Improving performance of open-pit mine production scheduling problem under grade uncertainty by hybrid algorithms

Abstract

One of the surface mining methods is open-pit mining, by which a pit is dug to extract ore or waste downwards from the earth’s surface. In the mining industry, one of the most significant difficulties is long-term production scheduling (LTPS) of the open-pit mines. Deterministic and uncertainty-based approaches are identified as the main strategies, which have been widely used to cope with this problem. Within the last few years, many researchers have highly considered a new computational type, which is less costly, i.e., meta-heuristic methods, so as to solve the mine design and production scheduling problem. Although the optimality of the final solution cannot be guaranteed, they are able to produce sufficiently good solutions with relatively less computational costs. In the present paper, two hybrid models between augmented Lagrangian relaxation (ALR) and a particle swarm optimization (PSO) and ALR and bat algorithm (BA) are suggested so that the LTPS problem is solved under the condition of grade uncertainty. It is suggested to carry out the ALR method on the LTPS problem to improve its performance and accelerate the convergence. Moreover, the Lagrangian coefficients are updated by using PSO and BA. The presented models have been compared with the outcomes of the ALR-genetic algorithm, the ALR-traditional sub-gradient method, and the conventional method without using the Lagrangian approach. The results indicated that the ALR is considered a more efficient approach which can solve a large-scale problem and make a valid solution. Hence, it is more effectual than the conventional method. Furthermore, the time and cost of computation are diminished by the proposed hybrid strategies. The CPU time using the ALR-BA method is about 7.4% higher than the ALR-PSO approach.

Full text

J. Cent. South Univ. (2020) 27: 2479−2493
DOI: https://doi.org/10.1007/s11771-020-4474-z
Improving performance of open-pit mine production scheduling
problem under grade uncertainty by hybrid algorithms
Kamyar TOLOUEI1, Ehsan MOOSAVI1, Amir Hossein BANGIAN TABRIZI1,
Peyman AFZAL1, Abbas AGHAJANI BAZZAZI2
1. Department of Petroleum and Mining Engineering, South Tehran Branch, Islamic Azad University,
Tehran, Iran;
2. Department of Mining Engineering, University of Kashan, Kashan, Iran
© Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: One of the surface mining methods is open-pit mining, by which a pit is dug to extract ore or waste
downwards from the earth’s surface. In the mining industry, one of the most significant difficulties is long-term
production scheduling (LTPS) of the open-pit mines. Deterministic and uncertainty-based approaches are identified as
the main strategies, which have been widely used to cope with this problem. Within the last few years, many researchers
have highly considered a new computational type, which is less costly, i.e., meta-heuristic methods, so as to solve the
mine design and production scheduling problem. Although the optimality of the final solution cannot be guaranteed,
they are able to produce sufficiently good solutions with relatively less computational costs. In the present paper, two
hybrid models between augmented Lagrangian relaxation (ALR) and a particle swarm optimization (PSO) and ALR and
bat algorithm (BA) are suggested so that the LTPS problem is solved under the condition of grade uncertainty. It is
suggested to carry out the ALR method on the LTPS problem to improve its performance and accelerate the
convergence. Moreover, the Lagrangian coefficients are updated by using PSO and BA. The presented models have
been compared with the outcomes of the ALR-genetic algorithm, the ALR-traditional sub-gradient method, and the
conventional method without using the Lagrangian approach. The results indicated that the ALR is considered a more
efficient approach which can solve a large-scale problem and make a valid solution. Hence, it is more effectual than the
conventional method. Furthermore, the time and cost of computation are diminished by the proposed hybrid strategies.
The CPU time using the ALR-BA method is about 7.4% higher than the ALR-PSO approach.
Key words: open-pit mine; long-term production scheduling; grade uncertainty; augmented Lagrangian relaxation;
particle swarm optimization algorithm; bat algorithm
Cite this article as: Kamyar TOLOUEI, Ehsan MOOSAVI, Amir Hossein BANGIAN TABRIZI, Peyman AFZAL,
Abbas AGHAJANI BAZZAZI. Improving performance of open-pit mine production scheduling problem under grade
uncertainty by hybrid algorithms [J]. Journal of Central South University, 2020, 27(9): 2479−2493. DOI:
https://doi.org/10.1007/s11771-020-4474-z.
1 Introduction
The open pit mine is actually a superficial
mining starting with the extraction of ore or waste
from the surface by dint of digging the cavity. The
mining operation is ended when the process
develops with deeper and deeper caverns. Another
foremost step in mining planning is long-term
production scheduling (LTPS) optimization process.
LTPS is especially important in mining projects,
because in addition to determining the actual value
of the project, medium-term and short-term
schedules are planned based on it. Therefore, in the
Received date: 2019-11-28; Accepted date: 2020-07-02
Corresponding author: Ehsan MOOSAVI, PhD, Assistant Professor; Tel: +98-9138375368; E-mail: se.moosavi@yahoo.com,
se_moosavi@azad.ac.ir; ORCID: https://orcid.org/0000-0002-5626-2689

