# Chicken Swarm Optimization Based on Elite Opposition-Based Learning

Authors: Chiwen Qu; Shi’an Zhao; Yanming Fu; Wei He
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Chicken swarm optimization is a new intelligent bionic algorithm, simulating the chicken swarm searching for food in nature. Basic algorithm is likely to fall into a local optimum and has a slow convergence rate. Aiming at these deficiencies, an improved chicken swarm optimization algorithm based on elite opposition-based learning is proposed. In cock swarm, random search based on adaptivet distribution is adopted to replace that based on Gaussian distribution so as to balance the global exploitation ability and local development ability of the algorithm. In hen swarm, elite opposition-based learning is introduced to promote the population diversity. Dimension-by-dimension greedy search mode is used to do local search for individual of optimal chicken swarm in order to improve optimization precision. According to the test results of 18 standard test functions and 2 engineering structure optimization problems, this algorithm has better effect on optimization precision and speed compared with basic chicken algorithm and other intelligent optimization algorithms.


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## Body

## 1. Introduction

Many problems in areas of scientific computing, engineering science, and business management can be concluded as global optimum problems. The consumption time of traditional accurate computation approaches for solving large-scale optimization problems increases exponentially. So this approach cannot meet the real requirements. To solve these problems, many scholars, by simulating life habits of creatures in the nature, presented intelligent swarm optimization algorithms including particle swarm optimization (PSO) [1], ant colony optimization (ACO) [2], artificial bee colony (ABC) [3], invasive weed colonization optimization (IWO) [4], firefly algorithm (FA) [5, 6], cuckoo search (CS) algorithm [7, 8], fish swarm algorithm (FSA) [9], bat algorithm (BA) [10, 11], monkey algorithm (MA) [12], krill herd (KH) algorithm, and flower pollination algorithm (FPA) [13]; all these have gained favorable results. As a new kind of burgeoning metaheuristic algorithm, intelligent swarm optimization algorithms have characteristics of high precision, fast convergence rate, and good stability and can obtain the exact solution or approximate solution of large-scale optimum problems within limited time.Chicken swarm optimization (CSO) is a new intelligent bionic algorithm proposed by Meng et al. [14] in 2014, which simulates chickens swarm hierarchy and their food search behavior. The whole chicken swarm is divided into cock swarm, hen swarm, and chick swarm. Chickens with highest fitness values and lowest fitness values are taken as cock swarm and chick swarm, respectively, and the rest are taken as hen swarm. When solving optimization problems, each chicken in the swarm corresponds to a solution. Different search strategies are adopted for different subswarm according to different population. In contrast with standard particle swarm optimization, differential evolution, and bat algorithm, chicken swarm has advantage in either searching precision or convergence rate [14, 15]. In basic chicken swarm algorithm, a random search strategy based on Gaussian distribution is adopted for particles of cock swarm. This search strategy has strong ability of local development, but its global development ability is weak to make it liable to fall into a local optimum. For hen swarm particles, the searching is done by common guidance of cocks of their own population and other populations, which helps hen particles close to global optimum. However, the population diversity will lack and the hen particles will fall into a local optimum when most cock particles (with cock particles of own race and other races) are in a local optimum. Consequently, the global optimal solution cannot be obtained and its search performance will be affected for basic chicken swarm algorithm.In this paper, a chicken swarm optimization algorithm on the basis of elite opposition-based learning (EOCSO) is presented to solve global optimum problems. A random search strategy based on dynamic adaptivet distribution is adopted in this algorithm for cock swarm to replace the random search based on Gaussian distribution. The local exploitation ability and global development ability are balanced. In order to improve the optimization precision and convergence rate of the algorithm, an opposition-based learning method is used to improve population diversity for hen swarm and a greedy dimension-by-dimension search mode is applied to individual of optimal chicken swarm for local search. Through experiments of 18 basic test functions and 2 engineering structure optimization measurement problems, the comparison among improved chicken swarm algorithm, basic chicken swarm algorithm, and other typical intelligent algorithms is conducted to show that improved chicken swarm algorithm has more excellent optimization precision, convergence rate, and robustness.


## 2. Chicken Swarm Algorithm

Chicken swarm optimization is a new intelligent bionic algorithm proposed according to various behaviors of cocks, hens, and chicks in the process of searching food. In this algorithm, chicken swarm in searching space is mapped as specific particle individual. Cock particle swarm, hen particle swarm, and chicken particle swarm are sorted according to fitness value of particle, and each subswarm uses different searching mode [16, 17].In this algorithm, several particles with best fitness are selected as cock particle swarm, which is given by(1)xi,jt+1=xi,jt+randn⁡0,σ2·xi,jt,where xi,jt+1 and xi,jt are the position of jth dimension of particle i in t+1 and t iterations, respectively, and randn⁡(0,σ2) is a random number of Gaussian distribution whose variance is σ2. The parameter σ2 can be calculated by(2)σ2=1,fiti<fitkexpfitk-fitifiti+ξ,fiti≥fitk,where i,k∈[1,rsize] and i≠k. rsize represents the number of cock swarms. fiti and fitk are the fitness values of cock particle i and k, respectively; ξ represents a number which is small enough.Moreover, most particles with good fitness are selected as hen swarm. Its random search is done via cocks of population of hen and that of others, which can be expressed as(3)xi,jt+1=xi,jt+S1·rand·xr1,jt-xi,jt+S2·rand·xr2,jt-xi,jt,where xr1,jt and xr2,jt are the position of cock individual r1 in the population of hen xi and cock individual r2 in the other population, respectively. rand is an uniform random number over [0,1]. S1 and S2 denote the weight calculated by(4)S1=exp⁡fiti-fitr1fiti+ξ,S2=exp⁡fitr2-fiti,where fitr1 and fitr2 are, respectively, the fitness value of cock individual r1 in the population of hen xi and cock individual r2 in the other population.All individuals, except for cock swarm and hen swarm, are defined as chick swarm. Its search mode follows that of hen swarm, which is given by(5)xi,jt+1=xi,jt+FL·xm,jt-xi,jt,FL∈0,2,where FL stands for a parameter, meaning that the chick would follow its mother to forage for food. xm,jt represents the position of the ith chick’s mother (m∈[1,N]).


## 3. Chicken Swarm Optimization Based on Elite Opposition-Based Learning

### 3.1. Random Search Strategy Based on Dynamic Adaptivet Distribution

In chicken swarm optimization, a random search mode on the basis of Gaussian distribution is adopted for cock particle searching. This search mode has strong ability in local development, but it is weak in global exploitation. Therefore, enlargement of search space for cock particle will effectively reduce the risk to fall into a local optimum. Thus, the search mode witht distribution is used for cock particle in order to balance the global development ability and local exploitation ability of the algorithm.Thet distribution, also called student distribution [18], owns degree of freedom parameter n; its probability density function is as follows:(6)ptx=Γn+1/2nπ·Γn/2·1+x2n-n+1n+1/2,-∞<x<+∞,where Γ is the Gamma function and Γ(n+1/2)=(2n)!π/(n)!4n.Apparently, thet distribution is defined as Cauchy distribution density function (t(n=1)=C(0,1)) when degree of freedom parameter n=1 and is defined as Gaussian distribution density function when n=∞ (t(n→∞)→N(0,1)). Thus, Cauchy distribution and Gaussian distribution are two special boundary cases of t distribution. The characteristics of Cauchy distribution and Gaussian distribution are integrated into t distribution, and different mutation range can be obtained via changing degree of freedom parameter n.In the algorithm, random search based on dynamic adaptivet distribution is selected for the position of cock particles xi=(xi,1,xi,2,…,xi,d), which is expressed as(7)xi,j′=xi,j+ψi,j·tn·xi,j,where xi,j′ is the position after random search of dynamic adaptive t distribution used by jth dimension of cock particle i, xi,j is the position of jth dimension of ith cock particle, t(n) is random function of t distribution with iteration n as its degree of freedom, and ψ is mutation control factor. Each dimension value performs differently in searching process of multidimensional objective function. Thus, the mutation control factor ψ has the same control strategy to each dimension, and it cannot meet the requirement of algorithm obviously. In this paper, dynamic adaptive mutation control factor is used to adjust mutation range, as given by(8)ψi,j=β·1n·∑k=1nxk,j-xbest,j,where β is proportionality coefficient, n is the number of cock members, (1/n)·∑k=1nxk,j is mean value of jth dimension of cock particle in the current iteration, and xbest,j is a mutated value of jth dimension of cock particle in optimal position of the current iteration.In random search of dynamic adaptivet distribution, iteration n is taken as degree of freedom parameter, and dynamic adaptive mutation control factor is used to adjust searching range. In the early stage of evolution, the search is conducted by t distribution (which is similar to that via Cauchy distribution) and maintains population diversity due to small n. Thus, the algorithm has good ability of global exploitation. With increase of n, random search of t distribution transits gradually from that of Cauchy distribution to that of Gaussian distribution, and random search of t distribution is similar to that of Gaussian distribution with good ability of local development in the later stage.


### 3.2. Elite Opposition-Based Learning

Elite opposition-based learning proposed by Tizhoosh [19] is a new technology applied to intelligent computing area, and it has been successfully applied in many intelligent algorithm optimizations [20–22]. Theoretically verified by Zhong and others [23], opposition-based learning can acquire a solution that closes to global optimal solution with a higher probability. In the paper, elite opposition-based learning is adopted to enlarge search range for hen swarm and enhance population diversity so as to improve searching performance of the algorithm. The basic thoughts are as follows.Set the optimal cock position and its opposite position intth iteration as xbestt=(xbest,1t,xbest,2t,…,xbest,dt) and opbestt=(opbest,1t,opbest,2t,…,opbest,dt), respectively; then(9)opbest,jt=k·laj+ubj-xbest,jt,where xbest,jt∈[laj,ubj], k is a coefficient which denotes (0,1) uniformly distributed random variable, and [laj,ubj] is the boundary of jth dimension of search space in the current population. The result is obtained by(10)laj=min⁡xi,j,ubj=max⁡xi,j,where i∈(1,2,…,size) and j∈(1,2,…,d). Equation (10) is applied to replace fixed boundary, which ensures that cocks generate opposite solution in decreased space range. Besides, threshold value of the opposite solution may go beyond the range of [laj,ubj] to form a nonfeasible solution. Traditional handling is to set the searching particle which goes beyond borders as boundary value. As a result, numerous searchers gather on the border. To avoid above problem, border buffering wall is used for handling, and its basic thoughts are as follows.Suppose thataj is upper bound of jth dimension of search range in the population and bj is the lower bound. If opposite solution of the optimal individual in the population is opbest,j′, the value after border buffering wall handling will be(11)opbest,jt=aj·1-sgnaj·L+sgnaj·L·αi,j·opbest,jtaj-bj·rand,opbest,jt>ajbj·1+sgnbj·L-sgnbj·L·αi,j·opbest,jtaj-bj·rand,opbest,jt<bj,where sgn is sign function and L∈[0,1] is a proper constant, which relates to the thickness of buffering wall.With border buffering wall to process off normal individual, search individuals are able to decide their values dynamically according to the practical searching conditions and can better overcome deficiencies brought by traditional handling. The search mode of hen swarm using elite opposition-based learning can be expressed as follows:(12)xi,jt+1=xi,jt+S1·rand·xr1,jt-xi,jt+S2·rand·xr2,jt-xi,jt+S3·rand·opbest,jt-xi,jt,where S1, S2, and S3 are weights, S3 and S2 have the same value, rand is an uniform random number over [0,1], and opbest,jt is the opposite solution of the optimal particle of the population of chick swarm particle i in tth iteration.


### 3.3. Local Search via Greedy Dimension-by-Dimension Search

In basic chicken swarm optimization algorithm, the cock swarm, the hen swarm, and the chick swarm use different random search modes to acquire comparatively high-quality solution and convergence rate. But the three subswarms are all assessed by overall upgrading assessment method. For high-dimensional global optimization problem, overall upgrading assessment method will affect the quality of solution and convergence rate due to the interference among each dimension [24]. Suppose that the objective global optimization function is f(xi)=∑j=13xi,j2 (sphere test function) and independent variable equals xi=(0.2,0.2,0.2). Then, f(xi)=0.12. In iterations, the independent variable is updated as xi′=(0.1,0.3,0.4) after overall upgrading. Then, f(xi′)=0.26, and xi′=(0.1,0.3,0.4) will be abandoned in chicken swarm optimization as f(xi′)>f(xi). The overall upgrading assessment method will cause a slow convergence rate for the algorithm as the first-dimensional contribution is ignored due to deterioration of the second- and third-dimensional contributions.The greedy dimension-by-dimension search mode in this paper makes full use of updated achievement of each dimension. The basic idea is as follows. (1) Suppose thatxi,j′ is the value after updating of the value of jth dimension of chicken swarm particle xi=(xi,1,xi,2,…,xi,j,…,xi,d) (fitness value is fit(xi)). Then, xi,j′ and the values of other dimensions of chicken swarm particle xi before updating are integrated into a new chicken swarm particle y=(xi,1,xi,2,…,xi,j′,…,xi,d). (2) Calculate its fitness value fit(y). If fit(y)<fit(xi), then the updated result will be saved. Otherwise, it will be abandoned and the updating of (j+1)th dimension will be conducted. The procedures are shown  in Algorithm 1.Algorithm 1
(1) Set the generation countert, the maximum generation max_gen, dynamic adaptive
change stepstep, max step step_max, the chicken individual xi=(xi,1,xi,2,…,xi,j,…,xd).
(2)step=(max_gen-t)max_gen⋅step_max (*)
(3) fork=1:d
(4)temp=xi;
(5)temp(k)=temp(k)+(2∗rand-1)∗step;
(6)    iff(temp)<f(xi)
(7)xi(k)=temp(k);
(8)  end
(9) endTo ensure that each dimension of chicken particles in initial period of iteration updates in a relatively big step-length and in a small step-length in later period of iteration, dynamic adaptive step-length updating as  (*) in Algorithm1 is adopted to further improve the algorithm’s convergence rate and quality of solution.


