An Adaptive Average Grasshopper Optimization Algorithm for Solving
Numerical Optimization Problems
NAJWAN OSMAN-ALI, JUNITA MOHAMAD-SALEH
School of Electrical and Electronic Engineering,
Universiti Sains Malaysia,
Nibong Tebal 14300, Penang,
MALAYSIA
Abstract: - The grasshopper optimization algorithm (GOA), inspired by the behavior of grasshopper swarms,
has proven efficient in solving globally constrained optimization problems. However, the original GOA
exhibits some shortcomings in that its original linear convergence parameter causes the exploration and
exploitation processes to be unbalanced, leading to a slow convergence speed and a tendency to fall into a local
optimum trap. This study proposes an adaptive average GOA (AAGOA) with a nonlinear convergence
parameter that can improve optimization performance by overcoming the shortcomings of the original GOA.
To evaluate the optimization capability of the proposed AAGOA, the algorithm was tested on the CEC2021
benchmark set, and its performance was compared to that of the original GOA. According to the analysis of the
results, AAGOA is ranked first in the Friedman ranking test and can produce better optimization results
compared to its counterparts.
Key-Words: - Grasshopper optimization algorithm, GOA, meta-heuristics, optimization, swarm intelligence.
Received: October 29, 2022. Revised: March 16, 2023. Accepted: April 11, 2023. Published: May 10, 2023.
1 Introduction
Optimization is important for producing fast and
accurate solutions to various problems. Most
optimization problems are challenging to solve
because of their nonlinearity, multimodal objective
landscape, and nonlinear constraints, [1].
Optimization techniques for solving such problems
can be divided into two main categories, [2],
traditional and nature-inspired. Traditional
algorithms for solving optimization problems
include gradient-based, interior-point, and trust-
region methods. Even with modern computers, these
algorithms can be computationally intensive when
computing derivatives, particularly for problems
with discontinuities in their objective functions, and
may not have derivatives in certain regions. Such
limitations of traditional methods have diverted
optimization research towards solutions based on
nature-inspired methods. Nature-inspired algorithms
work as global optimizers based on interacting
agents to generate search moves within the search
space.
In recent years, numerous nature-inspired
algorithms have been developed that can be
categorized as single- or population-based. A single-
agent algorithm generated a single solution for each
run. Examples of single-based algorithms are
Guided Local Search (GLS), [3], Variable
Neighborhood Search (VNS), [4], and Iterated Local
Search (ILS) [5]. Under population-based agents, all
algorithms emulate the behavior of nature, such as
swarming, physics, evolutionary, and human
behavior, where they generate a set of multiple
solutions in each run, [6]. Under the swarming
category, the algorithm’s source of information is
collective behavior in nature, such as bee movement
when collecting honey or deciding to move to a new
nest, or the movement of ants foraging for food.
Popular algorithms in this category include Particle
Swarm Optimization (PSO), [7], Artificial Bee
Colony (ABC), [8], [9], Bat algorithm (BA), [10],
and Ant Colony Optimization (ACO), [11]. It is also
possible to create an algorithm based on physics
phenomena such as the Gravitational Search
Algorithm (GSA), [12], Water Evaporation
Optimization (WEO), [13], and Thermal Exchange
Optimization (TEO), [14]. Another category
involves algorithms inspired by evolutionary
phenomena such as selection, recombination, and
mutation. Popular algorithms in this category
include the Genetic Algorithm (GA), [15], [16],
Evolution Strategy (ES), [17], and Differential
Evolution (DE), [18], [19]. The last category
comprises algorithms that emulate human behavior.
Examples of algorithms in this category are the
Teaching Learning-Based Algorithm (TLBA), [20],
Imperialist Competitive Algorithm (ICA), [21], and
Harmony Search (HS), [22].
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Although nature-inspired algorithms have
different approaches and methodologies for solving
optimization problems, they all have one common
set of phases during the search process: exploration
and exploitation. In the exploration phase, the
algorithm explores the search space as w idely as
possible for solutions, whereas during exploitation,
the algorithm focuses its searches on areas
surrounding promising regions for potential optimal
solutions. A key to nature-inspired algorithms is to
achieve a balance between exploration and
exploitation.
With numerous optimization algorithms, there is
no conclusion regarding which algorithm performs
best for all optimization problems, [23]. This
problem is highlighted in the “No Free Lunch (NFL)
theorem”, [24], which states that no single state-of-
the-art optimization algorithm can be expected to
perform better than any other algorithm on a ll
classes of optimization problems. Therefore, using
this inference, building a nature-inspired algorithm
should be based on the application on w hich the
algorithm is to be used, instead of building an all-
around working algorithm.
Among the various nature-inspired algorithms,
the Grasshopper Optimization Algorithm (GOA)
has been successfully applied to various
applications, such as optimizing parameters on
support vector machines, [25], solving optimization
problems in an automatic voltage regular system,
[26], and obtaining the values of seven unknown
parameters of a proton exchange membrane fuel-cell
stack, [27]. Although the GOA can produce well-
optimized solutions, it has several shortcomings,
[28]. One of these is its linear convergence
parameter, which causes the exploration and
exploitation phases to become unbalanced, leading
to a slow convergence speed and a tendency to fall
into one of the local optima traps. Various studies
have been conducted to overcome these
shortcomings, and enhanced GOA can be
categorized into variant and hybrid versions. Under
the variant category, the original GOA was
improved by integrating the Levy Flight, [29],
employing chaotic maps to balance exploration and
exploitation, [30], and using a natural selection
strategy and dynamic feedback mechanism, [31].
The modified algorithms are categorized as GOA
variants. In the hybrid category, the original GOA
algorithm or its variants are combined with other
nature-inspired algorithms, such as the Genetic
Algorithm (GA), [32], ABC, [33], and Grey Wolf
Optimizer (GWO), [34]. The hybrid strategy usually
provides a better ability for the algorithm to move
out of a local optimum trap or solve any movement
issues, but usually at the cost of additional
complexity and a longer computational time.
This work aims to overcome the disadvantages
of the GOA through two improvements, producing a
GOA variant referred to as AAGOA. The first
improvement to the AAGOA employs a modified
parameter convergence value that balances the
exploitation and exploration of the GOA, and the
second improvement is the implementation of an
adaptive average for enhanced fitness of
grasshopper agents. The proposed AAGOA was
tested using the CEC2021 real-parameter
optimization benchmark problems, [35], and
compared with the original GOA.
