Content uploaded by Ali Asghar Heidari
Author content
All content in this area was uploaded by Ali Asghar Heidari on Apr 09, 2020
Content may be subject to copyright.
Journal Pre-proof
Slime mould algorithm: A new method for stochastic optimization
Shimin Li,Huiling Chen,Mingjing Wang,Ali Asghar Heidari,
Seyedali Mirjalili
PII: S0167-739X(19)32094-1
DOI: https://doi.org/10.1016/j.future.2020.03.055
Reference: FUTURE 5560
To appear in: Future Generation Computer Systems
Received date : 6 August 2019
Revised date : 16 February 2020
Accepted date : 29 March 2020
Please cite this article as: S. Li, H. Chen, M. Wang et al., Slime mould algorithm: A new method
for stochastic optimization, Future Generation Computer Systems (2020), doi:
https://doi.org/10.1016/j.future.2020.03.055.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the
addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive
version of record. This version will undergo additional copyediting, typesetting and review before it
is published in its final form, but we are providing this version to give early visibility of the article.
Please note that, during the production process, errors may be discovered which could affect the
content, and all legal disclaimers that apply to the journal pertain.
©2020 Published by Elsevier B.V.
Please note this is the uncorrected proof of the SMA draft, for final file, please
refer to published SMA paper at https://doi.org/10.1016/j.future.2020.03.055
Slime Mould Algorithm: A New
Method for Stochastic Optimization
Shimin Li1, Huiling Chen1*, Mingjing Wang1, Ali Asghar Heidari2,3, Seyedali Mirjalili4
1College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou, Zhejiang 325035, China
simonlishimin@foxmail.com, chenhuiling.jlu@gmail.com, wangmingjing.style@gmail.com
2School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran 1439957131,
Iran
as_heidari@ut.ac.ir, aliasghar68@gmail.com
3Department of Computer Science, School of Computing, National University of Singapore, Singapore 117417,
Singapore
aliasgha@comp.nus.edu.sg, t0917038@u.nus.edu
4Institute for Integrated and Intelligent Systems, Griffith University, Nathan, QLD 4111, Australia
ali.mirjalili@gmail.com
Corresponding Author: Huiling Chen
E-mail: chenhuiling.jlu@gmail.com (Huiling Chen)
Abstract
In this paper, a new stochastic optimizer, which is called slime mould algorithm (SMA), is proposed
based upon the oscillation mode of slime mould in nature. The proposed SMA has several new
features with a unique mathematical model that uses adaptive weights to simulate the process of
producing positive and negative feedback of the propagation wave of slime mould based on
bio-oscillator to form the optimal path for connecting food with excellent exploratory ability and
exploitation propensity. The proposed SMA is compared with up-to-date metaheuristics in an
extensive set of benchmarks to verify the efficiency. Moreover, four classical engineering structure
problems are utilized to estimate the efficacy of the algorithm in optimizing engineering problems.
The results demonstrate that the proposed SMA algorithm benefits from competitive, often
outstanding performance on different search landscapes. Source codes of SMA are publicly available
at http://www.alimirjalili.com/SMA.html
Keywords
Slime mould optimization algorithm; Adaptive weight; Engineering design problems; Constrained
optimization
1 Introduction
Metaheuristic algorithms (MAs) have become prevalent in many applied disciplines in recent
decades because of higher performance and lower required computing capacity and time than
deterministic algorithms in various optimization problems [1]. Simple concepts are required to
achieve favorable results, and it is facile to transplant to different disciplines. Also, the lack of
randomness in the later stage of some deterministic algorithm makes it inclined to sink into local
optimum, and random factors in MAs can make the algorithm search for all optimal solutions in
search space, thus effectively avoiding local optimum. In linear problems, some gradient descent
algorithms such as [2] are more efficient than stochastic algorithms for the utilization of gradient
information. The convergence speed of MAs will be less than gradient descent algorithms and can be
considered as a drawback. In non-linear problems, however, MAs typically commence the
optimization process with randomly generated solutions and do not demand gradient information,
which makes the algorithm eminently suitable for practical problems when the derivative
information is unknown. In real-world scenarios, the solution space of many problems is often
indeterminate or infinite. It may be infeasible to find optimal solutions by traversing the solution
space under current circumstances. MAs detect the proximate optimal solution of the problem by
sampling the enormous solution space randomly in a certain way, to find or generate better solutions
for the optimization problem under limited circumstances or computational capacity.
MAs are typically inspired by real-world phenomena find better heuristic solutions for
optimization problems by simulating physical rules or biological phenomena. MAs can be divided
into two main categories: swam-based methods and evolutionary techniques. The first kind mainly
simulate physical phenomena, apply mathematical rules or methodologies including: Multi-Verse
Optimizer (MVO) [3], Gravitational Local Search Algorithm (GLSA) [4], Charged System Search
(CSS) [5], Gravitational Search Algorithm (GSA) [6], Sine Cosine Algorithm (SCA) [7], Simulated
Annealing (SA) [8], Teaching-Learning-Based Optimization (TLBO) [9], Central Force
Optimization (CFO) [10] and Tabu Search (TS) [11]. Nature-inspired methods mainly include two
types: evolutionary methods and intelligent swarm techniques. The inspiration of the evolutionary
algorithm (EA) originates from the process of biological evolution in nature. Compared with the
traditional optimization algorithm, it is a global optimization method with better robustness and
applicability.
Some of the widespread algorithms in the class of EA are Genetic Algorithm (GA) [12], Genetic
Programming (GP) [13], Evolution Strategy (ES) [14], Evolutionary Programming (EP) [15] and
Differential Evolution (DE) [16]. The application of ES and EP in scientific research and practical
problems is also becoming more and more extensive. Swarm Intelligence (SI) [17] includes a
collective or social intelligence that artificially simulates the decentralization of biological clusters in
nature or the collective behavior of self-organizing systems. In this class of algorithms, the
inspiration usually comes from biological groups in nature that have collective behavior and
intelligence to achieve a certain purpose. In general, SI algorithms are more advantageous than
evolutionary algorithms because SI algorithms are accessible to appliance than evolutionary
algorithms with less operators that need to be controlled. Moreover, the SI algorithm has a stronger
capability to record and utilize historical information than EA. Established and recent algorithms in
this class are: Particle Swarm Optimization (PSO) [18], Wasp Swarm Optimization (WSO) [19],
Bat-inspired Algorithm (BA) [20] , Grey Wolf Optimization (GWO) [21], Fruit Fly Optimization
(FOA) [22] , Moth Flame Optimization (MFO) [23], Ant Colony Optimization (ACO) [24], Harris
Hawk Optimizer (HHO) [25], and Artificial Bee Colony (ABC) [26]. A schematic design for the
classification of evolutionary and SI methods are shown in Figure 1.
Figure 1 classification of evolutionary and SI methods
Although different MAs have some distinctness, they all have two identical stages in the search
gradation: exploration and exploitation [27, 28]. Exploration phase refers to the process of searching
solution space as widely, randomly, and globally as possible. Exploitation phase refers to the
competence of the algorithm to search more accurately in the area acquired by the exploration phase,
and its randomness decreases while its precision increases. When the exploration ability of the
algorithm is dominant, it can search the solution space more randomly and produce more
differentiated solution sets to converge fleetly. When the exploitative ability of the algorithm is
dominant, it searches more locally to enhance the quality and precision of the solution sets. However,
when the exploration facility is improved, it will lead to reductions in the exploitation capability, and
vice versa. Another challenge is that the balance of these two abilities is not necessarily identical to
different problems. Therefore, it is relatively challenging to attain an appropriate balance between
the two phases that are efficient for all optimization problems.