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last decade, special attention has been paid to LTPS,
providing optimal algorithms to obtain the best
schedules by mining engineers. So far, two
categories of deterministic algorithms and
uncertainty-based algorithms for long-term
schedules have been developed. In deterministic
algorithms, all input data to the mathematical model
of production scheduling are assumed to be definite,
and uncertainty is not considered in these
parameters. Thus, the operational constraints and
different expectations of the mine cannot be met
when operating in the mine.
The main reason for this inefficiency is the
uncertainty of the tonnage and grade of the mineral.
In uncertainty-based algorithms, the uncertainty of
the input parameters is considered in the production
scheduling model. LTPS is planned to extract and
process the mineral for a period of more than 10
years. The main purpose of LTPS is to prepare
annual and multi-year development schedules for
all blocks in the mine-life. In fact, the goals of
LTPS are to maximize the project’s net present
value (NPV), minimize the risk of achieving
production targets, and maximize mining life.
The LTPS problem is regarded as a hard
integer programming problem and non-
deterministic polynomial-time (NP) classes. It is
difficult to achieve a suitable solution for the LTPS
problem for the sake of its size and NP-hardness.
Hence, several research efforts emphasize the
efficient LTPS algorithms in order to proliferate
profitability and diminish computational time.
Table 1 illustrates a number of models presented in
recent years.
Ta bl e 1 Review of presented models since 1969
Year Authors Model D1 U2 HMM3 Ref.
1969 JOHNSON Linear programming * [1]
1974 WILLIAMS Dynamic programming, integer programming,
network flow, parametric programming * [2]
1983 GERSHON Linear programming, mixed integer programming * [3]
1986 DAGDELEN and JOHNSON Lagrangian relaxation method * [4]
1992 RAVENSCROFT Conditional simulation * [5]
1994 DOWD Geostatistical simulation * [6]
1995 ELEVLI Operation research, artificial intelligence * [7]
1995 DENBY and SCHOFIELD Genetic algorithm * * [8]
1994 TOLWINSKI Dynamic programming * [9]
1999 AKAIKE and DAGDELEN 4D network relaxation * [10]
2000 WHITTLE Milawa * [11]
2002 JOHNSON et al Mixed integer programming * * [12]
2002 DIMITRAKOPOULOS et al Generalized sequential Gaussian simulation,
direct block simulation * [13]
2003 GODOY and DIMITRAKOPOULOS Simulated annealing algorithm * * [14]
2004 DIMITRAKOPOULOS and RAMAZAN Linear programming * [15]
2004 RAMAZAN and DIMITRAKOPOULOS Mixed integer programming * [16]
2004 RAMAZAN and DIMITRAKOPOULOS Mixed integer programming * [17]
2006 GHOLAMNEJAD et al Chance constrained programming * [18]
2007 GHOLAMNEJAD and OSANLOO Chance constrained integer programming * [19]
2007 RAMAZAN and DIMITRAKOPOULOS Stochastic integer programming * [20]
2009 BOLAND et al Mixed integer programming * [21]
2010 BLEY et al Integer programming * [22]
2010 KUMRAL Robust stochastic optimization * [23]
2012 LAMGHARI and DIMITRAKOPOULOS Tabu search * * [24]
2012 GHOLAMNEJAD and MOOSAVI Binary integer programming * [25]
to be continued

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Continued
Year Authors Model D1 U2 HMM3 Ref.
2013 NANJARI and GOLOSINSKI Dynamic programming, mining heuristic * * [26]
2013 SATTARVAND and NIEMANN-DELIUS Ant colony optimization * * [27]
2013 GOODFELLOW and
DIMITRAKOPOULOS Simulated annealing algorithm * * [28]
2013 DIMITRAKOPOULOS and JEWBALI Stochastic integer programming * [29]
2014 LEITE and DIMITRAKOPOULOS Stochastic integer programming * [30]
2014 MOOSAVI et al Lagrangian relaxation method, genetic algorithm * * [31]
2014 MOOSAVI et al Augmented Lagrangian relaxation method,
genetic algorithm * * [32]
2014 KOUSHAVAND et al Mixed integer linear programming * [33]
2014 ASAD et al Stochastic network flow,
Lagrangian relaxation method * * [34]
2014 LAMGHARI et al Variable neighbourhood descent algorithm * * [35]
2015 SHISHVAN and SATTARVAND Ant colony optimization * * [36]
2016 MOKHTARIAN and SATTARVAND Imperialist competitive algorithm * * [37]
2016 MOKHTARIAN and SATTARVAND
Commodity price distribution function,
median latin hypercube sampling method,
integer programming
* [38]
2016 GOODFELLOW and
DIMITRAKOPOULOS
Simulated annealing algorithm, particle swarm
optimization, differential evolution * * [39]
2016 LAMGHARI and
DIMITRAKOPOULOS
Rockafellar and wets progressive
Hedging algorithm * [40]
2016 LAMGHARI and
DIMITRAKOPOULOS
Tabu search heuristic incorporating a diversification
strategy, variable neighborhood descent heuristic,
very large neighborhood search heuristic,
network flow techniques, diversified local search
* * [41]
2017 BAKHTAVAR et al Stochastic chance-constrained programming * [42]
2018 KHAN Particle swarm optimization, bat algorithm * * [43]
2018 RAHIMI et al Logical mathematical algorithm * * [44]
2018 TAHERNEJAD et al Information gap decision theory * [45]
2018 JELVEZ et al Expected time incremental heuristic algorithm * * [46]
2018 KHAN and ASAD Mixed integer linear programming * [47]
2018 ALIPOUR et al Robust counterpart linear optimization,
genetic algorithm * * [48]
2019 CHATTERJEE and
DIMITRAKOPOULOS
Lagrangian relaxation method, sub-gradient method,
branch and cut algorithm * * [49]
2019 DIMITRAKOPOULOS and SENÉCAL Multi-neighborhood tabu search * * [50]
1 Deterministic, 2 Uncertainty, 3 Heuristic and Meta-Heuristic method.
Despite researchers’ efforts, the LTPS problem
has not been a well-solved problem. Most of the
proposed models have disadvantages such as not
considering the sources of uncertainty, not meeting
operational constraints, generating non-optimal
solutions in high computational time. With
methodological and technological advances,
researchers have the chance to challenge traditional
approaches for computational reasons. This will
lead to the development of new models with
significant features that can enhance the capabilities
of current solutions. The augmented Lagrangian
relaxation method is efficiently used to solve
constrained optimization problems [51−53]. It
mostly exploits the specific decomposable structure
of the initial problem to deal with large-scale hard
problems in different fields, such as production
scheduling optimization problems. To determine the
multiplier values relying on the previous
computation consequences, the authors apply the
sub-gradient (SG) method that is generally utilized.
According to the zigzag phenomenon and small

