Spiral Gaussian mutation sine cosine algorithm Framework and comprehensive performance optimization

Abstract

Sine Cosine Algorithm (SCA), as a recently viral population-based meta-heuristic, which is in the extensive application for a variety of optimization cases. Regardless of the concerns on its novelty, SCA updates the population based on a simple updating rule with a basic structure and few parameters. However, it is considered that SCA remains the weakness of low diversity, slow convergence speed, stagnation in local optimum, and low accuracy of solutions. In that case, an attempt is made to design a new SCA version to overcome these shortcomings, namely FGSCA, combined with two strategies, including spiral motion and Gaussian mutation. Spiral motion is inspired by Moth-flame Optimization (MFO), which is employed to strengthen the capacity of exploitation based on the original SCA. The Gaussian mutation is adopted to increase the population's diversity generated by SCA and strengthen local exploration capability. The principle is to update the population generated by SCA, respectively, and then choose the best from these two new populations by greedy selection. Combining the two strategies effectively strengthens the original SCA's performance and maintains a proper exploitation and exploration balance. To examine the performance of FGSCA, it is used for making comparisons with eight well-known meta-heuristic algorithms (MAs), six reported SCA variants, and ten improved MAs on 23 well-known benchmark test problems and 30 standard IEEE CEC2014 benchmark test problems. Results indicate that the overall performance of FGSCA is superior to twenty-four comparative MAs on 53 benchmark test problems. Besides, FGSCA is employed to resolve three practical engineering problems. Results exhibit that FGSCA harvests the best results among twenty-four comparative MAs. Thus, FGSCA is expected to be a mighty and efficient tool for dealing with various complex optimization problems.

Full text

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Spiral Gaussian Mutation Sine Cosine Algorithm: Framework and
Comprehensive Performance Optimization
Wei Zhou1, Pengjun Wang1*, Ali Asghar Heidari 2, Xuehua Zhao3, Huiling Chen4*
1 College of Electrical and Electronic Engineering, Wenzhou University, Wenzhou 325035, China
(zhouweiqaq@163.com,wangpengjun@wzu.edu.cn)
2 School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran,
Iran
(as_heidari@ut.ac.ir, aliasghar68@gmaill.com)
3School of Digital Media, Shenzhen Institute of Information Technology, Shenzhen 518172, China
(zhaoxh@sziit.edu.cn)
4College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou, Zhejiang 325035,
China
(chenhuiling.jlu@gmail.com)
Corresponding Author: Pengjun Wang, Huiling Chen
E-mail: wangpengjun@wzu.edu.cn (Pengjun Wang), chenhuiling.jlu@gmail.com (Huiling Chen)
Abstract:
Sine Cosine Algorithm (SCA), as a recently viral population-based meta-heuristic, which is in the
extensive application for a variety of optimization cases. Regardless of the concerns on its novelty, SCA
updates the population based on a simple updating rule with a basic structure and few parameters.
However, it is considered that SCA remains the weakness of low diversity, slow convergence speed,
stagnation in local optimum, and low accuracy of solutions. In that case, an attempt is made to design
a new SCA version to overcome these shortcomings, namely FGSCA, combined with two strategies,
including spiral motion and Gaussian mutation. Spiral motion is inspired by Moth-flame Optimization
(MFO), which is employed to strengthen the capacity of exploitation based on the original SCA. The
Gaussian mutation is adopted to increase the population's diversity generated by SCA and strengthen
local exploration capability. The principle is to update the population generated by SCA, respectively,
and then choose the best from these two new populations by greedy selection. Combining the two
strategies effectively strengthens the original SCA's performance and maintains a proper exploitation
and exploration balance. To examine the performance of FGSCA, it is used for making comparisons
with eight well-known meta-heuristic algorithms (MAs), six reported SCA variants, and ten improved
MAs on 23 well-known benchmark test problems and 30 standard IEEE CEC2014 benchmark test
problems. Results indicate that the overall performance of FGSCA is superior to twenty-four
comparative MAs on 53 benchmark test problems. Besides, FGSCA is employed to resolve three
practical engineering problems. Results exhibit that FGSCA harvests the best results among twenty-
four comparative MAs. Thus, FGSCA is expected to be a mighty and efficient tool for dealing with
various complex optimization problems.

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Keywords: Sine cosine algorithm; Spiral motion; Gaussian mutation; Global optimization; Engineering
application
1 Introduction
Optimization methods can be divided according to different aspects; one of the aspects is the
number of objectives, i.e., single-objective, multi-objective, many objective cases, or the relations of the
variables and their definitions, such as large scale tasks, robust optimization, memetic procedures, and
fuzzy optimization [1, 2]. It is usually transformed into a mathematical structure to find its minimum
or maximum. Therefore, to resolve such optimization problems, different solutions have been proposed
in various pieces of literature. However, these methods have certain limitations that can only be effective
for problems with mathematical characteristics such as continuity, differentiability, and multimodality.
Many practical problems with complex attribute space, such as smart grids [3, 4], photovoltaic power
generation systems [5, 6], and adaptive control [7-10], have no unique mathematical characteristics. In
such cases, decision-makers aim to manage one or more aspects of the problem or its parameters [11,
12]; stochastic approximated solvers can be utilized for a flexible and soft process. Many MAs have
been developed for this aim and applied to various optimization problems in recent years. MAs perform
better for non-differentiable and non-continuous optimization problems than traditional approaches
[13-15]. The same laws of nature primarily inspire these methods. For instance, applications of
differential evolution (DE), whale optimizer (WOA) [16], and teaching-learning-based optimizer
(TLBO) are widespread. Particle Swarm Optimization (PSO) [16], a popular optimization method,
mimics some animals' social behavior in nature, such as the group of fish and birds. Harris Hawks
Optimizer (HHO)
1
[17] simulates the cooperation and hunt behavior of Harris' hawks. Slime Mould
Algorithm (SMA)
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[18], a new stochastic optimizer, is based on the oscillation mode of slime in nature.
Unlike the algorithms mentioned above, Sine Cosine Algorithm (SCA) [19] is inspired by two
functions: sine and cosine functions in the mathematical theory. The emergence of SCA has attracted
widespread attention, and it is regarded as one of the solutions to many optimization problems.
However, for those complexes, high-dimensional and multi-modal problems, SCA's overall
performance is not behaving well as expected. Besides, as the dimensions gradually increase, the
possibility of falling into a local optimum goes up. Therefore, researchers try to design new SCA
versions to resolve various practical problems better. Zhu et al. [20] designed an orthogonal SCA with
kernel extreme learning machine (OLSCA-KELM) to evaluate the Sino cooperative education project.
Liu et al. [21] proposed a boosted covariance guided SCA with support vector machines (COSCA-SVM)
to predict cervical hyperextension injury. Tu et al. [22] developed a chaotic local search SCA with an
adaptive support vector machine framework (RF-CSCA-SVM) to predict fresh graduate students'
entrepreneurial intention. Lin et al. [23] designed novel chaos enhanced SCA with the fuzzy k-Nearest
Neighbor (CESCA-FKNN) framework to predict students' intentions for master programs. Chen et al.
[24] proposed an improved SCA with an opposition-based learning strategy (ISCA) for parameter
detection of the solar cells modules. Tan et al. [25] adopted a hybrid structural equation modeling-
artificial neural networks (SEM-ANN) approach to predict the drivers of behavioral intention. Bojan-
1
https://aliasgharheidari.com/HHO.html
2
https://aliasgharheidari.com/SMA.html

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Dragos et al. [26] proposed two fuzzy controllers based on grey wolf optimizer (GWO) for
electromagnetic actuated clutch systems. Precup et al. [27] designed an experiment-based approach to
teach optimization techniques (OT) in the systems engineering curricula.
Fan et al. [28] proposed an enriched fruit fly optimization with SCA (SCA-FOA) to solve three
engineering problems. Chen et al. [29] industrialized an advanced orthogonal learning-driven multi-
swarm SCA (OMGSCA) for resolving complex optimization cases. In work [30], the authors proposed
chaotic oppositional SCA to deal with global optimization problems. Kumar et al. [31] presented a novel
hybrid SCA combined with binary PSO (BPSO-SCA) to perform cluster analysis. Rizal et al. [32]
proposed a hybrid sine-spiral dynamic algorithm (HSSDA) based on the sine equation and a spiral
equation to optimize the autoregressive with the exogenous input model flexible link. Rizal et al. [33]
designed a hybrid spiral and sine cosine algorithm (SSCA) to resolve global optimization problems.
Chen et al. [34] designed a multi-strategy enhanced SCA (MSCA) to resolve three constrained practical,
real-world problems. Tawhid et al. [35] presented a new multi-objective optimization method based on
the basic SCA (MO-SCA) to tackle the multi-objective engineering design problems. Zhou et al. [36]
introduced a mutation operator of differential evolution to enhance SCA (SCDE) performance for
sensor nodes' location. Guo et al. [37] developed a hybrid SCA based on the optimal neighborhood and
quadratic interpolation strategy (QISCA) to resolve three representative engineering problems. Authors
of [38] designed a bare-bones sine cosine algorithm for coping with real-life global optimization. In
work [39], they suggested a memory-guided SCA (MG-SCA) for tackling four notable engineering cases.
Zhou et al. [40] designed a multi-core sine cosine algorithm with SSA, GWO, and levy fight strategy
(SGLSCA) to harvest the solutions to three practical engineering optimization problems. Ji et al. [41]
proposed an adaptive chaotic SCA (ASCA) for constrained and unconstrained optimization.
In this paper, we propose a boosted SCA variant to further strengthen the capability of global
exploration and local exploitation by drawing into two strategies, including spiral motion and Gaussian
mutation. The proposed enhanced SCA is named FGSCA. The spiral motion role is to update further
the population generated by SCA to strengthen local search capacity based on the original SCA.
Meanwhile, Gaussian mutation is employed to effectively increase the SCA population's diversity and
enhance local exploration ability. Finally, the best population is obtained through the way of greedy
selection. In short, the combination of the two strategies dramatically enhances the performance of
the basic SCA and stays a proper balance in exploitation and exploration.
To verify the proposed method's performance, 23 well-known benchmark test problems and 30
standard IEEE CEC2014 benchmark test problems are used to examine the performance of the
proposed FGSCA. The corresponding competitor contains eight well-known MAs: WOA [16], GWO
[16], MFO [16], BA [42], FA [42], PSO [43], ACO [43], six recent reported SCA variants: modified SCA
with neighborhood search and levy mutation (MSCA) [44], adaptive SCA with PSO (ASCA-PSO) [45],
modified SCA (m-SCA) [46], SCA with differential evolution (SCADE) [47], SCA based on Cauchy and
Gaussian strategy (CGSCA) [48], SCA with opposition-based learning (OBSCA) [49], and ten improved
MAs: enhanced associative learning-based exploratory whale optimizer (BMWOA) [50], A-C parametric
WOA (ACWOA) [51], improved WOA (IWOA) [52], cloud model bat algorithm (CBA) [53], bat
algorithm based on collaborative and dynamic learning of opposite population (CDLOBA) [54],
improved fruit fly optimization algorithm (IFOA) [55], adaptive mutation fruit fly optimization
(AMFOA) [56], GWO combined with opposition-based learning and levy flight (OBLGWO) [57],
ensemble particle swarm optimizer (EPSO) [58] and improved IJAYA (IJAYA) [59]. Table 1
demonstrates the motivation and limitation of the above six reported SCA variants and ten state-of-

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the-art MAs to further exhibit the advance and necessity of the research work about FGSCA. The final
comparative experimental results show that the comprehensive performance of FGSCA based on the
53 benchmark problems sets is better than the above eight well-known MAs, six reported SCA variants
and ten state-of-the-art MAs. In addition, FGSCA is also compared with the above twenty-four
algorithms to solve three practical engineering problems. The results show that FGSCA achieves the
best results among all competing algorithms. Thus, FGSCA can be expected to become a mighty and
efficient tool to deal with various complex and formidable optimization problems.
Table 1. The motivation and limitations of various involved algorithm variants for comparison with
the proposed FGSCA
Authors
Variants
Motivation
Limitation
Reported SCA variants
Qu et al.
[44]
MSCA
MSCA keeps the right balance in
exploring and exploiting and
performs a fast convergence rate.
The time complexity of MSCA compared
to other competing algorithms needs
further demonstration.
Issa et al.
[45]
ASCA-PSO
ASCA-PSO enables a good
population diversification and
preserves the best information on
the position.
ASCA-PSO is compared with nine basic
algorithms. Its performance compared to
some improved algorithms has yet to be
verified.
Gupta et al.
[46]
m-SCA
m-SCA can jump out from local
optima and exploit all the
promising search regions.
The superiority of m-SCA over other
improved SCA variants requires further
study.
Xu et al. [47]
SCADE
SCADE has more powerful
exploitation capabilities than the
original SCA.
SCADE has not yet performed
performance evaluation with other
competing algorithms on the CEC2017 test
set
Kumar et al.
[48]
CGSCA
CGSCA effectively inhibits the loss
of population diversity and avoids
falling into the local optimum.
CGSCA is developed to extract the
maximum power point of solar
photovoltaics; it remains further research in
multimodal functions, hybrid functions, and
composition functions.
Adb Elaziz
et al. [49]
OBSCA
The global exploration ability of
OBSCA is stronger than the
original SCA.
The local exploitation capabilities of
OBSCA are required to be further
improved.
Reported WOA variants
Heidari et al.
[50]
BMWOA
BMWOA aims to improve and
balance the capability of
exploring and exploiting.
The application of BMWOA in practical
engineering needs further confirmation.
Elhosseini et
al. [51]
ACWOA
ACWOA focus on a further
balance between exploration and
exploitation.
The performance of ACWOA relies on two
parameters, including A and C.
Tubishat et
al. [52]
IWOA
IWOA effectively alleviates the
disadvantage that the original
WOA is prone to fall into the local
The performance of IWOA on various
function test sets requires further validation.

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optimum.
Reported BA variants
Zhou et al.
[53]
CBA
CBA has more advantages in
exploring and exploiting capability
than the primary BA.
The performance of CBA in complex
engineering optimization problems requires
to be further confirmed.
Yong et al.
[54]
CDLOBA
CDLOBA has a special
convergence rate compared with
the primary BA.
The time complexity of CDLOBA and the
comparative experiment of improved
algorithms need to be further realized.
Reported FOA variants
Tian et al.
[55]
IFOA
IFOA adopts the inertia weight
function to balance the global
exploration and local fine-tuning.
The time complexity of IFOA and the
application of hybrid functions require to
study further.
Wang et al.
[56]
AMFOA
AMFOA adopts an adaptive
mutation strategy to improve
convergence speed as quickly as
possible.
The performance of AMFOA on various
function test sets is expected to be further
conducted.
Reported GWO variants
Heidari et al.
[57]
OBLGWO
The global exploration and local
exploitation capabilities have been
enhanced compared to the original
GWO.
The performance comparison experiments
between OBLGWO and other reported
improved algorithms are expected to be
further verified.
Reported PSO variants
Lymn et al.
[58]
EPSO
EPSO combines various PSO
variants to boost the diversity of
the population.
The computation complexity of EPSO is
relatively high.
Reported JAYA variants
Yu et al. [59]
IJAYA
IJAYA is designed to identify
unknown parameters in
photovoltaic models, and the
performance is quite splendid.
The performance of IJAYA on unimodal,
multimodal, hybrid, and composition
function tests are expected to be further
validated.
2 Related works
2.1 Spiral motion
As mentioned above, the introduction of spiral motion is inspired by MFO, which aims to
strengthen the capacity of exploitation based on the original SCA. As the name suggests, the core
principles of spiral motion are strictly related to MFO. It establishes the original MFO to exploit further
the population generated by SCA. Because of the powerful capabilities shown in resolving real-world
optimization problems, we consider the possibility of combining MFO to enhance the performance of
SCA. As we can see, the spiral motion of MFO is extensively applied in resolving real-world problems.
Xu et al. [60] proposed a boosted MFO with a multi-strategy to improve the original MFO. Elsakaan et

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al. [61] presented an improved MFO to resolve the non-convex economic dispatch problem with valve
point effects and emissions. Sayed et al. [62] designed a hybrid MFO to select the best discriminating
mitosis cells' features. Wang et al. [63] utilized MFO to realize the multi-energy system's minimum
operation cost. Said et al. [64] adopted MFO to cluster the abdominal image to gain the initial segmented
image. Xu et al. [65] proposed an enhanced MFO with Gaussian mutation and chaos strategy to apply
in financial stress prediction. Yue et al. [66] presented an improved MFO to evaluate the air target threat
in the gray neural network model.
2.2 Gaussian mutation
In contrast, the introduction of Gaussian mutation into SCA aims to strengthen the population
diversity and speed up the objective function convergence rate. As we can see, various MAs were
combined with Gaussian mutation to strengthen the primary method's performance in many kinds of
literature. Patwal et al. [67] combined Gaussian mutation with PSO to further reinforce its exploitation
capability. Xu et al. [68] designed an improved MFO technique with cultural learning and Gaussian
mutation to significantly promote MFO. Zhang et al. [69] proposed a new fruit fly optimization with a
Gaussian mutation operator and a chaotic local search strategy for feature selection problems. Chen et
al. [70] put forward a boosted bacterial foraging optimization with a chaotic design and a Gaussian
mutation to strengthen its performance.
Liu et al. [71] introduced Gaussian mutation into a shuffled frog-leaping algorithm to decrease the
likelihood of running into local optimality. Luo et al. [72] presented a hybrid WOA, including Gaussian
mutation, to strengthen the evolving population's diversity level. Chen et al. [73] presented a hybrid
genetic algorithm with a single random point and Gaussian mutation to speed up the method's
convergence. Luo et al. [74] designed an improved multi-strategy grasshopper optimization algorithm
with Gaussian mutation, Levy-flight strategy, and opposition-based learning to balance exploration and
exploitation. Feng et al. [75] promoted monarch butterfly optimization integrated with neighborhood
mutation with crowding and Gaussian mutation to prevent premature convergence. Han et al. [76]
combined adaptive Cauchy and Gaussian mutations with differential evolution algorithms to attain
global optimal solutions effectively. Zhu et al. [77] proposed an enhanced dandelion algorithm with
Gaussian mutation and Levy-flight strategy to examine three probability models. Kaur et al. [78]
designed a hybrid boost GWO with a multi-strategy, including Cauchy, Gaussian, and an opposition-
based learning strategy, to gain the best solution.
2.3 Algorithm classification
Compared with traditional methods, optimization algorithms cover a wide range, from land to
ocean, water to air, living animals and plants, microorganisms to inanimate natural phenomena, physical
and chemical mathematics, nonlinear science, and complex adaptive systems. There is still no unified
standard for classifying optimization algorithms, and there are different classification methods from
different angles. Thus, starting from the inspiration of optimization algorithms, this paper divides more
than some well-known algorithms published into four categories. The detailed classification results are
shown in Table 2.

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Table 2. Optimization algorithm classification and representative algorithms
Type
Representative algorithms
Human-based optimization
Teaching–Learning-Based Optimization (TLBO) [79]
Swarm-based optimization
Particle Swarm Optimization (PSO) [80], Slime Mould Algorithm
(SMA) [18], HHO, HGS
Physic-based optimization
Simulated Annealing (FA) [81], Sine Cosine Algorithm (SCA) [19],
Gravitational Search Algorithm (GSA) [82], INFO, RUN
Evolutionary algorithms
Genetic Algorithm (GA) [83], Differential Evolution (DE) [84],
Genetic Programming (GP) [85], Evolutionary Programming (EC)
[86].
2.4 Single-objective optimization
Optimization algorithms can effectively solve complex optimization problems such as single-
objective, multi-objective, high-dimensional, discrete, and complex constraints [87, 88]. Based on the
No Free Lunch (NFL) theorem [89], this paper mainly designs an improved sine-cosine optimization
algorithm to solve multiple single-objective engineering optimization problems. Single-objective
optimization means maximizing or minimizing an objective function. Besides, the problem can also
include a set of constraints. Constraints fall into two categories: equality constraints and inequality
constraints. The minimization model for single-objective optimization is generally as follows:
󰇛󰇜={󰇛󰇜}
（1）
󰇛󰇜
（2）
󰇛󰇜
（3）

（4）
where  is the number of variables,  is the number of inequality constraints,  is the number of
equality constraints,  is the lower bound of the  variable, and  is the upper bound of the
 variable.
With the complexity of dimension and constraint conditions, traditional methods are gradually
become weak in solving such problems. The swarm optimization algorithm is generally simple,
comprehensible, operable, and reliable, which is regarded as an effective solution.
3 Structure of SCA
Recently there are many meta-heuristics have been proposed, such as hunger games search (HGS)
[90], the weighted mean of vectors (INFO) [91], colony predation algorithm (CPA) [92], and Runge

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Kutta optimizer (RUN) [93]. SCA is a meta-heuristic method proposed in 2016 with shallow efficacy
[19]. We stay neutral about the originality of the SCA in this study, and we only focus on the
performance optimization of SCA. The SCA is a population-based technique that starts with the
process of searching for random solutions. As we all know, all random optimization emphasizes the
exploration and exploitation of attribute space. In SCA, two different mathematical expressions are
used to update the solution for the balance of exploration and exploitation. These expressions can be
shown below [19]:

󰇫
󰇛󰇜
 

󰇛󰇜
 
（1）
where 
 is the  dimension of the  position at the  generation population, , and
 are all random numbers,  is the  dimension of the terminal point at the  generation
population, and  expresses absolute value.
Four parameters need to be introduced. The variable  is to transform two different search
spaces, which realizes the alternation in the inner area of  and  or in its outer space randomly.
It decreases linearly from  to 0, balancing exploration and exploitation while searching space. Further,
the parameter 󰇟 󰇠 is a random number to update the next solution in the correct direction.
The parameter  is a lucky number from 0 to 1 that aims to enhance randomly () or weaken
() the effect of the target agent on the present agent. Finally, the parameter  represents that
the sine and cosine functions are selected in Eq. (1) with equal probability.
The detailed formula for parameter  is as follows [19]:
󰇛 
󰇜
（2）
where  indicates the current evaluation,  is the maximum number of evaluations, and
 is constant.
As can be seen in Fig. 1, it shows the current trend of the solution toward the target solution. The
core formula of SCA is introduced to two mathematical functions, including sine and cosine functions.
The effect on sine and cosine function depends on how to gain the next solution and define a place
between the current solution and the sensible solution in the whole search space. The alternating pattern
of sine and cosine functions surrounds two solutions of the equation and updates iteratively, ensuring
that the search space is fully utilized between the two solutions.

