A comparison of modified tree-seed algorithm for high-dimensional numerical functions

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ORIGINAL ARTICLE

A comparison of modified tree–seed algorithm for high-dimensional

numerical functions

Ays¸e Bes¸kirli

1 •

Durmus¸ O

¨ zdemir

1•

Hasan Temurtas¸

1

Received: 19 July 2018 / Accepted: 15 March 2019 / Published online: 26 March 2019 Ó Springer-Verlag London Ltd., part of Springer Nature 2019

Abstract

Optimization methods are used to solve many problems and, under certain constraints, can provide the best possible results.

They are inspired by the behavior of living things in nature and called metaheuristic algorithms. The population-based

tree–seed algorithm (TSA) is an example of these algorithms and is used to solve continuous optimization problems that

have recently emerged. This method, inspired by the relationship between trees and seeds, produces a certain number of

seeds for each tree during each iteration. In this study, during seed formation in the TSA, trees were selected using the

tournament selection method rather than by random means. Efforts were also made to enhance high-dimensional solutions,

utilizing problem dimensions, D, of 20, 50, 100 and 1000 by optimizing the search tendency parameter within the structure

of the algorithm, resulting in a modified TSA (MTSA). Empirical test data, convergence graphs and box plots were

obtained by applying the MTSA to numerical benchmark functions. In addition, the results of the current algorithms in the

literature were compared with the MTSA and the statistical test results were presented. The results from this analysis

demonstrated that the MTSA could achieve superior results to the original TSA.

Keywords Tree–seed algorithm

Metaheuristic algorithms Benchmark functions Optimization

1 Introduction

Researchers have developed numerous heuristic algorithms

to solve difficult engineering problems, many of which are

encountered in real life [

1 –

3 ]. It has been realized that the

modeling of these algorithms enables the solutions to

real-world problems to be reached more quickly and that the

results obtained are more representative of the actual

val-ues. This has led to the emergence of optimization

meth-ods; optimization is a method of determining the most

favorable result for a function with given constraints for

continuous or discontinuous problems [

4 ]. Values used in

optimization problems are generally considered to be

continuous variables [

5 ].

The creatures of nature that inspired science from the

past to present have great roles in the emergence of new

developments [

6 ]. Scientists have developed many

algo-rithms by observing the movements of living things in

nature, and these are called heuristic methods [

7 ,

8 ]. These

consist of six nature-inspired bases namely, techniques

which are biologically based, physics based, swarm based,

social based, music based and chemistry based [

9 ]. Genetic

algorithms (GA) [

10 ], differential evolution (DE)

algo-rithms [

11 ], multi-verse optimizer (MVO) [

12 ], harmony

search (HS) [

13 ] and artificial algae algorithms (AAA) [

14 ]

are biologically based algorithms; gravitational search

algorithms (GSA) [

15 ], water wave optimization (WWO)

[

16 ] and simulated annealing (SA) [

17 ] are physics-based

algorithms; particle swarm optimization (PSO) [

18 ], ant

colony optimization [

19 ], salp swarm algorithm (SSA)

[

20 ], grasshopper optimization algorithm (GOA) [

21 ] and

bat algorithms (BA) [

22 ] are swarm-based algorithms; tabu

search algorithms (TS) [

23 ] are social-based algorithms;

harmony

search

algorithms

[

24 ]

are

music-based

& Ays¸e Bes¸kirli

[email protected] Durmus¸ O¨ zdemir

[email protected] Hasan Temurtas¸

[email protected]

1 _{Department of Computer Engineering, Ku¨tahya Dumlupınar} University, 43100 Kutahya, Turkey

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algorithms; and artificial chemical reaction algorithms [

25 ]

are chemistry-based algorithms [

26 ]. In addition to these

methods, the TSA proposed by Kiran in 2015 [

27 ], as a

new population-based algorithm, has been used to solve

continuous optimization problems.

An examination of documented studies reveals that

algorithms produce more successful results when changes

are introduced to their parameters. Among other

alter-ations, Akay and Karaboga added a modification rate (MR)

parameter to the artificial bee colony (ABC) algorithm [

1 ].

This indicated that the ABC could be effective in solving

optimization problems when alterations were made to its

parameter values. Alavidoost et al. carried out Taguchi

design experiments by performing parameter control and

calibration of the GA [

28 ]. Comparison of the data

obtained with results from existing methods indicated that

these performance enhancements were successful. Beskirli

et al. estimated the energy demand, which is one of the

engineering problems, with the DE algorithm. [

29 ]. Kiran

and Findik added a MR parameter to the ABC algorithm,

producing the ABCMR, and the results obtained were

compared with those of the original ABC [

30 ]. These

experimental findings showed that this change was highly

beneficial in terms of optimum global convergence. Yilmaz

and Kucuksille altered the parameter values of the bat

algorithm and applied it to standard test functions and

constrained real-world problems. The researchers found

that the results obtained were more effective than those

derived from other documented methods [

2 ]. Cano et al.

proposed a distributed algorithm based on the MapReduce

framework and they stated that MapReduce was

suit-able for optimization methods [

31 ]. Cano et al. in their

another research evaluated the benefits of using a scalable

and distributed computing architecture for real-parameter

optimization problems [

32 ]. O

¨ zyo¨n et al. developed a new

method in GSA and applied them to the benchmark

func-tions. When they compared the results with the original

GSA, they suggested that the proposed method achieved

better results [

33 ]. Babalik et al. made discrete parameter

changes to the TSA and applied it to benchmark functions

[

34 ]. The resulting findings were compared with those

derived from PSO, ABC, GA and DE algorithms and

indicated that the suggested method achieved better results.

This study aimed to provide improved solutions to

high-dimensional problems. In the process of seed production in

TSA, trees are selected randomly. In modified TSA

(MTSA), the method used for the selection of trees was

changed and the tournament selection method was used.

Besides, the ST parameter is also optimized. The results

from this method were seen to be superior when compared

with those obtained using the original TSA. Moreover, the

convergence and box plot graphs for the modified and

original algorithms are presented in Figs.

2 ,

3 ,

4 ,

5 ,

6 and

7 of this paper. In addition, ABC, PSO, GSA, SSA, GOA,

MVO and TSA algorithms in the literature were run in the

benchmark functions. When obtained results were

com-pared with MTSA’s results, MTSA was found to be more

successful.

This document is organized as follows: Sect.

2 provides

a description of the TSA; the modifications made to the

TSA are explained in Sect.

3 ; Section

4 details the results

from the experimental procedures; Section

5 contains the

conclusion and future works from this study.

2 Tree–seed algorithm

The TSA proposed by Kiran, as a new population-based

algorithm in 2015, was used to solve continuous

opti-mization problems [

27 ]. The TSA has a specific

relation-ship between trees and seeds, which is attributable to

natural phenomena. In nature, tree seeds disperse on the

soil surface and grow over time to form new trees [

35 ]. If

the surfaces of trees are considered to be the research area,

the positions of trees and seeds represent possible solutions

for optimization problems [

36 ]. Therefore, the importance

of the position of the seeds increases due to the formation

of trees. The search field is defined by two separate

equa-tions. The first of these concerns the production of seeds for

the best position of the tree population and allows the local

search strength of the algorithm to be increased. In the

second equation, two different tree positions are used for

new seed production [

27 ].

S

i;j

¼ T

i;j

þ a

i;j

x B

j

T

r;j

ð1Þ

S

i;j

¼ T

i;j

þ a

i;j

x T

i;j

T

r;j

ð2Þ

where S

i;j

represents the jth dimension of the ith seed of the

tree; T

i;j

is the jth dimension of the ith tree; B

j

is the jth

dimension of the best tree position obtained; T

r;j

is the jth

dimension of the rth tree which is randomly selected from

the population; and a is the scaling factor which is

ran-domly generated in the [- 1,1] interval. One of these two

equations must be selected to determine the position of the

new seed, and this choice is controlled by the ST control

parameter in the [0, 1] interval. The ST value should be

high for a powerful local search and fast convergence;

however, a low value results in slow convergence and a

more powerful global search [

37 ].

By Kiran, the equations for the initialize phase of the

TSA algorithm are given as follows in his study [

35 ].