Despite the success of conventional and recent MAs, none of them can guarantee finding the
global optimum for all optimization problems. This has been proven logically the No-Free-Lunch
(NFL) theory [29]. This theorem motivated numerous researchers to design a new algorithm and
solve new classes of problems more efficiently. With the aspiration of proposing a more versatile and
efficient algorithm, this paper introduces a new meta-heuristic algorithm: slime mould algorithm
(SMA). This method is aroused by the diffusion and foraging conduct of slime mould. An overall set
of 33 benchmarks and four famous manufacturing design problems has rigorously verified the
effectiveness and robustness of SMA.
The remainder of the paper is structured as below. Section 2 illustrated the concept and elicitation
source of slime mould algorithm, and the mathematical model was established. Section 3 firstly gave
a qualitative analysis of the algorithm and made a comprehensive comparison of 33 benchmark
functions, then tested it on four engineering design problems. Section 4 summarized the whole work
and put forward some inspirations for future work.
2 Slime mould algorithm
In this section, the basic concept and conduct of slime mould will be introduced. Then a
mathematical model inspired by its behavior pattern will be established.
2.1 Originality
Before this article, some scholars have proposed similar naming algorithms, but the way of
designing the algorithm and usage scenarios are quite different from the algorithms proposed in this
paper. Monismith and Mayfield [30] solves the single-objective optimization problem by simulating
the five life cycles of amoeda Dictyostelium discoideum: a state of vegetative, aggregatice, mound,
slug, or dispersal while using ε-ANN to construct an initial position-based mesh. Li et al. [31]
proposed a method to construct wireless sensor networks by using two forms of slime mould tubular
networks to correspond to two different regional routing protocols. Qian. et al. [32] combined the
Physarum network with the ant colony system to improve the algorithm's competence to avoid local
optimal values to handle the Traveling Salesman Problem better. Inspired by the diffusion of slime
mould, Schmickland Crailsheim [33] proposed a bio-inspired navigation principle designed for
swarm robotics. Becker [34] generated inexpensive and fault-tolerant graphs by simulating the
foraging process of the slime mould Physarum polycephalum. As can be seen from the above
discussion, most of the modeled slime mould algorithms were used in graph theory and generation
networks. The algorithm used to optimize the problem [30] simulates the five life cycles of amoeda
Dictyostelium discoideum, but the experiments and proofs in the article are slightly less.
The SMA proposed in this paper mainly simulates the behavior and morphological changes of
slime mould Physarum polycephalum in foraging and does not model its complete life cycle. At the
same time, the use of weights in SMA is to simulate the positive and negative feedback generated by
slime mould during foraging, thus forming three different morphotype, is a brand new idea. This
paper also conducted a full experiment on the characteristics of the algorithm. The results in the next
sections demonstrate the superiority of the SMA algorithm.
2.2 Concept and elicitation
The slime mould mentioned in this article generally refers to Physarum polycephalum. Because it
was first classified as a fungus, thus it was named "slime mould" whose life cycle was originally
specified by Howard [35] in a paper published in 1931. Slime mould is a eukaryote that inhabits cool
and humid places. The main nutritional stage is Plasmodium, the active and dynamic stage of slime
mould, and also the main research stage of this paper. In this stage, the organic matter in slime mould
seeks food, surrounds it, and secretes enzymes to digest it. During the migration process, the front
end extends into a fan-shaped, followed by an interconnected venous network that allows cytoplasm
to flow inside [36], as shown in Figure 2. Because of their unique pattern and characteristic, they
can use multiple food sources at the same time to form a venous network connecting them. If there is
enough food in the environment, slime mould can even grow to more than 900 square centimeters
[36].
Owing to the feature of slime mould can be easily cultured on agar and oatmeal [37], they were
widely used as model organisms. Kamiya and his colleagues [38] were the first team to study the
detailed process of the cytoplasmic flow of slime mould. Their work is of great help to our
subsequent understanding of the way slime mould move and connects food in the environment. We
now cognize that when a vein approaches a food source, the bio-oscillator produces a propagating
wave [39] that increases the cytoplasmic flow through the vein, and the faster the cytoplasm flows,
the thicker the vein. Through this combination of positive-negative feedback, the slime can establish
the optimal path to connect food in a relatively superior way. Therefore, slime mould was also
mathematically modeled and applied in graph theory and path networks [40-42].
Figure 2 Foraging morphology of slime mould
The venous structure of slime mould develops along with the phase difference of the contraction
mode [39], so there are three correlations between the morphological changes of the venous structure
and the contraction mode of slime mould.
1) Thick veins form roughly along the radius when the contraction frequencies vary from outside
to inside.
2)When the contraction mode is unstable, anisotropy begins to appear.
3)When the contraction pattern of slime mould is no longer ordered with time and space, the
venous structure is no longer present.
Therefore, the relationship between venous structure and contraction pattern of slime mould is
consistent with the shape of naturally formed cells. The thickness of each vein is determined by the
flow feedback of the cytoplasm in the Physarum solver [43]. The raise in the flow of cytoplasm leads
to an increase in the diameter of veins. As the flow decreases, the veins contract because of the
decrease of the diameter. Slime mould can build a stronger route where food concentration is higher,
thus ensuring that they get the maximum concentration of nutrients. Recent studies have also
revealed that slime mould have the competence of making foraging arrangements based on
optimization theory [44]. When the quality of various food sources is different, slime mould can
choose the food source with the highest concentration. However, slime mould also needs to weigh
speed and risk in foraging. For instance, slime mould needs to make faster decisions in order to
avoid environmental damage to them. Experiments have shown that the quicker the decision-making
speed is, the possibilities of slime mould to find the prime food source is smaller [45]. Therefore,
when deciding the source of food, slime mould obviously needs to weigh the speed and accuracy.
Slime mould need to decide when to leave this area and search in another area when foraging.
When lacking complete information, the best way for a slime mould to estimate when to leave the
current position is to adopt heuristic or empirical rules based on the insufficient information
currently available. Experience has shown that when slime mould encounter high-quality food, the
probability of leaving the area is reduced [46]. However, due to its unique biological characteristics,
slime mould can utilize a variety of food sources at the same time. Therefore, even if the slime
mould has found a better source of food, it can still divide a component of the biomass to exploit
both resources simultaneously when higher quality food is found [43].
Slime mould can also dynamically adjust their search patterns according to the quality of foodstuff
provenience. When the quality of food sources is high, the slime mould will use the region-limited
search method [47], thus focusing the search on the food sources that have been found. If the
denseness of the food provenience initially found is low, the slime mould will leave the food source
to explore other alternative food sources in the region [48]. This adaptive search strategy can be
more reflected when different quality food blocks are dispersed in a region. Some of the mechanisms
and characteristics of the slime mould mentioned above will be mathematically modeled in the
subsequent sections.
2.3 Mathematical model
In this part, the mathematical model and method proposed will be described in details.
2.3.1 Approach food
Slime mould can approach food according to the odor in the air. To express its approaching behavior
in mathematical formulas, the following formulas are proposed to imitate the contraction mode:
(2.1)
where
is a parameter with a range of ,
decreases linearly from one to zero.
represents the current iteration,
represents the individual location with the highest odor
concentration currently found,represents the location of slime mould,
and
represent two
individuals randomly selected from slime mould,
represents the weight of slime mould.
The formula of is as follows:
(2.2)
where , represents the fitness of , represents the best fitness obtained in all
iterations.