The remainder of this paper is organized as
follows. Section 2 introduces the concept of the
original GOA. The proposed AAGOA is explained
in detail in Section 3. This is followed by Section 4,
which presents and compiles the experimental
results based on the CEC2021 benchmark or test
functions to assess AAGOA performance. Finally,
Section 5 c oncludes the study based on the
simulation results.
2 Grasshopper Optimization
Algorithm (GOA)
Grasshoppers behave differently, depending on their
environment. In a swarm, one grasshopper tends to
first move independently and then try its best to
evade the other grasshoppers. Only when a
grasshopper is triggered by other grasshoppers, such
as a touch on its leg will it become aggressive and
begin swarming. A swarm of grasshoppers moves to
find a source of food from one place to another,
[36]. This swarm intelligence behavior was used as
the basis for the computational GOA development.
In [37], the authors used the behavior of a swarm
of grasshoppers to develop a search strategy in the
search space for optimization problems. A key
aspect that enables the GOA to converge to a
solution is the interaction between the grasshoppers
and agents. In a sw arm, the GOA considers three
primitive corrective zones of behavior among
grasshopper agents: the attraction, comfort, and
repulsion zones. Fig. 1 illustrates how the GOA
implements interactions between grasshoppers.
Each grasshopper had a comfort zone represented
by a sphere, as shown in Fig. 1. Any grasshopper
within the radius of the comfort zone will have a
neutral force, that is, its attraction force will be the
same as the repulsion force. Grasshopper A was
used as a reference. Grasshopper B is within the
comfort zone; hence, it is not attracted to or repelled
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by Grasshopper A owing to a neutral force.
Grasshoppers that move closer to grasshopper A
within the comfort zone, such as grasshopper C,
repel themselves away from grasshopper A.
Grasshoppers that are outside the comfort zone,
such as grasshopper D, tend to be attracted to move
towards grasshopper A. All the explained
movements continue to be executed by a swarm of
grasshopper populations until they converge on a
solution.
A mathematical formulation can be used to
represent the natural behavior of the grasshopper’s
movement in a swarm. GOA’s main equation can be
expressed based on the position of the grasshopper
given by [37],
( )
1, 2
dd
Nji
dd
dd
i ji
j ji ij
xx
ub lb
X c c sx x T
d
= ≠

−
−
= −+



∑
(1)
where refers to the position of the i-th
grasshopper in all dimensions, and D is the
dimensionality of the search space, where d=1,2,,
D.
and are the current positions of the i-th
and j-th grasshoppers in the d-th dimension,
respectively. Where T denotes the best solution
currently available. Equation (1) involves two
terms: The first term involves a summation term in
the bracket multiplied by c, and the second term is
T. The first term considers the position of the other
grasshoppers and implements their interaction in the
natural environment within an area specified by ubd
and lbd, representing the upper and lower bounds in
the search space in the d-th dimension, respectively.
Dij is the distance between the two grasshoppers the
i-th and j-th and is calculated using
.
Parameter s in the first term represents the strength
of the social forces between two grasshoppers, given
by [37]:
( )
/rl r
s r fe e
−−
= −
(2)
Where f and l are the attraction intensity and
attraction length scale, respectively, and their best
values in the original GOA are f=0.5, l=1.5, and r is
the normalized value of
between 1 and 4,
[37]. The c parameter in (1) is a monotonically
decreasing coefficient that reduces linearly with
every iteration and plays a vital role in GOA
exploration and exploitation, as it controls the
shrinking and expansion of grasshoppers’ comfort,
repulsive, and attraction zones. In the original GOA,
a balance between exploration and exploitation was
implemented by linearly decreasing the value of
parameter c using the following equation, [37]:
max min
max
cc
c c iter L
−
= −
(3)
where cmax is the maximum value, cmin is the
minimum value, iter is the current iteration, and L is
the maximum number of iterations. The second term
in (1) simulates grasshoppers’ tendency to move
toward the food source. Iterating (1) for all
grasshoppers yields a converged solution.
Fig. 1: Primitive corrective patterns among
individuals in a swarm of grasshoppers
3 Adaptive Average Grasshopper
Optimization Algorithm (AAGOA)
First, AAGOA implements a nonlinear convergence
parameter to balance the exploration and
exploitation phases. Second, the best solution found
thus far in (1) was replaced with the adaptive
average value calculated from the current and
previous best solutions. Both modifications were
implemented concurrently using the modified
version of equation (1).
3.1 Modified Parameter Convergence Value
Parameter c in the original GOA starts from a value
close to unity at the beginning of the iteration and
decreases linearly to a s mall constant value to
support the exploitation phase. However, using a
linear decrement of the c value would cause the
algorithm to converge to a solution too quickly and
may miss possible optimal solutions in the search
space during the exploration phase.
To overcome this limitation, the version of the
GOA variant proposed in this study, AAGOA,
adopts a n onlinear decreasing value called cm.
Following the studies of [38], [39], this study
adopted a nonlinear approach to the cm parameter
value. The value of the cm parameter starts at unity,
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similar to that of c in the original GOA. However,
instead of using a single equation to calculate the cm
parameter value with increasing iterations, the
calculation of the cm parameter depends on the
following three conditions.
(4)
where
2
exp 0.5 , if 0.1
0.1, otherwise
iter
L
α
ασ



−≥


=

(5)
Equation (1) is updated using cm, and is represented
as follows:
( )
1, 2
dd
Nji
dd
dd
im m ji
j ji ij
xx
ub lb
X c c sx x T
d
= ≠

−
−
= −+



∑
(6)
In the proposed modification, the nonlinear cm
changes in three stages. In the first stage, which
occurred for the first 60% of the maximum
iterations, the cm value of every iteration was
changed based on (4) and (5). In these equations, σ
controls the rate of the decrease in cm. A smaller σ
value will decrease the decrement rate, causing the
cm value to remain high during the earlier executions
of the algorithm. Conversely, a l arge value of σ
causes a steeper change in cm. The selection of the σ
value depends on the nature of the optimization
problem, but a typical value between 3 a nd 5
produces acceptable results in most optimization
problems, [38]. Based on trial-and-error simulation
runs, this study used σ = 4 t o produce optimum
results. During the first stage, the value of cm is set
to 0.1 if the calculation produces a cm value less than
or equal to 0.1. The second stage occurs between
60% and 90% of the maximum number of iterations.