### 3.4. Algorithm Flow

Based on Sections3.1–3.3, the procedure of the chicken swarm optimization algorithm is shown in Algorithm 2.Algorithm 2
(1) Set the initial parameters, including the total population size popsize, the roosters accounts Nr,
the hens accountsNh, the mother hens accounts Nm, updating frequency of the
chicken swarmG and the maximum number of generations itermax et al.
(2) Generate a population x=(x1,x2,…,xi,…,xpopsize) of popsize chickens with random solutions.
(3) Calculate the fitness fitness(xi) and find the best solution xbest of the population.
(4) for iter=1:itermax
(5)  if iter%  G==1 or iter==1
(6)   Sort all population individuals according to their fitness.
(7)   Divide total population individuals into three subpopulations (called rooster population, hen
population, and chickens population) according to their sort criteria, and establish
the relationship between the chickens and its mother (hens).
(8)  end
(9)  Update the rooster population individuals according to Eq. (7) and calculate their fitness.
(10)     Update the hen population individuals according to Eq. (12) and calculate their fitness.
(11)     Update the chicken population individuals according to Eq. (5) and calculate their fitness.
(12)     Update the personal best position xi∗ and the global optimal position xbest.
(13)     Perform local search for the global optimal individual according to Section 3.3.
(14) end
(15) Output the best solution xbest


## 3.1. Random Search Strategy Based on Dynamic Adaptivet Distribution

In chicken swarm optimization, a random search mode on the basis of Gaussian distribution is adopted for cock particle searching. This search mode has strong ability in local development, but it is weak in global exploitation. Therefore, enlargement of search space for cock particle will effectively reduce the risk to fall into a local optimum. Thus, the search mode witht distribution is used for cock particle in order to balance the global development ability and local exploitation ability of the algorithm.Thet distribution, also called student distribution [18], owns degree of freedom parameter n; its probability density function is as follows:(6)ptx=Γn+1/2nπ·Γn/2·1+x2n-n+1n+1/2,-∞<x<+∞,where Γ is the Gamma function and Γ(n+1/2)=(2n)!π/(n)!4n.Apparently, thet distribution is defined as Cauchy distribution density function (t(n=1)=C(0,1)) when degree of freedom parameter n=1 and is defined as Gaussian distribution density function when n=∞ (t(n→∞)→N(0,1)). Thus, Cauchy distribution and Gaussian distribution are two special boundary cases of t distribution. The characteristics of Cauchy distribution and Gaussian distribution are integrated into t distribution, and different mutation range can be obtained via changing degree of freedom parameter n.In the algorithm, random search based on dynamic adaptivet distribution is selected for the position of cock particles xi=(xi,1,xi,2,…,xi,d), which is expressed as(7)xi,j′=xi,j+ψi,j·tn·xi,j,where xi,j′ is the position after random search of dynamic adaptive t distribution used by jth dimension of cock particle i, xi,j is the position of jth dimension of ith cock particle, t(n) is random function of t distribution with iteration n as its degree of freedom, and ψ is mutation control factor. Each dimension value performs differently in searching process of multidimensional objective function. Thus, the mutation control factor ψ has the same control strategy to each dimension, and it cannot meet the requirement of algorithm obviously. In this paper, dynamic adaptive mutation control factor is used to adjust mutation range, as given by(8)ψi,j=β·1n·∑k=1nxk,j-xbest,j,where β is proportionality coefficient, n is the number of cock members, (1/n)·∑k=1nxk,j is mean value of jth dimension of cock particle in the current iteration, and xbest,j is a mutated value of jth dimension of cock particle in optimal position of the current iteration.In random search of dynamic adaptivet distribution, iteration n is taken as degree of freedom parameter, and dynamic adaptive mutation control factor is used to adjust searching range. In the early stage of evolution, the search is conducted by t distribution (which is similar to that via Cauchy distribution) and maintains population diversity due to small n. Thus, the algorithm has good ability of global exploitation. With increase of n, random search of t distribution transits gradually from that of Cauchy distribution to that of Gaussian distribution, and random search of t distribution is similar to that of Gaussian distribution with good ability of local development in the later stage.


## 3.2. Elite Opposition-Based Learning

Elite opposition-based learning proposed by Tizhoosh [19] is a new technology applied to intelligent computing area, and it has been successfully applied in many intelligent algorithm optimizations [20–22]. Theoretically verified by Zhong and others [23], opposition-based learning can acquire a solution that closes to global optimal solution with a higher probability. In the paper, elite opposition-based learning is adopted to enlarge search range for hen swarm and enhance population diversity so as to improve searching performance of the algorithm. The basic thoughts are as follows.Set the optimal cock position and its opposite position intth iteration as xbestt=(xbest,1t,xbest,2t,…,xbest,dt) and opbestt=(opbest,1t,opbest,2t,…,opbest,dt), respectively; then(9)opbest,jt=k·laj+ubj-xbest,jt,where xbest,jt∈[laj,ubj], k is a coefficient which denotes (0,1) uniformly distributed random variable, and [laj,ubj] is the boundary of jth dimension of search space in the current population. The result is obtained by(10)laj=min⁡xi,j,ubj=max⁡xi,j,where i∈(1,2,…,size) and j∈(1,2,…,d). Equation (10) is applied to replace fixed boundary, which ensures that cocks generate opposite solution in decreased space range. Besides, threshold value of the opposite solution may go beyond the range of [laj,ubj] to form a nonfeasible solution. Traditional handling is to set the searching particle which goes beyond borders as boundary value. As a result, numerous searchers gather on the border. To avoid above problem, border buffering wall is used for handling, and its basic thoughts are as follows.Suppose thataj is upper bound of jth dimension of search range in the population and bj is the lower bound. If opposite solution of the optimal individual in the population is opbest,j′, the value after border buffering wall handling will be(11)opbest,jt=aj·1-sgnaj·L+sgnaj·L·αi,j·opbest,jtaj-bj·rand,opbest,jt>ajbj·1+sgnbj·L-sgnbj·L·αi,j·opbest,jtaj-bj·rand,opbest,jt<bj,where sgn is sign function and L∈[0,1] is a proper constant, which relates to the thickness of buffering wall.With border buffering wall to process off normal individual, search individuals are able to decide their values dynamically according to the practical searching conditions and can better overcome deficiencies brought by traditional handling. The search mode of hen swarm using elite opposition-based learning can be expressed as follows:(12)xi,jt+1=xi,jt+S1·rand·xr1,jt-xi,jt+S2·rand·xr2,jt-xi,jt+S3·rand·opbest,jt-xi,jt,where S1, S2, and S3 are weights, S3 and S2 have the same value, rand is an uniform random number over [0,1], and opbest,jt is the opposite solution of the optimal particle of the population of chick swarm particle i in tth iteration.


## 3.3. Local Search via Greedy Dimension-by-Dimension Search

In basic chicken swarm optimization algorithm, the cock swarm, the hen swarm, and the chick swarm use different random search modes to acquire comparatively high-quality solution and convergence rate. But the three subswarms are all assessed by overall upgrading assessment method. For high-dimensional global optimization problem, overall upgrading assessment method will affect the quality of solution and convergence rate due to the interference among each dimension [24]. Suppose that the objective global optimization function is f(xi)=∑j=13xi,j2 (sphere test function) and independent variable equals xi=(0.2,0.2,0.2). Then, f(xi)=0.12. In iterations, the independent variable is updated as xi′=(0.1,0.3,0.4) after overall upgrading. Then, f(xi′)=0.26, and xi′=(0.1,0.3,0.4) will be abandoned in chicken swarm optimization as f(xi′)>f(xi). The overall upgrading assessment method will cause a slow convergence rate for the algorithm as the first-dimensional contribution is ignored due to deterioration of the second- and third-dimensional contributions.The greedy dimension-by-dimension search mode in this paper makes full use of updated achievement of each dimension. The basic idea is as follows. (1) Suppose thatxi,j′ is the value after updating of the value of jth dimension of chicken swarm particle xi=(xi,1,xi,2,…,xi,j,…,xi,d) (fitness value is fit(xi)). Then, xi,j′ and the values of other dimensions of chicken swarm particle xi before updating are integrated into a new chicken swarm particle y=(xi,1,xi,2,…,xi,j′,…,xi,d). (2) Calculate its fitness value fit(y). If fit(y)<fit(xi), then the updated result will be saved. Otherwise, it will be abandoned and the updating of (j+1)th dimension will be conducted. The procedures are shown  in Algorithm 1.Algorithm 1
(1) Set the generation countert, the maximum generation max_gen, dynamic adaptive
change stepstep, max step step_max, the chicken individual xi=(xi,1,xi,2,…,xi,j,…,xd).
(2)step=(max_gen-t)max_gen⋅step_max (*)
(3) fork=1:d
(4)temp=xi;
(5)temp(k)=temp(k)+(2∗rand-1)∗step;
(6)    iff(temp)<f(xi)
(7)xi(k)=temp(k);
(8)  end
(9) endTo ensure that each dimension of chicken particles in initial period of iteration updates in a relatively big step-length and in a small step-length in later period of iteration, dynamic adaptive step-length updating as  (*) in Algorithm1 is adopted to further improve the algorithm’s convergence rate and quality of solution.


## 3.4. Algorithm Flow

Based on Sections3.1–3.3, the procedure of the chicken swarm optimization algorithm is shown in Algorithm 2.Algorithm 2
(1) Set the initial parameters, including the total population size popsize, the roosters accounts Nr,
the hens accountsNh, the mother hens accounts Nm, updating frequency of the
chicken swarmG and the maximum number of generations itermax et al.
(2) Generate a population x=(x1,x2,…,xi,…,xpopsize) of popsize chickens with random solutions.
(3) Calculate the fitness fitness(xi) and find the best solution xbest of the population.
(4) for iter=1:itermax
(5)  if iter%  G==1 or iter==1
(6)   Sort all population individuals according to their fitness.
(7)   Divide total population individuals into three subpopulations (called rooster population, hen
population, and chickens population) according to their sort criteria, and establish
the relationship between the chickens and its mother (hens).
(8)  end
(9)  Update the rooster population individuals according to Eq. (7) and calculate their fitness.
(10)     Update the hen population individuals according to Eq. (12) and calculate their fitness.
(11)     Update the chicken population individuals according to Eq. (5) and calculate their fitness.
(12)     Update the personal best position xi∗ and the global optimal position xbest.
(13)     Perform local search for the global optimal individual according to Section 3.3.
(14) end
(15) Output the best solution xbest


## 4. Results and Discussions

### 4.1. Parameter Setting

To test the algorithm’s performance, seven typical intelligent algorithms, bat algorithm (BA) [10, 11], particle swarm optimization (PSO) [1], differential evolution (DE) [25], ant colony optimization (ACO)  [2],  cuckoo search (CS) algorithm [7], flower pollination algorithm (FPA) [13], and chick swarm algorithm (CSO) [14], are selected to do contrast experiments with EOCSO algorithm. The parameters of EOCSO algorithm are set as follows. Population size is set as 100, and each of the scale of cock swarm and that of chick swarm occupies 15%, while the scale of hen swarm takes up 70%. All of the common parameters of these algorithms (including the population size, dimensions, and maximum number of generations) are set to be the same for a fair comparison. Other parameters are described in Section 3. The parameters set of all other five algorithms is shown in Table 1.Table 1
The parameters set of all other algorithms.
Algorithms
Parameters
BA
Q
m
i
n
=
0,  Qmax=2,  R0=0.1,  A=0.9,  α=0.95,  γ=0.9
PSO
c
1
=
1.49445,  c2=1.49445,  ϖ=0.729
DE
p
C
R
=
0.2,  βmin=0.2,  βmax=0.8
ACO
The intensification factorq=0.5, the deviation-distance ratio zeta=1
CS
p
a
=
0.25,  α=1
FPA
p
=
0.8
CSO
N
r
=
0.15
p
o
p
s
i
z
e,  Nh=0.7popsize,  Nm=0.15popsize,  G=10


