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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 cross-section 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.
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Ultimate Pit Limit Determination Using Flashlight Algorithm
Mohammadreza Sadeghi a, Hesam Dehghani a, Behshad Jodeiri Shokri a, *
Department of Mining Engineering, Hamedan University of Technology (HUT), Hamedan, Iran
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 cross-section 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 open-pit 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 meta-heuristic algorithm is a higher-level 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 time-consuming and difficult to
implement, especially when the problem dimensions are large. Unlike
rigorous algorithms, the heuristic and meta-heuristic 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, Askari-Nasab and Awuah-
Offei, in 2009, determined the ultimate pit limit with discounted block
values by using the intelligent open-pit 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 open-pit 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 Carlo-based 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
open-pit mining [17].
Article History:
Received: 18 January 2020,
Revised: 15 April 2020,
Accepted: 25 April 2020.
M. Sadeghi et al. / Int. J. Min. & Geo-Eng. (IJMGE), 55-1 (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.
Dynamic programming
This method was first introduced in 1965 by Lerchs and Grossmann
as the optimal design of open-pit 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.
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, non-overlying 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.
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
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 single-bottom or multi-bottom 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 component-based
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
Steps of the Flashlight algorithm
The steps to determine the ultimate limit in open-pit 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
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. & Geo-Eng. (IJMGE), 55-1 (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,
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
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
2. Exclude the positive blocks that, if fall in the ultimate limit,
decrease the limit value and may even make the pit
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
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
White List
Separated By
Stage 1
Stage 2
Stage 3
Stage 4
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.
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. & Geo-Eng. (IJMGE), 55-1 (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
Floating cone
Dynamic programming
Pit value
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
Floating cone
Dynamic programming
Pit value
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
Floating cone
Dynamic programming
Pit value
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 Sanandaj-Sirjan 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
10-11˚ 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. & Geo-Eng. (IJMGE), 55-1 (2021)
of the mine is about 7 million tons, with an average grade of 52%
Magnetite. The ore body is mined through open-pit 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 cross-section 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 two-bottom 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 cross-section (Value ×
10000 Rials)
Floating cone
Dynamic programming
Pit value
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 meta-heuristic. 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
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.
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... A hypothetical flashlight is applied, in which all blocks that were illuminated determine the ultimate pit limit for an iteration, then the positive blocks are evaluated and placed on a blocklist or in a whitelist, the first consisting of blocks that must be forgotten, while the second groups the blocks that can be in the main or replaced flashlights. An application in a cross section of an iron ore deposit showed that the flashlight heuristic reached the same value as a rigorous approach (the dynamic programming algorithm discussed below), obtaining great improvements compared to the floating cone and Korobov's algorithms [17,[28][29][30][31][32]. ...
Planning is one of the fundamental tasks for surface mine projects, and it can play a critical role in the success and results obtained after the development and operationalization of the project. This role will be more important when the miners need to go deeper to access the ore, especially when its grade tends to decrease. The quality of the surface mine planning decision depends on the available data, information, and experience of the decision makers, being imperative that robust decision criteria, new technologies and advanced analytics tools are employed in order to make the most of the huge amount of data generated at each instant. This chapter focuses on advanced analytics approaches such as machine learning and artificial intelligence as valuable decision-making tools to improve the surface mine planning process, as well as mathematical optimization techniques already successfully employed in solving ultimate pit and production scheduling problems, using rigorous algorithms, heuristics and metaheuristics. This chapter also covers stochastic mine planning and discusses future mine planning.