9
Figure 1. Impacts of Sine and Cosine in the next position.
Figure 2. The basic principle of SCA.
Fig. 2 shows in detail the influence of sine and cosine functions in a random number on the next
solution. It indicates how changing the value of the sine and cosine function to seek the next step of
the current solution as well. Two trends can be analyzed. One is toward the inner direction of the
current search space, and the other is in the direction of outside space. It is achieved by defining a
random parameter  from 0 to 2π.
The general steps of SCA begin with the optimization process of a set of initial random solutions.
With the number of evaluations gradually increasing, the best solution currently retained is regarded as
the target solution. The search space is slowly explored for the next solution by updating the formula,
including sine and cosine functions. The SCA pseudo-code is shown in Algorithm 1.
Algorithm 1. Pseudo-code of SCA
Initialize a set of search agents (X);
Do
Evaluate each search agent by the objective function;
Update the best agent gained so far ();

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Update , ,  and ;
Gain the next swarm by Eq. (1);
While ()
Return the best solution and position.
4 Proposed FGSCA
4.1 Spiral motion
Spiral motion is the central combination of MFO, attained from moths' navigation method [16,
94-98]. The core step is that the next location of the solution gained by SCA is further updated by using
spiral motion.
First, the population gained by SCA is presented as the initial moths, which is as follows [99]:






    

  
  
   

    





（3）
where  means the population gained by SCA, which is also regarded as moths of spiral motion.
Besides,  implies the size of moths, implies the size of each individual of moths, and 
means  dimension of  individual of . The individuals of  are regarded as
candidate solutions.
Then, the collection of flames is expressed in matrix form. The expression for flames is shown
below [16]:






    

  
  
   

    





（4）
where  means the sorted population gained by SCA,  means  dimension of
 individual of .
The distance between months and flames can be described as follows [16]:
 
（5）
where 󰇟󰇠, 󰇟󰇠.
For a better balance of exploration and exploitation,  is defined below, based on the adaptive
principle for numbers of flames [16].
󰇛 
󰇜
（6）
where parameter  is the current iteration, is the maximum number of the iteration, and 
represents the size of moths, and round means round off.

11
Finally, each dimension value of each individual of the new population  updated by spiral
motion is expressed as below [16]:
 󰇫󰇛󰇜 
󰇛󰇜 
（7）
where  is the  dimension of  individual of the new population gained by spiral motion,
 is a constant and equal to 1 that controls the shape of the logarithmic spiral function,  is defined
as below [16]:
󰇛󰇜
（8）
󰇛󰇛󰇜󰇜
（9）
where  indicates the current evaluation,  is the maximum number of assessments,  is
a linearly decreasing function,  is a random number from 0 to 1, and the range of  is from -2
to 1. There are two exceptional cases to be considered. The moth moves toward the flame when  is
equal to -1; the other is that the moth keeps away from the flame when  is equal to 1. For better view,
the simple diagram of spiral motion is shown in Fig. 3
The pseudo-code of spiral motion is expressed below, where  means the population size and
 represents the population dimension value.
Figure 3. The moth goes forward by logarithmic spiral in the search space.
Algorithm 2. Pseudo-code of spiral motion
Input: , , , ;
1. Define the parameter  and  by Eq. (6) and Eq. (9);
2. 
3. 
4. Define parameter  by Eq. (8);
5. Update each dimension value of each individual by Eq. (5) and Eq. (7);

12
6. End For
7. End For
Output: .
4.2 Gaussian mutation
Considering the possibility that SCA can fall into local optimality, we introduce Gaussian mutation
[100]. The so-called Gaussian mutation operation replaces the original population with a random
number that conforms to the mean value and the variance. According to the normal distribution
characteristics, Gaussian mutation also focuses on a local area near the original individual [101] and has
shown great effectiveness in many optimizers [102-108]. Gaussian mutation involves adding a random
value from the normal distribution to each vector element to create a new descendant. The Gaussian
density function is shown as follows [99]:
󰇛󰇜󰇛󰇜


（11）
where parameter  and  indicates the variance value and mean value. Furthermore, the normal
distribution becomes the standard normal distribution when the mean and variance values are equal to
0 and 1.
We break the rules of SCA individuals moving to another location in the search space and increase
diversity for the next generation of individuals through Gaussian mutation. The search agents are
mutated with each evaluation, which can be described as follows [99]:
󰇛󰇜
（12）
where  and  mean the  position and mutated position of the population  and , and 
is the mutation operator.  is a random number that satisfies a normal distribution, and  is
used to adjust the mutation operator, equal to 1. The pseudo-code of Gaussian mutation is expressed
as below, where  means the population size.
Algorithm 3. Pseudo-code of Gaussian mutation
Input: X, ;
1. 
2. Update the population  by Eq. (12);
3. End For
Output: G.
4.3 Framework of FGSCA
The overall framework of FGSCA is shown in Fig. 4. The original SCA is combined with two
strategies, including spiral motion and Gaussian mutation. The former is to enhance the capacity of the
ability of exploitation. The latter improvement is useful for the diversity of the population. Suppose
that the populations F and G are obtained by updating the spiral motion and Gaussian strategies,

13
respectively. The core of FGSCA is shown below [16, 19, 99]:
 󰇫 
 
（13）
where  and  are the -th individual of the -th generation population F and G,  and
 are the -th individual fitness value of the -th generation population F and G, and  is
the -th individual of the next  generation S.
The pseudo-code of FGSCA is shown below, where parameter  means the current
evaluations and  represents the generation of the current population. Fig. 4 shows the flowchart of
the proposed FGSCA, which is mainly composed of 5 parts. First, the initialization population includes
the definition of some vital parameters. Then, start the primary operation. Next, use the sine cosine
updating formula to obtain the new population X. Based on the population X and the given parameters,
the spiral motion is used to obtain the population F. Similarly, Gaussian mutation is used to obtain the
population G. Finally, based on the two populations F and G, the selected population S is obtained
through greedy selection. The main principle of greedy selection is to compare each individual's fitness
value in the population F and the population G one by one and use the individual with a small fitness
value as each individual in the final next-generation population X. The pseudo-code of FGSCA is
available at the website of https://github.com/zhouwei35/FGSCA.
Algorithm 4. Pseudo-code of FGSCA
Input：Population size , Max evaluations , Upper boundary , Lower boundary ,
Population dimension , Objective function ;
Output：;
Step 1: Initialization population
1) Define the parameter  and ;
2) Initialize population 󰇛󰇜;
3) Evaluate the population  and gain the fitness values;
4) ;
5) Sort population  by fitness values and obtain the  and ;
Step 2: Main operation
6) While 
7) If 
8) Evaluate the population ;
9) ;
10) Update the  and ;
11) End
Step 3: Sine cosine updating
12) Define the parameter  and ;
13) Update the new population  by Eq. (1);
Step 4: Spiral motion
14) Obtain the population  by Algorithm 2 according to the input of , ,
and ;
Step 5: Gaussian mutation
15) Define the parameter ;

14
16) Obtain the population  by Algorithm 3 according to the input of  and ;
17) Bring back the solutions of population  and population  going outside;
18) Evaluate the population  and population ;
19) ;
20) Obtain best population  by greedy selection in population  and population ;
21) Regard population  as the next generation population ;
22) ;
23) End while
24) Output .
Figure 4. The flowchart of FGSCA
4.4 Operation of FGSCA
The operation of FGSCA is described in detail in this section as shown in Fig. 5. Any optimization
algorithm is inseparable from the initialization population operation. The general flow chart of the
optimization algorithm is shown in Fig. 6. First, initialize the initial population , representing by a
matrix of rows and  columns. The number of rows  represents the number of individuals, and
the number of columns  represents the number of dimensions of the problem to be solved. The
optimal individual 
 can be obtained by evaluating the population . Then, the 1-th generation
population is obtained through the SCA update formula. Furthermore, new populations F and G are
obtained by means of spiral motion and Gaussian mutation strategy, respectively. Similarly, the optimal
individuals 
 and
of populations F and G is obtained by the same way. By comparison, the
smallest individual  among the optimal individuals in all current populations can be obtained.
Population S is obtained from populations F and G by greedy selection. Therefore, through the above

15
update methods, the current optimal individual  and the new population S are obtained in the
first generation update. Finally, the population S is updated as the initial population of the next
generation update. By repeating the above operations, the n-th generation population  and the
optimal individual  is finally obtained.
Besides, each method has 300,000 evaluation times to consume during a random test. When the
number of evaluations is exhausted, it means that the algorithm will end the current operation. For
FGSCA, the number of assessments is mainly consumed in two places. One is the evaluation of the
initial population or the newly obtained population, and the other is the evaluation of the two
populations gained by spiral motion and Gaussian mutation. Suppose there are 30 individuals in one
population, and each individual is evaluated with one assessment. It indicates that SCA consumes 30
evaluations to update each generation's population, while FGSCA consumes 90 checks to update each
generation's population. One operation process of FGSCA ends when 300,000 evaluation times are
exhausted.
Figure 5. The operation process of FGSCA

16
Figure 6. The general flow of optimization algorithm
4.5 Computational complexity
The computational complexity (CC) of an optimizer at hand can be determined when such an
algorithm cost to reach the final result. In this paper, the CC of the proposed FGSCA consists of six
parts: initialization step, fitness assessment step, updating step of agents, spiral motion stage, Gaussian
mutation level, and optimum population selection. As we know, the complexity only focuses on the
four aspects because fitness assessments vary for each case. The CC of the FGSCA to initialize the N
agents is O(N). The CC of the updating approach in the swarm is O(T×N)+O(T×N×D), counting
looking for the best agent and updating individuals' location. T states the maximum number of
valuations, and D signifies the dimension of specific problems. The CC of the spiral motion part is
O(T×N×D). The CC of the Gaussian mutation level is O(T×N). Lastly, the CC of the optimum
population selection idea is O(T×N). In brief, the CC of the basic SCA is O(N×(1+T+T×D)), and the
final CC of FGSCA is O(N×(1+T×(1+2×D+2×N))).
5 Experimental results
The best SCA version is selected in this section: FGSCA, using spiral motion and Gaussian
mutation update strategies on 23 well-known benchmark problems and 30 standard IEEE CEC2014
benchmark test problems. To test the performance of FGSCA, the proposed FGSCA is compared with
other competitive MAs. Meantime, FGSCA also conducts comparative experiments with other
improved MAs. Also, wall-clock time cost, diversity, and balance of FGSCA are analyzed, as well. All
the following experiments are conducted in the same environment, and 30 search agents are used for
300,000 evaluations. Each involved function is independently tested 30 times to reduce the impact of
randomness.

17
5.1 Benchmark function validation
To further evaluate the proposed method, 53 benchmark functions are adopted to test the
performance of FGSCA further. The detailed descriptions of 23 benchmark functions are listed in the
appendix (see Table A.1). Furthermore, 30 detailed descriptions of the CEC2014 benchmark functions
are listed in the appendix (see Table B.1). This paper adopts these 53 functions for experiments. In fact,
23 benchmark functions are divided into three types, including single-peak, multi-peak, and fixed-
dimensional functions. An additional composition function is added for CEC2014 benchmark functions.
There is only one global optimal solution for single-peak functions in the entire space, so they can be
used to test the proposed method's exploitation performance. There are many local optimums in space
as a whole for multi-peak functions. With the dimension increases, the local optimal solution may appear
more frequently. Therefore, the proposed FGSCA is tested on these benchmark functions, aiming to
evaluate the proposed method's exploration and exploitation capabilities more justly and fairly, as per
rules of fair comparison in artificial intelligence [109-112]. Fair comparison ensures that all methods
have been tested under the same condition without a specific bias in favor of a specific method, which
has been obeyed in different artificial intelligence research [113-116]. It is well accepted in different
optimization and machine learning research [117-120], and we also followed it in our optimization
experiments.
The proposed algorithm's performance is mainly evaluated when the simulation results are
obtained from the following two parts. One is the average result (Avg), and the other is the standard
deviation (Std). The experimental results of FGSCA are annotated in bold in the table. All experiments
are conducted in the same environment, using 30 search agents for 300,000 evaluations in the paper. It
is vital to independently test each function involved 30 times to minimize the impact of randomness.
A non-parametric statistical test with a significance level of 0.05 using the Wilcoxon signed-rank test is
adapted to test whether the performance of FGSCA is significantly better than other methods. The
subsequent simulation experiments are conducted on a PC with a frequency of 3.4GHz, and the
operating system is win10. All comparative tests in this paper are implemented in MATLAB R2016a.
Besides, the symbol "+/=/-" indicates whether FGSCA is superior, equal to, or lower than other related
methods. Besides, the average ranking of these test algorithms is listed at the end of the table. The
average value (ARV) is expressed as the average ranking value of each algorithm [65].
5.2 Influence of two strategies
This section investigates the impact of the random introduction of spiral motion and Gaussian
mutation on SCA performance. In that case, four different SCA variants are designed. In Table 3, "F"
and "G" respectively represent "spiral motion" and "Gaussian mutation", "1" means that SCA combines
this strategy, and "0" means that this strategy is not adopted. For example, FGSCA shows that SCA
combines spiral motion and Gaussian mutation. FSCA exhibits that SCA combines spiral motion
without Gaussian mutation. GSCA reveals that SCA combines Gaussian mutation without spiral
motion.
A total of four SCA variants are examined on 53 benchmark functions to assess these methods'
performance. The Friedman test is used to obtain the final average ranking of the involved techniques
to study the differences between different strategy combinations. Table 4 shows the final ranking and

18
ARV of the four SCA variants. As can be seen, FGSCA ranks first, with an ARV value of 1.33962.
SCA's ARV value combined with spiral motion and Gaussian mutation is 3.20755 and 2.13208,
respectively. It demonstrates that combining these two strategies simultaneously has the greatest effect
on the original SCA.
On the one hand, the use of spiral motion further effectively boosts the local exploitation ability.
On the other hand, the adoption of Gaussian mutation improves the population's diversity and avoids
falling into the local optimum. Thus, FGSCA with the best performance is used for subsequent
comparative experiments on 53 benchmark test problems.
Fig. 7 shows the convergence curves of some typical benchmark function tests related to FGSCA,
FSCA, GSCA, and SCA. It can be seen that FGSCA has the best overall performance on F3, F6, F12,
F21, F26, F32, F41, F46, and F50. For single-peak functions F3 and F6, FGSCA has the fastest
convergence speed and the best convergence value. It is because spiral motion and Gaussian mutation
significantly improved the exploitation capabilities of the original SCA. For multimodal function F12,
FGSCA has a stronger ability to jump out of local optimum than FSCA, GSCA, and SCA. For fixed-
dimensional function F21, FGSCA has a smaller optimized value than FSCA. In the CEC2014 function
test set, FGSCA is overall superior to the other three SCA variants. FGSCA shows fast convergence
speed and convergence value on the unimodal function F26. For the multimodal function F32, the
convergence speed of FGSCA is slightly inferior to that of FSCA, but the final convergence value is
the smallest among the four variants. For the fixed-dimensional functions F41 and the composition
functions F46 and F50, FGSCA gains a faster convergence speed and a better convergence value.
Table 3. Various SCA variants with two strategies
F
G
FGSCA
1
1
FSCA
1
0
GSCA
0
1
SCA
0
0
Table 4. Average ranking values of various SCA variants
Rank
ARV
FGSCA
1
1.33962
FSCA
4
3.20755
GSCA
2
2.13208
SCA
3
3.11321

19
Figure 7. Convergence curves of 9 benchmark functions
5.3 Separate analysis of FGSCA and SCA
Fig. 8(a) shows the mathematical representation of four types of benchmark functions, including
single-peak functions F1, multi-peak functions F10 and F32, fixed-dimensional functions F23 and F39,
and composition functions F46. Fig. 8(b) shows the historical search curve of FGSCA on these
functions. It can be seen that the positions of the individuals are scattered around the optimal solution
in the entire search space in the case of 1000 iterations. Fig. 8(c) shows the trajectory of FGSCA in the
first dimension during each iteration. It indirectly illustrates the individual's fluctuation state in the first
dimension during the search space while gaining the optimal value. Furthermore, FGSCA is more
volatile in the early stage and tends to be stable in the later stage than SCA, which means that it is highly
likely that the population will spread around the optimal solution. As for SCA, the individual's value in
the first dimension oscillates up and down, and it is likely to get stuck in the exploration stage rather
than in the exploitation stage. Fig. 8(d) shows the average fitness value of all groups in FGSCA during
1000 iterations. It can be seen that, as the number of iterations increases, the average fitness value of
FGSCA on F1, F10, F32, F39, and F46 gradually decreases. Even though FGSCA fluctuates somewhat
in F33, it tends to decline overall. From the curve point of view, the average fitness curve of FGSCA
not only converges faster than SCA, but the final convergence value is also the smallest. Fig. 8(e) shows
the convergence values of FGSCA and SCA in 1000 iterations. It is easy to obtain that FGSCA
converges faster than SCA, but the final optimized value is the smallest.