### 4.2. Test Function

To verify the optimization precision and convergence rate of the proposed algorithm, 18 standard test functions in [26, 27] are chosen for contrast experiments. f1~f7 are high-dimensional unimodal functions and are extremely hard to converge to a global optimum, which are used to inspect the searching precision. f8~f13 are high-dimensional multimodal functions with several local extreme points. So they are used to test the global searching performance and avoid premature of the algorithms [28–30]. f14~f18 are low-dimensional multimodal functions. The standard test functions are seen in Table 2.Table 2
Benchmark test functions.
Number
Name
Benchmark test functions
Dimension
Scope
Optimum
f
1
(
x
)
Sphere model
f
x
=
∑
i
=
1
D
x
i
2
30
[
-
100,100
]
0
f
2
(
x
)
Schwefel’s problem 2.22
f
x
=
∑
i
=
1
D
|
x
i
|
+
∏
i
=
1
D
|
x
i
|
30
[
-
10,10
]
0
f
3
(
x
)
Schwefel’s problem 1.2
f
x
=
∑
i
=
1
D
∑
j
=
1
i
x
i
2
30
[
-
100,100
]
0
f
4
(
x
)
Schwefel’s problem 2.21
f
x
=
m
a
x
i
=
1
D
x
i
30
[
-
100,100
]
0
f
5
(
x
)
Generalized Rosenbrock’s function
f
x
=
∑
i
=
1
n
-
1
100
x
i
+
1
-
x
i
2
2
+
1
-
x
i
2
30
[
-
30,30
]
0
f
6
(
x
)
Step function
f
x
=
∑
i
=
1
D
x
i
+
0.5
30
[
-
100,100
]
0
f
7
(
x
)
Quartic function, that is, noise
f
x
=
∑
i
=
1
D
i
⋅
x
i
4
+
r
a
n
d
o
m
0,1
30
[
-
1.28,1.28
]
0
f
8
(
x
)
Generalized Schwefel’s problem 2.26
f
x
=
∑
i
=
1
D
-
x
i
⋅
sin
⁡
x
i
30
[
-
500,500
]
-
418.9829
∗
n
f
9
(
x
)
Generalized Rastrigin’s function
f
x
=
∑
i
=
1
n
x
i
2
-
10
⋅
cos
⁡
2
⋅
π
⋅
x
i
+
10
30
[
-
5.12,5.12
]
0
f
10
(
x
)
Ackley’s function
f
(
x
)
=
-
20
exp
⁡
-
0.2
1
d
∑
i
=
1
d
x
i
2
-
exp
⁡
1
d
∑
i
=
1
d
cos
⁡
2
π
x
i
+
20
+
e
30
[
-
32,32
]
0
f
11
(
x
)
Generalized Griewank function
f
(
x
)
=
1
4000
⋅
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
c
o
s
x
i
i
+
1
30
[
-
600,600
]
0
f
12
(
x
)
Generalized penalized function
f
x
=
π
D
10
sin
2
⁡
π
⋅
y
1
+
∑
i
=
1
D
-
1
y
i
-
1
2
1
+
10
sin
2
⁡
π
⋅
y
i
+
1
+
y
n
-
1
2
+
∑
i
=
1
n
u
x
i
,
10,100,4 yi=1+xi+14 uxi,α,k,m=k(xi-α)m,xi>α0,-α≤xi≤αk(-xi-α)m,xi≤α
30
[
-
50,50
]
0
f
13
(
x
)
Generalized penalized Function
f
(
x
)
=
0.1
10
sin
2
⁡
(
3
π
⋅
x
1
)
+
∑
i
=
1
D
-
1
x
i
-
1
2
1
+
10
sin
2
⁡
3
π
⋅
x
i
+
1
+
x
n
-
1
2
+
∑
i
=
1
n
u
x
i
,
10,100,4
30
[
-
50,50
]
0
f
14
(
x
)
Shekel’s foxholes function
f
(
x
)
=
1
500
+
∑
j
=
25
25
1
j
+
∑
i
=
1
2
(
x
i
-
a
i
j
)
6
-
1
2
[
-
65.56,65.56
]
0.9980
f
15
(
x
)
Kowalik’s function
f
(
x
)
=
∑
i
=
1
11
a
i
-
x
1
(
b
i
2
+
b
i
x
2
)
b
i
2
+
b
i
x
3
+
x
4
2
4
[
-
5,5
]
0.0003075
f
16
(
x
)
Hartman’s function
f
(
x
)
=
-
∑
i
=
1
4
c
i
⋅
e
x
p
-
∑
j
=
1
3
a
i
j
(
x
j
-
p
i
j
)
2
3
[
0,1
]
-
3.8628
f
17
(
x
)
Hartman’s function
f
(
x
)
=
-
∑
i
=
1
4
c
i
⋅
e
x
p
-
∑
j
=
1
6
a
i
j
(
x
j
-
p
i
j
)
2
6
[
0,1
]
-
3.32
f
18
(
x
)
Shekel’s family
f
(
x
)
=
-
∑
i
=
1
10
(
x
-
a
i
)
(
x
-
a
i
)
T
+
C
i
-
1
10
[
0,10
]
-
10.5364


### 4.3. Influence of Dynamic Adaptivet Distribution Search Strategy on Bird Swarm Algorithm

Influence of dynamic adaptivet distribution search strategy on bird swarm algorithm contains two parts. The first part is the influence of different parameter values of variable control factor ψ on Gaussian distribution algorithm. The second part is the empirical comparison and analysis of CSO under dynamic adaptive t distribution (ATD-CSO) and dynamic adaptive Gaussian distribution (AGD-CSO) as well as the original algorithm (O-CSO). High-dimensional unimodal functions (f3,f5), high-dimensional multimodal functions (f11,f13), and low-dimensional multimodal functions (f15,f17) are selected as test functions.


#### 4.3.1. Impact Analysis of Control Factorψ

Other parameters remain invariant, 0.2, 0.5, and 0.8, and dynamic adaptive change is used as control factorψ for Gaussian distribution algorithm. Test function parameters are set the same as Section 4.1, and test result is assessed from optimal value, mean value, worst value, and standard variance. Table 3 shows the statistical result of different ψ on the test functions.Table 3
Influence of variousψ on the test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
ψ
=
0.2
Best
0.01631162
25.28779662
0
0.608008583
0.000307777
-
3.321995171
Mean
5.826567112
26.47886719
0
2.347145417
0.000428489
-
3.308813113
Worst
25.52896184
28.91557367
0
4.237078833
0.001223174
-
3.202409203
Std.
7.284604701
0.4586344
0
0.92771315
0.000277897
0.036625398
ψ
=
0.5
Best
0.530928436
25.99904272
0
0.69896625
0.00031048
-
3.321877282
Mean
132.4241294
27.1885086
0
1.34834884
0.00044856
-
3.315628525
Worst
729.2974459
28.55843288
0
1.96348532
0.001223763
-
3.201008218
Std.
197.5724792
0.696629183
0
0.287524396
0.000224541
0.026987801
ψ
=
0.8
Best
0.018825799
26.28800133
0
0.717391541
0.000315938
-
3.321995171
Mean
500.9268589
27.17671059
0
1.083153802
0.000534308
-
3.301356117
Worst
1228.354641
28.77558754
0
1.33573296
0.00071324
-
3.199239821
Std.
400.9524735
0.60589184
0
0.219814939
0.000140829
0.044548317
Original strategy
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
Dynamic adaptive
Best
1.455387
E
-
22
0.09980032
0
1.90772
E
-
12
0.000307486
-
3.321995172
Mean
2.932745
E
-
20
0.57800657
0
9.09553
E
-
12
0.000307486
-
3.321995172
Worst
9.453261
E
-
20
1.08886543
0
6.57632
E
-
10
0.000307486
-
3.321995172
Std.
5.187532
E
-
21
0.19345322
0
2.42964
E
-
12
0
5.264821
E
-
12As can be seen from Table3, search effect is better when ψ is smaller. But it is not quite ideal for Gaussian distribution with fixed ψ to improve algorithm performance. Through dynamic adaptive parameter setting, the algorithm can be ensured to maintain a large search step size in initial search stage. In later stage, dynamic adaptive step size of control factor ψ is reduced, while the local search ability is enhanced. The test result proved that dynamic adaptive control factor has more advantages than single fixed control factor.


#### 4.3.2. Comparison of ATD-CSO, AGD-CSO, and O-CSO

Table4 indicates the statistical result of dynamic adaptive t distribution, dynamic adaptive Gaussian distribution, and original CSO algorithm on three different kinds of test function. As the data shows, in terms of optimal values, 6 and 4 orders of magnitude are improved for f3 and f13 separately by ATD-CSO compared with AGD-CSO. For function f5, searching precision of ATD-CSO is higher than that of AGD-CSO. For f11, f15, and f17, optimal solution is searched by both ATD-CSO and AGD-CSO. Also, for mean value, worst value, and standard variance, f11 and f15 obtain the same through these two algorithms. But, for the standard variance for f17, ATD-CSO is better than AGD-CSO. Compared with O-CSO, ATD-CSO produces optimal mean value, worst value, and variance. It can be seen from Figures 1–6 that, in early evolution period, convergence rate of ATD-CSO is faster than that of AGD-CSO when calculating the above 6 functions. In later period, the convergence rate is similar, but the overall convergence rate of ATD-CSO is more optimal. However, the convergence rate of O-CSO is slower than that of ATD-CSO and AGD-CSO. Therefore, ATD-CSO has some advantages in terms of solution precision, robustness, and convergence rate generally.Table 4
Influence of different distributions on test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
Original CSO
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
AGD-CSO
Best
1.455387
E
-
22
0.09980032
0
1.90772
E
-
12
0.000307486
-
3.321995172
Mean
2.932745
E
-
20
0.57800657
0
9.09553
E
-
12
0.000307486
-
3.321995172
Worst
9.453261
E
-
20
1.08886543
0
6.57632
E
-
10
0.000307486
-
3.321995172
Std.
5.187532
E
-
21
0.19345322
0
2.42964
E
-
12
0
5.264821
E
-
12
ATD-CSO
Best
3.81028
E
-
28
0.050669099
0
5.43358
E
-
16
0.000307486
-
3.321995172
Mean
1.24542
E
-
27
0.18932938
0
5.16541
E
-
15
0.000307486
-
3.321995172
Worst
1.06346
E
-
27
0.67578053
0
4.81746
E
-
13
0.000307486
-
3.321995172
Std.
9.19088
E
-
28
0.037375926
0
4.52782
E
-
15
0
9.858438
E
-
14Figure 1
Convergence rates forf3(x).Figure 2
Convergence rates forf5(x).Figure 3
Convergence rates forf11(x).Figure 4
Convergence rates forf13(x).Figure 5
Convergence rates forf15(x).Figure 6
Convergence rates forf17(x).


### 4.4. The Influence of Dynamic Adaptivet Distribution and Opposition-Based Learning on CSO (ATD-EO-CSO)

As demonstrated in Table5, the best value, mean value, worst value, and variance of f3,f5,f13,f15 obtained by dynamic adaptive t distribution and elite opposition-based learning are apparently better than those obtained by original CSO. Target value can be searched on function f11 by both ATD-EO-CSO and O-CSO. Therefore, the O-CSO is suitable to solve this function, and the improvement of the solution after optimization is not obvious. For function f17, the best value can be searched by both. However, in terms of mean value, worst value, and variance, the results will be better based on cowork of both t distribution and elite opposition-based learning. Since searching can be done more widely via elite opposition-based learning, boundary buffering wall is adopted to deal with individuals crossing the border, avoiding the deficiency that individuals gather on boundary value to improve the diversity of population. As a result, ATD-EO-CSO is superior in solution precision.Table 5
Statistical results of the influence of dynamic adaptivet distribution and opposition-based learning on CSO.
Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
Original CSO
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
ATD-EO-CSO
Best
7.108233
E
-
42
2.844564
E
-
6
0
2.095481
E
-
25
0.000307486
-
3.321995172
Mean
1.039602
E
-
40
9.654632
E
-
6
0
2.006629
E
-
24
0.000307486
-
3.321995172
Worst
4.540331
E
-
39
3.239232
E
-
5
0
9.910062
E
-
24
0.000307486
-
3.321995172
Std.
3.919585
E
-
41
3.6784872
E
-
7
0
1.887675
E
-
25
0
6.455396
E
-
14


### 4.5. Influence of Local Search via Greedy Dimension-by-Dimension Search on Bird Swarm Algorithm

Influence of local search via greedy dimension-by-dimension search on bird swarm algorithm contains two parts. The first part is the influence of fixed step sizestep=0.1, step=0.3, step=0.6, and the adaptive step on algorithm. The second part is the comparison between adaptive step strategy and original CSO. Table 6 shows the statistical result by means of various step values. According to Table 6, when step value is small, higher search precision can be obtained, but it is liable to get into premature convergence. In case of larger step value, the range of search step size is larger, and it is liable to skip optimal solution range, which causes lower searching effect in later stage of evolution and reduced local search ability. Through adaptive step, the algorithm’s global research can be ensured to conduct in long step size at initial stage and in short step size at later stage, which improves the search efficiency of dimension-by-dimension search. Hence, adaptive step dimension-by-dimension search is superior to the fixed step size strategy in terms of search precision.Table 6
Influence of various step values on test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
s
t
e
p
=
0.1
Best
1.1009139
E
-
30
2.5022290
E
-
4
0
7.1919782
E
-
22
0.000307486
-
3.321995172
Mean
2.5675751
E
-
26
7.1700102
E
-
4
0
2.0929358
E
-
20
0.000307486
-
3.321995163
Worst
5.9338088
E
-
25
5.0003902
E
-
3
0
9.0510335
E
-
20
0.000307486
-
3.321995061
Std.
5.4742928
E
-
26
4.8413835
E
-
5
0
6.9769787
E
-
22
7.537751
E
-
10
9.6237934
E
-
7
s
t
e
p
=
0.3
Best
3.0856694
E
-
28
4.3623649
E
-
5
0
4.9527803
E
-
18
0.000307486
-
3.321995172
Mean
6.4211983
E
-
25
3.3217367
E
-
4
0
5.8674385
E
-
15
0.000307486
-
3.321995172
Worst
5.9480356
E
-
24
7.1641617
E
-
4
0
6.7902546
E
-
12
0.000307486
-
3.321995163
Std.
1.2289958
E
-
26
8.0512434
E
-
5
0
2.6579955
E
-
16
7.967684
E
-
10
5.616744
E
-
8
s
t
e
p
=
0.6
Best
4.0232351
E
-
26
9.7209661
E
-
4
0
5.2857901
E
-
25
0.000307486
-
3.321995172
Mean
1.3561952
E
-
22
3.5667080
E
-
3
0
9.2285451
E
-
20
0.000307499
-
3.321956549
Worst
7.4188975
E
-
21
8.5082852
E
-
3
0
2.1028715
E
-
18
0.000307495
-
3.321906894
Std.
6.2712862
E
-
22
2.4668710
E
-
4
0
7.0434808
E
-
20
4.460765
E
-
8
8.512414
E
-
5
Original CSO
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
Adaptivestep
Best
9.5838158
E
-
40
8.9257907
E
-
3
0
9.4224685
E
-
25
0.000307486
-
3.321995172
Mean
1.8439439
E
-
36
3.7218918
E
-
2
0
5.3242987
E
-
23
0.000307486
-
3.321995172
Worst
8.5706881
E
-
36
7.5851102
E
-
2
0
8.1986156
E
-
22
0.000307486
-
3.321995172
Std.
1.0682863
E
-
37
2.3586933
E
-
2
0
2.1204222
E
-
23
9.124585
E
-
18
8.884563
E
-
15