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This work attempts to estimate the amount of fly-rock in the Angoran mine in the Zanjan province (Iran) using the gene expression programming (GEP) predictive technique. For this, the input data including the fly-rock, mean depth of the hole, powder factor, stemming, explosive weight, number of holes, and booster is collected from the mine. Then using GEP, a series of intelligent equations are proposed in order to predict the fly-rock distance. The best GEP equation is selected based on some well-established statistical indices in the next stage. The coefficient of determination for the training and testing datasets of the GEP equation are 0.890 and 0.798, respectively. The model obtained from the GEP method is then optimized using the teaching-learning-based optimization (TLBO) algorithm. Based on the results obtained, the correlation coefficient of the training and testing data increase to 91% and 89%, which increases the accuracy of the equation. This new intelligent equation could forecast fly-rock resulting from mine blasting with a high level of accuracy. The capabilities of this intelligent technique could be further extended to the other blasting environmental issues. Keywords Blasting operations Fly-rock Gene expression programming Teaching-learning-based optimization algorithm
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Ultimate pit limit is an important aspect of open pit mining. The optimal ultimate outline determines the tonnage of extractable ore, the volume of waste to be removed, the location of the subsidiary facilities, the location of ore and waste stockpiles, the life time of the mine and the estimated net present value (NPV) of the entire mining operation. Traditionally, there are two major approaches to optimizing the ultimate pit limit. One seeks to determine the ultimate pit using undiscounted profit maximization and the other by determining the optimal mining sequence of all blocks and discounting the value of the blocks. The outline with the highest cumulative NPV will be chosen as the final pit limit. For each of these approaches, different algorithms are presented. The aim of this paper is to present an analytical investigation about the mathematical relationship between sets of blocks of ultimate pits generated by each of these approaches in an ore body. This investigation is in fact the mathematical proof of the theorem that a discounted ultimate pit is smaller than or equal to the undiscounted pit. The results show that the discounted pit is always a subset of the undiscounted pit.
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An integral connection exists among the mine production planning, the mined material destination, and the ultimate pit limit (UPL) in the mining engineering economy. This relation is reinforced by real information and the benefits it engenders in the mining economy. Hence, it is important to create optimizing algorithms to reduce the errors of economic calculations. In this work, a logical mathematical algorithm that considers the important designing parameters and the mining economy is proposed. This algorithm creates an optimizing repetitive process among different designing constituents and directs them into the maximum amount of the mine economical parameters. This process will produce the highest amount of ores and the highest degree of safety. The modeling produces a new relation between the concept of the cutoff grade, mine designing, and mine planning, and it provides the maximum benefit by calculating the destination of the ores. The proposed algorithm is evaluated in a real case study. The results show that the net present value of the mine production is increased by 3% compared to previous methods of production design and UPL.
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Mineral projects are composed of geological, operational and market uncertainties, and reducing these uncertainties is one of the objectives of engineering. Most surveys assess the impact of geological and operational uncertainties on the mining planning. The objective of this work is to study the impact of market uncertainty on the mineral activity. The influence of iron ore price simulation on mining sequencing will be evaluated. The price of iron ore has random behavior that is best represented by the Geometric Brownian Movement system. This study analyzed the historical series of iron ore in order to determine the percentage volatility and drift. Traditionally, a constant and deterministic price is used for the ore mined in all periods of a mineral project. The direct block scheduling methodology was adopted because it is able to apply the appropriate financial discount factor to the simulated probabilistic price. The proposed methodology was able to quantify the market uncertainty.
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Planning and design procedure of an open pit mining project just can be started after ultimate pit determination. In the carried out study in this paper it was shown that the most important factor in ultimate pit determination and in consequence in the whole planning and design procedure of an open pit mine is the metal price. Metal price fluctuations in recent years were exaggerated and imposed a high degree of uncertainty to the mine planning procedure while none of the existent algorithms of the pit limit determination consider the metal price uncertainty. Real Option Approach (ROA) is an efficient method of decision making in the condition of uncertainty. This approach usually used for evaluation of defined natural resources projects up to now. This study considering the metal price uncertainty used real option approach to prepare a methodology for determining the Ultimate Pit Limits (UPL). The study was carried out on a non-ferrous metallic cylindrical ore deposit but the achieved methodology can be adjusted for all kinds of the deposits. The achieved methodology was comprehensively described through the examples in a way that can be used by the mine planners.
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Determination of the pit limits is a vital task in the designing process of a mine. While overestimation increases stripping ratio, underestimation can result in loss of economic ore. The available methods of pit limit optimization are mostly based on the programming approaches (linear and/ or dynamic) and graph theory. In the large scale mines with probably varying wall slopes especially in 3D states performance of these approaches may be unsatisfactory. In this paper, a new 3D pit optimization algorithm has been developed and applied in an economic classified block model. In the first step economic block model of the Esfordi phosphate mine (Iran) was prepared. In the second step, the block model was divided into positive and negative regions with the help of a well trained feed-forward back-propagation neural network. For this, 3D blocks coordinates were considered as the neural network inputs whereas the blocks net economic values were set to be the model output. Finally, the optimum pit limit was determined on the classified block model considering constraints of wall slopes and impurities. This process was set to run in the MATLAB software. The results obtained have been compared to that from the modified Lerchs-Grossman's algorithm. The results are the same but in the presence of some kinds of ore impurity constraints the proposed algorithm generates a pit with more profit.
