An Improved Gradient-Based Optimization Algorithm for Solving Complex Optimization Problems.pdf

Citation: Altbawi, S.M.A.; Khalid,
S.B.A.; Mokhtar, A.S.B.; Shareef, H.;
Husain, N.; Yahya, A.; Haider, S.A.;
Moin, L.; Alsisi, R.H. An Improved
Gradient-Based Optimization
Algorithm for Solving Complex
Optimization Problems. Processes
2023, 11, 498. https://doi.org/
10.3390/pr11020498
Academic Editor: Olympia Roeva
Received: 23 December 2022
Revised: 28 January 2023
Accepted: 3 February 2023
Published: 7 February 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
processes
Article
An Improved Gradient-Based Optimization Algorithm for
Solving Complex Optimization Problems
Saleh Masoud Abdallah Altbawi 1,* , Saifulnizam Bin Abdul Khalid 1 , Ahmad Safawi Bin Mokhtar 1,
Hussain Shareef 2 , Nusrat Husain 3 , Ashraf Yahya 3 , Syed Aqeel Haider 4 , Lubna Moin 3
and Rayan Hamza Alsisi 5
1 Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia
2 Department of Electrical Engineering, United Arab Emirates University,
Al Ain P.O. Box 15551, United Arab Emirates
3 Department of Electronics & Power Engineering, Pakistan Navy Engineering College, National University of
Sciences and Technology (NUST), Islamabad 44000, Pakistan
4 Department of Computer & Information Systems Engineering, Faculty of Electrical & Computer Engineering,
NED University of Engineering and Technology, Karachi 75270, Pakistan
5 Department of Electrical Engineering, Faculty of Engineering, Islamic University of Madinah,
Madinah 41411, Saudi Arabia
* Correspondence: masoud@graduate.utm.my
Abstract: In this paper, an improved gradient-based optimizer (IGBO) is proposed with the target
of improving the performance and accuracy of the algorithm for solving complex optimization and
engineering problems. The proposed IGBO has the added features of adjusting the best solution by
adding inertia weight, fast convergence rate with modified parameters, as well as avoiding the local
optima using a novel functional operator (G). These features make it feasible for solving the majority
of the nonlinear optimization problems which is quite hard to achieve with the original version of
GBO. The effectiveness and scalability of IGBO are evaluated using well-known benchmark functions.
Moreover, the performance of the proposed algorithm is statistically analyzed using ANOVA analysis,
and Holm–Bonferroni test. In addition, IGBO was assessed by solving well-known real-world
problems. The results of benchmark functions show that the IGBO is very competitive, and superior
compared to its competitors in finding the optimal solutions with high convergence and coverage.
The results of the studied real optimization problems prove the superiority of the proposed algorithm
in solving real optimization problems with difficult and indefinite search domains.
Keywords: gradient-based optimizer; improve gradient-based optimizer; metaheuristic; inertia;
operator; engineering optimization problems
1. Introduction
In recent years, information technology has had a deep impact on human civiliza-
tion [1]. Due to this advancement, a massive amount of data needs to be analyzed, more
complicated real-world problems need to be solved, and enhancement of the computing
efficiency of computers is needed [2]. Artificial Intelligence (AI) has been a persistently
hot topic to deal with this development. AI refers to the simulation of human intelligence
in machines that are programmed to think like humans and mimic their actions [3]. The
term may also be applied to any machine that exhibits traits associated with a human mind
such as learning and problem-solving. AI methods in MG strategies contain Reasoning
and Learning (RL) and Swarm Intelligence (SI) methods [4]. The SI-based algorithms have
attained significant popularity among researchers and are considered one of the highest
encouraging categories of AI especially metaheuristic algorithms [5]. In general, meta-
heuristic optimization algorithms try to imitate the physical, biological, or even chemical
procedures that take place in the environment. However, some algorithms depend on
Processes 2023, 11, 498. https://doi.org/10.3390/pr11020498 https://www.mdpi.com/journal/processes

Processes 2023, 11, 498 2 of 26
mathematical theories [6]. The most generic ones are: Genetic Algorithm, which simulates
Darwin’s theory of evolution [7], Simulated Annealing (SA) algorithm, that is developed
from the thermodynamic process [8], Particle swarm optimization (PSO) algorithms, which
simulate the behaviour of fish school or bird flock [9], Differential Evolution (DE) algorithm,
that is applied in solving problems and functions by iteratively improving a candidate
solution based on an evolutionary process [10], Teaching Learning-Based Optimization
(TLBO), which is based on a teaching-learning process [11], Jaya algorithm, that based on
the concept that tries to reach the best solution and tries to avoid failure to move away
from the worst solution [12], Cuckoo Search (CS) Algorithm, which imitates the brood
parasitism conduct of some cuckoo species in conjunction with the Levy flight conduct of
certain birds and fruit flies [13], flower pollination algorithm (FPA) that brings its metaphor
from the pollination in the flowering cycle of some plants in nature [14]. African Vultures
Optimization Algorithm (AVOA) which imitates the nature of African vultures in foraging
and navigation [15].
Gradient-based optimization (GBO) is a newly developed metaheuristic optimization
to solve optimization problems. It contains search directions defined by the gradient of the
function at the current point. GBO algorithm is motivated by Newton’s gradient proce-
dure including two principal processes: gradient search rule process and local escaping
operator. The gradient-based approach uses the gradient search rule to improve exploring
phenomena and quickens the convergent rate of GBO to obtain the optimal position within
the search space. However, the local escaping process prevents GBO to avoid getting stuck
into the local optima [16].
In swarm-based algorithms, inertia weight is a concept utilized to balance the influence
of the current position and the attraction to the best-known position in the search space.
This helps the algorithm avoid getting trapped in local optima and to explore the search
space more effectively. The value of inertia weight is typically decreased over time to
increase the exploration [17].
In general, the optimization algorithm procedure steps, consist of parameter selection,
variables (search agents), initialization, exploration, exploitation, randomization formula
of the step search, selection of step, and terminating condition [18]. Each search agent in
the population interacts with other agents to locate the optimal solution [19]. Generally,
the swarm-based algorithms require some common control parameters like population
size and the number of iterations [20]. In addition, some algorithms have specific control
parameters besides the general parameters, known as hyper-parameters. These parameters
exist to improve the performance of the algorithm by tuning their values properly [21].
The search agents in a SI system are designed to have simple rules. There is no
central control to give order to how individual agents should perform [22]. The agent’s real
performance is local, with a degree of arbitrary. However, the relations between such agents
and the other parameters in the algorithm take the edge of the occurrence of “intelligent”
to mimic the global behavior, while the individual agents cannot find alone [23].
The major contributions of this paper are listed as follows:
• Utilizing modified inertia weight in the original version of GBO to adjust the accuracy
of the best solution. Whereas, the inertia weight in optimization algorithm, gives more
weight to previous solutions in order to converge faster but also allows for exploration
of new solutions.
• Modified parameters are utilized in GBO to boost the convergence speed and provide
the proper balance of global and local search capabilities.
• A novel operator (G) is introduced which supports the diversity search in the search
space. Whereas, G applied to move search agents toward better solutions lead-
ing to suitable performance both in the global search and local search using new
developed formula.
The prove the superiority of the proposed IGBO, its performance is compared with
GBO, CS, DE, FPA, PSO, TLBO, and AVOA using a wide range of benchmark functions on
a few real-world problems.

