3
g(e,i) =grade of element (metal), e, for increment, i
In addition to the cost/value coefficients, the primary
data inputs to the model are the grade tonnage informa-
tion for each increment in each panel, tons(m,l,p,i) and
g(e,i), and the total tons in each panel, tons(m,l,p) =∑i
tons(m,l,p,i).
Capacity constraint equations are defined for each time
period in terms of maximums or minimums for:
• Tons mined at each layback, mine or group of mines
• Sink rate (panels/year) for each layback
• Tons, grade, metal, recovered metal processed at each
destination or group of destinations
• Stockpile inventory
Example capacity constraint equation for total tons mined
from all mines in time period, t:
∑m ∑l ∑p W(m,l,p,t) MaxMineTons(t)
Example recovered metal constraint equation for a specific
metal, destination, time period (e, d, t):
∑m ∑i g(e,i)*yD(e,i,d)*X(m,i,d,t)
+∑s ∑i g(e,i)*yS(e,i,d)*Z(i,s,d,t)
≤ MaxDestRecMetal(e,d,t)
Additional constraints are created for each time period
to define a feasible mining sequence and ensure material
flows are not adding or subtracting tons:
• Uniform mining of increments within a panel: if a
panel is only partially mined during a period, then
all increments in the panel are mined in the same
proportion.
• Panel prerequisites: a panel cannot start mining until
its overlying panel is completely mined out or any
other defined prerequisites are mined out. Example:
panel Lay3P5 constrained by overlying panel Lay3P4
and by inner layback’s panel Lay2P5.
• Finite resources: mining limited by panel tonnage
input data no negative flows.
• Mine continuity: mining flows equal stockpile flows
plus process flows for each increment, time.
• Stockpile continuity: inventory =prior inventory +
inflows – outflows for each increment, time inven-
tories cannot be negative
Example mine continuity constraint equation for each
mine, increment, time period (m, i, t):
∑l∑pW(m,l,p,t)*[tons(m,l,p,i)/tons(m,l,p)]
=∑d X(m,i,d,t) +∑s Y(m,i,s,t)
The Peakfinder MILP model is similar to other MILP
production scheduling formulations. Many examples, with
varying assumptions, are presented in Dimitrakopoulos,
2018. The Peakfinder model is most similar to a formula-
tion used at Newmont’s Nevada Operations (Hoerger et al,
1999), but with expanded mine resolution, sink rate con-
straints, multi-metal revenue computations and changes to
the variable formulation to significantly reduce the number
of flow variables.
The Peakfinder model has several features that may
show differences to other specific MILP models: increment
destinations are computed during the optimization process
all constraints are explicitly defined and enforced with-
out use of penalties the model uses pre-defined laybacks
as inputs increment proportions are mined uniformly if
partial panels are mined during a period increment grades
are defined constants for all flows to maintain linearity,
the model segregates increments in stockpiles rather than
forcing blending during reclaim. Most importantly, note
that mine sequence, cutoff grades and process/stockpile
sequences are all outputs of the optimization process, not
inputs.
As with other mine engineering analyses, mining and
processing unit costs are key inputs. Unit costs should
include all spending, regardless of accounting classification,
to be incurred if the mine life, process life or site footprint
are extended. This typically includes operating costs, sus-
taining capital, site overhead, site administration, com-
munity development costs, reclamation costs and almost
any other spending that is not part of the initial site devel-
opment capital or part of a profit-based tax. The model
formulation allows for fixed yearly costs, but for opera-
tions constrained by mill capacity, the same schedules are
obtained when fixed yearly costs are included in the milling
unit cost.
The workflow used for optimizing a production sched-
ule consists of 1) layback designs and grade-tonnage com-
putations for each panel are designed in a mine design
software package 2) a C program, Peakfinder, formulates
the MILP optimization problem 3) CPLEX, a commercial
MILP solver, computes the optimum solution to within
0.01% of optimality 4) Peakfinder reads the CPLEX solu-
tion for reporting and analysis 5) Excel is used for graphing
and comparison of alternative plans. The process described
could also be implemented with commercially available
production scheduling software.
g(e,i) =grade of element (metal), e, for increment, i
In addition to the cost/value coefficients, the primary
data inputs to the model are the grade tonnage informa-
tion for each increment in each panel, tons(m,l,p,i) and
g(e,i), and the total tons in each panel, tons(m,l,p) =∑i
tons(m,l,p,i).