Traditional implementations of open pit optimisation algorithms are designed simply to find a set of nested open pit limits that maximise the undiscounted financial pay-off for a series of commodity prices using a single ‘estimated’ orebody model. Then, the maximum Net Present Value (NPV) open pit limit is derived by considering alternate (usually only best and worst-case) mining schedules for each open pit limit. Divorcing the open pit limit delineation from the NPV calculation in this two-step approach does not guarantee that an optimal NPV open pit solution will be found. A new open pit optimisation algorithm that considers the mining schedule is proposed. As a consequence, it can also account explicitly for commodity price cycles and uncertainty that can be modelled by stochastic simulation techniques. This state-of-the-art algorithm integrates Monte Carlo-based simulation and heuristic optimisation techniques into a global system that directly provides NPV optimal pit outlines. This new approach to open pit optimisation is demonstrated for a large copper deposit using multiple orebody models.
The Ultimate Pit Limit (UPL) design is an important step in open pit mining. The objective function for UPL determination is usually maximization of profit (economic aspect) that satisfies the mine owners and shareholders. However, from the viewpoint of Sustainable Development (SD), there are three aspects for mining activities including: environmental, social and economic. All aspects of SD must be properly considered during the UPL design. In addition to the mine owner, mining has other stakeholders that the mine designer must consider their interests and concerns. Integration of SD aspects in UPL design is a challenge and there is not any comprehensive method in this regard. This paper describes an attempt to address this challenge by developing a methodology using mathematical optimization and SD indicators. According to this model, it is possible to integrate the SD principles in UPL design. In that regard, the UPL problem is converted into a Multiple Attribute Decision Making (MADM) problem. Then the best solution is found by the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method. The proposed model is applied in a copper mine as a case study. The research findings show that the UPL selection based on SD concepts may lead to a larger pit than traditional method (profit optimization). It also contains greater ore and lower profit than the traditional UPL.
Open pit mine design optimization under uncertainty is one of the most critical and challenging tasks in the mine planning process. This paper describes the implementation of a minimum cut network flow algorithm for the optimal production phase and ultimate pit limit design under commodity price or market uncertainty. A new smoothing splines algorithm with sequential Gaussian simulation generates multiple commodity price scenarios, and a computationally efficient stochastic framework accommodates the joint representation and processing of the mining block economic values that result from these commodity price scenarios. A case study at an existing iron mining operation demonstrates the performance of the proposed method, and a comparison with conventional deterministic approach shows a higher cumulative metal production coupled with a 48% increase in the net present value (NPV) of the operation.
The problem of long-term production planning of open pit mines is a large combinatorial problem. Application of mathematical programming approaches suffer from reduced computational efficiency due to the large amount of decision variables. This paper presents a new metaheuristic approximation approach based on the Ant Colony Optimization (ACO) for the solution of the problem of open-pit mine production planning. It is a three-dimensional optimization procedure which has the capability of considering any type of objective function, non-linear constraints and real technical restrictions. The proposed process is programmed and tested through its application on a real scale Copper–Gold deposit. The study revealed that the ACO approach is capable to improve the value of the initial mining schedule regarding the current commercial tools considering penalties and without, in a reasonable computational time. Several variants of ACO were examined to find the most compatible variants and the best parameter ranges. Results indicated that the Max–Min Ant System (MMAS) and the Ant Colony System (ACS) are the best possible variants based on the required less amount of memory. It is also proved that the MMAS is the most explorative variant, while the ACS is the fastest method.
The ecological costs of open pit metal mining are quantified, which include lost value of direct eco-services, lost value of indirect eco-services, prevention and restoration costs, and cost of carbon emission from energy consumption. These ecological costs are incorporated in an iterative ultimate pit optimization algorithm. A case study is presented to demonstrate the influence of ecological costs on pit design outcome. The results show that it is possible to internalize ecological costs in mine designs. The pit optimization outcome shifts considerably to the conservative side and the profitability decreases substantially when ecological costs are accounted for.