Processes 2023, 11, 498 3 of 26
The rest of the article is organized as follows: In Section 2 background and related
works are presented. Section 3 presents a brief description of the GBO while the proposed
IGBO is described in Section 4. Section 5 describes the benchmark function and Section 6
explains the real-world problems. The obtained results and performance comparison
using benchmark functions against the different optimization algorithms are presented in
Section 7. In Section 8 the proposed IGBO is employed to solve a challenging real-world
optimization problem in the field of engineering. Finally, Section 9 summarizes the study.
2. Background and Related Works
The improvement of the optimization algorithm depends on the enhancement of the
procedure steps or advancements of hybrid algorithms [24]. Every year there is competition
in the algorithms on benchmarks function and real-world problems based on time, accuracy,
and result to decide which algorithm is efficient [25]. There are different approaches to
improving algorithms such as parameters tuning, opposite strategy, inertia weight, chaos
strategy, fuzzy logic strategy, adding new operators, or multi-objective theory [26].
Adding inertia weight to adjust the accuracy and convergence speed toward the
optimal solution [27]. There are different methods to implement inertia such as fixed
and arbitrary inertia weights, and Adaptive inertia weights [28]. Tuning parameters to
enhance convergence rate by many approaches. For example, with fixed values that are
appropriate to the search process, some approaches gradually change the parameters of
operators through the problem search process while, in some approaches, the mechanism
will update the parameter for some instances of problems. Some of these methods aim to
develop an adaptive mechanism to change the parameter value according to the search
process [29]. Adding extra operators to balance exploration and exploitation and enhance
diversity. There are different kinds of operators such as comparison/relational operators,
stream operators, subscript operators, function operators, and arithmetic operators [30].
Because of the robustness of GBO, it is applied to solve quite complex real-world
problems. The authors in [31] used GBO to find the optimal design automatic voltage
regulator (AVR) using the GBO algorithm. In [32], GBO is used to calculate the reliability
redundancy allocation problem of a series-parallel framework. In [33], GBO is employed in
the estimation of the parameters of solar cells and photovoltaic (PV) panels as an effective
and precise method. GBO in [34] is utilized in the calculation of the Economic Load
Dispatch (ELD) problem for different situations such as ELD with transmission losses, and
mixed economic and emission dispatch. In [35], GBO was utilized with Proton Exchange
Membrane Fuel Cell to estimate the optimal parameters of three distinct kinds of PEM
fuel cells. The scholars in [36] used an ensemble random vector functional link model
(ERVFL) incorporated with GBO to model the ultrasonic welding of a polymeric material
blend. ERVFL-GBO has the best outcome which indicates its high accuracy over other
tested methods.
Despite the effective performance, GBO is trapped in local solution when conducting
complicated non-linear functions, thus, it can decrease its accuracy. To overcome these
drawbacks, different variants of the GBO have been introduced. The researchers in [37] used
a multi-objective GBO algorithm-based Weighted multi-view Clustering (MO-GBO-WMV)
to find the consensus clustering among different partitioning generated from individual
views and compared this approach with some other methods to demonstrate the advanced
ability of this approach. In [38] introduced a multi-objective gradient-based optimizer
(MO-GBO) to handle the best solution for more than one objective, where needed. In [39],
by improving the GBO using Ca and Cb which are new chaotic numbers generated by
various chaos maps. Then IGBO is used to derive the parameters of PV modules. Similarly,
in [40] a novel random learning mechanism is designed to improve the performance of
GBO. After that, IGBO was utilized to extract parameters of four photovoltaic models.
In [41], an improved gradient-based optimizer denoted by (CL-GBO) to build DNA coding
sets that contain the Cauchy and Levy mutation operators, which are utilized as readers and
addresses of the libraries. In [42] the goal was to solve single and multi-Economic Emission

Processes 2023, 11, 498 4 of 26
Dispatch problems, using an elegant method depending on mix-object both Manta ray
foraging optimization (MRFO) with GBO, named (MRFO–GBO) to avoid trapped into
local optima as well as accelerate the solution process. In [43] enhance the performance of
Grey Wolf Optimizer (GWO) by a new procedure to use GBO to produce a new algorithm
called (G-GWO). This combination was applied to improve the algorithm’s exploitation
and exploration and also added Gaussian walk and Levy flight. These two are arbitrary
operators utilized to increase the diversity search of exploration in the G-GWO. In [44]
modified GBO algorithm using operator D to improve the stability between exploration
and exploitation phases in the search process. Similarly, in [45] used binary search, which is
an advanced type of search algorithm that finds and fetches data from a sorted list of items.
It is called the binary GBO (B-GBO) algorithm. Then B-GBO is used in feature selecting
problems of machine learning and data mining.
Optimization algorithms have gained a surge in popularity and have achieved sig-
nificant attention from both academic and industrial fields. Here the review some recent
optimization algorithms. Arithmetic Optimization Algorithm (AOA) has developed de-
pending on well-known Arithmetic theory [46]. In [47], the scholars introduced Aquila
Optimizer (AO), which mathematically models and mimics the nature of aquila during
the procedure of hunting the prey. Dwarf Mongoose Optimization (DMO) algorithm is
introduced in [48] which simulate the teamwork behaviors of the dwarf mongoose. The
authors in [49] have developed Ebola Optimization search Algorithm (EOSA) based on the
propagation behavior of the Ebola virus disease. Gazelle Optimization Algorithm (GOA)
have presented in [50], which is inspired by the survival ability in their predator-dominated
environment. The authors in [51] have developed prairie dog optimization (PDO) algo-
rithm, which mimic the behaviour of four prairie dog in foraging and burrow building.
Reptile Search Algorithm (RSA) has presented in [52], which is inspired by the nature
of Crocodiles in hunting. The authors in [53], introduced oppositional unified particle
swarm gradient-based optimizer based on mix of oppositional learning, unified particle
swarm algorithm, and GBO algorithm to solve the complex inverse analysis of structural
damage problems. In [54], the scholars have developed social engineering particle swarm
optimization algorithm, which consists of combination of social engineering optimizer and
particle swarm optimization to deal with structural health monitoring as objective function.
However, this combination of more than algorithm may lead to slower convergence and
inaccuracy in some optimization problems where is the speed is compulsory.
Summarizing the previous studies, the GBO algorithm has superiority over all modern
counterparts in solving problems in different fields. However, ordinary GBO has some
limitations such as:
GBO will still be trapped into local optima and suffer from the imbalance between
exploitation and exploration, premature convergence, and slow convergence speed under
some circumstances due to incomplete judgment standard and operators.
The main function of the local escaping operator (LEO) phase algorithm is to avoid
the occurrence of local optimal stagnation, but only when the random number is less than
0.5, it will enter the LEO phase.
The former GBO does not identify the optimal solution for discrete functions which
have discrete search spaces and decision variables such as feature selection problems.
There is only one guidance towards the best solution during the updating process,
which limits the exploitation capability and can lead to the propensity of falling into the
local optimal solution.
Former GBO does not have enough internal memory to save optimal solutions among
all generations, which leads to a lack the population diversity. Moreover, the performance
of the algorithm is affected significantly by the space domain of the objective function.
However, an intensive searching process may lead to the deterioration of multimodal
objective functions.

Processes 2023, 11, 498 5 of 26
3. Original Gradient-Based Optimizer Algorithm
GBO was inspired by Newton’s gradient-based method. It combines concepts from
gradient-based and population methods to find solutions. GBO uses two principles, the
gradient search rule (GSR) and the local escaping operator (LEO). The GSR improves the
exploration tendency and accelerates the convergence rate in the search space, while the
LEO allows GBO to escape from local optima [16].
Metaheuristic algorithms such as GBO usually generate several stochastic operators
without the demand to drop data and depend on steady performance. GBO mainly uses
two operators and a set of vectors to explore the entire search space [40].
3.1. Initialization
Every individual in the whole population is denoted as a vector for a simple imple-
mentation description, which can be defined as:
Xn,d = [Xn,1, Xn,2, . . . , Xn,d], n = 1, 2, . . . , N; d = 1, 2, . . . , D (1)
Therefore, the GBO size population is denoted by N with the D-variables.
So, the vector is created randomly and represented as below:
xn = Xmin + rand(0, 1) × (Xmax − Xmin) (2)
where Xmin, and Xmax are the lower and upper limits the variables X, and rand (0, 1) is an
arbitrary number in the boundary [0, 1].
3.2. Gradient Search Rule
The GSR operator is utilized in the gradient-based technique to enhance the explo-
ration feasibility and enhance the convergence speed to find the optimal solution in the
search domain. GSR expression is represented as
GSR = norm × ρ1 ×
2∆x × Xn
(xworst − xbest + ε)
(3)
where norm is a random value chosen within normal distribution, xworst and xbest are the
worst and optimal solution in the whole search progress of optimization, ∆x is the value of
increase, and ε is an arbitrary number chosen from the boundary (0 to 0.1).
To attain the balance between the exploration and exploitation phases and follow the
search feasibility development, the GSR will be improved depending on the mathematical
expression below:
ρ1 = 2 × rand × α − α (4)
∆x = rand(1 : N) × |step| (5)
step =
xbest − xm
r1

+ δ
2
(6)
δ = 2 × rand ×

xm
r1 + xm
r2 + xm
r3 + xm
r4
4
− xm
n

(7)
α = β × sin

3π
2
+ sin

β ×
3π
2

(8)
β = βmin + (βmax − βmin) ×

1 −
m
M
3
2
(9)
where rand(1 : N) is an arbitrary value with N population, r1, r2, r3, and r4 (r1 6= r2 6=
r3 6= r4 6= n) are integers random numbers chosen from [1, N]. rand is a random number
between [0, −1], βmin = 0.2, βmax = 1.2, m is the value of the current iteration, M is the total
iterations, and α is a function based on β.