Capacity constraint equations are defined for each time
period in terms of maximums or minimums for:
• Tons mined at each layback, mine or group of mines
• Sink rate (panels/year) for each layback
• Tons, grade, metal, recovered metal processed at each
destination or group of destinations
• Stockpile inventory
Example capacity constraint equation for total tons mined
from all mines in time period, t:
∑m ∑l ∑p W(m,l,p,t) MaxMineTons(t)
Example recovered metal constraint equation for a specific
metal, destination, time period (e, d, t):
∑m ∑i g(e,i)*yD(e,i,d)*X(m,i,d,t)
+∑s ∑i g(e,i)*yS(e,i,d)*Z(i,s,d,t)
≤ MaxDestRecMetal(e,d,t)
Additional constraints are created for each time period
to define a feasible mining sequence and ensure material
flows are not adding or subtracting tons:
• Uniform mining of increments within a panel: if a
panel is only partially mined during a period, then
all increments in the panel are mined in the same
proportion.
• Panel prerequisites: a panel cannot start mining until
its overlying panel is completely mined out or any
other defined prerequisites are mined out. Example:
panel Lay3P5 constrained by overlying panel Lay3P4
and by inner layback’s panel Lay2P5.
• Finite resources: mining limited by panel tonnage
input data no negative flows.
• Mine continuity: mining flows equal stockpile flows
plus process flows for each increment, time.
• Stockpile continuity: inventory =prior inventory +
inflows – outflows for each increment, time inven-
tories cannot be negative
Example mine continuity constraint equation for each
mine, increment, time period (m, i, t):
∑l∑pW(m,l,p,t)*[tons(m,l,p,i)/tons(m,l,p)]
=∑d X(m,i,d,t) +∑s Y(m,i,s,t)
The Peakfinder MILP model is similar to other MILP
production scheduling formulations. Many examples, with
varying assumptions, are presented in Dimitrakopoulos,
2018. The Peakfinder model is most similar to a formula-
tion used at Newmont’s Nevada Operations (Hoerger et al,
1999), but with expanded mine resolution, sink rate con-
straints, multi-metal revenue computations and changes to
the variable formulation to significantly reduce the number
of flow variables.
The Peakfinder model has several features that may
show differences to other specific MILP models: increment
destinations are computed during the optimization process
all constraints are explicitly defined and enforced with-
out use of penalties the model uses pre-defined laybacks
as inputs increment proportions are mined uniformly if
partial panels are mined during a period increment grades
are defined constants for all flows to maintain linearity,
the model segregates increments in stockpiles rather than
forcing blending during reclaim. Most importantly, note
that mine sequence, cutoff grades and process/stockpile
sequences are all outputs of the optimization process, not
inputs.
As with other mine engineering analyses, mining and
processing unit costs are key inputs. Unit costs should
include all spending, regardless of accounting classification,
to be incurred if the mine life, process life or site footprint
are extended. This typically includes operating costs, sus-
taining capital, site overhead, site administration, com-
munity development costs, reclamation costs and almost
any other spending that is not part of the initial site devel-
opment capital or part of a profit-based tax. The model
formulation allows for fixed yearly costs, but for opera-
tions constrained by mill capacity, the same schedules are
obtained when fixed yearly costs are included in the milling
unit cost.
The workflow used for optimizing a production sched-
ule consists of 1) layback designs and grade-tonnage com-
putations for each panel are designed in a mine design
software package 2) a C program, Peakfinder, formulates
the MILP optimization problem 3) CPLEX, a commercial
MILP solver, computes the optimum solution to within
0.01% of optimality 4) Peakfinder reads the CPLEX solu-
tion for reporting and analysis 5) Excel is used for graphing
and comparison of alternative plans. The process described
could also be implemented with commercially available
production scheduling software.