New Evolutionary Selection Operators for Snake Optimizer

Ruba Abu Khurma

, Moutaz Alazab

, J. J. Merelo

and Pedro A. Castillo

Department of Computer Science, Al-Ahliyya Amman University, Amman, Jordan

Department of Artiﬁcial Intelligence, Al-Balqa University, Al-Salt, Jordan

Department of Computer Architecture and Computer Technology, ETSIIT and CITIC,

University of Granada, Granada, Spain

Keywords:

Snake Optimizer, Evolutionary Operators, Selection Schemes.

Abstract:

Evolutionary algorithms (EA) adopt a Darwinian theory which is known as ”survival of the ﬁttest”. Snake

Optimizer (SO) is a recently developed swarm algorithm that inherits the selection principle in its structure.

This is applied by selecting the ﬁttest solutions and using them in deriving new solutions for the next iterations

of the algorithm. However, this makes the algorithm biased towards the highly ﬁtted solutions in the search

space, which affects the diversity of the SO algorithm. This paper proposes new selection operators to be

integrated with the SO algorithm and replaces the global best operator. Four SO variations are investigated by

individually integrating four different selection operators: SO-roulettewheel, SO-tournament, SO-linearrank,

and SO-exponentialrank. The performance of the proposed SO variations is evaluated. The experiments show

that the selection operators have a great inﬂuence on the performance of the SO algorithm. Finally, a parameter

analysis of the SO variations is investigated.

1 INTRODUCTION

Natural systems live in groups characterized by

decentralization and self-organization among their

members (Khurma et al., 2020). These swarm sys-

tems have their special relationships between swarm

members and their environment that control their

search for food and sustenance. Swarm Intelligence

researchers investigate the collective behavior of an-

imals and turn their social relationships into mathe-

matical methodologies (Abu Khurma et al., 2022).

The SO algorithm is a recently developed swarm

algorithm by Fatma (Hashim and Hussien, 2022). The

main inspiration for the SO algorithm comes from the

behavior of snakes in nature. Many environmental

factors inﬂuence the behavior of snakes. For exam-

ple, snake mating occurs when the temperature is cold

and there is food. Otherwise, snakes are looking for

food. The SO methodology translates this inspiration

into two phases of exploration and exploitation. Ex-

ploration is the situation where no food is found and

the temperature is cold so that the snakes (solutions)

search globally in the search space. The stage of ex-

ploitation includes several cases. If the food is avail-

able and the temperature is hot, the snakes eat the food

that is present. If food is available and the temperature

is cool, snakes enter the mating process. The mating

process has two modes, either ﬁghting mode or mat-

ing mode. The ﬁghting mode makes the snakes ﬁght

until the male gets the best female and the female gets

the best male. Mating mode between a pair of snakes

occurs depending on the amount of food. Mating may

result in the birth of new snakes.

The SO algorithm evaluates solutions at each it-

eration to get the best solutions in the male group

and female group which are called Snakemale

best

and

Snake f emale

best

respectively. The SO update proce-

dure for other solutions is guided by the positions of

the best solutions. During the iterations of the algo-

rithm, the re-position of solutions in the search space

depends on the distance from the best solutions. This

means that the search process is biased toward the

best solution. Changing the positions of solutions

concerning one point during the search affects the di-

versity of solutions and the exploration of the algo-

rithm. This may lead also to premature convergence

and stagnation in local minima.

In the literature, there have been vast studies that

investigate the effects of evolutionary selection op-

erators on the performance of swarm intelligent al-

gorithms in different applications. A previous study

(Khurma et al., 2021), integrated different selection

Khurma, R., Alazab, M., Merelo, J. and Castillo, P.

New Evolutionary Selection Operators for Snake Optimizer.

DOI: 10.5220/0011524300003332

In Proceedings of the 14th International Joint Conference on Computational Intelligence (IJCCI 2022), pages 82-90