20
Therefore, we comparatively analyze the agent trajectory, average fitness value, and convergence
value of FGSCA and SCA by screening six representative test functions from 53 benchmark functions.
The final result shows that FGSCA performs better than SCA. Specifically, FGSCA has a stronger
ability to obtain the optimal solution position than SCA. Not only does it converge faster, but the final
convergence value is the smallest, as well.
Figure 8. (a) Graphical representations of benchmark mathematical functions, (b) Search history of
FGSCA, (c) trajectory of FGSCA in the first dimension, (d) the average fitness of all group of FGSCA,
and (e) the convergence curve of FGSCA and SCA
5.4 Comparison with other reported well-known optimizers
In this section, the proposed method is compared with some famous MAs, such as WOA [16],
GWO [16], MFO [16], BA [42], FA [42], PSO [43], ACO [43]. The optional and specific parameter
settings are shown in Table 5 and Table 6. Each method runs in the same test environment. The detailed
experimental results are shown in Table 7. Table 7 shows the Avg index and Std index of involved
procedures based on 30 random runs. The test results of FGSCA are marked in bold. From the overall

21
performance perspective, FGSCA ranks first compared to other involved methods, and we give the
specific ranking at the end of Table 7. Therefore, we infer that FGSCA is superior to other methods in
53 benchmark functions.
Table 7 shows the average result of each method performing 30 random convergences, as well. In
principle, the smaller the Avg, the better the optimization result. For single-peak benchmark functions
F1-F7 and F24-F26, FGSCA showed the best results in 300,000 evaluations. The specific performance
is that the final optimization values of F1-F6 and F25 are the smallest. The main reason is that the
introduction of spiral motion and Gaussian mutation significantly promotes the original SCA's ability
to jump out of the local optimal.
For F7, the optimal solution obtained by ACO is slightly better than FGSCA, but within the error
range, the results of FGSCA are still satisfactory. For F24, BA is more competitive than FGSCA, but
FGSCA optimization results are superior to most other involved methods. For F26, FGSCA performs
better than WOA, GWO, MFO, FA, and ACO. For multimodal functions F8-F13 and F27-F33, FGSCA
has significant advantages over other MAs, as well. For example, the final convergence value of FGSCA
on F9-F13 and F30-F31 is the smallest. It shows that FGSCA has a pretty good early-stage exploration
and late-stage exploitation abilities. For fixed-dimensional functions F14-F23 and F34-F43, FGSCA
shows the smallest optimized value on F16-F17, F21-F23. On F15, F19, and F20, FGSCA has certain
competitiveness with WOA. For the composition Functions F44-F53, FGSCA obtained the smallest
optimized values on F46-F48 and F50-F53. This shows that the introduction of spiral motion enhances
the population's ability to jump out of the local optimal, and the introduction of the Gaussian strategy
improves the population's diversity. Fig. 9 shows the convergence curves of FGSCA, WOA, GWO,
MFO, BA, FA, PSO, and ACO in F3, F6, F10, F12, F21, F31, F42, F48, and F53.
Also, we get the Std of each method in the degree of dispersion in Table 7, as well. The Std reflects
the degree of dispersion of the current data. In theory, the smaller the Std, the better the stability of
the method. It is easy to see that FGSCA has the smallest Std on F1-F5, F9-F12, F16-F17, F21-F23,
F25, F27, F30, F46-F48, F50-F52, indicating that the stability of FGSCA is better than other methods
on these benchmark functions. It can be seen from the symbol "+/=/-" that FGSCA is significantly
better than GWO, ACO, and WOA in 36, 27, and 34 functions among the 53 benchmark functions,
equal to them in 10, 13, and 9 functions, respectively.
In short, through the analysis of the above series of experimental results, it can be inferred that
the introduction of spiral motion and Gaussian mutation has dramatically improved the perception
ability of the neighbors between individuals. The spiral motion aims to strengthen the original SCA's
local search ability, and the Gaussian mutation enhances the diversity of the population. Combining the
two strategies significantly improves the actual SCA's exploitation ability so that SCA has a strong ability
to jump out of the local optimum. Therefore, the proposed method resolves various benchmark
problems more effectively than other involved MAs.
Table 5. The optional parameters for involved algorithms
Optional parameter
Value
Population size
30
Random tests number
30
Maximum evaluations number
300000
Table 6. The parameters setting for involved comparative algorithms

22
Method
Random parameter
Fix parameter
FGSCA
󰇛󰇜; 󰇛󰇜

SCA
󰇛󰇜;󰇛󰇜 󰇛󰇜

WOA
󰇛󰇜; 󰇛󰇜
󰇛󰇜󰇛󰇜; 󰇛󰇜

GWO
󰇛󰇜
-
MFO
󰇛󰇜 󰇛󰇜

BA
-

FA
-
; ; 
PSO
-
; ;  
ACO
-
; ; 
MSCA
󰇛󰇜;󰇛󰇜 󰇛󰇜


ASCA-PSO
󰇛󰇜;󰇛󰇜 󰇛󰇜
; ;  ;
 ;  
m-SCA
󰇛󰇜;󰇛󰇜 󰇛󰇜

SCADE
󰇛󰇜;󰇛󰇜 󰇛󰇜
 ; ; 

CGSCA
󰇛󰇜;󰇛󰇜 󰇛󰇜
;
OBSCA
󰇛󰇜;󰇛󰇜 󰇛󰇜

BMWOA
󰇛󰇜; 󰇛󰇜; 
󰇛󰇜
󰇟󰇠;

ACWOA
󰇛󰇜; 󰇛󰇜
󰇛󰇜󰇛󰇜; 󰇛󰇜

OBLGWO
󰇛󰇜
-
IWOA
󰇛󰇜; 󰇛󰇜;
b; 
CBA
-
 ;  
CDLOBA
-
 ;  
IFOA
-

AMFOA
-

EPSO
PSO: 󰇛󰇜 
󰇟󰇠
-
CLPSO: 󰇛󰇜 
󰇛󰇜
-
FDRPSO: 󰇛󰇜
-
HPSO-TVAC: 󰇛󰇜 
󰇛󰇜
-
-
LIPS: 
IJAYA
󰇛󰇜
-

23
Figure 9. Convergence curves of FGSCA and other seven well-known MAs on 9 benchmark
functions
Table 7. Results of FGSCA compared with seven well-known MAs on 53 benchmark functions.
F1
F2
F3
Avg
Std
Avg
Std
Avg
Std
FGSCA
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
WOA
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
GWO
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
MFO
1.66667E+03
3.79049E+03
2.83333E+01
1.72374E+01
2.83333E+01
1.72374E+01
BA
6.25090E-01
3.74925E-01
3.33745E+00
2.01618E+00
3.33745E+00
2.01618E+00
FA
1.14702E+04
1.20123E+03
4.78936E+01
2.99825E+00
4.78936E+01
2.99825E+00
PSO
9.79668E+01
1.13349E+01
4.77855E+01
4.31660E+00
4.77855E+01
4.31660E+00
ACO
2.55251E-238
0.00000E+00
6.41003E-163
3.14346E-162
6.41003E-163
3.14346E-162
F4
F5
F6
Avg
Std
Avg
Std
Avg
Std
FGSCA
0.00000E+00
0.00000E+00
2.18998E+01
1.77239E-01
9.83112E-27
3.44502E-26
WOA
7.93286E+00
1.81200E+01
2.43157E+01
3.51631E-01
5.04029E-06
1.68181E-06
GWO
1.12976E-151
3.22595E-151
2.63009E+01
8.37899E-01
5.24501E-01
4.22036E-01

24
MFO
6.16875E+01
9.02527E+00
2.11405E+04
3.86472E+04
3.34675E+03
6.64410E+03
BA
4.38093E+00
3.92253E+00
3.11352E+02
5.85184E+02
7.18796E-01
4.35923E-01
FA
4.04014E+01
2.02261E+00
7.70174E+06
1.69736E+06
1.13121E+04
1.25572E+03
PSO
3.83005E+00
1.83854E-01
8.06318E+04
1.58426E+04
9.94661E+01
1.17229E+01
ACO
5.30030E+01
8.16520E+00
3.89160E+01
1.06762E+02
5.66169E-29
1.52289E-28
F7
F8
F9
Avg
Std
Avg
Std
Avg
Std
FGSCA
9.83112E-27
3.44502E-26
-1.07225E+04
1.15043E+03
0.00000E+00
0.00000E+00
WOA
5.04029E-06
1.68181E-06
-1.21242E+04
1.06812E+03
0.00000E+00
0.00000E+00
GWO
5.24501E-01
4.22036E-01
-6.21678E+03
7.93315E+02
0.00000E+00
0.00000E+00
MFO
3.34675E+03
6.64410E+03
-8.56482E+03
9.97359E+02
1.55412E+02
4.32661E+01
BA
7.18796E-01
4.35923E-01
-7.19557E+03
6.68214E+02
2.51729E+02
1.96922E+01
FA
1.13121E+04
1.25572E+03
-4.22463E+03
2.49397E+02
2.30960E+02
8.83327E+00
PSO
9.94661E+01
1.17229E+01
-6.62923E+03
1.43659E+03
3.46922E+02
1.48305E+01
ACO
5.66169E-29
1.52289E-28
-8.48170E+03
4.97165E+02
5.04111E+01
1.90190E+01
F10
F11
F12
Avg
Std
Avg
Std
Avg
Std
FGSCA
8.88178E-16
0.00000E+00
8.88178E-16
0.00000E+00
6.93276E-30
1.98433E-29
WOA
2.90138E-15
2.22422E-15
2.90138E-15
2.22422E-15
1.17416E-06
1.30518E-06
GWO
7.75676E-15
9.01352E-16
7.75676E-15
9.01352E-16
2.42549E-02
1.48164E-02
MFO
1.34471E+01
9.03426E+00
1.34471E+01
9.03426E+00
1.21202E-01
3.06658E-01
BA
2.26626E+00
7.03548E-01
2.26626E+00
7.03548E-01
8.28222E+00
3.16646E+00
FA
1.59241E+01
4.04957E-01
1.59241E+01
4.04957E-01
2.26671E+06
1.21387E+06
PSO
7.83176E+00
2.52340E-01
7.83176E+00
2.52340E-01
3.55058E+00
3.93647E-01
ACO
2.64383E+00
5.94030E+00
2.64383E+00
5.94030E+00
1.80178E-01
3.06056E-01
F13
F14
F15
Avg
Std
Avg
Std
Avg
Std
FGSCA
6.95845E-03
2.00901E-02
1.72551E+00
9.72058E-01
7.09536E-04
4.99930E-04
WOA
1.13337E-03
3.34443E-03
9.98004E-01
1.33415E-14
4.29618E-04
2.81050E-04
GWO
3.88147E-01
2.15787E-01
4.06763E+00
4.02160E+00
1.01121E-03
3.65890E-03
MFO
2.12533E-01
7.22925E-01
3.61445E+00
3.58583E+00
1.79275E-03
3.79592E-03
BA
1.65729E-01
1.04656E-01
2.28460E+00
1.58066E+00
2.25394E-03
4.92825E-03
FA
1.67780E+07
4.26236E+06
9.98007E-01
6.12338E-06
1.02161E-03
1.55416E-04
PSO
1.56658E+01
1.65068E+00
3.75885E+00
2.52317E+00
1.00357E-03
9.89986E-05
ACO
1.06609E-01
5.52865E-01
2.77408E+00
2.29512E+00
3.38790E-03
6.90552E-03
F16
F17
F18
Avg
Std
Avg
Std
Avg
Std
FGSCA
-1.03163E+00
6.45393E-16
3.97887E-01
0.00000E+00
3.00000E+00
2.16932E-15
WOA
-1.03163E+00
6.25270E-15
3.97887E-01
2.03296E-10
3.00000E+00
9.22165E-08
GWO
-1.03163E+00
2.73495E-11
3.97887E-01
1.35214E-09
3.00000E+00
7.79429E-08
MFO
-1.03163E+00
6.77522E-16
3.97887E-01
0.00000E+00
3.00000E+00
2.15990E-15
BA
-1.03154E+00
9.43284E-05
3.97945E-01
6.35542E-05
3.00836E+00
6.17503E-03
FA
-1.03155E+00
7.09002E-05
3.97946E-01
6.00150E-05
3.00195E+00
1.55012E-03
PSO
-1.03148E+00
1.69491E-04
3.97971E-01
7.72072E-05
3.01084E+00
1.05018E-02

25
ACO
-1.03163E+00
6.77522E-16
3.97887E-01
0.00000E+00
3.90000E+00
4.92950E+00
F19
F20
F21
Avg
Std
Avg
Std
Avg
Std
FGSCA
-3.86199E+00
2.40488E-03
-3.18765E+00
5.68915E-02
-1.01532E+01
6.73596E-15
WOA
-3.86277E+00
3.08836E-05
-3.21385E+00
1.64920E-01
-9.84416E+00
1.69268E+00
GWO
-3.86226E+00
1.99958E-03
-3.26954E+00
6.72377E-02
-9.64644E+00
1.54629E+00
MFO
-3.86278E+00
2.71009E-15
-3.25591E+00
7.10267E-02
-7.21839E+00
3.29894E+00
BA
-3.85420E+00
3.85503E-03
-3.01605E+00
7.61190E-02
-8.19916E+00
2.04589E+00
FA
-3.86242E+00
2.42307E-04
-3.24894E+00
4.28972E-02
-8.20394E+00
7.31258E-01
PSO
-3.85715E+00
3.73295E-03
-2.92985E+00
2.47356E-01
-7.01237E+00
1.45289E+00
ACO
-3.86278E+00
2.71009E-15
-3.22866E+00
5.37026E-02
-5.57118E+00
3.59638E+00
F22
F23
F24
Avg
Std
Avg
Std
Avg
Std
FGSCA
-1.04029E+01
1.14267E-15
-1.05364E+01
2.03340E-15
1.47459E+06
5.83882E+05
WOA
-1.04029E+01
3.08985E-07
-1.05364E+01
7.89084E-07
3.03331E+07
1.31370E+07
GWO
-1.04029E+01
8.89362E-07
-1.03561E+01
9.87348E-01
6.81817E+07
4.69681E+07
MFO
-7.29707E+00
3.42522E+00
-7.55100E+00
3.53052E+00
1.21580E+08
1.16536E+08
BA
-8.10289E+00
2.16619E+00
-9.51619E+00
9.63492E-01
7.50579E+05
3.27235E+05
FA
-8.31108E+00
7.72133E-01
-8.79606E+00
8.17216E-01
2.56598E+08
5.53529E+07
PSO
-7.31299E+00
1.29828E+00
-7.90376E+00
1.08397E+00
8.43936E+06
2.41774E+06
ACO
-6.56512E+00
3.68363E+00
-8.19463E+00
3.64655E+00
2.03772E+06
1.92821E+06
F25
F26
F27
Avg
Std
Avg
Std
Avg
Std
FGSCA
1.87203E+04
1.41517E+04
7.98434E+03
2.75875E+03
4.95044E+02
3.39540E+01
WOA
4.27040E+06
7.67817E+06
3.34670E+04
2.64496E+04
5.88972E+02
5.43347E+01
GWO
2.20876E+09
2.80628E+09
2.83413E+04
9.78714E+03
6.39679E+02
6.46250E+01
MFO
1.08040E+10
6.97316E+09
9.47446E+04
4.82276E+04
1.18932E+03
1.07282E+03
BA
5.69437E+05
2.99934E+05
4.07210E+02
1.61090E+02
4.40653E+02
4.17664E+01
FA
1.54753E+10
1.70197E+09
6.13868E+04
7.90099E+03
1.54102E+03
2.04856E+02
PSO
1.45252E+08
1.39360E+07
9.55335E+02
9.96640E+01
4.65659E+02
3.58511E+01
ACO
3.68831E+07
1.44288E+08
2.06379E+04
1.69006E+04
4.81234E+02
4.13032E+01
F28
F29
F30
Avg
Std
Avg
Std
Avg
Std
FGSCA
5.20557E+02
2.45019E-01
6.19564E+02
4.06885E+00
7.00017E+02
1.75208E-02
WOA
5.20275E+02
1.42975E-01
6.36388E+02
2.71372E+00
7.01013E+02
9.02255E-02
GWO
5.20920E+02
1.48659E-01
6.14613E+02
2.71866E+00
7.19487E+02
1.48724E+01
MFO
5.20306E+02
1.69967E-01
6.23745E+02
4.04872E+00
7.82099E+02
4.52823E+01
BA
5.20959E+02
6.73179E-02
6.33703E+02
3.68324E+00
7.00667E+02
1.97083E-01
FA
5.20923E+02
6.71066E-02
6.33581E+02
1.04562E+00
8.35835E+02
1.42589E+01
PSO
5.20948E+02
4.47367E-02
6.22438E+02
2.65945E+00
7.02307E+02
1.26853E-01
ACO
5.20924E+02
5.65295E-02
6.14757E+02
2.88932E+00
7.05844E+02
2.13830E+01
F31
F32
F33
Avg
Std
Avg
Std
Avg
Std
FGSCA
8.48466E+02
2.25067E+01
1.05466E+03
2.43421E+01
4.90747E+03
1.30229E+03

26
WOA
9.93290E+02
3.63958E+01
1.14010E+03
5.78060E+01
4.86204E+03
7.08131E+02
GWO
8.85660E+02
1.67586E+01
1.00776E+03
3.72043E+01
3.07770E+03
4.91414E+02
MFO
9.38289E+02
3.60325E+01
1.10563E+03
4.64604E+01
4.59302E+03
7.68677E+02
BA
1.03832E+03
4.77084E+01
1.19144E+03
6.92278E+01
5.21936E+03
8.07427E+02
FA
1.02390E+03
1.15847E+01
1.16057E+03
1.27044E+01
7.57258E+03
3.07554E+02
PSO
9.72809E+02
1.72845E+01
1.11643E+03
3.23259E+01
5.03395E+03
4.97916E+02
ACO
8.67440E+02
1.73174E+01
9.64623E+02
2.22546E+01
3.00047E+03
4.80195E+02
F34
F35
F36
Avg
Std
Avg
Std
Avg
Std
FGSCA
5.55110E+03
1.01293E+03
1.20093E+03
5.20529E-01
1.30053E+03
1.21806E-01
WOA
5.91850E+03
7.42347E+02
1.20155E+03
4.23329E-01
1.30051E+03
1.32078E-01
GWO
4.14070E+03
8.63891E+02
1.20189E+03
1.00857E+00
1.30047E+03
4.61292E-01
MFO
5.48795E+03
8.31725E+02
1.20049E+03
1.67885E-01
1.30213E+03
1.43903E+00
BA
5.74744E+03
7.97526E+02
1.20128E+03
3.22395E-01
1.30051E+03
1.08607E-01
FA
7.89586E+03
4.49251E+02
1.20253E+03
2.40642E-01
1.30278E+03
2.87880E-01
PSO
5.76258E+03
6.03213E+02
1.20249E+03
2.66868E-01
1.30038E+03
7.85609E-02
ACO
4.07464E+03
1.68093E+03
1.20226E+03
4.08162E-01
1.30047E+03
1.03620E-01
F37
F38
F39
Avg
Std
Avg
Std
Avg
Std
FGSCA
1.40055E+03
3.09458E-01
1.51755E+03
7.98906E+00
1.61206E+03
7.61454E-01
WOA
1.40031E+03
1.56096E-01
1.57243E+03
2.17567E+01
1.61271E+03
4.15091E-01
GWO
1.40367E+03
5.92648E+00
1.93499E+03
1.33500E+03
1.61088E+03
7.18619E-01
MFO
1.43334E+03
1.94066E+01
2.73304E+05
5.36069E+05
1.61278E+03
5.48360E-01
BA
1.40036E+03
1.61321E-01
1.52878E+03
5.54374E+00
1.61328E+03
3.95113E-01
FA
1.44046E+03
4.19480E+00
1.36777E+04
4.57723E+03
1.61285E+03
2.33443E-01
PSO
1.40029E+03
1.38737E-01
1.51675E+03
1.11761E+00
1.61196E+03
4.59557E-01
ACO
1.40056E+03
2.63967E-01
1.52322E+03
3.96231E+01
1.61140E+03
5.39552E-01
F40
F41
F42
Avg
Std
Avg
Std
Avg
Std
FGSCA
6.90990E+05
4.36529E+05
2.81125E+05
5.03690E+05
1.91586E+03
1.23019E+01
WOA
4.66638E+06
2.86503E+06
1.20883E+04
2.70593E+04
1.95282E+03
3.74913E+01
GWO
8.91238E+05
7.91010E+05
1.40996E+07
2.47132E+07
1.93748E+03
2.25195E+01
MFO
5.29594E+06
1.18385E+07
3.38206E+07
1.43467E+08
1.97671E+03
6.64732E+01
BA
1.06913E+05
6.74647E+04
1.00327E+05
5.38408E+04
1.93125E+03
3.26394E+01
FA
6.56299E+06
2.22242E+06
3.14150E+08
9.84934E+07
2.00914E+03
1.04518E+01
PSO
2.23845E+05
1.17377E+05
1.98912E+06
6.34957E+05
1.91667E+03
2.40799E+00
ACO
2.95620E+05
5.73648E+05
7.61269E+03
6.52317E+03
1.91558E+03
1.91099E+01
F43
F44
F45
Avg
Std
Avg
Std
Avg
Std
FGSCA
5.26978E+04
3.57472E+04
3.47235E+05
3.20667E+05
2.93703E+03
2.81539E+02
WOA
3.13511E+04
2.25498E+04
1.15711E+06
1.03152E+06
3.03663E+03
2.52987E+02
GWO
1.74196E+04
1.04255E+04
4.58340E+05
8.47068E+05
2.56987E+03
1.53334E+02
MFO
5.87791E+04
4.06444E+04
4.64265E+05
5.66616E+05
2.97376E+03
2.30777E+02
BA
2.41600E+03
1.42619E+02
5.85748E+04
3.54953E+04
3.22627E+03
2.98747E+02