During this phase, cm is set to a co nstant value of
0.1. The third and last stages occurred at more than
90% of the maximum iteration, where the value of
cm was set to a constant of 0.05.
Fig. 2 compares the changes in the c value
obtained using the original GOA and the proposed
nonlinear cm method applied to AAGOA. Based on
this graph, the value of cm in the proposed method
supports more exploration during the first 50
iterations. Subsequently, as it approaches 250
iterations, the cm values converge rapidly to the
target position. As it reaches a steady position, the
constant value in the second stage allows the
algorithm to search globally within the entire space.
During the last stage, the search space becomes
narrower because of the smaller cm value. This
provides an opportunity for the algorithm to search
intensely around the local optimal solution position.
This signifies a more focused exploitation stage.
Therefore, by using nonlinear cm equations, the
algorithm should be able to improve its exploration
and exploitation. These changes can also accelerate
the convergence of the algorithm.
Fig. 2: Variation in convergence parameter values
for the GOA and AAGOA
3.2 Adaptive Average for Target Fitness
In the original GOA, the i-th grasshopper position is
calculated using Equation (1). Currently, the
equation works by adding the current best position,
T, found throughout the dimensional space. The
second proposed improvement uses the predicted
position, P, instead of T. Hence, (1) can be modified
as follows:
( )
1, 2
dd
Nji
dd
dd
im m ji
j ji ij
xx
ub lb
X c c sx x P
d
= ≠

−
−
= −+



∑
(7)
where the predicted position can be calculated using
the following formula:
PTA= +
(8)
where
󰆹 is the unit difference between the current
best position and the mean of the previous k number
of T values represented by
and can be calculated
using the following equation:
w
w
TT
ATT
−
=−
(9)
where
1
1
n
wi
ink
TT
k
=−+
=∑
(10)
Variable n in (10) represents the total number of T
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currently recorded. Equations (7) to (10) are
executed if the total number of recorded best
positions is equal to or greater than k. Otherwise, the
value of the predicted position is set to the current
best position.
The purpose of using the predicted position is to
improve the exploitation process by considering past
search trends and providing a possible target
position that is better than the current position.
However, considering different types of
optimization problems, using only the predicted
position in certain types of problems will not
produce better results than using the current best
position, as in the original GOA. Hence, the best
way to obtain benefits from using both the predicted
position and the current best position is to
interchange both methods using probability. Based
on the trial-and-error method, the best results were
achieved when the change was implemented using
0.1 probability, i.e., a 10% chance of using P instead
of T when calculating the grasshopper’s next
position.
3.3 AAGOA Implementation
The pseudocode for the proposed AAGOA is
presented in Algorithm 1. Table 1 lists the
parameter settings used to simulate these
algorithms. These parameters were based on the
original GOA research to test the performance of the
algorithm, [37]. Additional parameter values, such
as σ and k, are specific to the proposed modification
of the GOA.
Algorithm 1 Pseudocode for AAGOA
1.
Generate the initial population of Grasshoppers
(= 1,2, ,)
randomly with a selected D
value.
2.
Set parameter setting values
3.
Set probability value of adaptive average
implementation
4.
Evaluate the fitness () of each grasshopper .
5.
= best solution
6.
while (( <) do
7.
Update cm using equations (4) and (5)
8.
for = to (all N grasshoppers in the
population)
do
9.
Normalize the distance between grasshoppers
10.
Calculate the possibility with a 10% chance of
executing equation (7)
11.
if the possibility is fulfilled then
12.
Update the position of the current search agent
using equation (7)
13.
else
14.
Update the position of the current search agent
using equation (6)
15.
end if
16.
Bring back the current search agent if it goes
outside the boundaries
17.
end for
18.
Update if there is a better solution
19.
Update record of
20.
if the number of records is larger than k
21.
Calculate the predicted position using equation
(8)
22.
else
23.
Predicted position =
24.
end if
25.
 = + 1
26.
end while
27.
Return the best solution of
Table 1. Parameter settings of selected algorithms
Algorithm
Parameter setting
GOA
=30, =500  = 1,  =
0.00004, = 0.5, = 1.5
AAGOA
=30, =500,  = 1,  =
0.00004, = 0.5, = 1.5, = 4, =10
The procedure of the algorithm starts with the
initialization of , and settings of the parameter
values for, ,  ,  , a nd . One run loop
iterates the entire procedure for a preset number of
maximum iterations while updating the cm value
using equations (4) and (5) and the current best
position.
Referring to the pseudocode, the nested
repetition loop involves the execution of equations
(1) and (2) for each grasshopper in the population.
Probability is calculated to determine which
equation will be implemented during a particular
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iteration. The best solution was updated at the end
of each repetition loop. At the beginning of the
iteration, the total number of recorded was less
than k. Therefore, the value of was assigned to
the value of . The process stops after the execution
of the 500-th iterations, the maximum preset
iteration number for each simulation run. This
maximum number of iterations followed most
previously published studies. Thirty grasshopper
agents were used for each simulation. In addition,
30 independent simulations were run for every
parameter setting, and all runs used 20 dimensions,
that is D=20.
4 Performance Evaluation
All algorithms in this work were implemented and
executed on MATLAB R2021a using a workstation
with an Intel(R) Core(TM) i7-9750H CPU @
2.60GHz 2.59 GHz processor and 32GB RAM. The
performance of the proposed AAGOA is assessed in
this section using four experiments. The first
experiment evaluated the GOA and AAGOA based
on the average values, standard deviations, and best
values using ten standard benchmark functions for
five different operators and generated a total of 50
standard functions. The second test strictly tested
the convergence performance of the AAGOA and
GOA. The third experiment aimed to test the
AAGOA using Wilcoxon's nonparametric ranking
test and Friedman's ranking test. The fourth test
presents the complexity of the AAGOA.