The formula of
is as follows:
(2.3)
(2.4)
The formula of
is listed as follows:
(2.5)
(2.6)
where indicates that ranks first half of the population, denotes the random value
in the interval of , denotes the optimal fitness obtained in the current iterative process,
denotes the worst fitness value obtained in the iterative process currently, denotes
the sequence of fitness values sorted(ascends in the minimum value problem).
Figure 3 visualizes the effects of Eq. (2.1). The location of searching individual can be
updated according to the best location
currently obtained, and the fine-tuning of parameters
,
and
can change the location of the individual. Figure 3 is also used to illustrate the position
change of the searching individual in three-dimensional space. in the formula can make
individuals form search vectors at any angle, that is, search solution space in any direction, so that
the algorithm has the possibility of finding the optimum solution. Therefore, Eq. (2.1) enables the
searching individual to search in all possible directions near the optimal solution, thus simulating the
circular sector structure of slime mould when approaching food. It is also applicable to extend this
concept to Hyper-dimensional space.
Figure 3 Possible locations in 2-dimention and 3-dimention
Figure 4 Assessment of fitness
2.3.2 Wrap food
This part simulates the contraction mode of venous tissue structure of slime mould mathematically
when searching. The higher the concentration of food contacted by the vein, the stronger the wave
generated by the bio-oscillator, the faster the cytoplasm flows, and the thicker the vein. Eq. (2.5)
mathematically simulated the positive and negative feedback between the vein width of the slime
mould and the food concentration that was explored. The component in Eq. (2.5) simulates the
uncertainty of venous contraction mode. is used to alleviate the change rate of numerical value
so that the value of contraction frequency does not change too much. simulates the slime
mould to adjust their search patterns according to the quality of food. When the food concentration is
content, the bigger the weight near the region is; when the food concentration is low, the weight of
the region will be reduced, thus turning to explore other regions. Figure 4 shows the process of
evaluating fitness values for slime mould.
Based on the above principle, the mathematical formula for updating the location of slime mould
is as follows:
(2.7)
where and denote the lower and upper boundaries of search range, and denote the
random value in [0,1]. The value of will be discussed in the parameter setting experiment.
2.3.3 Grabble food
Slime mould mainly depends on the propagation wave produced by the biological oscillator to
change the cytoplasmic flow in veins, so that they tend to be in a better position of food
concentration. On the purpose of simulating the variations of venous width of slime mould, we used
,
and
to realize the variations.
mathematically simulates the oscillation frequency of slime mould near one at different food
concentration, so that slime mould can approach food more quickly when they find high-quality food,
while approach food more slowly when the food concentration is lower in individual position, thus
improving the efficiency of slime mould in choosing the optimal food source.
The value of
oscillates randomly between and gradually approaches zero as the
increasement of iterations. The value of
oscillates between [-1,1] and tends to zero eventually.
The trend of the two values is shown as Figure 5. Synergistic interaction between
and
mimics the selective behavior of slime mould. In order to find a better source of food, even if slime
mould has found a better source of food, it will still separate some organic matter for exploring other
areas in an attempt to find a higher quality source of food, rather than investing all of it in one
source.
Figure 5 Trends of
and
Moreover, the oscillation process of
simulates the state of slime mould deciding whether to
approach the food source or find other food sources. Meanwhile, the process of probing food is not
smooth. During this period, there may be various obstacles, such as light and dry environment,
which restrict the spread of slime mould. However, it also improves the possibility of slime mould to
find higher quality food and evades the trapping of local optimum.
The pseudo code of the SMA is shown in Algorithm 1. The intuitive and detailed process of SMA
is shown in Figure 6.
There are still many mechanisms that can be added to the algorithm, or more comprehensive
simulation of the life cycle of slime mould. However, to enhance the extensibility of the algorithm,
we simplify the process and operators of the algorithm, leaving only the simplest algorithm as
possible.
Algorithm 1 Pseudo-code of SMA
Initialize the parameters popsize, ;
Initialize the positions of slime mould ;
While (
Calculate the fitness of all slime mould;
Calculate the W by Eq. (2.5);
For
;
;
End
;
End While
Return ;
2.3.4 Computational complexity analysis
SAM mainly consists of the subsequent components: initialization, fitness evaluation, and sorting,
weight update, and location update. Among them, N denotes the number of cells of slime mould, D
denotes the dimension of functions, and T denotes the maximum number of iterations. The
computation complexity of initialization is , the computation complexity of fitness evaluation
and sorting is , the computational complexity of weight update is , the
complexity of location update is . Therefore, the total complexity of SMA is
.
Figure 6 Flowchart of SMA
3 Experimental results and analyses
In this sector, we compared the SMA with some competitive MAs in an all-inclusive set of 33
benchmarks. The experimentations were ran on the operating system of Windows Server 2012 R2
Datacenter with 128 GB RAM and CPU of Intel (R) Xeon (R) E5-2650 v4 (2.20 GHz). The
algorithms for comparison were coded by MATLAB R2018b.
3.1 Qualitative analysis
The qualitative analysis results of SMA in handling unimodal functions and multimodal functions
are presented in Figure 7 to intuitively analyze the position and fitness changes of slime mould
during foraging. The figure is comprised of four concernment indicators: search history, the
trajectory of the slime mould in the 1st dimension, the average fitness of slime mould, and
convergence curve. Search history represents the location and distribution of slime mould in the
iteration process. The trajectory of slime mould reveals the behavior of the position change of slime
mould in the first part of the first dimension. Average fitness indicates the variation trend of the
average fitness of the slime mould colony changes with the iteration process. Convergence curve
shows the optimal fitness value in the slime mould during the iteration process.
From the search history subplot, the slime mould in different benchmark functions put up a similar
cross-type search trajectory clustered near the optimal value, thus accurately searching in reliable
search areas and reflecting fast convergence. Meanwhile, the distribution of slime mould is mainly
concentrated in multiple regions with local optimum, which shows the tradeoff of slime mould
between multiple local optimums.
The trajectory of the first slime mould in the first dimension can be used as a representative of
other parts of slime mould, revealing the primary exploratory behavior of slime mould. The fast
oscillation in the prophase and the slight oscillation in the anaphase can ensure the fast convergence
of slime mould and the accurate search near the optimal solution [49]. As can be perceived from the
figure, the position curve of slime mould has very large amplitude in the early iteration process, even
up to 50% of the exploration space. In the later iteration period, if the function is smooth, the
amplitude of the position of slime mould begins to decrease; if the amplitude of the function changes
significantly, the position amplitude also changes greatly. This reflects the high adaptability and
robustness of slime mould in different functions.
By observing the average fitness curve, the variation tendency of the fitness of slime mould during
the iterative procedure can be visually observed. Although the average fitness curve of slime mould
is oscillating, the average fitness value tends to decrease, and the oscillation frequency decreases
inversely proportional to iterations, thus ensuring the rapid convergence of slime mould in the
prophase and the precise search in the anaphase.
Convergence curve reveals the average fitness of the optimal fitness value searched by slime
mould varies with iterations. By observing the downtrend of the curve, we can observe the
convergence rate of slime mould and the time when it switches between the exploration and
exploration gradation.
Figure 7 Qualitative analysis
3.2 Benchmark function validation
In this section, SMA was assessed on a comprehensive set of functions from 23 benchmarks and
CEC 2014. These functions cover unimodal, multimodal, hybrid, and composite functions, as shown
in Tables 1-3. Some composite functions of CEC 2014 are shown in Figure 8. Dim denotes the
dimension of function; Range denotes the definition domain of the function, and denotes the
optimal value of the function. The MAs used for comparison include well-regarded and recent ones:
WOA [50], GWO [21], MFO [23], BA [20], SCA [7], FA[51], PSO[18], SSA [52], MVO [3], ALO
[53], PBIL [54], DE [55] and advanced MAs: AGA[56], BLPSO [57], CLPSO [58], CBA [59],
RCBA [60], CDLOBA [61], m_SCA [62], IWOA [63], LWOA [64], and CSSA [65]. The parameter
setup of traditional MAs is detailed in Table 4. The parameter selection was based on the parameters
used by the original author in the article or the parameters widely used by various researchers.