27
FA
1.87349E+04
5.93867E+03
1.74850E+06
6.41111E+05
2.98555E+03
1.00527E+02
PSO
2.33343E+03
6.28461E+01
1.03332E+05
5.58083E+04
2.88464E+03
2.05111E+02
ACO
2.01590E+04
2.17146E+04
1.40055E+05
1.31258E+05
2.48732E+03
2.01667E+02
F46
F47
F48
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.50000E+03
0.00000E+00
2.60000E+03
0.00000E+00
2.70000E+03
0.00000E+00
WOA
2.63308E+03
8.18910E+00
2.60611E+03
4.45244E+00
2.71513E+03
1.66420E+01
GWO
2.63715E+03
1.31346E+01
2.60000E+03
7.33035E-04
2.71233E+03
3.96458E+00
MFO
2.66615E+03
3.39099E+01
2.67464E+03
2.69669E+01
2.71456E+03
6.26078E+00
BA
2.61525E+03
2.87004E-03
2.66298E+03
2.48340E+01
2.73080E+03
1.52866E+01
FA
2.73490E+03
1.92281E+01
2.70263E+03
6.17574E+00
2.73375E+03
4.99084E+00
PSO
2.61598E+03
5.21133E-01
2.62927E+03
6.08947E+00
2.71213E+03
6.86572E+00
ACO
2.61602E+03
2.23251E+00
2.64394E+03
2.54780E+00
2.70741E+03
3.74868E+00
F49
F50
F51
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.70055E+03
1.31123E-01
2.90000E+03
0.00000E+00
3.00000E+03
0.00000E+00
WOA
2.70046E+03
1.27011E-01
3.83397E+03
3.35300E+02
5.19426E+03
6.62882E+02
GWO
2.75024E+03
5.06231E+01
3.38002E+03
1.01692E+02
3.95344E+03
2.80136E+02
MFO
2.70242E+03
1.27200E+00
3.64800E+03
1.90081E+02
3.97717E+03
2.76492E+02
BA
2.70383E+03
1.81823E+01
3.78349E+03
4.32563E+02
5.27148E+03
7.77092E+02
FA
2.70224E+03
3.16663E-01
3.80652E+03
2.28349E+01
4.23985E+03
1.57885E+02
PSO
2.79044E+03
3.05250E+01
3.44653E+03
2.76590E+02
7.25465E+03
1.14131E+03
ACO
2.75497E+03
9.72126E+01
3.42841E+03
8.32715E+01
3.93721E+03
2.95746E+02
F52
F53
Overall Rank
Avg
Std
Avg
Std
Rank
+/=/-
ARV
FGSCA
3.10000E+03
0.00000E+00
7.06385E+03
9.76438E+03
1
~
2.30187
WOA
4.15501E+06
4.82829E+06
7.20479E+04
4.95793E+04
4
34/9/10
3.92453
GWO
1.35783E+06
1.87189E+06
5.63045E+04
3.48586E+04
2
36/10/7
3.73585
MFO
3.01624E+06
3.55935E+06
5.62318E+04
5.19774E+04
7
38/3/12
5.35849
BA
3.88106E+07
3.35442E+07
3.12467E+04
6.91379E+04
5
40/7/6
4.90566
FA
3.56300E+06
1.29954E+06
1.84226E+05
4.36109E+04
8
48/3/2
6.71698
PSO
3.41029E+04
7.47822E+04
1.29125E+04
6.23833E+03
6
41/7/5
4.962261
ACO
2.40792E+06
4.48226E+06
1.03552E+04
8.97741E+03
3
27/13/13
3.77358
5.5 Comparison with other reported SCA variants
In this section, FGSCA is compared with six reported SCA variants to assess the performance on
53 benchmark test problems, namely MSCA [44], ASCA-PSO [45], m-SCA [46], SCADE [47], CGSCA
[48], OBSCA [49]. As can be seen, Table 8 shows the detailed experimental results of FGSCA and the
other six SCA variants. First of all, from the perspective of Avg, the comprehensive performance of
FGSCA ranks first, which means that FGSCA surpasses the six SCA variants recently reported in the
accuracy of the solution. Specifically, FGSCA obtains the smallest Avg in unimodal functions F1-F4,
F7, and F24-F26. For F5 and F6, SCADE and MSCA obtained the smallest Avg, respectively, while

28
FGSCA harvests the second smallest Avg. Since the search space of the unimodal function has only
one optimal solution and does not contain other local solutions, it can be explained that the combination
of spiral motion and Gaussian mutation greatly enhances the global exploration ability of SCA. In short,
the accuracy of FGSCA's solution on the unimodal function is quite satisfactory. For multimodal
functions F8-F13 and F27-F33, FGSCA acquires the third smallest Avg on F8, F29, and F33, gains the
second smallest Avg on F12, F13, F31, and F27-F28, and achieves the smallest Avg on F9-F11, F30,
and F32. On the whole, the performance of FGSCA on multimodal functions is superior to other SCA
variants. The combination of spiral motion and Gaussian mutation effectively enhances the local
exploitation capabilities of the original SCA. For the hybrid functions F14-F23 and F34-F43, the Avg
optimized by FGSCA promotes 13 functions in the top three and eight functions in the first. Thus,
FGSCA also has a strong advantage over other SCA variants in the performance of hybrid functions.
For the composition functions F44-F53, FGSCA gains the smallest Avg on F46-F52, which shows that
FGSCA has excellent advantages in solving such problems, as well. In general, the comprehensive
performance of FGSCA surpasses the other six reported SCA variants in terms of unimodal functions,
multimodal functions, hybrid functions and composition functions.
From the perspective of Std, FGSCA still performed very well on 53 benchmark test problems.
FGSCA harvested the smallest Std on the unimodal functions F1-F5, F7, F24-F26, which indicates that
the stability of FGSCA on the unimodal function is better than the other six SCA variants. For the
multimodal functions F9-F11, F27, and F30, the Std obtained by FGSCA is the smallest, demonstrating
that the combination of spiral motion and Gaussian variation effectively improves the stability of SCA
in solving multimodal functions. For the hybrid functions F16-F19, F21-F23, and F37, the stability of
FGSCA is still highly satisfactory. For the composition functions F46-F48 and F50-F52, the stability of
FGSCA is still not inferior to the other six SCA variants. In short, FGSCA has broader potential than
the other six SCA variants in various complex issues.
Fig. 10 shows the convergence curves of FGSCA and other six SCA variants on nine benchmark
functions, namely F1, F7, F19, F22, F26, F32, F38, F46, and F51. For the convergence curves of
unimodal functions F1, F7, and F26, the convergence speed and convergence accuracy of FGSCA are
better than the other six SCA variants. For the multimodal function F32, the convergence speed of
FGSCA is slightly worse than that of MSCA, but the convergence accuracy is better than that of MSCA.
For the hybrid functions F22 and F38, the convergence accuracy of FGSCA is better than other SCA
variants. For the composition functions, F46 and F51, with spiral motion and Gaussian mutation, the
convergence speed, and accuracy of FGSCA are highly commendable.
In general, the advantage of FGSCA in the recently reported SCA variants is undeniable, which is
mainly due to the combination of spiral motion and Gaussian variation. Based on the population gained
by SCA, the spiral movement further promotes other individuals in the population to accelerate to move
closer to the leader in a spiral manner. Meantime, Gaussian mutation is introduced to increase the
population's diversity and strive for individuals in the population to move closer to the leader from
different positions. In this way, other individuals are expected to find a better position than the leader
on the way, which helps the population avoid falling into a local optimum.

29
Figure 10. Convergence curves of FGSCA and other six SCA variants on 9 benchmark functions
Table 8. Results of FGSCA compared with six SCA variants on 53 benchmark functions
F1
F2
F3
Avg
Std
Avg
Std
Avg
Std
FGSCA
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
MSCA
7.23629E-46
1.96672E-45
2.94596E-22
8.45678E-22
1.29866E+02
9.26519E+01
ASCA-
PSO
7.30548E+01
8.63825E+00
4.34936E+01
6.18014E+00
2.07555E+02
2.79772E+01
m-SCA
0.00000E+00
0.00000E+00
3.53109E-320
0.00000E+00
8.93744E-199
0.00000E+00
SCADE
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
CGSCA
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
OBSCA
2.47995E-107
7.81681E-107
2.72406E-88
6.14901E-88
2.08908E-28
4.32853E-28
F4
F5
F6
Avg
Std
Avg
Std
Avg
Std
FGSCA
0.00000E+00
0.00000E+00
2.19742E+01
1.01897E-01
1.74507E-28
2.02495E-28
MSCA
1.87905E+00
4.96336E-01
3.69911E+01
2.93075E+01
3.09218E-29
1.31233E-29
ASCA-
3.57191E+00
2.57867E-01
6.57027E+04
1.16158E+04
8.44492E+01
8.97106E+00

30
PSO
m-SCA
5.39062E-165
0.00000E+00
2.74303E+01
9.83207E-01
2.29109E+00
5.14596E-01
SCADE
3.18632E-190
0.00000E+00
1.97978E+01
1.32424E+01
1.68504E-07
1.06837E-07
CGSCA
4.34506E-289
0.00000E+00
2.80509E+01
7.83667E-01
4.07517E+00
2.58420E-01
OBSCA
1.67385E-22
5.27802E-22
2.80094E+01
1.15621E-01
3.95684E+00
1.71800E-01
F7
F8
F9
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.35839E-05
1.42184E-05
-1.10002E+04
1.27118E+03
0.00000E+00
0.00000E+00
MSCA
7.29190E-02
1.86686E-02
-1.04188E+2
90
6.55350E+04
2.48937E+00
1.26205E+00
ASCA-
PSO
3.14673E+01
6.33517E+00
-6.91345E+03
8.66142E+02
2.88385E+02
1.54919E+01
m-SCA
7.82318E-05
4.79716E-05
-6.72190E+03
7.24713E+02
0.00000E+00
0.00000E+00
SCADE
2.53682E-04
1.01954E-04
-1.25695E+04
1.43465E-02
0.00000E+00
0.00000E+00
CGSCA
3.24597E-05
1.59442E-05
-4.57814E+03
2.70016E+02
0.00000E+00
0.00000E+00
OBSCA
8.73232E-04
6.36142E-04
-4.17072E+03
5.90567E+02
0.00000E+00
0.00000E+00
F10
F11
F12
Avg
Std
Avg
Std
Avg
Std
FGSCA
8.88178E-16
0.00000E+00
0.00000E+00
0.00000E+00
2.74108E-27
7.08126E-27
MSCA
9.36140E-14
1.86080E-14
6.16903E-02
5.71348E-02
4.06520E-30
4.91222E-30
ASCA-
PSO
7.14244E+00
6.58886E-01
1.00233E+00
1.40623E-02
3.42465E+00
3.72889E-01
m-SCA
4.00108E+00
8.43501E+00
0.00000E+00
0.00000E+00
1.38334E-01
4.92877E-02
SCADE
8.88178E-16
0.00000E+00
0.00000E+00
0.00000E+00
4.55642E-09
2.59316E-09
CGSCA
8.88178E-16
0.00000E+00
0.00000E+00
0.00000E+00
4.00898E-01
2.93248E-02
OBSCA
4.08562E-15
1.12347E-15
0.00000E+00
0.00000E+00
3.90969E-01
4.27447E-02
F13
F14
F15
Avg
Std
Avg
Std
Avg
Std
FGSCA
4.39495E-03
5.67385E-03
2.07392E+00
3.06903E+00
5.57342E-04
4.32728E-04
MSCA
6.32710E-02
8.04082E-02
9.98004E-01
2.45480E-16
1.06466E-03
2.34421E-04
ASCA-
PSO
1.16324E+01
2.33304E+00
9.98004E-01
2.42408E-08
1.10949E-03
2.50098E-04
m-SCA
1.47526E+00
2.12439E-01
9.98004E-01
6.24285E-13
4.90626E-04
3.86087E-04
SCADE
4.79627E-08
2.87070E-08
9.98004E-01
1.47716E-12
3.10948E-04
2.16555E-06
CGSCA
2.15989E+00
8.16468E-02
9.98005E-01
1.30603E-06
4.19484E-04
2.87637E-04
OBSCA
2.18182E+00
8.91821E-02
1.59327E+00
9.58388E-01
5.83890E-04
1.74368E-04
F16
F17
F18
Avg
Std
Avg
Std
Avg
Std
FGSCA
-1.03163E+00
0.00000E+00
3.97887E-01
0.00000E+00
3.00000E+00
2.36848E-15
MSCA
-1.03163E+00
2.09346E-16
3.97887E-01
0.00000E+00
3.00000E+00
1.69581E-08
ASCA-
PSO
-1.03139E+00
1.85750E-04
3.98186E-01
3.94905E-04
3.00739E+00
4.11088E-03
m-SCA
-1.03163E+00
1.83235E-10
3.97887E-01
4.58246E-08
3.00000E+00
2.03782E-08
SCADE
-1.03163E+00
1.21779E-06
3.97937E-01
4.14695E-05
3.00019E+00
1.44572E-04

31
CGSCA
-1.03162E+00
4.15335E-06
3.97995E-01
1.15845E-04
3.00000E+00
8.61249E-07
OBSCA
-1.03163E+00
1.47509E-07
3.98041E-01
1.76115E-04
3.00000E+00
4.95579E-07
F19
F20
F21
Avg
Std
Avg
Std
Avg
Std
FGSCA
-3.86278E+00
9.00430E-16
-3.22396E+00
7.20521E-02
-1.01532E+01
1.02558E-15
MSCA
-3.86042E+00
3.80714E-03
-3.22398E+00
8.81887E-02
-4.31426E+00
3.54918E+00
ASCA-
PSO
-3.85889E+00
3.58560E-03
-3.15343E+00
3.13019E-02
-7.88955E+00
2.42327E+00
m-SCA
-3.86278E+00
7.14496E-07
-3.24787E+00
6.38082E-02
-1.01532E+01
1.03635E-05
SCADE
-3.85583E+00
2.37750E-03
-3.27297E+00
3.26998E-02
-1.01532E+01
1.22226E-05
CGSCA
-3.85477E+00
1.26743E-04
-3.03871E+00
1.68120E-01
-2.93526E+00
2.93272E+00
OBSCA
-3.86234E+00
3.25906E-04
-3.25450E+00
1.79553E-02
-9.78070E+00
2.84935E-01
F22
F23
F24
Avg
Std
Avg
Std
Avg
Std
FGSCA
-1.04029E+01
1.77636E-15
-1.05364E+01
1.87244E-15
1.91918E+06
1.22108E+06
MSCA
-6.60188E+00
4.19049E+00
-6.32408E+00
3.30677E+00
6.49282E+06
8.07603E+06
ASCA-
PSO
-8.42617E+00
2.36030E+00
-7.39518E+00
1.64983E+00
2.48391E+07
1.41754E+07
m-SCA
-9.87551E+00
1.66783E+00
-1.05364E+01
1.48402E-05
6.00943E+07
3.84850E+07
SCADE
-1.04029E+01
3.58583E-05
-1.05364E+01
2.02410E-05
4.49921E+08
9.31119E+07
CGSCA
-5.62026E+00
3.01585E+00
-5.49593E+00
2.88169E+00
2.62839E+08
5.15911E+07
OBSCA
-9.95441E+00
2.87051E-01
-1.01140E+01
3.08532E-01
4.03614E+08
1.32320E+08
F25
F26
F27
Avg
Std
Avg
Std
Avg
Std
FGSCA
1.90726E+04
1.34674E+04
7.45392E+03
2.63648E+03
4.74354E+02
5.37464E+01
MSCA
2.45010E+04
1.71627E+04
2.27336E+04
1.81999E+04
4.68437E+02
5.62354E+01
ASCA-
PSO
8.69362E+08
1.43786E+09
2.28012E+04
4.67668E+03
5.44202E+02
6.15917E+01
m-SCA
5.98812E+09
3.34848E+09
2.66254E+04
6.91123E+03
7.54039E+02
1.44431E+02
SCADE
2.93111E+10
4.08623E+09
5.58480E+04
5.98389E+03
2.69798E+03
5.09914E+02
CGSCA
1.85734E+10
1.72033E+09
4.27469E+04
5.28940E+03
1.66382E+03
2.64506E+02
OBSCA
2.43563E+10
2.38705E+09
4.99994E+04
9.90311E+03
2.36518E+03
8.10377E+02
F28
F29
F30
Avg
Std
Avg
Std
Avg
Std
FGSCA
5.20531E+02
2.58146E-01
6.22275E+02
4.30485E+00
7.00020E+02
2.28448E-02
MSCA
5.20008E+02
1.47010E-02
6.17454E+02
3.41311E+00
7.00079E+02
7.54218E-02
ASCA-
PSO
5.20956E+02
4.55911E-02
6.24489E+02
3.06280E+00
7.05365E+02
4.47934E+00
m-SCA
5.20603E+02
1.21013E-01
6.20315E+02
2.41017E+00
7.43866E+02
2.48906E+01
SCADE
5.20943E+02
4.43235E-02
6.34718E+02
1.75211E+00
9.16330E+02
2.88880E+01
CGSCA
5.20914E+02
9.72654E-02
6.33783E+02
2.72193E+00
8.88400E+02
3.56462E+01
OBSCA
5.20948E+02
4.04887E-02
6.32212E+02
1.33473E+00
9.04121E+02
3.46032E+01
F31
F32
F33
Avg
Std
Avg
Std
Avg
Std

32
FGSCA
8.48171E+02
2.20762E+01
1.05073E+03
2.64061E+01
4.25357E+03
1.41530E+03
MSCA
8.21392E+02
8.01815E+00
1.05528E+03
3.26385E+01
1.39200E+03
2.27568E+02
ASCA-
PSO
9.56752E+02
2.52296E+01
1.09831E+03
2.77279E+01
5.23792E+03
4.85135E+02
m-SCA
9.39172E+02
1.95605E+01
1.06255E+03
2.84867E+01
4.20043E+03
7.88250E+02
SCADE
1.07322E+03
1.01914E+01
1.21264E+03
8.67257E+00
7.38341E+03
3.42477E+02
CGSCA
1.04775E+03
1.18298E+01
1.18384E+03
1.30506E+01
6.81166E+03
4.21850E+02
OBSCA
1.06183E+03
1.62459E+01
1.20107E+03
1.98586E+01
6.31385E+03
4.14069E+02
F34
F35
F36
Avg
Std
Avg
Std
Avg
Std
FGSCA
6.00085E+03
1.12414E+03
1.20093E+03
7.40190E-01
1.30062E+03
1.40228E-01
MSCA
4.33417E+03
6.75522E+02
1.20012E+03
5.62825E-02
1.30060E+03
8.28716E-02
ASCA-
PSO
6.28574E+03
1.06145E+03
1.20259E+03
2.56888E-01
1.30056E+03
9.51845E-02
m-SCA
4.41643E+03
3.91171E+02
1.20081E+03
3.54560E-01
1.30069E+03
2.46009E-01
SCADE
8.22967E+03
2.87041E+02
1.20265E+03
2.82145E-01
1.30394E+03
2.71542E-01
CGSCA
8.09224E+03
3.55807E+02
1.20244E+03
3.03725E-01
1.30330E+03
3.06083E-01
OBSCA
7.29369E+03
3.75807E+02
1.20227E+03
3.38227E-01
1.30346E+03
2.37570E-01
F37
F38
F39
Avg
Std
Avg
Std
Avg
Std
FGSCA
1.40051E+03
2.73715E-01
1.51885E+03
7.63346E+00
1.61259E+03
2.99068E-01
MSCA
1.40037E+03
2.89499E-01
1.56581E+03
8.65907E+01
1.61153E+03
4.05549E-01
ASCA-
PSO
1.40306E+03
8.31827E+00
1.51926E+03
1.48914E+00
1.61234E+03
2.38720E-01
m-SCA
1.41628E+03
7.84572E+00
2.71835E+03
2.65013E+03
1.61171E+03
7.53259E-01
SCADE
1.48370E+03
1.57411E+01
2.39175E+04
7.65249E+03
1.61271E+03
1.48936E-01
CGSCA
1.45123E+03
6.95623E+00
5.94501E+03
2.95543E+03
1.61288E+03
1.27004E-01
OBSCA
1.46197E+03
1.38118E+01
2.01301E+04
1.54860E+04
1.61289E+03
1.64285E-01
F40
F41
F42
Avg
Std
Avg
Std
Avg
Std
FGSCA
6.92084E+05
4.82937E+05
2.61865E+07
8.15512E+07
1.91449E+03
2.88594E+00
MSCA
1.50319E+06
8.92132E+05
7.30093E+03
8.31677E+03
1.91442E+03
5.68171E+00
ASCA-
PSO
5.49402E+05
2.33061E+05
4.37242E+06
8.94481E+05
1.91630E+03
1.58709E+00
m-SCA
1.16339E+06
4.87015E+05
8.25355E+06
1.20467E+07
1.94274E+03
2.54889E+01
SCADE
1.56794E+07
6.36184E+06
1.97489E+08
1.31960E+08
2.01488E+03
1.70675E+01
CGSCA
6.07666E+06
3.45025E+06
1.99402E+08
1.42833E+08
1.99408E+03
2.85217E+01
OBSCA
1.05856E+07
5.91006E+06
1.54974E+08
1.38422E+08
2.00932E+03
1.00541E+01
F43
F44
F45
Avg
Std
Avg
Std
Avg
Std
FGSCA
3.19977E+04
1.28909E+04
2.86052E+05
1.97438E+05
2.93196E+03
2.35577E+02
MSCA
4.58379E+04
1.34736E+04
2.00783E+06
1.25258E+06
3.11133E+03
2.91459E+02
ASCA-
PSO
7.43329E+03
2.31329E+03
2.20544E+05
1.73904E+05
2.75496E+03
1.10106E+02