ISBN: 978-989-758-611-8; ISSN: 2184-3236

operators with the Moth Flame Optimizer (MFO) and

use them with Levy ﬂight operator to enhance the fea-

ture selection process. The results showed improved

performance in the diagnosis of disease. The au-

thors in (Al-Betar et al., 2018a), improved the original

greedy selection operator of the Grey Wolf Optimizer

(GWO) by using other less-biased selection operators

extracted from evolutionary algorithms. Six GWO

variations are produced and studied on 23 mathe-

matical benchmark functions. The results show that

TGWO achieved the best results. In (Al-Betar et al.,

2012), the authors used new selection operators with

the Harmony Search (HS) algorithm in memory con-

sideration to replace the original random selection

The results on benchmark functions showed that the

selection operators affected the performance of the

HS algorithm. The authors in (Al-Betar et al., 2018b),

studied the effect of the selection operators on the Bat

algorithm. They replaced the global-best selection

with other evolutionary operators. Their evaluation on

25 IEEE-CEC2005 functions showed competitive re-

sults. In (Awadallah et al., 2019), The authors studied

the effect of different selection operators on the Ar-

tiﬁcial Bee Colony (ABC) algorithm. The evaluation

results on benchmark functions showed the effects of

the selection operators on ABC algorithm. In (Shehab

et al., 2016), the authors studied the effect of replac-

ing the global best selection mechanism of the Parti-

cle Swarm Optimization (PSO) algorithm with other

selection operators. The results showed a direct effect

of the selection operators on the performance of the

PSO. Bottom-line, the literature has many research

papers that investigates the effect of different selec-

tion schemes on the performance of several swarm in-

telligent algorithms. According to the No-Free-Lunch

theorem, (Adam et al., 2019), no swarm algorithm

has the same performance in tackling all optimization

problems. Therefore, there is always a chance for re-

searchers to suggest new algorithms and experiment

them with different optimization problems.

In this paper, new evolutionary selection opera-

tors are borrowed from the Genetic Algorithm (GA)

and integrated with the SO algorithm. These opera-

tors are Roulette wheel, tournament, linear rank, and

exponential rank. Each operator is integrated individ-

ually with the SO algorithm and substitutes the global

best standard operator. Four SO variations are pro-

duced using these operators: SO-roulettewheel, SO-

tournament, SO-linearrank, and SO-exponentialrank.

The performance of each SO variation is investigated

and evaluated using the standard benchmark mathe-

matical functions.

The remaining parts of this paper are organized as

follows: Section 2 presents the SO algorithm. Sec-

tion 3 discusses the proposed selection operators inte-

grated with SO. Section 4 analyzes and discusses the

experimental results. Finally, Section 5 summarizes

the paper and determines some future works.

2 SNAKE OPTIMIZER (SO)

This section presents the mathematical model of a re-

cently published SO algorithm (Hashim and Hussien,

2022). The following points explains in detail SO

steps:

• Initializing solutions: SO starts by initiating a

set of random solutions in the search space using

Eq.(1). These solutions compose the snakes pop-

ulation to be optimized in the next steps.

Snake

= Snake

min

+ random × (Snake

max

−

Snake

min

) (1)

where Snake

is the location in the search space of

the i

solution in the swarm. random is a random

number ∈ [0,1]. Snake

max

and Snake

min

are the

minimum and the maximum values respectively

for the studied problem.

• Division of solutions: the population is divided

into two parts (50% male and 50% female) using

Eq. (2) and Eq. (3)

Num

male

≈ Num/2 (2)

Num

f emale

≈ Num − Num

male

(3)

where Num is the size of the population (all

snakes). Num

male

is the number of the male so-

lutions. Num

f emale

is the number of female solu-

tions.

• Evaluate solutions: get the best solution from

the male group (Snakebest

male

), female group

(Snakebest

f emale

) and ﬁnd the location of the food

f ood

. Two other concepts are deﬁned which are

the temperature (Temperature) and the quantity

of food (Qantity) as in Eq.(4) and Eq.(5) respec-

tively.

Temperature = Exp(

−Curiter

Totiter

) (4)

where Curiter is the current iteration and Totiter

is the number of all iterations.

Qantity = Const

× Exp(

Curiter − Totiter

Totiter

) (5)

where Const

is a constant value equal 0.5.

New Evolutionary Selection Operators for Snake Optimizer

• Exploring the search space (food is not found):

this depends on using a speciﬁed threshold value.

If Quantity < 0.25, the solutions search globally

by updating their locations with respect to a spec-

iﬁed random location in the search space. This

modeled by Eq.(6)-Eq.(9)

Snakemale

(iter + 1) = Snakemale

rand

(iter)±

Const

× ABmale × ((Snake

max

− Snake

min

)

× rand + Snake

min

) (6)

where Snakemale

is i

male solution,

Snakemale

rand

is the location of random male

solution, rand is a random number ∈ [0,1] and

male

is the ability of the male solution to ﬁnd

the food and can be computed using Eq.(7):

ABmale = Exp(−

Fitnessmale

rand

Fitnessmale

) (7)

where Fitnessmale

rand

is the ﬁtness of

Snakemale

rand

and Fitnessmale

is the ﬁt-

ness of i

solution the in male group and Const

is a constant equals 0.05.

