301
the representation of the true grade variability, based on generating scenarios with negative
pairwise correlation. This technique is tested in a particular stochastic framework, which
generates a single production plan based on multiple scenarios.
Scenario reduction has been studied in the last decades with different approaches. One of
them is based on selecting a small set that represents the characteristics of a larger set. Usually,
the size of the small set is fixed beforehand, and the objective is finding the best subset that
minimizes the difference between them considering a relevant attribute for a particular problem. This has been used by Heitsch & Römisch (2003), where they proposed a heuristic to find
such subset, defining the relevant distance and adding or subtracting a single scenario aiming to
minimize it. A similar methodology was used by Dupačová et al. (2003) and Heitsch & Römisch
(2009) in power management. Armstrong et al. (2013) proposed a random search procedureto
find a representative subset in an stochastic mine planning problem. They used a proxy variable
(ore tonnes above certain cut-off grades) to characterize the differences between scenarios, and
the random search procedure proved its usefulness reducing the number of scenarios needed for
the stochastic formulation in a real deposit. A similar approach, using proxy variables related to
the expected response of a transfer function, has been used by Deutsch & Srinivasan (1996) and
McLennan & Deutsch (2005). These proxy variables are used to construct a ranking of realizations to then select a small subset to characterize the uncertainty on the response. This allows
to process only a subset of the complete scenarios, typically linked to some key percentiles to
represent the variability in the response (Deutsch 2007 and Pereira et al. 2017).
Another approach has been the use of variance reduction techniques, which are procedures
to reduce the variance on some estimation without increasing the number of realizations, based
on modifications on the algorithm used to generate the scenarios. There are many techniques
under this category, and more details can be found in Cheng (1986) and James (1985). In this
work, we will use a variance reduction technique known as antithetic variates (Hammersley &
Morton 1956), which is based on generating scenarios with negative correlation, expecting that
maximizing the difference among the scenarios allows a better representation of the expected
variability of the uncertain parameter. An extension of this methodology to a sequential simulation algorithm was proposed by Guthke & Bárdossy (2012). This extension allows to generate
an arbitrary number of scenarios negatively correlated in a geostatistical framework. This was
used in Nelis et al. (2018) in an stochastic mine planning problem, which achieved a significant
variance reduction in the NPV estimation of the mine schedule. In this work, we will use this
variance reduction technique in a real case study, using a widely studied stochastic framework,
based on the minimization of the deviations from the production targets considering multiple
grade scenarios. This framework was first proposed as an optimization problem in Ramazan
& Dimitrakopoulos (2007), but many extensions has been proposed such as multiple elements
(Benndorf & Dimitrakopoulos 2013), different processing streams (de Freitas et al. 2015) and
mining complexes (Goodfellow & Dimitrakopoulos 2016). Specifically, the performance of
the antithetic random fields technique will be tested in an optimization model based on Leite
& Dimitrakopoulos (2014). This optimization model, along with the antithetic random fields
algorithm will be detailed in the next section.
2 METHODS
2.1 Antithetic random fields
The antithetic random fields technique is based on the work of Guthke & Bárdossy (2012).
They propose a methodology to generate an arbitrary number of scenarios with negative
correlation in a sequential simulation algorithm, modifying the generation of the random
numbers needed in each scenario. For a conventional sequential Gaussian simulation, each
node in the simulation is visited in a random order, and a random number is generated to
obtain the simulated value for that node. A new random path is generated for the following scenarios, and the same node is simulated using a different, iid random number. In the
algorithm proposed by Guthke & Bárdossy (2012), the same random path is used for all the
the representation of the true grade variability, based on generating scenarios with negative
pairwise correlation. This technique is tested in a particular stochastic framework, which
generates a single production plan based on multiple scenarios.
Scenario reduction has been studied in the last decades with different approaches. One of
them is based on selecting a small set that represents the characteristics of a larger set. Usually,
the size of the small set is fixed beforehand, and the objective is finding the best subset that
minimizes the difference between them considering a relevant attribute for a particular problem. This has been used by Heitsch & Römisch (2003), where they proposed a heuristic to find
such subset, defining the relevant distance and adding or subtracting a single scenario aiming to
minimize it. A similar methodology was used by Dupačová et al. (2003) and Heitsch & Römisch
(2009) in power management. Armstrong et al. (2013) proposed a random search procedureto
find a representative subset in an stochastic mine planning problem. They used a proxy variable
(ore tonnes above certain cut-off grades) to characterize the differences between scenarios, and
the random search procedure proved its usefulness reducing the number of scenarios needed for
the stochastic formulation in a real deposit. A similar approach, using proxy variables related to
the expected response of a transfer function, has been used by Deutsch & Srinivasan (1996) and
McLennan & Deutsch (2005). These proxy variables are used to construct a ranking of realizations to then select a small subset to characterize the uncertainty on the response. This allows
to process only a subset of the complete scenarios, typically linked to some key percentiles to
represent the variability in the response (Deutsch 2007 and Pereira et al. 2017).
Another approach has been the use of variance reduction techniques, which are procedures
to reduce the variance on some estimation without increasing the number of realizations, based
on modifications on the algorithm used to generate the scenarios. There are many techniques
under this category, and more details can be found in Cheng (1986) and James (1985). In this
work, we will use a variance reduction technique known as antithetic variates (Hammersley &
Morton 1956), which is based on generating scenarios with negative correlation, expecting that
maximizing the difference among the scenarios allows a better representation of the expected
variability of the uncertain parameter. An extension of this methodology to a sequential simulation algorithm was proposed by Guthke & Bárdossy (2012). This extension allows to generate
an arbitrary number of scenarios negatively correlated in a geostatistical framework. This was
used in Nelis et al. (2018) in an stochastic mine planning problem, which achieved a significant
variance reduction in the NPV estimation of the mine schedule. In this work, we will use this
variance reduction technique in a real case study, using a widely studied stochastic framework,
based on the minimization of the deviations from the production targets considering multiple
grade scenarios. This framework was first proposed as an optimization problem in Ramazan
& Dimitrakopoulos (2007), but many extensions has been proposed such as multiple elements
(Benndorf & Dimitrakopoulos 2013), different processing streams (de Freitas et al. 2015) and
mining complexes (Goodfellow & Dimitrakopoulos 2016). Specifically, the performance of
the antithetic random fields technique will be tested in an optimization model based on Leite
& Dimitrakopoulos (2014). This optimization model, along with the antithetic random fields
algorithm will be detailed in the next section.
2 METHODS
2.1 Antithetic random fields
The antithetic random fields technique is based on the work of Guthke & Bárdossy (2012).
They propose a methodology to generate an arbitrary number of scenarios with negative
correlation in a sequential simulation algorithm, modifying the generation of the random
numbers needed in each scenario. For a conventional sequential Gaussian simulation, each
node in the simulation is visited in a random order, and a random number is generated to
obtain the simulated value for that node. A new random path is generated for the following scenarios, and the same node is simulated using a different, iid random number. In the
algorithm proposed by Guthke & Bárdossy (2012), the same random path is used for all the