Content uploaded by Behshad Jodeiri Shokri
Author content
All content in this area was uploaded by Behshad Jodeiri Shokri on Apr 27, 2021
Content may be subject to copyright.
* Corresponding author. Tel: +989128694889, Fax: +988118411407.
Email address:
b.jodeiri@hut.ac.ir (B. Jodeiri Shokri).
Journal Homepage: ijmge.ut.ac.ir
Ultimate Pit Limit Determination Using Flashlight Algorithm
Mohammadreza Sadeghi a, Hesam Dehghani a, Behshad Jodeiri Shokri a, *
a
Department of Mining Engineering, Hamedan University of Technology (HUT), Hamedan, Iran
A B S T R A C T
In this paper, the flashlight (FL) algorithm, which is categorized as a heuristic method, has been suggested to determine the ultimate pit limit
(UPL). In order to apply the suggested algorithm and other common algorithms, such as the dynamic programming, the Korobov, and the
floating cone, and to validate the capability of the proposed method, the ultimate pit limit was determined in a crosssection of the Korkora
reserve, which is located in Kurdistan province, northwestern of Iran and consists of 3080 blocks. The comparison of the FL algorithm and
other methods revealed that same as high accuracy dynamic programming methods, the proposed algorithm could find the optimum value,
while the Korobov and the floating cone algorithms failed to determine the optimum limit.
Keywords :
Heuristic algorithm; Ultimate pit limit; Optimization; Flashlight algorithm
1. Introduction
Determining the ultimate pit limit (UPL) is one of the most
challenging topics in surface mining that should be investigated in the
preliminary stages of mining operations. It should be determined when
some parameters, such as the profitability of the mining project, are
proven after the exploration stage. Moreover, some critical procedures
in openpit planning, including designing, locating, and performing the
feasibility study, will be conducted once an economic block model has
been proven by considering its financial aspects, such as determining
the grade of minerals, their prices, and operating costs.
The objective function in the designing stage is to maximize the final
value of the pit by observing the technical and economic considerations.
Due to the importance of setting the ultimate pit limit, many researchers
have tried to present various methods. Generally, these methods can be
categorized into the three following groups:
1. Rigorous algorithms are applied to find the optimum solution to
the problem. For instance, the dynamic programming method
presented by Lerchs and Grossmann algorithm is considered as
one of the most important algorithms in this category [1].
2. Heuristic algorithms, such as the floating cone and Korobov
algorithms, are used for finding an approximate solution by
applying the block search regardless of being close to the true
solution or not [2, 3].
3. A metaheuristic algorithm is a higherlevel procedure or heuristic
designed to find, generate, or select a heuristic (partial search
algorithm) that may provide a sufficiently good solution to an
optimization problem, especially with incomplete or imperfect
information or limited computation capacity. The ant colony
algorithm is one of these algorithms that has been applied in this
field [4].
Although rigorous algorithms can achieve the optimal solution in 3D
block modelings, they are usually timeconsuming and difficult to
implement, especially when the problem dimensions are large. Unlike
rigorous algorithms, the heuristic and metaheuristic algorithms are
very fast. The most important drawback of the algorithms, as mentioned
before, is the uncertainty of the optimal solution.
Despite the great importance of the ultimate pit limit, there is limited
research on how exactly it is determined. As mentioned, Pana developed
the floating cone algorithm in 1965 [2]. Then, in 2012, Elahi et al.
developed the floating cone II and III algorithms for eliminating the
floating cone problems [5]. Although the presented algorithms
modified the floating cone method, the optimal solution could not
achieve with them. Akbari et al. (2008) determined the ultimate pit limit
considering the metal price uncertainties and using the real option
valuation approach [6]. Also, Underwood and Tolwinski (1998)
determined the ultimate pit limit using a network flow algorithm [7].
Unlike the mentioned research works, most researchers have
examined the effects of changes in economic and environmental
parameters on the final pit limit. For instance, AskariNasab and Awuah
Offei, in 2009, determined the ultimate pit limit with discounted block
values by using the intelligent openpit simulator (IOPS) [8]. In 2011,
Sayadi et al. used ANNs to determine the ultimate pit limit [9].
Dimitrakopoulos (2011) explained the importance of uncertainties in
open pit design [10]. The ecological costs of optimizing the ultimate pit
outline in openpit metal mines have been considered by Xu et al. (2014)
[11]. Chatterjee et al. (2016) determined the production phase and the
ultimate pit limit under the uncertainty of commodity price [12].
