304
production target at scenario s ∈ S. Since the same block may present a different amount
of ore or waste tonnes in different scenarios, each simulation is associated with a different
deviation constraint. Similarly, Equation (9) accounts for the excess of production of resource
r from the upper production target in each scenario. Since not every resource is allowed to
deviate from the production target, Equation (10) represents the capacity constraint for such
resources. Equation (11) represents the precedences constraints, which establish a spatial order
of the extraction to maintain the pit stability. Equation (12) states that each block can be
extracted once. Finally, Equation (13) establishes the bounds for the deviations variables.
2.3 Convergence analysis
The performance of the antithetic random fields technique (Section 2.1) in the stochastic mine planning problem (Section 2.2) is evaluated with the following methodology. An
instance of problem (7) is the set of parameters used to define the objective function and
the constraints for a particular case study, such as prices, costs, attributes, etc. More relevant
for this study, an instance is defined for the set of scenarios used to represent the grade variability. If the number of scenarios, S , , is a good representation of this variability, every
instance of problem (7) should have a similar optimal objective function value. Equivalently,
the dispersion in the objective function value among different instances with the same sample
size S gives information about the representation of the grade variability for that sample
size. Therefore, the evaluation is based on defining several instances of problem (7) with different sample size and different simulation algorithms, and studying their dispersion. The
comparison between the results obtained with different simulation algorithms is an indicator
of the performance of such algorithms representing the true grade variability of the deposit.
For this case study, the procedure to compare the performance of conventional and antithetic simulations is described as follows:
1. Scenario generation: The scenarios necessary to define the instances are generated according to each simulation algorithm:
i. For the conventional sequential Gaussian algorithm, 600 independent scenarios are
generated
ii. For the antithetic scenarios, 600 realizations are generated in total, but given the nature
of this algorithm they are generated in negative-correlated sets of size m. Firstly, 300
sets of m = 2 are generated (α = −1), which are noted ARF2. Then, 60 sets of m = 10
are generated (
. ),
1
. 1 which are noted ARF10
2. Instance definition: Using the previously generated scenarios, 30 instances are defined for
each simulation algorithm and sample size. The sample sizes S chosen for this analysis were 2, 10 and 20. Depending on the simulation algorithm, the definition of these
instances is different, which is detailed next:
i. Conventional simulations: 30 sets of S = 2 are randomly selected from the pool of
scenarios, each scenario picked individually. Equivalently, 30 sets of S = 10 and 30
sets of S = 20 are selected randomly.
ii. ARF2: For the sample size S = 2, 30 sets of paired realizations are selected. For
S = 10, each one of the 30 instances is made of 5 sets of ARF2, each set picked randomly. Finally, for S = 20, 30 instances are defined, each one with10 sets of ARF2.
iii. ARF10: For S = 2, 30 pairs of realizations are picked randomly, each one of these
pairs is selected from a single set of ARF10. For S = 10, each instance is defined with
a single set of ARF10. Similarly, for S = 20, each one of the 30 instances is defined
by 2 sets of ARF10 picked randomly.
3. Statistical analysis: For each case, the 30 instances are solved and the optimal values of the
objective function is obtained. The mean value of these 30 instances is an estimator of the
true objective function value, while the standard deviation is a measure of the precision of
this estimation for each sample size and simulation algorithm. The performance of this
antithetic random fields technique is evaluated comparing the standard deviation obtained
by using ARF2 and ARF10, and the standard deviation of the conventional simulations.
production target at scenario s ∈ S. Since the same block may present a different amount
of ore or waste tonnes in different scenarios, each simulation is associated with a different
deviation constraint. Similarly, Equation (9) accounts for the excess of production of resource
r from the upper production target in each scenario. Since not every resource is allowed to
deviate from the production target, Equation (10) represents the capacity constraint for such
resources. Equation (11) represents the precedences constraints, which establish a spatial order
of the extraction to maintain the pit stability. Equation (12) states that each block can be
extracted once. Finally, Equation (13) establishes the bounds for the deviations variables.
2.3 Convergence analysis
The performance of the antithetic random fields technique (Section 2.1) in the stochastic mine planning problem (Section 2.2) is evaluated with the following methodology. An
instance of problem (7) is the set of parameters used to define the objective function and
the constraints for a particular case study, such as prices, costs, attributes, etc. More relevant
for this study, an instance is defined for the set of scenarios used to represent the grade variability. If the number of scenarios, S , , is a good representation of this variability, every
instance of problem (7) should have a similar optimal objective function value. Equivalently,
the dispersion in the objective function value among different instances with the same sample
size S gives information about the representation of the grade variability for that sample
size. Therefore, the evaluation is based on defining several instances of problem (7) with different sample size and different simulation algorithms, and studying their dispersion. The
comparison between the results obtained with different simulation algorithms is an indicator
of the performance of such algorithms representing the true grade variability of the deposit.
For this case study, the procedure to compare the performance of conventional and antithetic simulations is described as follows:
1. Scenario generation: The scenarios necessary to define the instances are generated according to each simulation algorithm:
i. For the conventional sequential Gaussian algorithm, 600 independent scenarios are
generated
ii. For the antithetic scenarios, 600 realizations are generated in total, but given the nature
of this algorithm they are generated in negative-correlated sets of size m. Firstly, 300
sets of m = 2 are generated (α = −1), which are noted ARF2. Then, 60 sets of m = 10
are generated (
. ),
1
. 1 which are noted ARF10
2. Instance definition: Using the previously generated scenarios, 30 instances are defined for
each simulation algorithm and sample size. The sample sizes S chosen for this analysis were 2, 10 and 20. Depending on the simulation algorithm, the definition of these
instances is different, which is detailed next:
i. Conventional simulations: 30 sets of S = 2 are randomly selected from the pool of
scenarios, each scenario picked individually. Equivalently, 30 sets of S = 10 and 30
sets of S = 20 are selected randomly.
ii. ARF2: For the sample size S = 2, 30 sets of paired realizations are selected. For
S = 10, each one of the 30 instances is made of 5 sets of ARF2, each set picked randomly. Finally, for S = 20, 30 instances are defined, each one with10 sets of ARF2.
iii. ARF10: For S = 2, 30 pairs of realizations are picked randomly, each one of these
pairs is selected from a single set of ARF10. For S = 10, each instance is defined with
a single set of ARF10. Similarly, for S = 20, each one of the 30 instances is defined
by 2 sets of ARF10 picked randomly.
3. Statistical analysis: For each case, the 30 instances are solved and the optimal values of the
objective function is obtained. The mean value of these 30 instances is an estimator of the
true objective function value, while the standard deviation is a measure of the precision of
this estimation for each sample size and simulation algorithm. The performance of this
antithetic random fields technique is evaluated comparing the standard deviation obtained
by using ARF2 and ARF10, and the standard deviation of the conventional simulations.