Processes 2023, 11, 498 6 of 26
The step is a mathematical equation, that signifies the step size, which is based on xbest
and xm
r1, while ρ2 is represented by:
ρ2 = 2 × rand × α − α (10)
For more enhance of the exploitation of the surrounding area of xn, by using the
direction of movement (DM). The DM can be determined by:
DM = rand × ρ2 × (xbest − xn) (11)
The GSR can be expressed as:
GSR = rand × ρ1 ×
2∆x × xn
(ypn − yqn + ε)
In which
ypn = rand ×

[zn+1 + xn]
2
+ rand × ∆x

(12)
yqn = rand ×

[zn+1 + xn]
2
− rand × ∆x

(13)
zn+1 = xn − rand ×

2∆x × xn
xworst − xbest + ε

(14)
Concerning the GSR and DM, Equations. The position X1m
n at any iteration is calcu-
lated using GSR and DM as follows:
X1m
n = xm
n − GSR + DM (15)
Full expression of X1m
n can be written as
X1m
n = xm
n − norm × ρ1 ×
2∆x × xm
n
(ypm
n − yqm
n + ε)
+ rand × ρ2 × (xm
r1 − xm
r2) (16)
By substituting the position of the best vector (xbest) with the current vector (xm
n ) in
Equation (1), the new vector (X2m
n ) can be calculated as follows:
X2m
n = xbest − randn × ρ1 ×
2∆x × xm
n
(ypm
n − yqm
n + ε)
+ rand × ρ2 × (xm
r1 − xm
r2) (17)
based on the positions X1m
n , X2m
n and the existing position (Xm
n ), at the next iteration, the
new coming solution (xm+1
n ) can be calculated as:
xm+1
n = ra × (rb × X1m
n + (1 − rb) × X2m
n ) + (1 − ra) × X3m
n (18)
where ra and rb are arbitrary values within the range [0, 1].
3.3. Local Escaping Operator Process
The LEO operator provides an important feature of the GBO to escape local solutions
and improve the convergence rate. It helps the algorithm escape from local minima
or saddle points, which can trap the algorithm and prevent it from finding the global
minimum.
There are areas in the search space that can prevent the algorithm from finding the
global minimum. The operator works by introducing random variations or perturbations to
the model’s parameters at each iteration. This helps the algorithm explore other regions of
the parameter space and potentially discover a more optimal solution. As the optimization
process continues, the amount of noise added is usually reduced to allow the algorithm to
arrive at a more accurate result.

Processes 2023, 11, 498 7 of 26
By using several positions (X1m
n , X2m
n ), so LEO produces an optimal solution with ad-
vanced performance (Xm
LEO). The pseudo code of the operator LEO is shown in Algorithm 1.
Algorithm 1: Operator LEO
i f rand pr
i f rand 0.5
Xm
LEO = Xm+1
n + f1 × u1 × xbest − u2 × xm
k

+ f2 × ρ1 × (u3 × (X2m
n − X1m
n ) + u2 × xm
r1 − xm
r2

)/2
Xm+1
n = Xm
LEO
else
Xm
LEO = xbest + f1 × u1 × xbest − u2 × xm
k

+ f2 × ρ1 × (u3 × (X2m
n − X1m
n ) + u2 × xm
r1 − xm
r2

)/2
Xm+1
n = Xm
LEO
End
End
where pr is a probability value, pr = 0.5, the values f1, and f2 are represent distribution arbi-
trary values between [−1; 1], u1, u2, and u3 are three random numbers, created as follows:
u1 =

2 × rand if µ1 0.5
1 otherwise
(19)
u2 =

rand if µ1 0.5
1 otherwise
(20)
u3 =

rand if µ1 0.5
1 otherwise
(21)
where rand is an arbitrary value in the range of [0, 1], and µ1 is a random number from the
interval [0, 1]. The previous equations can be written as:
u1 = L1 × 2 × γ1 + (1 − L1) (22)
u2 = L1 × γ2 + (1 − L1) (23)
u3 = L1 × γ3 + (1 − L1) (24)
where γ1, γ2, and γ3 are equal to rand. L1 is a binary parameter with a value of 0 or 1. If
parameter µ1 is less than 0.5, the value of L1 is 1, otherwise, it is 0. To determine the solution
xm
k in Equation (25), the following scheme is suggested.
xm
k =

xrand if µ2 0.5
xm
p otherwise
(25)
xrand = Xmin + rand(0, 1)×(Xmax − Xmin) (26)
4. Improved Gradient-Based Optimizer Algorithm
In this study, an Improved gradient-based optimizer (IGBO) is introduced to solve
different real problems accurately. The main motives for enhancing the original GBO are
described in the following sections.
4.1. Varying Inertia Weight
The inertia weight concept was introduced in many works of literature. During the
last few decades, researchers have developed many inertias weigh strategies, and it has an
important role in optimization processes using population-based metaheuristic algorithms.
It provides a good balance between the local search and the global search capabilities of the
algorithm. A mathematical equation is used to generate a new inertia weight [55].
In this paper time-varying, inertia weight strategy has been used in which the value
of the inertia weight is determined based on the iteration number. The linearly decreasing

Processes 2023, 11, 498 8 of 26
inertia weight was introduced by many researchers and was shown to be effective in
improving the fine-tuning characteristic of algorithms [28]. The weight parameter ω is
computed as follows:
ω(Itr) =

MaxItr − Itr
MaxItr
2
∗ (ωmax − ωmin) + ωmin (27)
where ω(Itr): The inertia weight at iteration Itr
ωmin: The minimum inertia weight (final), ωmax: The maximum inertia weight (initial),
MaxItr: Iteration by which inertial weight should be at final.
Different optimization problems need different Inertia boundary values. The need
just to adjust set the minimum and the maximum boundaries (ωmin, ωmax) which are
mentioned in some literature such as (0 and 1), (−1 and 1), or (0.9 and 1.2). In this study,
for some optimization problems the suitable values for ωmin and ωmax are 0.4 and 0.9
respectively, and 0.8 and 0.2 for the other optimization problem. In each generation, the
inertia weight ω(Itr) is generated automatically using Equation (27) within the selected
range [56]. In this study, the magnitude of inertia weight is selected by the trial-and-error
technique, so (ωmin = 0.8, ωmax = 1.2). Then Inertia weight will be used in the first
position generated by LEO.
So, the first position xm+1
n can be defined as:
xm+1
n = ω(Itr) × ra × (rb × X1m
n + (1 − rb) × X2m
n ) + (1 − ra) × X3m
n (28)
The new strategy holds the simple idea that the particles should visit more positions
in the search space at the beginning and have better local search ability at the end.
4.2. Adaptive Parameters
The tuning of the parameters has a huge impact on the optimization performance
of the metaheuristic approach [57]. For different problems, they need different values of
parameters; some problems need to enhance the diversity and the convergence speed. In
another aspect, some optimization problems need to escape the higher level of diversity
search that may lead to rash convergence and decreased convergence speed [58]. GBO has
very important parameters to find globally the best solutions. However, GBO uses a fixed
and random (rand) value. To escape the awkward parameters and achieve fine-tuning, a
new parameter-setting-free technique is presented. The mechanism delivers the issue of
setting five key parameters inside GSR and LEO [59].
4.2.1. Adaptive Parameters in GSR
The original GBO, the parameters ra and rb designed to have a random magnitude
in the interval between [0, 1]. However, by the tests, we could find advanced optimal by
giving high value to ra and low values to rb value within the domain [0, 1]. That can be
done by changing the values of these parameters with mathematical equations used in
IGBO. These mathematical equations extract the best values of the parameters, so with the
increase in the number of iterations, the values of ra will raise using Equation (27), and
the value of rb will decrease using Equation (28). That would help boost exploration to a
larger extent and is likely to select the solutions from the entire feasible range [60]. Figure 1
show the visualization of adaptive parameters in GSR. The IGBO proposed the following
equations to set the adjustment:
ra =
Itr
MaxItr
(29)
rb = 1 −

Itr
MaxItr

(30)

