2
This paper uses a case study to demonstrate how a
MILP production scheduling model can be used to esti-
mate the benefits of different types of improvement proj-
ects. Other models could also be used to estimate value at
a minimum, a value estimation model should account for
finite resources and allow for cutoffs to be varied over time
to balance processing and stockpiling with the ore grades
available at any given time. The model used, nicknamed
Peakfinder, also accounts for flexibility to reallocate min-
ing between laybacks and incorporates vertical advance
constraints. For perspective, the Peakfinder model can also
be used to compare the value of capacity constraint expan-
sions against the value of cost reductions, mine planning
changes, geostatistical block estimation changes and mine
selectivity changes.
PEAKFINDER PRODUCTION
SCHEDULING MODEL
The Peakfinder MILP production schedule optimization
model is designed to compute mining and processing
sequences that maximize the net present value (NPV) of
cash flow generated from a set of mines and destinations
(processing facilities or waste dumps). The model will be
described using open pit terminology, but it could also be
used for operations that include underground mines.
The Peakfinder model works with one or more mines
subdivided into laybacks and panels with the tonnage in
each panel split into many increments to provide metallur-
gical and grade resolution. Increments can be defined as his-
togram bins for single metal deposits or as multivariate bins
or k-means clusters for multi-product mines (Goodfellow
and Dimitrakopoulos, 2014). Each panel typically con-
sists of one or more benches. The optimization problem
is defined in terms of maximizing NPV by choosing ton-
nage flows for each time period (Figure 1): the tonnages
to be mined from each mine, layback, panel, W(m,l,p,t)
the increment tonnages to be sent from each mine to each
destination, X(m,i,d,t) and/or to each stockpile, Y(m,i,s,t)
and the increment tonnages to be sent from each stockpile
to each destination, Z(i,s,d,t).
An overview of the MILP mathematical formulation is
presented here. The objective function is expressed as:
Maximize Discounted Net Surplus:
∑m ∑l ∑p ∑t pv(t)*MineVal(m,l,p,t)*W(m,l,p,t)
+∑m ∑i ∑d ∑t pv(t)*DestVal(m,i,d,t)*X(m,i,d,t)
+∑m ∑i ∑s ∑t pv(t)*StockVal(m,i,s,t)*Y(m,i,s,t)
+∑i ∑s ∑d ∑t pv(t)*ReclaimVal(i,s,d,t)*Z(i,s,d,t)
With present value factor, pv(t) =1/(1+discount rate)t–1
Value coefficients for each flow variable are computed
using multiple lookup tables for costs and for prices and
recoveries for e=1 to E elements:
MineVal(m,l,p,t) =–cM(m,l,p)
DestVal(m,i,d,t) =–cX(m,d) +–cDD(i,d)
+∑e (pr(e,t)–cK(e,i,d))
*(1–rD(e,i,d))*yD(e,i,d)*g(e,i)
StockVal(m,i,s,t) =–cS(m,s)
ReclaimVal(i,s,d,t) =–cR(s,d) +–cDS(i,d)
+∑e (pr(e,t) – cK(e,i,d))
*(1 – rD(e,i,d))*yS(e,i,d)*g(e,i)
where:
pr(e,t) =Metal price ($/unit of metal) for element e
cM(m,l,p) =Mining cost (to mine exit) ($/ton)
cX(m,d) =Ex-mine hauling cost (from mine exit to
destination) ($/ton)
cDD(i,d) =Destination processing cost ($/ton) for
direct feed
cDS(i,d) =Destination processing cost ($/ton) for
stockpiled feed
cS(m,s) =Stockpiling cost (including haulage from
mine exit to stockpile) ($/ton)
cR(s,d) =Reclaiming cost (including haulage cost
from stockpile to destination) ($/ton)
yD(e,i,d) =Yield (recovery) from direct feed
yS(e,i,d) =Yield (recovery) from stockpile reclaim feed
rD(e,i,d) =Royalty (NSR proportion)
cK(e,i,d) =Marketing cost, including smelting, refin-
ing, freight ($/unit of metal)
Figure 1. Peakfinder MILP model overview
This paper uses a case study to demonstrate how a
MILP production scheduling model can be used to esti-
mate the benefits of different types of improvement proj-
ects. Other models could also be used to estimate value at
a minimum, a value estimation model should account for
finite resources and allow for cutoffs to be varied over time
to balance processing and stockpiling with the ore grades
available at any given time. The model used, nicknamed
Peakfinder, also accounts for flexibility to reallocate min-
ing between laybacks and incorporates vertical advance
constraints. For perspective, the Peakfinder model can also
be used to compare the value of capacity constraint expan-
sions against the value of cost reductions, mine planning
changes, geostatistical block estimation changes and mine
selectivity changes.