Figure 8 Illustration of CEC 2014 composite functions
Table 1
Unimodal and multimodal test functions of 23 standard benchmarks
Functions
Dim
Range
n
[-100,100]
0
n
[-10,10]
0
n
[-100,100]
0
n
[-100,100]
0
n
[-30,30]
0
n
[-100,100]
0
n
[-128,128]
0
n
[-500,500]
-418.9829*n
n
[-5.12,5.12]
0
n
[-32,32]
0
n
[-600,600]
0
n
[-50,50]
0
n
[-50,50]
0
Table 2
Unimodal and simple multimodal functions of CEC2014
Functions
Dim
Range
n
[-100,100]
100
n
[-100,100]
200
n
[-100,100]
500
n
[-100,100]
600
n
[-100,100]
1300
n
[-100,100]
1400
n
[-100,100]
1500
n
[-100,100]
1600
Table 3
Hybrid and Composition functions of CEC 2014
Functions
Dim
Range
n
[-100,100]
1700
n
[-100,100]
1800
n
[-100,100]
1900
n
[-100,100]
2000
n
[-100,100]
2100
n
[-100,100]
2200
n
[-100,100]
2300
n
[-100,100]
2400
n
[-100,100]
2500
n
[-100,100]
2600
n
[-100,100]
2700
n
[-100,100]
2800
Table 4
Parameter settings of counterparts
Algorithm
Parameter settings
WOA
GWO
MFO
BA
SCA
FA
PSO
SSA
MVO
ALO
PBIL
DE
All algorithms were performed under the same conditions to achieve fairness in comparative
experiments. Among them, the population was set to 30, the dimension and the iteration time was set
to 30 and 1000 respectively. To reduce the impacts of random factors in the algorithm on the results,
all the compared algorithms were run individually 30 times in each function and averaged as the
final running result. On the purpose of measuring experiment results, Standard deviation (STD),
Average results (AVG) and Median (MED) were employed to evaluate the results. Note that best
results will be bolded (take one in the case of juxtaposition).
3.2.1 Exploitation competence analysis
The data in Table 5 demonstrates that SMA ranked first or tied first on average when solving F1-5,
F7, and F14. The convergence curves of F2 and F5 in Figure 9 can be visually observed that SMA
has the fastest convergence trend among all the comparative functions. The data in Table 6
demonstrates that SMA can still exhibit significant advantages even when compared to a modified
Ma, such as ranking first among other unimodal functions other than F5 and F14. These functions
are unimodal functions in the benchmarks, reflecting SMA's efficient exploration capability.
Moreover, in order to more fairly evaluate the local search efficiency of the algorithm, an evaluation
version of the experiment has been added. The data in Table 7 demonstrate the experimental results
obtained by 300,000 evaluations of the SMA with 10 other participants on the unimodal functions. In
the experimental results, the values obtained by SMA were still better than those of other algorithms
on F1-5 and F7. At the same time, the median values of the solutions were also consistent with the
ranking of the optimal values, indicating the stability of the SMA.
Table 5
Comparison results on unimodal functions with traditional algorithms during 1000 iterations
F1
F2
F3
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
0.000000
0.000000
1.08E-64
5.330E-207
0.000000
5.93E-58
0.00000
0.00000
8.22E-02
SCA
0.015244
0.029989
9.36E+01
1.150E-05
2.743E-05
8.06E-03
3261.99676
2935.03792
2.75E+04
SSA
1.231E-08
3.536E-09
1.83E+02
0.848146
0.941518
8.90E+00
236.62194
155.54710
2.94E+03
GWO
4.223E-59
1.081E-58
4.39E-46
1.128E-34
9.149E-35
7.07E-28
4.027E-15
1.418E-14
1.50E-09
MFO
2000.0006
4068.3807
2.04E+03
33.666839
20.253973
3.42E+01
24900.5554
14138.0477
2.91E+04
WOA
4.322E-153
2.276E-152
2.34E-54
5.032E-104
1.591E-103
3.42E-34
20802.2782
10554.3925
5.30E+04
GOA
7.670196
6.676643
1.27E+03
9.540510
14.128406
3.09E+01
1794.1195
1103.3922
7.64E+03
DA
1158.4940
600.8920
1.19E+03
14.313148
5.649106
1.45E+01
9612.3629
6188.5858
9.64E+03
ALO
1.050E-05
7.825E-06
7.10E+00
28.698940
42.100743
3.02E+01
1275.7431
596.2918
1.73E+03
MVO
0.318998
0.112060
9.40E+02
0.388930
0.137834
1.39E+01
48.11246
21.77526
4.61E+03
PBIL
46908.0000
4218.6045
4.84E+04
95.200000
5.892134
9.80E+01
54824.1
6552.855378
6.02E+04
PSO
128.803704
15.368375
1.42E+02
86.075426
65.298810
1.12E+02
406.96260
71.30926
6.06E+02
DE
3.030E-12
3.454E-12
4.01E-04
3.723E-08
1.196E-08
2.24E-03
24230.5748
4174.3788
3.00E+04
F4
F5
F6
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
2.301E-197
0.000000
1.31E-25
0.42779
0.63700
9.89E+00
0.000879
0.000415
5.97E-01
SCA
20.532489
11.046644
7.53E+01
532.7126
1907.4456
1.58E+06
4.550121
0.357049
3.37E+01
SSA
8.254602
3.287966
1.62E+01
135.5698
174.1213
7.77E+03
0.000000
0.000000
2.04E+02
GWO
1.776E-14
2.228E-14
9.01E-12
27.10029
0.86432
2.73E+01
0.726058
0.278337
9.75E-01
MFO
64.420279
8.689356
6.47E+01
5348258
20289785
5.35E+06
1656.708
5277.651
1.68E+03
WOA
45.706343
26.935040
4.61E+01
27.26543
0.57447
2.73E+01
0.100557
0.110525
1.01E-01
GOA
12.596514
4.317304
2.35E+01
1631.1583
2241.1368
2.58E+05
4.884661
4.512327
1.36E+03
DA
23.631736
8.191777
2.37E+01
127371
96386
1.31E+05
1330.292
632.470
1.34E+03
ALO
12.133214
3.585375
1.32E+01
298.8031
431.1446
5.00E+02
0.000012
0.000011
7.49E+00
MVO
1.076968
0.310884
1.40E+01