Snake f emale

(iter + 1) =

Snake f emale

rand

(iter) ±Const

× AB f emale × ((Snake

max

− Snake

min

) × rand+

Snake

min

) (8)

where Snake f emale

is i

female solution,

Snake f emale

rand

is the location of random female

solution, rand is a random number ∈ [0,1] and

f emale

is the ability of the female solution to

ﬁnd the food and can be computed using Eq.(9):

AB f emale = Exp(−

Fitness f emale

rand

Fitness f emale

) (9)

where Fitness f emale

rand

is the ﬁtness of

Snake f emale

rand

and Fitness f emale

is the

ﬁtness of i

solution the in male group and

Const

is a constant equals 0.05.

• Exploiting the search space (Food is found) If

the quantity of food is greater than a speciﬁed

threshold Quantity > 0.25 then the temperature is

checked. If Temperature > 0.6 (hot), The solu-

tions will move to the food only.

Snake

(i, j)

(iter + 1) = L

f ood

Const

× Temperature × rand×

f ood

− Snake

(i, j)

(iter)) (10)

where Snake

(i, j)

is the location of a solution (male

or female), L

f ood

is the location of the best solu-

tions, and Const

is constant value and equals 2.

If Temperature > 0.6 (cold), The snake will be in

the ﬁght mode or mating mode Fight Mode.

Snakemale

(iter + 1) = Snakemale

(iter)±

Const

× FAMtimesrand × (Snake f emale

best

−

Snakemale

(iter)) (11)

where Snakemale

is the i

male location,

Snake f emale

best

is the location of the best solu-

tion in female group, and FAM is the ﬁghting abil-

ity of male solution.

Snake f emale

(itert + 1) =

Snake f emale

(iter + 1)

±Const3 × FAF × rand × (Snakemale

best

−

Snake f emale

(iter + 1)) (12)

where Snake f emale

is the i

female location,

Snakemale

best

is the location of the best solution

in the male group, and FAF is the ﬁghting ability

of the female solution.

FAM and FAF can be computed from the follow-

ing equations:

FAM = Exp(−

Fitness f emale

best

Fitness

) (13)

FAF = Exp(−

Fitnessmale

best

Fitness

) (14)

where Fitness f emale

best

is the ﬁtness of the best

solution of the female group, Fitnessmale

best

the ﬁtness of the best solution of male group, and

Fitness

is the solution ﬁtness.

Mating mode.

Snakemale

(iter + 1) = Snakemale

(iter)±

Const

× MAm × rand × (Quantity×

Snake f emale

(iter) − Snakemale

(iter)) (15)

Snake f emale

(iter + 1) =

Snake f emale

(iter) ±Const

× MA f m × rand×

(Quantity × Snakemale

(iter)

− Snake f emale

(iter)) (16)

where Snake f emale

is the location of the i

so-

lution in female group and Snakemale

is the lo-

cation of the i

solution in male group and MAm

and MAf are the ability of male and female for

mating respectively and they can be computed as

follow:

MAm = Exp(−

Fitness f emale

Fitnessmale

) (17)

ECTA 2022 - 14th International Conference on Evolutionary Computation Theory and Applications

MA f = Exp(−

Fitnessmale

Fitness f emale

) (18)

If Egg hatch, select worst male solution and worst

female solution and replace them

Snakemale

worst

= Snake

min

+ rand×

(Snake

max

− Snake

min

) (19)

Snake f emale

worst

= Snake

min

+ rand×

(Snake

max

− Snake

min

) (20)

where Snakemale

worst

is the worst solution in the

male group, Snake f emale

worst

is the worst solu-

tion in female group. The diversity factor opera-

tor ± gives chance to increase or decrease loca-

tions’ solution to give high probability to change

the the locations of solutions in the search space

in all possible directions.

Algorithm 1: Continuous SO Algorithm.

Input: Dim, UB, LB, Num, Totiter, and Curiter

Output:Best Snake

Initialize the Snakes randomly

while Curiter ≤ Totiter do

Evaluate each Snake in groups Num

male

and

Num

f emale

Find best male Snakemale

best

Find best female Snake f emale

best

Deﬁne Tempreture using Eq. (4).

Deﬁne food Quantity Quantity using Eq. (5).

if (Quantity < 0.25) then

Perform exploration using Eq. (6) and Eq. (8)

else if (Temperature > 0.6) then

Perform exploitation Eq. (10)

else if (rand > 0.6) then

Snakes in Fight Mode Eq. (11) and Eq. (12)

else

Snakes in Mating Mode Eq. (15) and Eq. (16)

Change the worst male Snake and female

Snake Eq. (19) and Eq. (20)

end if

end while

Return best Snake

3 THE PROPOSED SELECTION

OPERATORS

3.1 Roulette Wheel Selection

This method selects solutions based on their ﬁtness

value in proportion to the ﬁtness of other solutions in

the population. Thus, the probability to select a so-

lution depends on the absolute value of its ﬁtness in

relation to the ﬁtness of other solutions in the popula-

tion. The selection probability Prob

for the solution

i is computed by Eq 21. In Algorithm 2, Random

is a random value from a uniform random distribu-

tion U (0,1); Total prob is the accumulative selec-

tion probabilities of solution S

which is deﬁned by

Total prob =

∑

i=1

Prob

f (S

)

∑

popsize

j=1

f (S

)

(21)

Algorithm 2: Pseudocode for the Roulette wheel selection.