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steps, the sub-gradient procedure may join
gradually on large problems. The use of this method
has been increased in recent years by incorporating
meta-heuristic algorithms. They can be simply used
for the solution of hard optimization problems and
they are responsible for great modeling flexibility.
The results in different industries demonstrate the
high efficiency of this model. Particle swarm
optimization (PSO) and bat algorithm (BA) are
among the most widely used meta-heuristic
algorithms that have features such as rapid
convergence and the production of optimal
solutions in a logical time for solving various
optimization problems [54−56].
The present paper scrutinizes the use of
meta-heuristic methods in solving the long-term
production scheduling optimization problem in
deterministic and uncertainty conditions. To solve
the LTPS problem under grade uncertainty
condition, we introduce two hybrid models between
the augmented Lagrangian relaxation (ALR)
method and the particle swarm optimization (PSO)
algorithm and similarly between ALR and bat
algorithm (BA). In this research, it is suggested to
enhance the convergence rate by the ALR method
on the LTPS problem. Furthermore, also the PSO
and BA are practiced to bring up to date Lagrange
coefficients. The presented models have been
compared with the outcomes of combining ALR
with genetic algorithm (GA), traditional sub-
gradient method (SG), and the conventional method
without using the Lagrangian approach (Conv.). In
terms of average net present value, average ore
grade, and CPU time, results illustrate that ALR-BA
generates the best outcomes while satisfying
constraints. The results indicate that the advanced
versions have significantly improved in comparison
with the conventional method.
The next part of this paper is provided as
below. According to the condition of grade
uncertainty, the objective functions and their
associated restrictions are modeled in Sections 2
and 3. A summary of the methodology and hybrid
models is divulged in Sections 4 to 7 and the
proposed models will be also advanced. Next, an
assessment of the results is displayed in Section 8.
Moreover, this section includes data collection and
preparation. Authentication of the recognized
models is realized. Finally, Section 9 carries out the
conclusion.
2 Traditional formulation of LTPS
problem
Long-term production scheduling model has
been practiced to project production aims and ore
material current over several years. Holistically, it
takes a basic image of the production and expressed
as a linear problem.
2.1 Objective function
To reflect decision-making determinations, the
simplest way is to signify a complete space
optimization model for each period of the
scheduling horizon. Since the obtainability of
restrains is intermingled into the model, the LTPS
problem is contributed:
1
Maximize = (1+ )
t
NT t
nn
t
nt
NV
Z
X
r
=
´
åå (1)
In the constructed model, the following
indications were recognized: n is the block
identification number, n=1, 2, …, N; N is the total
number of blocks to be scheduled; t is the
scheduling periods index, t=1, 2, …, T; T is the
total number of scheduling periods; t
n
NV is the net
value to be generated by mining block n in period t;
r is the discount rate in each period; t
n
X
is the
binary variable as follows:
1, if is mined during period
=0, otherwise
t
n
nt
Xì
ï
í
ï
î
2.2 Constraints
Constraints are based on Ref. [16] as follows.
2.2.1 Grade blending constraints
In the production scheduling, one of the most
substantial complications is the ore grade which has
to be put to one side steady while directing to the
processing plant. Henceforth, the grade of ore that
is being directed to the mill has to definite between
two limits.
1) Upper bound constraints: It is worth
mentioning that the average grade of the material
led to the mill should be a slighter quantity or
equivalent to the certain grade value, Gmax, for each
period, t:
max
1
() 0
Nt
nnn
n
gG OX
=
-´´£
å (2)