PEAKFINDER PRODUCTION
SCHEDULING MODEL
The Peakfinder MILP production schedule optimization
model is designed to compute mining and processing
sequences that maximize the net present value (NPV) of
cash flow generated from a set of mines and destinations
(processing facilities or waste dumps). The model will be
described using open pit terminology, but it could also be
used for operations that include underground mines.
The Peakfinder model works with one or more mines
subdivided into laybacks and panels with the tonnage in
each panel split into many increments to provide metallur-
gical and grade resolution. Increments can be defined as his-
togram bins for single metal deposits or as multivariate bins
or k-means clusters for multi-product mines (Goodfellow
and Dimitrakopoulos, 2014). Each panel typically con-
sists of one or more benches. The optimization problem
is defined in terms of maximizing NPV by choosing ton-
nage flows for each time period (Figure 1): the tonnages
to be mined from each mine, layback, panel, W(m,l,p,t)
the increment tonnages to be sent from each mine to each
destination, X(m,i,d,t) and/or to each stockpile, Y(m,i,s,t)
and the increment tonnages to be sent from each stockpile
to each destination, Z(i,s,d,t).
An overview of the MILP mathematical formulation is
presented here. The objective function is expressed as:
Maximize Discounted Net Surplus:
∑m ∑l ∑p ∑t pv(t)*MineVal(m,l,p,t)*W(m,l,p,t)
+∑m ∑i ∑d ∑t pv(t)*DestVal(m,i,d,t)*X(m,i,d,t)
+∑m ∑i ∑s ∑t pv(t)*StockVal(m,i,s,t)*Y(m,i,s,t)
+∑i ∑s ∑d ∑t pv(t)*ReclaimVal(i,s,d,t)*Z(i,s,d,t)
With present value factor, pv(t) =1/(1+discount rate)t–1
Value coefficients for each flow variable are computed
using multiple lookup tables for costs and for prices and
recoveries for e=1 to E elements:
MineVal(m,l,p,t) =–cM(m,l,p)
DestVal(m,i,d,t) =–cX(m,d) +–cDD(i,d)
+∑e (pr(e,t)–cK(e,i,d))
*(1–rD(e,i,d))*yD(e,i,d)*g(e,i)
StockVal(m,i,s,t) =–cS(m,s)
ReclaimVal(i,s,d,t) =–cR(s,d) +–cDS(i,d)
+∑e (pr(e,t) – cK(e,i,d))
*(1 – rD(e,i,d))*yS(e,i,d)*g(e,i)
where:
pr(e,t) =Metal price ($/unit of metal) for element e
cM(m,l,p) =Mining cost (to mine exit) ($/ton)
cX(m,d) =Ex-mine hauling cost (from mine exit to
destination) ($/ton)
cDD(i,d) =Destination processing cost ($/ton) for
direct feed
cDS(i,d) =Destination processing cost ($/ton) for
stockpiled feed
cS(m,s) =Stockpiling cost (including haulage from
mine exit to stockpile) ($/ton)
cR(s,d) =Reclaiming cost (including haulage cost
from stockpile to destination) ($/ton)
yD(e,i,d) =Yield (recovery) from direct feed
yS(e,i,d) =Yield (recovery) from stockpile reclaim feed
rD(e,i,d) =Royalty (NSR proportion)
cK(e,i,d) =Marketing cost, including smelting, refin-
ing, freight ($/unit of metal)
Figure 1. Peakfinder MILP model overview