307
than both the conventional and ARF types of simulations, allowing a better sampling with
the same number of scenarios. The reduction in the standard deviation for ARF2 in this
sample size is 54% compared to the conventional algorithm and 20% compared to ARF10.
Finally, for a sample size of S = 20, ARF2 achieves the lower standard deviation as well,
with a reduction of 47% compared to both conventional and ARF10 simulations.
A possible explanation for these results is related to the nature of the optimization problem. Since the value of the extraction is estimated prior to the simulation process, the geological scenarios are only used to estimate the deviations from the production targets. Since
the deviations only consider the classification of the material as ore or waste, the differences
among different scenarios are based on whether each block is above or below the cut-off
grade. Therefore, a favorable way to sample for this optimization model is focusing on the
cases when each block belong to each category, and not necessarily on the complete range
of grades for each block. This could explain the performance of ARF2, compared with the
rest of the simulation algorithms. Different case studies for this optimization model should
be tested to confirm this behavior.
An unexpected result is the difference between the average values of the objective function
for each type of simulation and sample size. For every sample size, the average value is lower
for ARF2 and higher for the conventional simulation. Moreover, the average value tends to
decrease when the sample size is larger. This behavior was not seen in previous works, where
the average value was similar for every simulation type (Nelis et al. 2018). The apparent bias
is larger with smaller sample sizes, where the average value for conventional algorithm is 22%
higher than the average value of ARF2 with S = 2. This difference decreases with larger sample sizes: for S = 10, the difference is 11% between the conventional simulations and ARF2
and for S = 20 gets to 7%. The differences between ARF10 and Conventional are smaller: 4%
for S = 2, 6% for S = 10 and 5% for S = 20. Since the true value of the objective function
is unknown, it is not possible to establish whether the Antithetic Simulations present a bias.
However, it can be seen that the difference between the average value among different simulation types gets smaller with larger sample sizes, so it is expected this difference in the expected
value tends to zero with even larger sample sizes. This also could be an indication that even a
sample size of S = 20 is not large enough to guarantee convergence for this case study.
4 CONCLUSIONS
The performance of a variance reduction technique was presented in this work. The results
showed that the antithetic random fields implementation achieved a variance reduction in
the estimation of the objective value of a particular long-term stochastic optimization model
in a case study using real data. A difference between the expected value estimation using this
variance reduction technique compared to conventional simulation algorithm was found,
but it approached to zero with larger sample sizes. This variance reduction technique shows
promising results, and could lead to solve larger cases with fewer scenarios.
ACKNOWLEDGMENTS
The authors acknowledge the support of the Natural Sciences and Engineering Council of
Canada (NSERC), funding reference number RGPIN-2017-04200 and RGPAS-2017-507956,
and the support of CONICYT through Grant “Fondo Basal FB0809”.
REFERENCES
Armstrong, M., Ndiaye, A., Razanatsimba, R. & Galli, A. 2013. Scenario reduction applied to geostatistical
simulations. Mathematical Geosciences 45: 165–182.
Benndorf, J. & Dimitrakopoulos R. 2013. Stochastic long-term production scheduling of iron ore deposits:
Integrating joint multi-element geological uncertainty. Journal of Mining Science 49(1): 68–81.
than both the conventional and ARF types of simulations, allowing a better sampling with
the same number of scenarios. The reduction in the standard deviation for ARF2 in this
sample size is 54% compared to the conventional algorithm and 20% compared to ARF10.
Finally, for a sample size of S = 20, ARF2 achieves the lower standard deviation as well,
with a reduction of 47% compared to both conventional and ARF10 simulations.
A possible explanation for these results is related to the nature of the optimization problem. Since the value of the extraction is estimated prior to the simulation process, the geological scenarios are only used to estimate the deviations from the production targets. Since
the deviations only consider the classification of the material as ore or waste, the differences
among different scenarios are based on whether each block is above or below the cut-off
grade. Therefore, a favorable way to sample for this optimization model is focusing on the
cases when each block belong to each category, and not necessarily on the complete range
of grades for each block. This could explain the performance of ARF2, compared with the
rest of the simulation algorithms. Different case studies for this optimization model should
be tested to confirm this behavior.
An unexpected result is the difference between the average values of the objective function
for each type of simulation and sample size. For every sample size, the average value is lower
for ARF2 and higher for the conventional simulation. Moreover, the average value tends to
decrease when the sample size is larger. This behavior was not seen in previous works, where
the average value was similar for every simulation type (Nelis et al. 2018). The apparent bias
is larger with smaller sample sizes, where the average value for conventional algorithm is 22%
higher than the average value of ARF2 with S = 2. This difference decreases with larger sample sizes: for S = 10, the difference is 11% between the conventional simulations and ARF2
and for S = 20 gets to 7%. The differences between ARF10 and Conventional are smaller: 4%
for S = 2, 6% for S = 10 and 5% for S = 20. Since the true value of the objective function
is unknown, it is not possible to establish whether the Antithetic Simulations present a bias.
However, it can be seen that the difference between the average value among different simulation types gets smaller with larger sample sizes, so it is expected this difference in the expected
value tends to zero with even larger sample sizes. This also could be an indication that even a
sample size of S = 20 is not large enough to guarantee convergence for this case study.
4 CONCLUSIONS
The performance of a variance reduction technique was presented in this work. The results
showed that the antithetic random fields implementation achieved a variance reduction in
the estimation of the objective value of a particular long-term stochastic optimization model
in a case study using real data. A difference between the expected value estimation using this
variance reduction technique compared to conventional simulation algorithm was found,
but it approached to zero with larger sample sizes. This variance reduction technique shows
promising results, and could lead to solve larger cases with fewer scenarios.
ACKNOWLEDGMENTS
The authors acknowledge the support of the Natural Sciences and Engineering Council of
Canada (NSERC), funding reference number RGPIN-2017-04200 and RGPAS-2017-507956,
and the support of CONICYT through Grant “Fondo Basal FB0809”.
REFERENCES
Armstrong, M., Ndiaye, A., Razanatsimba, R. & Galli, A. 2013. Scenario reduction applied to geostatistical
simulations. Mathematical Geosciences 45: 165–182.
Benndorf, J. & Dimitrakopoulos R. 2013. Stochastic long-term production scheduling of iron ore deposits:
Integrating joint multi-element geological uncertainty. Journal of Mining Science 49(1): 68–81.