33
m-SCA
8.36913E+03
3.03675E+03
3.45031E+05
3.80614E+05
2.63854E+03
1.93263E+02
SCADE
2.44182E+04
5.65098E+03
2.29442E+06
2.44402E+06
3.10445E+03
1.66625E+02
CGSCA
1.97461E+04
5.30497E+03
1.70765E+06
7.11166E+05
3.02415E+03
1.05078E+02
OBSCA
3.29377E+04
1.48224E+04
1.84820E+06
8.79286E+05
3.06108E+03
1.61403E+02
F46
F47
F48
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.50000E+03
0.00000E+00
2.60000E+03
0.00000E+00
2.70000E+03
0.00000E+00
MSCA
2.61837E+03
8.86012E+00
2.64302E+03
2.89817E+01
2.70112E+03
4.99938E-01
ASCA-
PSO
2.62511E+03
7.63144E+00
2.63997E+03
1.16620E+01
2.71572E+03
1.16582E+01
m-SCA
2.63763E+03
5.22726E+00
2.60000E+03
4.41064E-04
2.71323E+03
2.70091E+00
SCADE
2.50000E+03
0.00000E+00
2.60000E+03
4.24280E-06
2.70000E+03
0.00000E+00
CGSCA
2.50000E+03
0.00000E+00
2.60000E+03
1.38687E-06
2.70000E+03
0.00000E+00
OBSCA
2.69643E+03
2.36027E+01
2.60000E+03
3.17194E-04
2.70000E+03
7.24861E-06
F49
F50
F51
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.70058E+03
1.46898E-01
2.90000E+03
0.00000E+00
3.00000E+03
0.00000E+00
MSCA
2.70070E+03
1.36211E-01
3.41666E+03
2.72282E+02
3.29803E+03
5.27978E+01
ASCA-
PSO
2.70062E+03
1.62416E-01
3.46660E+03
3.07046E+02
4.40042E+03
4.75323E+02
m-SCA
2.70066E+03
1.66013E-01
3.14435E+03
3.34620E+01
3.90026E+03
1.52843E+02
SCADE
2.70379E+03
4.65477E-01
3.20754E+03
2.25070E+02
4.70024E+03
1.21102E+03
CGSCA
2.70287E+03
5.39146E-01
2.90000E+03
0.00000E+00
3.00000E+03
0.00000E+00
OBSCA
2.70412E+03
2.42752E-01
3.24560E+03
3.17683E+01
5.39553E+03
2.93684E+02
F52
F53
Overall Rank
Avg
Std
Avg
Std
Rank
+/=/-
ARV
FGSCA
3.10000E+03
0.00000E+00
5.55687E+03
7.45308E+03
1
~
2.05660
MSCA
3.13837E+03
4.94881E+01
3.97651E+03
1.99073E+02
3
23/9/21
3.60377
ASCA-
PSO
5.57254E+06
7.96109E+06
5.59446E+04
4.71065E+04
6
39/2/12
5.13208
m-SCA
1.88281E+06
4.71902E+06
4.73776E+04
2.96519E+04
2
33/4/16
3.54717
SCADE
1.53039E+07
1.25337E+07
5.41109E+05
2.03294E+05
5
36/2/15
4.94340
CGSCA
3.10000E+03
0.00000E+00
5.43152E+04
1.08240E+05
4
35/1/17
4.75472
OBSCA
1.93401E+07
9.44605E+06
4.25754E+05
1.91499E+05
7
45/0/8
5.64151
5.6 Comparison with other reported improved optimizers
In this section, FGSCA is compared with other ten improved MAs to further evaluate the
performance, including BMWOA [50], ACWOA [51], IWOA [52], CBA [53], CDLOBA [54], IFOA [55],
AMFOA [56], OBLGWO [57], EPSO [58] IJAYA [59]. The detailed comparison results are shown in
Table 9. From the perspective of Avg, the comprehensive convergence accuracy of FGSCA is better
than the other ten improved MAs. Specifically, for the unimodal functions F1-F4, the Avg values
optimized by FGSCA are 0.00000E+00. Other unimodal functions such as F5-F7 and F24-F26, even

34
though the Avg values obtained by FGSCA are slightly inferior to BMWOA, CDLOBA and EPSO, the
overall Avg values are quite satisfactory. The performance of FGSCA is still the best among those
improved MAs on multimodal functions; for example, FGSCA has strong adaptability when solving F9,
F10, and F11 problems. When solving the problems of F27 and F29-F32, EPSO is more competitive
than FGSCA. For the hybrid functions F16, F17, F19, F21, and F38, FGSCA harvested the smallest
Avg. However, on hybrid functions, F34, F35, F39, and F42, EPSO, CDLOBA, EPSO, and IJAYA have
certain competitiveness with FGSCA, respectively. For the composition functions F46-F48 and F50-
F52, FGSCA still has a strong advantage over the other ten types of improved MAs.
From the analysis of Std, the stability of FGSCA is better than the other ten improved MAs in
unimodal functions F1-F4, multimodal functions F9-F11 and F30, hybrid function F22, and
composition functions F46-F48 and F50-F52. For unimodal functions, F5-F7 and F24-F26, IFOA,
EPSO, and OBLGWO have certain competitiveness with FGSCA, but the overall stability is acceptable.
For other multimodal functions like F12, F13, F27-F29, and F31-F33, the performance of FGSCA is
not the most stable, but the final optimal Avg is satisfactory. For the hybrid functions F14-F21, F23,
and F34-F43, the stability of FGSCA is not the best, but it is better than most of the improved MAs.
For the hybrid functions F44, F45, F49, and F53, the comprehensive stability of FGSCA rank at the
forefront. Therefore, FGSCA surpasses the other ten improved MAs in terms of stability.
Fig. 11 shows the convergence curves of FGSCA and the other ten improved MAs on nine
benchmark functions, namely F1, F4, F11, F17, F19, F21, F38, F46, and F52. For the unimodal
functions F1 and F4 and the multimodal function F11, the convergence speed of FGSCA is the fastest
compared to other MAs. For the hybrid functions F17, F19, F21, and F38 and the composition
functions F46 and F52, FGSCA is superior to other MAs in the final convergence results. It can be seen
that the spiral motion and Gaussian mutation effectively improve the convergence value of the original
SCA on various types of functions.
In short, FGSCA has superior overall performance compared with the other ten improved MAs,
which is mainly reflected in the convergence speed and convergence results. For unimodal problems,
multi-peak problems, hybrid problems, and composition problems, FGSCA can show strong strength
in these ten improved MAs. Spiral motion and Gaussian mutation improve the original SCA's optimal
solution performance and convergence speed. It maintains a proper balance in the trend of
diversification and intensification. Thus, it can be concluded that FGSCA solves various benchmark
problems more effectively than other improved MAs.

35
Figure 11. Convergence curves of FGSCA and other ten improved MAs on 9 benchmark functions
Table 9. Results of FGSCA compared with ten improved MAs on 53 benchmark functions.
F1
F2
F3
Avg
Std
Avg
Std
Avg
Std
FGSCA
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
BMWOA
7.55704E-05
7.80595E-05
4.25496E-03
3.08607E-03
1.99643E-02
3.89674E-02
ACWOA
5.03361E-02
1.87983E-02
1.02531E+00
2.09178E-01
1.73043E+01
6.40542E+00
OBLGWO
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
1.53865E+02
3.10081E+02
IWOA
0.00000E+00
0.00000E+00
0.00000E+00
0.00000E+00
2.58586E+01
2.69399E+01
CBA
3.28293E-10
6.11797E-10
2.30849E+01
4.05924E+01
1.34164E+01
6.29913E+00
CDLOBA
2.40786E-04
7.47540E-05
1.43286E+01
4.40908E+01
6.50478E-04
2.69173E-04
IFOA
3.68590E-09
7.53693E-11
3.10894E-04
6.27772E-06
6.13166E-08
1.32076E-08
AMFOA
3.85044E-09
2.01046E-09
2.76200E-04
7.31863E-05
8.47803E-07
5.10518E-07
EPSO
5.14703E-36
7.93390E-36
2.75028E-22
5.11036E-22
6.11636E+02
4.16557E+02
IJAYA
2.85043E-01
7.58778E-02
4.06191E+01
1.23585E+01
1.70171E+04
2.10147E+03
F4
F5
F6
Avg
Std
Avg
Std
Avg
Std

36
FGSCA
0.00000E+00
0.00000E+00
2.19151E+01
1.28931E-01
8.52511E-28
1.53730E-27
BMWOA
2.87511E-03
3.05162E-03
1.09032E-03
2.32392E-03
1.72629E-04
4.56310E-04
ACWOA
3.89507E-01
1.77521E-01
3.99868E+01
1.86920E+01
5.30360E-02
1.47133E-02
OBLGWO
8.31134E-78
1.43601E-77
2.87413E+01
5.06936E-02
1.15897E+00
5.27203E-01
IWOA
1.09131E-04
2.70514E-04
2.36418E+01
5.49630E-01
3.06016E-06
1.31181E-06
CBA
6.44470E+00
7.18704E+00
3.60891E+01
2.85218E+01
1.44641E-08
3.03643E-08
CDLOBA
3.77173E+01
9.92091E+00
8.68107E+01
1.08691E+02
2.46840E-04
8.46285E-05
IFOA
1.12465E-05
3.10524E-08
2.84370E+01
1.54895E-04
6.24972E+00
1.22668E-01
AMFOA
9.84395E-06
2.53179E-06
2.82579E+01
2.59768E-01
7.50026E+00
5.50594E-05
EPSO
2.97492E-05
3.73476E-05
2.38194E+01
1.91402E+01
5.95605E-29
1.61997E-29
IJAYA
3.79770E+01
8.25883E+00
5.52877E+02
8.73067E+02
2.81041E-01
1.40234E-01
F7
F8
F9
Avg
Std
Avg
Std
Avg
Std
FGSCA
4.03462E-05
3.18333E-05
-1.11162E+04
9.42982E+02
0.00000E+00
0.00000E+00
BMWOA
8.94122E-04
5.46741E-04
-1.25695E+04
1.33401E-03
9.49857E-05
1.66780E-04
ACWOA
9.71315E-02
3.19897E-02
-1.24645E+04
1.26549E+02
1.08063E+02
2.29132E+01
OBLGWO
7.15326E-06
6.62776E-06
-1.25695E+04
1.64194E-03
0.00000E+00
0.00000E+00
IWOA
1.69359E-04
2.29552E-04
-1.22626E+04
6.91400E+02
0.00000E+00
0.00000E+00
CBA
5.01742E-02
3.14807E-02
-7.51897E+03
5.21737E+02
1.10333E+02
3.33522E+01
CDLOBA
1.68358E-02
1.02841E-02
-6.74167E+03
6.22479E+02
1.53212E+02
5.67331E+01
IFOA
4.99281E-05
2.06087E-05
-2.25355E+03
4.36759E+02
7.43146E-07
2.08881E-08
AMFOA
2.68304E-05
1.15708E-05
-2.08456E+02
1.27659E+02
6.13289E-07
3.65770E-07
EPSO
3.85111E-02
8.99598E-03
-1.15864E+04
2.73806E+02
1.40311E+02
1.91514E+01
IJAYA
2.08054E-01
4.73383E-02
-6.89918E+03
6.76040E+02
1.50993E+02
2.91915E+01
F10
F11
F12
Avg
Std
Avg
Std
Avg
Std
FGSCA
8.88178E-16
0.00000E+00
0.00000E+00
0.00000E+00
7.55088E-30
1.11966E-29
BMWOA
3.52046E-03
3.72947E-03
2.00860E-04
2.96326E-04
1.39683E-06
3.97825E-06
ACWOA
6.70725E-01
8.33411E-01
1.28093E-02
9.18727E-03
1.41669E+00
1.70444E+00
OBLGWO
8.88178E-16
0.00000E+00
0.00000E+00
0.00000E+00
7.40632E-02
3.48694E-02
IWOA
2.66454E-15
1.87244E-15
1.97286E-03
4.15916E-03
4.90875E-07
2.72946E-07
CBA
1.52362E+01
3.50631E+00
1.25090E-02
1.24545E-02
1.70114E+01
6.04392E+00
CDLOBA
1.97872E+01
5.57016E-01
5.46991E+01
5.82040E+01
1.64293E+01
8.59654E+00
IFOA
4.46163E-05
1.00899E-06
1.76966E-10
7.88344E-12
9.28406E-01
3.56385E-02
AMFOA
3.45307E-05
7.37466E-06
2.01610E-10
1.25927E-10
1.66902E+00
1.30827E-05
EPSO
1.26121E-14
4.75173E-15
9.11130E-03
8.52821E-03
3.71424E-30
1.28600E-30
IJAYA
3.70265E+00
5.45244E+00
1.68268E-01
5.26172E-02
5.73437E+01
1.45042E+01
F13
F14
F15
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.19747E-03
4.63268E-03
2.37134E+00
3.06180E+00
7.28644E-04
5.47094E-04
BMWOA
2.81686E-05
7.00967E-05
9.98004E-01
1.81299E-16
3.99055E-04
2.89566E-04
ACWOA
2.65015E-02
1.10077E-02
9.98004E-01
1.29459E-11
5.58662E-04
2.74245E-04
OBLGWO
5.64425E-01
3.23968E-01
2.87130E+00
3.17982E+00
6.11576E-04
2.06846E-04
IWOA
7.87026E-06
2.43104E-06
9.98004E-01
1.51685E-15
5.82557E-04
4.42100E-04

37
CBA
1.54008E+01
2.11244E+01
2.28434E+00
1.68162E+00
8.70263E-03
1.00403E-02
CDLOBA
3.20579E+01
1.60648E+01
1.88948E+00
1.35550E+00
2.70563E-03
6.20762E-03
IFOA
2.57710E-01
8.14950E-01
9.98004E-01
7.60966E-10
3.21790E-04
1.08521E-05
AMFOA
5.92917E-01
8.03955E-02
1.26705E+01
3.34953E-15
3.27875E-04
1.67975E-05
EPSO
3.13934E-29
1.31066E-29
9.98004E-01
0.00000E+00
3.07486E-04
2.46442E-19
IJAYA
1.04952E+02
1.21198E+01
9.98004E-01
0.00000E+00
9.04648E-04
4.16284E-04
F16
F17
F18
Avg
Std
Avg
Std
Avg
Std
FGSCA
-1.03163E+00
1.04673E-16
3.97887E-01
0.00000E+00
3.00000E+00
1.66821E-15
BMWOA
-1.03163E+00
2.66864E-16
3.97887E-01
6.83391E-14
3.00000E+00
5.43291E-15
ACWOA
-1.03163E+00
2.01704E-06
3.97891E-01
4.19819E-06
3.00011E+00
1.30037E-04
OBLGWO
-1.02682E+00
1.01279E-02
3.97888E-01
1.05796E-06
1.41843E+01
1.44545E+01
IWOA
-1.03163E+00
3.10510E-15
3.97887E-01
6.76286E-11
3.00000E+00
3.52779E-10
CBA
-1.03163E+00
6.09320E-07
3.97888E-01
8.43311E-07
3.00002E+00
2.87620E-05
CDLOBA
-1.03160E+00
2.87085E-05
3.97902E-01
1.22336E-05
3.00244E+00
2.37014E-03
IFOA
-1.01581E+00
1.65763E-02
6.47648E-01
2.10117E-01
3.01543E+00
3.22843E-03
AMFOA
-4.77957E-01
3.33015E-01
1.68275E+00
1.34023E+00
6.00010E+02
2.74049E-03
EPSO
-1.03163E+00
0.00000E+00
5.56021E+01
7.48978E-15
3.00000E+00
9.70696E-16
IJAYA
-1.03163E+00
0.00000E+00
3.97887E-01
0.00000E+00
3.00000E+00
5.92119E-16
F19
F20
F21
Avg
Std
Avg
Std
Avg
Std
FGSCA
-3.86278E+00
9.36222E-16
-3.21944E+00
8.00301E-02
-1.01532E+01
1.02558E-15
BMWOA
-3.86278E+00
2.61473E-15
-3.28633E+00
5.74308E-02
-1.01532E+01
3.77658E-12
ACWOA
-3.86262E+00
1.26098E-04
-3.24635E+00
6.36444E-02
-8.62108E+00
2.46106E+00
OBLGWO
-3.83309E+00
4.48015E-02
-2.93938E+00
2.31254E-01
-8.10235E+00
3.11893E+00
IWOA
-3.86199E+00
2.49236E-03
-3.26304E+00
1.00072E-01
-1.01532E+01
1.78876E-07
CBA
-3.86277E+00
1.37500E-05
-3.24515E+00
6.65563E-02
-4.91822E+00
3.61146E+00
CDLOBA
-3.85866E+00
3.44170E-03
-3.11242E+00
3.71799E-02
-5.32966E+00
3.34405E+00
IFOA
-3.77111E+00
4.79345E-02
-2.81912E+00
8.50568E-03
-1.00941E+01
4.92240E-02
AMFOA
-3.55533E+00
3.79088E-01
-1.86957E+00
5.14318E-01
-3.60611E+00
6.89348E-01
EPSO
-3.86278E+00
9.36222E-16
-3.31011E+00
3.75973E-02
-1.01532E+01
0.00000E+00
IJAYA
-3.86278E+00
9.36222E-16
-3.32167E+00
5.18144E-04
-1.01532E+01
3.50302E-15
F22
F23
F24
Avg
Std
Avg
Std
Avg
Std
FGSCA
-1.04029E+01
1.18424E-15
-9.76343E+00
2.44437E+00
1.83773E+06
1.24607E+06
BMWOA
-1.04029E+01
4.75467E-12
-1.05364E+01
3.32245E-12
9.35821E+07
3.88175E+07
ACWOA
-9.33564E+00
2.23920E+00
-8.91019E+00
2.61015E+00
3.05521E+06
1.24886E+06
OBLGWO
-7.01362E+00
3.16886E+00
-6.62938E+00
3.30202E+00
8.69037E+08
3.00169E+08
IWOA
-1.04029E+01
7.53882E-08
-1.05364E+01
8.46039E-07
1.84769E+07
8.70723E+06
CBA
-6.95447E+00
3.75571E+00
-6.10780E+00
3.87860E+00
4.44772E+06
1.62810E+06
CDLOBA
-7.53668E+00
3.68030E+00
-6.21421E+00
3.76943E+00
5.49552E+05
3.78973E+05
IFOA
-1.03499E+01
4.92099E-02
-1.03650E+01
3.35401E-01
1.48552E+09
9.61572E+07
AMFOA
-3.61500E+00
7.54514E-01
-3.67804E+00
8.92897E-01
2.20773E+09
1.51914E+08
EPSO
-1.04029E+01
2.36848E-15
-1.05364E+01
1.87244E-15
3.63354E+06
1.99319E+06

38
IJAYA
-1.04029E+01
1.77636E-15
-1.05364E+01
1.18424E-15
1.46463E+07
1.00131E+07
F25
F26
F27
Avg
Std
Avg
Std
Avg
Std
FGSCA
1.23155E+04
1.46567E+04
8.47107E+03
3.78516E+03
4.95564E+02
4.44498E+01
BMWOA
1.35301E+08
5.37923E+07
4.74952E+04
7.08452E+03
6.65684E+02
5.74225E+01
ACWOA
4.72614E+05
9.42206E+04
7.95593E+02
3.04593E+02
4.79873E+02
5.32505E+01
OBLGWO
5.60907E+10
1.26035E+10
1.08411E+05
2.30093E+04
1.04557E+04
2.91849E+03
IWOA
1.77441E+06
8.94874E+05
2.66449E+04
1.76171E+04
5.71441E+02
5.66645E+01
CBA
3.53473E+04
7.03871E+04
5.94976E+03
5.96176E+03
5.13507E+02
3.04794E+01
CDLOBA
1.07611E+04
1.32127E+04
6.65852E+04
1.73069E+04
5.04170E+02
6.67895E+01
IFOA
7.69016E+10
3.59852E+09
8.55121E+04
9.57436E+02
1.09015E+04
1.17873E+03
AMFOA
9.31404E+10
1.06926E+09
1.78353E+07
1.86772E+07
2.33019E+04
3.18620E+03
EPSO
3.20385E+03
2.39462E+03
2.97727E+03
2.67182E+03
4.47667E+02
5.52602E+01
IJAYA
9.16116E+05
2.78983E+05
3.33061E+04
6.86222E+03
5.39522E+02
3.50258E+01
F28
F29
F30
Avg
Std
Avg
Std
Avg
Std
FGSCA
5.20490E+02
2.20945E-01
6.18320E+02
3.01328E+00
7.00017E+02
1.11451E-02
BMWOA
5.20991E+02
6.72560E-02
6.32791E+02
2.93495E+00
7.02210E+02
4.30780E-01
ACWOA
5.20449E+02
8.95933E-02
6.31090E+02
3.20543E+00
7.00671E+02
4.45794E-02
OBLGWO
5.21034E+02
7.19856E-02
6.40191E+02
2.17338E+00
1.11826E+03
1.13151E+02
IWOA
5.20175E+02
1.43776E-01
6.29971E+02
2.87986E+00
7.00899E+02
1.43335E-01
CBA
5.20161E+02
1.60793E-01
6.41135E+02
2.25706E+00
7.00018E+02
2.55366E-02
CDLOBA
5.20893E+02
1.56785E-01
6.35232E+02
2.64908E+00
7.00013E+02
1.18195E-02
IFOA
5.21147E+02
2.91428E-02
6.42352E+02
7.91417E-01
1.43390E+03
3.33043E+01
AMFOA
5.21191E+02
9.47765E-02
6.49022E+02
7.39047E-01
1.65996E+03
1.13378E+01
EPSO
5.20899E+02
7.97275E-02
6.07736E+02
1.89286E+00
7.00013E+02
2.25639E-02
IJAYA
5.20435E+02
1.15858E-01
6.28963E+02
2.28310E+00
7.00750E+02
8.78224E-02
F31
F32
F33
Avg
Std
Avg
Std
Avg
Std
FGSCA
8.51554E+02
2.35812E+01
1.05138E+03
2.17731E+01
4.81597E+03
1.08880E+03
BMWOA
9.57841E+02
1.74006E+01
1.11289E+03
3.61913E+01
5.34486E+03
6.07272E+02
ACWOA
8.76844E+02
2.21883E+01
1.12285E+03
4.83017E+01
2.03322E+03
3.77244E+02
OBLGWO
1.10928E+03
1.85230E+01
1.22672E+03
3.10202E+01
8.56752E+03
4.71926E+02
IWOA
9.24214E+02
2.13567E+01
1.12900E+03
5.55587E+01
2.78847E+03
6.17766E+02
CBA
1.01632E+03
5.04525E+01
1.16416E+03
5.81461E+01
5.36176E+03
6.45863E+02
CDLOBA
1.06036E+03
4.69881E+01
1.20916E+03
6.57812E+01
5.70599E+03
8.99412E+02
IFOA
1.20654E+03
1.93967E+01
1.26273E+03
5.70378E+00
9.92775E+03
2.35849E+02
AMFOA
1.28114E+03
2.01261E+01
1.29983E+03
1.82285E+00
1.06058E+04
3.70154E+02
EPSO
8.38915E+02
3.58164E+01
1.07004E+03
2.38512E+01
1.36719E+03
1.94518E+02
IJAYA
9.21262E+02
1.80559E+01
1.03289E+03
3.77583E+01
4.82358E+03
4.62345E+02
F34
F35
F36
Avg
Std
Avg
Std
Avg
Std
FGSCA
5.40482E+03
1.06070E+03
1.20073E+03
6.13010E-01
1.30050E+03
9.02596E-02
BMWOA
7.38577E+03
5.64369E+02
1.20232E+03
8.57716E-01
1.30061E+03
1.01032E-01