In the initial phase of the search with TSA, the first tree

positions, which are possible solutions for optimization

problems, are given in Eq.

3 .

T

i;j

¼ L

j;min

þ r

i;j

H

j;max

L

j;min

(3)

Here, L

j;min

is the lower bound of the search space.

H

j;max

is the higher bound of the search space. r

i;j

is a

random number generated for each dimension and position

in the range of [0, 1].

The best solution selected from the population for

minimization is shown in Eq.

4 .

B

¼ min f T

i

!

n

o

i

¼ 1; 2; . . .; N

ð4Þ

Here, N denotes the number of trees in the population.

When new seed locations are created for a tree, the

number of seeds (NS) depends on population size and this

may be more than one. The 10% of the population size is

the minimum number of seeds produced for a tree. The

25% of the population is also, the maximum number of

seeds obtained from a tree. The number of seed production

in TSA is obtained by completely random method. The

flow chart of the TSA is shown in Fig.

1 . ST which is the

control parameter of TSA is seen how it is used in Fig.

1 . If

the number randomly generated in the [0–1] range is less

than the ST value then Eq.

1 is used, otherwise Eq.

2 is

used.

3 Modification of TSA

As the complexity of problems increases, it is often

dif-ficult for an algorithm to obtain the best solution [

38 ],

and it becomes necessary to introduce changes to produce

effective results. Thus, the aim of this study was to

extend both the local and global search space of the

algorithm by altering the selection method for the trees in

the TSA. The tournament selection method was proposed

instead of the use of randomly selected trees to generate

seeds. This procedure operates by selecting the stronger

of randomly selected values [

39 ]. A row vector with a

random permutation of the trees from 1 to N is created

and trees are selected which exhibit the best fitness. The

advantage of tournament selection is that the population

will not be able to select worse trees, and will not,

therefore, participate in the determination of the next tree

position. The trees selected by the tournament selection

method cannot be deleted from the population; in other

words, there is a possibility that a tree may be reselected.

Each dimension of each seed is updated by a randomly

selected tree from the tree or population with the best

fitness value and this process is influenced by the value of

the ST parameter which can assume a value of between 0

and 1. In this study, the value of the ST parameter was

optimized, and the MTSA method was derived from

changes made to the TSA seed-selection method. In

Eqs. 5 and 6 presented below, the modification of the

selection method of the trees to produce higher quality

seeds was visualized. The seeds produced according to

these formulas are selected according to the value of the

ST parameter. If the number randomly selected in the

[0–1] range is less than the ST value, then Eq. 5 is used,

otherwise Eq. 6 is used.

ð5Þ

ð6Þ

where S

i;j

represents the jth dimension of the ith seed of

the tree; T

i;j

is the jth dimension of the ith tree; B

j

is the jth

dimension of the best tree position obtained; T

t;j

is the jth

dimension of the tth tree which is selected from the

pop-ulation with tournament method; and a is the scaling factor

which is randomly generated in the [- 1,1] interval.

In Eqs. 5 and 6, the seed production step was

reformu-lated. At the selection stage, in the original TSA formula,

random selection method (T

r

) used the trees selection

process. Instead in MTSA, a new method was suggested;

the tournament selection method (T

t

) was used in the

selection of trees. In this way, the quality of the seed

production is increased and more quality solutions are

obtained in the search space.

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4 Experimental studies

4.1 Performance analysis and comparisons

on the benchmark functions of TSA

and MTSA

Fifteen benchmark test functions were used to compare the

performance of the MTSA and original TSA methods, and

these are shown in Table

1 .

The initial population values of the TSA and MTSA

were determined as 10, 20, 30, 40 and 50. The stopping

criterion of the algorithm was influenced by the maximum

number of function evaluations (MaxFEs), which was

calculated in accordance with Eq.

7 .

MaxFEs

¼ D 10;000

ð7Þ

To measure the performance of the MTSA in

high-di-mensional problems, D was set to values of 20, 50, 100 and

1000. Moreover, the ST parameter was optimized using

values of 0.1, 0.5 and 0.9. Each method was executed 30

times and the stability of the algorithm was then assessed.

Based on these variables, the TSA and MTSA algorithms

were run and the best, mean and standard deviation values

which were obtained are shown in Tables

2 ,

3 ,

4 ,

5 ,

6 ,

7 ,

8 ,

9 and

10 . The convergence graphs of the function results

and the box plots, representing the stability of the

algo-rithm, are shown in Figs.

2 ,

3 ,

4 ,

5 ,

6 and

7 .

Table

2 shows the results obtained for the various

functions when the ST and D had values of 0.1 and 20,

respectively. These findings show that the TSA gave

superior performance when the population values were 10

Fig. 1 Framework of the TSA

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and 20 for the F11 function and 10 for the F12 and F14

functions. The MTSA achieved better results for other

population values.

Table

3 shows the results obtained for the various

functions when the ST and D had values of 0.1 and 50,

respectively. These findings show that the TSA gave

superior results for the F7, F10, F11, F12, F14 and F15

functions when the population was 10. The MTSA

achieved better performance for other population values.

Table

4 shows the results obtained for the various

functions when the ST and D had values of 0.1 and 100,

respectively. These findings show that the TSA gave better

results for the F7 and F14 functions when the population

was 10. However, the MTSA achieved superior

perfor-mance for other population values.

In summary, Tables

2 ,

3 and

4 contain the results

obtained for the various functions when the ST value of the

algorithm was 0.1 and D was set to 20, 50 and 100. Under

these conditions, the functions F1 to F15 were solved using

both the TSA and MTSA algorithms. These tables show

that the TSA gives better results for some functions when

the population value is 10; however, for the remaining

functions, the results show that the MTSA provides

supe-rior performance to the TSA.

Figure

2 shows the convergence graphs and box plots of

the F1 (sphere) function when the ST and D had values of

0.1 and 20, respectively. Data were collated for a total of

30 executions. Both algorithms exhibited similar

conver-gence behavior; however, more favorable results were

obtained for the MTSA. Examination of the box plots

shows that the TSA exhibited an unstable pattern for

populations of 40 and 50; the best results for this algorithm

were obtained for populations of 10, 20 and 30. The MTSA

showed slight instability for populations of 20 and 50 but

Table 1 Benchmark functions

Fn. Name C Search range Function

F1 Sphere US _½_{100; 100}D f1ð Þ ¼x P N i¼1 x2 i F2 Elliptic UN _½_{100; 100}D f2ð Þ ¼x P n i¼1 106 ði1Þ= n1ð Þ x2 i F3 SumSquares US _½_{10; 10}D f3ð Þ ¼x Pn i¼1 ix2 i F4 SumPower MS _½_{10; 10}D f4ð Þ ¼x Pn i¼1 xi j jðiþ1Þ F5 Schwefel2.22 UN _½_{10; 10}D f5ð Þ ¼x P n i¼1 xi j j þQn i¼1 xi j j F6 Schwefel2.21 UN _½_{100; 100}D _f 6ð Þ ¼ maxx ifj j;xi 1 i ng F7 Alpine MS _½_{10; 10}D f7ð Þ ¼x P n i¼1 xisin xð Þ þ 0:1xi i j j F8 Quartic US _½_{1:28; 1:28}D f8ð Þ ¼x P n i¼1 ix4 i F9 QuarticWN US _½_{1:28; 1:28}D f9ð Þ ¼x Pn i¼1 ix4 i þ random 0; 1½ Þ F10 Rosenbrock UN _½_{10; 10}D f10ð Þ ¼x P N1 i¼1 ½100 xiþ1 x2i 2 þ xi 1Þ2 i F11 Rastrigin MS _½_{5:12; 5:12}D f11ð Þ ¼x P N i¼1 x2 i 10 cos 2pxð iÞ þ 10 F12 Non-Continuous Rastrigin MS _½_{5:12; 5:12}D f12ð Þ ¼x P n i¼1 y2 i 10 cos 2pyð iÞ þ 10 yi¼ xi; xj j\i 1 2 round 2xð iÞ 2 ; j j xi 1 2 8 > < > : F13 Griewank MN _½_{600; 600}D f13ð Þ ¼x P N i¼1 x2 i 4000 QN i¼1 cos xiffi i p þ 1 F14 Schwefel2.26 MS _½_{500; 500}D f14ð Þ ¼ 418:98 n x P n i¼1 xisin ffiffiffiffiffiffi xi j j p F15 Ackley MN _½_{32; 32}D y f15ð Þ ¼ 20 þ e 20 exp 0:2x ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N PN i¼1 x2 i s ! exp 1 N PN i¼1 cos 2pxð iÞ