Set Random ∼ U (0,1)

Set F = False

Set Total prob = 0

Set L = 0

while i ≤ popsize and not (F) do

Total prob = Total prob + Prob

if Total prob > Random then

L = i

F = True

end if

i = i + 1

end while

3.2 Tournament Selection

Algorithm 3 shows the tournament selection method,

which begins by selecting a set of random solutions

from the population. The number of selected solu-

tions is called the tournament size. The procedure

proceeds by selecting the best solution in tournament

T . This process is repeated n times.

Algorithm 3: Pseudocode for the Tournament selection.

Select L random solutions from the population.

Select the ﬁrst-best solution from tournament with

probability Prob.

Select the second-best solution with probability

Prob × (1 − Prob).

Select the third-best solution with probability Prob ×

((1 − Prob)

And so on...

3.3 Linear Rank Selection

This selection method overcomes the shortcomings of

the Roulette wheel method by adopting the rank of

solutions instead of their ﬁtness values. Eq 22 shows

that the probability of selection for a solution depends

on its rank. The highest rank n is given to the best

solution while the lowest rank 1 is given for the worst

New Evolutionary Selection Operators for Snake Optimizer

solution. Once all solutions in the swarm are ranked,

Algorithm 4 is applied.

Prob

rank

n × (n − 1)

(22)

Algorithm 4: Pseudocode for the Linear rank selection.

Set X

= 0

for i = 1 to popsize do

= X

i−1

+ Prob

end for

for i = 1 to popsize do

Generate a Random ∈ [0, popsize]

for 1 ≤ j ≤ popsize do

if Prob

≤ Random then

Select the j

solution

end if

end for

3.4 Exponential Rank Selection

This method sorts the ranked solutions depending on

their probabilities by using exponentially weighted as

in Eq 23. the value of X is between 0 and 1. If X = 1,

the difference in the selection probability between the

best and the worst solutions is ignored. If X = 0,

this continuously increases the difference in the se-

lection probability in such a way it produces an expo-

nential curve along the ranked solution. Algorithm 5

shows the steps of exponential ranking, it differs from

the linear ranking in the computation of the selection

probabilities.

Prob

rank

∑

popsize

j=1

rank

(23)

Algorithm 5: Pseudocode for the Exponential rank selec-

tion.

Set X

= 0

for i = 1 to popsize do

= X

i−1

+ Prob

end for

for i = 1 to popsize do

Generate a Random ∈ M

for 1 ≤ j ≤ popsize do

if Prob

< r then

Select the j

solution

end if

end for

4 ANALYSIS AND DISCUSSIONS

OF THE EXPERIMENTAL

RESULTS

This section perform the experiments on the four gen-

erated SO variations listed below to evaluate the effect

of different selection operators on the performance of

the SO algorithm. Each variation adopts a speciﬁc

selection operator that is integrated with the SO algo-

rithm:

1. SO-Global-best: it uses the SO algorithm with

Global-best selection operator.

2. SO-Roulettewheel: it uses the SO algorithm with

the Roulette wheel selection operator.

3. SO-Tournament: It uses the SO algorithm with the

tournament selection operator.

4. SO-Linearrank: it uses the SO algorithm with the

linear rank selection operator.

5. SO-Exponentialrank: it uses the SO algorithm

with the exponential rank selection operator.

All the experiments are conducted using a com-

puter with processor 11th Gen Intel(R) Core(TM)

16-1135G7@2.40GHz with 16 GB of RAM and 64-

bit for Microsoft Windows 10 Pro. The MATLAB

(R2010a) is used to implement the source code.

Different parameter settings are used to evaluate

the SO with different selection operators. The inves-

tigated parameters are the number of dimensions and

population size for each SO version as follows: num-

ber of dimensions Dim = (10, 20, and 30) (Trelea,

2003), population size = (30, 50, and 80) (Richards

and Ventura, 2003). Each run is iterated 100,000

times.

The optimal parameter setting for each version

is shown in Table 1. The experiments are then per-

formed using ﬁve scenarios with different parameter

settings, as shown in Table 1. Each scenario studies

the capability of the two parameters, and each of these

parameters includes a set of values. For example the

ﬁrst scenario shows the SO-Global-best with its opti-

mal value of each parameter, as shown Dim=30, and

population size= 50, These values are identiﬁed as an

optimal value for the experiments, and so on for all

scenarios.

