Each summary and critique must be a minimum of 2.0 pages in length, 1.5- spaced, #12 font, and be mechanically and grammatically sound. ? Each paper

T H E O R Y A N D A P P L I C A T I O N S O F S O F T C O M P U T I N G M E T H O D S

Species co-evolutionary algorithm: a novel evolutionary algorithm based on the ecology and environments for optimization

Wuzhao Li1,4 • Lei Wang1 • Xingjuan Cai1 • Junjie Hu3 • Weian Guo2

� The Natural Computing Applications Forum 2015

Abstract In classic evolutionary algorithms (EAs), so-

lutions communicate each other in a very simple way so the

recombination operator design is simple, which is easy in

algorithms’ implementation. However, it is not in accord

with nature world. In nature, the species have various kinds

of relationships and affect each other in many ways. The

relationships include competition, predation, parasitism,

mutualism and pythogenesis. In this paper, we consider the

five relationships between solutions to propose a co-evo-

lutionary algorithm termed species co-evolutionary algo-

rithm (SCEA). In SCEA, five operators are designed to

recombine individuals in population. A set including sev-

eral classical benchmarks are used to test the proposed

algorithm. We also employ several other classical EAs in

comparisons. The comparison results show that SCEA

exhibits an excellent performance to show a huge potential

of SCEA in optimization.

Keywords Evolutionary algorithm � Recombination operator � Species co-evolution algorithm � Optimization

1 Introduction

Optimization is a classical topic and plays a very active

role in many fields, such as science, engineering, finance,

medicine and military [1, 2]. Even in our daily life, various

kinds of optimization problems are very common, includ-

ing minimizing charging cost of electrical vehicle, maxi-

mizing profits from our investments and reducing time cost

in path planning [3, 4]. As a feasible and very effective

approach in dealing with optimization problem, evolu-

tionary algorithms (EAs) exhibit their dramatic perfor-

mances and nowadays draw worldwide attentions to

develop the algorithms [5–7]. In EAs family, there are

many famous algorithms including genetic algorithms

(GA) [8, 9], particle swarm optimization (PSO) [10, 11],

evolutionary strategies (ES) [12], ant colony optimization

(ACO) [13] and differential evolution (DE) [14, 15]. The

algorithms have been implemented in many different areas.

They can not only show an excellent performance in

common optimization problems, but also be very effective

to specific problems such as non-deterministic polynomial

(NP) problems and multiobjective optimization [16]. Due

to the exclusive advantages, say robust and reliable per-

formance, global search capability and no or little infor-

mation requirement, the researches on them have been

developed in current decades [17].

Most ideas in EAs are proposed by simulating nature

world. Inspired from various kinds of nature phenomena,

professionals abstracted the mechanism of nature to pro-

pose novel evolutionary algorithms for solving optimiza-

tion problems. Several examples are given as follows.

Genetic algorithm (GA) simulates producing generations of

chromosomes according to the genetic mechanism in bio-

logical process. Simulated annealing (SA) is proposed by

the idea of annealing in metallurgy [18]. The essence of

& Weian Guo [email protected]

1 Department of Electronics and Information,

Tongji University, Shanghai 201804, China

2 Sino-German College Applied Sciences of Tongji University,

Shanghai 201804, China

3 Center for Electric Power and Energy, Department of

Electrical Engineering, Technical University of Denmark,

2800 Copenhagen, Denmark

4 Jiaxing Vocational Technical College,

Jiaxing 314036, Zhejiang, China

123

DOI 10.1007/s00521-015-1971-3

Received: 5 March 2015 / Accepted: 9 June 2015 / Published online: 27 June 2015

Neural Comput & Applic (2019) 31:2015–2024

particle swarm optimization (PSO) is from the behavior of

birds flocking [10, 11]. Ant colony optimization (ACO)

mimics the ecological behavior of ants in finding food [19].