Processes 2023, 11, 498 9 of 26
x = (
Itr
MaxItr
) × (1 − (
Itr
MaxItr
)) × X1 + 1 − (1 −
Itr
MaxItr
) × X2
+ 1 − (
Itr
MaxItr
) × X3
γ , γ and γ
γ
γ and γ
γ =
Itr
MaxItr
γ = 1 − (
Itr
MaxItr
)
γ = 1 − (
Itr
MaxItr
)
u u u
u = L × 2 × (
Itr
MaxItr
) + (1 − L )
u = L × (1 − (
Itr
MaxItr
)) + (1 − L )
u = L × (1 − (
Itr
MaxItr
)) + (1 − L )
Figure 1. Visualization of adaptive parameters in GSR.
So, the new position at the next iteration xm+1
n can be defined as:
xm+1
n =

Itr
MaxItr

×

1 −

Itr
MaxItr

× X1m
n +

1 −

1 −

Itr
MaxItr

× X2m
n

+

1 −

Itr
MaxItr

× X3m
n (31)
4.2.2. Adaptive Parameters in LEO
Similarly, in standard GBO the parameters γ1, γ2 and γ3 set to have a random value
in the range between [0, 1]. Whereas, to avoid the local optima, these parameters tend to
have different approaches to achieve that goal. The parameter γ1 is likely to increase its
rate steadily along with the increase in iterations. In both parameters γ2 and γ3 tend to step
down the values of them gradually with the increase in iterations, to make the algorithm
modify the solutions to get the best one. The best solutions keep on adding the number of
iterations until the last [61]. Therefore, the randomness equations of the parameters can be
calculated as:
γ1 =
Itr
MaxItr
(32)
γ2 = 1 −

Itr
MaxItr

(33)
γ3 = 1 −

Itr
MaxItr

(34)
So, LEO uses the three random numbers u1, u2, and u3, will be generated as follows:
u1 = L1 × 2 ×

Itr
MaxItr

+ (1 − L1) (35)
u2 = L1 ×

1 −

Itr
MaxItr

+ (1 − L1) (36)
u3 = L1 ×

1 −

Itr
MaxItr

+ (1 − L1) (37)
4.3. Adaptive Operator G
Increasing the range of random integers used in a special operator in optimization
algorithm can assist the diversity and exploration of the search domain. This modification led

Processes 2023, 11, 498 10 of 26
to a higher probability of finding the optimal solution in the case of optimization functions,
and that depends on the nature of both the specific algorithm and optimization problem [62].
The operator G enhances the feasibility of the search in the search domain. Moreover,
it improves the balancing between exploration and exploitation processes to search for the
global optimal solution by increasing the G value throughout iterations instead of being
chosen as a constant value or random between (0,1). Subsequently, the operating time of
the proposed IGBO is decreased in contrast with the former GBO [63].
This method is enhancing the performance of optimization algorithms seeking to
obtain the best solution and decreasing the search space. The convergence of the approach
according to how the movement of solutions in the search domain [64]. In the GBO
algorithm, the direction of movement (DM) is used to converge near the area of the solution.
Therefore, the suggestion is to change the DM randomly with extra values to discover more
areas in the search domain [44]. Hence, the DM of IGBO is utilized to be
DM = G × ρ2 × (xbest − xn) (38)
Here, the value of the operator G changes significantly between specific limits.
The operator G can be expressed as
G = g1 + ((g2 − g1) ∗ (rand)) (39)
where g1 and g2 are real numbers chosen in this study to be (1 and 10). This range selected
by trial method according to the nature of IGBO algorithm and nature of the selected
optimization functions used in this research.
This operator improves the search process by discovering more areas in the search
space. Moreover, it optimizes the balance between exploitation and exploration. The
pseudo code of the IGBO algorithm is shown in Algorithm 2.
Algorithm 2. Pseudo code of the IGBO algorithm
1 Step 1. Initialization
2 Assign values for parameters pr, ε, ωmin, ωmax , g1 , g2 and M
3 Generate an initial population X0 = [x0,1, x0,2, . . . , x0,D]
4 Evaluate the objective function value f(X0), n = 1, . . . , N
5 Specify the best and worst solutions xm
best and xm
worst
6 Step 2. Main loop
7 While mM (iterations) do
8 for n = 1 : N (particles) do
9 for i = 1 : D (dimensions) do
10 Select randomly r1 6= r2 6= r3 6= r4 6= n in the range of [1, N]
11 Calculate the operator G using Equation (37)
12 Compute the direction of movement (DM) using Equation (36)
13 Gradient search Rule
14 Calculate the parameters ra and rb using Equations (27) and (28).
15 Calculate the position xm+1
n,i using Equation (29)
16 end
17 Local escaping operator
18 Calculate the parameters γ1, γ2 and γ3 using Equations (30)–(32)
19 Created random numbers u1, u2, and u3 using Equations (33)–(35)
20 if rand pr then
21 Calculate the position xm
LEO
22 At the next iteration calculate the new coming position Xm+1
n
23 end
24 Update the positions xm
best and xm
worst
25 end
26 m=m+1
27 end
28 Step 3. return xm
best

Processes 2023, 11, 498 11 of 26
5. Test Process
This process examines IGBO and its counterparts with benchmark functions along
with Friedman rank, and Real-world problems to verify their performance.
5.1. Parameter Settings
In all quantitative experiments, the initial parameters of the competitor algorithms
were set according to the reference papers, as shown in Table 1.
Table 1. Initial parameters of the competitor algorithms.
Algorithm Ref.
Parameter
Name Value
IGBO
Probability Parameter (pr) pr = 0.5
Minimum Balance Parameter (βmin) βmin = 0.2
Maximum Balance Parameter (βmax) βmax = 1.2
Inertia weights (ωmin, ωmax)
(ωmin = 0.7,
ωmax = 1.2)
GBO [16]
Probability Parameter (pr) pr = 0.5
Minimum Balance Parameter (βmin) βmin = 0.2
Maximum Balance Parameter (βmax) βmax = 1.2
CS [13] Discovery rate of alien eggs/solutions (p) p = 0.25
DE [65]
Scale Factor (F) F = 0.5
Crossover Probability rate (Cr) Cr = 0.5
FPA [66] Probability switch (p) p = 0.8
PSO [67]
Cognitive Constant (C1) C1 = 2
Social Constant (C2) C2 = 2
Minimum Inertia weight (ωmin) ωmin =0.7
Maximum Inertia weight (ωmax) ωmax =0.9
Maximum Velocity (Vmax) Vmax =0.002
TLBO [65]
Teaching factor (Tf ) Tf = 1, 2
Teaching step (Ts)
Ts chosen randomly
between [0, 1]
AVOA [15]
Probability parameter for selecting the first
best vulture (L1)
(L1 = 0.8)
Probability parameter for selecting the
second-best vulture (L2)
(L2 = 0.2)
A parameter that determines the disruption
of the exploration and exploitation
phases (k)
(k = 2.5)
A parameter to determine the probability of
selecting the mechanisms in the exploration
phase (p1)
(p1 = 0.6)
selecting the mechanisms in the exploitation
phase of the first part (p2)
(p2 = 0.4)
selecting the mechanisms in the exploitation
phase of the second phase (p3)
(p3 = 0.6)

Processes 2023, 11, 498 12 of 26
5.2. Benchmark Test Functions
The quality of optimization algorithms is commonly examined by using conventional
standard literature benchmark functions [68]. There are various categories of these test
functions. That variety in functions, to achieve the best representation of a wider range
of real-world large-scale optimization functions and compare various metaheuristic algo-
rithms with more convenience and flexibility [69]. Some optimization algorithms, besides
the IGBO algorithm, will be tested using different benchmark functions, which are cate-
gorized into two groups: (i) Unimodal benchmark functions in Table 2, (ii) Multimodal
benchmark functions in Table 3. In every table, (Dim) denotes the number of dimensions of
the functions, while the lower and upper bounds of the variables are denoted by (Range),
and the global minimum of the function is represented by (fmin) [70].
Table 2. Unimodal benchmark functions.
f. No. Name Dim Range fmin
F1 Sphere 30 [−100, 100] 0
F2 Schwefel’s 2.20 30 [−100, 100] 0
F3 Schwefel’s 2.21 30 [−100, 100] 0
Table 3. Multimodal benchmark functions.
f. No. Name Dim Range fmin
F4 Qing 30 [−500, 500] 0
f5 Alpine N. 1 30 [−10, 10] 0
F6
Xin-She Yang N.
1
30 [−5, 5] 0
F7 Salomon 30 [−100, 100] 0
F8
Xin-She Yang N.
2
30 [−2 pi, 2 pi] 0
F9 Penalized 30 [−50, 50] 0
5.3. Statistical Analysis
In this section, measurement criteria and statical tests are applied to prove the signifi-
cance of the IGBO algorithm different from the others. These tests are analysis of variance
(ANOVA), and Holm–Bonferroni test. These two statistical tests were conducted on results
obtained by every algorithm from 50 independent runs with 1000 iterations each.
5.3.1. Measurement Criteria
To evaluate the results five measured criteria were used. These measurements were
the worst, best, mean, median, and standard deviation (SD) [16].
5.3.2. Wilcoxon Signed-Rank Test
It is a non-parametric statistical test, which utilized to evaluate if there is a significant
difference between two correlated samples or repeated measurements on the same sample.
It is employed to determine whether there are a statistically significant difference two
related sets of observations.
5.3.3. Friedman Rank Test
The Friedman test is a non-parametric statistical test established by Milton Fried-
man [64]. The test makes a comparison among the mean ranks in the related groups
and identifies how the groups differed. Moreover, tells you which group of data was
rated best versus worst. The Friedman test is widely supported by many statistical soft-
ware packages [65]. IBM SPSS software has been used in this work to calculate Friedman
Rank Test.