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Table 2 Analysis results of functions for ST = 0.1 and D = 20

TSA MTSA

Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 F1

Best 1.74E-188 2.47E-84 1.80E-50 3.13E-35 4.99E-26 5.95E2241 3.22E2128 1.18E283 3.39E260 1.21E244 Mean 1.18E-183 8.73E-82 3.45E-49 1.37E-34 3.12E-25 1.72E-233 2.30E-125 1.58E-81 2.06E-58 4.21E-43 Std. 0.00E?00 1.77E-81 6.83E-49 1.02E-34 2.19E-25 0.00E?00 3.59E-125 2.36E-81 3.17E-58 4.33E-43 F2

Best 6.50E-240 1.87E-106 2.53E-65 1.57E-48 9.43E-36 8.36E2294 1.24E2176 1.76E2124 1.72E298 3.72E279 Mean 1.55E-217 5.75E-99 3.33E-60 3.09E-42 4.01E-33 1.78E-249 3.64E-16 2.45E?00 1.85E-08 7.67E-15 Std. 0.00E?00 1.84E-98 1.50E-59 1.46E-41 1.09E-32 0.00E?00 1.96E-15 1.32E?01 9.96E-08 3.12E-14 F5

Best 7.73E-130 8.02E-57 1.93E-34 2.83E-24 6.79E-18 2.25E2172 3.82E287 2.89E256 5.78E240 1.13E229 Mean 7.39E-127 4.45E-56 6.04E-34 8.63E-24 1.62E-17 2.65E-168 1.32E-85 6.59E-55 3.09E-39 5.27E-29 Std. 2.24E-126 3.98E-56 3.94E-34 4.68E-24 5.40E-18 0.00E?00 3.08E-85 1.13E-54 2.27E-39 3.59E-29 F6

Best 8.95E-11 6.27E-06 9.23E-04 2.46E-02 9.33E-02 1.21E209 2.70E208 1.41E206 3.45E206 1.06E205 Mean 1.04E-08 5.62E-05 3.63E-03 4.73E-02 1.79E-01 2.41E-06 2.89E-06 3.83E-01 2.30E-01 1.06E-01 Std. 1.87E-08 3.94E-05 1.73E-03 1.71E-02 5.57E-02 2.41E-06 7.80E-06 1.03E?00 4.16E-01 2.11E-01 F7

Best 9.55E-276 1.30E-161 5.21E-21 1.24E-07 2.98E-04 0.00E100 9.55E2259 2.41E2152 4.06E278 4.82E235 Mean 4.24E-08 3.12E-16 5.51E-05 1.75E-03 7.22E-03 3.45E?06 2.04E-02 2.37E-02 7.59E-03 7.59E-03 Std. 2.28E-07 1.60E-15 2.47E-04 2.46E-03 2.90E-03 1.86E?07 2.53E-02 4.04E-02 1.95E-02 2.44E-02 F8

Best 7.98E-04 1.07E-03 1.99E-03 2.34E-03 2.57E-03 7.33E204 8.52E204 1.12E203 1.27E203 1.36E203 Mean 2.67E-03 3.25E-03 4.18E-03 5.55E-03 6.11E-03 2.31E-03 2.34E-03 2.66E-03 2.58E-03 2.87E-03 Std. 1.07E-03 1.18E-03 1.35E-03 1.44E-03 1.88E-03 1.27E-03 9.62E-04 1.19E-03 6.78E-04 6.80E-04 F10

Best 4.06E-03 6.20E?00 1.09E?01 1.32E?01 1.39E?01 1.58E203 5.80E203 1.07E101 8.78E100 9.64E203 Mean 7.81E?00 1.07E?01 1.29E?01 1.38E?01 1.45E?01 6.70E?00 1.12E?01 1.68E?01 2.00E?01 1.72E?01 Std. 1.21E?01 1.65E?00 1.20E?00 2.47E-01 2.23E-01 4.40E?00 1.12E?01 9.85E?00 1.40E?01 1.02E?01 F11

Best 2.98E100 3.02E100 1.72E?01 2.59E?01 4.37E?01 3.98E?00 3.98E?00 2.98E100 9.95E201 9.95E201 Mean 1.00E?01 1.51E?01 4.38E?01 6.43E?01 7.71E?01 1.26E?01 8.13E?00 6.33E?00 4.51E?00 4.81E?00 Std. 4.08E?00 1.41E?01 1.50E?01 1.61E?01 1.26E?01 4.29E?00 2.72E?00 2.29E?00 2.31E?00 1.99E?00 F12

Best 6.00E100 9.01E?00 3.07E?01 4.24E?01 4.44E?01 1.00E?01 8.00E100 8.00E100 7.40E100 6.04E100 Mean 1.55E?01 3.09E?01 5.61E?01 6.66E?01 6.96E?01 1.54E?01 1.20E?01 1.13E?01 1.31E?01 1.26E?01 Std. 5.02E?00 1.43E?01 1.33E?01 9.33E?00 1.09E?01 4.34E?00 3.68E?00 3.19E?00 4.86E?00 3.78E?00

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Table 2 (continued)

TSA MTSA

Best 0.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E?00 0.00E100 0.00E100 0.00E100 0.00E100 0.00E100 Mean 4.27E-03 2.56E-04 4.53E-06 3.71E-04 2.07E-03 8.77E-03 1.48E-03 4.20E-03 1.48E-03 6.25E-04 Std. 6.39E-03 1.33E-03 1.72E-05 1.47E-03 7.42E-03 1.23E-02 3.40E-03 5.62E-03 3.40E-03 2.34E-03 F14

Best 1.18E102 1.18E?02 2.37E?02 1.49E?03 1.99E?03 2.37E?02 1.18E102 1.18E102 1.18E102 1.18E102 Mean 6.74E?02 8.22E?02 1.84E?03 2.54E?03 3.23E?03 9.11E?02 6.45E?02 5.20E?02 4.07E?02 4.30E?02 Std. 3.17E?02 5.95E?02 9.83E?02 4.84E?02 4.40E?02 3.00E?02 3.01E?02 1.80E?02 2.00E?02 2.15E?02 F15

Best 2.22E-15 2.22E-15 2.22E-15 2.22E-15 7.55E-14 2.22E215 2.22E215 2.22E215 2.22E215 2.22E215 Mean 2.44E-15 2.29E-15 2.22E-15 2.44E-15 2.51E-13 3.85E-02 2.22E-15 2.22E-15 2.22E-15 2.22E-15 Std. 6.66E-16 3.99E-16 0.00E?00 6.66E-16 9.54E-14 2.07E-01 0.00E?00 0.00E?00 0.00E?00 0.00E?00 Best values are highlighted in bold

TSA MTSA

Best 1.15E-42 4.56E-08 6.76E?01 3.94E?03 3.80E?04 3.22E281 4.65E235 8.02E226 7.62E218 6.62E215 Mean 3.56E-30 1.07E-01 2.77E?03 6.78E?04 4.73E?06 1.60E-51 5.62E-26 1.70E-15 1.57E-11 2.23E-08 Std. 1.92E-29 3.42E-01 4.81E?03 1.32E?05 8.25E?06 8.60E-51 2.32E-25 6.38E-15 4.35E-11 6.26E-08 F5

Best 1.91E?01 3.75E?01 5.40E?01 6.10E?01 5.82E?01 1.87E101 1.83E101 2.09E101 2.57E101 2.32E101 Mean 3.43E?01 5.41E?01 6.68E?01 7.13E?01 7.32E?01 3.21E?01 3.05E?01 3.29E?01 3.61E?01 3.76E?01 Std. 1.01E?01 8.64E?00 6.30E?00 4.32E?00 4.59E?00 8.50E?00 6.48E?00 4.53E?00 6.74E?00 7.98E?00 F7

Best 1.28E215 4.40E-12 3.93E-02 6.67E-02 9.24E?00 5.16E-15 8.88E216 4.44E216 1.75E224 3.49E215 Mean 7.05E-05 2.21E-04 3.97E?00 1.06E?01 1.95E?01 1.11E-02 7.59E-15 5.05E-15 1.24E-13 2.40E-12 Std. 3.77E-04 8.53E-04 6.01E?00 8.04E?00 5.84E?00 6.00E-02 1.32E-14 4.39E-15 6.52E-13 5.50E-12

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revealed a more stable pattern for populations of 10, 20 and

40, when compared to the TSA.