4.1 CEC2021 Benchmark Test Functions
In this study, CEC2021 benchmark functions were
employed to test the efficacy of the proposed
AAGOA over its original algorithm. Table 2
presents the CEC2021 benchmark function suites.
There are ten functions in CEC2021 with four
different types: Function F1 represents unimodal
functions, functions F2 to F4 represent basic
functions, and functions F5 to F7 represent hybrid
functions. Finally, functions F8 to F10 represent
composition functions. where Nf represents the
number of basic functions forming a particular
function. The search space is defined by the upper
and lower bounds defined as [-100,100] D, where D
represents the dimension number that can be used
with the function. These benchmark functions of
CEC2021 are single-objective bounds constrained
by transformations in bias, shift, and rotation.
Table 2. Summary of CEC2021 bound-constrained
real-parameter benchmark functions
Types No Functions
Unimodal
Functions
F1
Shifted and Rotated Bent Cigar
Function
Basic
Functions
F2
Shifted and Rotated Schwefel’s
Function
F3
Shifted and Rotated Lunacek bi-
Rastrigin Function
F4
Expanded Rosenbrock’s plus
Griewangk’s Function
Hybrid
Functions
F5 Hybrid Function 1 (Nf = 3)
F6 Hybrid Function 2 (Nf = 4)
F7 Hybrid Function 3 (Nf = 5)
Composition
Functions
F8 Composition Function 1 (Nf = 3)
F9 Composition Function 2 (Nf = 4)
F10 Composition Function 3 (Nf = 5)
Search range: [-100,100] D, D = 10, D = 20
Because the operators parameterize the
benchmark functions, this suggests that the
proposed algorithm can be evaluated by testing it
with various possible configurations of operators on
all benchmark test functions. Different
transformations of bias, shift, and rotation can be
represented by binary parameters, with 1 indicating
activated and 0 indicating deactivation. The
investigated transformations were (000), (010),
(011), (100), and (110). For each parameter setting
and transformation, the results were evaluated based
on the mean of the best values from the 30
independent trial runs. The best algorithm is the one
that is able to produce the lowest mean results for
most of the benchmark functions.
4.2 Experimental Results
The benchmark functions for the unimodal, basic,
hybrid, and composition functions use 30 search
agents over 500 iterations. The presented results
were recorded based on 3 0 independent trials with
random initial conditions to calculate statistical
results. These results include the mean fitness
(mean), which represents the average performance
and reliability of the algorithm; the standard
deviation of fitness (std), which represents the
stability of the algorithm; and best fitness (best),
which represents the best optimization ability of the
algorithm.
The CEC2021 benchmark test functions were
optimized using the proposed AAGOA and the
original GOA. The optimization results for all
compared algorithms on a ll benchmark test
functions are presented in Table 3, Table 4, Table 5,
Table 6, and Table 7. To present a qualitative
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evaluation of the proposed algorithm, the
convergence curves of each benchmark function
type for the (000) and (111) transformations are
presented in Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig.
8, Fig. 9 and Fig. 10.
There is only one extreme point in the unimodal
test function F1. This type of function is suitable for
benchmarking the exploitation behavior of the
algorithm. Based on Table 3 until Table 7, AAGOA
outperforms GOA in terms of best, mean, and std
values for all different types of transformations,
indicating reliable and better performance compared
to GOA with better stability. Fig. 3 and Fig. 7 show
that AAGOA can converge quickly compared to
GOA, especially during the iteration where the
convergence parameter cm is steeper. Based on this
result, it can be concluded that AAGOA has good
exploitation capability. The main reason for the
better results compared to the GOA is the adaptive
average for target fitness which provides a better
target position.
The basic functions, F2 to F4 in CEC2021, have
several local optima that are used to evaluate the
exploration ability. The results in Table 3 until
Table 7 show that AAGOA works best in 10 out of
the 15 benchmark tests. Although the AAGOA does
not achieve the best results in the remaining five
benchmark functions, most of the results were close
to the GOA. Fig. 4 shows that the AAGOA is able
to quickly converge towards a better fitness position
compared to the GOA. A similar result is shown in
Fig. 8; however, in the beginning, the GOA had
better results than AAGOA. As cm becomes steeper
and with the use of an adaptive average to predict
better target fitness, the AAGOA convergence
capability significantly increases. These results
indicate that AAGOA has a competitive exploration
ability.
Hybrid function, F5 to F7 consists of
combinations of basic functions that are unimodal or
multimodal. It can be used to evaluate the
performance of both the exploitation and
exploration of the algorithm. The results in Table 3
until Table 7 show that the AAGOA performs better
than the GOA. AAGOA outperformed GOA i n
terms of the best, mean, and std values for all
different types of transformation, indicating reliable
and better performance compared to GOA. AAGOA
did not achieve good std values in functions F6(010)
and F6(110), but the difference compared to the
GOA std values was close. The convergence curves
in Fig. 5 and Fig. 9 show that the AAGOA
converges faster than the GOA. It was concluded
that AAGOA has good exploitation and exploration
for hybrid functions.
The ability of an algorithm to avoid local optima
can be evaluated using the composition functions F8
to F10. They are suitable for benchmarking
exploration and exploitation simultaneously for a
large number of local optima. The AAGOA r esults
in Table 3 until Table 7 manage to achieve better
results than the GOA in five out of 15 functions.
Although it was less than half of the functions, the
difference between the GOA and AAGOA results
was small. Fig. 6 and Fig. 10 indicate that AAGOA
still converges better than GOA despite having a
higher fitness value in F8(000) compared to GOA in
the early iteration, but AAGOA significantly
produces better fitness values with increasing
iterations. Based on these results, AAGOA has
acceptable exploitation and exploration capabilities
for composition functions.
