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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.

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version of record. This version will undergo additional copyediting, typesetting and review before it

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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

5.210E+02

6.14E+02

3.27E+00

6.210E+02

1.30E+03

3.11E-01

1.301E+03

MFO

5.20E+02

1.73E-01

5.203E+02

6.23E+02

3.53E+00

6.231E+02

1.30E+03

1.26E+00

1.302E+03

WOA

5.20E+02

1.61E-01

5.204E+02

6.36E+02

4.15E+00

6.363E+02

1.30E+03

1.05E-01

1.301E+03

GOA

5.20E+02

7.96E-02

5.210E+02

6.17E+02

3.63E+00

6.250E+02

1.30E+03

6.95E-02

1.301E+03

MVO

5.20E+02

4.14E-02

5.210E+02

6.10E+02

3.97E+00

6.214E+02

1.30E+03

1.24E-01

1.301E+03

PSO

5.21E+02

4.59E-02

5.210E+02

6.23E+02

3.42E+00

6.231E+02

1.30E+03

7.31E-02

1.300E+03

DE

5.21E+02

4.46E-02

5.206E+02

6.20E+02

2.07E+00

6.226E+02

1.30E+03

4.04E-02

1.300E+03

AGA

5.00E+02

4.82E-01

5.005E+02

6.00E+02

1.68E-02

6.000E+02

1.30E+03

2.53E-02

1.300E+03

F19

F20

F21

Algorithm

AVG

STD

MED

AVG

STD

MED

AVG

STD

MED

SMA

1.40E+03

3.13E-01

1.401E+03

1.51E+03

1.83E+00

1.517E+03

1.61E+03

7.14E-01

1.612E+03

SCA

1.44E+03

7.88E+00

1.466E+03

4.96E+03

4.20E+03

2.681E+04

1.61E+03

2.17E-01

1.613E+03

SSA

1.40E+03

2.22E-01

1.400E+03

1.51E+03

2.10E+00

1.523E+03

1.61E+03

6.27E-01

1.612E+03

GWO

1.40E+03

7.60E+00

1.410E+03

1.89E+03

7.48E+02

1.960E+03

1.61E+03

6.66E-01

1.612E+03

MFO

1.43E+03

2.55E+01

1.435E+03

3.14E+05

5.20E+05

3.141E+05

1.61E+03

5.10E-01

1.613E+03

WOA

1.40E+03

1.23E-01

1.400E+03

1.57E+03

2.49E+01

1.575E+03

1.61E+03

5.51E-01

1.613E+03

GOA

1.40E+03

3.31E-01

1.401E+03

1.51E+03

2.20E+00

1.531E+03

1.61E+03

7.46E-01

1.612E+03

MVO

1.40E+03

3.32E-01

1.401E+03

1.51E+03

2.26E+00

1.527E+03

1.61E+03

5.89E-01

1.612E+03

PSO

1.40E+03

9.78E-02

1.400E+03

1.52E+03

1.38E+00

1.517E+03

1.61E+03

4.48E-01

1.612E+03

DE

1.40E+03

1.24E-01

1.400E+03

1.51E+03

1.10E+00

1.513E+03

1.61E+03

2.18E-01

1.612E+03

AGA

1.40E+03

1.21E-02

1.400E+03

1.50E+03

7.70E-03

1.500E+03

1.60E+03

9.19E-03

1.600E+03

3.2.3 Analysis of avoiding locally optimal solutions

All functions in Tables 11-12, as fix-dimension multimodal functions, have multiple local optima,

which are challenging for MAs, thus can discriminate the overall efficacy of algorithms in

exploration and exploration. According to the data in Tables 11-12, SMA ranked first in AVG on F28,

F29, F30, F32, and F33, which show a very potential comprehensive ability. It can also be seen from

the optimum curve of F28-33 in Figures 9 that SMA achieves superior solutions faster than other

counterparts, thus well coordinating the ability of exploration and exploration. The statistics of

Tables 13-14 illustrate that SMA can also maintain certain advantages in composition functions

compared with the advanced algorithm, which further reflects that SMA can avert falling into local

optimum with fast convergence. F25, F32, and F33 in Figure 10 also intuitively incarnate the

performance preponderance of SMA in composition functions.

Table 11

Comparison results on the Hybrid functions of CEC 2014 with traditional algorithms

F22

F23

F24

Algorithm

AVG

STD

MED

AVG

STD

MED

AVG

STD

MED

SMA

1981009

955714

3.35E+06

23768.042

9648.796

2.41E+05

1916.4612