39
ACWOA
5.63006E+03
5.88791E+02
1.20075E+03
3.74703E-01
1.30058E+03
7.12024E-02
OBLGWO
8.49989E+03
8.37627E+02
1.20306E+03
4.93262E-01
1.30672E+03
5.57607E-01
IWOA
5.78113E+03
5.71399E+02
1.20103E+03
3.64450E-01
1.30049E+03
1.72581E-01
CBA
5.96810E+03
6.35533E+02
1.20111E+03
3.55409E-01
1.30037E+03
7.46590E-02
CDLOBA
5.67921E+03
4.51284E+02
1.20030E+03
1.58880E-01
1.30050E+03
1.23162E-01
IFOA
9.23940E+03
3.87530E+02
1.20323E+03
3.33078E-01
1.30830E+03
2.67498E-01
AMFOA
1.14200E+04
2.77281E+02
1.20813E+03
1.37735E-01
1.30983E+03
3.14144E-01
EPSO
5.23704E+03
1.85644E+03
1.20232E+03
3.14324E-01
1.30038E+03
7.30500E-02
IJAYA
5.85092E+03
5.97199E+02
1.20183E+03
6.54860E-01
1.30051E+03
1.14573E-01
F37
F38
F39
Avg
Std
Avg
Std
Avg
Std
FGSCA
1.40037E+03
1.86070E-01
1.51557E+03
6.55414E+00
1.61211E+03
4.84625E-01
BMWOA
1.40031E+03
5.11528E-02
1.55783E+03
1.47278E+01
1.61252E+03
4.53346E-01
ACWOA
1.40028E+03
4.43465E-02
1.51962E+03
5.05041E+00
1.61257E+03
2.92621E-01
OBLGWO
1.57566E+03
4.03422E+01
9.40301E+04
5.38330E+04
1.61265E+03
4.49280E-01
IWOA
1.40027E+03
4.71325E-02
1.56210E+03
2.32507E+01
1.61246E+03
6.25961E-01
CBA
1.40034E+03
1.51446E-01
1.57063E+03
1.40003E+01
1.61332E+03
3.47782E-01
CDLOBA
1.40028E+03
5.66645E-02
1.68273E+03
5.01307E+01
1.61323E+03
2.80953E-01
IFOA
1.66714E+03
7.05545E+00
2.73292E+05
4.08563E+04
1.61401E+03
8.41936E-02
AMFOA
1.73412E+03
1.02623E+01
5.75697E+05
6.74650E+04
1.61445E+03
1.87418E-01
EPSO
1.40032E+03
4.60714E-02
1.52006E+03
1.30884E+00
1.61156E+03
5.85215E-01
IJAYA
1.40026E+03
4.52442E-02
1.52285E+03
5.53936E+00
1.61246E+03
4.15723E-01
F40
F41
F42
Avg
Std
Avg
Std
Avg
Std
FGSCA
5.86163E+05
3.93855E+05
3.57190E+05
5.66988E+05
1.91397E+03
2.75212E+00
BMWOA
6.12063E+06
3.63200E+06
8.58900E+04
6.85344E+04
1.92851E+03
1.83884E+01
ACWOA
5.55783E+05
3.20708E+05
1.02630E+04
4.52472E+03
1.91374E+03
2.04570E+00
OBLGWO
1.04439E+08
6.95069E+07
1.13209E+09
1.02303E+09
2.21974E+03
1.02757E+02
IWOA
1.84399E+06
1.27383E+06
7.66107E+03
4.55364E+03
1.96003E+03
4.74407E+01
CBA
3.22475E+05
1.69227E+05
9.60539E+03
8.30182E+03
1.92315E+03
1.93052E+01
CDLOBA
3.70678E+04
3.23400E+04
8.55219E+03
7.45693E+03
1.97700E+03
4.29310E+01
IFOA
1.39248E+08
3.29046E+07
6.44063E+09
9.61479E+08
2.27731E+03
2.71050E+00
AMFOA
9.06452E+08
1.54634E+08
1.29875E+10
2.99227E+08
2.54224E+03
3.94426E+01
EPSO
4.05543E+05
2.26176E+05
3.93038E+03
2.15487E+03
1.91434E+03
1.90615E+01
IJAYA
8.56542E+05
3.55568E+05
1.09559E+04
6.54950E+03
1.91115E+03
1.43179E+00
F43
F44
F45
Avg
Std
Avg
Std
Avg
Std
FGSCA
3.35645E+04
1.80877E+04
1.51779E+06
3.74278E+06
2.96864E+03
3.28127E+02
BMWOA
2.34457E+04
1.38113E+04
1.06917E+06
5.96554E+05
2.90981E+03
2.66076E+02
ACWOA
2.94290E+03
3.26446E+02
3.53250E+05
2.95032E+05
2.98328E+03
2.91565E+02
OBLGWO
1.88977E+05
1.03486E+05
4.21921E+07
3.90872E+07
3.32771E+03
3.95189E+02
IWOA
2.31535E+04
7.40662E+03
1.07955E+06
9.33774E+05
3.04228E+03
2.19977E+02
CBA
3.12078E+03
6.16684E+02
1.19392E+05
3.96065E+04
3.42988E+03
3.84237E+02
CDLOBA
2.74299E+04
1.51012E+04
2.66363E+04
1.66367E+04
3.13831E+03
3.28871E+02

40
IFOA
3.44880E+05
8.16704E+04
7.88061E+07
3.36835E+07
3.23561E+04
1.37620E+04
AMFOA
1.53480E+08
2.75953E+08
1.08975E+09
3.73232E+08
1.63310E+06
1.00150E+06
EPSO
2.56741E+03
1.99005E+02
8.46724E+04
6.01049E+04
2.41680E+03
5.83175E+01
IJAYA
6.20882E+03
1.89663E+03
1.64227E+05
6.54947E+04
2.54036E+03
9.29270E+01
F46
F47
F48
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.50000E+03
0.00000E+00
2.60000E+03
0.00000E+00
2.70000E+03
0.00000E+00
BMWOA
2.50052E+03
5.05096E-01
2.60014E+03
1.57951E-01
2.70001E+03
6.60651E-03
ACWOA
2.61548E+03
1.52656E-01
2.60319E+03
7.83266E-01
2.71533E+03
1.14182E+01
OBLGWO
2.93484E+03
2.11936E+02
2.60045E+03
2.42997E-01
2.70000E+03
0.00000E+00
IWOA
2.61958E+03
1.51495E+00
2.60149E+03
1.04057E+00
2.72256E+03
1.31097E+01
CBA
2.61588E+03
2.41255E-01
2.66014E+03
1.63278E+01
2.74026E+03
1.73735E+01
CDLOBA
2.61709E+03
4.23627E+00
2.71246E+03
3.56947E+01
2.71842E+03
8.58623E+00
IFOA
2.50000E+03
2.51269E-05
2.60000E+03
1.49596E-04
2.70000E+03
2.70763E-07
AMFOA
2.50000E+03
3.54460E-04
2.60002E+03
2.91279E-03
2.70000E+03
4.09629E-06
EPSO
2.61524E+03
5.02742E-13
2.63226E+03
7.65030E+00
2.70719E+03
3.43694E+00
IJAYA
2.61573E+03
1.19006E-01
2.62560E+03
1.27510E+01
2.70986E+03
2.32815E+00
F49
F50
F51
Avg
Std
Avg
Std
Avg
Std
FGSCA
2.70062E+03
1.41729E-01
2.90000E+03
0.00000E+00
3.00000E+03
0.00000E+00
BMWOA
2.70058E+03
1.15966E-01
2.90007E+03
9.09296E-02
3.00010E+03
1.71690E-01
ACWOA
2.70054E+03
7.50925E-02
3.53919E+03
3.80129E+02
4.63689E+03
4.92260E+02
OBLGWO
2.80000E+03
0.00000E+00
4.16319E+03
9.89726E+01
5.62560E+03
2.28708E+03
IWOA
2.71042E+03
3.14741E+01
3.65368E+03
2.87269E+02
4.81964E+03
5.68649E+02
CBA
2.70143E+03
2.85543E+00
4.07368E+03
3.59882E+02
5.58398E+03
9.04871E+02
CDLOBA
2.70103E+03
1.66194E+00
3.89689E+03
2.95711E+02
5.18979E+03
6.65755E+02
IFOA
2.79161E+03
2.65406E+01
2.90000E+03
2.51013E-06
3.00000E+03
7.62372E-06
AMFOA
2.80000E+03
1.04356E-08
2.90000E+03
7.25216E-05
3.00000E+03
1.29032E-04
EPSO
2.76025E+03
5.15257E+01
3.18277E+03
5.75426E+01
3.67508E+03
3.56965E+01
IJAYA
2.70047E+03
8.04710E-02
3.70559E+03
5.42289E+01
3.83017E+03
1.49750E+02
F52
F53
Overall Rank
Avg
Std
Avg
Std
Rank
+/=/-
ARV
FGSCA
3.10000E+03
0.00000E+00
5.96928E+03
7.27122E+03
1
~
3.11321
BMWOA
2.76316E+05
2.77940E+05
3.79808E+04
3.22381E+04
4
37/3/13
5.28302
ACWOA
6.14747E+06
4.23976E+06
1.01648E+04
3.26110E+03
5
32/5/16
5.52830
OBLGWO
1.13720E+08
1.26999E+08
1.90221E+06
1.03699E+06
10
42/2/9
7.50943
IWOA
3.81945E+06
4.97133E+06
2.30565E+04
1.02496E+04
3
29/4/20
5.09434
CBA
2.97947E+07
3.30131E+07
1.61989E+04
1.26335E+04
7
34/4/15
6.88679
CDLOBA
1.65515E+07
1.12565E+07
3.52126E+04
4.32477E+04
8
36/3/14
7.16981
IFOA
4.97781E+03
1.74572E+01
3.32704E+03
1.66779E+00
9
47/0/6
7.47170
AMFOA
4.85689E+03
5.06884E+02
3.30638E+03
2.82743E+01
11
50/0/3
8.41509
EPSO
3.90907E+03
1.47925E+02
4.75236E+03
7.95727E+02
2
17/9/27
3.43396
IJAYA
2.62663E+06
4.21979E+06
5.92945E+03
7.04217E+02
6
28/6/19
5.60377

41
5.7 Wall-clock time cost
This section focuses on the time consumed by each method from start to finish. The test
environment uses 30 search agents to search for 300,000 evaluations. The detailed experimental results
are shown in Tables 10-12, where the table results are listed in units of seconds. As you can see, Tables
10-12 show detailed wall-clock time consumed by FGSCA, FSCA, GSCA, SCA, WOA, GWO, MFO,
BA, FA, PSO, ACO, MSCA, ASCA-PSO, m-SCA, SCADE, CGSCA, OBSCA, BMWOA, ACWOA,
OBLGWO, IWOA, CBA, CDLOBA, IFOA, AMFOA, EPSO, and IJAYA. The scale diagram of the
wall-clock time is drawn to vividly describe these methods' wall-clock time. As shown in Fig. 12, FGSCA
consumes more wall-clock time overall than other FSCA, GSCA, SCA, WOA, GWO, BA, MFO, BA,
PSO, and ACO. It demonstrates that FGSCA has a higher overall complexity than these algorithms.
Compared with the six SCA variants, FGSCA has lower time complexity than MSCA, ASCA-PSO, m-
SCA, SCADE, CGSCA, and OBSCA, which shows that FGSCA has more efficient characteristics.
Besides, the overall clock consumption of FGSCA is lower than that of other improved MAs, including
BMWOA, ACWOA, OBLGWO, IWOA, CBA, CDLOBA, IFOA, AMFOA, EPSO, and IJAYA. In
short, FGSCA has certain advantages in accuracy and efficiency compared to well-known MAs, SCA
variants, and improved MAs. It can be inferred that FGSCA is expected to become an accurate and
efficient tool to deal with various complex optimization problems.
There is a trade-off requiring to be considered between solution accuracy and time consumption
when facing a specific optimization problem. Suppose the decision-makers pay more attention to the
final solution's quality when time and resources are not limited [121]. In that case, FGSCA is obviously
a better choice to solve the optimization problem. Well-known MAs such as PSO is more suitable when
decision-makers pay more attention to time consumption. Considering FGSCA has more time-
consuming, parallel computing can be adopted to reduce the time consumed. Thus, FGSCA is a more
sensible choice to deal with complex problems than other improved MAs because it has a more
impressive performance accuracy and time complexity.
Table 10. The wall-clock time cost of SCA variants and involved MAs on 53 benchmark functions
F
FGSCA
FSCA
GSCA
SCA
WOA
GWO
BA
FA
PSO
ACO
F1
6.30E+01
4.62E+01
4.40E+01
3.02E+01
3.42E+01
4.29E+01
5.15E+01
5.69E+01
3.14E+02
3.07E+01
F2
8.52E+01
7.09E+01
6.94E+01
6.29E+01
6.46E+01
6.76E+01
6.93E+01
7.46E+01
5.26E+02
5.44E+01
F3
1.29E+02
1.14E+02
1.05E+02
9.32E+01
8.63E+01
9.45E+01
9.48E+01
9.62E+01
2.48E+02
9.31E+01
F4
7.70E+01
5.99E+01
5.68E+01
4.44E+01
4.64E+01
5.25E+01
5.33E+01
5.65E+01
3.64E+02
4.36E+01
F5
9.96E+01
7.16E+01
6.97E+01
4.18E+01
4.53E+01
5.49E+01
5.85E+01
5.92E+01
4.16E+02
4.01E+01
F6
6.99E+01
4.58E+01
4.35E+01
3.16E+01
4.23E+01
5.12E+01
6.01E+01
6.12E+01
3.98E+02
3.61E+01
F7
1.13E+02
1.04E+02
9.95E+01
7.92E+01
8.18E+01
9.21E+01
8.82E+01
9.84E+01
5.42E+02
8.31E+01
F8
9.86E+01
7.05E+01
6.63E+01
4.53E+01
4.75E+01
5.69E+01
6.24E+01
5.92E+01
3.04E+02
4.26E+01
F9
8.75E+01
6.93E+01
6.75E+01
4.72E+01
5.21E+01
5.67E+01
5.54E+01
6.07E+01
4.84E+02
4.32E+01
F10
1.18E+02
9.23E+01
8.98E+01
6.42E+01
7.25E+01
7.71E+01
7.56E+01
8.13E+01
3.54E+02
7.16E+01
F11
9.53E+01
7.46E+01
7.25E+01
5.32E+01
6.29E+01
6.85E+01
7.57E+01
7.83E+01
3.62E+02
6.12E+01
F12
2.37E+02
2.07E+02
2.01E+02
1.92E+02
1.98E+02
1.99E+02
2.04E+02
2.15E+02
4.86E+02
1.76E+02
F13
1.97E+02
1.69E+02
1.57E+02
1.39E+02
1.42E+02
1.58E+02
1.83E+02
2.11E+02
4.26E+02
1.27E+02
F14
4.21E+02
3.62E+02
3.52E+02
3.03E+02
2.83E+02
2.69E+02
3.24E+02
3.37E+02
5.72E+02
2.92E+02

42
F15
6.50E+01
5.43E+01
5.23E+01
4.39E+01
3.64E+01
3.92E+01
4.59E+01
4.77E+01
2.69E+02
3.79E+01
F16
4.26E+01
3.51E+01
3.31E+01
2.68E+01
2.89E+01
2.93E+01
2.86E+01
3.33E+01
2.09E+02
3.12E+01
F17
6.40E+01
4.83E+01
4.33E+01
4.04E+01
3.52E+01
3.87E+01
4.52E+01
4.93E+01
2.02E+02
4.15E+01
F18
4.87E+01
3.94E+01
3.69E+01
3.29E+01
3.35E+01
3.28E+01
3.67E+01
3.89E+01
2.56E+02
4.02E+01
F19
8.67E+01
7.24E+01
6.84E+01
6.38E+01
6.58E+01
6.78E+01
7.29E+01
7.58E+01
2.95E+02
6.64E+01
F20
8.18E+01
7.57E+01
7.27E+01
5.92E+01
5.57E+01
5.74E+01
6.51E+01
6.82E+01
3.49E+02
6.45E+01
F21
1.63E+02
1.37E+02
1.26E+02
1.12E+02
1.21E+02
1.43E+02
1.25E+02
1.49E+02
3.06E+02
1.35E+02
F22
1.76E+02
1.42E+02
1.38E+02
1.25E+02
1.27E+02
1.36E+02
1.49E+02
1.58E+02
4.04E+02
1.24E+02
F23
2.48E+02
2.04E+02
1.99E+02
1.87E+02
1.82E+02
1.92E+02
1.84E+02
1.98E+02
4.66E+02
1.88E+02
F24
7.11E+01
5.65E+01
5.15E+01
4.34E+01
4.22E+01
4.73E+01
4.76E+01
5.28E+01
3.13E+02
5.01E+01
F25
6.17E+01
5.13E+01
4.71E+01
3.59E+01
3.96E+01
4.24E+01
4.52E+01
5.18E+01
2.62E+02
4.14E+01
F26
7.38E+01
5.99E+01
5.69E+01
4.53E+01
4.87E+01
5.07E+01
4.93E+01
5.26E+01
2.87E+02
4.65E+01
F27
9.43E+01
7.84E+01
7.24E+01
6.23E+01
6.54E+01
6.90E+01
6.82E+01
7.22E+01
4.16E+02
6.69E+01
F28
9.67E+01
8.33E+01
7.82E+01
6.59E+01
6.81E+01
7.28E+01
7.47E+01
8.17E+01
3.79E+02
7.31E+01
F29
3.34E+02
2.98E+02
2.88E+02
2.61E+02
2.81E+02
2.94E+02
2.93E+02
3.04E+02
5.12E+02
2.76E+02
F30
7.50E+01
5.84E+01
5.34E+01
4.29E+01
4.42E+01
4.79E+01
5.27E+01
5.61E+01
2.42E+02
4.67E+01
F31
6.06E+01
5.47E+01
4.97E+01
3.36E+01
3.55E+01
3.82E+01
3.56E+01
4.14E+01
2.47E+02
3.73E+01
F32
5.64E+01
4.22E+01
3.82E+01
3.24E+01
3.62E+01
3.91E+01
4.13E+01
4.46E+01
2.24E+02
4.18E+01
E33
5.96E+01
4.62E+01
3.92E+01
3.06E+01
3.11E+01
3.22E+01
3.45E+01
3.76E+01
2.15E+02
3.42E+01
F34
7.28E+01
7.54E+01
6.24E+01
4.94E+01
5.15E+01
5.21E+01
5.52E+01
6.27E+01
2.69E+02
5.28E+01
F35
1.51E+02
1.21E+02
1.07E+02
9.08E+01
9.24E+01
9.69E+01
1.04E+02
1.11E+02
5.89E+02
1.03E+02
F36
1.72E+02
1.49E+02
1.38E+02
1.26E+02
1.19E+02
1.32E+02
1.35E+02
1.47E+02
4.25E+02
1.22E+02
F37
8.14E+01
7.32E+01
6.63E+01
5.35E+01
5.12E+01
6.02E+01
5.92E+01
6.62E+01
3.51E+02
5.82E+01
F38
7.56E+01
6.20E+01
5.60E+01
4.57E+01
5.52E+01
5.78E+01
5.63E+01
5.96E+01
3.40E+02
5.46E+01
F39
7.24E+01
5.81E+01
5.41E+01
4.40E+01
4.28E+01
5.11E+01
4.98E+01
5.38E+01
2.82E+02
5.15E+01
F40
7.24E+01
6.32E+01
5.82E+01
4.68E+01
4.26E+01
4.94E+01
5.37E+01
5.82E+01
3.42E+02
5.38E+01
F41
9.20E+01
7.81E+01
7.01E+01
5.93E+01
6.32E+01
6.77E+01
6.23E+01
6.81E+01
3.21E+02
5.86E+01
F42
9.17E+01
7.89E+01
6.89E+01
6.15E+01
6.21E+01
6.68E+01
6.82E+01
7.26E+01
3.53E+02
6.42E+01
F43
1.50E+02
1.33E+02
1.23E+02
1.14E+02
1.06E+02
1.16E+02
1.18E+02
1.25E+02
3.81E+02
1.21E+02
F44
7.31E+01
6.09E+01
5.19E+01
4.13E+01
4.32E+01
4.68E+01
4.42E+01
4.73E+01
2.94E+02
4.58E+01
F45
9.88E+01
8.21E+01
7.81E+01
6.14E+01
6.41E+01
6.63E+01
6.65E+01
6.98E+01
2.92E+02
6.57E+01
F46
9.56E+01
7.61E+01
6.60E+01
5.86E+01
6.24E+01
6.53E+01
6.22E+01
6.97E+01
3.97E+02
6.95E+01
F47
2.01E+02
1.73E+02
1.64E+02
1.35E+02
1.42E+02
1.58E+02
1.51E+02
1.49E+02
5.02E+02
1.41E+02
F48
1.91E+02
1.66E+02
1.56E+02
1.37E+02
1.41E+02
1.53E+02
1.46E+02
1.51E+02
5.06E+02
1.44E+02
F49
1.84E+02
1.65E+02
1.55E+02
1.33E+02
1.47E+02
1.36E+02
1.41E+02
1.49E+02
4.21E+02
1.34E+02
F50
4.42E+02
4.17E+02
3.97E+02
3.77E+02
3.82E+02
3.75E+02
3.86E+02
3.99E+02
6.47E+02
4.29E+02
F51
4.67E+02
4.36E+02
4.16E+02
3.97E+02
3.92E+02
4.16E+02
4.17E+02
4.39E+02
6.33E+02
4.17E+02
F52
1.67E+02
1.38E+02
1.27E+02
1.08E+02
1.03E+02
1.17E+02
1.15E+02
1.21E+02
3.52E+02
1.20E+02
F53
1.74E+02
1.64E+02
1.50E+02
1.31E+02
1.34E+02
1.42E+02
1.45E+02
1.51E+02
3.63E+02
1.49E+02
Sum
6.97E+03
5.97E+03
5.60E+03
4.84E+03
4.93E+03
5.24E+03
5.39E+03
5.73E+03
1.97E+04
5.09E+03
Table 11. The wall-clock time cost of SCA variants on 53 benchmark functions
F
MSCA
ASCA-PSO
m-SCA
SCADE
CGSCA
OBSCA
F1
2.68E+02
9.44E+01
7.05E+01
2.13E+02
2.10E+02
5.71E+01