Processes 2023, 11, 498 13 of 26
6. Evaluation of the Igbo Algorithm on Real-World Engineering Problems
In this section, three real-world design problems are tested. The results obtained
by IGBO were compared with those of different well-known optimizers suggested in
previous studies. The population and the maximum number of iterations were 30 and 500,
respectively, for all problems.
6.1. Three-Bar Truss Design Problem
This engineering problem illustrates the truss’s form and the forces applied to the
structure as it is explained in Figure 2. This problem has two design parameters (x1, x2).
The purpose of this problem is to minimize the total weight of the structure. It also has
various constraints, such as deflection, buckling, and stress [15]; mathematically stated
as follows:
Minimize Fitness
→
x

=

2
√
2xA1 + xA2

× l (40)
𝑥 , 𝑥
Minimize Fitness (𝑥
⃗) = 2√2𝑥 + 𝑥 × l
𝑔 (𝑥
⃗) =
√2𝑥 + 𝑥
√2𝑥 + 2𝑥 𝑥
𝑃 − 𝜎 ≤ 0
𝑔 (𝑥
⃗) =
𝑥
√2𝑥 + 2𝑥 𝑥
𝑃 − 𝜎 ≤ 0,
𝑔 (𝑥
⃗) =
1
√2𝑥 + 𝑥
𝑃 − 𝜎 ≤ 0,
0 ≤ 𝑥 , 𝑥 ≤ 1, l = 100 cm, 𝑝 = 2 , σ = 2
h x x
t x t x
Considering: X = [x , x , x , x ] = [h, l, t , t ]
Figure 2. Three-Bar Truss Design.
Subject to:
g1
→
x

=
√
2xA1 + xA2
√
2x2
A1 + 2xA1xA2
P − σ ≤ 0 (41)
g2
→
x

=
xA2
√
2x2
A1 + 2xA1xA2
P − σ ≤ 0, (42)
g3
→
x

=
1
√
2xA2 + xA1
P − σ ≤ 0, (43)
where, 0 ≤ xA1, xA2 ≤ 1, l = 100 cm, p = 2 kN
cm2 , σ = 2 kN
cm2
6.2. I-Beam Design Problem
It is also a difficult engineering structure optimization problem. The objective of
this structure is to reduce the vertical deflection of the I-beam so that the design problem
identifies the optimal geometric parameters related to the cross-section as displayed in
Figure 3. The design variables in this problem are: length (h or x1), height (l or x2), and the
thicknesses (tw or x3 and tf or x4) [71]. This optimization problem and its constraints are
explained in the equations as follows:
Considering : X = [x1, x2, x3, x4] = [h, l, tw, tf] (44)
Minimize Fitness =
5000
1
12 tw(h − 2tf)3
+ 1
6 lt3
f + 2ltf

h−tf
2
2
(45)

Processes 2023, 11, 498 14 of 26
g (x) = 2lt + t (h − 2t ) ≤ 300
g (x) =
180000x
t (h − 2t ) + 2lt [4t + 3h(h − 2t )]
+
15000x
(h − 2t )t + 2t l
≤ 6
10 ≤ h ≤ 80 10 ≤ l ≤ 50, 0.9 ≤ t ≤ 5, 0.9 ≤ t ≤ 5
G (s) = K + K s ʎ
+ K sµ
𝐺 (𝑠) =
𝐾
1 + 𝑆𝑇
𝐺 (𝑠) =
𝐾
1 + 𝑆𝑇
𝐺 (𝑠) =
𝐾
1 + 𝑆𝑇
𝐺 (𝑠) =
𝐾
1 + 𝑆𝑇
Figure 3. I-beam design.
Subject to:
g1(x) = 2ltf + tw(h − 2tf) ≤ 300 (46)
g2(x) =
180000x1
tw(h − 2tf)3
+ 2ltf

4t2
f + 3h(h − 2tf)
+
15000x2
(h − 2tf)t3
w + 2tfl3
≤ 6 (47)
The variables are subject to: 10 ≤ h ≤ 80, 10 ≤ l ≤ 50, 0.9 ≤ tw ≤ 5, 0.9 ≤ tf ≤ 5
6.3. Automatic Voltage Regulator Design Problem (AVR)
AVR is one of the real-world problems which the researchers tried to solve in different
aspects [72]. Moreover, AVR is one of the main components in any power system which is
used to control the outcome’s voltage under different conditions of the operating process.
The objective function of AVR is to estimate the optimal parameters of the Fractional Order
Proportional Integrator Derivative (FOPID) controller [73]. The mathematical representa-
tion of all the AVR parts using Laplace transformation. The output transfer function of all
parts of IGBO-based FOPID controlling in the AVR system is shown in Figure 4. Moreover,
the parts are the FOPID controller, amplifier, exciter, generator, and sensor [74].
GFOPID(s) = KP + KIs−L
+ KDsµ
(48)
GA(s) =
KA
1 + STA
(49)
GE(s) =
KE
1 + STE
(50)
GG(s) =
KG
1 + STG
(51)
GS(s) =
KS
1 + STS
(52)
Figure 4. IGBO-based FOPID-AVR FOPID controller.