Richmond (2018), considering the importance of uncertainty in open
pit design, integrated a Monte Carlobased simulation and the heuristic
optimization techniques into a global system that directly provides NPV
optimal pit outlines [13]. In 2018, Burgarelli et al. developed a new
approach based on Brownian motion for direct block scheduling under
marketing uncertainties [14]. Rahimi et al. (2018) presented a new
algorithm to optimize mine production planning, mined material
destination, and ultimate pit limit [15]. Adibi et al. (2018) used the
Technique for Order of Preference by Similarity to Ideal Solution
(TOPSIS) method to design the ultimate pit limit by considering the
sustainable development parameters [16]. Saleki et al. (2019) found a
mathematical relationship between the ultimate pit limits, which are
generated by discounted and undiscounted block value maximization in
openpit mining [17].
Article History:
Received: 18 January 2020,
Revised: 15 April 2020,
Accepted: 25 April 2020.
M. Sadeghi et al. / Int. J. Min. & GeoEng. (IJMGE), 551 (2021)
In this paper, a heuristic flashlight algorithm has been developed to
determine the ultimate pit limit. This developed algorithm has a high
speed and accuracy and achieves the optimum solution in all problems
based on programming logic. Unlike other traditional heuristic
algorithms that only consider the ultimate limit value in the last step,
this algorithm optimizes the solution at all steps by checking the
ultimate limit value.
2. Ultimate limit determination methods
Due to the importance of the ultimate pit limit, several most common
methods have been introduced and discussed in this section.
2.1.
Dynamic programming
This method was first introduced in 1965 by Lerchs and Grossmann
as the optimal design of openpit mines. This is a rigorous method that
finds the optimum solution only in 2D mode. This method examines the
final solution using numerical calculations and graphical solutions. This
algorithm is mainly based on determining the critical path.
2.2.
Floating cone
Pana introduced this algorithm in 1965. The algorithm includes a cone
with the blocks of positive economic values in a heuristic manner. Since
each mineral block should pay back the mining costs of the uppermost
level of tailing blocks, thus only the cones consisting of mineral blocks
and tailing of positive values can be mined. The floating cone method is
one of the easiest and fastest algorithms that can only determine the
simple, optimum solution, nonoverlying examples, and in most cases,
it contains errors, but it is still used for the rules of thumb due to the
simplicity. One of the drawbacks of this method is the overlying and
formation of larger cones with fewer profits. To eliminate the defects of
this method, the floating cone II and floating cone III algorithms were
presented [5].
Floating cone II works in a way that a cone is formed in each row for
each positive number, and the cones are arranged in a descending order.
It then mines the cones from large to small and plots a diagram of the
changes in the ultimate limit value. Then, according to the diagram, it
mines the cones until the value of the ultimate limit is increasing, and
this process is repeated for all the rows. Floating cone III has the same
steps as floating cone II, except that all positive numbers are checked at
once instead of searching in a row. Despite the improvement in the final
solution, these methods still fail to introduce the optimum solution.
2.3.
Korobov algorithm
Korobov first introduced this method in 1974. This algorithm also acts
in a heuristic manner, and by finding the positive blocks, assigns the
value of the blocks to the upper negative blocks to finally determine
where the optimum limit is located. Since the assignment of positive
values to negative blocks does not follow a specific trend, the Korobov
algorithm is also stuck in the local optimum. As a result, in some cases,
the solution obtained by this algorithm is far away from the optimum
solution.
3. Flashlight algorithm
Due to the importance of the ultimate pit limit designing and the
mentioned problems in different algorithms, the flashlight (FL)
algorithm has been developed in the present paper as a new method.
This algorithm is based on the fact that the designer determines the
parameters in the pit bottom blocks, and other parts of the pit are
determined based on the conditions of the wall slope and the
topography of the area.
Therefore, the singlebottom or multibottom pit should always place
in the orebody, because further wastes extraction is not economical.
Now, the question arises as to what positive block can be used as the
bottom or within the optimum ultimate limit?
The solution to this question largely depends on the objective
function used in the problem. In other words, the pit should be designed
to maximize the final value of the existing blocks. The mining of each
block requires the extracting of a cone from other blocks on top of the
block, and the block can place within the optimal limit only when its
mining can help improve the final value and does not reduce the value.
Generally, the decision should be made according to the flowchart
shown in figure 1 for the feasibility study of block mining.