More function evaluations are needed when the

the number of dimensions increases. At the same

time, increasing the computations increases the al-

gorithm’s reliability. The most important issue of

this work is to balance between reliability and cost.

Therefore, the optimal value for the number of dimen-

sions should be between 10 and 30 and should not be

ECTA 2022 - 14th International Conference on Evolutionary Computation Theory and Applications

Table 1: Parameters scenarios.

Scenario Number Selection operator Dim Population size

Scenario1 Global-best 30 50

Scenario2 SO-Roulettewheel 30 50

Scenario3 SO-Tournament 30 30

Scenario4 SO-Linearrank 30 30

Scenario5 SO-Exponentialrank 30 50

more than 30 when the problem becomes more com-

plex. The results achieved in this work are consistent

with (Jin et al., 2013).

For the population size parameter, assigning a

value of 50 is recommended when dealing with high

dimensional problems. However, selecting a popula-

tion size of [30, 50] is recommended for lower dimen-

sional problems. The values of population size are

consistent with previous studies (Li-Ping et al., 2005).

This study uses 14 global benchmark functions

that include both unimodal and multimodal functions.

These are used commonly to solve minimization op-

timization problems (Civicioglu and Besdok, 2013).

The purpose of adopting these functions is to evaluate

the performance of the SO algorithm.

Table 2 and Table 3 show the optimal solutions ob-

tained by the SO variations using the 14 benchmark

functions. The target form using benchmark func-

tions is to get the minimum solution and this depend

on each benchmark. For most of a benchmark the

best value is near Zero. However, the best value for

other benchmark is near (- 450) like shifted bench-

mark functions. All selection operators try to be near

to the best solution, but SO-Tournament obtained the

ﬁrst rank. The SO-Exponentialrank got the worst so-

lution, SO-Roulettewheel, SO-Global-best and SO-

Linearrank are respectively among them.

Table 2 and Table 3 summarize the results of the

SO variations using the 14 benchmark functions in

each scenario, as shown in Table 1. The results in Ta-

bles 2 and 3 are arranged from scenario1 to scenario5

to save the best value for each parameter, which

means in scenario5 each of the selection schemes has

the best values of parameters. Each SO version im-

plemented 30 runs, and the values in the table re-

fer to the average and standard deviations (within

the parentheses). The optimal solutions appear in

bold font. The results show that SO-Tournament ob-

tains the optimal results for all the benchmark func-

tions. SO-Global-best and SO-Roulettewheel obtaine

the eight best results for the Sphere, Schwefel prob-

lem 2.22, Step, Rosenbrock, Rotated hyper-ellipsoid,

Rastrigin, Ackley, and Griewank benchmark func-

tions. SO-Linearrank obtains the best results for

most of the benchmark functions. By contrast, SO-

Exponentialrank obtains poor results when compared

with the other selection operators, especially for the

Rotated hyper-ellipsoid, Rastrigin, Shifted Sphere,

and Shifted Rosenbrock benchmark functions.

5 CONCLUSION AND FUTURE

WORK

This study investigates the effect of integrating dif-

ferent evolutionary selection operators in the struc-

ture of the SO optimizer. The work is done by

replacing the original global-best solution scheme

by four other selection operators. The inte-

gration of selection schemes with SO produces

four variations of the SO algorithm namely SO-

Roulettewheel, SO-Tournament, SO-Linearrank and

SO-Exponentialrank. The SO variations aim is to ap-

ply the survival of the ﬁttest selection principle in the

search space. The experiments were performed on

the global mathematical benchmark functions. The

results proved that integrating the selection operators

in the search space of the SO algorithm is capable

to improve the balance between the global and local

search phases and to alleviate the premature conver-

gence due to entrapment in local minima. Further-

more, the SO-Tournament achieved the best results,

followed by SO-Roulettewheel and SO-Globalbest,

whose results were very close. SO-Linearrank and

SO-Exponentialrank come in the fourth and the last

place respectively. This study investigates the effect

of selection schemes for the ﬁrst time on the SO algo-

rithm. For future, we intend to explore this study by

considering the search time. Also, we intend to apply

the SO with selection operators in speciﬁc domains to

solve some real-life problems.

ACKNOWLEDGMENTS

This work is supported by the Ministerio espa

nol de

Econom

ıa y Competitividad under project PID2020-

115570GB-C22 (DemocratAI::UGR).

New Evolutionary Selection Operators for Snake Optimizer

Table 2: The Average and Standard Deviation Results of Benchmark Functions.