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where gn is the average grade of block n and On is
the ore tonnage in block n.
2) Lower bound constraints: Notably, the
average grade of the material directed to the mill
should be more or the same as the fixed value, Gmax,
for each period, t:
min
1
() 0
Nt
nnn
n
gG OX
=
-´´³
å (3)
2.2.2 Reserve constraints
Basically, restrictions are affected for each
block so as to specify that all measured blocks in
the model cannot be mined more than once.
1
1, =1, 2, 3, ,
Tt
n
t
X
nN
=
£
å (4)
2.2.3 Processing capacity constraint
Total tons of the treated ore should be less than
the processing capacity, PCmax, in every period, t:
max
1
()
Nt
nn
n
OX PC
=
´£
å (5)
2.2.4 Mining capacity constraint
The whole available mining capacity, MCmax,
should be more than the whole quantity of material
(waste and ore) to be mined for each period, t:
max
1
(+)
Nt
nn n
n
OW X MC
=
´£
å (6)
where Wn is the tonnage of waste material within
block n.
2.2.5 Slope constraints
These constraints verify that before mining a
specified block, all of the overlying blocks should
be mined. The following two methods are applied
to implement these constraints:
1) Using one constraint for each block per
period:
11
0, =1, 2, , ,
lt
tr
ky
yr
YX X y l
==
-£
åå 
=1, 2, , , =1, 2, , kNtT
(7)
where k is the index of a block considered
extraction at period t; Y is the total number of
blocks overlying block k; y is the counter for the
Y-overlying blocks.
2) Using Y-constraints for each block at each
period:
1
0, =1, 2, , ,
t
tr
ky
r
X
Xk N
=
-£
å
=1, 2, , tT (8)
3 LTPS model considering grade
uncertainty
3.1 Importance of uncertainty in LTPS
In engineering projects, various sources of
uncertainty complicate decisions. Principally, the
risks associated with a project arise from the
uncertainties in that project, which can affect goals.
Due to the importance of this subject, the
classification of uncertainty sources in mining
projects was provided by DIMITRAKOPOULOS
[57]:
1) The uncertainty of the grade and the
uncertainty of tonnage bring about the uncertainty
of the ore deposit model.
2) Technical uncertainty, such as extraction,
reveals slope constraints, drilling capacity, etc.
Economic uncertainties include capital costs,
operating costs, and product prices.
3) Among the uncertainties, grade uncertainty
directs into a large share of probabilities.
3.2 LTPS model via grade uncertainty
The indicator kriging (IK) is one of the applied
procedure methods for grade estimation in mining
projects. This method was proposed by JOURNEL
[58] to estimate the resources. The indicator method
is binary data encoding related to the cut-off value,
Zc. For the Z(x), ik(x)=1, if Z(x)≥Zc, and otherwise,
ik(x)=0. In fact, it is a nonlinear conversion of data
to the binary systems. Values between 0 and 1 for
each estimation point provide a set of indicators-
converted quantity using kriging that can be
expounded as the proportion of the block overhead
the determined cut-off on data support and the
probability that the grade is overhead the
determined indicator [25]. In the optimization
process of this study, this probability is
contemplated as the probability index (PIn) for
block n. The high probability blocks have less risk
than low probability ones.
Actually, in the current section, an integer
programming-based model with considering grade
uncertainty has been developed. In this method, a
weight based on indicator kriging is assigned to
each block (PIn), which indicates the probability

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made from n for each block in the block model.
This approach establishes the objective function in
such a way that the initial production periods are
allocated to the higher-certainty mineral blocks.
Subsequently, the other objective function is
presented to the objective function of the traditional
model as follows:
21
Maximize = (1+ )
t
NT t
nnn
t
nt
NV
Z
PI X
r
=
´´
åå (9)
This objective function is subject to the
constraints (2) to (8).
4 ALR function for LTPS problem
Now, the ALR method is measured as one of
the possible methods for making out the projected
problem.
4.1 LR scheme
The Lagrangian relaxation (LR) method is
recognized as one of mathematical means for a
mixed-integer programming problem. In the
presentation [59−63] of this technique in LTPS,
system limitations are relaxed by Lagrangian
multipliers and presented to the objective function.
Next, the relaxed problem is rankled into a more
practicable sub-problem for distinct units and
solved via dynamic programming. Due to the
violations of system restraints, the coefficients are
promoted by dint of a sub-gradient method. The
convergence standard is achieved if the duality gap
is within a certain limit.
LR relaxes the system constraints as a result of
Lagrangian multipliers. Then the relaxed problem is
split into some smaller sub-problems. The constant
Lagrangian function can be made by dint of
assigning non-negative Lagrangian multipliers λt, μt
and ʋt in terms of processing type at period t to the
constraints. The LTPS problem is illuminated
through the Lagrangian relaxation method by
relaxing or momentarily ignoring the preventing
constraints and solving the problem as if they have
never been. While maximizing due to the control
variable t
n
X
in the LTPS problem, this is done
over the dual optimization process, which strives to
affect the constrained optimum by lessening the
Lagrangian function L due to the Lagrangian
multipliers λt, μt and ʋt:
j*=Min j(λ, µ, ʋ), where j(λ, µ, ʋ)=Max L(X, λ, µ, ʋ).
λ, µ, ʋ X
4.2 ALR scheme
For the time being, the ALR method is
identified as one of the possible methods for
solving the proposed problem. Considering the
augmented Lagrangian function proposed by
ANDREANI et al [64], an ALR technique is
practiced, which can efficiently produce viable
solution for the main problem. For the following
constrained optimization problem, assume f, g, h to
admit continuous first derivatives as follows:
Minf(x)
s.t. h(x)=0, g(x)≤0, x
Î
Ω={x|H(x)=0, G(x)≤0}.
Consider the following augmented Lagrangian
function as:
2
(, , , )=()+ ()+ +
2
i
ii
Lx f x h x