Table 3. Optimized results for (000) transformation
Functions
Criteria
AAGOA
GOA
F1
best
1.27E+03
1.02E+05
mean
3.88E+03
6.19E+05
std
2.73E+03
4.91E+05
rank
1
2
F2
best
1.45E+03
1.63E+03
mean
2.43E+03
2.63E+03
std
4.35E+02
5.38E+02
rank
1
2
F3
best
7.26E+01
4.31E+01
mean
1.25E+02
9.88E+01
std
3.37E+01
3.60E+01
rank
2
1
F4
best
2.45E+00
3.21E+00
mean
5.66E+00
7.43E+00
std
2.07E+00
2.53E+00
rank
1 2
F5
best
4.40E+03
3.06E+04
mean
5.39E+04
3.53E+05
std
9.31E+04
2.09E+05
rank
1
2
F6
best
7.95E+01
2.09E+02
mean
5.15E+02
6.39E+02
std
2.61E+02
2.78E+02
rank
1
2
F7
best
1.68E+03
5.31E+03
mean
1.44E+04
8.77E+04
std
2.47E+04
7.54E+04
rank
1
2
F8
best
2.69E+02
3.95E+02
mean
1.63E+03
1.80E+03
std
6.91E+02
9.09E+02
rank
1
2
F9
best
3.65E+00
5.83E+00
mean
3.81E+01
1.73E+01
std
2.92E+01
1.49E+01
rank
2 1
F10
best
5.05E+01
5.10E+01
mean
7.71E+01
7.02E+01
std
1.73E+01
1.46E+01
rank
2
1
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Table 4. Optimized results for (010) transformation
Functions
Criteria
AAGOA
GOA
F1
best
1.77E+03
2.88E+04
mean
1.21E+04
5.18E+05
std
7.24E+03
4.30E+05
rank
1
2
F2
best
1.40E+03
1.45E+03
mean
2.41E+03
2.49E+03
std
4.70E+02
6.13E+02
rank
1
2
F3
best
6.03E+01
7.62E+01
mean
1.27E+02
1.14E+02
std
3.48E+01
2.82E+01
rank
2
1
F4
best
2.85E+00
2.72E+00
mean
6.79E+00
7.08E+00
std
4.17E+00
2.47E+00
rank
1
2
F5
best
3.20E+03
1.05E+04
mean
1.09E+05
6.44E+05
std
1.45E+05
6.04E+05
rank
1
2
F6
best
2.09E+02
3.36E+02
mean
6.66E+02
6.88E+02
std
2.67E+02
2.25E+02
rank
1 2
F7
best
4.16E+03
9.40E+03
mean
4.77E+04
2.25E+05
std
4.66E+04
1.93E+05
rank
1
2
F8
best
1.00E+02
1.03E+02
mean
7.79E+02
9.30E+02
std
1.19E+03
1.45E+03
rank
1
2
F9
best
4.49E+02
4.44E+02
mean
5.06E+02
5.12E+02
std
5.71E+01
5.82E+01
rank
1
2
F10
best
4.86E+02
4.21E+02
mean
5.07E+02
5.00E+02
std
3.14E+01
2.98E+01
rank
2
1
Table 5. Optimized results for (011) transformation
Functions
Criteria
AAGOA
GOA
F1
best
1.40E+03
1.82E+05
mean
5.51E+03
9.28E+05
std
3.99E+03
8.48E+05
rank
1
2
F2
best
1.39E+03
1.79E+03
mean
2.46E+03
2.60E+03
std
5.49E+02
4.77E+02
rank
1
2
F3
best
7.63E+01
5.92E+01
mean
1.26E+02
1.27E+02
std
4.18E+01
3.51E+01
rank
1
2
F4
best
3.54E+00
2.85E+00
mean
9.93E+00
7.32E+00
std
6.06E+00
2.42E+00
rank
2
1
F5
best
3.92E+03
5.51E+04
mean
1.41E+05
4.80E+05
std
1.39E+05
4.37E+05
rank
1
2
F6
best
1.62E+02
1.62E+02
mean
6.28E+02
7.54E+02
std
2.68E+02
2.86E+02
rank
1
2
F7
best
2.70E+03
1.20E+04
mean
5.35E+04
1.28E+05
std
6.60E+04
1.39E+05
rank
1
2
F8
best
1.00E+02
1.04E+02
mean
1.83E+03
1.78E+03
std
1.58E+03
1.51E+03
rank
2
1
F9
best
4.38E+02
4.27E+02
mean
4.97E+02
4.94E+02
std
4.46E+01
5.90E+01
rank
2
1
F10
best
4.02E+02
4.01E+02
mean
4.60E+02
4.47E+02
std
3.09E+01
3.61E+01
rank
2
1
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Table 6. Optimized results for (110) transformation
Functions
Criteria
AAGOA
GOA
F1
best
1.77E+03
2.88E+04
mean
1.21E+04
5.18E+05
std
7.24E+03
4.30E+05
rank
1
2
F2
best
1.40E+03
1.45E+03
mean
2.41E+03
2.49E+03
std
4.70E+02
6.13E+02
rank
1
2
F3
best
6.03E+01
7.62E+01
mean
1.27E+02
1.14E+02
std
3.48E+01
2.82E+01
rank
2
1
F4
best
2.85E+00
2.72E+00
mean
6.79E+00
7.08E+00
std
4.17E+00
2.47E+00
rank
1
2
F5
best
3.20E+03
1.05E+04
mean
1.09E+05
6.44E+05
std
1.45E+05
6.04E+05
rank
1
2
F6
best
2.09E+02
3.36E+02
mean
6.66E+02
6.88E+02
std
2.67E+02
2.25E+02
rank
1
2
F7
best
4.16E+03
9.40E+03
mean
4.77E+04
2.25E+05
std
4.66E+04
1.93E+05
rank
1
2
F8
best
1.00E+02
1.03E+02
mean
7.79E+02
9.30E+02
std
1.19E+03
1.45E+03
rank
1
2
F9
best
4.49E+02
4.44E+02
mean
5.06E+02
5.12E+02
std
5.71E+01
5.82E+01
rank
1
2
F10
best
4.86E+02
4.21E+02
mean
5.07E+02
5.00E+02
std
3.14E+01
2.98E+01
rank
2
1
Table 7. Optimized results for (111) transformation
Functions
Criteria
AAGOA
GOA
F1
best
1.40E+03
1.82E+05
mean
5.51E+03
9.28E+05
std
3.99E+03
8.48E+05
rank
1
2
F2
best
1.39E+03
1.79E+03
mean
2.46E+03
2.60E+03
std
5.49E+02
4.77E+02
rank
1
2
F3
best
7.63E+01
5.92E+01
mean
1.26E+02
1.27E+02
std
4.18E+01
3.51E+01
rank
1
2
F4
best
3.54E+00
2.85E+00
mean
9.93E+00
7.32E+00
std
6.06E+00
2.42E+00
rank
2
1
F5
best
3.92E+03
5.51E+04
mean
1.41E+05
4.80E+05
std
1.39E+05
4.37E+05
rank
1
2
F6
best
1.62E+02
1.62E+02
mean
6.28E+02
7.54E+02
std
2.68E+02
2.86E+02
rank
1
2
F7
best
2.70E+03
1.20E+04
mean
5.35E+04
1.28E+05
std
6.60E+04
1.39E+05
rank
1
2
F8
best
1.00E+02
1.04E+02
mean
1.83E+03
1.78E+03
std
1.58E+03
1.51E+03
rank
2
1
F9
best
4.38E+02
4.27E+02
mean
4.97E+02
4.94E+02
std
4.46E+01
5.90E+01
rank
2
1
F10
best
4.02E+02
4.01E+02
mean
4.60E+02
4.47E+02
std
3.09E+01
3.61E+01
rank
2
1