### 4.6. Analysis of Experimental Results

In order to avoid being influenced by random factors, the experimental test for each case is conducted by 20 trials independently. The algorithm’s searching performance is assessed according to the best value, mean value, standard deviation, and the worst value produced by the test result. The number of iterations off14~f18 and f1~f13 is 50 and 1000 in each trial, respectively. The algorithm is examined by Matlab 2012a on the platform with Win 8 OS, Intel Core i5-4210U 2.4 GHZ CPU, and 4 GB memory. The test statistical results are shown in Tables 7, 8, and 9.Table 7
Test statistical results of functionsf1~f7.
Benchmark function
BA
PSO
DE
ACO
CS
FPA
CSO
EOCSO
f
1
(
x
)
Best
9.41672
E
-
06
6.774921774
3.036519
E
-
12
4.98686
E
-
08
0.040549145
6.34628
E
-
06
1.1893
E
-
55
1.19
E
-
201
Mean
1.20474
E
-
05
20.80098597
5.713553
E
-
12
1.8224
E
-
07
0.08183638
0.000732691
3.08046
E
-
52
7.0639
E
-
198
Worst
0.000014224
31.55593153
9.811158
E
-
12
3.76894
E
-
07
0.120709249
0.009162134
3.16469
E
-
51
7.2029
E
-
197
Std.
1.40714
E
-
06
9.102140636
2.040785
E
-
12
8.35921
E
-
08
0.021679322
0.002091063
8.83269
E
-
52
0
f
2
(
x
)
Best
0.012593909
6.594522786
3.042127
E
-
08
4.92993
E
-
05
1.479134511
1.97228
E
-
07
1.91405
E
-
43
6.5805
E
-
116
Mean
31.7282206
15.68696047
4.943835
E
-
08
0.000507759
2.180229997
7.11331
E
-
06
6.5601
E
-
41
2.13
E
-
112
Worst
131.069574
35.29663724
7.117019
E
-
08
0.00191464
3.029290054
1.51764
E
-
05
8.03444
E
-
40
9.5751
E
-
112
Std.
48.01800899
11.87133052
1.339655
E
-
08
0.000549138
0.403631239
4.85023
E
-
06
1.78694
E
-
40
2.8682
E
-
112
f
3
(
x
)
Best
2.27
E
-
05
4.13
E
+
02
15255.144282
11951.29445
5.99
E
+
02
2.57
E
-
02
7.00
E
+
01
4.49
E
-
51
Mean
3.94
E
+
02
9.17
E
+
02
24696.821583
16566.43552
8.59
E
+
02
6.70
E
-
02
8.06
E
+
02
9.55
E
-
49
Worst
4.77
E
+
03
1.75
E
+
03
29125.784537
22644.44883
1.10
E
+
03
1.10
E
-
01
1.67
E
+
03
1.28
E
-
47
Std.
1081.678915
411.6200929
3515.8752513
3287.33213
122.9938329
0.023267666
430.7691621
2.85739
E
-
48
f
4
(
x
)
Best
6.573087982
5.90965909
1.2612253015
3.058888831
5.273452649
12.24579118
0.000935694
1.41944
E
-
17
Mean
12.41978556
9.027042499
2.0432637030
5.055981092
6.116103354
15.33881514
3.027225289
2.20256
E
-
16
Worst
22.65352526
12.9169796
2.5042954408
7.296151756
7.215730774
19.26838273
10.74127991
7.30302
E
-
16
Std.
3.804254343
2.121495266
0.3514754143
0.953357307
0.52178179
2.184822605
3.480451504
1.96521
E
-
16
f
5
(
x
)
Best
0.196158304
343.6027646
25.837681251
17.1433613
0.013792405
2.12808
E
-
06
26.50669099
9.53224
E
-
09
Mean
1.836717634
530.442542
39.933721918
17.86118196
0.59160704
0.398901513
27.28932938
0.000161047
Worst
20.41255589
1427.570578
70.708954223
19.24540999
1.749078665
3.986765311
28.77578053
0.001214443
Std.
4.605435894
299.0234027
13.215780402
0.617903448
0.57410589
1.227005992
0.537375926
0.000359017
f
6
(
x
)
Best
7.69118
E
-
06
13.55667233
1.888545
E
-
12
5.85581
E
-
08
0
5.47537
E
-
06
1.671870558
0
Mean
9.79475
E
-
06
19.24687067
5.975184
E
-
12
1.68807
E
-
07
3.77265
E
-
10
0.000265874
2.125011694
0
Worst
1.13079
E
-
05
25.44278333
1.122444
E
-
11
4.86966
E
-
07
1.85832
E
-
09
0.000603906
2.710216862
0
Std.
9.19879
E
-
07
4.68251131
2.832052
E
-
12
9.4266
E
-
08
4.55015
E
-
10
0.000212809
0.301038821
0
f
7
(
x
)
Best
0.021225243
0.198906064
0.0171466731
0.01217843
0.001333226
0.029750255
0.000690044
0.001333226
Mean
0.039176522
0.294912344
0.0240711071
0.024336378
0.003336074
0.108161359
0.002471843
0.003336074
Worst
0.0606201
0.372732671
0.0301412708
0.034138264
0.004764046
0.160329105
0.005679536
0.004764046
Std.
0.011065883
0.061817637
0.0041777349
0.007196828
0.000889975
0.044585434
0.001443844
0.000889975Table 8
Test statistical results of functionsf8~f13.
⁢Benchmark function
BA
PSO
DE
ACO
CS
FPA
CSO
EOCSO
f
8
(
x
)
Best
-
8502.897133
-
7963.943031
-
12569.479219
-
6090.034663
-
9009.250145
-
10694.82285
-
9133.943944
-
12569.48662
Mean
-
7060.92851
-
6682.009356
-
12546.705396
-
8060.242464
-
8449.968964
-
10097.82587
-
8186.965487
-
12474.61802
Worst
-
5969.754314
-
5622.614213
-
12369.793372
-
351.75903
-
7986.143265
-
9466.344181
-
7087.508251
-
12405.53547
Std.
779.3469624
760.027236
53.131673
569.1183589
345.8135405
423.049839
596.4376456
51.6979597
f
9
(
x
)
Best
30.84598172
71.56741319
47.02959855
108.4758259
6.691249469
15.91943794
0
0
Mean
75.66883074
134.1736791
60.42457292
124.3756751
13.42928534
22.83514081
0
0
Worst
146.2602458
177.1407232
71.54930622
138.2030082
20.70524753
27.467636
0
0
Std.
32.32801039
35.8886515
7.058839227
7.737668073
4.409105246
3.170192621
0
0
f
10
(
x
)
Best
11.60989873
4.914039621
4.5281
E
-
07
6.54539
E
-
05
4.44089
E
-
15
2.013315503
4.44089
E
-
15
4.44089
E
-
15
Mean
13.64745922
5.590179783
6.07484
E
-
07
0.00013903
4.74731
E
-
15
2.981722685
5.50671
E
-
15
4.70507
E
-
15
Worst
15.09860799
6.074485544
7.84382
E
-
07
0.00022114
5.09361
E
-
15
4.298104668
7.99361
E
-
15
7.99361
E
-
15
Std.
0.891126943
0.311563718
1.11025
E
-
07
3.80777
E
-
05
2.25609
E
-
16
0.762341832
1.67035
E
-
15
7.99919
E
-
16
f
11
(
x
)
Best
44.79000242
0.997496962
6.55065
E
-
12
0.192490967
0
0.000152641
0
0
Mean
79.03694557
1.203784605
1.42400
E
-
10
0.360243829
0
0.016635909
0
0
Worst
136.5821255
1.333771315
5.74106
E
-
10
0.540731319
0
0.029692265
0
0
Std.
23.79302299
0.092691068
1.63674
E
-
10
0.092281983
0
0.009192596
0
0
f
12
(
x
)
Best
5.506766196
2.320391097
1.49415
E
-
13
1.11888
E
-
06
1.02781
E
-
09
2.91491
E
-
06
0.072486935
1.57054
E
-
32
Mean
2371.120996
5.817999746
5.97228
E
-
13
8.9491
E
-
06
0.024315241
0.288022193
0.127682199
8.79042
E
-
30
Worst
23844.45981
9.370164248
2.12638
E
-
12
2.61683
E
-
05
0.103669022
5.455285401
0.294365491
5.22
E
-
29
Std.
6390.937613
2.137993761
4.2672
E
-
13
7.29249
E
-
06
0.033905983
1.217902188
0.063713849
1.37918
E
-
29
f
13
(
x
)
Best
64.83329341
7.038580766
9.44877
E
-
13
9.89913
E
-
06
4.1161
E
-
08
0.153346238
0.809497151
1.34978
E
-
32
Mean
571341.8789
13.34788267
3.57122
E
-
12
6.3923
E
-
05
0.056026931
11.23615156
1.07318734
4.63024
E
-
28
Worst
5468139.556
20.53950899
1.56888
E
-
11
0.000309668
0.393967438
24.58405568
1.574427431
9.1725
E
-
27
Std.
1242194.366
4.368603518
3.0416
E
-
12
6.7996
E
-
05
0.097188974
8.579601122
0.193250844
2.05001
E
-
27Table 9
Test statistical results of functionsf14~f18.
Benchmark function
BA
PSO
DE
ACO
CS
FPA
CSO
EOCSO
f
14
(
x
)
Best
0.998003838
0.998003838
0.998003838
1.000647439
0.998003838
0.998003838
0.998003838
0.998003838
Mean
6.991491327
0.998003839
0.998003838
1.937812339
0.998003838
0.998003838
0.998003838
0.998003838
Worst
20.15348696
0.99800384
0.998003838
3.969893097
0.998003838
0.998003838
0.998003838
0.998003838
Std.
5.441619771
5.35015
E
-
10
0
0.73957407
0
0
0
0
f
15
(
x
)
Best
0.000307834
0.000307498
0.000368258
0.001325092
0.000483732
0.000307486
0.000330135
0.000307486
Mean
0.001478932
0.000510263
0.000629779
0.002483753
0.000707371
0.000307486
0.000615881
0.000307486
Worst
0.007775518
0.000780325
0.000764669
0.004487698
0.001227785
0.000307486
0.001262543
0.000307486
Std.
0.001522132
0.000197339
0.000107155
0.000907677
0.000195684
1.0842
E
-
19
0.000208736
0
f
16
(
x
)
Best
-
3.862782147
-
3.862781836
-
3.862782148
-
3.862782111
-
3.862782148
-
3.862782148
-
3.862782148
-
3.862782148
Mean
-
3.862782125
-
3.862776258
-
3.862782148
-
3.862781771
-
3.862782148
-
3.862782148
-
3.862782148
-
3.862782148
Worst
-
3.862782102
-
3.862763876
-
3.862782148
-
3.862780215
-
3.862782148
-
3.862782148
-
3.862782146
-
3.862782148
Std.
1.44804
E
-
08
4.86058
E
-
06
2.27813
E
-
15
4.04355
E
-
07
2.27813
E
-
15
2.27813
E
-
15
4.9224
E
-
10
2.27813
E
-
15
f
17
(
x
)
Best
-
3.321994333
-
3.321950303
-
3.321995172
-
3.317012353
-
3.32199515
-
3.321995172
-
3.321995172
-
3.321995172
Mean
-
3.262546901
-
3.228484991
-
3.321440415
-
3.24287723
-
3.286327027
-
3.321995172
-
3.291752652
-
3.321995172
Worst
-
3.203098591
-
3.132218821
-
3.311000677
-
2.949849459
-
3.203099953
-
3.321995172
-
3.200111233
-
3.321995172
Std.
0.060990736
0.081326677
0.002457311
0.08989745
0.055899237
4.55626
E
-
16
0.053478449
9.11252
E
-
16
f
18
(
x
)
Best
-
10.5364095
-
10.53508305
-
10.53640982
-
10.43395927
-
10.53640982
-
10.53640982
-
10.53640982
-
10.53640981
Mean
-
6.282649281
-
9.273746833
-
10.53640982
-
9.464159333
-
10.53640982
-
10.53640982
-
10.53536203
-
10.53636531
Worst
-
2.427335106
-
2.425929198
-
10.53640982
-
5.139776016
-
10.53640982
-
10.53640982
-
10.52081005
-
10.53595233
Std.
3.666609304
2.64050751
3.99158
E
-
13
1.221326502
3.64501
E
-
15
0.00010276
0.003534247
1.8225
E
-
15The test statistical results of functionsf1~f7 are shown in Table 7. As can be seen from Table 7, the best value, mean value, standard variance, and the worst value of EOCSO are all superior to those of other intelligent algorithms (including BA, PSO, DE, ACO, CS, FPA, and CSO). Particularly, global minimum of the test function f5(x) lies in the bottom of parabola, which is quite reliable to fall into a local optimum in the searching process. The optimal result of EOCSO is 9.53224E-09, which is 8, 11, 10, 10, 7, 3, and 10 orders of magnitude higher than that of BA, PSO, DE, ACO, CS, FPA, and CSO, respectively. The mean value of EOCSO is 1.61047E-5, which is 5, 7, 4, 4, 4, 4, and 6 orders of magnitude higher than that of the compared algorithms, respectively. Besides, EOCSO algorithm is also superior to seven other intelligent algorithms in terms of the worst value and standard variance. It comes to a conclusion that EOCSO algorithm has high precision and good robustness in searching high-dimensional unimodal function.The test statistical result of functionsf8~f13 is demonstrated in Table 8, from which we can find that EOSCO algorithm can obtain global optimal extreme value for f8, f7, and f11, which is free from interference of local extreme value. For other algorithms, only CSO can obtain global optimum in searching of f9 and f11. For f10, the result is almost equivalent to that of EOCSO, CSO, and CS. Therefore, the original CSO is suitable to solve this function; the improvement of the solution after optimization is not obvious. With regard to f12~f13, EOSCO does better than five other algorithms in precision and standard variance of its solution, which reflects the advantage of EOCSO.In Table9, the test statistical result of low-dimension multimodal functions f14~f18 is described. EOCSO and seven other algorithms can be seen to be able to get the global optimum in searching process, but EOCSO has fixed advantages in its mean value, worst value, and standard variance. “Error” on y-axis is the error between actual global minimum and converged minimum. The convergence graphs of the 6 algorithms are expressed in Figures 7–24. In EOCSO algorithm, its convergence rate of all 18 functions is more excellent than that in the other five algorithms, especially for f1~f4, f6, and f8~f13. The convergence curve is quite smooth and drops rapidly, reflecting its good convergence rate.Figure 7
Convergence rates forf1(x).Figure 8
Convergence rates forf2(x).Figure 9
Convergence rates forf3(x).Figure 10
Convergence rates forf4(x).Figure 11
Convergence rates forf5(x).Figure 12
Convergence rates forf6(x).Figure 13
Convergence rates forf7(x).Figure 14
Convergence rates forf8(x).Figure 15
Convergence rates forf9(x).Figure 16
Convergence rates forf10(x).Figure 17
Convergence rates forf11(x).Figure 18
Convergence rates forf12(x).Figure 19
Convergence rates forf13(x).Figure 20
Convergence rates forf14(x).Figure 21
Convergence rates forf15(x).Figure 22
Convergence rates forf16(x).Figure 23
Convergence rates forf17(x).Figure 24
Convergence rates forf18(x).