407.9465
615.3290
8.63E+04
0.323756
0.097394
9.34E+02
PBIL
79.666667
4.088110
8.00E+01
143346156
31547349
1.51E+08
45881.833
4850.932
4.77E+04
PSO
4.498158
0.329339
4.79E+00
154736
36039
1.85E+05
132.779
15.189
1.45E+02
DE
1.965929
0.430531
1.32E+01
46.12942
27.29727
1.40E+02
3.096E-12
1.461E-12
4.11E-04
F7
F14
F15
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
8.839E-05
7.118E-05
4.08E-04
9549563
6529870
2.97E+07
22233.8245
14144.9575
5.47E+07
SCA
0.024382
0.020732
6.04E-01
425718766
116756947
7.06E+08
2.689E+10
5.427E+09
3.97E+10
SSA
0.095541
0.050530
1.59E-01
20297116
8153518
6.91E+07
11222.8121
11173.7583
3.37E+08
GWO
0.000869
0.000435
1.46E-03
88751868
66700399
1.29E+08
2.254E+09
1.759E+09
3.98E+09
MFO
4.620163
13.076256
4.77E+00
87010749
137363574
1.00E+08
1.341E+10
7.685E+09
1.35E+10
WOA
0.000986
0.001147
2.66E-03
160431438
69271930
1.62E+08
2.154E+09
1.086E+09
2.17E+09
GOA
0.024028
0.011253
2.96E-02
33807500
14819986
1.28E+08
17667580
11032455
2.34E+09
DA
0.326978
0.138556
3.31E-01
305164519
121919102
3.05E+08
6.363E+09
2.751E+09
6.37E+09
ALO
0.103373
0.034257
1.06E-01
12505761
5184932
1.69E+07
12378
9058
1.25E+07
MVO
0.020859
0.009584
1.42E-01
14860094
6244884
5.89E+07
566570
210025
1.45E+09
PBIL
282.1349
43.2693
2.93E+02
574020990
128317251
7.02E+08
4.961E+10
5.107E+09
5.32E+10
PSO
111.0068
21.5378
1.11E+02
17174833
5483990
2.16E+07
191733286
23903821
2.09E+08
DE
0.026937
0.006322
5.44E-02
100597441
31636302
1.78E+08
1601.8022
3314.1727
1.97E+05
Table 6
Comparison results on the unimodal functions with advanced algorithms
F1
F2
F3
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
0.000000
0.000000
4.72E-37
4.20E-187
0.000000
1.24E-66
0.000000
0.000000
1.19E-02
BLPSO
2208.3313
397.7883
5.00E+03
17.665054
1.905407
3.35E+01
13540.48
1672.45
1.82E+04
CLPSO
596.7364
150.3595
5.15E+03
11.846531
1.669288
4.09E+01
16836.42
3085.75
2.71E+04
CBA
0.113583
0.454545
4.38E-01
305804
1652847
5.73E+05
73.709725
31.029467
2.54E+02
RCBA
0.201488
0.052889
5.31E-01
10.958358
28.471304
2.77E+01
95.544912
43.376020
7.44E+02
CDLOBA
0.005957
0.002133
1.88E-02
3781.932
15086.168
1.24E+04
1.791342
6.166318
3.50E+02
m_SCA
2.521E-46
1.378E-45
8.14E-04
3.478E-33
1.420E-32
2.01E-06
8.991E-16
3.188E-15
5.82E+03
IWOA
8.130E-146
4.370E-145
1.00E-53
2.385E-102
6.585E-102
1.44E-33
15410.3
7420.1
3.62E+04
LWOA
6.743E-07
7.589E-07
1.55E-01
2.801E-07
3.833E-07
6.54E-02
43293.10
13505.91
9.25E+04
CSSA
0.017344
0.027805
1.74E-02
0.061732
0.027609
6.21E-02
2.926441
3.133898
2.95E+00
F4
F5
F6
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
8.84E-183
0.00000
1.80E-36
1.27571
4.90297
1.22E+01
0.000880
0.000407
9.26E-01
BLPSO
27.66310
2.40967
3.54E+01
520889
178483
2.75E+06
2207.564
410.182
5.20E+03
CLPSO
42.44490
4.41014
5.61E+01
113820
39571
2.95E+06
563.251
138.054
5.26E+03
CBA
17.03820
7.72324
2.20E+01
197.6163
360.2440
2.58E+02
0.001823
0.007886
1.16E-01
RCBA
9.00594
3.41186
1.49E+01
148.2466
122.4613
2.29E+02
0.187352
0.054118
4.62E-01
CDLOBA
46.10460
7.48538
4.81E+01
138.1210
178.6248
2.29E+02
0.005940
0.001899
1.79E-02
m_SCA
2.248E-13
1.223E-12
1.53E+01
27.62609
0.84321
3.34E+01
2.540097
0.499546
4.06E+00
IWOA
13.12456
16.19609
2.26E+01
26.57003
0.66075
2.70E+01
0.036361
0.069578
6.17E-02
LWOA
11.12439
14.63066
2.69E+01
25.63874
6.59153
2.90E+01
0.009637
0.002992
4.25E-01
CSSA
0.03301
0.01983
3.45E-02
0.17508
0.16603
1.76E-01
0.030982
0.062573
3.11E-02
F7
F14
F15
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
8.21E-05
7.16E-05
3.24E-04
9689581
7904687
3.20E+07
15808.97
10533.48
5.40E+07
BLPSO
0.59346
0.17290
1.50E+00
1.72E+08
3.74E+07
2.98E+08
3.718E+09
5.932E+08
8.78E+09
CLPSO
0.26201
0.05157
1.74E+00
1.77E+08
6.19E+07
4.28E+08
1.985E+09
4.391E+08
1.47E+10
CBA
0.47023
0.31242
7.47E-01
1.15E+07
5802441
1.80E+07
513564.79
1056309.50
2.80E+06
RCBA
0.61360
0.25709
1.02E+00
5943596
2275351
1.06E+07
372942.94
107512.69
8.44E+05
CDLOBA
26.93780
39.54585
6.71E+01
4469831
2849244
1.07E+07
18462.13
9920.05
3.57E+04
m_SCA
0.00071
0.00053
2.02E-02
1.15E+08
6.69E+07
3.52E+08
1.048E+10
4.703E+09
2.38E+10
IWOA
0.00185
0.00236
3.92E-03
9.34E+07
4.72E+07
1.19E+08
1.047E+09
8.576E+08
1.43E+09
LWOA
0.00650
0.00439
3.44E-02
8.81E+07
3.31E+07
4.11E+08
3.334E+08
1.326E+08
2.21E+10
CSSA
0.00019
0.00016
6.78E-04
1.68E+09
2.36E+08
1.68E+09
8.837E+10
6.958E+09
8.84E+10
Table 7
Comparison results on unimodal functions during 3E5 evaluations
F1
F2
F3
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
0.00000
0.00000
2.150E-268
0.00000
0.00000
1.999E-141
0.00000
0.00000
7.427E-244
SCA
5.33E-52
2.92E-51
1.325E-19
3.28E-60
9.54E-60
1.256E-28
2.65E+00
1.03E+01
2.763E+03
SSA
3.97E-09
7.20E-10
6.629E+01
2.20E-01
5.24E-01
4.818E+00
6.21E-08
1.97E-08
5.697E+02
GWO
0.00000
0.00000
0.000E+00
0.00000
0.00000
1.002E-286
8.62E-174
0.00000
1.908E-125
MFO
1.67E+03
3.79E+03
1.667E+03
3.53E+01
2.45E+01
3.533E+01
1.58E+04
1.08E+04
1.579E+04
WOA
0.00000
0.00000
0.000E+00
0.00000
0.00000
0.000E+00
2.15E+01
5.44E+01
1.755E+03
GOA
1.37E-03