 
æö
ç÷
´
ç÷
èø
å
2
()+
2
j
j
jgx



æö
ç÷
´
ç÷
ç÷
èø
å (10)
4.3 ALR for LTPS problem reformulation
When the coupling constraints (4)−(6) are
relaxed, the above reformulation contributes to
decomposing the resulted model (9) into a number
of sub-problems. To solve the open pit mines LTPS
problem, the augmented Lagrangian relaxation
method is employed in the present paper.
The novel maximization objective is alike to
the minimization of a reviewed objective function.
At this point, the objective is specified as:
2
Min ( )
NT
nt
Z
X-
åå (11)
Explicitly, equality and inequality constraints
(4)−(6) are relaxed and the subsequent augmented
Lagrangian relaxation problem is acquired:
2
( , , , , )=Min ( )+ 2
NT
nt
LX Z X


-´
åå
(
)
(
)
(
2
1+ +
2
t
TT
tt
nnn
tt
XOX


æö
ç÷
-´´-
ç÷
èø
åå
2
max ++
2
t
PC



ö
÷´
÷
ø
(
)
2
max
++
t
Tt
nn n
t
OW X MC


æö
ç÷
´-
ç÷
èø
å (12)
According to the objective function, we have:

J. Cent. South Univ. (2020) 27: 2479−2493
2485
1
(, , , , )=Min (1+ )
t
NT NT n
t
nt n t
NV
LX r

=
-´
åå åå
2
2
max
+(1)++
22
() ++
2
t
T
tt
nn n
t
t
Tt
nn
t
PI X X
OX PC





æö
ç÷
´´ - ´
ç÷
èø
æö
ç÷
´- ´
ç÷
èø
å
å
(
)
2
max
++
t
Tt
nn n
t
OW X MC


æö
ç÷
´-
ç÷
èø
å (13)
The adjustment of Lagrangian multipliers
should be carefully completed so as to curtail the
Lagrangian function based on the Lagrangian
multipliers. To reach a quick solution, a
combination of sub-gradient method and various
meta-heuristics are needed to be used by most
references to adjust Lagrangian multipliers to alter
Lagrangian multipliers. In this research, the
Lagrangian multipliers are adjusted and the
performance of augmented Lagrangian relaxation
method is improved through using the particle
swarm optimization (PSO) algorithm and bat
algorithm (BA).
5 PSO methodology
In 1995, the PSO methodology was first
introduced by EBERHART et al [65] and
KENNEDY et al [66]. This was an optimization
method based on probability rules. Researchers
have pondered the social behavior of bird or fish
groups while searching for food so that the
population can be guided to a promising area for
space search. Certain sensible processes are
practiced for the manners of the creatures of the
ruling body. Birds modify their physical movements
by escaping missions to search for food. Hence,
every one of the group member supposedly uses the
previous experiences and other detections from
members in order to find food. This kind of
corporation is considered a positive movement
within a competitive search for food. The PSO is
grounded on the idea of sharing information among
the group members. In PSO, a particle is denoted to
each answer to a problem which is the situation of a
bird in the search space. All particles include a
degree of ability that the quality of action optimizes
it. Furthermore, each particle embraces a factor
called velocity which identifies it in the search
range [67−69].
The PSO starts with a group of inadvertent
replies. Next, it searches for the location and
velocity of each particle so as to determine the best
answer in the problem space. The two most
remarkable values indicate that each particle is
identified at each step of population movement.
As a result, the first step is recognized as the
finest answer in terms of suitability ever obtained
for each particle. This is actually the personal best
and is termed pbest. The global best, identified as
gbest, is another best value ever attained by means
of the PSO. To search for new solutions, swarm of
particles is randomly initialized over the searching
space and moves through D-dimensional space.
Authorize i
k
x
and i
k

respectively to be the
position and velocity of the i-th particle in the
searching space at the k-th iteration, then its
velocity and position of this particle at the (k+1)th
iteration are updated using the following equations:
111 22
=+( )+( )--
ii ii gi
kk kk kk
wcrp xcrp x