The idea of biogeography-based optimization draws from

the philosophy of island biogeography [20, 21]. In addition,

different algorithms are hybridized to propose novel

framework which combines the advantages from each

individual algorithm [22]. Due to EAs’ dramatic perfor-

mances, they now have been implemented to many appli-

cations and gain a big success. All the achievements show

that EAs are successful in handling optimization problem,

which also demonstrate that nature phenomena are feasible

and useful to supply us new solutions in the development

of artificial intelligence.

More than two centuries ago, Charles Darwin had a very

extensive experience when he collected and investigated

the different kinds of life forms during a long journey. On

the basis of his investigations, Darwin had the idea that

each species was developed from same or similar ancestors

so that the species have very similar features. In 1838, he

summarized his ideas and described the evolutionary pro-

cess which is nowadays termed natural selection. He

believed that the evolutionary process made the similar

features happen [23]. After that, Darwin published this

famous idea about the evolution caused by natural selection

in ‘‘On the Origin of Species’’ in 1859. In this book, we

know that different species are actually outcomes of con-

tinuing natural evolutionary process [24]. During the pro-

cess, there exist several kinds of relationship between

species and their environments. By direct or indirect

communication between species and environments, species

evolved and gradually fit the environments. The relation-

ship models include competition, predation and parasitism.

This can help environment keep balance of species number

and also can help species fit environments, which can be

considered as an optimization outcome. Hence, it is rea-

sonable to mimic the process and abstract the inherent

mechanism to propose novel EAs. Inspired from the rela-

tionships, in this paper, we propose a novel evolutionary

algorithm which is named species co-evolution algorithm

(SCEA). In SCEA, the relationships between species

include competition, predation, parasitism, mutualism and

pythogenesis. In future work, novel relationships can also

be adopted in design of SCEA. Since many operators are

employed in SCEA, the algorithm can provide more ways

for solutions to combine their features so that the algorithm

is very helpful to avoid local premature and stagnation.

The rest of this paper is organized as follows. In Sect. 2,

we show the mathematical model for the novel algorithm in

detail. The basic idea to design SCEA is also illustrated in

this section. In Sect. 3, we established SCEA’s model and

presented the flowchart of SCEA. We employ 14 classical

and widely used benchmarks in Sect. 4 to conduct a

numerical optimization test. Several other famous EAs

including GA, ACO and PSO are employed in compar-

isons. The results are analyzed and discussed in this sec-

tion. We conclude in Sect. 5 and propose our future work.

2 Mathematical models

There exist many relationship forms in nature so that

species have different ways to communicate and interact

with each other as well as environments. The different

ways provide us new ideas to propose a novel evolutionary

algorithm. In this section, we summarize several main

forms among species in nature and illustrate the mathe-

matical models of species.

2.1 Competition operator

Competition is very common among species. Even in one

population, different individuals may fight each other for

food, water, mating, etc. Hence, competition is a feasible

solution to select winner to enjoy all resources. By assuming

that competition happens between two individuals, the

mathematical models of competition are given as follows.

dN1

dt ¼ r1N1

K1 � N1 � aN2 K1

� � ð1Þ

dN2

dt ¼ r2N2

K2 � N2 � bN1 K2

� � ð2Þ

where a is the competitive index of specie B to specie A. A large value of a means a competitive ability of B in competition with A. b is the competitive index of specie A to specie B. The larger b is, the better performance A has. N1 presents the population scale of A, while N2 is the

population scale of B. K1 and K2 are constants. r1 and r2 are

the increasing rate of species A and B to enlarge their

population scale.

2.2 Predation operator

In nature, the predators will predate their prey for survive.

From the view of energy, the prey will be killed and lost its

energy, while predator will occupy the prey’s energy. This

is also a common nature phenomenon in nature. The

mathematical models of predation are listed as follows.

dN

dt ¼ r1N ð3Þ

dP

dt ¼ �r2P ð4Þ

dN

dt ¼ ðr1 � ePÞN ð5Þ

123

Neural Comput & Applic (2019) 31:2015–20242016

where N presents the population scale of prey, P is the

population scale of predator. r1 is the population increasing

rate of prey. e represents the probability that prey can escape from predator.