7.51E-04
7.244E+02
4.93E-01
5.10E-01
1.954E+01
1.15E+02
3.94E+02
2.836E+03
MVO
3.11E-03
7.04E-04
5.957E+02
3.84E-02
1.30E-02
1.113E+01
3.70E-01
1.10E-01
1.613E+03
PSO
1.01E+02
1.43E+01
1.113E+02
4.69E+01
3.54E+00
5.156E+01
1.85E+02
2.76E+01
2.205E+02
DE
1.46E-159
3.86E-159
4.314E-76
2.02E-94
2.33E-94
1.359E-45
1.39E+03
7.73E+02
6.275E+03
AGA
2.38E-02
2.48E-02
5.567E-02
1.18E-02
3.99E-03
1.701E-02
4.51E-02
4.92E-02
8.333E-02
F4
F5
F6
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
0.00000
0.00000
2.648E-131
2.22E-03
9.67E-04
1.837E-01
9.61E-06
4.23E-06
1.583E-02
SCA
4.46E-03
1.34E-02
1.490E+01
2.73E+01
6.99E-01
2.793E+01
3.70E+00
2.72E-01
4.367E+00
SSA
3.72E-01
7.06E-01
7.726E+00
7.27E+01
9.68E+01
2.160E+03
3.86E-09
9.08E-10
6.799E+01
GWO
1.79E-152
8.68E-152
2.593E-126
2.61E+01
9.13E-01
2.632E+01
4.64E-01
2.81E-01
6.100E-01
MFO
6.54E+01
1.03E+01
6.536E+01
2.69E+06
1.46E+07
2.686E+06
2.99E+03
7.91E+03
2.990E+03
WOA
3.68E+00
7.91E+00
4.832E+00
2.44E+01
3.14E-01
2.437E+01
5.89E-06
2.44E-06
5.896E-06
GOA
2.45E+00
2.03E+00
1.366E+01
1.52E+02
3.50E+02
6.639E+04
1.52E-03
7.49E-04
7.702E+02
MVO
8.89E-02
3.43E-02
9.891E+00
6.68E+01
9.45E+01
3.591E+04
3.05E-03
7.30E-04
6.130E+02
PSO
3.81E+00
2.16E-01
3.993E+00
8.98E+04
1.83E+04
1.085E+05
9.85E+01
8.65E+00
1.094E+02
DE
3.54E-15
5.37E-15
7.076E-07
3.08E+01
1.81E+01
3.259E+01
0.00000
0.00000
0.000E+00
AGA
3.17E-02
2.19E-02
6.531E-02
5.10E-02
6.04E-02
1.262E-01
1.58E-02
1.69E-02
1.145E-01
F7
F14
F15
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
9.53E-06
8.25E-06
5.830E-05
2.15E+06
7.66E+05
9.335E+06
1.09E+04
1.28E+04
5.209E+06
SCA
2.43E-03
2.30E-03
1.570E-02
2.35E+08
5.63E+07
3.955E+08
1.65E+10
3.59E+09
2.586E+10
SSA
8.58E-03
4.21E-03
2.034E-02
1.72E+06
6.73E+05
2.440E+07
1.21E+04
9.72E+03
1.130E+08
GWO
6.07E-05
4.25E-05
9.191E-05
5.78E+07
3.28E+07
8.364E+07
2.18E+09
2.05E+09
3.621E+09
MFO
3.64E+00
5.34E+00
3.660E+00
9.51E+07
1.18E+08
9.580E+07
1.05E+10
7.21E+09
1.054E+10
WOA
1.38E-04
1.36E-04
3.663E-04
2.67E+07
1.08E+07
2.686E+07
4.45E+06
7.57E+06
4.481E+06
GOA
1.70E-03
9.63E-04
2.530E-03
1.31E+07
9.07E+06
4.304E+07
2.27E+07
1.24E+08
1.157E+09
MVO
2.99E-03
1.04E-03
6.692E-02
2.78E+06
1.07E+06
2.863E+07
1.55E+04
1.05E+04
9.453E+08
PSO
1.02E+02
2.89E+01
1.022E+02
8.12E+06
2.06E+06
1.019E+07
1.51E+08
1.61E+07
1.643E+08
DE
2.48E-03
6.04E-04
4.437E-03
2.05E+07
6.27E+06
3.310E+07
8.91E+02
1.81E+03
9.373E+02
AGA
1.77E-04
1.22E-04
3.056E-04
1.73E+02
8.34E+01
2.952E+02
2.40E+02
5.14E+01
2.971E+02
3.2.2 Exploration competence analysis
The data in Table 8 represents that SMA is still competitive in multimodal functions. In F8-F11
and F20-21, the AVG of SMA was the smallest or the smallest in parallel compared with other
algorithms. From the convergence curves of F8 and F21 in Figure 9, it can be observed that SMA
can search for the highest accuracy fitness value in these two multimodal functions, while some
algorithms fail to obtain a superior solution after a certain amount of iterations. This is due to local
optima stagnation, which illustrates that SMA can still show better exploration ability in case of
preferable exploration. From the data in Table 9, it can be seen that the results of SMA in F9-F11,
F17, and F20-21 are optimal, and only slightly lower than other algorithms in F8, F18, and F19,
which indicates that SMA can still maintain its advantages over advanced algorithms and reflect
SMA's capability to avoid local optimum solutions. Figure 10 also shows that SMA can find a
superior solution at a relatively fast convergence tendency in multimodal functions such as F9-11,
F17, and F21. Table 10 illustrates the experimental results of SMA with 10 other comparators on the
multimodal function. Among them, SMA obtained the best average and median results on F8-F11
compared with other algorithms, and AGA obtained the best average and median on F16-21.
Compared with AGA, SMA has a greater advantage in unimodal functions, while AGA has a
preferable performance in multimodal functions.
Table 8
Results on multimodal functions with traditional algorithms during 1000 iterations
F8
F9
F10
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
-12569.4
0.1
-1.26E+04
0.00000
0.00000
9.96E-01
8.882E-16
0.00000
8.88E-16
SCA
-3886.1
225.6
-3.82E+03
18.35521
21.43693
7.22E+01
11.32308
9.66101
1.42E+01
SSA
-7816.8
842.3
-6.98E+03
56.61307
12.89967
1.38E+02
2.25688
0.72068
5.03E+00
GWO
-6088.7
859.4
-3.83E+03
0.06990
0.38287
1.12E-01
0.00000
0.00000
1.62E-14
MFO
-8711.6
827.4
-8.71E+03
162.06619
49.63022
1.63E+02
15.79421
6.91218
1.60E+01
WOA
-11630.6
1277.5
-1.15E+04
0.00000
0.00000
0.00000
3.967E-15
2.030E-15
4.09E-15
GOA
-7430.4
761.2
-5.33E+03
86.74360
31.98704
2.35E+02
4.63913
1.06742
9.76E+00
DA
-5631.8
590.7
-5.62E+03
155.13449
38.31121
1.56E+02
8.64831
1.22491
8.72E+00
ALO
-5610.1
438.7
-5.61E+03
80.88997
20.29005
8.49E+01
2.00733
0.77081
2.90E+00
MVO
-7744.9
693.4
-5.59E+03
112.71842
24.57189
2.33E+02
1.14572
0.70341
7.70E+00
PBIL
-4046.4
331.0
-3.87E+03