Figure 1: Decision making on extracting the block
The FL algorithm considers that mining a positive block and its upper
cone is only economical when it ultimately improves the solution. The
negative blocks cannot be mined on their own unless they are in the
cone, which is above a positive block, and the value of the positive block
is high enough to continue to be economical despite the presence of
upper negative blocks. Since FL is a holistic algorithm, the probability
of getting stuck in the local optimum is greatly reduced. Besides,
checking the final solution in each run of the algorithm will make the
optimal solution accessible. However, in the componentbased
algorithms, such as floating cone or Korobov, either the mining of
blocks with positive value is ignored because of the overlying of the
cones or the final value will be greater than the actual value due to the
failure to detect effective negative blocks. For solving this problem, all
positive blocks in this algorithm are found in the first step and placed in
the ultimate limit. Then, each block will be determined from bottom to
top and from right to left (or left to right) whether its mining helps
improve the final solution.
In this method, the optimum limit is determined using a hypothetical
flashlight. At the end of the process, all blocks that have been lit up by
the flashlight are considered as the ultimate pit limit. The positive blocks
in this algorithm can be placed in two groups, i.e., blacklist and white
list. The blacklist blocks cannot take either the main or replaced
flashlight. The white list blocks can take the main or replaced flashlights,
but for reducing the running time, they will not operate during the
process.
3.1.
Steps of the Flashlight algorithm
The steps to determine the ultimate limit in openpit mines by the
flashlight algorithm are as follows.
1. Start
2. From bottom to top and from right to left (or left to right), the
flashlights on all positive blocks are placed in the dark.
3. While the algorithm does not reach the last block in the top right
(or top left), steps 4 to 16 will be continued:
4. All numbers in light illumination are summed and recorded in the
UPL
total
.
5. Move from bottom to top and from right to left (or left to right)
to find the positive block.
6. If the block has the main flashlight, go to 7 otherwise go to 8.
7. Turn off the main flashlight and from the bottom to top and from
right to left (or left to right) and put the replaced flashlight on
M. Sadeghi et al. / Int. J. Min. & GeoEng. (IJMGE), 551 (2021)
the blocks, which are in the dark and are not on the blacklist, then
go to 9.
8. If the block is in the white list, go to 3; otherwise, turn off all the
flashlights that have illuminated the mentioned block; give the
replace flashlights to the all positive blocks which are in the dark
and are not in the blacklist from right to left (or left to right).
9. The sum of the value in the light is placed in the UPL rep1,
and
the counter i=0 and j is the number of replacing flashlights.
10. if i<j, go to 11; otherwise, go to 3.
11. if UPL total<UPL rep1, turn off the main flashlight and put it in
the blacklist. Change the replace flashlights to the main and go to
3.
12. From the bottom to top and from the right to left move to the
next replaced flashlight that is not on the white list.
13. Turn off the replaced flashlight and put the sum of values in the
light area in the UPL rep2.
14. If the UPL rep1 <= the UPL rep2
,
place the replaced flashlight in
the blacklist; otherwise, placed it in the white list.
15. Turn on the replaced flashlight again.
16. i=i+1 and go to 10.
17. The sum of the illuminated area value is the ultimate pit value.
18. End
In general, two constraints should be considered to obtain the
appropriate results in the FL algorithm.
1. Include the positive blocks that increase the value of the ultimate
limit.
2. Exclude the positive blocks that, if fall in the ultimate limit,
decrease the limit value and may even make the pit
uneconomical.
If an algorithm comprehensively covers the two objectives of the
mentioned constraints, this algorithm determines the optimum ultimate
limit. The FL algorithm considers objective 1 in step 2, as it starts by
examining all positive numbers and considers objective 2 in the steps 5
and 6, so that if a positive block decreases the ultimate limit value, it is
removed from the calculation.
The advantage of the FL algorithm over other common mentioned
algorithms is that it examines the ultimate limit value at each step and
improves the final solution at each step with an overall look, but other
algorithms such as Korobov and floating cone calculate the final value
after going through the defined steps.
A hypothetical example is presented for the implementation steps of
the FL algorithm in figure 2. It should be noted that the vertical
movement in this algorithm must be from bottom to top while there is
no restriction on moving horizontally, i.e., from right to left or left to
right. Although the movements in this example were supposed to be
from bottom to top and from left to right, if the horizontal movement
had been done in the opposite direction, from right to left, the final
results would not have changed.
Figure (2.A) shows a hypothetical economic block model. In the first
step, to find the UPL, the algorithm moves from the bottom left to the
top right and illuminates a major flashlight in the block if it encounters
a positive block in the dark.