2.3 Parasitism operator

Parasitism is also common in ecosystem. An example is

given that in human body, parasites may be found in

intestines and stomach. In this subsection, we use Nichol-

son–Bailey model to illustrate parasitism model. For host,

the model of population scale is given as follows,

Ntþ1 ¼ FNte�apt ð6Þ

where F is the population increasing rate of host and et -ap

is

the proportion of population that has not been hosted. N is

the population scale of host.

For parasite, the model of population scale is given as

follows,

Ptþ1 ¼ FPtð1 � e�aptÞ ð7Þ

where P is the population scale of parasite.

In above equations, the value of a often can be obtained by experiments or statistic:

a ¼ 1=pð Þ lnðN=SÞ ð8Þ

where S is the population that has not been hosted.

Although there are far more relationships between spe-

cies and environments in this nature, it is not feasible to

exhibit by mathematical models. Hence, related parts will

be explained in next section.

3 Species co-evolution algorithm

We summarized that there are mainly five living modes in

nature. They are predation, parasitism, competition,

mutualism and saprophytes. First, we consider the situation

that the fitness of two species is overmatched.

3.1 Predation

If one creature is much superior or stronger than another

one, the weaker may be preyed by the stronger. For

example, for lion and rabbit, we consider lion can prey

rabbit. Then, we think the weaker’s feature will be replaced

by stronger’s. In optimization algorithm, we consider the

stronger as the individual with high fitness and the weaker

with low fitness. Let A be the stronger and B be the weaker,

and then, we can get the following pseudo-codes. If the

predation occurs, then

B:feature = A:feature

3.2 Parasitism

In some cases, although the finesses between two species

are quite different, the weaker can survive by some par-

ticular methods, such as parasitism. For example, although

worms are lower organisms, they can live in human being’s

intestines. Worms can absorb nutrient from human and

excrete the excreta to human beings. So this is an inter-

changeable progress. In this process, by assuming that A is

the stronger one, while B is the weaker species, we present

the operator of parasitism as follows,

Swap ðA:feature, B:feature)

where swap operator means to exchange the individual

feature.

In addition, we consider that the fitness of two species is

almost near. In this case, there are mainly following two

living conditions.

3.3 Competition

Two strong species will fight for food, spouse and region.

Since they have the near fitness, they are counterparts so

that it is difficult to distinguish which one is better. So in

this case, we usually adopt probability to decide which

specie can win in fight. Stronger a specie is, more likely it

can win and vice versa. Let A be the stronger and B be the

weaker. A.fitness presents A’s fitness, and B.fitness pre-

sents B’s fitness. As well, we define ‘pa’ as the probability

that A can win. ‘Pb’ is the probability that B can win.

Therefore, we know

pa = a:fitness/ a:fitness + b:fitnessð Þ ð9Þ pb = b:fitness/ a:fitness + b:fitnessð Þ ð10Þ

We define ‘rand’ as a random number between 0 and 1. If

‘rand’ is less than pa, then we decide A wins. B will be

preyed by A. If ‘rand’ is not less than A, then we decide A

loses. A will be preyed by B. Hence, in this process, we can

deal with competition as follows,

IF rand < pa b.feature = a.feature ;

ELSE a.feature = b.feature ;

END

3.4 Mutualism

If two species are both low organisms, maybe they cannot

survive by themselves. Hence, they usually will cooperate

with each other, but not fight. In this case, we consider the

123

Neural Comput & Applic (2019) 31:2015–2024 2017

two species can generate two new species by interchanging

their features. If the two new features are stronger than

older, then we think the older two species will be replaced

by the two new species. Otherwise, the older ones can

retain. In this case, we define A’ as the new specie gen-

erated by A and B’ as the new specie generated by B. Then,

we know if mutualism occurs,

[A’,B’]=Swap(A.feature, B.feature) IF A'.fitness > A.fitness

A = A'; END IF B'.fitness > B.fitness

B = B'; END

3.5 Pythogenesis

In the former subsections, we investigate four relationships.