Figure

3 shows the convergence graphs and box plots of

the F7 (alpine) function when the ST and D had values of

0.1 and 20, respectively. Examination of the convergence

graph for the TSA shows that the values recorded for

populations of 30, 40 and 50 were close to each other, but

better convergence occurred for populations of 10 and 20,

thereby indicating superior performance. The MTSA

obtained better results than the TSA for all populations and

these values are not close to each other. The box plots show

that the MTSA exhibited greater stability for all

popula-tions, with the exception of 20; therefore, it obtained better

results on an overall basis.

Table

5 shows the results obtained when the ST and

D were set to 0.5 and 20, respectively. Comparison of the

findings for the two algorithms shows that with the

exception of the values obtained for the F6 function and

population of 10, the MTSA produced the best solution for

all functions.

When the ST is 0.5 and D is 50, the TSA returns better

values for the F6 and F7 functions and population of 10.

For all other population values, the MTSA obtained better

solutions for all functions (Table

6 ).

Table

7 shows the results obtained when the ST and

D were set to values of 0.5 and 100, respectively. The TSA

achieved better performance using the F7 function with a

Table 3 (continued)

TSA MTSA

Best 7.81E202 3.48E?01 4.27E?01 4.79E?01 8.98E?01 4.73E-01 9.61E202 2.69E101 3.73E101 3.92E101 Mean 4.26E?01 4.90E?01 4.96E?01 7.18E?01 2.56E?02 4.08E?01 5.30E?01 7.23E?01 5.70E?01 5.89E?01 Std. 2.60E?01 2.15E?01 1.50E?01 3.63E?01 8.64E?01 2.88E?01 2.55E?01 3.10E?01 2.31E?01 2.31E?01 F11

Best 0.00E?00 0.00E?00 1.98E-13 2.22E-08 2.12E-05 0.00E100 0.00E100 0.00E100 0.00E100 0.00E100 Mean 7.39E-04 0.00E?00 1.54E-12 1.25E-07 5.99E-05 2.14E-03 9.86E-04 4.93E-04 9.04E-04 2.47E-04 Std. 2.78E-03 0.00E?00 2.12E-12 1.30E-07 3.85E-05 5.50E-03 2.51E-03 1.84E-03 2.81E-03 1.33E-03 F14

Best 4.44E215 1.35E-13 8.85E-08 2.91E-05 7.24E-04 6.66E-15 4.44E215 4.44E215 6.66E215 1.05E212 Mean 6.88E-15 4.78E-13 1.72E-07 4.82E-05 1.20E-03 8.44E-15 6.29E-15 6.66E-15 9.77E-15 2.33E-12 Std. 1.20E-15 3.00E-13 4.79E-08 1.24E-05 2.67E-04 4.15E-15 1.01E-15 8.11E-16 1.23E-15 1.06E-12 Best values are highlighted in bold

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TSA MTSA

Best 1.13E-19 3.65E?00 2.66E?03 3.15E?04 9.20E?04 8.81E258 5.10E220 6.32E212 1.10E206 7.73E205 Mean 1.70E-16 1.88E?01 5.93E?03 4.52E?04 1.20E?05 1.41E-54 2.99E-17 2.14E-10 1.62E-05 1.49E-03 Std. 6.41E-16 9.43E?00 1.95E?03 7.85E?03 1.53E?04 6.01E-54 9.14E-17 2.82E-10 2.45E-05 1.52E-03 F3

Best 1.16E-22 1.19E-04 1.95E-01 2.82E?00 8.81E?00 1.62E256 1.98E221 1.44E214 6.89E210 1.33E207 Mean 3.99E-21 3.68E-04 3.45E-01 4.18E?00 1.54E?01 2.53E-54 2.15E-20 1.74E-13 2.65E-09 2.47E-07 Std. 5.52E-21 1.68E-04 8.95E-02 1.28E?00 4.20E?00 4.66E-54 2.47E-20 1.72E-13 1.35E-09 4.21E-07 F6

Best 9.54E?01 9.41E?01 9.13E?01 9.28E?01 9.36E?01 9.16E101 9.28E101 8.73E101 9.26E101 9.28E101 Mean 9.74E?01 9.62E?01 9.57E?01 9.60E?01 9.58E?01 9.69E?01 9.63E?01 9.58E?01 9.59E?01 9.57E?01 Std. 8.15E-01 1.04E?00 1.38E?00 1.08E?00 9.27E-01 1.53E?00 1.27E?00 1.85E?00 1.25E?00 1.19E?00 F7

Best 1.93E214 1.27E-03 8.23E-01 1.52E?01 4.80E?01 4.68E-14 2.13E214 2.70E213 1.91E208 2.11E206 Mean 3.03E-12 2.92E-02 9.01E?00 5.06E?01 8.01E?01 4.84E-13 6.52E-14 9.22E-12 5.40E-07 3.58E-05 Std. 1.25E-11 2.88E-02 1.23E?01 1.94E?01 1.69E?01 6.51E-13 2.68E-14 1.30E-11 8.41E-07 4.80E-05 F8

Best 2.09E?02 2.23E?02 3.00E?02 5.36E?02 6.47E?02 1.94E102 1.54E102 1.71E102 1.58E102 1.67E102 Mean 2.83E?02 3.10E?02 6.03E?02 8.82E?02 9.96E?02 2.96E?02 2.32E?02 2.13E?02 2.18E?02 2.18E?02 Std. 4.57E?01 8.03E?01 2.15E?02 1.84E?02 1.28E?02 4.54E?01 3.46E?01 3.02E?01 3.03E?01 2.47E?01

(10)

population of 10, while the MTSA obtained more

suc-cessful solutions in other cases.

Figure

4 shows the convergence graphs and box plots of

the TSA and MTSA for the F1 (sphere) function when the

ST and D had values of 0.5 and 20, respectively. For a

population of 10, it can be seen that the MTSA presented

faster convergence in comparison with other population

values and obtained better results than the other methods.

The box plot graphs show that the TSA and MTSA

pro-vided comparable stability.

Figure

5 shows the convergence graphs and box plots of

the TSA and MTSA for the F7 (alpine) function when the

ST and D had values of 0.5 and 20, respectively. It can be

seen that the convergence graphs of the TSA were close to

each other for populations of 20, 30, 40 and 50, whereas

faster convergence behavior was observed for a population

of 10. The MTSA exhibited faster convergence for all

population values, with the exception of 40 and 50.

Examination of the box plots for both algorithms shows

that the MTSA exhibited greater stability.

Table

8 shows the results obtained when the ST and

D were set to values of 0.9 and 20, respectively, for both

algorithms. Examination of the results, for all population

values, indicates that the MTSA produced more favorable

results than the TSA.

Table

9 contains the results obtained for both algorithms

when the ST and D were set to values of 0.9 and 50,

respectively. These findings indicate that the MTSA

obtained superior results in comparison with the TSA.

Table

10 shows the results obtained when the ST and

D were set to values of 0.9 and 100, respectively. These

findings indicate that the TSA performed successfully for

the F4 function and population of 10; however, the MTSA

obtained superior results under other conditions.

Figure

6 shows the convergence graphs and box plots of

the TSA and MTSA for the F1 (sphere) function when the

ST and D had values of 0.9 and 20, respectively. It can be

seen that the convergence graph of the TSA started to

resemble that of the MTSA due to the increased ST value;

overall, however, it was observed that the MTSA obtained

better results. Although the box plots were comparable for

both algorithms, the MTSA produced a more stable graph.

Figure

7 shows the convergence graphs and box plots of

the TSA and MTSA for the F7 (alpine) function when the

ST and D had values of 0.9 and 20, respectively. It can be

seen that the MTSA produced rapid convergence to obtain

its best result in 100,000 FEs, while the TSA reached its

optimum solution in 170,000 FEs and therefore, fell

behind. Although the box plots were similar for both

algorithms, the median value for the MTSA indicated

greater stability.