### 4.7. Structural Design Examples

In order to validate the performance of proposed method for constraint problems, EOCSO is examined by solving constrained engineering design problems, such as speed reducer design problem and pressure vessel design problem.


#### 4.7.1. Speed Reducer Design Problem

Speed reducer design problem is proposed by the famous scholar Mezura-Montes, which is a classic constrained optimization and is used to verify the design engineering constrained optimization algorithm performance, as shown in Figure25. There are 7 variables and 11 inequality constraints of the problem. The mathematical model is represented as follows:(13)minfx=0.7854x1x223.3333x32+14.9334x3-43.0934-1.508x1x62+x72+7.477x63+x73+0.7854x4x62+x5x72s.t.g1x=27x1x22x3-1≤0,g2x=397.5x1x22x32-1≤0,g3x=1.93x43x2x3x64-1≤0,g4x=1.93x53x2x3x74-1≤0,g5x=745x4/x2x32+16.9×1061/2110x63-1≤0,g6x=745x5/x2x32+157.5×1061/285x73-1≤0,g7x=x2x340-1≤0,g8x=5x2x1-1≤0,g9x=x112x2-1≤0,g10x=1.5x6+1.9x4-1≤0,g11x=1.5x6+1.7x5-1≤0,where 2.6≤x1≤3.6, 0.7≤x2≤0.8, 17≤x3≤28,7.3≤x4,x5≤8.3, 2.9≤x6≤3.9, and 5.0≤x7≤5.5. This case study has been previously solved using other methods such as PSO-DE [31], MBA [32], HEAA [33], and HGA [34]. The best results of the various methods for solving the speed reducer design problem are shown in Table 7. Table 8 shows the statistical results of the different methods.Figure 25
Speed reducer design problem.It can be seen from Table10 that the optimal solution of the EOCSO algorithm is better than the other 4 kinds of algorithms. From Table 11, the optimal value, the average value, and the worst value of EOCSO are better than those of the other 4 algorithms. In terms of stability, the standard deviation of EOCSO is smaller than those of MBA and HEAA algorithms by 6 and 4 orders of magnitude, respectively.Table 10
Comparison of the best solutions obtained by different methods for speed reducer design problem.
EOCSO
PSO-DE [31]
MBA [32]
HEAA [33]
HGA [34]
x
1
3.500000000000003
3.5000000
3.500000
3.5000228993
3.500000
x
2
0.700000000000000
0.700000
0.700000
0.7000003924
0.700000
x
3
17.00000000000000
17.000000
17.000000
17.0000128592
17
x
4
7.300000000000000
7.300000
7.300033
7.3004277414
7.300000
x
5
7.715319911478278
7.800000
7.715772
7.7153774494
7.71533234
x
6
3.350214666096451
3.350214
3.350218
3.3502309666
3.35021511
x
7
5.286654464980222
5.2866832
5.286654
5.2866636970
5.28666404
g
1
(
x
)
-
0.073915280397874
-
0.07391528
-
0.07391528
-
0.07392283
-
0.07391528
g
2
(
x
)
-
0.197998527141950
-
0.19799853
-
0.19799853
-
0.19800568
-
0.19799853
g
3
(
x
)
-
0.499172248102422
-
0.49917185
-
0.49916745
-
0.49909455
-
0.49917251
g
4
(
x
)
-
0.904643904556068
-
0.90147170
-
0.90462711
-
0.90464255
-
0.90464413
g
5
(
x
)
-
3.33066907388
e
-
15
5.96466298
e
-
07
-
2.93020139
e
-
06
-
1.3895524
e
-
05
-
3.97499993
e
-
07
g
6
(
x
)
0
1.68865328
e
-
08
3.505341306
e
-
07
-
5.2295669
e
-
06
-
5.43110185
e
-
06
g
7
(
x
)
-
0.702500000000000
-
0.7025000
-
0.70250000
-
0.7024996
-
0.70250000
g
8
(
x
)
-
8.881784197001
e
-
16
0
0
-
6.1140457
e
-
06
0
g
9
(
x
)
-
0.583333333333333
-
0.58333333
-
0.58333333
-
0.58333078
-
0.58333333
g
10
(
x
)
-
0.051325753541825
-
0.05132589
-
0.05132936
-
0.05137799
-
0.05132566
g
11
(
x
)
-
4.21884749358
e
-
15
-
0.01085237
-
5.86590687
e
-
005
-
6.14133272
e
-
06
-
2.45744437
e
-
07
f
(
x
)
2994.341315684048
2996.348167
2994.482453
2994.499107
2994.47Table 11
Comparison of statistical results for speed reducer design problem by the various algorithms.
Algorithm
Best
Mean
Worst
Std.
EOCSO
2994.3413156840
2994.3413169852
2994.341318013
8
1.201279
1
E
-
06
PSO-DE [29]
2996.348167
2996.348174
2996.348204
6.4
E
-
06
MBA [34]
2994.482453
2996.769019
2999.652444
1.56
HEAA [30]
2994.499107
2994.613368
2994.752311
7.0
E
-
02
HGA [25]
2994.47
N
A
N
A
N
A


#### 4.7.2. Pressure Vessel Design Problem

Another benchmark structural optimization problem is the pressure vessel design problem proposed by Kannan and Kramer. Figure26 shows a pressure vessel design problem which has four variables (x1,x2,x3,x4) and four nonlinear inequality constraints (g1,g2,g3,g4). The objective function of the problem can be expressed as follows:(14)minfx=0.6224x1x3x4+1.7881x2x32+3.1661x12x4+19.84x12x3s.t.g1x=-x1+0.0193x3≤0g2x=-x2+0.0095x3≤0g3x=-πx32x4-43πx33+129000≤0g4x=x4-240≤0,where 0≤x1,x2≤100 and 10≤x3,x4≤200. The approaches have previously been applied to solve this problem including many different numerical optimization techniques, such as the new modification approach on bat algorithm (EBA) [35], the cuckoo search (CS) algorithm  [7],  interior search algorithm (ISA) [36], the mine blast algorithm (MBA) [32], the improved ant colony optimization (IACO) [37], an effective hybrid cuckoo search algorithm for constrained global optimization (HCS-LSAL) [38], the hybrid Nelder-Mead simplex search and particle swarm optimization (NM-PSO) [39], an improved accelerated PSO algorithm [40], and crow search algorithm (CSA) [41].Figure 26
The pressure vessel design problem.The best solutions obtained by the various methods are reported in Table12. Table 13 shows the statistical results. It is shown in Tables 12 and 13 that EOCSO performance for the pressure vessel design problem surpassed the other 10 methods in terms of the minimum obtained value, solution average, and the standard deviation.Table 12
Comparison of the best solutions for pressure vessel design problem by different algorithms.
Algorithm
x
1
x
2
x
3
x
4
g
1
x
g
2
x
g
3
x
g
4
x
f
x
EOCSO
0.778169
0.384649
40.31962
200
-
7.20
E
-
11
-
1.12
E
-
09
-
4.29
E
-
04
-
40
5885.3328
CS  [7]
0.8125
0.4375
42.098446
176.636596
—
—
—
—
6059.714
MBA [32]
0.7802
0.3856
40.4292
198.4964
0
0
-
86.3645
-
41.5035
5889.3216
HCS-LSAL [38]
0.8125
0.4375
42.09844
176.6366
-
2.01
E
-
09
-
0.0331588
-
0.002495
-
63.3634
6059.7143
NM-PSO [39]
0.8036
0.3972
41.6392
182.412
3.65
E
-
05
3.79
E
-
05
-
1.5914
-
57.5879
5930.3137
CSA [41]
0.8125
0.4375
42.098445
176.636598
-
4.02
E
+
09
-
0.0358808
-
7.12
E
+
04
-
63.3634
6059.7144Table 13
Statistical results of different approaches for pressure vessel design problem.
Algorithm
Best
Mean
Worst
Std.
EOCSO
5885.3328
5885.3328
5885.33279
5.14
E
-
06
EBA [35]
6059.71
6173.67
6370.77
142.33
CS  [7]
6059.714
6447.736
6495.347
502.693
ISA [36]
6059.71
6410.08
7332.84
384.6
MBA [32]
5889.32
6200.64
6392.5
160.34
IACO [37]
6059.73
6081.78
6150.13
67.2418
HCS-LSAL [38]
5930.3137
5946.7901
5960.0557
9.1614
NM-PSO [39]
6059.7143
6087.3225
6137.4069
2.21
E
-
02
IAPSO [40]
6059.7143
6068.7539
6090.5314
14.0057
CSA [41]
6059.714363
6342.499106
7332.841621
384.9454163


## 4.1. Parameter Setting

To test the algorithm’s performance, seven typical intelligent algorithms, bat algorithm (BA) [10, 11], particle swarm optimization (PSO) [1], differential evolution (DE) [25], ant colony optimization (ACO)  [2],  cuckoo search (CS) algorithm [7], flower pollination algorithm (FPA) [13], and chick swarm algorithm (CSO) [14], are selected to do contrast experiments with EOCSO algorithm. The parameters of EOCSO algorithm are set as follows. Population size is set as 100, and each of the scale of cock swarm and that of chick swarm occupies 15%, while the scale of hen swarm takes up 70%. All of the common parameters of these algorithms (including the population size, dimensions, and maximum number of generations) are set to be the same for a fair comparison. Other parameters are described in Section 3. The parameters set of all other five algorithms is shown in Table 1.Table 1
The parameters set of all other algorithms.
Algorithms
Parameters
BA
Q
m
i
n
=
0,  Qmax=2,  R0=0.1,  A=0.9,  α=0.95,  γ=0.9
PSO
c
1
=
1.49445,  c2=1.49445,  ϖ=0.729
DE
p
C
R
=
0.2,  βmin=0.2,  βmax=0.8
ACO
The intensification factorq=0.5, the deviation-distance ratio zeta=1
CS
p
a
=
0.25,  α=1
FPA
p
=
0.8
CSO
N
r
=
0.15
p
o
p
s
i
z
e,  Nh=0.7popsize,  Nm=0.15popsize,  G=10


