2
process grade uncertainty and its interaction with cut-off
grade adjustments needed to balance process capacity with
material mined above cut-off.
CONDITIONAL SIMULATION
Conditional Simulation, CS, is a geostatistical method for
evaluating recoverable resources. Recoverable resources
are defined as the proportion of tons and metal above a
cut-off grade that will be recovered from future selective
mining when new information, such as blasthole assays, is
available during the final ore control process [Journel and
Huijbregts, 1978]. The “simulation” part of CS refers to
providing multiple equally probable orebody realizations.
The “conditional” part of CS refers to each simulation being
conditional to all available drillhole data and reproducing
the input composite data’s histogram and spatial variabil-
ity [Ortiz, 2020]. Once a point model is created by con-
ditional simulation, a change of support process is used to
create blocks which demonstrate the ore control selectivity
which will be achieved via ore control of blasthole samples.
CS models can simultaneously capture variability
(i.e., the distribution of grades within each simulation)
and uncertainty (i.e., the distribution of each block’s grade
across all simulations). The uncertainty modeled by a set
of multiple orebody realizations can be translated to risk
analysis and expressed in statements such as “there is 80%
confidence that 2025s tonnage mined above a 0.03 oz/ton
cut-off will be within plus or minus 10% of 2.5 million
tons”.
Sequential Gaussian Simulation (SGS) is a common
method for creating conditional simulations [Rossi and
Deutsch, 2014]. Starting with a normal scores transforma-
tion allows the SGS method to be applied to non-Gaussian
data. The SGS method is implemented in commercially
available mining software packages, including Hexagon’s
MinePlan. The simulated points can then be converted to
ore control sized blocks with grades estimated via ordinary
kriging of the simulated points.
OPEN PIT MINE PRODUCTION
SCHEDULING
A traditional workflow for open pit production schedul-
ing starts with an ultimate pit subdivided into phases or
laybacks which are often guided by nested pits based on a
series of varying metal prices, processing cost and/or other
economic parameters. A mine schedule defines how many
tons will be mined from each phase in each time period
and a process schedule defines what material will be sent to
which process destination, stockpile or waste dump in each
time period [Clark and Dagdelen, 2023].
Direct block scheduling methods do not require prede-
termination of phase designs [Aras et al, 2019 Goodfellow
and Dimitrakopoulos, 2016]. These methods may create
challenges for designing haul road access or maintain-
ing minable widths but may suggest alternative mining
sequences and phase designs not identified from a tradi-
tional workflow.
MILP Production Schedule Optimization
In this paper, Mixed Integer Linear Programming
(MILP) models are used to create production schedules.
Deterministic and stochastic optimization models are used
to solve for the mining, processing and stockpiling ton-
nage flows that maximize the Net Present Value generated
from the orebody (Figure 1). Each possible tonnage flow
is assigned a value based on metal prices, discount rates,
increment grades and a database of cost and recovery infor-
mation. The optimum tonnage flows are computed using a
MILP solver with input formulated to ensure that the solu-
tion satisfies mining, processing, continuity and geometry
constraints.
The deterministic optimization computes the ton-
nage flows based on a fixed distribution of tons and grade
by material type for each bench of each phase [Hoerger,
2024]. For stochastic optimization, the grade/tonnage dis-
tributions for each panel are modeled as unknown with
the range of uncertainty provided by multiple conditional
simulation realizations. The stochastic optimization uses a
two-stage stochastic optimization with recourse approach
[Wolsey, 2021]—the first stage variables are the tons
mined the second stage (or recourse) variables are the pro-
cess and stockpile tonnages which adapt to the grade/ton-
nage distributions of each orebody realization [Hoerger and
Dagdelen, 2024].
STOCHASTIC EVALUATION /RISK
ANALYSIS
Regardless of how a mine plan is generated, conditional
simulations can be used to perform a stochastic evaluation
of the production schedule. Using the stochastic optimiza-
tion model with a predefined mine plan (i.e., all W min-
ing flow variables in Figure 1 are known constants), the
stochastic optimization model will optimize the processing
and stockpiling tonnage flows to adapt to each orebody
realization. For each realization, statistics are captured for
yearly and life of mine tons, grades, cash flows, and any
other metric of interest. This stochastic evaluation process
allows a risk analysis to be prepared for critical plan metrics
and enables plan vs plan risk comparisons.
process grade uncertainty and its interaction with cut-off
grade adjustments needed to balance process capacity with
material mined above cut-off.
