427
waste dump at the time of extraction. Otherwise, the plan generated by an optimizer will be
biased on the preselected destinations which will again underestimate the potential profit.
Due to the large scale of block by block mine production scheduling problems, the pushback
concept became attractive since it allows scheduling to be done from the sub-pits or nested pits
which results in less variables in the optimization model. The pushback design method used to
determine a production schedule in an open pit mine starts with determining the ultimate pit
limits. Once the ultimate pit limit is obtained, mine planning continues with the goal of finding
the optimal extraction sequence of the blocks, which in the end, results in nested pits called as
pushbacks. The need for dividing the pit into sub pits arises due to the scale of a block by block
production scheduling problem hence, sub-pits or pushbacks are used to schedule the blocks
quarterly or yearly. Many heuristic techniques were developed in order to sequence the pushbacks so that the schedules from the pushbacks will be in compliance with the resource capacity
and mill blending requirements. Dagdelen and Francois-Bongarcon (1982) determined a series
of pushbacks by varying the price of the commodity, cutoff grade, mining or processing costs.
Gershon (1987) proposed an algorithm which generates cones with the shape of a pit expanding towards the bottom of the pit with the vertex positioned on each block. The blocks are
ranked based on their positional weight and the highest rank block is scheduled first. Whittle’s
approach (1988) is a commonly used pit parametrization technique which varies the block
values incrementally and generates nested pits by implementing the LG algorithm. Seymour
(1995) proposed a parameterization algorithm which generates pits as a function of pit volume
and pit value. The algorithm is a modified version of the LG algorithm by adding the parametrized variables and allowing sub trees that represent the small pits to be formed (Meagher
et al., 2014). The branches are categorized as strong or weak based on the threshold value of
their strength which is calculated by dividing the cumulative value of the nodes in the branch
by the cumulative mass of the nodes in the branch. Strong branches form the small pits which
are sequenced based on their strength value. Ramazan and Dagdelen (1998) developed a minimum stripping ratio pushback design algorithm which is a modified version of the Seymour’s
algorithm in 1995. The goal of the algorithm is to find pits with the minimum stripping ratio
which will lead to a schedule where the ore blocks are mined as soon as possible. Somrit (2011)
introduced a phase design algorithm which uses Lagrange parameters to determine the size of
the pits in compliance with the annual resource capacity requirements. The usage of Lagrange
parameters is similar to the Whittle’s pit parametrization technique. The author uses linear
interpolation to determine the Lagrange parameters which result in “gap” problems. The methods proposed in the literature follow the incremental fashion of obtaining pushbacks that fail
to incorporate the operational requirements such as multi capacity, multi destinations, blending requirements, truck hours, stockpiles and also time value of money. Moreover, the size of
the pit has a non-linear relationship with the value of the pit which makes it extremely hard to
determine. But the common approach attempts to find the pit size with an assumption of a linear relationship which in the end results with a gap problem. The presence of a gap problem will
prevent pushbacks to meet with the capacity requirements. So far, researchers have not found
an approach to overcome the stated problems which will in the end produce poorly designed
pushbacks. Since many production schedule plans highly depend on the design of pushbacks,
poor designs will prevent the schedules from achieving a maximum NPV or even obtaining a
solution that meets all of the desired production requirements.
2 METHODOLOGY
2.1 Case study (McLaughlin deposit)
The case study will demonstrate the implementation of the new integer solution algorithm
proposed by Aras (2018) on scheduling a data set referred to as the McLaughlin Deposit for
10 years. The schematic description of the assumed mining complex is given in Figure 1. The
economic parameters that will be used to derive the block values are given in Table 1. The
blocks will be initially subjected to a break-even cutoff grade 0.03 oz/t which will separate
waste dump at the time of extraction. Otherwise, the plan generated by an optimizer will be
biased on the preselected destinations which will again underestimate the potential profit.
Due to the large scale of block by block mine production scheduling problems, the pushback
concept became attractive since it allows scheduling to be done from the sub-pits or nested pits
which results in less variables in the optimization model. The pushback design method used to
determine a production schedule in an open pit mine starts with determining the ultimate pit
limits. Once the ultimate pit limit is obtained, mine planning continues with the goal of finding
the optimal extraction sequence of the blocks, which in the end, results in nested pits called as
pushbacks. The need for dividing the pit into sub pits arises due to the scale of a block by block
production scheduling problem hence, sub-pits or pushbacks are used to schedule the blocks
quarterly or yearly. Many heuristic techniques were developed in order to sequence the pushbacks so that the schedules from the pushbacks will be in compliance with the resource capacity
and mill blending requirements. Dagdelen and Francois-Bongarcon (1982) determined a series
of pushbacks by varying the price of the commodity, cutoff grade, mining or processing costs.
Gershon (1987) proposed an algorithm which generates cones with the shape of a pit expanding towards the bottom of the pit with the vertex positioned on each block. The blocks are
ranked based on their positional weight and the highest rank block is scheduled first. Whittle’s
approach (1988) is a commonly used pit parametrization technique which varies the block
values incrementally and generates nested pits by implementing the LG algorithm. Seymour
(1995) proposed a parameterization algorithm which generates pits as a function of pit volume
and pit value. The algorithm is a modified version of the LG algorithm by adding the parametrized variables and allowing sub trees that represent the small pits to be formed (Meagher
et al., 2014). The branches are categorized as strong or weak based on the threshold value of
their strength which is calculated by dividing the cumulative value of the nodes in the branch
by the cumulative mass of the nodes in the branch. Strong branches form the small pits which
are sequenced based on their strength value. Ramazan and Dagdelen (1998) developed a minimum stripping ratio pushback design algorithm which is a modified version of the Seymour’s
algorithm in 1995. The goal of the algorithm is to find pits with the minimum stripping ratio
which will lead to a schedule where the ore blocks are mined as soon as possible. Somrit (2011)
introduced a phase design algorithm which uses Lagrange parameters to determine the size of
the pits in compliance with the annual resource capacity requirements. The usage of Lagrange
parameters is similar to the Whittle’s pit parametrization technique. The author uses linear
interpolation to determine the Lagrange parameters which result in “gap” problems. The methods proposed in the literature follow the incremental fashion of obtaining pushbacks that fail
to incorporate the operational requirements such as multi capacity, multi destinations, blending requirements, truck hours, stockpiles and also time value of money. Moreover, the size of
the pit has a non-linear relationship with the value of the pit which makes it extremely hard to
determine. But the common approach attempts to find the pit size with an assumption of a linear relationship which in the end results with a gap problem. The presence of a gap problem will
prevent pushbacks to meet with the capacity requirements. So far, researchers have not found
an approach to overcome the stated problems which will in the end produce poorly designed
pushbacks. Since many production schedule plans highly depend on the design of pushbacks,
poor designs will prevent the schedules from achieving a maximum NPV or even obtaining a
solution that meets all of the desired production requirements.
2 METHODOLOGY
2.1 Case study (McLaughlin deposit)
The case study will demonstrate the implementation of the new integer solution algorithm
proposed by Aras (2018) on scheduling a data set referred to as the McLaughlin Deposit for
10 years. The schematic description of the assumed mining complex is given in Figure 1. The
economic parameters that will be used to derive the block values are given in Table 1. The
blocks will be initially subjected to a break-even cutoff grade 0.03 oz/t which will separate