In summary, the MTSA and original TSA methods were

applied to 15 different benchmark functions. During the

subsequent analysis, the performance of both algorithms

was examined for different values of the population

num-ber, ST parameter and D, and the results are shown in

Tables

2 ,

3 ,

4 ,

5 ,

6 ,

7 ,

8 ,

9 and

10 . The general findings

from this evaluation show that the MTSA produced

supe-rior results to those of the original TSA when the ST value

is increased. At a constant ST parameter of 0.9, comparison

of both algorithms when the population value for the TSA

was 10 indicates that the MTSA obtained more successful

results for all population values, except when using the F4

and F10 functions. As a result of ST parameter

Table 4 (continued)

TSA MTSA

Best 0.00E?00 1.61E-02 1.92E?00 8.84E?00 3.04E?01 0.00E100 0.00E100 1.01E213 7.64E209 3.51E207 Mean 2.47E-04 2.78E-01 2.95E?00 1.71E?01 4.56E?01 2.15E-16 1.07E-03 2.13E-03 2.47E-04 1.17E-03 Std. 1.33E-03 1.87E-01 7.55E-01 4.68E?00 7.51E?00 2.61E-16 2.75E-03 4.70E-03 1.33E-03 2.97E-03 F14

Best 2.29E-12 3.75E-02 3.10E?00 6.45E?00 8.88E?00 1.33E214 4.22E212 6.35E208 5.85E206 1.22E204 Mean 1.16E-10 1.02E-01 4.51E?00 7.77E?00 1.04E?01 3.85E-02 4.95E-11 1.86E-07 3.72E-05 3.88E-04 Std. 1.56E-10 4.73E-02 5.34E-01 5.94E-01 5.38E-01 2.07E-01 7.11E-11 1.12E-07 2.18E-05 1.50E-04 Best values are highlighted in bold

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Table 5 Analysis results of functions for ST = 0.5 and D =2 0 TSA MTSA Pop = 1 0 Pop = 2 0 Pop = 3 0 Pop = 4 0 Pop = 5 0 Pop = 1 0 Pop = 2 0 Pop = 3 0 Pop = 4 0 Pop = 5 0 F1 Best 1.40E -149 9.04E -64 6.93E -38 2.18E -26 3.80E -19 2.22E 2 205 1.99E 2 100 1.85E 2 64 2.08E 2 45 1.90E 2 33 Mean 2.19E -144 4.68E -62 5.56E -37 1.17E -25 1.60E -18 1.26E -198 4.11E -98 9.31E -63 1.56E -44 6.38E -33 Std. 1.13E -143 7.29E -62 8.93E -37 8.00E -26 1.23E -18 0.00E ? 00 6.04E -98 2.39E -62 1.57E -44 4.67E -33 F2 Best 9.22E -148 1.59E -61 2.04E -35 6.19E -24 1.98E -16 7.90E 2 202 1.77E 2 98 6.18E 2 62 6.05E 2 43 3.60E 2 31 Mean 5.30E -143 1.44E -59 1.17E -34 4.35E -23 4.70E -16 6.24E -197 7.61E -95 2.52E -60 1.68E -41 5.88E -30 Std. 1.19E -142 2.14E -59 9.59E -35 3.38E -23 1.99E -16 0.00E ? 00 1.34E -94 3.78E -60 2.16E -41 6.73E -30 F3 Best 2.18E -150 3.68E -65 1.51E -39 1.18E -27 1.09E -20 2.51E 2 207 5.97E 2 102 4.76E 2 66 5.98E 2 47 6.81E 2 35 Mean 4.10E -146 3.50E -63 3.93E -38 7.92E -27 8.39E -20 4.44E -200 1.47E -98 5.65E -64 1.18E -45 5.98E -34 Std. 1.59E -145 6.84E -63 5.05E -38 5.35E -27 4.19E -20 0.00E ? 00 4.75E -98 8.28E -64 1.10E -45 4.51E -34 F4 Best 5.62E -222 6.77E -90 4.19E -54 9.23E -38 1.80E -28 4.19E 2 312 3.18E 2 167 1.41E 2 115 2.70E 2 85 2.69E 2 66 Mean 4.88E -203 1.48E -80 1.38E -49 2.15E -34 1.84E -25 3.20E -238 4.10E -144 2.27E -32 2.34E -28 1.36E -22 Std. 0.00E ? 00 7.97E -80 4.43E -49 9.26E -34 5.63E -25 0.00E ? 00 2.21E -143 1.22E -31 1.26E -27 7.30E -22 F5 Best 1.00E -91 2.22E -38 9.58E -23 9.37E -16 9.40E -12 9.40E 2 130 1.74E 2 62 2.69E 2 40 1.53E 2 28 3.74E 2 21 Mean 2.50E -89 1.75E -37 2.88E -22 2.03E -15 2.43E -11 8.87E -127 2.15E -61 1.76E -39 6.68E -28 8.06E -21 Std. 5.01E -89 1.53E -37 1.96E -22 9.97E -16 9.28E -12 2.47E -126 4.08E -61 1.42E -39 4.37E -28 4.30E -21 F6 Best 3.26E 2 12 1.84E -05 3.40E -03 4.62E -02 2.81E -01 5.08E -12 2.14E 2 10 5.75E 2 08 1.09E 2 05 1.12E 2 04 Mean 1.52E -10 9.09E -05 1.15E -02 1.04E -01 3.94E -01 1.05E -07 1.72E -07 1.43E -01 1.90E -03 1.85E -03 Std. 2.55E -10 6.49E -05 4.64E -03 3.13E -02 7.46E -02 3.69E -07 4.27E -07 6.73E -01 9.99E -03 7.27E -03 F7 Best 3.41E -130 5.35E -14 1.57E -03 9.35E -03 1.65E -02 1.76E 2 285 4.46E 2 147 6.69E 2 55 7.99E 2 23 2.12E 2 14 Mean 1.41E -06 5.70E -04 8.75E -03 1.63E -01 7.97E -01 7.26E -08 8.66E -08 2.17E -10 3.82E -08 1.27E -07 Std. 4.73E -06 9.99E -04 5.22E -03 3.91E -01 1.01E ? 00 3.18E -07 4.66E -07 1.17E -09 2.06E -07 3.86E -07 F8 Best 3.18E -193 3.69E -85 3.01E -53 1.25E -38 1.75E -29 2.47E 2 259 8.80E 2 142 1.85E 2 95 1.41E 2 69 1.84E 2 53 Mean 1.04E -185 1.53E -81 2.67E -51 7.46E -37 3.22E -28 9.31E -242 9.98E -134 1.32E -90 4.00E -25 1.76E -51 Std. 0.00E ? 00 4.50E -81 5.92E -51 1.51E -36 3.89E -28 0.00E ? 00 3.06E -133 3.01E -90 2.15E -24 3.86E -51