In competition and predation models, death bodies will be

appeared. These bodies can be handled by saprophytic

creatures. All the discarded features will be used by these

saprophytic species at some probabilities. Let ‘rand’ be a

random number between 0 and 1. And we set ‘decision’ as a

small fixed number between 0 and 1. We define A as

saprophytic creature. Then, we design this model as follows,

IF rand < decision A.feature = death body

END

3.6 Design of SCEA

Based on the modeling above, we design species co-evo-

lution algorithm as follows:

Step1: Generate a population. Set parameters including

population size N, threshold value for decision in

evolutionary process (D is to decide whether a

difference between fitness is large. H is used to

decide whether the species are high species),

termination conditions and so on

Step 2: If the termination conditions are satisfied,

terminate the algorithm. Otherwise, go to Step 3

Step 3: Randomly select individuals in pairs. After that,

the population will be divided into N/2 pairs

Step 4: In each pair, do the comparisons of species’

fitness. If the difference between two species is

larger than D, go to Step 6. Otherwise, go to

Step 6

Step 5: Generate a random between domain [0,1]. If the

random value is larger than 0.5, do predation

operator. After that, the loser species will be

handled by pythogenesis operator. If the random

value is not larger than 0.5, do parasitism

operator. Go to Step 7

Step 6: In a pair, compare the fitness with H. If the

fitness is larger than H, the two species are higher

creatures and do the competition operator. The

loser in competition will be handled by

pythogenesis operator. If the fitness is smaller

than H, the two species are lower creatures and

do the mutualism operator

Step 7: Based on Step 5 to Step 6, a new population has

been generated. Go to Step 2

The flowchart of SCEA can be found in Fig. 1. There

are total five operators in SCEA, which can help algorithm

enhance the adaptive ability in dealing with different

optimization problems. Figure 1 is only a basic design of

SCEA. However, in future, many other strategies can be

combined in this algorithm to enhance its optimization

ability such as elitism strategy.

4 Analysis of numerical simulations

To investigate the performances of our design, we conduct

the numerical comparisons. In this comparison, we employ

14 classical numerical benchmarks. In addition, we also

employ several well-known evolutionary algorithms to

compare with SCEA. As given in Table 1, we present the

function names, dimensions and domains of each bench-

mark. In addition, the characters of the 14 benchmarks can

be found in Table 2. According to Table 2, we know that

the benchmarks have different types so that the comparison

results are feasible and reasonable to reflect algorithms’

optimization ability. Other details including the mathe-

matical modeling of the benchmarks can be found in paper

[25–27].

In comparison, genetic algorithm (GA), particle swarm

optimization (PSO), ant colony optimization (ACO), dif-

ferential evolution (DE), evolutionary strategy (ES) and

population-based incremental learning (PBIL) are

employed in comparisons. The parameters of each algo-

rithm are setting as follows. In GA, we employ single

crossover operator. The crossover rate is set as 0.8, and the

mutation rate is 0.05. The roulette wheel selection is

employed. In particle swarm optimization, we set inertial

constant as 0.2. The cognitive constant is 1, while the

social constant for particle communication is set as 1. For

PBIL, the learning rate is set as 0.5. The elitism parameter

is 1, while mutation rate is 0. For ACO, the initial

123

Neural Comput & Applic (2019) 31:2015–20242018

pheromone value is set as 1e-5. The update constant and

decay rate for pheromone are set as 15 and 0.2, respec-

tively. For DE, the weighting factor is set as 0.5, while the

crossover rate is 0.5. For ES, the standard deviation is set as

1 for solutions to change. For SCEA, the value D is set as

0.7 and the value H is set as 0.5.

The numerical comparison results are shown as in

Tables 3 and 4. In comparison, we run 50 Monte Carlo

Start

End Is Termination

Condition Satisfied?