150.36667
19.01267
1.55E+02
18.44223
0.19901
1.85E+01
PSO
-6728.1
650.2
-6.72E+03
369.24464
18.68261
3.73E+02
8.41508
0.41051
8.75E+00
DE
-12409.8
149.2
-9.93E+03
59.28367
6.07679
8.60E+01
4.638E-07
1.383E-07
5.66E-03
F11
F12
F13
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
0.00000
0.00000
0.00000
0.001195
0.001422
1.42E-02
0.001577
0.003000
1.45E-01
SCA
0.23534
0.22480
1.29E+00
2.290194
2.958865
3.48E+07
518.6869
2782.8453
1.78E+07
SSA
0.01009
0.01067
2.75E+00
5.542545
3.122247
2.17E+01
1.010473
4.701096
9.51E+01
GWO
0.00028
0.00156
3.30E-04
0.037303
0.019955
5.70E-02
0.488377
0.174343
6.85E-01
MFO
22.63478
42.31343
2.82E+01
0.470607
0.782326
3.78E+02
6792.354
37201.162
8.22E+03
WOA
0.00000
0.00000
0.00000
0.005205
0.003512
5.21E-03
0.181197
0.166955
1.81E-01
GOA
0.83124
0.15983
1.29E+01
6.489011
2.717562
4.07E+03
26.3886
16.5919
1.36E+05
DA
9.87794
4.37600
1.00E+01
306.688
1096.994
3.10E+02
4.571E+04
1.022E+05
4.73E+04
ALO
0.00994
0.01271
1.07E+00
9.456697
3.198074
1.28E+01
2.193406
7.919110
3.25E+00
MVO
0.57543
0.08747
8.98E+00
1.294524
1.103471
1.27E+01
0.081286
0.043182
1.78E+03
PBIL
416.755
48.474
4.25E+02
2.667E+08
7.771E+07
2.99E+08
5.860E+08
9.982E+07
6.40E+08
PSO
1.03228
0.00489
1.04E+00
4.80322
0.86670
5.16E+00
23.191583
4.195613
2.88E+01
DE
9.761E-11
2.126E-10
7.56E-03
3.633E-13
3.399E-13
5.03E-05
1.691E-12
1.165E-12
2.44E-04
F16
F17
F18
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
521.0056
0.109097
5.21E+02
618.2822
3.265441
6.23E+02
1300.6543
0.117872
1.30E+03
SCA
521.0427
0.053484
5.21E+02
636.9826
2.244227
6.40E+02
1303.9293
0.374149
1.30E+03
SSA
520.0584
0.107997
5.21E+02
622.8313
4.728569
6.28E+02
1300.5756
0.148959
1.30E+03
GWO
521.0410
0.054652
5.21E+02
616.6474
2.512406
6.24E+02
1300.6905
0.549189
1.30E+03
MFO
520.2870
0.170908
5.20E+02
622.7437
2.701796
6.23E+02
1301.3678
1.019364
1.30E+03
WOA
520.7787
0.119860
5.21E+02
637.7305
2.887311
6.38E+02
1300.5741
0.260727
1.30E+03
GOA
520.1390
0.082631
5.21E+02
622.1088
4.176909
6.30E+02
1300.5707
0.149671
1.30E+03
DA
520.9891
0.094995
5.21E+02
637.2321
2.789804
6.37E+02
1301.4935
1.087595
1.30E+03
ALO
520.0494
0.093898
5.21E+02
626.0851
3.620101
6.27E+02
1300.4614
0.100828
1.30E+03
MVO
520.5350
0.102963
5.21E+02
614.4619
3.437751
6.25E+02
1300.6110
0.114900
1.30E+03
PBIL
521.0393
0.043185
5.21E+02
640.6707
1.407127
6.41E+02
1305.2666
0.311548
1.31E+03
PSO
521.0618
0.054837
5.21E+02
624.8413
3.071015
6.26E+02
1300.5438
0.095901
1.30E+03
DE
520.7948
0.090515
5.21E+02
629.2747
1.350482
6.32E+02
1300.5363
0.050040
1.30E+03
F19
F20
F21
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
1400.6670
0.361757
1.40E+03
1510.9564
3.012250
1.52E+03
1611.4845
0.567778
1.61E+03
SCA
1473.0029
15.520309
1.51E+03
16869
13476.33
1.26E+05
1613.2141
0.241155
1.61E+03
SSA
1400.4157
0.238649
1.40E+03
1513.1155
4.171347
1.53E+03
1612.2034
0.537832
1.61E+03
GWO
1407.2551
8.107508
1.42E+03
1949.1287
920.5966
2.05E+03
1611.7755
0.656408
1.61E+03
MFO
1430.1235
20.716796
1.43E+03
208671
416720.09
2.17E+05
1612.6679
0.536141
1.61E+03
WOA
1405.0142
6.261895
1.41E+03
1727.0908
122.1192
1.73E+03
1612.8485
0.463174
1.61E+03
GOA
1400.4834
0.331069
1.40E+03
1519.1245
6.359294
2.07E+03
1612.5397
0.510917
1.61E+03
DA
1422.6359
10.796483
1.42E+03
9188.8893
11460.10
9.19E+03
1613.1921
0.298363
1.61E+03
ALO
1400.2530
0.047583
1.40E+03
1513.5362
4.828335
1.52E+03
1612.6442
0.572926
1.61E+03
MVO
1400.5551
0.403115
1.40E+03
1512.5460
3.700993
1.54E+03
1612.2971
0.526756
1.61E+03
PBIL
1525.2857
13.420862
1.54E+03
1435558
748053.04
1.65E+06
1613.3661
0.212279
1.61E+03
PSO
1400.3217
0.095276
1.40E+03
1519.8378
1.631079
1.52E+03
1612.5422
0.412383
1.61E+03
DE
1400.4031
0.089745
1.40E+03
1517.1531
1.278695
1.52E+03
1612.5367
0.196986
1.61E+03
Table 9
Comparison results on the multimodal functions with advanced algorithms
F8
F9
F10
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
-12569.4
0.068790
-1.25E+04
0.00000
0.00000
0.00000
8.88E-16
0.00000
8.88E-16
BLPSO
-4544.5
400.3510
-3.87E+03
207.3039
17.0015
2.30E+02
10.22852
0.69752
1.30E+01
CLPSO
-8295.7
351.9193
-6.10E+03
139.7601
15.8072
2.17E+02
8.16910
0.64983
1.43E+01
CBA
-7355.4
720.5161
-7.32E+03
133.1773
40.7382
1.44E+02
14.91852
3.56105
1.50E+01
RCBA
-7248.6
814.7588
-7.24E+03
77.4955
14.5193
1.07E+02
6.76084
6.62622
9.76E+00
CDLOBA
-7236.3
600.1951
-7.23E+03
243.8551
62.2823
2.72E+02
19.57830
0.77234
1.97E+01
m_SCA
-5925.7
986.2730
-3.94E+03
0.00000
0.00000
1.11E+01
5.35800
9.03538
1.34E+01
IWOA
-11252.0
1780.6529
-1.12E+04
0.00000
0.00000
0.00000
3.73E-15
2.17E-15
3.73E-15
LWOA
-10775.8
1141.9779
-1.02E+04
5.12692
18.79066
2.12E+01
4.81E-05
2.84E-05
1.03E-01
CSSA
-12569.5
0.000239
-1.26E+04
7.14583
39.06861
7.15E+00
0.03173
0.03027
3.21E-02
F11
F12
F13
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
0.00000
0.00000
0.00000
0.00095
0.00101
2.68E-02
0.00135
0.00211
1.16E-01
BLPSO
21.49704
3.65806
4.49E+01
4441.072
7073.234
3.24E+05
378616.22
235965.32
3.39E+06
CLPSO
6.33968
0.91129
