146
of a crown pillar together with capital and operational development requirements were not
incorporated into this model.
King et al. (2016) incorporated crown and sill pillar placement into their OPUG transition
studies to separate the open pit from the underground mine. In their model, the location
of the crown pillar was simulated, and preselected to divide the deposit into OP and UG
mining zones before running an optimization for the OP mining zone and planning for the
UG mining zone. MacNeil and Dimitrakopoulos (2017) investigated the transition decision
at an operating open pit mine within the context of a mining complex comprising five
producing pits, four stockpiles and one processing plant. In their research, MacNeil and
Dimitrakopoulos (2017) priori identified the crown pillar envelope for a gold deposit and
evaluated four crown pillar locations within this envelope leading to four distinct candidate
transition depths. Decomposing the OPUG optimization process into scenarios has the
tendency to compromise the global optimal solution.
A mathematical model that solves the transition problem was developed by Whittle et al.
(2018) after modifying the normal pit optimization model based on the maximum graph closure
algorithm (Lerchs and Grossman, 1965). The modifications allow the algorithm to account for
the underground mining value of a block, and the requirement for a specified 2D crown pillar location above the underground mine. The algorithm of Whittle et al. (2018) is based on
what they called the “opportunity cost approach”, thus, if a given block is mined by open pit
method, its open pit value is gained while its value that would have been obtained by extracting
it using underground mining methods is lost. According to Whittle et al. (2018), the optimization
approach does not control the mining sequence with time and produces a near optimal value
for the mining project. Future works were therefore recommended to improve on their model.
In this research study, we have developed, implemented, and tested a MILP optimization
framework for evaluating the extraction strategy for a deposit. The MILP model maximizes
the Net Present Value (NPV) of the resource and determines the best extraction strategy for
a given ore body amenable to different mining options. The model further determines the
mining and processing schedule, the location of the required crown pillar, and the schedule
for underground capital and operating developments (shaft/decline, levels, ore drives and
crosscuts) for the optimal extraction option.
3 ASSUMPTIONS AND NOTATIONS
It is assumed that the size of the Selected Mining Units (SMUs) for open pit mining is
equivalent to the stope sizes for underground mining. In the MILP framework, the SMUs are
represented by mining blocks in general, or mining-cuts in specific relation to open pit mining
or mining-stopes for underground mining. The location of each mining block or mining-cut or
mining-stope is represented by the coordinates of the centroid. It is assumed that a crown pillar
is required for the exploitation of the orebody by underground mining. The size of the crown
pillar is estimated to be one vertical length of a stope or bench; thus, one bench or level in the
block model will represent the crown pillar. For underground mining, ore extraction is achieved
by a retreating method. Some of the notation of indices and parameters are as follows:
3.1 Indices
A general parameter f can take four indices in the format of f k l
f f
a t
,
, . Where:
j {
}
J
index for open pit mining-cuts in the model.
p {
}
P index for underground mining-stopes in the model.
3.2 Parameters
v j
op t
,
the open pit (op) discounted revenue generated by selling the final product within
mining-cut j in period t minus the discounted extra cost of extracting mining-cut j as
ore and processing it.
of a crown pillar together with capital and operational development requirements were not
incorporated into this model.
King et al. (2016) incorporated crown and sill pillar placement into their OPUG transition
studies to separate the open pit from the underground mine. In their model, the location
of the crown pillar was simulated, and preselected to divide the deposit into OP and UG
mining zones before running an optimization for the OP mining zone and planning for the
UG mining zone. MacNeil and Dimitrakopoulos (2017) investigated the transition decision
at an operating open pit mine within the context of a mining complex comprising five
producing pits, four stockpiles and one processing plant. In their research, MacNeil and
Dimitrakopoulos (2017) priori identified the crown pillar envelope for a gold deposit and
evaluated four crown pillar locations within this envelope leading to four distinct candidate
transition depths. Decomposing the OPUG optimization process into scenarios has the
tendency to compromise the global optimal solution.
A mathematical model that solves the transition problem was developed by Whittle et al.
(2018) after modifying the normal pit optimization model based on the maximum graph closure
algorithm (Lerchs and Grossman, 1965). The modifications allow the algorithm to account for
the underground mining value of a block, and the requirement for a specified 2D crown pillar location above the underground mine. The algorithm of Whittle et al. (2018) is based on
what they called the “opportunity cost approach”, thus, if a given block is mined by open pit
method, its open pit value is gained while its value that would have been obtained by extracting
it using underground mining methods is lost. According to Whittle et al. (2018), the optimization
approach does not control the mining sequence with time and produces a near optimal value
for the mining project. Future works were therefore recommended to improve on their model.
In this research study, we have developed, implemented, and tested a MILP optimization
framework for evaluating the extraction strategy for a deposit. The MILP model maximizes
the Net Present Value (NPV) of the resource and determines the best extraction strategy for
a given ore body amenable to different mining options. The model further determines the
mining and processing schedule, the location of the required crown pillar, and the schedule
for underground capital and operating developments (shaft/decline, levels, ore drives and
crosscuts) for the optimal extraction option.
3 ASSUMPTIONS AND NOTATIONS
It is assumed that the size of the Selected Mining Units (SMUs) for open pit mining is
equivalent to the stope sizes for underground mining. In the MILP framework, the SMUs are
represented by mining blocks in general, or mining-cuts in specific relation to open pit mining
or mining-stopes for underground mining. The location of each mining block or mining-cut or
mining-stope is represented by the coordinates of the centroid. It is assumed that a crown pillar
is required for the exploitation of the orebody by underground mining. The size of the crown
pillar is estimated to be one vertical length of a stope or bench; thus, one bench or level in the
block model will represent the crown pillar. For underground mining, ore extraction is achieved
by a retreating method. Some of the notation of indices and parameters are as follows:
3.1 Indices
A general parameter f can take four indices in the format of f k l
f f
a t
,
, . Where:
j {
}
J
index for open pit mining-cuts in the model.
p {
}
P index for underground mining-stopes in the model.
3.2 Parameters
v j
op t
,
the open pit (op) discounted revenue generated by selling the final product within
mining-cut j in period t minus the discounted extra cost of extracting mining-cut j as
ore and processing it.