+ (14)
11
=+
iii
kkk
xx

++
(15)
where r1 and r2 demonstrate accidental numbers
between 0 and 1, respectively; c1 and c2 are
constants; i
k
p demonstrates the best position of the
i-th particle, and
g
k
p correlates with the global
best position in the swarm up to the kth iteration.
The PSO algorithm pseudocode is shown in
Figure 1.
Objective function: f(x), x=(x1, x2, …, xD)
Initialize particle position and velocity for each particle
and set k−l.
Initialize the particle’s best known position to its initial
position i. e. =.
ii
kk
p
x
do
Update the best known position ()
i
k
p
of each particle
and swarm’s best known position ().
g
k
p
Calculate particle velocity according to the velocity Eq.
(14)
Update particle position according to the position Eq. (15)
While maximum iterations or minimum error criterial
is not attained
Figure 1 Pseudocode of PSO algorithm
6 Bat algorithm
Group-behavior-based collective intelligence
is one of the most robust optimization techniques.
YANG [70] proposed an algorithm inspired by the

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collective behavior of bats within a natural
environment predicated on the reception of
reflected sound by bats. They can track the exact
direction and location of their prey by the
pulse-echo technique (sending soundwaves and
receiving their reflection). They can also draw a
sound image of barriers that connect their loci and
identify them well when soundwaves return to the
bat wave sender. This system enables bats to detect
mobile objects like trees and insects. Microbats are
constantly sending short-lived loud beats of sound
in the range of 25−150 kHz and listening to the
echo that rebounds from nearby objects to catch
prey or evade obstacles. They naturally emit 10−20
pulses per second (PPS) and can increase it to
approximately 200 PPS as they get closer to their
target. YANG [70] provided the following paths for
transferring these specific characteristics of bats to
an optimization algorithm:
1) All bats rehearse their echolocation
capability in determining how distant they are from
a particular object and somehow they differentiate
between food (prey) and the background barrier.
They all really benefit from echolocation abilities.
2) Bats can modify loudness A0 and
wavelengths λ of emitted sound pulses to find food;
therefore, they can fly unwittingly at velocity ʋi in
position xi at frequency fmin. They can also change
wavelengths and rate or frequency of their issued
pulse as per the distance that they have with their
prey.
3) Loudness fluctuates between a big positive
value A0 and the lowest constant value Amin.
The current position of each bat is considered
to be a feasible solution to the optimization problem
[71−73].
Pursuant to regulation, velocity t
i

and
position t
i
x
for each i-th virtual bat in the t-th
iteration and frequency fi can be measured as
follows:
min max min
=+( )-
i
ff f f

(16)
1
=+( )
*
tt t
ii i
x
x

-- (17)
1
=+
tt t
ii i
xx

- (18)
In the equation above, [0, 1]

Î represents a
uniformly distributed random vector and x* is the
current optimal position, chosen in each iteration
after being compared with the position of the virtual
bats. Frequency f is typically considered to be fmin=
0 and fmax=100. In each iteration, a solution in local
search is selected as the optimal solution, and the
new position of each bat is locally updated with
random steps:
xnew=xold+
Î
t
A
(19)
In the equation above,
Î
Î
[−1, 1] denotes a
random number and t
A
denotes the mean loudness
of the bats in the t-th iteration. Furthermore,
loudness Ai and pulse rate r that is transmitted every
time are updated as follows:
+1tt
ii
A
A

= (20)
+1t
i
r=r0i[1−exp(−γt)] (21)
In the above equation, α and γ are constants
and r>0 and 0<α<1. When t→∞, we have
+1 0t
ii
rr® and +1 0.
t
i
A®Figure 2 depicts the
pseudocode of the bat algorithm.
Objective function f(x), x=(x1, …, xd)T
Initialize the bat population xi (i=1, 2, …, n) and vi
Define pulse frequence fi at xi
Initialize pulse rates ri and the loudness Ai
while (t<Max munber of iterations)
Generate new solutions by adjusting frequency,
and updating velocities and locations/solutions [Eqs. (16) to
(18)]
If (rand>ri)
Select a solution among the best solutions
Generate a local solution around the selected best
solution
end if
Generate a new solution by flying randomly
if (rand<Ai & f(xi)<f(x*))
Accept the new solutions
Increase ri and reduce Ai
end if
Rank the bats and find the current best x*
end while
Postprocess results and visualization
Figure 2 Pseudocode of bat algorithm
7 Framework of proposed models
In the present research, two steps are required
for the hybrid methods. The first one states the
Lagrangian function which brings up-to-date the
Lagrange multipliers. The second step is the precise
global extension of the stated ALR function, in
which the PSO, BA and GA are utilized to find out