4.95E+01
20.05685
8.11078
5.40E+05
11963.83
13926.90
4.89E+06
CBA
0.22145
0.11045
7.77E-01
15.33572
7.52799
1.59E+01
43.5008
21.1814
4.59E+01
RCBA
0.02800
0.00947
6.72E-02
13.56632
4.54840
1.47E+01
0.09299
0.03609
2.19E-01
CDLOBA
145.5030
96.9037
1.74E+02
20.17146
6.03281
2.08E+01
35.8588
11.9314
3.85E+01
m_SCA
0.00000
0.00000
5.52E-02
0.19369
0.16449
9.82E-01
1.58065
0.19641
2.40E+00
IWOA
0.00264
0.01100
3.70E-03
0.00930
0.02578
1.18E-02
0.16079
0.13761
2.07E-01
LWOA
0.02455
0.04926
4.54E-01
0.00063
0.00024
1.78E-02
0.01660
0.01442
2.05E-01
CSSA
0.02723
0.03762
2.74E-02
5.98E-05
5.33E-05
6.03E-05
0.00090
0.00086
9.06E-04
F16
F17
F18
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
521.0127
0.069163
5.21E+02
619.4282
2.915833
6.24E+02
1300.6589
0.145401
1.30E+03
BLPSO
521.0920
0.070988
5.21E+02
629.3125
1.805214
6.34E+02
1300.9286
0.138697
1.30E+03
CLPSO
521.0176
0.059879
5.21E+02
629.7237
1.356299
6.35E+02
1300.6655
0.089057
1.30E+03
CBA
520.3188
0.287026
5.20E+02
641.6516
3.410418
6.42E+02
1300.5091
0.134277
1.30E+03
RCBA
520.3774
0.123562
5.21E+02
640.2023
3.196174
6.41E+02
1300.4976
0.123416
1.30E+03
CDLOBA
521.0056
0.064721
5.21E+02
636.2815
2.936580
6.37E+02
1300.5098
0.146951
1.30E+03
m_SCA
520.9230
0.085023
5.21E+02
625.2555
2.906023
6.37E+02
1301.7144
0.980372
1.30E+03
IWOA
520.7061
0.096424
5.21E+02
634.7725
3.121824
6.36E+02
1300.5275
0.096831
1.30E+03
LWOA
520.7827
0.071113
5.21E+02
633.6692
3.853306
6.40E+02
1300.6093
0.123410
1.30E+03
CSSA
521.0604
0.088972
5.21E+02
644.9713
1.825103
6.45E+02
1309.5241
0.830936
1.31E+03
F19
F20
F21
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
1400.6565
0.361610
1.40E+03
1510.5477
2.46585
1.52E+03
1611.5995
0.70239
1.61E+03
BLPSO
1410.4409
2.902210
1.43E+03
1802.5795
180.2212
4.48E+03
1613.0067
0.23416
1.61E+03
CLPSO
1403.5324
2.812311
1.45E+03
1952.4155
304.9825
4.26E+04
1613.0049
0.22798
1.61E+03
CBA
1400.3048
0.092093
1.40E+03
1562.3666
18.85652
1.56E+03
1613.5381
0.36317
1.61E+03
RCBA
1400.2943
0.060668
1.40E+03
1538.9490
7.61211
1.54E+03
1613.6523
0.32500
1.61E+03
CDLOBA
1400.3181
0.058475
1.40E+03
1753.9951
117.6904
1.76E+03
1613.5741
0.25668
1.61E+03
m_SCA
1426.1725
10.27231
1.46E+03
4997.7533
4929.0634
1.55E+04
1612.5383
0.51908
1.61E+03
IWOA
1400.2787
0.143274
1.40E+03
1625.8982
78.1816
1.67E+03
1612.9124
0.55626
1.61E+03
LWOA
1400.3289
0.095342
1.47E+03
1572.8452
27.80344
1.26E+04
1612.8272
0.52137
1.61E+03
CSSA
1680.8338
17.75465
1.68E+03
232677.12
39953.5
2.33E+05
1613.1690
0.24750
1.61E+03
Table 10
Comparison results on multimodal functions during 3E5 evaluations
F8
F9
F10
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
-1.26E+04
2.48E-04
-1.257E+04
0.00000
0.00000
0.000E+00
8.88E-16
0.00000
8.882E-16
SCA
-4.41E+03
2.15E+02
-4.288E+03
0.00000
0.00000
3.499E+00
1.26E+01
9.43E+00
1.610E+01
SSA
-7.79E+03
7.06E+02
-7.419E+03
6.54E+01
1.50E+01
9.676E+01
1.81E+00
8.07E-01
3.901E+00
GWO
-6.38E+03
7.23E+02
-4.403E+03
0.00000
0.00000
0.000E+00
7.64E-15
1.08E-15
7.638E-15
MFO
-8.37E+03
7.59E+02
-8.366E+03
1.65E+02
3.28E+01
1.651E+02
1.58E+01
7.02E+00
1.576E+01
WOA
-1.21E+04
9.04E+02
-1.207E+04
0.00000
0.00000
0.000E+00
3.38E-15
2.12E-15
3.375E-15
GOA
-7.56E+03
6.06E+02
-6.158E+03
1.04E+02
4.22E+01
1.742E+02
2.71E+00
8.89E-01
7.415E+00
MVO
-8.18E+03
7.17E+02
-6.424E+03
8.27E+01
2.44E+01
1.772E+02
1.08E-01
3.58E-01
6.771E+00
PSO
-7.07E+03
8.27E+02
-7.067E+03
3.43E+02
1.69E+01
3.469E+02
7.78E+00
2.41E-01
8.041E+00
DE
-1.24E+04
1.31E+02
-1.243E+04
3.32E-02
1.82E-01
3.317E-02
7.64E-15
1.08E-15
7.994E-15
AGA
-8.38E+02
9.72E-03
-8.379E+02
9.94E-03
0.00000
1.655E-02
1.64E-02
0.00000
1.644E-02
F11
F12
F13
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
0.00000
0.00000
0.000E+00
7.55E-06
8.36E-06
2.780E-04
6.77E-06
3.68E-06
2.418E-03
SCA
8.03E-11
4.36E-10
6.453E-02
3.27E-01
5.08E-02
6.351E+03
1.98E+00
1.11E-01
2.375E+00
SSA
1.18E-02
1.10E-02
1.577E+00
1.41E+00
1.70E+00
6.100E+00
5.06E-03
6.75E-03
3.688E+00
GWO
2.49E-04
1.36E-03
2.514E-04
2.56E-02
1.20E-02
3.778E-02
4.01E-01
1.95E-01
5.442E-01
MFO
3.31E+01
5.55E+01
3.312E+01
2.29E-01
4.75E-01
2.288E-01
6.15E-01
1.11E+00
6.152E-01
WOA
6.58E-04
2.52E-03
6.577E-04
1.09E-06
4.07E-07
1.087E-06
3.84E-04
2.00E-03
3.836E-04
GOA
1.81E-02
1.51E-02
7.615E+00
1.93E+00
1.50E+00
1.380E+01
9.33E-01
3.86E+00
5.700E+03
MVO
2.76E-02
1.33E-02
6.603E+00
1.64E-01
5.09E-01
7.007E+00
4.06E-03
5.30E-03
3.389E+01
PSO
1.02E+00
1.27E-02
1.022E+00
3.38E+00
3.70E-01
3.822E+00
1.57E+01
1.83E+00
1.729E+01
DE
0.00000
0.00000
0.000E+00
1.57E-32
5.57E-48
1.571E-32
1.35E-32
5.57E-48
1.350E-32
AGA
2.14E-02
1.37E-02
3.063E-02
2.17E-02
2.82E-02
5.744E-02
1.13E-02
9.89E-03
1.987E-02
F16
F17
F18
Algorithm
AVG
STD
MED
AVG
STD
MED
AVG
STD
MED
SMA
5.21E+02
2.27E-01
5.210E+02
6.15E+02
3.06E+00
6.188E+02
1.30E+03
1.26E-01
1.301E+03
SCA
5.21E+02
5.60E-02
5.210E+02
6.33E+02
2.39E+00
6.364E+02
1.30E+03
3.71E-01
1.304E+03
SSA
5.20E+02
1.07E-01
5.210E+02
6.19E+02
4.24E+00
6.234E+02
1.30E+03
1.45E-01
1.301E+03
GWO
5.21E+02
5.11E-02