Randomly Select Species in Pairs

Fitness Difference Large?

Predation or Parasitism

Predation Operator

Parastism

Pythogenesis

Are the Individuals higher creatures?

Competition

Mutualism

Pythogenesis

Generate New Population

Yes

Yes

Yes No

No

Predation

Parastism

No

Initialization

Fig. 1 Flowchart of species co-evolution algorithm

Table 1 Benchmark functions Function index Function name Dimension Domain

F1 Ackley’s function 20 [-30, 30]D

F2 Fletcher–Powell 20 [-p, p]D

F3 Generalized Griewank’s function 20 [-600, 600]D

F4 Generalized penalized function 1 20 [-50, 50]D

F5 Generalized penalized function 2 20 [-50, 50]D

F6 Quartic function 20 [-1.28, 1.28]D

F7 Generalized Rastrigin’s function 20 [-5.12, 5.12]D

F8 Generalized Rosenbrock’s function 20 [-2.048, 2.048]D

F9 Schwefel’ problem 1.2 20 [-65.535, 65.535]D

F10 Schwefel’ problem 2.21 20 [-100, 100]D

F11 Schwefel’ problem 2.22 20 [-10, 10]D

F12 Schwefel’ problem 2.26 20 [-512, 512]D

F13 Sphere model 20 [-5.12, 5.12]D

F14 Step function 20 [-200, 200]D

123

Neural Comput & Applic (2019) 31:2015–2024 2019

simulations for each benchmark, and in each Monte Carlo

simulation, the limited generation is 100. In Table 3, we

present the best results for each algorithm, while in

Table 4, the mean results are shown. The best result is

collected by the best performance of each algorithm in the

50 runs. The mean result is an average value of 50 best

results for each algorithm. For convenience, for each

benchmark, we use bold font to mark the best performance

in Tables 3 and 4.

According to Table 3, we know that for F1, F2, F3, F7,

F8, F9, F10, F11, F13, F14, SCEA outperforms than other

algorithms. For F4 and F12, ACO performs the best. SCEA

is arranged second and fourth. For F5, GA performs the

best, and SCEA performs the second. For F6, DE performs

the best and SCEA performs the second as well. To sum

up, we know SCEA performs the best in most cases of

these benchmarks.

According to the results in Table 4, we found that SCEA

wins 13 times except F5. For F5, SCEA performs the

second best. Since in 100 runs, the mean results demon-

strate an average performance, we can draw the conclusion

that SCEA has an excellent performance in dealing with

optimization problems.

5 Applications in mechanical design

To show the ability of SCEA in applications, we employ

several classical mechanical design problems as

applications.

5.1 Design a gear train

The first application is to design a compound gear train

arrangement which is shown in Fig. 2. The gear ratio for a

reduction gear train is defined as the ratio of angular

velocity of the output shaft to that of the input shaft. The

overall gear ratio between input and output is shown as

follows,

ratio ¼ xoutput xinput

¼ TdTb

TaTf ð11Þ

where xoutput and xinput are the angular velocities of the output and input shafts, respectively, and T denotes the

number of teeth on each gearwheel. The ratio should be

designed as close as possible to 1/6.931. The number of

teeth in each gear should be an integer and lie between 12

and 60. This problem can be described mathematically as

follows.

Minimize:

F Xð Þ ¼ 1

6:931 � TaTb

TcTd

� �2 ¼

1

6:931 � x1x2

x3x4

� �2 ð12Þ

Subject to:

12 � xi � 60

where i 2 {1, 2, 3, 4} and xi 2 Z. Z represents integer set, and X ¼ x1; x2; x3; x4½ �T¼ Ta; Tb; Tc; Td½ �T .

Table 5 shows the comparison results. In this table, we

use the previous work in comparisons [28–34]. It is obvi-

ous that the results in paper [30] and SCEA have the same

best performances. The two algorithms outperform others.