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a new stochastic method near to the ideal maximum.
Figure 3 shows the flowchart of the suggested
approach.
Figure 3 Flowchart of proposed models
8 Numerical results
The presented model was expanded, executed,
and assessed in MATLAB R2019a environment.
The proposed experiment has been executed on an
Intel Quad-Core, 3.5 GHz, and 32 GB RAM PC
and MS Windows 7. All the developed formulations
are tested on the numerical experiments on the
artificial data set including 2150 blocks. In
conclusion, the enactment by ALR-BA is actually
better than other methods from the view of the
duality gap (Table 2).
The duality gap will be applied to evaluate the
modality of the solutions generated by the
meta-heuristic algorithms. The duality gap among
the best solutions produced by the various kinds of
the meta-heuristic algorithms, i.e., Zapprox and the
optimal solution found by MATLAB, i.e., ZMATLAB
is computed using the following equation:
MATLAB approx
approx
Gap 100%
ZZ
Z
-
=´
(22)
To compare the suggested mathematical model
for LTPS, the push-back data of an iron ore of
central Iranian iron ore body are selected as a case
study. The presented model was implemented in the
CHADORMALU mine. Also, its deposit has been
Ta bl e 2 Numerical results for synthetic data set
containing 2150 blocks
Iteration Method Duality gap
1
BA 0.172
PSO 0.196
GA 0.238
SG 2.75
ALR–BA 0.076
ALR-PSO 0.089
ALR-GA 0.091
ALR-SG 1.65
2
BA 0.082
PSO 0.095
GA 0.101
SG 1.23
ALR–BA 0.054
ALR-PSO 0.067
ALR-GA 0.085
ALR-SG 0.755
3
BA 0.038
PSO 0.043
GA 0.052
SG 0.763
ALR–BA 0.017
ALR-PSO 0.029
ALR-GA 0.038
ALR-SG 0.301
recogniszed as the major iron ore one in the central
part of Iran. CHADORMALU is located at the
center of PERSIA (IRAN) Desert, at the north of
grey CHAH-MOHAMMAD Mountains. 400
million tons of resource and 320 million tons of
reserves are divided between northern and southern
ore bodies by averaged Fe- and P-content of 55.2%,
0.9%, respectively.
Four push-backs are scheduled for the
CHADORMALU mine. The mathematical model
presented in this paper is practiced in the second
push-back. The 3D view of the second push-back is
illustrated in Figure 4. This push-back includes
6854 blocks of which 2754 are ore blocks and 4100
are waste blocks. The tonnage of waste and ore
presented in the aforementioned push-back shall be
103.8 and 110.2 million tons, respectively (with a
waste ratio of 0.94). The technical and economic
parameters and also, the number of model variables
for the second push-back of CHADORMALU mine
are illustrated in Tables 3 and 4.
Figures 5 and 6 show the numerical results of

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2488
Figure 4 A 3D view of second push-back in
CHADORMALU mine [25]
Ta bl e 3 Technical and economic parameters
Parameter Value
Iron ore price/(USD∙t−1) 61
Ore mining cost/(USD∙m−3) 6.2
Waste mining cost/(USD∙m−3) 5.9
Processing cost/(USD∙m−3) 28.65
Discount rate/% 10
Cut-off grade/% 52.8
Mining capacity/(Mt∙year−1) 25
Processing capacity/(Mt∙year−1) 8.1
Mining recovery/% 90
Processing recovery factor/% 76
Overall slope/(°) 52
Life of second push-back of
CHADORMALU mine/year 12
Ta bl e 4 Number of model variables for second
push-back of CHADORMALU mine
Variable Value
Reserve constraints 6854
Iron ore grade constraints 12
Processing capacity constraints 12
Mining capacity constraints 12
Wall slope constraints 82248
Binary constraints 6854
Total 95992
the proposed model for the CHADORMALU
push-back data set including 12 planning periods
with deterministic assumption and considering
grade uncertainty. The comparison of the average
net present value of the total of 12 years for all
presented models in deterministic and uncertainty-
based conditions is shown in Figure 7. As disclosed
Figure 5 Comparison of NPV for CHADORMALU
mine obtained by presented models in deterministic
condition
Figure 6 Comparison of NPV for CHADORMALU
mine obtained by presented models with considering
grade uncertainty
Figure 7 Comparison of average NPV in total 12 years
obtained by presented models in deterministic and
uncertainty-based conditions
in Figure 7, with considering grade uncertainty, the
average net present value using the ALR–BA
method is 4.056 M$ and the average net present
values through the ALR-PSO, ALR-GA, ALR-SG,

J. Cent. South Univ. (2020) 27: 2479−2493
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BA, PSO, GA, SG, and conventional methods are
3.996, 3.787, 3.741, 3.947, 3.914, 3.681, 3.589 and
3.527 M$, respectively.
Additionally, the average grades of the ore for
the case study in deterministic and uncertainty-
based conditions are displayed in Figures 8 and 9.
Also, the comparison of the average ore grade of
the total of 12 years for all presented models with
assumed deterministic condition and concerning
with grade uncertainty is demonstrated in Figure 10.
In uncertainty-based condition, the average grade of
ore in the total of 12 years by the ALR–BA method
is 54.84% and for the ALR-PSO, ALR-GA,
ALR-SG, BA, PSO, GA, SG, and conventional
methods are 54.71%, 54.03%, 53.90%, 54.56%,
54.45%, 53.75%, 53.58% and 53.49%, respectively.
The ore productions from the assessed methods, in
conditions of deterministic and uncertainty, are
shown in Figures 11 and 12. According to the
Figure 8 Comparison of average grade of ore for
CHADORMALU mine obtained by presented models in
deterministic condition
Figure 9 Comparison of average grade of ore for
CHADORMALU mine obtained by presented models
with considering grade uncertainty
Figure 10 Comparison of average grade of ore in total
12 years obtained by presented models in deterministic
and uncertainty-based conditions
Figure 11 Ore production obtained by presented models
in deterministic condition
Figure 12 Ore production obtained by presented models
in uncertainty-based condition
obtained results, the suggested method (ALR-BA),
while satisfying the constraints of mining capacity
and processing capacity, generates the best
outcomes in comparison with other methods.
Furthermore, Table 5 shows the CPU time and
the duality gap of each method. The results show