As shown in figure (2.B), once a flashlight is placed in the block (4,4),
there is no other positive block in the dark, and thus, the illuminated
area is a cone whose apex is the block (4,4), and the value of this cone is
+12. This value is set to the UPL total. The algorithm then moves again
from the bottom left to the top right and turns off the block (4,4), which
is the first major flashlight after reaching the block and provides an
alternative flashlight to all positive numbers going into the dark.
As depicted in figure (2.C), two replaced flashlights are placed on the
blocks (3,3) and (3,5). As there are two replaced flashlights, the
algorithm can just check these flashlights twice. The illuminated area
has a value of 5, which is set to UPL rep1, as presented in stage 1 in
Table 1. By comparing the numerical values of UPL total and UPL rep1,
it was found that the illuminated area B has a more economical value
than area C. Therefore, the algorithm will check the replaced flashlights
individually.
The algorithm moves from the bottom left to the top right, and as it
is shown in figure (2.D), the flashlight on block (3,3) is turned off. As it
is presented in Table 2 stage1, UPL rep2 is calculated as 0. After
comparing UPL rep1 and UPL rep2, it is obvious that turning off this
flashlight is better, and block (3,3) is moved to the blacklist. The
replaced flashlight is turned on again and the algorithm moves to the
next positive block. As it is shown in Figure 2.E, replaced flashlight on
block (3,5) is turned off. In Table (1) stage 2, the value of UPL rep2 is
calculated as +9. By comparing this amount with UPL rep1, it was found
out that turning off this flashlight would increase the value, and
therefore this block moves to the blacklist.
After checking all possible positive blocks, the algorithm again
surveys block (4,4) and turns it on, as shown in figure (2.F). In the next
step, the algorithm turns off flashlight on the block (4,4), and because
of that, blocks (3,3) and (3,5) are in the blacklist and cannot accept the
replaced flashlight.
As found in figure (2.G), the only block that can accept the flashlight
is (2,4). Based on the algorithm principles, this block can only check one
time. With comparing UPL rep1 (+14) and UPL total (+12) in this stage,
it is concluded that the value of the pit is increased (Table 1 stage 3).
Therefore, the main flashlight is turned off and added to the blacklist.
The replaced flashlight is changed to the main flashlight, and UPL rep1
changed to UPL total. As seen in figure (2.H), the main flashlight is
turned off again. It is concluded from Table 1 stage 4 that UPL rep1 is
+1, and it is less than UPL total. For surveying the replaced flashlight, if
this flashlight is turned off, the value of UPL rep2 is 0, and therefore,
this block moves to the white list (Table 1 stage 4).
As block (1,5) is in the white list, the condition of this block will not
be checked again. As represented in figure (2.I), the best result is +14,
and the apex of the cone is placed on the block (2,5).
Table 1: Stages of the FL algorithm
Parameter
total
UPL
rep1
UPL
Replace
FL
Counter
rep2
UPL
BlackList
Separated By
Comma
White List
Separated By
Comma
Stage 1
+12
5
+2
0
(3,3)

Stage 2
+12
5
+2
+9
(3,3),(3,5)

Stage 3
+14



(3,3),(3,5),(4,4)

Stage 4
+14
+1
+1
0
(3,3),(3,5),(4,4)
(1,5)
4. Numerical analysis
In order to compare the FL algorithm with other heuristic algorithms,
different scenarios are discussed in this section, where previous
algorithms are to be unable to compute the optimum limit.
4.1.
Floating cone algorithm
This algorithm has two major drawbacks: overlying and the formation
of a larger cone with less profit. Figure (2.A) shows the hypothetical
block model. Along with the FL algorithm, some various algorithms,
including dynamic programming, floating cone, and Korobov, have also
been used to determine the UPL in the hypothetical block model. The
results have been given in Table 2.
A) Overlying
Overlying occurs when a group of negative blocks is shared between
two cones, so that none of the cones have a positive value on their own,
but if the two cones are formed at the same time, the total value of limit
will be positive. According to figures (3.B) and (3.C), the floating cone
algorithm forms a cone after reaching block (3.3) whose final value is 1
and does not mine the cone. Like conditions occur for the block (3.4)
and this cone is not mined. However, as shown in figure (3.D), the
optimum ultimate limit is the combination of the two cones, and the
floating cone algorithm cannot achieve this result.
The results from the study of this example by the floating cone,
Korobov, dynamic programming, and the FL algorithms are presented
in Table 2.