Benchmark Function Selection Technique

Scenario1

Scenario2 Scenario3 Scenario4 Scenario5

Sphere

SO-Global-best

5.86E-07

(8.88E-07)

3.16E-07

(1.10E-08)

0.36E-07

(0.81E-07)

2.00E-07

(9.66E-09)

1.70E-08

(8.74E-09)

SO-Roulettewheel

8.40E-07

(3.06E-07)

7.12E-07

(1.34E-09)

5.47E-07

(3.14E-07)

3.76E-10

(0.872E-11)

1.40E-07

(3.06E-07)

SO-Tournament

0.00E+00

(0.00E+00)

5.82E-17

(1.10E-15)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

SO-Linearrank

2.27E+05

(8.37E+04)

3.14E-05

(2.40E-02)

3.70E-04

(6.20E-04)

8.76E-08

(7.07E-08)

4.46E-08

(0.12E-09)

SO-Exponentialrank

2.14E-02

(0.08E-02)

2.13E-03

(1.00E-02)

0.45E-03

(0.77E-02)

1.32E-04

(0.01E-04)

0.36E-07

(0.81E-07)

Schwefel’s problem 2.22

SO-Global-best

0.0026

(0.0036)

0.0025

(0.0042)

6.88E-05

(5.54E-04)

6.77E-07

(1.32E-06)

7.32E-08

(0.01E-10)

SO-Roulettewheel

0.0001

(0.0001)

4.74E-07

(0.0001)

3.65E-05

(2.66E-05)

1.33E-08

(7.64E-07)

8.12E-10

(3.54E-09)

SO-Tournament

3.22E-26

(0.00E+00)

3.46E-394

(0.00E+00)

1.46E-394

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

SO-Linearrank

0.0000

(0.0731)

6.24E-03

(5.73E-03)

8.65E-04

(4.80E-03)

0.01E-06

(7.72E-05)

4.12E-07

(2.76E-07)

SO-Exponentialrank

1.07E+03

(2.58 E+02 )

1.06E+03

(2.70E+02)

0.003E+03

(3.148E+03)

0.81E+02

(2.76E+03)

0.56E+03

(0.43E+03)

Step

SO-Global-best

6.17E-06

(5.40E-06)

8.40E-07

(0.10E-06)

5.10E-07

(2.46E-05)

7.83E-07

(1.44E-08)

0.64E-09

(1.43E-08)

SO-Roulettewheel

0.40E-07

(1.05E-07)

0.40E-07

(1.07E-07)

1.06E-07

(1.61E-07)

5.52E-08

(0.83E-09)

7.67E-11

(5.48E-10)

SO-Tournament

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

SO-Linearrank

7.10E-04

(0.12E-03)

7.10E-04

(0.88E-03)

4.58E-04

(0.48E+00)

0.11E-05

(5.71E-03)

3.87E-07

(1.62E-07)

SO-Exponentialrank

2.27E+03

(8.36E+03)

2.08E+03

(1.00E+03)

0.35E+02

(5.00E+02)

0.02E+02

(1.52E+02)

1.28E+01

(1.20E+00)

Rosenbrock

SO-Global-best

0.02423

(0.8008)

0.11E-05

(0.27E-05)

1.44E-05

(6.58E-06)

5.53E-06

(3.22E-06)

0.54E-07

(1.26E-08)

SO-Roulettewheel

0.20E-07

(1.15E-07)

7.20E-06

(1.15E-06)

5.47E-06

(3.14E-07)

1.24E-08

(3.56E-07)

2.07E-08

(5.18E-07)

SO-Tournament

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

SO-Linearrank

0.5586

(1.1032)

2.30E-02

(0.48E+01)

2.74E-04

(4.10E-04)

3.21E-05

(7.83E-06)

1.71E-08

(6.54E-07)

SO-Exponentialrank

0.35E+04

(5.60E+03)

2.17E+03

(5.00E+03)

0.41E+03

(1.00E+03)

1.06E+02

(0.38E+03)

0.05E+02

(1.00E+03)

Rotated hyper-ellipsoid

SO-Global-best

4.44E-06

(8.05E-06)

8.10E-06

(2.13E-06)

4.44E-06

(8.05E-06)

4.58E-06

(1.33E-06)

4.48E-07

(1.28E-07)

SO-Roulettewheel

6.50E-07

(0.12E-06)

6.50E-07

(0.12E-06)

2.540E-07

(6.21E-08)

5.73E-08

(6.45E-07)

4.58E-09

(2.38E-10)

SO-Tournament

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

SO-Linearrank

0.05E-04

(0.0078)

1.28E-04

(0.08E+01)

0.05E-06

(2.30E-06)

0.45E-05

(4.10E-06)

2.74E-08

(4.30E-06)

SO-Exponentialrank

0.70E+06

(5.45E+05)

0.38E+03

(2.18E+06)

2.13E+03

(1.00E+02)