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Ta bl e 5 General information about solution found by
MATLAB for proposed models
Method Number of
blocks, N
Number of
periods, T
Duality
gap
CPU
time/min
ALR-BA 6854 12 0.049 29.41
BA 6854 12 0.127 81.08
ALR-PSO 6854 12 0.053 31.59
PSO 6854 12 0.193 89.22
ALR-GA 6854 12 0.077 35.26
GA 6854 12 0.214 91.36
ALR-SG 6854 12 0.121 38.11
SG 6854 12 0.351 112.52
Conv. 6854 12 2.463 231.14
that the suggested method (ALR-BA) has the
lowest CPU time and duality gap compared to other
methods, and its CPU time is about 7.4% better
than ALR-PSO. The difference in the results
indicates the capability of the proposed method and
the weakness of the previous methods.
9 Conclusions
The aim of this research is to present a
mathematical model for long-term production
planning and achieve the highest revenue in the
grade uncertainty situation. To make the long-term
production planning, grade uncertainty was applied
as an input feature to the mathematical model of
production planning. A long-term production
development optimization model grade uncertainty
is employed as the binary integer programming.
Principally, the possibility index for each block
of ore was determined to point out the grade
uncertainty. For the next step, getting the best out of
the net present value with physical and operational
constraints was practiced to model the objective
function. Blocks with higher probabilities have less
risk than the blocks with lower probability. The
proposed model has considered grade uncertainty in
the mining sequence. The present paper suggested
the hybrid method of the augmented Lagrangian
relaxation-bat algorithm in order to solve the
long-term production problem in open-pit mines as
it is hard to solve the production planning models in
the open-pit mines. This research also presented a
new approach due to the optimization of Lagrange
coefficients and comparing its performance with the
traditional SG method in the Lagrangian relaxation
method.
The results of the case study prove that the
augmented Lagrangian relaxation method can carry
out a suitable solution to the main problem. The
hybrid strategy can produce a more effective
solution to the near-optimal solution in comparison
with the conventional method. Moreover, it was
specified that the stable convergence property and
prevention of early convergence are identified as
the main advantages of the method suggested in this
research. In terms of average net present value,
average ore grade and CPU time, results illustrate
that ALR-BA generates the best outcomes while
satisfying constraints. The CPU time by the
ALR-BA hybrid method was 201.73 min less than
the conventional method and also was 2.18, 5.85,
8.70, 51.67, 59.81, 61.95 and 83.11 min less than
the ALR-PSO, ALR-GA, ALR-SG, BA, PSO, GA,
and SG procedures, respectively.
Contributors
The main research targets were expanded by
Ehsan MOOSAVI and Kamyar TOLOUEI. Kamyar
TOLOUEI and Ehsan MOOSAVI conducted the
literature review and wrote the first draft of the
manuscript. Ehsan MOOSAVi and Kamyar
TOLOUEI established the models and calculated
the results. Amir Hossein BANGIAN TABRIZI,
Peyman AFZAL and Abbas AGHAJANI BAZZAZI
analyzed the calculated results and edited the draft
of manuscript. All authors replied to reviewers &
apos; comments and revised the final version.
Conflict of interest
Kamyar TOLOUEI, Ehsan MOOSAVI, Amir
Hossein BANGIAN TABRIZI, Peyman AFZAL,
Abbas AGHAJANI BAZZAZI declare that they
have no conflict of interest.
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(Edited by YANG Hua)
中文导读
混合算法改善品位不确定露天矿生产调度问题的性能
摘要：露天采矿工艺是地表采矿的一种方法，通过开挖坑洞从地表向下开采矿石或废物。工业生产过
程中，露天矿的长期生产调度(LTPS)问题是最大的生产难题之一，而基于确定性方法和不确定性的方
法被认为是解决此类问题的主要策略。在过去几年中，许多研究人员充分探究了一种成本较低的新型
计算法，即元启发式方法，用以解决矿山设计和生产调度问题。该方法尽管无法保证最终方案的最优
性，但能够以相对较低的计算成本推算出足够优秀的解决方案。本文提出了增强拉格朗日松弛(ALR)
与粒子群优化(PSO)，以 及 ALR 和蝙蝠算法(BA)的两种混合算法模型，以解决不确定品位条件下的露
天矿生产调度问题。该混合模型采用 ALR 方法解决露天矿生产调度问题，以提高其计算性能并加快
收敛速度，并通过 PSO 或BA 更新拉格朗日系数。所提出的计算模型与 ALR 遗传算法、ALR 传统次
梯度法和常规方法(未使用拉格朗日方法)的计算结果进行了比较，结果表明：相比于常规方法，ALR
法可以更加有效地解决大规模问题，并提出合理的解决方案。此外，混合算法可以降低计算时间和成
本，ALR-BA 方法的 CPU 运算时间比 ALR-PSO 方法大约高 7.4%。
关键词：露天矿；长期生产调度；品位不确定性；拉格朗日松弛；粒子群算法