Processes 2023, 11, 498 15 of 26
7. Results and Analysis
The parameters of IGBO and other algorithms are selected according to the subsection
parameter settings. Then, they have been tested and executed using the MATLAB (R2021a)
desktop computer running Windows 10 Enterprise 64-bit with an Intel ® Core TM (Santa
Clara, CA, USA) i5-8500 CPU processor and 8.00 GB RAM. All results are stored based on
50 population sizes and 50 independent runs with 1000 iterations for every run, then the
results are compared using the obtained results.
In benchmark test functions, the results found by IGBO are compared with seven well-
known algorithms (GBO, CS, DE, PSO, FPA, TLBO, AVOA). In addition, the results of IGBO
in real-world problems are compared with the result of the same counterpart algorithms.
7.1. Benchmark Test Functions Results
This section describes the results of IGBO and the chosen algorithms using variant
benchmark functions. Further, the results computes and compares the descriptive statistics
in terms of best, worst, median, mean value, and standard deviation of all the algorithms.
The best result for each function is highlighted in boldface.
The results of IGBO for unimodal benchmark functions are compared with other
algorithms in Table 4. IGBO algorithm obtained the lowest value in the results in the
functions (f1, f2) excluding the function (f3). This shows that IGBO has least variation in
results of compared to the competitor’s algorithms. Hence, IGBO is a better choice.
Table 4. Unimodal Benchmark Functions Results.
f. No Statistics IGBO GBO CS DE FPA PSO TLBO AVOA
f1
Best 0 3.16E-281 1.02 × 10−17
4.29 × 10−12
1.75 × 10+04
5.13 × 10−13
7.08 × 10−180
2.67 × 10−261
Worst 0 3.45 × 10−267
1.33 × 10+03
9.24 × 10−11
4.53 × 10+04
9.39 × 10−11
3.49 × 10−176
3.41 × 10−258
Median 0 1.47 × 10−272
2.77 × 10−10
2.83 × 10−11
2.69 × 10+04
9.84 × 10−12
4.60 × 10−178
1.35 × 10+260
Mean 0 1.43 × 10−268
3.18 × 10+01
3.20 × 10−11
2.79 × 10+04
2.07 × 10−12
3.19 × 10−177
1.92 × 10−259
Std 0 8.54 × 10−270
1.90 × 10+02
1.83 × 10−11
6.25 × 10+03
2.23 × 10−11
2.71 × 10−178
1.96 × 10−125
f2
Best 0 7.07 × 10−140
3.78 × 10−07
1.10 × 10−06
3.18 × 10+02
1.44 × 10−06
4.81 × 10−89
9.34 × 10−137
Worst 0 5.95 × 10−134
5.10 × 10+02
4.15 × 10−06
7.83 × 10+02
3.37 × 10−04
3.59 × 10−87
1.16 × 10−111
Median 0 5.91 × 10−137
2.33 × 10+00
2.04 × 10−06
5.64 × 10+02
6.61 × 10−06
5.91 × 10−88
7.70 × 10−121
Mean 0 6.62 × 10−135
2.79 × 10+01
2.15 × 10−06
5.73 × 10+02
2.63 × 10−05
8.23 × 10−88
2.32 × 10−113
Std 0 1.51 × 10−134
7.85 × 10+01
6.28 × 10−07
8.34 × 10+01
6.28 × 10−05
6.91 × 10−88
2.34 × 10−116
f3
Best 1.67 × 10−121
3.87 × 10−130
1.05 × 10+01
8.88 × 10−01
7.81 × 10+01
9.33 × 10−03
9.66 × 10−73
8.52 × 10−120
Worst 3.12 × 10−85
7.71 × 10−123
6.06 × 10+01
1.81 × 10+00
9.22 × 10+01
4.73 × 10−02
1.58 × 10−70
2.52 × 10−121
Median 1.62 × 10−111
1.04 × 10−125
2.57 × 10+01
1.24 × 10+00
8.68 × 10+01
1.78 × 10−02
1.03 × 10−71
1.58 × 10−123
Mean 1.62 × 10−267
3.37 × 10−124
2.79 × 10+01
1.26 × 10+00
8.63 × 10+01
1.95 × 10−02
1.66 × 10−71
9.28 × 10−122
Std 3.35 × 10+00
1.81 × 10−123
1.23 × 10+01
2.35 × 10−01
3.10 × 10+00
7.96 × 10−03
2.39 × 10−71
2.72 × 10−121
Table 5 illustrate p-values obtained from Wilcoxon rank-sum statistical test with 5%
accuracy. By looking at the results, it is evident that IGBO has achieved excellent results with
significant differences between the proposed IGBO approach and other optimization methods.
Table 5. Wilcoxon Signed Ranks test for unimodal benchmark functions.
IGBO vs. GBO CS DE FPA PSO TLBO AVOA
Z −2.097 −4.486 −5.170 −4.867 −6.245 −1.743 −3.341
p-values 6.21 × 10−05 9.04 × 10−12 5.56 × 10−16 3.55 × 10−14 1.29 × 10−17 3.49 × 10−19 2.70 × 10−09
By examining the results of unimodal functions using Friedman mean ranking test,
it is obvious that the proposed IGBO algorithm performed better than other counterpart
algorithms and had been able to achieve the appropriate score of 3.826 in Friedman test as
it is clear from Figure 5.

Processes 2023, 11, 498 16 of 26
− − − − − − −
× 10 × 10 × 10 × 10 × 10 × 10 × 10
5.54 × 10 7.80 × 10 2.93 × 10 2.39 × 10 5.92 × 10 2.71 × 10 1.26 × 10
6.29 × 10 31.783 73,933,906 1.67 × 10 6.82 × 10 2.88 × 10
2.91 × 10 2.3485 0.0000138 7.87 × 10 1.01 × 10 1.70 × 10 1.68 × 10
2.17 × 10 3.863 14,822,130 8.37 × 10 3.22 × 10 5.84 × 10
9.48 × 10 5.3199 10,455,346 3.49 × 10 9.65 × 10 4.07 × 10
1.83 × 10 5.60 × 10 2.60 × 10 1.78× 10 1.75 × 10 9.80 × 10
5.54 × 10 1.80 × 10 36.36 4.85 × 10 1.15 × 10 8.90 × 10
2.67 × 10 1.20 × 10 5.8848 2.11 × 10 2.67 × 10 3.20 × 10
1.44 × 10 4.20 × 10 8.6883 4.57 × 10 3.57 × 10 1.90 × 10
7.63 × 10 2.50 × 10 8.7131 8.49 × 10 2.90 × 10 8.29 × 10
5.54 × 10 5.00 × 10 6.31 × 10 1.02 × 10 4.71 × 10 1.01 × 10
5.54 × 10 2.58 × 10 79,748,031 5.21 × 10 1.74 × 10 1.78 × 10 8.67 × 10
5.54 × 10 4.00 × 10 0.72962 2.22 × 10 1.41 × 10 4.51 × 10
5.54 5.22 × 10 16,042,004 5.65 × 10 3.90 × 10 8.27 × 10
Figure 5. Friedman Mean rank test for unimodal functions.
Multimodal benchmark functions are complicated tasks to test the exploration ability
of the optimization algorithms to find the main optimal region of search domain.
The results of IGBO for multimodal benchmark functions are compared with other
algorithms in Table 6. Analysis of the results of this table shows that IGBO with its high
exploration power, has provided the global optimal for (f4, f5, f6, f8), which indicates the
more effective performance of IGBO.
Table 6. Multimodal Benchmark Functions Results.
f. No Statistics IGBO GBO CS DE FPA PSO TLBO AVOA
f4
Best 5.54 × 10−14
7.80 × 10−11
2.93 × 10−16 430.3679 2.39 × 10+10
5.92 × 10−10
2.71 × 10−20
1.26 × 10−06
Worst 6.29 × 10−09
31.783 73, 933, 906 1186.599 1.67 × 10+11
6.82 × 10−07
2.88 × 10−12
8.9331
Median 2.91 × 10−12
2.3485 0.0000138 874.0392 7.87 × 10+10
1.01 × 10−08
1.70 × 10−17
1.68 × 10−05
Mean 2.17 × 10−10
3.863 14, 822, 130 865.4393 8.37 × 10+10
3.22 × 10−08
5.84 × 10−14
0.40381
Std 9.48 × 10−10
5.3199 10, 455, 346 146.055 3.49 × 10+10
9.65 × 10−08
4.07 × 10−13
1.6162
f5
Best 1.83 × 10−158
5.60 × 10−145
2.60 × 10−08
0.021702 28.1595 1.78×10−08
1.75 × 10−90
9.80 × 10−127
Worst 5.54 × 10−149
1.80 × 10−134
36.36 0.028265 53.1178 4.85 × 10−06
1.15 × 10−88
8.90 × 10−116
Median 2.67 × 10−154
1.20 × 10−139
5.8848 0.025524 44.5526 2.11 × 10−07
2.67 × 10−89
3.20 × 10−122
Mean 1.44 × 10−152
4.20 × 10−136
8.6883 0.025537 44.4585 4.57 × 10−07
3.57 × 10−89
1.90 × 10−117
Std 7.63 × 10−150
2.50 × 10−135
8.7131 0.001555 5.0097 8.49 × 10−07
2.90 × 10−89
8.29 × 10−91
f6
Best 5.54 × 10−129
5.00 × 10−116
6.31 × 10−11
1.02 × 10−10
256,286.14 0.002556 4.71 × 10−55
1.01 × 10−86
Worst 5.54 × 10−119
2.58 × 10−60
79, 748, 031 5.21 × 10−08
1.74 × 10+10
1.4607 1.78 × 10−13
8.67 × 10−03
Median 5.54 × 10−125
4.00 × 10−100
0.72962 2.22 × 10−09
52,558,177 0.046811 1.41 × 10−28
4.51 × 10−65
Mean 5.54 5.22 × 10−62
16, 042, 004 5.65 × 10−09
487,357,738 0.12426 3.90 × 10−15
8.27 × 10−53
Std 5.54 × 10−14
3.64 × 10−61
11, 276, 834 9.88 × 10−09
244,656,464 0.25385 2.52 × 10−14
7.53 × 10−08
f7
Best 0.1997 2.00 × 10−130
4.4226 0.29987 13.0581 2.2999 0.29987 0.3275
Worst 1.1327 9.00 × 10−101
17.0238 0.40037 22.4805 4.3999 2.58987 1.86584
Median 0.199873 3.40 × 10−120
8.4499 0.31895 17.8066 2.9999 0.39884 2.7887
Mean 0.234792 1.80 × 10−102
8.8499 0.33611 17.8965 3.0799 0.46187 0.847516
Std 0.03587 1.30 × 10−101
2.9535 0.040328 1.9846 0.51627 0.014142 0.048512
f8
Best 1.015 × 10−12
3.51 × 10−12
2.64 × 10−11 7.13 × 10−11
5.64 × 10−11
3.51 × 10−12 2.79 × 10−11
9.01 × 10−12
Worst 8.72 × 10−12
3.45 × 10−12
1.05 × 10−11
2.32 × 10−11
1.05 × 10−11
2.71 × 10−06
1.12 × 10−10
3.58 × 10−11
Median 2.29 × 10−12
3.47 × 10−12
1.73 × 10−11
2.65 × 10−11
2.87 × 10−11
6.54 × 10−10
6.02 × 10−10
7.18 × 10−11
Mean 2.68 × 10−12
3.49 × 10−12
1.82 × 10−11
2.73 × 10−11
2.95 × 10−11
8.40 × 10−10
7.01 × 10−10
8.17 × 10−11
Std 5.29 × 10−13
1.68 × 10−13
1.48 × 10−12
1.28 × 10−12
7.45 × 10−13
6.57 × 10−07
7.08 × 10−10
6.84 × 10−12
f9
Best 1.52 × 10−18
9.45 × 10−06
1.24 × 10−16
9.53 × 10−12
2.63 × 10−5
1.52 × 10−18
6.29 × 10−24
1.40 × 10−10
Worst 2.55 × 10−14
0.10393 8.93 × 10−08
2.42 × 10−10
0.21576 1.5588 1.45 × 10−18
3.31 × 10−09
Median 5.19 × 10−17
3.40 × 10−05
3.29 × 10−11
5.25 × 10−11
2.14 × 10−03
0.10367 6.62 × 10−22
9.63 × 10−10
Mean 9.97 × 10−16
0.004205 1.46 × 10−10
6.66 × 10−11
0.014017 0.17841 8.04 × 10−20
1.16 × 10−09
Std 3.81 × 10−15
0.020548 0.005938 5.15 × 10−11
6.72 × 10−02
0.28147 2.69 × 10−19
7.09 × 10−09
Non-parametric Wilcoxon sign rank test has archived significant results when it is
applied on multimodal benchmark functions, which shown in Table 7. The post hoc analysis
confirms the effectiveness of the proposed method and it is statistically significant.