M. Sadeghi et al. / Int. J. Min. & GeoEng. (IJMGE), 551 (2021)
Figure 2: Example steps of the FL algorithm
Figure 3: Overlying problem in the floating cone algorithm
Table 2: Results of the UPL from different algorithms in overlying problem
Algorithm
Korbov
Floating cone
Dynamic programming
Flashlight
Pit value
+3
0
+3
+3
B) Extending the ultimate pit beyond the optimal pit limits
According to figure 4, the economic block model specified in section
A is solved by the floating cone algorithm as the first cone is mined
according to figure (4.B), whose value is + 2, and then the second cone
in figure (4.C) is mined, and the final value of the pit is +1. As seen, the
floating cone introduces a larger limit with a lower final value as the
ultimate limit, which is the main disadvantage of this method. The
results of the evaluation of this example with different algorithms are
presented in Table 3.
Figure 4: Formation of a larger cone with lower profit by floating cone algorithm
Table 3: Results of different algorithms in extending the ultimate pit beyond the
optimal pit limits problem
Algorithm
Korbov
Floating cone
Dynamic programming
Flashlight
Pit value
+2
+1
+2
+2
4.2.
The Korobov algorithm
Since the process of assigning positive blocks to negative blocks does
not follow a particular trend, the Korobov algorithm is unable to
determine the optimum limit in some cases. For instance, the results
revealed that the Korobov algorithm could not find the ultimate limit in
the hypothetical block model of Figure (5.A).
Figure 5: The Korobov algorithm error
As depicted in Figure (5.A), there is no optimum limit in this block
model. However, after reaching block (3.3), the value of this block was
given with the Korobov algorithm to blocks (2.3), (2.4), and (1.3).
Eventually, the specified limit in Figure (5.B) is defined as the ultimate
limit with a value of +3. The results from the study of this example with
different algorithms are presented in Table 4.
Table 4: Results from different algorithms in the Korbov problem
Algorithm
Korbov
Floating cone
Dynamic programming
Flashlight
Pit value
+3



4.3.
Case study
The related data of the Korkora skarn iron ore mine, which is located
in the Kurdistan province, northwestern Iran, has been used to evaluate
the performance of the FL algorithm. The geographical location of the
mine is shown in Figure 6.
The mine is a part of the SanandajSirjan belt and is located in a
mountainous area, about 2540 m above the sea level. The Korkora ore
body has a length of 758 m, with a width ranging from 64 to 349 m, and
a thickness ranging from 7 to 37 m. The slope of the iron horizon is about
1011˚ to the south. The mineral deposit is hosted by a rhyolite unit,
located in the center, with a fault of a steep slope. The oldest rocks in
the Korkora mine are the Cretaceous limestones. Magnetite and
hematite are the main products of the mineralization. The total reserve
M. Sadeghi et al. / Int. J. Min. & GeoEng. (IJMGE), 551 (2021)
of the mine is about 7 million tons, with an average grade of 52%
Magnetite. The ore body is mined through openpit mining. The height
and slope of the benches are 10 m and 90°, respectively.
Figure 6: Geographical location of Korkora mine
In this paper, the crosssection consisting of 3080 blocks of this limit
was explored using different methods to evaluate the accuracy and
validity of the FL algorithm compared to other heuristic and rigorous
algorithms. A Matlab code was developed for the implementation of the
FL algorithm. Table 4 contains four different values that represent the
UPL obtained by applying the floating cone, Korobov, dynamic
programming, and FL algorithms, respectively.
Since the dynamic programming algorithm always shows the
optimum ultimate limit in 2D mode and comparing the sections A, B,
C, and D, it is found that the Korobov and the floating cone algorithms
were unable to find the optimum twobottom limit obtained in section
C by the dynamic programming algorithm.The FL algorithm, however,
managed to escape the local optima and reach the optimum ultimate
limit. The values of the limits are specified in Table 5.
Table 5: Results from different algorithms in Korkora crosssection (Value ×
10000 Rials)
Algorithm
Korbov
Floating cone
Dynamic programming
Flashlight
Pit value
+38260
+36254
+48354
+48354
As seen, the numerical value obtained from the floating cone
algorithm is much lower than the value obtained by the dynamic
programming algorithm. However, the FL algorithm determines a value
exactly equal to the optimum value obtained by the dynamic
programming algorithm.