The simulation results show that the proposed algorithm

SCEA is competitive in dealing with practical problems.

5.2 Optimization in pressure vessel design

Pressure vessel design is a classical optimization problem

with constraints. In this subsection, we use this problem to

test SCEA’s ability to deal with constrains. For the design

Table 2 Characters of each benchmark

Function Multimodal Separable Regular

Ackley’s function H H

Fletcher–Powell H

Generalized Griewank’s function H H

Generalized penalized function 1 H H

Generalized penalized function 2 H H

Quartic function H H

Generalized Rastrigin’s function H H H

Generalized Rosenbrock’s function H

Schwefel’ problem 1.2 H

Schwefel’ problem 2.21

Schwefel’ problem 2.22 H

Schwefel’ problem 2.26 H H

Sphere model H H

Step function H

123

Neural Comput & Applic (2019) 31:2015–20242020

of a cylindrical pressure vessel, both ends should be cov-

ered by hemispherical heads. An illustration is shown in

Fig. 3. The variables are also marked in the figure. The

optimization target is to minimize the total cost including

forming and welding cost, materials cost. To archive this

goal, it is necessary to optimize the thicknesses of the shell

and the head, the inner radius and the length of the cylin-

drical section. The mathematical models are shown in (14),

where the parameters Ts, Th, R and L shown in Fig. 3 are

described by x1, x2, x3 and x4, respectively.

Minimize:

F Xð Þ ¼ 0:6224×1; x3; x4 þ 1:7781×2; x23 þ 3:1661x 2 1; x4

þ 19:84×21; x3 ð13Þ

Table 3 Numerical comparison in 100 Monte Carlo simulations of SCEA, PSO, PBIL, ACO, DE, ES, GA for best results

Benchmarks SCEA PBIL PSO

F1 1.03E?01 1.93E?01 1.61E?01

F2 7.14E?04 4.18E?05 3.71E?05

F3 8.80E?00 2.69E?02 9.53E?01

F4 4.73E?03 8.18E?07 2.50E?06

F5 6.08E?04 3.13E?08 1.73E?07

F6 3.82E-01 2.29E?01 2.07E?00

F7 4.37E?01 2.35E?02 1.69E?02

F8 1.55E?02 1.48E?03 8.77E?02

F9 6.18E?02 5.33E?03 4.67E?03

F10 3.69E?03 7.69E?03 9.63E?03

F11 9.90E?00 6.25E?01 4.09E?01

F12 5.04E?01 7.12E?01 4.24E?01

F13 1.66E?00 6.68E?01 1.61E?01

F14 8.75E?02 2.42E?04 1.05E?04

ACO DE ES GA

1.46E?01 1.31E?01 1.91E?01 1.60E?01

6.46E?05 1.57E?05 7.76E?05 2.77E?05

1.30E?01 2.35E?01 1.01E?02 4.14E?01

6.88E?02 1.87E?04 4.31E?07 5.38E?05

3.00E?08 9.90E?05 1.81E?08 3.20E?04

8.68E-01 3.40E-01 1.37E?01 8.21E-01

1.59E?02 1.42E?02 2.14E?02 1.06E?02

2.12E?03 1.98E?02 2.44E?03 5.42E?02

1.09E?03 2.78E?03 4.06E?03 1.27E?03

9.07E?03 1.09E?04 1.47E?04 1.04E?04

5.35E?01 2.12E?01 8.27E?01 3.42E?01

4.08E?01 5.55E?01 6.79E?01 5.03E?01

2.70E?01 3.74E?00 6.52E?01 7.34E?00

1.45E?03 2.41E?03 1.71E?04 2.92E?03

Table 4 Numerical comparison in 100 Monte Carlo simulations of SCEA, PSO, PBIL, ACO, DE, ES, GA for mean results

