Mining Goes Digital – Mueller et al. (Eds)
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
323
Incorporating grade uncertainty into sublevel stope sequencing
Y.A. Sari & M. Kumral
Department of Mining and Materials Engineering, McGill University, Montreal, QC, Canada
ABSTRACT: Uncertainty in mineral deposits causes deviations from the long-term production schedules of mines. As underground mining has high development and operational
costs, it is crucial to incorporate the uncertainty in the planning stage. In this research, a
new mixed integer linear program that solves the sublevel stope sequencing problem while
accounting for the block grade uncertainty is proposed. The new linear programming model
incorporates chance constrained programming to manage the risk of the project and to stabilize the expected net present value. In a case study, the effect of varying levels of risk is
studied by generating stope sequences for different risk levels. The trade-off between risk and
expected net present value is studied. It was found that at higher risk levels the stopes with
higher average grade are extracted earlier while at lower risk levels stopes with lower standard
deviation are prioritized.
1 INTRODUCTION
Sublevel stope sequencing problem is the long-term planning for sublevel stoping underground mining technique. Conventional stope sequencing approach focuses on ordering
of the stope extraction sequence such that mine stability and production requirement constraints will be respected and the profit will be maximized. The expected revenue from a
block is calculated from the block tonnage and grade. This approach assumes the ideal condition where the stope average grades are accurately known. In reality, grade information
is obtained through estimation or simulation of sparse drill hole samples. Therefore, the
estimated/simulated grades deviate from the actual grades. Underground mining has high
development and operational costs. Thus, when the plan is made based on the estimated/
simulated grades, important loss may occur if the deviation is high. Grade uncertainty can
be incorporated into stope sequence planning to minimize this kind of loss. In this paper, a
strategy for sublevel stope sequencing that accounts for grade uncertainty is proposed.
2 LITERATURE REVIEW
As the stope sequencing problem does not have as many variables as stope layout planning problem, most approaches make use of the mixed integer linear programming (MILP)
method. Chanda (1990) and Trout (1995) are the earliest adapters of MILP to stope sequencing. The objective is profit maximization while accounting for physical stability, minimum
production and capacity constraints. However, due to limited computational resources they
were not able to obtain results for larger cases.
Nehring, Topal, and Little (2010) combined development, drilling and backfilling
phases to reduce the number of variables as the phases have a linear flow. Terblanche and
Bley (2015) proposed smoothing the grade data just enough to dissociate feasible regions
which allows selective mining. As an alternative to MILP, Manchuk (2008) developed a
simulated annealing approach to the problem. This approach starts with a preliminary
sub-optimal sequence and at each iteration, it introduces a random change to the sequence.
If the resulting sequence is feasible, the updated net present value (NPV) is calculated and
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
323
Incorporating grade uncertainty into sublevel stope sequencing
Y.A. Sari & M. Kumral
Department of Mining and Materials Engineering, McGill University, Montreal, QC, Canada
ABSTRACT: Uncertainty in mineral deposits causes deviations from the long-term production schedules of mines. As underground mining has high development and operational
costs, it is crucial to incorporate the uncertainty in the planning stage. In this research, a
new mixed integer linear program that solves the sublevel stope sequencing problem while
accounting for the block grade uncertainty is proposed. The new linear programming model
incorporates chance constrained programming to manage the risk of the project and to stabilize the expected net present value. In a case study, the effect of varying levels of risk is
studied by generating stope sequences for different risk levels. The trade-off between risk and
expected net present value is studied. It was found that at higher risk levels the stopes with
higher average grade are extracted earlier while at lower risk levels stopes with lower standard
deviation are prioritized.
1 INTRODUCTION
Sublevel stope sequencing problem is the long-term planning for sublevel stoping underground mining technique. Conventional stope sequencing approach focuses on ordering
of the stope extraction sequence such that mine stability and production requirement constraints will be respected and the profit will be maximized. The expected revenue from a
block is calculated from the block tonnage and grade. This approach assumes the ideal condition where the stope average grades are accurately known. In reality, grade information
is obtained through estimation or simulation of sparse drill hole samples. Therefore, the
estimated/simulated grades deviate from the actual grades. Underground mining has high
development and operational costs. Thus, when the plan is made based on the estimated/
simulated grades, important loss may occur if the deviation is high. Grade uncertainty can
be incorporated into stope sequence planning to minimize this kind of loss. In this paper, a
strategy for sublevel stope sequencing that accounts for grade uncertainty is proposed.
2 LITERATURE REVIEW
As the stope sequencing problem does not have as many variables as stope layout planning problem, most approaches make use of the mixed integer linear programming (MILP)
method. Chanda (1990) and Trout (1995) are the earliest adapters of MILP to stope sequencing. The objective is profit maximization while accounting for physical stability, minimum
production and capacity constraints. However, due to limited computational resources they
were not able to obtain results for larger cases.
Nehring, Topal, and Little (2010) combined development, drilling and backfilling
phases to reduce the number of variables as the phases have a linear flow. Terblanche and
Bley (2015) proposed smoothing the grade data just enough to dissociate feasible regions
which allows selective mining. As an alternative to MILP, Manchuk (2008) developed a
simulated annealing approach to the problem. This approach starts with a preliminary
sub-optimal sequence and at each iteration, it introduces a random change to the sequence.
If the resulting sequence is feasible, the updated net present value (NPV) is calculated and