## 4.2. Test Function

To verify the optimization precision and convergence rate of the proposed algorithm, 18 standard test functions in [26, 27] are chosen for contrast experiments. f1~f7 are high-dimensional unimodal functions and are extremely hard to converge to a global optimum, which are used to inspect the searching precision. f8~f13 are high-dimensional multimodal functions with several local extreme points. So they are used to test the global searching performance and avoid premature of the algorithms [28–30]. f14~f18 are low-dimensional multimodal functions. The standard test functions are seen in Table 2.Table 2
Benchmark test functions.
Number
Name
Benchmark test functions
Dimension
Scope
Optimum
f
1
(
x
)
Sphere model
f
x
=
∑
i
=
1
D
x
i
2
30
[
-
100,100
]
0
f
2
(
x
)
Schwefel’s problem 2.22
f
x
=
∑
i
=
1
D
|
x
i
|
+
∏
i
=
1
D
|
x
i
|
30
[
-
10,10
]
0
f
3
(
x
)
Schwefel’s problem 1.2
f
x
=
∑
i
=
1
D
∑
j
=
1
i
x
i
2
30
[
-
100,100
]
0
f
4
(
x
)
Schwefel’s problem 2.21
f
x
=
m
a
x
i
=
1
D
x
i
30
[
-
100,100
]
0
f
5
(
x
)
Generalized Rosenbrock’s function
f
x
=
∑
i
=
1
n
-
1
100
x
i
+
1
-
x
i
2
2
+
1
-
x
i
2
30
[
-
30,30
]
0
f
6
(
x
)
Step function
f
x
=
∑
i
=
1
D
x
i
+
0.5
30
[
-
100,100
]
0
f
7
(
x
)
Quartic function, that is, noise
f
x
=
∑
i
=
1
D
i
⋅
x
i
4
+
r
a
n
d
o
m
0,1
30
[
-
1.28,1.28
]
0
f
8
(
x
)
Generalized Schwefel’s problem 2.26
f
x
=
∑
i
=
1
D
-
x
i
⋅
sin
⁡
x
i
30
[
-
500,500
]
-
418.9829
∗
n
f
9
(
x
)
Generalized Rastrigin’s function
f
x
=
∑
i
=
1
n
x
i
2
-
10
⋅
cos
⁡
2
⋅
π
⋅
x
i
+
10
30
[
-
5.12,5.12
]
0
f
10
(
x
)
Ackley’s function
f
(
x
)
=
-
20
exp
⁡
-
0.2
1
d
∑
i
=
1
d
x
i
2
-
exp
⁡
1
d
∑
i
=
1
d
cos
⁡
2
π
x
i
+
20
+
e
30
[
-
32,32
]
0
f
11
(
x
)
Generalized Griewank function
f
(
x
)
=
1
4000
⋅
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
c
o
s
x
i
i
+
1
30
[
-
600,600
]
0
f
12
(
x
)
Generalized penalized function
f
x
=
π
D
10
sin
2
⁡
π
⋅
y
1
+
∑
i
=
1
D
-
1
y
i
-
1
2
1
+
10
sin
2
⁡
π
⋅
y
i
+
1
+
y
n
-
1
2
+
∑
i
=
1
n
u
x
i
,
10,100,4 yi=1+xi+14 uxi,α,k,m=k(xi-α)m,xi>α0,-α≤xi≤αk(-xi-α)m,xi≤α
30
[
-
50,50
]
0
f
13
(
x
)
Generalized penalized Function
f
(
x
)
=
0.1
10
sin
2
⁡
(
3
π
⋅
x
1
)
+
∑
i
=
1
D
-
1
x
i
-
1
2
1
+
10
sin
2
⁡
3
π
⋅
x
i
+
1
+
x
n
-
1
2
+
∑
i
=
1
n
u
x
i
,
10,100,4
30
[
-
50,50
]
0
f
14
(
x
)
Shekel’s foxholes function
f
(
x
)
=
1
500
+
∑
j
=
25
25
1
j
+
∑
i
=
1
2
(
x
i
-
a
i
j
)
6
-
1
2
[
-
65.56,65.56
]
0.9980
f
15
(
x
)
Kowalik’s function
f
(
x
)
=
∑
i
=
1
11
a
i
-
x
1
(
b
i
2
+
b
i
x
2
)
b
i
2
+
b
i
x
3
+
x
4
2
4
[
-
5,5
]
0.0003075
f
16
(
x
)
Hartman’s function
f
(
x
)
=
-
∑
i
=
1
4
c
i
⋅
e
x
p
-
∑
j
=
1
3
a
i
j
(
x
j
-
p
i
j
)
2
3
[
0,1
]
-
3.8628
f
17
(
x
)
Hartman’s function
f
(
x
)
=
-
∑
i
=
1
4
c
i
⋅
e
x
p
-
∑
j
=
1
6
a
i
j
(
x
j
-
p
i
j
)
2
6
[
0,1
]
-
3.32
f
18
(
x
)
Shekel’s family
f
(
x
)
=
-
∑
i
=
1
10
(
x
-
a
i
)
(
x
-
a
i
)
T
+
C
i
-
1
10
[
0,10
]
-
10.5364


## 4.3. Influence of Dynamic Adaptivet Distribution Search Strategy on Bird Swarm Algorithm

Influence of dynamic adaptivet distribution search strategy on bird swarm algorithm contains two parts. The first part is the influence of different parameter values of variable control factor ψ on Gaussian distribution algorithm. The second part is the empirical comparison and analysis of CSO under dynamic adaptive t distribution (ATD-CSO) and dynamic adaptive Gaussian distribution (AGD-CSO) as well as the original algorithm (O-CSO). High-dimensional unimodal functions (f3,f5), high-dimensional multimodal functions (f11,f13), and low-dimensional multimodal functions (f15,f17) are selected as test functions.


### 4.3.1. Impact Analysis of Control Factorψ

Other parameters remain invariant, 0.2, 0.5, and 0.8, and dynamic adaptive change is used as control factorψ for Gaussian distribution algorithm. Test function parameters are set the same as Section 4.1, and test result is assessed from optimal value, mean value, worst value, and standard variance. Table 3 shows the statistical result of different ψ on the test functions.Table 3
Influence of variousψ on the test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
ψ
=
0.2
Best
0.01631162
25.28779662
0
0.608008583
0.000307777
-
3.321995171
Mean
5.826567112
26.47886719
0
2.347145417
0.000428489
-
3.308813113
Worst
25.52896184
28.91557367
0
4.237078833
0.001223174
-
3.202409203
Std.
7.284604701
0.4586344
0
0.92771315
0.000277897
0.036625398
ψ
=
0.5
Best
0.530928436
25.99904272
0
0.69896625
0.00031048
-
3.321877282
Mean
132.4241294
27.1885086
0
1.34834884
0.00044856
-
3.315628525
Worst
729.2974459
28.55843288
0
1.96348532
0.001223763
-
3.201008218
Std.
197.5724792
0.696629183
0
0.287524396
0.000224541
0.026987801
ψ
=
0.8
Best
0.018825799
26.28800133
0
0.717391541
0.000315938
-
3.321995171
Mean
500.9268589
27.17671059
0
1.083153802
0.000534308
-
3.301356117
Worst
1228.354641
28.77558754
0
1.33573296
0.00071324
-
3.199239821
Std.
400.9524735
0.60589184
0
0.219814939
0.000140829
0.044548317
Original strategy
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
Dynamic adaptive
Best
1.455387
E
-
22
0.09980032
0
1.90772
E
-
12
0.000307486
-
3.321995172
Mean
2.932745
E
-
20
0.57800657
0
9.09553
E
-
12
0.000307486
-
3.321995172
Worst
9.453261
E
-
20
1.08886543
0
6.57632
E
-
10
0.000307486
-
3.321995172
Std.
5.187532
E
-
21
0.19345322
0
2.42964
E
-
12
0
5.264821
E
-
12As can be seen from Table3, search effect is better when ψ is smaller. But it is not quite ideal for Gaussian distribution with fixed ψ to improve algorithm performance. Through dynamic adaptive parameter setting, the algorithm can be ensured to maintain a large search step size in initial search stage. In later stage, dynamic adaptive step size of control factor ψ is reduced, while the local search ability is enhanced. The test result proved that dynamic adaptive control factor has more advantages than single fixed control factor.


### 4.3.2. Comparison of ATD-CSO, AGD-CSO, and O-CSO

Table4 indicates the statistical result of dynamic adaptive t distribution, dynamic adaptive Gaussian distribution, and original CSO algorithm on three different kinds of test function. As the data shows, in terms of optimal values, 6 and 4 orders of magnitude are improved for f3 and f13 separately by ATD-CSO compared with AGD-CSO. For function f5, searching precision of ATD-CSO is higher than that of AGD-CSO. For f11, f15, and f17, optimal solution is searched by both ATD-CSO and AGD-CSO. Also, for mean value, worst value, and standard variance, f11 and f15 obtain the same through these two algorithms. But, for the standard variance for f17, ATD-CSO is better than AGD-CSO. Compared with O-CSO, ATD-CSO produces optimal mean value, worst value, and variance. It can be seen from Figures 1–6 that, in early evolution period, convergence rate of ATD-CSO is faster than that of AGD-CSO when calculating the above 6 functions. In later period, the convergence rate is similar, but the overall convergence rate of ATD-CSO is more optimal. However, the convergence rate of O-CSO is slower than that of ATD-CSO and AGD-CSO. Therefore, ATD-CSO has some advantages in terms of solution precision, robustness, and convergence rate generally.Table 4
Influence of different distributions on test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
Original CSO
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
AGD-CSO
Best
1.455387
E
-
22
0.09980032
0
1.90772
E
-
12
0.000307486
-
3.321995172
Mean
2.932745
E
-
20
0.57800657
0
9.09553
E
-
12
0.000307486
-
3.321995172
Worst
9.453261
E
-
20
1.08886543
0
6.57632
E
-
10
0.000307486
-
3.321995172
Std.
5.187532
E
-
21
0.19345322
0
2.42964
E
-
12
0
5.264821
E
-
12
ATD-CSO
Best
3.81028
E
-
28
0.050669099
0
5.43358
E
-
16
0.000307486
-
3.321995172
Mean
1.24542
E
-
27
0.18932938
0
5.16541
E
-
15
0.000307486
-
3.321995172
Worst
1.06346
E
-
27
0.67578053
0
4.81746
E
-
13
0.000307486
-
3.321995172
Std.
9.19088
E
-
28
0.037375926
0
4.52782
E
-
15
0
9.858438
E
-
14Figure 1
Convergence rates forf3(x).Figure 2
Convergence rates forf5(x).Figure 3
Convergence rates forf11(x).Figure 4
Convergence rates forf13(x).Figure 5
Convergence rates forf15(x).Figure 6
Convergence rates forf17(x).


## 4.3.1. Impact Analysis of Control Factorψ

Other parameters remain invariant, 0.2, 0.5, and 0.8, and dynamic adaptive change is used as control factorψ for Gaussian distribution algorithm. Test function parameters are set the same as Section 4.1, and test result is assessed from optimal value, mean value, worst value, and standard variance. Table 3 shows the statistical result of different ψ on the test functions.Table 3
Influence of variousψ on the test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
ψ
=
0.2
Best
0.01631162
25.28779662
0
0.608008583
0.000307777
-
3.321995171
Mean
5.826567112
26.47886719
0
2.347145417
0.000428489
-
3.308813113
Worst
25.52896184
28.91557367
0
4.237078833
0.001223174
-
3.202409203
Std.
7.284604701
0.4586344
0
0.92771315
0.000277897
0.036625398
ψ
=
0.5
Best
0.530928436
25.99904272
0
0.69896625
0.00031048
-
3.321877282
Mean
132.4241294
27.1885086
0
1.34834884
0.00044856
-
3.315628525
Worst
729.2974459
28.55843288
0
1.96348532
0.001223763
-
3.201008218
Std.
197.5724792
0.696629183
0
0.287524396
0.000224541
0.026987801
ψ
=
0.8
Best
0.018825799
26.28800133
0
0.717391541
0.000315938
-
3.321995171
Mean
500.9268589
27.17671059
0
1.083153802
0.000534308
-
3.301356117
Worst
1228.354641
28.77558754
0
1.33573296
0.00071324
-
3.199239821
Std.
400.9524735
0.60589184
0
0.219814939
0.000140829
0.044548317
Original strategy
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
Dynamic adaptive
Best
1.455387
E
-
22
0.09980032
0
1.90772
E
-
12
0.000307486
-
3.321995172
Mean
2.932745
E
-
20
0.57800657
0
9.09553
E
-
12
0.000307486
-
3.321995172
Worst
9.453261
E
-
20
1.08886543
0
6.57632
E
-
10
0.000307486
-
3.321995172
Std.
5.187532
E
-
21
0.19345322
0
2.42964
E
-
12
0
5.264821
E
-
12As can be seen from Table3, search effect is better when ψ is smaller. But it is not quite ideal for Gaussian distribution with fixed ψ to improve algorithm performance. Through dynamic adaptive parameter setting, the algorithm can be ensured to maintain a large search step size in initial search stage. In later stage, dynamic adaptive step size of control factor ψ is reduced, while the local search ability is enhanced. The test result proved that dynamic adaptive control factor has more advantages than single fixed control factor.


## 4.3.2. Comparison of ATD-CSO, AGD-CSO, and O-CSO

Table4 indicates the statistical result of dynamic adaptive t distribution, dynamic adaptive Gaussian distribution, and original CSO algorithm on three different kinds of test function. As the data shows, in terms of optimal values, 6 and 4 orders of magnitude are improved for f3 and f13 separately by ATD-CSO compared with AGD-CSO. For function f5, searching precision of ATD-CSO is higher than that of AGD-CSO. For f11, f15, and f17, optimal solution is searched by both ATD-CSO and AGD-CSO. Also, for mean value, worst value, and standard variance, f11 and f15 obtain the same through these two algorithms. But, for the standard variance for f17, ATD-CSO is better than AGD-CSO. Compared with O-CSO, ATD-CSO produces optimal mean value, worst value, and variance. It can be seen from Figures 1–6 that, in early evolution period, convergence rate of ATD-CSO is faster than that of AGD-CSO when calculating the above 6 functions. In later period, the convergence rate is similar, but the overall convergence rate of ATD-CSO is more optimal. However, the convergence rate of O-CSO is slower than that of ATD-CSO and AGD-CSO. Therefore, ATD-CSO has some advantages in terms of solution precision, robustness, and convergence rate generally.Table 4
Influence of different distributions on test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
Original CSO
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
AGD-CSO
Best
1.455387
E
-
22
0.09980032
0
1.90772
E
-
12
0.000307486
-
3.321995172
Mean
2.932745
E
-
20
0.57800657
0
9.09553
E
-
12
0.000307486
-
3.321995172
Worst
9.453261
E
-
20
1.08886543
0
6.57632
E
-
10
0.000307486
-
3.321995172
Std.
5.187532
E
-
21
0.19345322
0
2.42964
E
-
12
0
5.264821
E
-
12
ATD-CSO
Best
3.81028
E
-
28
0.050669099
0
5.43358
E
-
16
0.000307486
-
3.321995172
Mean
1.24542
E
-
27
0.18932938
0
5.16541
E
-
15
0.000307486
-
3.321995172
Worst
1.06346
E
-
27
0.67578053
0
4.81746
E
-
13
0.000307486
-
3.321995172
Std.
9.19088
E
-
28
0.037375926
0
4.52782
E
-
15
0
9.858438
E
-
14Figure 1
Convergence rates forf3(x).Figure 2
Convergence rates forf5(x).Figure 3
Convergence rates forf11(x).Figure 4
Convergence rates forf13(x).Figure 5
Convergence rates forf15(x).Figure 6
Convergence rates forf17(x).