1.22E+02

(1.72E-02)

5.83E+00

(0.05E+02)

Schwefel’s problem 2.26

SO-Global-best

-559.687870

(1.084706)

-23675.928

(1.136271)

-23560.799

(1.837641)

-23668.768

(1.023145)

-23661.314

(1.156238)

SO-Roulettewheel

-955.3679

(01.8273)

-3628.645

(545.183740)

-23669.613

(8.422817)

-23671.654

(1.323535)

-23674.796

(0.840532)

SO-Tournament

-941.927358

(146.046462)

-23678.54

(2.03102)

-23641.597

(0.000016)

-23641.512

(0.41E-04)

-23674.989

(0.07E-04)

SO-Linearrank

-9865.935499

(288.744633)

-23672.53

(0.752077)

-23675.928

(1.136271)

-23664.454

(1.743721)

-23677.965

(0.087465)

SO-Exponentialrank

-593.4963

(662.4268)

-9876.7576

(300.067265)

-9876.757

(300.067)

-9976.757

(130.421)

-22894.654

(112.384)

Rastrigin

SO-Global-best

8.58E-04

(0.61E-04)

5.83E-04

(8.80E-05)

5.10E-06

(2.46E-05)

2.25E-07

(5.23E-06)

0.48E-07

(2.83E-07)

SO-Roulettewheel

5.76E-07

(0.20E-07)

5.74E-08

(0.20E-07)

7.06E-08

(6.61E-08)

1.43E-08

(0.77E-08)

3.47E-09

(3.48E-08)

SO-Tournament

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

SO-Linearrank

5.58E-03

(3.72E-03)

0.28E-03

(0.28E-03)

4.58E-04

(0.48E-03)

2.00E-04

(7.11E-03)

3.34E-05

(1.04E-04)

SO-Exponentialrank

0.77E+05

(1.54E+05)

0.83E+05

(3.08E+05)

7.35E+04

(5.00E+03)

1.083E+02

(2.32E+04)

6.58E+02

(1.04E+02)

ECTA 2022 - 14th International Conference on Evolutionary Computation Theory and Applications

Table 3: The Average and Standard Deviation Results of Benchmark Functions.

Benchmark Function Selection Technique

Scenario1

Scenario2 Scenario3 Scenario4 Scenario5

Ackley

SO-Global-best

4.54E-03

(3.15E-03)

1.35E-03

(4.23E-03)

0.01E-03

(5.43E-04)

6.55E-04

(1.27E-05)

5.34E-05

(8.21E-05)

SO-Roulettewheel

8.26E-04

(1.60E-03)

4.47E-05

(6.74E-05)

0.03E-05

(1.64E-05)

7.01E-05

(2.44E-06)

2.32E-07

(7.45E-08)

SO-Tournament

3.77E-04

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00

SO-Linearrank

6.1207

(4.1086)

3.4124

(6.56E-02)

2.3824

(1.15E-03)

0.5586

(0.00E-03)

0.1738

(5.55E-3)

SO-Exponentialrank

18.5480

(1.7650)

17.1142

(0.681)

17.1344

(6.18E-02)

16.3562

(2.21E-04)

12.8726

(0.16E-04)

Griewank

SO-Global-best

5.24E-05

(0.27E-06)

1.27E-05

(4.78E-05)

0.54E-06

(6.07E-06)

2.12E-08

(7.52E-07)

3.43E-08

(6.24E-07)

SO-Roulettewheel

1.27E-06

(4.78E-06)

4.70E-07

(6.33E-05)

4.70E-07

(6.33E-05)

5.74E-08

(7.20E-07)

6.15E-09

(4.57E-09)

SO-Tournament

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

0.00E+00

(0.00E+00)

SO-Linearrank

0.0452

(0.0721)

5.05E-03

(5.08E-04)

6.02E-05

(0.75E-04)

0.14E-06

(6.32E-05)

0.01E-07

(1.34E-05)

SO-Exponentialrank

0.23E+03

(0.65E+03)

3.85E+02

(5.54E+02)

2.76E+02

(1.86E+02)

2.03E+02

(1.86E+02)

1.00E+00

(0.01E+00)

Camel-Back

SO-Global-best

-0.9386

(0.086)

-8.99E-01

(2.13E-01)

-9.18E-01

(3.73E-02)

-9.01E-01

(2.83E-02)

-8.78E-01

(2.15E-01)

SO-Roulettewheel

-0.8784

(0.1064)

-8.99E-01

(4.010E-01)

-9.55E-01

(8.763E-01)

-9.85E-01

(8.76E-02)

-9.20E-01

(1.01E-02)

SO-Tournament

-0.8467

(0.00E+00)

-6.45E-01

(2.01E-01)