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Table 5 (continued) TSA MTSA Pop = 1 0 Pop = 2 0 Pop = 3 0 Pop = 4 0 Pop = 5 0 Pop = 1 0 Pop = 2 0 Pop = 3 0 Pop = 4 0 Pop = 5 0 F9 Best 1.41E -03 2.08E -03 2.92E -03 2.60E -03 4.32E -03 7.15E 2 04 1.02E 2 03 1.84E 2 03 1.13E 2 03 2.38E 2 03 Mean 2.64E -03 4.58E -03 6.44E -03 7.97E -03 9.80E -03 2.19E -03 2.82E -03 3.60E -03 4.18E -03 4.89E -03 Std. 6.53E -04 1.54E -03 1.81E -03 2.56E -03 3.06E -03 7.81E -04 9.67E -04 7.29E -04 1.50E -03 1.50E -03 F10 Best 2.40E ? 00 1.23E ? 01 1.29E ? 01 1.46E ? 01 1.43E ? 01 1.66E 2 01 4.20E 1 00 1.09E 1 01 1.06E 1 01 1.14E 1 01 Mean 1.19E ? 01 1.36E ? 01 1.45E ? 01 1.51E ? 01 1.54E ? 01 1.24E ? 01 1.57E ? 01 1.35E ? 01 1.38E ? 01 1.42E ? 01 Std. 1.07E ? 01 5.67E -01 4.62E -01 2.69E -01 2.84E -01 1.67E ? 01 1.46E ? 01 1.23E ? 00 9.66E -01 8.30E -01 F11 Best 9.95E -01 3.55E ? 00 2.33E ? 01 4.70E ? 01 6.24E ? 01 1.78E 2 15 0.00E 1 00 0.00E 1 00 0.00E 1 00 9.54E 2 12 Mean 4.89E ? 00 2.62E ? 01 5.58E ? 01 7.05E ? 01 8.59E ? 01 5.07E ? 00 9.29E -01 9.65E -01 2.86E ? 00 7.59E ? 00 Std. 4.57E ? 00 1.27E ? 01 1.33E ? 01 1.07E ? 01 8.57E ? 00 2.84E ? 00 9.94E -01 2.52E ? 00 4.80E ? 00 9.26E ? 00 F12 Best 5.00E ? 00 1.41E ? 01 2.79E ? 01 4.20E ? 01 4.67E ? 01 2.00E 1 00 2.00E 1 00 3.00E 1 00 6.31E 1 00 9.00E 1 00 Mean 9.61E ? 00 2.96E ? 01 4.88E ? 01 5.65E ? 01 6.51E ? 01 7.77E ? 00 6.59E ? 00 1.13E ? 01 1.37E ? 01 1.71E ? 01 Std. 3.31E ? 00 7.61E ? 00 9.87E ? 00 8.45E ? 00 9.57E ? 00 3.07E ? 00 3.61E ? 00 5.04E ? 00 3.38E ? 00 4.58E ? 00 F13 Best 0.00E ? 00 0.00E ? 00 4.63E -14 1.52E -06 5.11E -06 0.00E 1 00 0.00E 1 00 0.00E 1 00 0.00E 1 00 0.00E 1 00 Mean 9.28E -04 5.04E -03 1.96E -02 7.67E -02 1.30E -01 8.86E -04 1.92E -09 7.75E -12 0.00E ? 00 7.77E -17 Std. 2.80E -03 1.40E -02 2.57E -02 7.40E -02 9.04E -02 2.50E -03 1.04E -08 4.17E -11 0.00E ? 00 4.19E -16 F14 Best 2.55E -04 2.55E -04 1.80E ? 03 1.82E ? 03 2.61E ? 03 2.55E 2 04 2.55E 2 04 2.55E 2 04 2.55E 2 04 2.55E 2 04 Mean 2.01E ? 02 1.45E ? 03 2.58E ? 03 2.99E ? 03 3.25E ? 03 2.61E ? 02 2.37E ? 01 1.05E ? 01 1.62E ? 02 3.35E ? 02 Std. 1.30E ? 02 7.19E ? 02 3.96E ? 02 3.42E ? 02 3.06E ? 02 1.48E ? 02 4.74E ? 01 3.20E ? 01 3.18E ? 02 5.26E ? 02 F15 Best 2.22E -15 2.22E -15 2.22E -15 2.02E -13 6.10E -10 2.22E 2 15 2.22E 2 15 2.22E 2 15 2.22E 2 15 2.22E 2 15 Mean 2.29E -15 2.22E -15 2.74E -15 1.29E -12 2.52E -09 2.59E -15 2.22E -15 2.22E -15 2.29E -15 3.26E -15 Std. 3.99E -16 0.00E ? 00 9.39E -16 2.12E -12 2.36E -09 1.01E -15 0.00E ? 00 0.00E ? 00 3.99E -16 1.11E -15 Best values are highlighted in bold

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TSA MTSA

Best 3.28E-56 1.72E-14 1.40E-05 3.00E-02 3.23E?00 1.52E2110 1.76E246 8.31E230 4.95E220 5.61E215 Mean 7.82E-53 1.30E-13 3.66E-05 8.18E-02 6.11E?00 2.62E-106 8.69E-45 2.49E-28 5.17E-19 4.87E-14 Std. 2.31E-52 1.85E-13 1.88E-05 2.42E-02 1.83E?00 9.49E-106 1.35E-44 4.82E-28 6.27E-19 3.70E-14 F3

Best 2.51E-53 7.15E-12 4.05E-01 1.29E?03 1.56E?05 9.99E286 1.83E247 4.99E230 2.55E224 6.48E221 Mean 9.51E-41 1.72E-06 4.12E?01 4.35E?04 7.27E?06 5.92E?01 4.45E-01 4.58E?02 4.56E-03 7.24E-04 Std. 3.77E-40 6.76E-06 6.33E?01 7.24E?04 1.05E?07 3.19E?02 2.39E?00 2.46E?03 1.58E-02 3.75E-03 F5

Best 4.44E216 1.55E-04 1.93E-01 9.65E?00 2.12E?01 1.05E-15 2.22E216 2.47E216 7.16E212 3.90E208 Mean 7.09E-06 1.62E-01 1.21E?01 2.25E?01 3.16E?01 3.17E-03 9.98E-08 4.48E-07 2.98E-05 1.34E-03 Std. 2.67E-05 4.78E-01 7.36E?00 5.67E?00 4.55E?00 1.48E-02 5.33E-07 2.41E-06 1.36E-04 3.41E-03 F8

Best 3.80E?01 7.80E?01 1.11E?02 2.21E?02 2.20E?02 3.50E101 2.70E101 4.21E101 4.26E101 5.04E101 Mean 6.74E?01 1.82E?02 2.49E?02 3.17E?02 3.58E?02 4.99E?01 6.30E?01 6.71E?01 9.86E?01 1.05E?02

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optimization, it was therefore found that a ST value of 0.9

allowed the best solutions to be obtained.

In this study, trees were selected using tournament

selection rather than the random selection method utilized

by the original TSA. The former technique provides a

recognized means of obtaining better quality and more

powerful results in high-dimensional problems. The

effi-ciency of the tournament selection method is clearly seen

when the population and D values are increased, and this is

apparent when examining the tables resulting from this

study. In analyses, the performance of the MTSA increases

in comparison with that of the TSA when D of the

opti-mization problem becomes larger.

The results of TSA and MTSA algorithms are obtained

in Table

11 , when the ST values 0.1, 0.5 and 0.9 and the

problem dimension value 1000D. When Table

11 is

eval-uated in terms of MTSA, there were few results that did not

achieve a better solution by comparison of TSA. These are:

in F1 function, when population value was 10, in F10

function when population value was 10 and ST values were

0.1 and 0.5, population value 20 and ST value was 0.5,

population value 30 and ST value was 0.1. Also, in F13,

function population value was 10 and ST value was 0.1,

another given bad result. In all other cases, MTSA has

found a better solution than TSA.

4.2 Performance analysis and comparisons

with statistical test results

Statistical result, which is the focus of our current work, is

the Wilcoxon rank sum test [

40 ]. Wilcoxon rank sum test is

a rank-based test, where the p value is evaluated by

cal-culating the rank for the two samples, which is compared to

the rank of all possible permutations of the samples.

In this study, it is aimed to evaluate the importance of

MTSA results by performing a nonparametric statistical

test called Wilcoxon rank sum test. In addition, the results

are presented with p value to prove that MTSA provides a

significant improvement compared to other algorithms

[

40 ,

41 ]. P value shows the possible amount of error when

we decide that there is a statistically significant difference.

Therefore, if the p value in a test result is less than 0.05, it

means that there is a significant difference in the

compar-ison result. As the p value becomes smaller, the evidence of

statistically significant difference increases. There is a

statistically significant difference in p value between 0.01

and 0.05. The p value has a high level of difference in the

range of 0.001 to 0.01. If the p value is smaller than 0.001,

there is a very high statistically significant difference [

42 ].

Table

12 shows the Wilcoxon test results of the TSA

and MTSA algorithms for the values of 20, 50, 100 and

1000 dimensions of all functions by taking the population

number 50 and ST values 0.1, 0.5 and 0.9.

In the statistical analysis, there is a significant difference

if the p value is less than 0.05, but if the p value is above

0.05, there is no significant difference between the values

[

33 ]. According to the results of Wilcoxon test analysis in

Table

12 , there was a significant difference between

MTSA and TSA.