5. Conclusions
The methods applied to determine the UPL are divided into three
general groups: rigorous, heuristic, and metaheuristic. The
disadvantages of rigorous methods include low calculation speed and
long data processing time. Although the heuristic methods have faster
calculation speeds, they are getting stuck in local optimums and cannot
specify the optimum limit. In this paper, the FL algorithm was
introduced to overcome these shortcomings, and the following results
were obtained:
1. The FL algorithm is capable of solving the problems of the well
known heuristic algorithms such as the floating cone and the Korobov,
and in the case where these algorithms were unable to determine the
optimal ultimate limit, the FL algorithm would give the optimum
solution.
2. The FL is based on the movement from bottom to top, and unlike
other heuristic algorithms, it examines the total values of the limit in
each step of the algorithm to finally obtain the optimum value.
3. Various methods determined the ultimate limit of Korkora mine,
and ultimately it was found that the value obtained from the FL
algorithm is equal to the optimum value obtained from the dynamic
programming method, which indicated the high accuracy of this
algorithm. However, the Korobov and floating cone algorithms failed to
determine the optimum limit.
4. The proposed algorithm can be used in a variety of software for
designing of the UPL.
REFERENCES
[1] Lerchs, H., Grossmann., I.F. (1964). Optimum Design of OpenPit
Mines.
Joint C.O.R.S. and O.R.S.A. Conference
,
Montreal
.
[2] Pana, M.T. (1965). The simulation approach to openpit design.
5th APCOM, Tucson Arizona
,
Tucson Arizona
.
[3] Korobov, S. (1974). Method for determining optimal open pit
limits (Montreal: Ecole Polytechnique de l’Université de
Montréal. Technical report EP74.
[4] Shishvan, M. S., & Sattarvand, J. (2015). Long term production
planning of open pit mines by ant colony optimization.
European Journal of Operational Research, 240(3), 825836.
[5] Elahi, E., Kakaie, R., & Yusefi, A. (2012). A new algorithm for
optimum open pit design: Floating cone method III. Journal of
Mining and environment, 2(2), 118125.
[6]. Akbari, A. D., OSANLOU, M., & Shirazi, M. A. (2008).
Determination of ultimate pit limits in open mines using real
option approach. INTERNATIONAL JOURNAL OF
ENGINEERING SCIENCE, 19, 2338.
[7]. Underwood, R., & Tolwinski, B. (1998). A mathematical
programming viewpoint for solving the ultimate pit
problem. European Journal of Operational Research, 107(1), 96
107.
[8] AskariNasab, H., & AwuahOffei, K. (2009). Open pit
optimisation using discounted economic block values. Mining
Technology, 118(1), 112.
[9] Sayadi, A. R., Fathianpour, N., & Mousavi, A. A. (2011). Open pit
optimization in 3D using a new artificial neural network.
Archives of Mining Sciences, 56(3), 389403.
[10] Dimitrakopoulos, R. (2011). Stochastic optimization for strategic
mine planning: a decade of developments. Journal of Mining
Science, 47(2), 138150.
[11] Xu, X. C., Gu, X. W., Qing, W. A. N. G., Liu, J. P., & Jun, W. A. N.
G. (2014). Ultimate pit optimization with ecological cost for
open pit metal mines. Transactions of Nonferrous Metals
Society of China, 24(5), 15311537
[12] Chatterjee, S., Sethi, M. R., & Asad, M. W. A. (2016). Production
phase and ultimate pit limit design under commodity price
uncertainty. European Journal of Operational Research, 248(2),
658667.
[13] Richmond, A. (2018). Direct net present value open pit
optimization with probabilistic models. In Advances in Applied
Strategic Mine Planning (pp. 217228). Springer, Cham.
[14] Burgarelli, H. R., Souza, F. R., Nader, A. S., Torres, V. F. N.,
Câmara, T. R., Ortiz, C. E. A., & Galery, R. (2018). Direct block
scheduling under marketing uncertainties. REMInternational
Engineering Journal, 71(2), 275280.
[15] Rahimi, E., Moosavi, E., Shirinabadi, R. & Gholinejad, M. (2018).
Optimized algorithm in mine production planning, mined
material destination, and ultimate pit limit. Journal of Central
South University. 25(6), 14751488.
M. Sadeghi et al. / Int. J. Min. & GeoEng. (IJMGE), 551 (2021)
[16] Adibi, N., Ataeepour, M., Rahmanpour, M. (2015). Integration
of sustainable development concepts in open pit mine design.
Journal of Cleaner Production, 108, 10371049.
[17] Saleki, M., Kakaie, R., & Ataei, M. (2019). Mathematical
relationship between ultimate pit limits generated by
discounted and undiscounted block value maximization in open
pit mining. Journal of Sustainable Mining, 18(2), 9499.