## 4.4. The Influence of Dynamic Adaptivet Distribution and Opposition-Based Learning on CSO (ATD-EO-CSO)

As demonstrated in Table5, the best value, mean value, worst value, and variance of f3,f5,f13,f15 obtained by dynamic adaptive t distribution and elite opposition-based learning are apparently better than those obtained by original CSO. Target value can be searched on function f11 by both ATD-EO-CSO and O-CSO. Therefore, the O-CSO is suitable to solve this function, and the improvement of the solution after optimization is not obvious. For function f17, the best value can be searched by both. However, in terms of mean value, worst value, and variance, the results will be better based on cowork of both t distribution and elite opposition-based learning. Since searching can be done more widely via elite opposition-based learning, boundary buffering wall is adopted to deal with individuals crossing the border, avoiding the deficiency that individuals gather on boundary value to improve the diversity of population. As a result, ATD-EO-CSO is superior in solution precision.Table 5
Statistical results of the influence of dynamic adaptivet distribution and opposition-based learning on CSO.
Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
Original CSO
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
ATD-EO-CSO
Best
7.108233
E
-
42
2.844564
E
-
6
0
2.095481
E
-
25
0.000307486
-
3.321995172
Mean
1.039602
E
-
40
9.654632
E
-
6
0
2.006629
E
-
24
0.000307486
-
3.321995172
Worst
4.540331
E
-
39
3.239232
E
-
5
0
9.910062
E
-
24
0.000307486
-
3.321995172
Std.
3.919585
E
-
41
3.6784872
E
-
7
0
1.887675
E
-
25
0
6.455396
E
-
14


## 4.5. Influence of Local Search via Greedy Dimension-by-Dimension Search on Bird Swarm Algorithm

Influence of local search via greedy dimension-by-dimension search on bird swarm algorithm contains two parts. The first part is the influence of fixed step sizestep=0.1, step=0.3, step=0.6, and the adaptive step on algorithm. The second part is the comparison between adaptive step strategy and original CSO. Table 6 shows the statistical result by means of various step values. According to Table 6, when step value is small, higher search precision can be obtained, but it is liable to get into premature convergence. In case of larger step value, the range of search step size is larger, and it is liable to skip optimal solution range, which causes lower searching effect in later stage of evolution and reduced local search ability. Through adaptive step, the algorithm’s global research can be ensured to conduct in long step size at initial stage and in short step size at later stage, which improves the search efficiency of dimension-by-dimension search. Hence, adaptive step dimension-by-dimension search is superior to the fixed step size strategy in terms of search precision.Table 6
Influence of various step values on test functions.
⁢Benchmark function
f
3
(
x
)
f
5
(
x
)
f
11
(
x
)
f
13
(
x
)
f
15
(
x
)
f
17
(
x
)
s
t
e
p
=
0.1
Best
1.1009139
E
-
30
2.5022290
E
-
4
0
7.1919782
E
-
22
0.000307486
-
3.321995172
Mean
2.5675751
E
-
26
7.1700102
E
-
4
0
2.0929358
E
-
20
0.000307486
-
3.321995163
Worst
5.9338088
E
-
25
5.0003902
E
-
3
0
9.0510335
E
-
20
0.000307486
-
3.321995061
Std.
5.4742928
E
-
26
4.8413835
E
-
5
0
6.9769787
E
-
22
7.537751
E
-
10
9.6237934
E
-
7
s
t
e
p
=
0.3
Best
3.0856694
E
-
28
4.3623649
E
-
5
0
4.9527803
E
-
18
0.000307486
-
3.321995172
Mean
6.4211983
E
-
25
3.3217367
E
-
4
0
5.8674385
E
-
15
0.000307486
-
3.321995172
Worst
5.9480356
E
-
24
7.1641617
E
-
4
0
6.7902546
E
-
12
0.000307486
-
3.321995163
Std.
1.2289958
E
-
26
8.0512434
E
-
5
0
2.6579955
E
-
16
7.967684
E
-
10
5.616744
E
-
8
s
t
e
p
=
0.6
Best
4.0232351
E
-
26
9.7209661
E
-
4
0
5.2857901
E
-
25
0.000307486
-
3.321995172
Mean
1.3561952
E
-
22
3.5667080
E
-
3
0
9.2285451
E
-
20
0.000307499
-
3.321956549
Worst
7.4188975
E
-
21
8.5082852
E
-
3
0
2.1028715
E
-
18
0.000307495
-
3.321906894
Std.
6.2712862
E
-
22
2.4668710
E
-
4
0
7.0434808
E
-
20
4.460765
E
-
8
8.512414
E
-
5
Original CSO
Best
7.00
E
+
01
26.50669099
0
0.809497151
0.000330135
-
3.321995172
Mean
8.06
E
+
02
27.28932938
0
1.07318734
0.000615881
-
3.291752652
Worst
1.67
E
+
03
28.77578053
0
1.574427431
0.001262543
-
3.200111233
Std.
430.7691621
0.537375926
0
0.193250844
0.000208736
0.053478449
Adaptivestep
Best
9.5838158
E
-
40
8.9257907
E
-
3
0
9.4224685
E
-
25
0.000307486
-
3.321995172
Mean
1.8439439
E
-
36
3.7218918
E
-
2
0
5.3242987
E
-
23
0.000307486
-
3.321995172
Worst
8.5706881
E
-
36
7.5851102
E
-
2
0
8.1986156
E
-
22
0.000307486
-
3.321995172
Std.
1.0682863
E
-
37
2.3586933
E
-
2
0
2.1204222
E
-
23
9.124585
E
-
18
8.884563
E
-
15


## 4.6. Analysis of Experimental Results

In order to avoid being influenced by random factors, the experimental test for each case is conducted by 20 trials independently. The algorithm’s searching performance is assessed according to the best value, mean value, standard deviation, and the worst value produced by the test result. The number of iterations off14~f18 and f1~f13 is 50 and 1000 in each trial, respectively. The algorithm is examined by Matlab 2012a on the platform with Win 8 OS, Intel Core i5-4210U 2.4 GHZ CPU, and 4 GB memory. The test statistical results are shown in Tables 7, 8, and 9.Table 7
Test statistical results of functionsf1~f7.
Benchmark function
BA
PSO
DE
ACO
CS
FPA
CSO
EOCSO
f
1
(
x
)
Best
9.41672
E
-
06
6.774921774
3.036519
E
-
12
4.98686
E
-
08
0.040549145
6.34628
E
-
06
1.1893
E
-
55
1.19
E
-
201
Mean
1.20474
E
-
05
20.80098597
5.713553
E
-
12
1.8224
E
-
07
0.08183638
0.000732691
3.08046
E
-
52
7.0639
E
-
198
Worst
0.000014224
31.55593153
9.811158
E
-
12
3.76894
E
-
07
0.120709249
0.009162134
3.16469
E
-
51
7.2029
E
-
197
Std.
1.40714
E
-
06
9.102140636
2.040785
E
-
12
8.35921
E
-
08
0.021679322
0.002091063
8.83269
E
-
52
0
f
2
(
x
)
Best
0.012593909
6.594522786
3.042127
E
-
08
4.92993
E
-
05
1.479134511
1.97228
E
-
07
1.91405
E
-
43
6.5805
E
-
116
Mean
31.7282206
15.68696047
4.943835
E
-
08
0.000507759
2.180229997
7.11331
E
-
06
6.5601
E
-
41
2.13
E
-
112
Worst
131.069574
35.29663724
7.117019
E
-
08
0.00191464
3.029290054
1.51764
E
-
05
8.03444
E
-
40
9.5751
E
-
112
Std.
48.01800899
11.87133052
1.339655
E
-
08
0.000549138
0.403631239
4.85023
E
-
06
1.78694
E
-
40
2.8682
E
-
112
f
3
(
x
)
Best
2.27
E
-
05
4.13
E
+
02
15255.144282
11951.29445
5.99
E
+
02
2.57
E
-
02
7.00
E
+
01
4.49
E
-
51
Mean
3.94
E
+
02
9.17
E
+
02
24696.821583
16566.43552
8.59
E
+
02
6.70
E
-
02
8.06
E
+
02
9.55
E
-
49
Worst
4.77
E
+
03
1.75
E
+
03
29125.784537
22644.44883
1.10
E
+
03
1.10
E
-
01
1.67
E
+
03
1.28
E
-
47
Std.
1081.678915
411.6200929
3515.8752513
3287.33213
122.9938329
0.023267666
430.7691621
2.85739
E
-
48
f
4
(
x
)
Best
6.573087982
5.90965909
1.2612253015
3.058888831
5.273452649
12.24579118
0.000935694
1.41944
E
-
17
Mean
12.41978556
9.027042499
2.0432637030
5.055981092
6.116103354
15.33881514
3.027225289
2.20256
E
-
16
Worst
22.65352526
12.9169796
2.5042954408
7.296151756
7.215730774
19.26838273
10.74127991
7.30302
E
-
16
Std.
3.804254343
2.121495266
0.3514754143
0.953357307
0.52178179
2.184822605
3.480451504
1.96521
E
-
16
f
5
(
x
)
Best
0.196158304
343.6027646
25.837681251
17.1433613
0.013792405
2.12808
E
-
06
26.50669099
9.53224
E
-
09
Mean
1.836717634
530.442542
39.933721918
17.86118196
0.59160704
0.398901513
27.28932938
0.000161047
Worst
20.41255589
1427.570578
70.708954223
19.24540999
1.749078665
3.986765311
28.77578053
0.001214443
Std.
4.605435894
299.0234027
13.215780402
0.617903448
0.57410589
1.227005992
0.537375926
0.000359017
f
6
(
x
)
Best
7.69118
E
-
06
13.55667233
1.888545
E
-
12
5.85581
E
-
08
0
5.47537
E
-
06
1.671870558
0
Mean
9.79475
E
-
06
19.24687067
5.975184
E
-
12
1.68807
E
-
07
3.77265
E
-
10
0.000265874
2.125011694
0
Worst
1.13079
E
-
05
25.44278333
1.122444
E
-
11
4.86966
E
-
07
1.85832
E
-
09
0.000603906
2.710216862
0
Std.
9.19879
E
-
07
4.68251131
2.832052
E
-
12
9.4266
E
-
08
4.55015
E
-
10
0.000212809
0.301038821
0
f
7
(
x
)
Best
0.021225243
0.198906064
0.0171466731
0.01217843
0.001333226
0.029750255
0.000690044
0.001333226
Mean
0.039176522
0.294912344
0.0240711071
0.024336378
0.003336074
0.108161359
0.002471843
0.003336074
Worst
0.0606201
0.372732671
0.0301412708
0.034138264
0.004764046
0.160329105
0.005679536
0.004764046
Std.
0.011065883
0.061817637
0.0041777349
0.007196828
0.000889975
0.044585434
0.001443844
0.000889975Table 8
Test statistical results of functionsf8~f13.
⁢Benchmark function
BA
PSO
DE
ACO
CS
FPA
CSO
EOCSO
f
8
(
x
)
Best
-
8502.897133
-
7963.943031
-
12569.479219
-
6090.034663
-
9009.250145
-
10694.82285
-
9133.943944
-
12569.48662
Mean
-
7060.92851
-
6682.009356
-
12546.705396
-
8060.242464
-
8449.968964
-
10097.82587
-
8186.965487
-
12474.61802
Worst
-
5969.754314
-
5622.614213
-
12369.793372
-
351.75903
-
7986.143265
-
9466.344181
-
7087.508251
-
12405.53547
Std.
779.3469624
760.027236
53.131673
569.1183589
345.8135405
423.049839
596.4376456
51.6979597
f
9
(
x
)
Best
30.84598172
71.56741319
47.02959855
108.4758259
6.691249469
15.91943794
0
0
Mean
75.66883074
134.1736791
60.42457292
124.3756751
13.42928534
22.83514081
0
0
Worst
146.2602458
177.1407232
71.54930622
138.2030082
20.70524753
27.467636
0
0
Std.
32.32801039
35.8886515
7.058839227
7.737668073
4.409105246
3.170192621
0
0
f
10
(
x
)
Best
11.60989873
4.914039621
4.5281
E
-
07
6.54539
E
-
05
4.44089
E
-
15
2.013315503
4.44089
E
-
15
4.44089
E
-
15
Mean
13.64745922
5.590179783
6.07484
E
-
07
0.00013903
4.74731
E
-
15
2.981722685
5.50671
E
-
15
4.70507
E
-
15
Worst
15.09860799
6.074485544
7.84382
E
-
07
0.00022114
5.09361
E
-
15
4.298104668
7.99361
E
-
15
7.99361
E
-
15
Std.
0.891126943
0.311563718
1.11025
E
-
07
3.80777
E
-
05
2.25609
E
-
16
0.762341832
1.67035
E
-
15
7.99919
E
-
16
f
11
(
x
)
Best
44.79000242
0.997496962
6.55065
E
-
12
0.192490967
0
0.000152641
0
0
Mean
79.03694557
1.203784605
1.42400
E
-
10
0.360243829
0
0.016635909
0
0
Worst
136.5821255
1.333771315
5.74106
E
-
10
0.540731319
0
0.029692265
0
0
Std.
23.79302299
0.092691068
1.63674
E
-
10
0.092281983
0
0.009192596
0
0
f
12
(
x
)
Best
5.506766196
2.320391097
1.49415
E
-
13
1.11888
E
-
06
1.02781
E
-
09
2.91491
E
-
06
0.072486935
1.57054
E
-
32
Mean
2371.120996
5.817999746
5.97228
E
-
13
8.9491
E
-
06
0.024315241
0.288022193
0.127682199
8.79042
E
-
30
Worst
23844.45981
9.370164248
2.12638
E
-
12
2.61683
E
-
05
0.103669022
5.455285401
0.294365491
5.22
E
-
29
Std.
6390.937613
2.137993761
4.2672
E
-
13
7.29249
E
-
06
0.033905983
1.217902188
0.063713849
1.37918
E
-
29
f
13
(
x
)
Best
64.83329341
7.038580766
9.44877
E
-
13
9.89913
E
-
06
4.1161
E
-
08
0.153346238
0.809497151
1.34978
E
-
32
Mean
571341.8789
13.34788267
3.57122
E
-
12
6.3923
E
-
05
0.056026931
11.23615156
1.07318734
4.63024
E
-
28
Worst
5468139.556
20.53950899
1.56888
E
-
11
0.000309668
0.393967438
24.58405568
1.574427431
9.1725
E
-
27
Std.
1242194.366
4.368603518
3.0416
E
-
12
6.7996
E
-
05
0.097188974
8.579601122
0.193250844
2.05001
E
-
27Table 9
Test statistical results of functionsf14~f18.
Benchmark function
BA
PSO
DE
ACO
CS
FPA
CSO
EOCSO
f
14
(
x
)
Best
0.998003838
0.998003838
0.998003838
1.000647439
0.998003838
0.998003838
0.998003838
0.998003838
Mean
6.991491327
0.998003839
0.998003838
1.937812339
0.998003838
0.998003838
0.998003838
0.998003838
Worst
20.15348696
0.99800384
0.998003838
3.969893097
0.998003838
0.998003838
0.998003838
0.998003838
Std.
5.441619771
5.35015
E
-
10
0
0.73957407
0
0
0
0
f
15
(
x
)
Best
0.000307834
0.000307498
0.000368258
0.001325092
0.000483732
0.000307486
0.000330135
0.000307486
Mean
0.001478932
0.000510263
0.000629779
0.002483753
0.000707371
0.000307486
0.000615881
0.000307486
Worst
0.007775518
0.000780325
0.000764669
0.004487698
0.001227785
0.000307486
0.001262543
0.000307486
Std.
0.001522132
0.000197339
0.000107155
0.000907677
0.000195684
1.0842
E
-
19
0.000208736
0
f
16
(
x
)
Best
-
3.862782147
-
3.862781836
-
3.862782148
-
3.862782111
-
3.862782148
-
3.862782148
-
3.862782148
-
3.862782148
Mean
-
3.862782125
-
3.862776258
-
3.862782148
-
3.862781771
-
3.862782148
-
3.862782148
-
3.862782148
-
3.862782148
Worst
-
3.862782102
-
3.862763876
-
3.862782148
-
3.862780215
-
3.862782148
-
3.862782148
-
3.862782146
-
3.862782148
Std.
1.44804
E
-
08
4.86058
E
-
06
2.27813
E
-
15
4.04355
E
-
07
2.27813
E
-
15
2.27813
E
-
15
4.9224
E
-
10
2.27813
E
-
15
f
17
(
x
)
Best
-
3.321994333
-
3.321950303
-
3.321995172
-
3.317012353
-
3.32199515
-
3.321995172
-
3.321995172
-
3.321995172
Mean
-
3.262546901
-
3.228484991
-
3.321440415
-
3.24287723
-
3.286327027
-
3.321995172
-
3.291752652
-
3.321995172
Worst
-
3.203098591
-
3.132218821
-
3.311000677
-
2.949849459
-
3.203099953
-
3.321995172
-
3.200111233
-
3.321995172
Std.
0.060990736
0.081326677
0.002457311
0.08989745
0.055899237
4.55626
E
-
16
0.053478449
9.11252
E
-
16
f
18
(
x
)
Best
-
10.5364095
-
10.53508305
-
10.53640982
-
10.43395927
-
10.53640982
-
10.53640982
-
10.53640982
-
10.53640981
Mean
-
6.282649281
-
9.273746833
-
10.53640982
-
9.464159333
-
10.53640982
-
10.53640982
-
10.53536203
-
10.53636531
Worst
-
2.427335106
-
2.425929198
-
10.53640982
-
5.139776016
-
10.53640982
-
10.53640982
-
10.52081005
-
10.53595233
Std.
3.666609304
2.64050751
3.99158
E
-
13
1.221326502
3.64501
E
-
15
0.00010276
0.003534247
1.8225
E
-
15The test statistical results of functionsf1~f7 are shown in Table 7. As can be seen from Table 7, the best value, mean value, standard variance, and the worst value of EOCSO are all superior to those of other intelligent algorithms (including BA, PSO, DE, ACO, CS, FPA, and CSO). Particularly, global minimum of the test function f5(x) lies in the bottom of parabola, which is quite reliable to fall into a local optimum in the searching process. The optimal result of EOCSO is 9.53224E-09, which is 8, 11, 10, 10, 7, 3, and 10 orders of magnitude higher than that of BA, PSO, DE, ACO, CS, FPA, and CSO, respectively. The mean value of EOCSO is 1.61047E-5, which is 5, 7, 4, 4, 4, 4, and 6 orders of magnitude higher than that of the compared algorithms, respectively. Besides, EOCSO algorithm is also superior to seven other intelligent algorithms in terms of the worst value and standard variance. It comes to a conclusion that EOCSO algorithm has high precision and good robustness in searching high-dimensional unimodal function.The test statistical result of functionsf8~f13 is demonstrated in Table 8, from which we can find that EOSCO algorithm can obtain global optimal extreme value for f8, f7, and f11, which is free from interference of local extreme value. For other algorithms, only CSO can obtain global optimum in searching of f9 and f11. For f10, the result is almost equivalent to that of EOCSO, CSO, and CS. Therefore, the original CSO is suitable to solve this function; the improvement of the solution after optimization is not obvious. With regard to f12~f13, EOSCO does better than five other algorithms in precision and standard variance of its solution, which reflects the advantage of EOCSO.In Table9, the test statistical result of low-dimension multimodal functions f14~f18 is described. EOCSO and seven other algorithms can be seen to be able to get the global optimum in searching process, but EOCSO has fixed advantages in its mean value, worst value, and standard variance. “Error” on y-axis is the error between actual global minimum and converged minimum. The convergence graphs of the 6 algorithms are expressed in Figures 7–24. In EOCSO algorithm, its convergence rate of all 18 functions is more excellent than that in the other five algorithms, especially for f1~f4, f6, and f8~f13. The convergence curve is quite smooth and drops rapidly, reflecting its good convergence rate.Figure 7
Convergence rates forf1(x).Figure 8
Convergence rates forf2(x).Figure 9
Convergence rates forf3(x).Figure 10
Convergence rates forf4(x).Figure 11
Convergence rates forf5(x).Figure 12
Convergence rates forf6(x).Figure 13
Convergence rates forf7(x).Figure 14
Convergence rates forf8(x).Figure 15
Convergence rates forf9(x).Figure 16
Convergence rates forf10(x).Figure 17
Convergence rates forf11(x).Figure 18
Convergence rates forf12(x).Figure 19
Convergence rates forf13(x).Figure 20
Convergence rates forf14(x).Figure 21
Convergence rates forf15(x).Figure 22
Convergence rates forf16(x).Figure 23
Convergence rates forf17(x).Figure 24
Convergence rates forf18(x).