Processes 2023, 11, 498 17 of 26
Table 7. Wilcoxon Signed Ranks test for multimodal benchmark functions.
IGBO vs. GBO CS DE FPA PSO TLBO AVOA
Z −4.189 −4.803 −7.574 −7.861 −3.986 −8.956 −5.122
p-Value 1.23 × 10−11 1.34 × 10−12 3.62 × 10−15 1.82 × 10−17 6.90 × 10−09 8.48 × 10−18 1.054 × 10−13
The obtained Friedman test results for multimodal functions are sown in Figure 6. The
overall rank demonstrates that IGBO algorithm is superior to its counterparts. It obtained
the lower value with 3.193.
2.55 × 10 0.10393 8.93 × 10 2.42 × 10 0.21576 1.45 × 10 3.31 × 10
5.19 × 10 3.40 × 10 3.29 × 10 5.25 × 10 2.14 × 10 6.62 × 10 9.63 × 10
9.97 × 10 0.004205 1.46 × 10 6.66 × 10 0.014017 8.04 × 10 1.16 × 10
3.81 × 10 0.020548 0.005938 5.15 × 10 6.72 × 10 2.69 × 10 7.09 × 10
− − − − − − −
1.23 × 10 1.34 × 10 3.62 × 10 1.82 × 10 6.90 × 10 8.48 × 10 1.054 × 10
Figure 6. Friedman Mean rank test for multimodal functions.
To identify the nature of the algorithms with the functions, Figure 7 shown the 3D
design, which allows for a more visual and intuitive understanding of the function’s
behavior and properties. The convergence curve is demonstrated in Figure 8, the evaluation
of convergence capability shows the robustness of IGBO against different algorithms.
Figure 7. 3D design of the functions.

Processes 2023, 11, 498 18 of 26
Figure 8. Convergence curve of the functions.
7.2. Comparison of Computational Time
The modifications to improve original GBO, make it able to find optimal from the entire
feasible range with the proper balance of global and local search capabilities Moreover,
they effect to boost the convergence speed of IGBO compared to its counterparts. Each
algorithm runs 1000 iterations and the average of the elapsed time is considered a criterion
for computational time. The proposed algorithm needs less time to find the best solution
measured in seconds. Table 8 illustrates the comparison of computational time between
IGBO and other algorithms.
Table 8. Comparison of computational time between IGBO and other algorithms.
Benchmark
Test Functions
Elapsed Time (s)
IGBO GBO CS DE FPA PSO TLBO AVOA
Unimodal
functions
6.2186 7.9960 12.6258 8.2360 11.1468 9.05327 9.6487 8.9960
Multimodal
functions
19.5034 20.6731 21.6094 22.1904 22.6831 20.0217 23.536 21.8772
7.3. Result of Real-World Problems
This part show results of the parameter values of the maximum function evaluations
(MFEs), that compare the proposed IGBO algorithm against counterpart algorithms. All
algorithms are used to solve real-world problems as mentioned before, with 50 numbers of
population and 30 runs containing 1000 iterations.

Processes 2023, 11, 498 19 of 26
7.3.1. Three-Bar Truss Design Results
Table 9 shows the results of the comparative algorithms for solving the three-bar
truss design problem, and Figure 9 shows the convergence curve and best positions of the
three-bar truss design using IGBO.
Table 9. Comparison of best solutions for the three-bar truss design.
Optimal Cost IGBO GBO CS DE FPA PSO TLBO AVOA
MFEs 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000
Best Weight 263.8258 263.9861 264.1753 264.6827 265.1275 264.8643 266.9627 264.5241
Variables
x(1) 0.78868 0.78869 0.78677 0.78868 0.78868 0.78865 0.78871 0.78678
x(2) 0.40825 0.40825 0.41366 0.40825 0.40825 0.40831 0.40815 0.40832
𝒙(𝟏)
𝒙(𝟐)
Figure 9. Convergence curve and best positions of three-bar truss design using IGBO.
The non-parametric Wilcoxon sign rank test has achieved significant results when
applied to three-bar truss problem, as demonstrated in Table 10. Moreover, In Friedman
test, IGBO algorithm has achieved the smallest rank compared with other intelligent
optimization algorithms. These tests confirm the effectiveness of the proposed method and
it is statistically significant.
Table 10. Wilcoxon and Friedman tests for three-bar truss problem.
Test Type
Wilcoxon Friedman
Z p-Value Mean Rank Overall Rank
IGBO vs. – – – –
IGBO – – 1.233 1
GBO 3.014 1.66 × 10−23 2.862 2
CS −3.682 7.45 × 10−29 4.355 4
DE −5.062 3.08 × 10−35 6.509 5
FPA −7.608 5.21 × 10−39 8.680 7
PSO −6.853 7.94 × 10−36 9.754 8
TLBO −7.911 6.18 × 10−41 7.953 6
AVOA −3.849 9.21 × 10−34 3.561 3

Processes 2023, 11, 498 20 of 26
7.3.2. I-Beam Design Results
Table 11 shows the results of the comparative algorithms for solving the I-beam design
problem, and Figure 10 shows the convergence curve and best positions of I-beam design
using IGBO.
Table 11. Comparison of the best solutions for I-beam design.
Optimal Cost IGBO GBO CS DE FPA PSO TLBO AVOA
MFEs 30,000 30,000 30,000 30,000 30,000 30,000 30,000 30,000
Best Weight 0.013069 0.013074 0.013175 0.01593 0.013082 0.013075 0.013096 0.013075
Variables
(h) 79.98 80 80 80 80 80 80 80
(b) 49.99 50 50 50 50 50 50 50
(tw) 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
(tf) 2.32179 2.32179 2.32181 2.32179 2.32181 2.32179 2.32179 2.32179
× 10
− × 10
− × 10
− × 10
− × 10
− × 10
− × 10
𝒉
𝒃
𝒕𝒘
𝒕𝒇
Figure 10. Convergence curve and best positions of I-beam design using IGBO.
Table 12 shows Wilcoxon sign rank test and Friedman mean rank test for I-beam
problem. Wilcoxon test has produced significant outcomes when used on I-Beam Design
results. Furthermore, the overall ranking using Friedman test proved that IGBO algorithm
is superior to other algorithms.
Table 12. Wilcoxon and Friedman tests for I-beam problem.
Test Type
Wilcoxon Friedman
Z p-Value Mean Rank Overall Rank
IGBO vs. – – – –
IGBO – – 2.617 1
GBO −5.231 1.27 ×10−31 4.108 3
CS −8.573 7.95 ×10−41 7.247 8
DE −9.161 2.66 ×10−46 4.976 4
FPA −7.042 3.81 ×10−37 5.862 6
PSO −6.268 5.91 ×10−33 5.354 5
TLBO −7.843 1.78 ×10−38 6.059 7
AVOA −6.715 2.04 ×10−34 3.865 2