-9.89E-01

(6.03E-04)

-9.89E-01

(6.03E-04)

-9.87E-01

(9.76E-04)

SO-Linearrank

0.6278

(1.1030)

0.42E+00

(2.52E+00)

0.01E+00

(0.02E+03)

0.01E+00

(0.03E+03)

-9.99E-01

(0.65E-03)

SO-Exponentialrank

6.11E+03

(7.17E+03)

8.23E+03

(0.04E+04)

4.75E+03

(8.63E+02)

4.75E+03

(8.63E+02)

2.76E+02

(8.63E-02)

Shifted Sphere

SO-Global-best

0.63E+04

(5.10E+03)

4701.5781

(1650.0532)

176.1707

(1074.8632)

-440.967

(1.5315)

-448.735

(0.2537)

SO-Roulettewheel

0.74E+04

(5.00E+03)

-445.9375

(0.0171)

-447.5478

(0.6462)

-442.8980

(0.3451)

-449.0970

(1.3456)

SO-Tournament

1.53E+04

(0.02E+04)

-449.9887

(0.0076)

-447.8877

(0.0008)

-448.6540

(0.3256)

-449.9690

(0.0652)

SO-Linearrank

2.16E+05

(7.83E+04)

4701.5681

(1650.0532)

2454.5217

(1065.5807)

642.4270

(6.22E+05)

631.6450

(1.28E+05)

SO-Exponentialrank

0.83E+06

(3.80E+06)

4.70E+05

(2.65E+06)

1.46E+05

(1.07E+06)

8.45E+04

(5.47E+05)

5.21E+04

(5.54E+04)

Shifted Schwefel’s problem 1.2

SO-Global-best

4.30E+04

(0.15E+04)

835487.6950

(208027.653)

-439.9446

(0.041105)

-440.9780

(0.8532)

-449.772

(0.06543)

SO-Roulettewheel

4.03E+04

(0.28E+04)

-48.8595

(255.0173)

-449.7793

(255.01733)

-441.799

(0.21E-02)

-449.9310

(1.58E-03)

SO-Tournament

6.87E+04

(2.24E+04)

-447.0061

(1.8581)

-449.8697

(5.3633)

-448.473

(0.27E-03)

-449.959

(6.08E-03)

SO-Linearrank

0.76E+06

(8.07E+05)

-449.9446

(0.04110)

-216.7574

(0.0411)

-421.4110

(7.30E-03)

-439.654

(5.54E-03)

SO-Exponentialrank

4.60E+04

(0.40E+04)

5487.5840

(8927.6540)

1532.7480

(6475.0145)

3074.0130

(2767.3530)

435.0010

(447.2040)

Shifted Rosenbrock

SO-Global-best

1.01E+12

(1.13E+11)

404.0800

(104.5541)

408.4310

(252.2410)

400.2102

(104.5421)

401.0367

(201.4681)

SO-Roulettewheel

1.03E+12

(1.50E+11)

386.0000

(105.7600)

487.6550

(115513)

468.004

(1.75E+03)

454.6333

(1.00E+03)

SO-Tournament

1.38E+12

(5.43E+11)

475.7140

(110.4870)

490.4560

(122.8890)

420.1020

(321.5670)

378.5490

(202.321)

SO-Linearrank

1.16E+13

(5.31E+12)

478.3000

(308.8300)

595.4780

(108.3210)

570.1234

(281.7543)

543.4710

(218.5000)

SO-Exponentialrank

1.12E+12

(1.50E+11)

1405.7900

(1.76E+8)

1100.5431

(1.45E+7)

980.6789

(1.61E+05)

870.3210

(1.60E+05)

Shifted Rastrigin

SO-Global-best

1322.0560

(25.7743)

-329.9980

(0.2045)

-329.1239

(0.9590)

-302.9870

(0.2169)

-319.8900

(0.4321)

SO-Roulettewheel

133.0134

(20.6035)

-329.9620

(0.184239)

-429.8760

(0.1742)

-320.8876

(1.4310)

-323.7890

(0.2310)

SO-Tournament

132.5543

(16.8017)

-220.0990

(12.0074)

-328.7689

(1.7653)

-329.7890

(3.50E-03)

-429.8890

(2.61E-03)

SO-Linearrank

7.68E+03

(170.0245)

-329.967

(0.0731)

-389.0990

(1.6091)

-289.8860

(0.2361)

-320.9780

(0.6538)

SO-Exponentialrank

922.6654

(46.2210)

1025.7654

(1765.2100)

1743.3210

(1542.9900)

-201.6541

(38.5210)

-281.5678

(3.7651)

New Evolutionary Selection Operators for Snake Optimizer

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ECTA 2022 - 14th International Conference on Evolutionary Computation Theory and Applications