
ore tonnage in mining-cut j.
o p
o
ore tonnage in mining-stope p.
r
processing recovery; the proportion of mineral commodity recovered.
sp
selling price of mineral commodity in present value terms.
sc
selling cost of mineral commodity in present value terms.
pc extra cost in present value terms per tonne of ore for mining and processing in period t.
cm l
t
cost per bench or level l in present value terms of mining a tonne of rock material by
open pit mining in period t.
cm
t
cost in present value terms of mining a tonne of rock material by underground mining
in period t.
4 THE INTEGRATED MILP MODEL
An integrated MILP model is formulated to determine the time and sequence of extraction
of ore and waste blocks over the mine life for open pit and/or underground mining. The
proposed MILP model interrogates the orebody and determines the best mining option that
produces an optimal extraction sequence to maximize the Net Present Value (NPV) of the
mining project. The mining options could either be open pit mining or underground mining
or a combination of both open pit and underground mining. The model further determines
the capital and operational development schedules required to extract the orebody by underground mining. The NPV of the extraction strategy is maximized in the presence of technical, geotechnical, geological, and economic constraints to enforce the mining sequence, grade
blending requirements, capital and operational developments, and mining and processing
capacities.
4.1 Modeling the economic block value
The economic block values are defined based on the SMUs, thus, mining-cuts for open pit
mining and mining-stopes for underground mining. The value of the block is a function
of the recovered quantity of mineral present in the block (processing recovery), the discounted revenue from selling the commodity, and the discounted mining, processing and
selling costs.
The discounted revenues generated by selling the final product within block k being
extracted in period t by open pit mining v j
op t
, and underground mining v p
ug t
, are respectively
given in Equations (1) and (2). Similarly, the discounted costs of mining all the material
within block k being extracted in period t by open pit mining q j
op t
, and underground mining
q p
ug t
, are respectively given in Equations (3) and (4).
v
o g r
o pc
j
op t
j
j
g
j
j
J
j
J
,
(
)
sp sc
=
×
o j
× ×
r
−
)
sc
×
⎛
⎝
⎜
⎛ ⎛
⎜
⎝ ⎝
⎜ ⎜
⎞
⎠
⎟
⎞ ⎞
⎟
⎠ ⎠
⎟ ⎟
=
=
∑
⎛ ⎛
⎝ ⎝
∑
1
1
(1)
v
o g r
O pc
p
ug t
p
p
o g
p
O
p
P
p
p
,
(
)
sp sc
=
×
o
× ×
r
−
)
sc
⎛
⎝
⎜
⎛ ⎛
⎜
⎝ ⎝
⎜ ⎜
⎞
⎠
⎟
⎞ ⎞
⎟
⎠ ⎠
⎟ ⎟
=
=
∑
⎛ ⎛
⎝ ⎝
∑
1
1
(2)