CONDITIONAL SIMULATION
Conditional Simulation, CS, is a geostatistical method for
evaluating recoverable resources. Recoverable resources
are defined as the proportion of tons and metal above a
cut-off grade that will be recovered from future selective
mining when new information, such as blasthole assays, is
available during the final ore control process [Journel and
Huijbregts, 1978]. The “simulation” part of CS refers to
providing multiple equally probable orebody realizations.
The “conditional” part of CS refers to each simulation being
conditional to all available drillhole data and reproducing
the input composite data’s histogram and spatial variabil-
ity [Ortiz, 2020]. Once a point model is created by con-
ditional simulation, a change of support process is used to
create blocks which demonstrate the ore control selectivity
which will be achieved via ore control of blasthole samples.
CS models can simultaneously capture variability
(i.e., the distribution of grades within each simulation)
and uncertainty (i.e., the distribution of each block’s grade
across all simulations). The uncertainty modeled by a set
of multiple orebody realizations can be translated to risk
analysis and expressed in statements such as “there is 80%
confidence that 2025s tonnage mined above a 0.03 oz/ton
cut-off will be within plus or minus 10% of 2.5 million
tons”.
Sequential Gaussian Simulation (SGS) is a common
method for creating conditional simulations [Rossi and
Deutsch, 2014]. Starting with a normal scores transforma-
tion allows the SGS method to be applied to non-Gaussian
data. The SGS method is implemented in commercially
available mining software packages, including Hexagon’s
MinePlan. The simulated points can then be converted to
ore control sized blocks with grades estimated via ordinary
kriging of the simulated points.
OPEN PIT MINE PRODUCTION
SCHEDULING
A traditional workflow for open pit production schedul-
ing starts with an ultimate pit subdivided into phases or
laybacks which are often guided by nested pits based on a
series of varying metal prices, processing cost and/or other
economic parameters. A mine schedule defines how many
tons will be mined from each phase in each time period
and a process schedule defines what material will be sent to
which process destination, stockpile or waste dump in each
time period [Clark and Dagdelen, 2023].
Direct block scheduling methods do not require prede-
termination of phase designs [Aras et al, 2019 Goodfellow
and Dimitrakopoulos, 2016]. These methods may create
challenges for designing haul road access or maintain-
ing minable widths but may suggest alternative mining
sequences and phase designs not identified from a tradi-
tional workflow.
MILP Production Schedule Optimization
In this paper, Mixed Integer Linear Programming
(MILP) models are used to create production schedules.
Deterministic and stochastic optimization models are used
to solve for the mining, processing and stockpiling ton-
nage flows that maximize the Net Present Value generated
from the orebody (Figure 1). Each possible tonnage flow
is assigned a value based on metal prices, discount rates,
increment grades and a database of cost and recovery infor-
mation. The optimum tonnage flows are computed using a
MILP solver with input formulated to ensure that the solu-
tion satisfies mining, processing, continuity and geometry
constraints.
The deterministic optimization computes the ton-
nage flows based on a fixed distribution of tons and grade
by material type for each bench of each phase [Hoerger,
2024]. For stochastic optimization, the grade/tonnage dis-
tributions for each panel are modeled as unknown with
the range of uncertainty provided by multiple conditional
simulation realizations. The stochastic optimization uses a
two-stage stochastic optimization with recourse approach
[Wolsey, 2021]—the first stage variables are the tons
mined the second stage (or recourse) variables are the pro-
cess and stockpile tonnages which adapt to the grade/ton-
nage distributions of each orebody realization [Hoerger and
Dagdelen, 2024].
STOCHASTIC EVALUATION /RISK
ANALYSIS
Regardless of how a mine plan is generated, conditional
simulations can be used to perform a stochastic evaluation
of the production schedule. Using the stochastic optimiza-
tion model with a predefined mine plan (i.e., all W min-
ing flow variables in Figure 1 are known constants), the
stochastic optimization model will optimize the processing
and stockpiling tonnage flows to adapt to each orebody
realization. For each realization, statistics are captured for
yearly and life of mine tons, grades, cash flows, and any
other metric of interest. This stochastic evaluation process
allows a risk analysis to be prepared for critical plan metrics
and enables plan vs plan risk comparisons.