Benchmarks SCEA PBIL PSO

F1 1.24E?01 4.87E?01 2.34E?01

F2 1.02E?05 7.90E?05 5.43E?05

F3 1.21E?01 5.09E?02 1.34E?02

F4 6.79E?03 1.56E?08 8.55E?06

F5 7.45E?04 6.09E?08 6.45E?07

F6 5.64E-01 5.53E?01 3.80E?00

F7 6.15E?01 3.14E?02 4.02E?02

F8 3.43E?02 5.82E?03 1.84E?03

F9 9.42E?02 8.43E?03 7.53E?03

F10 5.37E?03 9.48E?03 1.94E?04

F11 1.58E?01 7.33E?01 6.54E?01

F12 6.17E?01 9.21E?01 8.45E?01

F13 2.50E?00 9.71E?01 3.41E?01

F14 1.12E?03 4.21E?04 2.18E?04

ACO DE ES GA

3.28E?01 2.44E?01 3.61E?01 2.64E?01

9.47E?05 4.64E?05 8.54E?05 9.43E?05

2.03E?01 2.41E?01 3.42E?02 5.91E?01

8.47E?03 2.49E?04 7.37E?07 6.87E?05

4.15E?08 1.93E?06 2.17E?08 5.15E?04

1.84E?00 6.45E-01 2.58E?01 1.89E?00

8.38E?01 2.67E?02 5.32E?02 3.55E?02

5.35E?03 3.88E?02 5.54E?03 6.98E?02

2.34E?03 4.24E?03 6.95E?03 3.64E?03

1.32E?04 3.54E?04 2.43E?04 2.33E?04

8.53E?01 4.85E?01 1.39E?02 5.83E?01

8.45E?01 6.22E?01 8.91E?01 6.44E?01

5.23E?01 6.46E?00 4.23E?01 9.09E?00

2.61E?03 5.21E?03 2.35E?04 4.45E?03

Fig. 2 Compound gear train

123

Neural Comput & Applic (2019) 31:2015–2024 2021

Subject to:

g1ðXÞ ¼ �x1 þ 0:0193×3 � 0 g2ðXÞ ¼ �x2 þ 0:00954×3 � 0

g3ðXÞ ¼ �px23x4 � 4

3 px33 þ 1296000 � 0

g4ðXÞ ¼ x4 � 240 � 0

where the domain of decision variables is designed as

follows,

0:0625 � xi � 6:1875; ði ¼ 1; 2Þ

and

10 � xi � 200; ði ¼ 3; 4Þ

where x1 and x2 are the integers multiples of 0.0625, and

the x3 and x4 are real numbers.

To deal with constraints, we employ penalty function to

calculate the violation. For each constraint, we do a nor-

malization of the violations to avoid bias. Then, for each

solution, its fitness is combined by both optimization

objective and the sum of violations. Then, we obtain the

results shown in Table 6.

It is noted that for this problem, Fu and Loh once present

the optimal value 7197.7 in paper [29, 35]. Nevertheless,

there are no details about decision variables in that paper.

Hence, we do not cite the results in this paper. In addition,

in paper [36], Sarvari and Zamanifar published their results

7195.99. However, the description of the optimization

results is very inaccuracy, so we do not cite it at all. The

numerical test results are shown in Table 6, where we also

employ some previous work. By comparisons, we find that

SCEA performs the best which validates that the proposed

algorithm is very competitive in dealing with optimization

problems as well as it can be well implemented in engi-

neering optimization problems.

6 Conclusions

In this paper, we

Collepals.com Plagiarism Free Papers

Are you looking for custom essay writing service or even dissertation writing services? Just request for our write my paper service, and we'll match you with the best essay writer in your subject! With an exceptional team of professional academic experts in a wide range of subjects, we can guarantee you an unrivaled quality of custom-written papers.

Get ZERO PLAGIARISM, HUMAN WRITTEN ESSAYS

Why Hire Collepals.com writers to do your paper?

Quality- We are experienced and have access to ample research materials.

We write plagiarism Free Content

Confidential- We never share or sell your personal information to third parties.

Support-Chat with us today! We are always waiting to answer all your questions.