Processes 2023, 11, 498 21 of 26
7.3.3. Automatic Voltage Regulator Design Results
Due to the novelty performance and advanced ability in tuning, the proposed study
optimization of the AVR system contains a FOPID controller. Figure 11 shows step response
of the AVR-based IGBO-FOPID controller.
− × 10
− × 10
− × 10
− × 10
− × 10
− × 10
− × 10
ʎ
𝑉 (𝑠)
𝑉 (𝑠)
=
0.032251𝑠 .
+ 3.2251𝑠 .
+ 0.19585𝑠 .
+ 19.585𝑠 .
+ 0.056251𝑠 + 5.6251
0.0004𝑠 . +0.0454𝑠 . + 0.555𝑠 . + 3.2251𝑠 . + 1.51𝑠 . +20.585𝑠 . + 5.6251
Figure 11. Step response of AVR-based IGBO-FOPID controller.
The FOPID parameters estimated at the end of the IGBO search process are shown in
Table 13.
Table 13. Optimized FOPID parameters.
Parameter Value
KP 1.508478
KI 0.992512
L 1.051341
KD 0.621713
µ 1.219410
The transfer function model for the proposed GBO-based AVR system with the incor-
poration of optimized variables is given in the equation below:
Vre f (s)
Vm(s)
=
0.032251s3.5708 + 3.2251s2.5708 + 0.19585s2.3513 + 19.585s1.3513 + 0.056251s + 5.6251
0.0004s5.3513 + 0.0454s4.3513 + 0.555s3.3513 + 3.2251s2.5708 + 1.51s2.3513 + 20.585s1.3513 + 5.6251
(53)
To validate the effectiveness of the proposed optimal AVR design, its dynamic response
is compared with that of the previously designed AVRs under identical operating conditions
as shown in Figure 12.
Figure 12. Comparison of proposed AVR design with other methods.

Processes 2023, 11, 498 22 of 26
Table 14 provides the quantitative evaluation of the dynamic response based on some
of the very important dynamic response indicators such as percentage overshoot, settling
time, and peak time.
Table 14. Dynamic response evaluation of proposed GBO-based AVR design.
PID/FOPID
Tuning Method
Peak Value
(Pv)
Percentage
Overshoot
(%Mp)
Rise Time
(tr)
Peak Time
(tp)
Settling
Time (ts)
IGBO-FOPID
(Proposed)
1.084 8.36 0.0899 0.176 0.932
GBO-FOPID [31] 1.110 11.3 0.0885 0.16 0.653
JOA-FOPID [74] 1.130 13.2 0.0827 0.1750 0.453
SSA-FOPID [75] 1.15 15.5 0.0981 0.209 0.551
DE-PID [76] 1.3285 32.8537 0.1516 0.3655 2.6495
PSO- PID [76] 1.3006 30.0634 0.1610 0.3824 3.3994
ABC-PID [76] 1.2501 25.0071 0.1557 0.3676 3.0939
BBO-PID [77] 1.1552 15.5187 0.1485 0.3165 1.4457
PSA-PID [78] 1.1693 16.93 0.1438 0.3159 0.8039
Conversely, Table 15 describes stability criterion results with the best AVR design-
based algorithms. The stability indicators are Phase Margin (PM), Delay Margin (DM),
Bandwidth (BW), and Phase Gain (PG). As can be seen from the proposed GBO tuned
FOPID-AVR provides the most stable design among the considered AVRs with the highest
PM and BW values.
Table 15. Comparative AVR designs are based on stability evaluation indicators.
PID/FOPID-Tuning Method PM DM BW PG
IGBO-FOPID (Proposed) 95.9 0.0866 19.3 0.971
GBO-FOPID [31] 91.1 0.0753 21.27 1.2
JOA-FOPID [74] 90.3 0.0765 20.60 1.24
SSA-FOPID [75] 89.3 0.0910 17.01 1.21
DE-PID [76] 58.4 0.0920 12.8 4.2
PSO- PID [76] 62.2 0.1030 12.182 3.75
ABC-PID [76] 69.4 0.111 12.879 2.87
BBO-PID [77] 81.6 0.122 14.284 1.56
PSA-PID [78] 79.69 0.115 14.636 1.68
In summary, the results of the real-world problems, demonstrate that IGBO can deal
with different challenging problems and various combinatorial optimization problems.
Thus, IGBO is the most powerful optimization algorithm with the lowest computational
costs and high convergence speed to get the optimal solution.
8. Discussion
In this section, a comprehensive understanding of the performance of the proposed
IGBO algorithm and its significance in comparison to the other algorithms studied in this
manuscript. The key findings from this study are summarized as follows:
• The initial comparison that was made between IGBO and several state-of-the-art algo-
rithms such as GBO, CS, DE, FPA, PSO, TLBO, and AVOA using unimodal benchmark
functions with dimensions of 30. Whereas, the optimization based on 50 population
sizes and 50 independent runs with 1000 iterations for every run. the proposed IGBO
algorithm demonstrated outstanding performance in most cases.
• The second comparison between IGBO and the same its counterparts’ algorithms
using the multimodal benchmark functions along with 30 dimensions. The results
revealed a remarkable performance from the version of IGBO.

Processes 2023, 11, 498 23 of 26
• The implementation of evolutionary algorithms competitors to several real-world
constraint mechanical design optimization problems. The IGBO algorithm has shown
great performance when applied to solving real-world constraint optimization prob-
lems, and as such, it has been successfully utilized to address automatic voltage
regulator design.
It is worthwhile to mention that, despite the excellent optimizing capability of the
IGBO algorithm, there are some limitations associated with the proposed IGBO such as
solutions for the optimal temporary spatial-variation constraint sample are very difficult to
achieve and comparatively greater computational complexity than its competing algorithms.
9. Conclusions and Future Works
In this manuscript, an Improved Gradient Optimization (IGBO) algorithm has been
presented for solving real-world engineering and optimization problems. The proposed
IGBO performance was examined using benchmark test functions to verify its effective-
ness. To validate its superior optimization capabilities, its performance is compared with
the seven most widely used metaheuristic-based optimization algorithms. The results
demonstrate that the proposed algorithm is better than its competing algorithms in terms
of achieving the most optimal solution to the benchmark functions and real-world engi-
neering designs. Moreover, the statistical test analysis shows that the IGBO algorithm has
better performance than its original version. In addition, the solution to three real-world
problems was also examined and compared with other well-known algorithms to justify the
performance of the proposed IGBO algorithm. The optimization results showed that IGBO
has a high exploration ability in the scanning search domain, escaping local areas, and
finding the main optimal area. IGBO is superior than the seven competitor algorithms and
provides far more competitive optimization results in solving unimodal and multimodal
benchmark functions. Moreover, IGBO performance in evaluating three design problems
showed its high ability to solve real- world optimization problems.
Finally, multi-objective and binary forms of IGBO may be established as future works
for solving multi-objective and discrete optimization problems. In addition, the utilize
of a chaotic map in every iteration will improve the performance to avoid local optima
and accelerate convergence. Furthermore, using IGBO for solving real-world optimization
problems in different applications and domains can be another valuable future work.
Author Contributions: Conceptualization, S.M.A.A.; Methodology, S.M.A.A.; Software, S.M.A.A.;
Validation, S.M.A.A.; Writing—original draft, S.M.A.A.; Validation, N.H., A.Y.,S.A.H., L.M. and
R.H.A.; Writing—(review editing), N.H., A.Y., S.A.H., L.M. and R.H.A.; Supervision, A.S.B.M.,
S.B.A.K. and H.S. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: Not applicable.
Acknowledgments: The first author thanks the Libyan government for supporting this research
with a scholarship provided by the Ministry of Higher Education and Scientific Research. The
authors would like to acknowledge the facilities provided by Universiti Teknologi Malaysia for the
accomplishment of this work. The authors would like to thank the editor and anonymous reviewers
for improving this paper with their valuable comments and suggestions.
Conflicts of Interest: The authors declare no conflict of interest.
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