Fig. 3: Convergence curves for unimodal function
F1(000)
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Fig. 4: Convergence curves for basic function
F2(000)
Fig. 5: Convergence curves for hybrid function
F7(000)
Fig. 6: Convergence curves for composition
function F8(000)
Fig. 6: Convergence curves for unimodal function
F1(111)
Fig. 7: Convergence curves for basic function
F2(111)
Fig. 8: Convergence curves for hybrid function
F7(111)
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Fig. 10: Convergence curves for composition
function F8(111)
In addition to evaluating the statistical
performance based on the best, mean, and std results
using benchmark functions, non-parametric multiple
comparisons were used to further verify the validity
of the results using the lowest mean value for all
transformations. Two statistical analyses were used:
the Friedman ranking test and Wilcoxon signed-rank
test. The Friedman ranking test ranks the algorithms
from best to worst. The best algorithm received the
lowest rank, while the worst algorithm received the
highest rank. The Wilcoxon signed-rank test with a
95% confidence interval was used to validate the
significance of the improvement provided by the
proposed algorithm.
The rank values for each benchmark function
obtained using the Friedman ranking test are listed
in Table 3 until Table 7. Table 8, Table 9, Table 10,
Table 11, a nd Table 12 display the results for the
Wilcoxon signed-rank test, and Table 13 together
with Table 14 display a summary result for the
Friedman ranking test and Wilcoxon signed-rank
test, respectively, based on the various
transformations.
From the statistical results in Table 13, it is clear
that AAGOA p erforms best with a F riedman
ranking test sum ranking value of 65, compared to
GOA with a sum ranking value of 85. AAGOA
ranked better for F1 to F7 for most transformations,
with lesser results for F8 to F10.
The mean and standard deviations are only used
to compare the overall performance of the
algorithm. Wilcoxon signed-rank test was used to
prove that the results were statistically significant.
Based on t he Wilcoxon signed-rank test, the
AAGOA can improve 19 out of 50 benchmark
functions while maintaining 29 similar results to the
original GOA. Although AAGOA had two lesser
results compared to the original GOA, the
improvements were significantly greater. Based on
the Wilcoxon signed-rank test, the improvements
provided by AAGOA in terms of best, mean, and
std with better ranking using the Friedman ranking
test were statistically significant.
Table 8. Wilcoxon signed-rank test results for (000)
transformation.
Functions
Criteria
AAGOA
F1
p-values
1.73E-06
h
+
F2
p-values
1.16E-01
h
=
F3
p-values
1.25E-02
h
-
F4
p-values
2.26E-03
h
+
F5
p-values
2.88E-06
h
+
F6
p-values
6.83E-03
h
+
F7
p-values
2.16E-05
h
+
F8
p-values
2.80E-01
h
=
F9
p-values
1.04E-03
h
-
F10
p-values
9.78E-02
h
=
Table 9. Wilcoxon signed-rank test results for (010)
transformation.
Functions
Criteria
AAGOA
F1
p-values
1.73E-06
h
+
F2
p-values
4.65E-01
h
=
F3
p-values
1.11E-01
h
=
F4
p-values
1.85E-01
h
=
F5
p-values
1.36E-05
h
+
F6
p-values
9.92E-01
h
=
F7
p-values
1.73E-06
h
+
F8
p-values
2.05E-04
h
+
F9
p-values
3.82E-01
h
=
F10
p-values
3.29E-01
h
=
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Table 10. Wilcoxon signed-rank test results for
(011) transformation.
Functions
Criteria
AAGOA
F1
p-values
1.73E-06
h
+
F2
p-values
1.92E-01
h
=
F3
p-values
3.93E-01
h
=
F4
p-values
1.47E-01
h
=
F5
p-values
1.13E-05
h
+
F6
p-values
9.78E-02
h
=
F7
p-values
2.60E-05
h
+
F8
p-values
5.44E-01
h
=
F9
p-values
7.97E-01
h
=
F10
p-values
1.71E-01
h
=
Table 11. Wilcoxon signed-rank test results for
(110) transformation.
Functions
Criteria
AAGOA
F1
p-values
1.73E-06
h
+
F2
p-values
4.65E-01
h
=
F3
p-values
1.11E-01
h
=
F4
p-values
1.85E-01
h
=
F5
p-values
1.36E-05
h
+
F6
p-values
9.92E-01
h
=
F7
p-values
1.73E-06
h
+
F8
p-values
2.05E-04
h
+
F9
p-values
3.82E-01
h
=
F10
p-values
3.29E-01
h
=
Table 12. Wilcoxon signed-rank test results for
(111) transformation.
Functions
Criteria
AAGOA
F1
p-values
1.73E-06
h
+
F2
p-values
4.65E-01
h
=
F3
p-values
1.11E-01
h
=
F4
p-values
1.85E-01
h
=
F5
p-values
1.36E-05
h
+
F6
p-values
9.92E-01
h
=
F7
p-values
1.73E-06
h
+
F8
p-values
2.05E-04
h
+
F9
p-values
3.82E-01
h
=
F10
p-values
3.29E-01
h
=
Table 13. Summary of Friedman ranking test
Operators
GOA
AAGOA
000
13
17
010
12
18
011
14
16
110
12
18
111
14
16
Sum
85
65
Rank
2
1
Table 14. Wilcoxon signed-rank test summary.