Table

13 shows the comparison results of ABC, PSO,

GSA, SSA, GOA, MVO and TSA with MTSA. For each

function, first row indicates the best result of the 25 run, the

second row indicates the mean result of the 25 run, the

Table 6 (continued)

TSA MTSA

Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 Std. 2.25E?01 5.71E?01 5.37E?01 4.50E?01 3.48E?01 1.03E?01 2.28E?01 2.42E?01 4.60E?01 4.56E?01 F13

Best 0.00E?00 8.88E-16 1.11E-06 1.91E-03 1.95E-01 0.00E100 0.00E100 0.00E100 0.00E100 0.00E100 Mean 2.47E-04 1.58E-10 1.27E-04 3.12E-02 3.59E-01 3.29E-04 3.70E-18 0.00E?00 0.00E?00 5.73E-14 Std. 1.33E-03 5.96E-10 5.28E-04 2.97E-02 1.21E-01 1.77E-03 1.99E-17 0.00E?00 0.00E?00 2.52E-13 F14

Best 6.66E-15 8.87E-09 2.31E-04 1.49E-02 2.64E-01 6.66E215 6.66E215 8.88E215 5.28E212 1.49E209 Mean 7.18E-15 1.96E-05 6.41E-04 5.29E-02 1.00E?00 7.77E-15 7.33E-15 9.99E-15 1.31E-11 3.43E-09 Std. 9.39E-16 1.96E-05 4.57E-04 5.54E-02 6.97E-01 1.49E-15 1.02E-15 1.11E-15 5.96E-12 1.51E-09 Best values are highlighted in bold

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TSA MTSA

Best 2.07E-21 1.56E?00 5.98E?03 8.19E?04 2.68E?05 1.04E258 8.27E220 9.79E212 2.42E206 6.66E204 Mean 1.28E-19 8.02E?00 1.16E?04 1.38E?05 4.46E?05 1.46E-55 2.67E-18 1.47E-10 1.87E-05 3.07E-03 Std. 1.43E-19 5.49E?00 3.29E?03 2.80E?04 9.18E?04 3.05E-55 2.70E-18 1.20E-10 1.69E-05 3.00E-03 F3

Best 1.08E-01 9.73E?16 4.73E?25 1.18E?34 3.25E?38 2.31E205 3.21E104 2.19E107 4.62E110 5.79E108 Mean 1.39E?08 4.29E?24 7.24E?31 4.03E?37 7.25E?42 2.09E?53 1.54E?40 3.84E?32 7.16E?33 4.49E?29 Std. 4.57E?08 1.39E?25 3.35E?32 6.98E?37 2.23E?43 7.39E?53 7.34E?40 2.04E?33 3.85E?34 2.42E?30 F5

Best 6.01E-22 4.66E-04 6.35E-01 7.71E?00 3.26E?01 2.46E252 3.35E220 2.62E213 9.34E209 9.37E207 Mean 1.32E-20 8.30E-04 1.11E?00 1.25E?01 4.23E?01 2.20E-50 3.88E-19 2.15E-12 3.24E-08 3.31E-06 Std. 1.63E-20 2.34E-04 3.71E-01 2.72E?00 6.89E?00 3.86E-50 3.26E-19 1.73E-12 1.78E-08 1.72E-06 F6

Best 1.48E214 2.84E-02 5.07E?00 3.20E?01 5.00E?01 3.21E-14 8.24E215 3.60E210 9.71E206 2.28E204 Mean 2.21E-07 3.54E?00 4.64E?01 7.33E?01 9.39E?01 5.53E-05 1.10E-11 4.71E-06 2.65E-03 2.66E-02 Std. 1.14E-06 1.12E?01 2.12E?01 1.59E?01 1.45E?01 2.98E-04 5.16E-11 1.94E-05 5.92E-03 3.04E-02 F8

Best 3.86E-23 4.63E-04 2.99E-01 7.53E?00 2.86E?01 7.68E253 2.50E222 4.03E216 2.30E210 3.78E208 Mean 4.29E-20 2.67E-03 2.47E?00 2.52E?01 5.54E?01 4.25E-47 1.93E-20 2.40E-13 8.17E-09 1.48E-06 Std. 1.13E-19 2.67E-03 1.71E?00 1.20E?01 1.63E?01 1.50E-46 2.54E-20 3.07E-13 1.37E-08 3.43E-06 F9

Best 1.41E102 1.84E?02 3.72E?02 3.90E?02 8.58E?02 1.44E?02 1.25E102 1.21E102 1.07E102 1.53E102 Mean 2.17E?02 3.74E?02 6.80E?02 8.58E?02 9.36E?02 2.01E?02 1.67E?02 1.86E?02 2.24E?02 2.34E?02

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third row indicates the standard deviation of the 25 run, the

fourth row indicates the p value and signs value of the

algorithms. All these algorithms were fixed at 200,000

MaxFEs and operated under the same conditions.

The optimum results in Table

13 are highlighted in bold

type. As can be seen from this table, MTSA provides the

best and mean values in 11 of the 15 benchmark functions

which are: F1, F2, F3, F4, F5, F7, F10, F12, F13, F14 and

F15. In addition, the ABC showed the second, GSA and

SSA showed the third most effective performance on

benchmark functions. PSO, GOA, MVO and TSA are far

behind in terms of average and best fitness values. MTSA

showed that five out of eight unimodal functions were

outperforms according to ABC, PSO, GSA, SSA, GOA,

MVO, TSA. Compared to ABC, PSO, GSA, SSA, GOA,

MVO, TSA for the multimodal benchmark functions,

MTSA provides the best results in six of the seven

multi-modal benchmark functions. MTSA’s good performance is

ensured by choosing the trees using the tournament

selec-tion method during seed producselec-tion. Therefore, it can be

stated that the algorithm makes high exploration. This

exploration is necessary to explore the search space, thus

avoiding the local optimum and approaching the global

optimum. In addition, the proposed MTSA could

signifi-cantly improve the performance of TSA. The p values

obtained by the Wilcoxon test and the pair-wise

compar-ison of the best score for 200.000 MaxFEs with 5%

sig-nificance level from all the statistical tests used are

presented in the table. It was generated as a pair-wise

comparison like as: MTSA to ABC, MTSA to PSO, MTSA

to GSA, MTSA to SSA, MTSA to GOA, MTSA to MVO,

MTSA to TSA. For MTSA, since the p value of most of the

functions is smaller than 0.05, it is seen that there is a

significant difference between MTSA and other algorithms.

In addition to this test, sign indicators are used. If the

results are statistically different (p \ 0.05), it was marked

‘‘?’’ and if the results were not statistically different (p

C 0.05), marking was performed as ‘‘-’’. Figure

8 shows

the MTSA and other algorithms status of according to their

rank numbers.

Table

13 presented that the Wilcoxon statistical test was

applied to the results of the algorithms included in the

study and the results are given in Table

14 .

Therefore, when the p values of the Wilcoxon test

results in Table

14 were examined, a statistically

signifi-cant difference was found between ABC–MTSA, PSO–

MTSA, GSA–MTSA and GOA–MTSA. There was a

sta-tistically significant difference between SSA–MTSA and

MVO–MTSA. There was a very high difference between

TSA and MTSA.

4.3 Performance assessment of MTSA

on multilevel thresholding problem

The preferred image thresholding method for real-world

problems is performed and based using image brightness.