43
F2
2.79E+02
1.13E+02
9.14E+01
2.39E+02
2.36E+02
7.25E+01
F3
3.23E+02
6.13E+02
6.32E+02
9.99E+02
7.30E+02
5.70E+02
F4
2.70E+02
9.50E+01
7.00E+01
2.09E+02
2.10E+02
5.58E+01
F5
2.78E+02
1.25E+02
1.04E+02
2.60E+02
2.43E+02
8.67E+01
F6
2.72E+02
9.59E+01
7.44E+01
2.13E+02
2.13E+02
5.85E+01
F7
2.77E+02
1.20E+02
9.70E+01
2.54E+02
2.37E+02
8.24E+01
F8
2.78E+02
1.16E+02
9.56E+01
2.44E+02
2.34E+02
7.70E+01
F9
2.75E+02
1.01E+02
7.28E+01
2.12E+02
2.15E+02
6.08E+01
F10
2.77E+02
1.16E+02
9.08E+01
2.34E+02
2.28E+02
7.38E+01
F11
2.75E+02
1.28E+02
1.06E+02
2.64E+02
2.39E+02
8.85E+01
F12
2.90E+02
2.16E+02
2.03E+02
4.05E+02
3.34E+02
1.81E+02
F13
2.92E+02
2.15E+02
2.05E+02
4.13E+02
3.37E+02
1.79E+02
F14
1.04E+02
7.05E+02
7.43E+02
1.18E+03
8.33E+02
6.75E+02
F15
6.99E+02
7.29E+01
5.67E+01
2.08E+02
2.01E+02
5.04E+01
F16
3.78E+02
4.81E+01
3.67E+01
1.74E+02
1.79E+02
2.91E+01
F17
3.79E+02
4.48E+01
3.27E+01
1.75E+02
1.74E+02
2.73E+01
F18
3.74E+02
4.67E+01
3.06E+01
1.70E+02
1.72E+02
2.54E+01
F19
5.78E+02
1.02E+02
9.22E+01
2.61E+02
2.28E+02
7.98E+01
F20
9.78E+02
1.06E+02
9.33E+01
2.60E+02
2.30E+02
8.21E+01
F21
8.51E+02
2.42E+02
2.37E+02
4.63E+02
3.64E+02
2.19E+02
F22
9.27E+02
3.05E+02
3.18E+02
5.72E+02
4.40E+02
2.89E+02
F23
1.03E+02
4.20E+02
4.32E+02
7.24E+02
5.41E+02
3.95E+02
F24
2.79E+02
1.23E+02
9.99E+01
2.53E+02
2.47E+02
8.37E+01
F25
3.44E+02
1.43E+02
1.11E+02
2.91E+02
2.68E+02
8.57E+01
F26
2.77E+02
1.12E+02
8.61E+01
2.32E+02
2.24E+02
7.14E+01
F27
2.73E+02
1.12E+02
8.63E+01
2.32E+02
2.25E+02
6.98E+01
F28
2.68E+02
1.14E+02
8.78E+01
2.36E+02
2.27E+02
7.24E+01
F29
2.99E+02
3.55E+02
3.51E+02
6.11E+02
4.66E+02
3.14E+02
F30
2.73E+02
1.14E+02
9.21E+01
2.41E+02
2.33E+02
7.66E+01
F31
2.74E+02
1.11E+02
8.82E+01
2.36E+02
2.25E+02
7.09E+01
F32
2.72E+02
1.12E+02
8.99E+01
2.38E+02
2.30E+02
7.43E+01
E33
2.76E+02
1.25E+02
9.85E+01
2.53E+02
2.36E+02
8.29E+01
F34
2.75E+02
1.25E+02
1.04E+02
2.56E+02
2.41E+02
8.56E+01
F35
3.01E+02
1.71E+02
1.45E+02
3.28E+02
3.00E+02
1.28E+02
F36
4.92E+02
2.08E+02
1.65E+02
4.52E+02
4.09E+02
1.30E+02
F37
4.05E+02
1.56E+02
1.30E+02
3.39E+02
3.20E+02
9.75E+01
F38
4.83E+02
1.99E+02
1.55E+02
4.34E+02
4.10E+02
1.29E+02
F39
4.91E+02
1.97E+02
1.55E+02
4.18E+02
4.14E+02
1.36E+02
F40
4.56E+02
2.07E+02
1.69E+02
4.16E+02
4.04E+02
1.45E+02
F41
5.78E+02
2.36E+02
1.86E+02
5.00E+02
4.82E+02
1.56E+02
F42
4.63E+02
2.71E+02
2.36E+02
5.19E+02
4.52E+02
1.98E+02
F43
4.36E+02
1.87E+02
1.48E+02
3.87E+02
3.56E+02
1.16E+02
F44
4.30E+02
1.88E+02
1.43E+02
3.80E+02
3.71E+02
1.21E+02
F45
4.21E+02
1.83E+02
1.48E+02
3.88E+02
3.55E+02
1.24E+02

44
F46
4.17E+02
2.58E+02
2.27E+02
4.85E+02
4.38E+02
1.96E+02
F47
4.08E+02
2.35E+02
1.99E+02
4.41E+02
3.88E+02
1.73E+02
F48
4.07E+02
2.44E+02
2.02E+02
4.47E+02
3.98E+02
1.76E+02
F49
6.11E+02
8.56E+02
8.28E+02
1.46E+03
1.07E+03
7.64E+02
F50
6.40E+02
9.41E+02
9.17E+02
1.53E+03
1.11E+03
8.36E+02
F51
6.46E+02
4.40E+02
3.85E+02
8.01E+02
6.77E+02
3.34E+02
F52
5.67E+02
4.23E+02
3.77E+02
7.89E+02
6.15E+02
3.41E+02
F53
5.58E+02
3.28E+02
2.84E+02
6.37E+02
5.42E+02
2.54E+02
Sum
2.13E+04
1.17E+04
1.05E+04
2.26E+04
1.94E+04
9.16E+03
Table 12. The wall-clock time cost of improved MAs on 53 benchmark functions
F
BMWOA
ACWOA
OBLGWO
IWOA
CBA
CDLOBA
IFOA
AMFOA
EPSO
IJAYA
F1
4.11E+02
1.73E+02
7.01E+01
1.30E+02
2.27E+02
1.70E+02
1.22E+02
1.05E+02
6.77E+02
3.95E+02
F2
5.02E+02
2.17E+02
1.04E+02
1.69E+02
2.93E+02
2.39E+02
1.68E+02
1.44E+02
7.82E+02
4.65E+02
F3
1.53E+03
7.49E+02
6.32E+02
6.84E+02
7.63E+02
1.08E+03
7.55E+02
7.66E+02
1.60E+03
1.82E+03
F4
3.95E+02
1.79E+02
7.37E+01
1.30E+02
2.34E+02
1.85E+02
1.32E+02
1.06E+02
6.15E+02
3.90E+02
F5
5.36E+02
2.29E+02
1.13E+02
1.72E+02
2.83E+02
2.48E+02
1.77E+02
1.47E+02
7.56E+02
4.80E+02
F6
4.17E+02
1.77E+02
6.88E+01
1.24E+02
2.36E+02
1.93E+02
1.36E+02
1.05E+02
7.13E+02
3.99E+02
F7
4.01E+02
1.61E+02
7.64E+01
1.24E+02
2.19E+02
1.93E+02
1.23E+02
1.11E+02
6.66E+02
4.32E+02
F8
3.98E+02
1.71E+02
8.44E+01
1.29E+02
2.24E+02
1.98E+02
1.26E+02
1.27E+02
1.07E+03
4.27E+02
F9
3.49E+02
1.28E+02
5.16E+01
9.90E+01
1.89E+02
1.53E+02
1.03E+02
8.57E+01
9.13E+02
3.85E+02
F10
4.97E+02
2.10E+02
9.29E+01
1.61E+02
2.61E+02
2.47E+02
1.64E+02
1.36E+02
7.71E+02
4.74E+02
F11
4.86E+02
2.24E+02
1.20E+02
1.75E+02
2.91E+02
2.38E+02
1.63E+02
1.51E+02
7.99E+02
5.14E+02
F12
7.00E+02
3.12E+02
2.07E+02
2.56E+02
3.41E+02
3.66E+02
2.42E+02
2.33E+02
8.24E+02
7.75E+02
F13
5.41E+02
2.51E+02
1.65E+02
2.05E+02
2.95E+02
3.20E+02
2.01E+02
1.88E+02
7.15E+02
7.03E+02
F14
1.71E+03
8.69E+02
7.93E+02
8.15E+02
9.13E+02
1.25E+03
7.63E+02
7.57E+02
1.42E+03
2.09E+03
F15
3.32E+02
1.26E+02
5.00E+01
9.55E+01
1.87E+02
1.38E+02
8.65E+01
8.28E+01
4.98E+02
3.72E+02
F16
2.30E+02
8.18E+01
2.23E+01
4.58E+01
1.18E+02
8.42E+01
4.75E+01
4.47E+01
3.65E+02
2.99E+02
F17
1.93E+02
8.49E+01
2.13E+01
4.95E+01
1.22E+02
8.37E+01
8.05E+01
4.73E+01
7.20E+02
3.00E+02
F18
2.36E+02
1.14E+02
3.06E+01
7.60E+01
1.76E+02
1.18E+02
7.10E+01
6.99E+01
5.34E+02
3.11E+02
F19
3.92E+02
1.24E+02
7.17E+01
9.68E+01
1.76E+02
1.66E+02
9.12E+01
9.52E+01
4.47E+02
4.37E+02
F20
4.75E+02
1.60E+02
9.33E+01
1.36E+02
2.34E+02
2.12E+02
1.21E+02
1.22E+02
6.14E+02
4.54E+02
F21
7.55E+02
3.05E+02
2.44E+02
2.74E+02
3.55E+02
4.17E+02
2.68E+02
2.72E+02
7.05E+02
8.30E+02
F22
9.00E+02
3.38E+02
2.87E+02
3.39E+02
4.21E+02
5.13E+02
3.58E+02
3.63E+02
8.12E+02
1.02E+03
F23
1.26E+03
5.98E+02
5.17E+02
5.48E+02
6.64E+02
8.22E+02
5.19E+02
5.12E+02
1.06E+03
1.47E+03
F24
4.33E+02
1.85E+02
9.22E+01
1.38E+02
2.40E+02
2.07E+02
1.41E+02
1.21E+02
1.04E+03
4.28E+02
F25
3.59E+02
1.43E+02
7.00E+01
1.06E+02
1.88E+02
1.57E+02
1.06E+02
8.67E+01
6.01E+02
3.93E+02
F26
3.31E+02
1.36E+02
6.57E+01
9.99E+01
1.83E+02
1.52E+02
9.94E+01
8.72E+01
1.10E+03
3.89E+02
F27
3.12E+02
1.26E+02
5.93E+01
9.36E+01
1.79E+02
1.49E+02
1.03E+02
8.20E+01
7.01E+02
4.00E+02
F28
3.70E+02
1.45E+02
7.35E+01
1.07E+02
2.08E+02
1.79E+02
1.15E+02
9.49E+01
2.11E+03
4.21E+02
F29
8.19E+02
4.19E+02
3.23E+02
3.57E+02
4.50E+02
5.44E+02
3.40E+02
3.29E+02
9.21E+02
1.09E+03
F30
3.40E+02
1.40E+02
6.98E+01
1.05E+02
1.86E+02
1.57E+02
1.00E+02
8.53E+01
5.38E+02
4.03E+02
F31
3.95E+02
1.71E+02
7.48E+01
1.24E+02
2.08E+02
1.71E+02
1.19E+02
9.48E+01
1.58E+03
4.20E+02
F32
3.76E+02
1.45E+02
7.10E+01
1.10E+02
1.95E+02
1.82E+02
1.26E+02
1.07E+02
1.61E+03
4.27E+02

45
E33
3.70E+02
1.60E+02
7.71E+01
1.26E+02
2.22E+02
2.01E+02
1.39E+02
1.10E+02
1.52E+03
4.33E+02
F34
3.66E+02
1.47E+02
7.45E+01
1.13E+02
2.01E+02
1.71E+02
1.08E+02
9.72E+01
2.00E+03
4.48E+02
F35
4.63E+02
1.91E+02
1.17E+02
1.55E+02
2.51E+02
2.40E+02
1.58E+02
1.46E+02
2.19E+03
5.34E+02
F36
3.40E+02
1.35E+02
6.53E+01
9.99E+01
1.80E+02
1.57E+02
1.08E+02
8.74E+01
1.26E+03
4.10E+02
F37
4.37E+02
1.82E+02
8.96E+01
1.43E+02
2.51E+02
2.12E+02
1.55E+02
1.26E+02
1.25E+03
4.59E+02
F38
5.00E+02
2.12E+02
9.77E+01
1.59E+02
2.69E+02
2.27E+02
1.49E+02
1.29E+02
1.01E+03
4.84E+02
F39
3.76E+02
1.52E+02
7.26E+01
1.11E+02
2.11E+02
1.75E+02
1.13E+02
9.78E+01
1.89E+03
4.34E+02
F40
3.82E+02
1.70E+02
7.91E+01
1.23E+02
2.20E+02
1.88E+02
1.26E+02
1.08E+02
1.57E+03
4.52E+02
F41
3.39E+02
1.38E+02
6.60E+01
1.01E+02
1.85E+02
1.57E+02
1.08E+02
9.48E+01
7.65E+02
4.61E+02
F42
4.26E+02
1.92E+02
1.12E+02
1.58E+02
2.59E+02
2.54E+02
1.65E+02
1.50E+02
1.12E+03
5.93E+02
F43
3.91E+02
1.49E+02
7.09E+01
1.09E+02
1.97E+02
1.60E+02
1.09E+02
9.13E+01
1.79E+03
4.46E+02
F44
3.21E+02
1.34E+02
6.32E+01
1.03E+02
1.79E+02
1.51E+02
1.03E+02
8.61E+01
1.57E+03
4.61E+02
F45
3.39E+02
1.39E+02
6.98E+01
1.06E+02
1.91E+02
1.61E+02
1.10E+02
9.09E+01
1.76E+03
4.66E+02
F46
4.98E+02
2.15E+02
1.34E+02
1.76E+02
2.61E+02
2.63E+02
1.69E+02
1.55E+02
9.41E+02
6.40E+02
F47
4.80E+02
2.07E+02
1.26E+02
1.74E+02
2.79E+02
2.61E+02
1.71E+02
1.56E+02
7.22E+02
6.52E+02
F48
4.38E+02
1.95E+02
1.23E+02
1.67E+02
2.55E+02
2.62E+02
1.70E+02
1.55E+02
1.07E+03
6.30E+02
F49
1.14E+03
5.95E+02
4.62E+02
5.37E+02
6.48E+02
7.76E+02
5.09E+02
4.92E+02
1.57E+03
1.44E+03
F50
1.25E+03
6.24E+02
5.05E+02
5.43E+02
6.84E+02
7.78E+02
4.70E+02
4.58E+02
1.38E+03
1.55E+03
F51
5.75E+02
2.56E+02
1.70E+02
2.19E+02
3.34E+02
3.40E+02
2.06E+02
1.92E+02
1.15E+03
7.69E+02
F52
5.12E+02
2.53E+02
1.69E+02
2.14E+02
3.24E+02
3.37E+02
1.96E+02
1.80E+02
1.01E+03
7.58E+02
F53
4.08E+02
1.83E+02
1.09E+02
1.50E+02
2.38E+02
2.31E+02
1.52E+02
1.31E+02
2.15E+03
6.64E+02
Sum
2.77E+04
1.22E+04
7.74E+03
1.01E+04
1.51E+04
1.52E+04
9.96E+03
9.19E+03
4.51E+04
3.28E+04

46
Figure 12. Time consumption of involved methods on 53 benchmark function
5.8 Diversity and balance analysis
The exploration and exploration curves are plotted in the entire space to further study the
promotion effect of the two strategies on the original SCA in the exploration and exploitation stage.
The experiment results are shown in Fig. 13, which shows the proportion of F1, F7, F14, F21, F27,
F35, F42, F49, and F51 in the exploration and exploitation phase of the whole search space. The
proportion of the exploration stage is high in the early stage, while the proportion of the exploitation
is high in the middle and later stages. It is necessary to consider that there are multiple regions of locally
optimal solutions when FGSCA is adopted to resolve a certain test function.
FGSCA first conducts quicker exploration, quickly locks in areas where there may be an optimal

47
solution, and then conducts in-depth exploitation. The relatively large proportion of FGSCA explores
in the initial stage, which indicates that FGSCA seeks the best solution or improves the existing solution
in the entire search space as much as possible to avoid deteriorating into the local optimal solution.
FGSCA is basically in the exploitation stage in the later stage, which aims to quickly converge to the
minimum in a small local space. Therefore, FGSCA maintains a balance of exploration and exploitation
during search space.
Also, a comparative experiment on the diversity of FGSCA and SCA is conducted, as well. It is
tested in the same environment, and each function involved is independently tested 30 times to
moderate the special effects of randomness. As shown in Fig. 14, nine representative test functions
from 53 benchmark functions, including single-peak functions F1, F6, multi-peak functions F10, F15,
F27, fixed-dimensional functions F15, F36, F43, and composition functions F51, are selected to
compare and analyze the diversity of FGSCA and SCA. Diversity represents the average Euclidean
distance calculated between the agents. It can be easily seen that the average Euclidean distance of
FGSCA is closer than the basic SCA, which means that the combination of two strategies effectively
upsurges the convergence trends of SCA. It can be inferred that a member of the population can use
some of the finest gen obtainable in the entire evolutionary cluster. The proposed FGSCA can
effectively guide the population to the best area of the search space. In summary, we can determine
that the spiral motion and Gaussian mutation can well enhance SCA's performance.
Figure 13. Exploration and exploitation phases of 9 benchmark functions for FGSCA