Operators
+
=
-
000
5
3
2
010
4
6
0
011
3
7
0
110
4
6
0
111
3
7
0
Sum
19
29
2
4.3 Algorithm Complexity
The complexity of the algorithms is measured by the
amount of time and space required to solve a
problem for a given input size. The complexities of
the AAGOA and GOA algorithms were calculated
using the method described by CEC2021, [35].
Table 15 shows the computational complexity of
both algorithms in 20 dimensions where 0 is the
time computed by running the following codes:
0.55x=
for i = 1:200000
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( )
; / 2; * ; ( );
log( ); exp( ); 2 ;
xxxxx xxxxsqrtx
x xx xx xx
=+= = =
= = = +
end
where t1 is the time required to execute 200,000
evaluations of the benchmark F1 in 20 dimensions.
t2 is the execution time of the algorithm for F1,
using the same number of evaluations. 2
is the
mean value of t2 over the five runs.
The procedure was applied to both the GOA and
AAGOA. It can be observed in the last row of the
table that the computational cost of AAGOA is
higher than that of GOA. This is due to the
implementation of an adaptive average for the target
fitness that adds to the complexity of the AAGOA.
Although having a higher computational cost,
AAGOA can produce significantly better results
than GOA.
Table 15. Computational complexity of AAGOA
and GOA for 20 dimensions
Variable
AAGOA
GOA
0
t
8.49E-03 8.49E-03
1
t
2.44E+02 2.44E+02
2
t
7.67E+03 3.93E+03
( )
21 0
ttt−
8.74E+05 4.34E+05
5 Conclusion
This study proposes a modified version of the GOA,
referred to as AAGOA. The proposed improvements
to the GOA introduced a nonlinear parameter
convergence value and an adaptive average for the
target fitness. Based on the optimization results
using the CEC2021 benchmark functions, the
AAGOA produced better results than the GOA on
most tested benchmark functions in different
transformation combinations (i.e., bias, shift, and
rotation). The nonlinear convergence parameter
helps the algorithm explore the search space
efficiently at the beginning of the iteration and
focuses on the local optimum towards convergence.
The adaptive average for the target fitness provides
the capability to move away from the local optimum
entrapment by introducing the predicted target
fitness based on t he previous target fitness. Future
research should focus on reducing the complexity of
the AAGOA and implementing the modification
performed in AAGOA with other GOA variants to
further improve the optimization results using
improved benchmark functions.
References:
[1] X. S. Yang, “Nature-Inspired Optimization
Algorithms: Challenges and Open Problems”,
Journal of Computational Science, Vol.46,
2020, pp. 101104.
[2] X. S. Yang, M. Karamanoglu, “Nature-
Inspired Computation and Swarm
Intelligence”, Academic Press, 2020, pp. 3-18.
[3] C. Voudouris and E. Tsang, “Guided local
search and its application to the traveling
salesman problem,” Eur. J. Oper. Res., Vol.
113, No. 2, Mar. 1999, pp. 469–499.
[4] N. Mladenović and P. Hansen, “Variable
neighborhood search,” Comput. Oper. Res.,
Vol. 24, No. 11, Nov. 1997, pp. 1097–1100.
[5] T. Stützle, R. Ruiz, “Iterated Local Search.”
In: Martí, R., Pardalos, P., Resende, M. (eds)
Handbook of Heuristics. Springer, 2018.
[6] Y. Meraihi, A. B. Gabis, S. Mirjalili, and A.
Ramdane-Cherif, “Grasshopper optimization
algorithm: theory, variants, and applications,”
IEEE Access, 9, 2021, pp. 50001–50024.
[7] R. Eberhart and J. Kennedy, “A new
optimizer using particle swarm theory,” in
Proc. 6th Int. Symp. Micro Mach. Hum. Sci.,
1995, pp. 39–43.
[8] D. Karaboga and B. Basturk, “Artificial bee
colony (ABC) optimization algorithm for
solving constrained optimization problems,”
in P. Melin, O., Castillo, L. T., Aguilar, J.
Kacprzyk, and W. Pedrycz, (eds) Foundations
of Fuzzy Logic and Soft Computing (Lecture
Notes in Computer Science), Vol. 4529,
Berlin, Germany: Springer, 2007.
[9] L. Wang, X. Zhang and X. Zhang, "Antenna
Array Design by Artificial Bee Colony
Algorithm With Similarity Induced Search
Method," in IEEE Transactions on Magnetics,
Vol. 55, No. 6, June 2019, pp. 1-4.
[10] E. S. Ali, "Power System Stabilizers Design
Using BAT Algorithm," WSEAS Transactions
on Systems and Control, Vol. 17, 2022 , pp.
466-476.
[11] A. Colorni, M Dorigo, V. Maniezzo
“Distributed optimization by ant colonies” in
Proceedings of the first European conference
on artificial life, 1991. pp. 134–42
[12] E. Rashedi, H. Nezamabadi-Pour, and S.
Saryazdi, “GSA:Agravitational search
algorithm,” Information Sciences, Vol. 179,
No. 13, Jun. 2009, pp. 2232–2248.
[13] A. Kaveh and T. Bakhshpoori, “Water
evaporation optimization: A novel physically
inspired optimization algorithm,” Computers
& Structures, Vol. 167, Apr. 2016, pp. 69–85.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.13
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Volume 18, 2023
[14] A. Kaveh and A. Dadras, “A novel meta-
heuristic optimization algorithm: Thermal
exchange optimization,” Advances in
Engineering Software, Vol. 110, A ug. 2017,
pp. 69–84.
[15] J. H. Holland, “Genetic algorithms,” Scientific
American, Vol. 267, No. 1, 1992, pp. 66–73.
[16] Saad Babesse, Fouad Inel, "Optimized
Genetic Algorithms Reduced Order Model
Based RST Roll Control of Antiroll Bar
Dedicated to Semi-active Suspension,"
WSEAS Transactions on Systems and Control,
vol. 17, 2022, pp. 504-514.
[17] H. G. Beyer and H. P. Schwefel, “Evolution
strategies–A comprehensive introduction,”
Natural Computing, Vol. 1, No. 1, 2002, pp.