Images consist of homogeneous gray level regions that

imply effective partitioning possibilities. In these

prob-lems, multilevel thresholding (at least two thresholds) is

generally used. The selection of the threshold number is

determined according to the problem type. In the multilevel

threshold of images, the histogram of the image is divided

into different groups and a single density value is assigned

to each group. There are studies on nonparametric methods

Table 7 (continued)

TSA MTSA

Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 Std. 3.84E?01 1.50E?02 1.56E?02 1.22E?02 4.94E?01 4.13E?01 3.35E?01 3.83E?01 9.96E?01 5.73E?01 F13

Best 0.00E?00 1.97E-02 1.87E?00 1.45E?01 5.93E?01 0.00E100 0.00E100 5.74E214 8.31E209 2.23E206 Mean 4.11E-04 6.06E-02 3.05E?00 2.88E?01 9.02E?01 5.75E-04 0.00E?00 1.02E-12 5.79E-08 3.01E-05 Std. 2.21E-03 4.53E-02 7.69E-01 7.90E?00 1.68E?01 2.18E-03 0.00E?00 2.37E-12 4.26E-08 6.01E-05 F14

Best 1.17E-12 3.72E-02 2.89E?00 8.51E?00 1.25E?01 1.33E214 3.07E212 3.04E208 1.30E205 2.18E204 Mean 9.84E-10 4.10E-01 6.05E?00 1.21E?01 1.53E?01 1.91E-14 1.92E-11 1.42E-07 4.86E-05 6.03E-04 Std. 3.63E-09 1.20E?00 3.15E?00 3.79E?00 2.46E?00 4.00E-15 2.01E-11 1.14E-07 2.07E-05 2.71E-04 Best values are highlighted in bold

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TSA MTSA

Best 1.22E-07 1.82E-03 6.91E-02 3.04E-01 1.22E?00 6.34E210 2.16E205 1.27E203 1.21E202 6.45E202 Mean 6.35E-07 7.65E-03 1.57E-01 5.88E-01 1.56E?00 2.04E-08 7.88E-05 2.81E-03 2.55E-02 1.09E-01 Std. 4.87E-07 3.50E-03 3.76E-02 1.34E-01 2.44E-01 5.23E-08 4.39E-05 1.16E-03 8.93E-03 2.50E-02 F7

Best 2.26E-06 9.09E-03 3.13E-01 2.51E?00 3.62E?00 1.39E207 1.01E206 5.26E205 1.88E204 2.99E203 Mean 2.42E-04 1.11E-01 1.56E?00 3.91E?00 5.74E?00 2.75E-05 1.10E-04 4.38E-04 2.96E-03 7.35E-03 Std. 2.52E-04 1.02E-01 4.31E-01 7.17E-01 8.79E-01 4.92E-05 1.12E-04 2.30E-04 1.45E-03 3.06E-03 F8

Best 5.35E?00 1.16E?01 1.18E?01 1.48E?01 1.59E?01 4.01E100 7.98E100 7.97E100 9.87E100 1.03E101 Mean 1.33E?01 1.40E?01 1.50E?01 1.64E?01 1.76E?01 1.49E?01 1.28E?01 1.25E?01 1.27E?01 1.29E?01 Std. 2.53E?00 1.05E?00 1.01E?00 7.07E-01 1.06E?00 1.25E?01 1.73E?00 2.12E?00 1.53E?00 1.25E?00 F11

Best 0.00E?00 9.06E?00 3.19E?01 4.40E?01 6.25E?01 0.00E100 0.00E100 1.17E209 4.78E201 2.15E100 Mean 3.98E-01 1.48E?01 3.73E?01 5.54E?01 7.40E?01 3.98E-01 3.52E-13 2.09E-01 2.83E?00 5.71E?00 Std. 6.08E-01 3.13E?00 3.40E?00 6.83E?00 5.49E?00 5.51E-01 1.83E-12 4.65E-01 1.12E?00 1.50E?00 F12

Best 0.00E?00 8.86E?00 1.86E?01 2.60E?01 3.91E?01 0.00E100 0.00E100 4.92E203 3.26E100 5.74E100 Mean 5.00E-01 1.28E?01 2.51E?01 3.72E?01 5.05E?01 8.33E-01 6.69E-02 1.71E?00 5.95E?00 7.75E?00

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in the literature about the image threshold [

27 ]. Threshold

values are determined by optimizing certain criteria, such

as maximizing inter-class variance or different entropy

measurements. It has been shown by Akay using ABC to

maximize the criterion of variance among classes and to be

better than the entropy criterion [

43 ]. In this study, class

variance criterion was maximized to evaluate the

perfor-mance of selected thresholds.

4.3.1 Problem formulation

Image segmentation is usually done by dividing the image

into a histogram by selecting the appropriate threshold

levels. Finding the optimal threshold levels is a complex

process, because an image histogram can contain some

large valleys and peaks at different heights. The method,

proposed by Otsu, can solve these problems by dividing the

histogram into different classes according to the pixel

probabilities [

44 ]. This section describes the formulations

for multilevel thresholding. When L is a gray level in an

image, the threshold value t is between 0 and L - 1 and I is

a given image [

45 ]. In Formula

8 , the multilevel

thresh-olding can be defined as follows:

P

0

¼ M x; y

f

ð

Þ 2 Ij0 M x; y

ð

Þ t

0

1 g

P

1

¼ M x; y

f

ð

Þ 2 Ijt

0

M x; y

ð

Þ t

1

1 g

. . .

P

n

¼ M x; y

f

ð

Þ 2 Ijt

n1

M x; y

ð

Þ L 1

g

ð8Þ

The maximization process is calculated by dividing the

image into different classes of histograms with the

fol-lowing formulas.

t

¼ arg max f

½

b

ð Þ

t

ð9Þ

f

b

ð Þ ¼

t

X

n i¼0

r

i

ð10Þ

r

n

¼ x

n

ð

l

n

l

T

Þ

2

ð11Þ

l

n

¼

1 x

n

X

L1 i¼tn

i

p

i

ð12Þ

x

n

¼

X

L1 i¼tn

p

i

ð13Þ

p

i

¼ x

i

=X

ð14Þ

Here, x

i

is the number of pixels at the i level. X is the total

number of pixels in each level. In Eq.

14 , p

i

i it is the normal

value of the gray level. In Eq.

13 , the estimate of the

prob-ability of occurrence of classes is calculated. In Eq.

12 , n

determines the average density of the class. In addition, l

_T

in

Eq.

11 gives the average density of the original image, and

generally in Eq.

11 , calculates the variance of the class

n. Heuristic methods are used to solve multilevel

thresh-olding problems. The reason for this is to minimize the high

cost of calculation. MTSA, which gives better quality results

in benchmark functions, is used in maximizing the

multi-level thresholding problem for the images given in Fig.

9 .

The results of TSA and ABC methods were compared with

those of the proposed method. The Cameraman and Lena

pictures, which are frequently used in image processing

problems, are included in the MATLAB library. All the

other images and multilevel thresholding coding were taken

from Kiran’s study [

24 ]. It is aimed to obtain the most

appropriate 2, 3, 4 and 5 thresholds using the ABC, TSA and

Table 8 (continued)

TSA MTSA

Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 Pop = 10 Pop = 20 Pop = 30 Pop = 40 Pop = 50 Std. 7.19E-01 1.60E?00 3.83E?00 5.89E?00 6.03E?00 6.87E-01 2.49E-01 1.28E?00 1.16E?00 9.42E-01 F13

Best 0.00E?00 3.10E-13 1.52E-02 1.03E-01 1.73E-01 0.00E100 0.00E100 0.00E100 7.77E216 2.83E212 Mean 4.97E-03 1.06E-02 6.48E-02 2.02E-01 3.03E-01 6.24E-04 3.04E-04 6.68E-04 7.01E-04 6.15E-04 Std. 8.65E-03 9.63E-03 2.46E-02 5.62E-02 5.45E-02 2.19E-03 1.34E-03 2.13E-03 1.92E-03 1.60E-03 F14

Best 2.55E-04 6.96E?02 1.45E?03 2.11E?03 2.31E?03 2.55E204 2.55E204 2.55E204 1.25E201 3.81E100 Mean 3.16E?01 1.02E?03 1.82E?03 2.46E?03 2.81E?03 3.16E?01 2.72E-04 4.44E?00 2.00E?02 3.62E?02 Std. 5.24E?01 1.87E?02 1.86E?02 1.62E?02 2.39E?02 6.06E?01 7.19E-05 2.30E?01 1.24E?02 1.56E?02 F15

Best 2.22E-15 2.22E-15 1.41E-11 1.95E-07 5.69E-05 2.22E215 2.22E215 2.22E215 5.62E213 7.58E210 Mean 2.59E-15 4.67E-10 7.98E-07 2.28E-05 1.18E-02 2.59E-15 3.18E-15 4.14E-15 1.63E-12 2.29E-09 Std. 8.28E-16 2.51E-09 4.14E-06 4.43E-05 5.33E-02 8.28E-16 1.10E-15 7.55E-16 8.64E-13 1.11E-09 Best values are highlighted in bold