48
Figure 14. Diversity of 9 benchmark functions for FGSCA and SCA
6 Discussions and limitations
This paper proposes a combination of spiral motion and Gaussian mutation to improve the
performance of SCA. In principle, the purpose of the spiral motion is to enhance the original SCA's
exploitation ability, and Gaussian mutation increases the original SCA population's diversity. These two
strategies work together on the population obtained using SCA. In detail, the initialized population is
updated by SCA to obtain a new population. The population is re-updated by way of spiral motion and
Gaussian mutation, and two new populations are gained. To fully absorb the advantages of the two
strategies, the greedy selection principle is adopted to screen the two populations' best population.
Finally, the obtained population is regarded as the guide of the next generation. The combination of
the two strategies is not accidental, and the random promotion effect of the two strategies on SCA is
tested based on comparative experiments. The results show that the simultaneous introduction of spiral
motion and Gaussian mutation has the most significant SCA effect. To further illustrate the advantages
of FGSCA compared to SCA, we separately analyze and discuss the search history, agent trajectory,
average fitness value, and convergence value under the iterative framework. Also, FGSCA is compared
with seven well-known MAs on 53 benchmark functions. To further highlight the progressive nature
of FGSCA, FGSCA is also compared with other improved MAs to test the potential of the algorithm.
The analysis of the results shows that the overall performance of FGSCA ranks first. The solution's
convergence speed and superiority are more prominent regardless of multimodal functions, multimodal
functions, hybrid functions, and composition functions. Finally, systematic experiments on the wall-cost

49
time, diversity, and balance of FGSCA are also conducted.
From the perspective of time complexity, FGSCA consumes more time than the well-known MAs
when solving various optimization problems but has lower time consumption than the reported SCA
variants and other improved MAs. Therefore, FGSCA is worthy of recognition in terms of its
effectiveness in solving practical problems. From the robustness perspective, FGSCA can always obtain
a robust and stable solution in optimization problems with different characteristics like unimodal,
multimodal, hybrid, and composition problems. It is mainly due to the combination of spiral motion
and Gaussian variation that can always balance exploration and exploitation. Besides, the diversity of
the FGSCA population is superior to the original SCA.
From the above experiments, FGSCA is exceptionally superior in these aspects. It has a better
solution and faster convergence speed when solving 53 benchmark functions. However, considering all
the practical advantages, FGSCA has some limitations compared to other algorithms as well. First of
all, the two strategies are effectively introduced. Even though the original SCA's performance is
improved, it dramatically increases the time of resolving it. Therefore, there must be a good trade-off
between efficiency and accuracy when dealing with practical engineering problems. Besides, the
parameter settings will affect the performance of FGSCA, including the parameters , ,  and 
of the original SCA, the parameters  and  of the spiral motion, and the parameter  of the
Gaussian mutation. Therefore, the decision-maker needs to adjust the method appropriately to find the
ideal solution according to the nature of the research problem.
In the future, the potential of the proposed FGSCA can be explored in more practical problems
including the drug discovery [122, 123], location-based services [124, 125], information retrieval services
[126-128] Also, it is better to join the proposed FGSCA with parallel commuting for a better
performance on more complex problems such as regression tasks [129, 130], kayak cycle phase
segmentation [131], recommender system [132-134], human motion capture [135], image classification
[136], image-to-image translation [137] and bionic electronic skin sensing [138, 139].
7 FGSCA for the engineering problems
In this section, we have investigated the efficacy of the FGSCA in three engineering cases,
including welded beam design (WBD) [140], I-beam design (IBD) [40], and pressure vessel design (PVD)
[107].
7.1 WBD problem
WBD is a problem that can be resolved by FGSCA, which aims to make the minimum value of
the fabrication cost for the welded beam. The constraints of optimization contain bucking load (),
shear stress ( ), bending stress in the beam (), and deflection rate (). It is comprised of four
parameters, including the thickness of the weld (), the length of the bar (), the height of the bar (),
and the thickness of the bar (). The related model of WBD is shown below.
Consider 󰇟󰇠󰇟󰇠
Objective 󰇛󰇜󰇛󰇜

50
Subject to 󰇛󰇜󰇛󰇜 ,
󰇛󰇜󰇛󰇜 ,
󰇛󰇜󰇛󰇜,
󰇛󰇜,
󰇛󰇜󰇛󰇜,
󰇛󰇜,
󰇛󰇜󰇛󰇜,
Variable ranges: ,
,
,
.
where 󰇛󰇜󰇛󰆒󰇜󰆒󰆒󰆒
󰇛󰆒󰆒󰇜,
󰆒
, 󰆒󰆒
,󰇛
󰇜,


󰇛
󰇜,
󰇝󰇟

󰇛
󰇜󰇠󰇞,
󰇛󰇜

, 󰇛󰇜

,
󰇛󰇜



󰇛

󰇜,
,  ,
, ,
, .
Table 13. Results of FGSCA and involved methods for WBD problem
Method
Optimum values for parameters
Optimum cost




FGSCA
0.2046
3.3319
9.1025
0.2048
1.7245
SCA
0.1970
3.6718
9.0722
0.2149
1.8147
WOA
0.2080
3.1565
9.4727
0.2112
1.8021
GWO
0.2118
3.3601
8.9088
0.2180
1.7982
MFO
0.2069
3.6248
8.7123
0.2220
1.8113
BA
0.1768
4.2868
9.1329
0.2074
1.8147
FA
0.1990
3.8212
8.8001
0.2178
1.8107
PSO
0.1814
4.2457
8.9201
0.2115
1.8109
ACO
0.1938
3.5491
9.2832
0.2116
1.8058
MSCA
0.2046
3.4718
9.0139
0.2138
1.7808
ASCA-PSO
0.1983
3.7547
8.8409
0.2156
1.7910
m-SCA
0.1992
3.6056
8.9535
0.2137
1.7783

51
SCADE
0.1902
3.4671
9.3458
0.2093
1.7827
CGSCA
0.1851
3.6524
9.3124
0.2081
1.7837
OBSCA
0.2114
3.1390
9.2752
0.2131
1.7847
BMWOA
0.1975
3.4377
9.1712
0.2112
1.7731
ACWOA
0.2019
3.2798
9.6312
0.2031
1.7739
OBLGWO
0.1976
3.4069
9.3207
0.2054
1.7501
IWOA
0.1928
3.4255
9.3400
0.2054
1.7491
CBA
0.1866
3.6743
9.0682
0.2112
1.7702
CDLOBA
0.1667
4.2075
9.1497
0.2069
1.7877
IFOA
0.1880
3.9545
8.9767
0.2107
1.7880
AMFOA
0.1879
3.5924
9.2679
0.2105
1.7914
EPSO
0.1987
3.4394
9.1098
0.2081
1.7409
IJAYA
0.1729
4.0800
9.0050
0.2076
1.7609
This problem can be solved by FGSCA and other algorithms involved in this section. The detailed
results are listed in Table 13. It is easy to see that FGSCA harvested an optimum cost of 1.7245. When
h, l, t, and b are set as 0.2046, 3.3319, 9.1025, and 0.2048; it indicates that the final result is the lowest.
SCA harvested an optimum cost of 1.8147. It can be seen that the combination of spiral motion and
Gaussian mutation has dramatically improved the ability of SCA to handle the WBD problem. Besides,
compared with other well-known MAs, including WOA, GWO, MFO, BA, FA, PSO, and ACO, FGSCA
performs better and gains the minimum value. FGSCA can still reap the smallest optimum costs
compared with the other six SCA variants. Also, FGSCA is compared with the other ten improved MAs,
including BMWOA, ACWOA, OBLGWO, IWOA, CBA, CDLOBA, IFOA, AMFOA, EPSO, IJAYA,
to assess the performance of FGSCA further. Results show that the cost obtained by FGSCA is the
highest priority among similarly improved MAs.
In general, FGSCA has enormous advantages over other methods involved. Compared with
twenty-four comparative MAs, FGSCA has a less optimum cost when solving the WBD problem. It is
exposed that the FGSCA intensely improves the rates and results of the basic SCA, and it can be
regarded as a satisfactory answer to the WBD problem
7.2 IBD problem
Real-world applications often have multiple restrictions, and decision-makers must consider
minimizing the costs concerning physical constraints on the engineering case [141-146]. FGSCA is also
a choice to tackle this engineering problem when faced with IBD problem. This problem comprises
four parameters: length, height, and two thicknesses, aiming to reach the prime cost. The related model
of the problem is shown below:
Consider 󰇟󰇠󰇟󰇠
Objective 󰇛󰇜
󰇛󰇜
 

󰇛
󰇜
Subject to 󰇛󰇜󰇛󰇜,
Variable ranges: ,

52
,
,
.
Table 14. Results of FGSCA and involved algorithms for IBD problem
Algorithm
Optimal values for parameters
Optimum cost




FGSCA
50
80
1.764702441
5
0.006625959
SCA
50
80
3.482012179
5
0.006701735
WOA
50
80
2.215979174
5
0.006707870
GWO
50
80
1.859991795
5
0.006707698
MFO
50
80
2.674707971
5
0.006704098
BA
50
80
2.935881266
5
0.006712526
FA
50
80
1.970790194
5
0.006710108
PSO
50
80
4.681998061
5
0.006705121
ACO
50
80
1.866043874
5
0.006699431
MSCA
50
80
1.688090073
5
0.006645241
ASCA-PSO
50
80
4.258517525
5
0.006661151
m-SCA
50
80
1.687023018
5
0.006645512
SCADE
50
80
4.047615913
5
0.006647413
CGSCA
50
80
2.543466098
5
0.006648714
OBSCA
50
80
2.211436848
5
0.006651659
BMWOA
50
80
1.972215865
5
0.006644578
ACWOA
50
80
2.782649262
5
0.006641900
OBLGWO
50
80
1.858253037
5
0.006641331
IWOA
50
80
1.692226051
5
0.006644190
CBA
50
80
1.692746263
5
0.006644068
CDLOBA
50
80
1.668401162
5
0.006650217
IFOA
50
80
1.663679038
5
0.006651411
AMFOA
50
80
1.726576939
5
0.006650619
EPSO
50
80
1.714443859
5
0.006638597
IJAYA
50
80
1.680634566
5
0.006647125
The final comparative experimental results of FGSCA and other methods to resolve the IBD
problem are shown in Table 14. It can be known that FGSCA obtained the smallest optimum cost in
these methods. Specifically, FGSCA received an optimum cost of 0.006625959 when , , , 
are equal to 50, 80, 1.764702441, and 5. Even though SCA harvested the optimum cost of 0.006701735,
it is slightly worse than FGSCA. ACO achieved an optimum cost of 0.006699431, followed closely by
SCA, MFO, PSO, GWO, WOA, FA, and BA. FGSCA has a lower optimum cost than MSCA, ASCA-
PSO, m-SCA, SCADE, CGSCA, and OBSCA. Besides, EPSO has a certain degree of competitiveness
with FGSCA, and the optimum cost of the other nine improved MAs is more than that of FGSCA.
All in all, compared with these well-known Mas, SCA variants, and improved MAs, the superiority
of FGSCA in solving the IBD problem cannot be ignored. Therefore, it indicates that FGSCA can
effectively resolve this problem and offer practical and splendid solutions for the IBD problem.

53
7.3 PVD problem
Another engineering problem worth studying is the PVD problem. We want to minimize the
cylindrical vessel's cost as much as we can due to the limited potential of methods. The four parameters
are: the thickness of the head (), the width of the shell (), the length of the tubular section without
allowing for the head () and the internal radius (). The mathematical model of this case is presented
below:
Consider 󰇟󰇠󰇟󰇠
Objective 󰇛󰇜



Subject to 󰇛󰇜,
󰇛󰇜,
󰇛󰇜


,
󰇛󰇜,
Variable ranges: ,
,
,
.
Table 15. Results of FGSCA and tested algorithms for PVD problem
Algorithm
Optimum values for parameters
Optimum cost




FGSCA
0.8750
0.04375
45.2530
141.2125
6102.9011
SCA
1.1250
0.5625
58.0485
46.6883
6912.5975
WOA
1.0625
0.5625
54.4092
68.8737
6903.8253
GWO
1.1250
0.5625
57.4394
48.7111
6896.4955
MFO
0.9375
0.5625
45.6943
137.3791
6930.3243
BA
0.8750
0.4375
41.5128
197.4422
6913.5276
FA
1.1250
0.5625
58.2109
46.7792
6944.9364
PSO
1.0625
0.5000
51.9351
88.7486
6926.4600
ACO
1.0000
0.5625
50.3255
97.6262
6898.5762
MSCA
0.8750
0.4375
43.9743
160.5798
6407.1483
ASCA-PSO
0.9375
0.5000
47.5414
119.9715
6500.3248
m-SCA
0.8750
0.4375
44.5885
153.6161
6326.5175
SCADE
0.9375
0.5000
48.2465
114.6866
6458.5468
CGSCA
0.9375
0.5000
48.4121
115.1900
6502.3570
OBSCA
0.8750
0.5000
43.5672
159.4350
6518.6421
BMWOA
0.9375
0.5000
48.5649
112.9899
6460.0042
ACWOA
0.9375
0.5000
48.3941
114.3307
6472.6221
OBLGWO
0.8125
0.5625
42.0725
180.1706
6531.3525
IWOA
0.8750
0.4375
44.0799
156.3524
6313.4805

54
CBA
1.0000
0.5000
51.6855
85.7172
6429.2770
CDLOBA
0.9375
0.5000
48.5373
112.5566
6441.8349
IFOA
0.8750
0.4375
42.6370
170.1072
6424.0974
AMFOA
0.8750
0.4375
43.9963
162.5928
6464.0187
EPSO
0.8125
0.4375
40.4181
199.0234
6284.1031
IJAYA
0.8125
0.4375
40.9337
199.2033
6379.4925
Table 15 shows the final results of FGSCA and other involved methods to resolve the PVD
problem. It is easy to see that the optimization results of FGSCA are minimal. It means that FGSCA
has the highest potential in solving PVD problems. Specifically, when, , , and  are equal to
0.8750, 0.04375, 45.2530, and 141.2125; the optimum cost of FGSCA is 6102.9011. Compared with
SCA, the optimum cost of FGSCA is much less than SCA. It can also be seen from Table 15 that
FGSCA shows more substantial advantages than the well-known MAs in solving this problem. Besides,
FGSCA has less optimum cost than improved SCA variants such as MSCA, ASCA-PSO, m-SCA,
SCADE, CGSCA, and OBSCA. FGSCA performs better than improved MAs such as BMWOA,
ACWOA, OBLGWO, IWOA, CBA, CDLOBA, IFOA, AMFOA, EPSO, and IJAYA.
Compared with the well-known MAs, SCA variants, and improved MAs, FGSCA performs better
in solving PVD problems. According to the three cases, the final results reveal that FGSCA can resolve
applied problems with better exploratory and exploitative trends and obtain solutions than other rivals.
Meanwhile, we can conclude that the proposed FGSCA can have an excellent practical future for real-
world cases and engineering challenges that have constraints.
8 Conclusion and future directions
This paper introduces two effective strategies in the original SCA, including spiral motion and
Gaussian mutation, to enhance SCA performance. The introduction of spiral motion enhances the
communication between populations in the neighborhood and improves SCA's exploitation ability.
Gaussian mutation effectively improves the diversity of the population and enhances the local
exploration ability of SCA. To test the performance of FGSCA, it is used for making comparisons with
eight well-known MAs; six reported SCA variants and ten improved MAs on 23 well-known benchmark
test problems and 30 standard IEEE CEC2014 benchmark test problems. Results show the overall
performance of FGSCA is superior to twenty-four comparative MAs on 53 benchmark test problems.
Besides, FGSCA is employed to resolve three practical engineering problems. Results exhibit that
FGSCA obtains the best results among twenty-four comparative MAs. The extra experiment on the
wall-clock time cost is conducted, as well. Though FGSCA takes more time than other well-known
MAs, it performs more satisfactorily on the solution's accuracy. Based on the analysis of diversity and
balance, it is further concluded that FGSCA has more promising potential than the basic SCA. Besides,
FGSCA shows rapid convergence and high accuracy in practical engineering problems. However,
FGSCA consumes more time than the original SCA in solving those practical problems, and its
performance is affected by various parameters. FGSCA is expected to be a powerful and efficient
technology to deal with multiple complex and challenging optimization problems.
There are still more works that still need to be explored in our future work. First, FGSCA can be
used in discrete and multi-objective versions to complete discrete optimization tasks. It is foreseeable

55
that FGSCA will be applied to more practical problems with its convenient advantages. Also, MAs'
multiple core strategies can be combined with the original SCA, which is worth further research in the
future.
Acknowledgments
This research is supported by the Zhejiang Provincial Natural Science Foundation of China
(LJ19F020001), and National Natural Science Foundation of China (62076185, 7180313, U1809209),
and Guangdong Natural Science Foundation (2021A1515011994).
Appendix
Table A.1
Description of 23 benchmark functions.
Function
Search range
Optimum value
󰇛󰇜


[-100,100]
󰇝󰇞
󰇛󰇜 

  


[-10,10]
󰇝󰇞
󰇛󰇜

 
 
[-100,100]
󰇝󰇞
󰇛󰇜󰇝󰇞
[-100,100]
󰇝󰇞
󰇛󰇜󰇟󰇛󰇜󰇛󰇜󰇠


[-30,30]
󰇝󰇞
󰇛󰇜󰇛󰇟󰇠󰇜


[-100,100]
󰇝󰇞
󰇛󰇜󰇟󰇠


[-128,128]
󰇝󰇞
󰇛󰇜 󰇛󰇜


[-500,500]
󰇝󰇞

󰇛󰇜󰇟󰇛󰇜󰇠


[-5.12, 5.12]
󰇝󰇞
󰇛󰇜󰇧


 󰇨
󰇡
󰇛󰇜

 󰇢
[-32,32]
󰇝󰇞
󰇛󰇜


 󰇡
󰇢

 
[-600,600]
󰇝󰇞
󰇛󰇜
󰇥󰇛󰇜󰇛󰇜󰇟


󰇛󰇜󰇠󰇛󰇜󰇦󰇛󰇜


[-50,50]
󰇝󰇞

56


󰇛󰇜󰇱 󰇛󰇜 
 
󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜󰇟


󰇛󰇜󰇠󰇛󰇜󰇟󰇛󰇜󰇠
󰇛󰇜


[-50,50]
󰇝󰇞
󰇛󰇜󰇧
 
󰇛󰇜



 󰇨
[-65,65]
󰇝󰇞
󰇛󰇜

 󰇛󰇜

[-5,5]
󰇝󰇞
󰇛󰇜




[-5,5]
󰇝󰇞
󰇛󰇜󰇡

󰇢󰇡

󰇢
[-5,5]
󰇝󰇞
󰇛󰇜󰇟󰇛󰇜󰇛


󰇜󰇠󰇟󰇛󰇜
󰇛

󰇜󰇠
[-2,2]
󰇝󰇞
󰇛󰇜

 󰇧 

 󰇛󰇠󰇜󰇨
[1,3]
󰇝󰇞
󰇛󰇜

 󰇧 

 󰇛󰇠󰇜󰇨
[0,1]
󰇝󰇞
󰇛󰇜󰇟󰇛󰇜󰇛󰇜󰇠


[0,10]
󰇝󰇞
󰇛󰇜󰇟󰇛󰇜󰇛󰇜󰇠


[0,10]
󰇝󰇞
󰇛󰇜󰇟󰇛󰇜󰇛󰇜󰇠


[0,10]
󰇝󰇞
Table B.1
Summary of CEC2014 benchmark problem.
Type
No.
Function
Optimum value
Unimodal
Functions
24
Rotated High Conditioned Elliptic Function
󰇝󰇞
25
Rotated Bent Cigar Function
󰇝󰇞
26
Rotated Discus Function
󰇝󰇞
Multimodal
Functions
27
Shifted and Rotated Rosenbrock’s Function
󰇝󰇞
28
Shifted and Rotated Ackley’s Function
󰇝󰇞
29
Shifted and Rotated Weierstrass Function
Function
󰇝󰇞
30
Shifted and Rotated Griewank’s Function
󰇝󰇞

57
31
Shifted Rastrigin’s Function
󰇝󰇞
32
Shifted and Rotated Rastrigin’s Function
󰇝󰇞
33
Shifted Schwefel’s Function
󰇝󰇞
Hybrid
Functions
34
Shifted and Rotated Schwefel’s Function
󰇝󰇞
35
Shifted and Rotated Katsuura Function
󰇝󰇞
36
Shifted and Rotated Happycat Function
󰇝󰇞
37
Shifted and Rotated Hgbat Function
󰇝󰇞
38
Shifted and Rotated Expanded Griewank’s
Plus Rosenbrock’s Function
󰇝󰇞
39
Shifted and Rotated Expanded Scaffer’s
Function
󰇝󰇞
40
Hybrid Function 1 (N=3)
󰇝󰇞
41
Hybrid Function 2 (N=3)
󰇝󰇞
42
Hybrid Function 3 (N=4)
󰇝󰇞
43
Hybrid Function 4 (N=4)
󰇝󰇞
Composition
Functions
44
Hybrid Function 5 (N=5)
󰇝󰇞
45
Hybrid Function 6 (N=5)
󰇝󰇞
46
Composition Function 1 (N=5)
󰇝󰇞
47
Composition Function 2 (N=3)
󰇝󰇞
48
Composition Function 3 (N=3)
󰇝󰇞
49
Composition Function 4 (N=5)
󰇝󰇞
50
Composition Function 5 (N=5)
󰇝󰇞
51
Composition Function 6 (N=5)
󰇝󰇞
52
Composition Function 7 (N=3)
󰇝󰇞
53
Composition Function 8 (N=3)
󰇝󰇞
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