3–52.
[18] R. Storn and K. Price, “Differential
evolution–A simple and efficient heuristic for
global optimization over continuous spaces,”
Journal of global optimization, vol. 11, no. 4,
1997, pp. 341–359.
[19] Z. Zhang, Y. Cai and D. Zhang, "Solving
Ordinary Differential Equations With
Adaptive Differential Evolution," in IEEE
Access, Vol. 8, 2020, pp. 128908-128922.
[20] R. V. Rao, V. J. Savsani, and D. P. Vakharia,
“Teaching-learning-based optimization: A
novel method for constrained mechanical
design optimization problems,” Computer-
Aided Design, Vol. 43, No. 3, Mar. 2011, pp.
303–315.
[21] E. Atashpaz-Gargari and C. Lucas,
“Imperialist competitive algorithm: An
algorithm for optimization inspired by
imperialistic competition,” in Proc. IEEE
Congress on Evolutionary Computation, Sep.
2007, pp. 4661–4667.
[22] Z.W. Geem, J. H. Kim, and G. V.
Loganathan, “A new heuristic optimization
algorithm: Harmony search,” Simulation, Vol.
76, No. 2, Feb. 2001, pp. 60–68.
[23] N. Lynn, and P. N. Suganthan, “Ensemble
particle swarm optimizer,” Applied Soft
Computing, Vol. 55, 2017, pp.533–548.
[24] D. H. Wolpert, W. G. Macready, “No free
lunch theorems for optimization,” IEEE
transactions on evolutionary computation,
Vol. 1, No. 1, 1997, pp. 67–82.
[25] I. Aljarah, A. M. Al-Zoubi, H. Faris, M. A.
Hassonah, S. Mirjalili, and H. Saadeh,
“Simultaneous feature selection and support
vector machine optimization using the
grasshopper optimization algorithm,”
Cognitive Computation, Vol. 10, No. 3, 2018,
pp. 478–485.
[26] H. Baran and S. Ekinci, “Grasshopper
optimization algorithm for automatic voltage
regulator system,” in Proceedings of the 2018
5th International Conference on Electrical
and Electronic Engineering (ICEEE), IEEE,
Istanbul, Turkey, May 2018, pp. 152–156
[27] A. A. El‐Fergany, “Electrical characterization
of proton exchange membrane fuel cells stack
using grasshopper optimizer.” IET Renewable
Power Generation, Vol. 12 No.1, 2018, pp.9–
17.
[28] H. Feng, H. Ni, R. Zhao, and X. Zhu, “An
enhanced grasshopper optimization algorithm
to the bin packing problem,” Journal of
Control Science and Engineering, 2020, pp.
1–19.
[29] Z. Elmi, and, M.Ö. Efe, “Multi-objective
grasshopper optimization algorithm for robot
path planning in static environments,” in 2018
IEEE International Conference on Industrial
Technology (ICIT), IEEE, Feb. 2018, pp. 244–
249.
[30] S. Arora, and P. Anand, “Chaotic grasshopper
optimization algorithm for global
optimization,” Neural Computing and
Applications, Vol. 31, No. 8, 2019, pp.4385–
4405.
[31] J. Wu, H. Wang, N. Li, P. Yao, Y. Huang, Z.
Su, and Y. Yu, “Distributed trajectory
optimization for multiple solar-powered
UAVs target tracking in urban environment
by Adaptive Grasshopper Optimization
Algorithm,” Aerospace Science and
Technology, Vol. 70, 2017, pp.497–510.
[32] M. M. Annie Alphonsa and N. Mohana
Sundaram, “A reformed grasshopper
optimization with genetic principle for
securing medical data,” Journal of
Information Security and Applications, Vol.
47, Aug. 2019, pp. 410–420.
[33] B. P. Dahiya, S. Rani, and P. Singh, “Lifetime
improvement in wireless sensor networks
using hybrid grasshopper meta-heuristic,”
Proceedings of ICRIC 2019: Recent
Innovations in Computing, Springer
International Publishing, 2020, pp. 305–320.
[34] T. C. Teng, M. C. Chiang, and C. S. Yang, “A
hybrid algorithm based on GWO and GOA
for cycle traffic light timing optimization,” in
Proc. IEEE International Conference on
Systems, Man and Cybernetics (SMC), Bari,
Italy, Oct. 2019, pp. 774–779.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.13
Najwan Osman-Ali, Junita Mohamad-Saleh
E-ISSN: 2224-2856
134
Volume 18, 2023
[35] A. Mohamed, A. Hadi, A. Mohamed, P.
Agrawal, A. Kumar, P. Suganthan, “Problem
definitions and evaluation criteria for the CEC
2021 special session and competition on
single objective bound constrained numerical
optimization,” Tech. Rep., November 2020,
[online] Available: https://github.com/P-N-
Suganthan/2021-SO-BCO
[36] A.V. Latchininsky, “Locusts and remote
sensing: a review,” Journal of Applied Remote
Sensing, Vol. 7, N o. 1, 2013, pp. 075099-
075099.
[37] S. Saremi, S. Mirjalili, A. Lewis,
“Grasshopper optimization algorithm: theory
and application,” Advances in engineering
software, Vol. 105, 2017, pp. 30–47.
[38] P. Mishra, V. Goyal, and A. Shukla, “An
improved grasshopper optimization algorithm
for solving numerical optimization problems,”
in Advances in Intelligent Computing and
Communication. Lecture Notes in Networks
and Systems, Vol. 109, S pringer, Singapore,
2020, pp. 179–188.
[39] R. Zhao, H. Ni, H. Feng, and X. Zhu, “A
dynamic weight grasshopper optimization
algorithm with random jumping,” in Advances
in Computer Communication and
Computational Sciences: Proceedings of IC4S
2018, Vol. 924, Springer, Singapore, 2019,
pp. 401–413.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Najwan Osman-Ali carried out the investigation,
formal analysis, methodology, software, and
writing-original draft.
-Junita Mohamad-Saleh is responsible for
supervision, project administration, writing, review,
and editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.13
Najwan Osman-Ali, Junita Mohamad-Saleh
E-ISSN: 2224-2856
135
Volume 18, 2023