## 4.7. Structural Design Examples

In order to validate the performance of proposed method for constraint problems, EOCSO is examined by solving constrained engineering design problems, such as speed reducer design problem and pressure vessel design problem.


### 4.7.1. Speed Reducer Design Problem

Speed reducer design problem is proposed by the famous scholar Mezura-Montes, which is a classic constrained optimization and is used to verify the design engineering constrained optimization algorithm performance, as shown in Figure25. There are 7 variables and 11 inequality constraints of the problem. The mathematical model is represented as follows:(13)minfx=0.7854x1x223.3333x32+14.9334x3-43.0934-1.508x1x62+x72+7.477x63+x73+0.7854x4x62+x5x72s.t.g1x=27x1x22x3-1≤0,g2x=397.5x1x22x32-1≤0,g3x=1.93x43x2x3x64-1≤0,g4x=1.93x53x2x3x74-1≤0,g5x=745x4/x2x32+16.9×1061/2110x63-1≤0,g6x=745x5/x2x32+157.5×1061/285x73-1≤0,g7x=x2x340-1≤0,g8x=5x2x1-1≤0,g9x=x112x2-1≤0,g10x=1.5x6+1.9x4-1≤0,g11x=1.5x6+1.7x5-1≤0,where 2.6≤x1≤3.6, 0.7≤x2≤0.8, 17≤x3≤28,7.3≤x4,x5≤8.3, 2.9≤x6≤3.9, and 5.0≤x7≤5.5. This case study has been previously solved using other methods such as PSO-DE [31], MBA [32], HEAA [33], and HGA [34]. The best results of the various methods for solving the speed reducer design problem are shown in Table 7. Table 8 shows the statistical results of the different methods.Figure 25
Speed reducer design problem.It can be seen from Table10 that the optimal solution of the EOCSO algorithm is better than the other 4 kinds of algorithms. From Table 11, the optimal value, the average value, and the worst value of EOCSO are better than those of the other 4 algorithms. In terms of stability, the standard deviation of EOCSO is smaller than those of MBA and HEAA algorithms by 6 and 4 orders of magnitude, respectively.Table 10
Comparison of the best solutions obtained by different methods for speed reducer design problem.
EOCSO
PSO-DE [31]
MBA [32]
HEAA [33]
HGA [34]
x
1
3.500000000000003
3.5000000
3.500000
3.5000228993
3.500000
x
2
0.700000000000000
0.700000
0.700000
0.7000003924
0.700000
x
3
17.00000000000000
17.000000
17.000000
17.0000128592
17
x
4
7.300000000000000
7.300000
7.300033
7.3004277414
7.300000
x
5
7.715319911478278
7.800000
7.715772
7.7153774494
7.71533234
x
6
3.350214666096451
3.350214
3.350218
3.3502309666
3.35021511
x
7
5.286654464980222
5.2866832
5.286654
5.2866636970
5.28666404
g
1
(
x
)
-
0.073915280397874
-
0.07391528
-
0.07391528
-
0.07392283
-
0.07391528
g
2
(
x
)
-
0.197998527141950
-
0.19799853
-
0.19799853
-
0.19800568
-
0.19799853
g
3
(
x
)
-
0.499172248102422
-
0.49917185
-
0.49916745
-
0.49909455
-
0.49917251
g
4
(
x
)
-
0.904643904556068
-
0.90147170
-
0.90462711
-
0.90464255
-
0.90464413
g
5
(
x
)
-
3.33066907388
e
-
15
5.96466298
e
-
07
-
2.93020139
e
-
06
-
1.3895524
e
-
05
-
3.97499993
e
-
07
g
6
(
x
)
0
1.68865328
e
-
08
3.505341306
e
-
07
-
5.2295669
e
-
06
-
5.43110185
e
-
06
g
7
(
x
)
-
0.702500000000000
-
0.7025000
-
0.70250000
-
0.7024996
-
0.70250000
g
8
(
x
)
-
8.881784197001
e
-
16
0
0
-
6.1140457
e
-
06
0
g
9
(
x
)
-
0.583333333333333
-
0.58333333
-
0.58333333
-
0.58333078
-
0.58333333
g
10
(
x
)
-
0.051325753541825
-
0.05132589
-
0.05132936
-
0.05137799
-
0.05132566
g
11
(
x
)
-
4.21884749358
e
-
15
-
0.01085237
-
5.86590687
e
-
005
-
6.14133272
e
-
06
-
2.45744437
e
-
07
f
(
x
)
2994.341315684048
2996.348167
2994.482453
2994.499107
2994.47Table 11
Comparison of statistical results for speed reducer design problem by the various algorithms.
Algorithm
Best
Mean
Worst
Std.
EOCSO
2994.3413156840
2994.3413169852
2994.341318013
8
1.201279
1
E
-
06
PSO-DE [29]
2996.348167
2996.348174
2996.348204
6.4
E
-
06
MBA [34]
2994.482453
2996.769019
2999.652444
1.56
HEAA [30]
2994.499107
2994.613368
2994.752311
7.0
E
-
02
HGA [25]
2994.47
N
A
N
A
N
A


### 4.7.2. Pressure Vessel Design Problem

Another benchmark structural optimization problem is the pressure vessel design problem proposed by Kannan and Kramer. Figure26 shows a pressure vessel design problem which has four variables (x1,x2,x3,x4) and four nonlinear inequality constraints (g1,g2,g3,g4). The objective function of the problem can be expressed as follows:(14)minfx=0.6224x1x3x4+1.7881x2x32+3.1661x12x4+19.84x12x3s.t.g1x=-x1+0.0193x3≤0g2x=-x2+0.0095x3≤0g3x=-πx32x4-43πx33+129000≤0g4x=x4-240≤0,where 0≤x1,x2≤100 and 10≤x3,x4≤200. The approaches have previously been applied to solve this problem including many different numerical optimization techniques, such as the new modification approach on bat algorithm (EBA) [35], the cuckoo search (CS) algorithm  [7],  interior search algorithm (ISA) [36], the mine blast algorithm (MBA) [32], the improved ant colony optimization (IACO) [37], an effective hybrid cuckoo search algorithm for constrained global optimization (HCS-LSAL) [38], the hybrid Nelder-Mead simplex search and particle swarm optimization (NM-PSO) [39], an improved accelerated PSO algorithm [40], and crow search algorithm (CSA) [41].Figure 26
The pressure vessel design problem.The best solutions obtained by the various methods are reported in Table12. Table 13 shows the statistical results. It is shown in Tables 12 and 13 that EOCSO performance for the pressure vessel design problem surpassed the other 10 methods in terms of the minimum obtained value, solution average, and the standard deviation.Table 12
Comparison of the best solutions for pressure vessel design problem by different algorithms.
Algorithm
x
1
x
2
x
3
x
4
g
1
x
g
2
x
g
3
x
g
4
x
f
x
EOCSO
0.778169
0.384649
40.31962
200
-
7.20
E
-
11
-
1.12
E
-
09
-
4.29
E
-
04
-
40
5885.3328
CS  [7]
0.8125
0.4375
42.098446
176.636596
—
—
—
—
6059.714
MBA [32]
0.7802
0.3856
40.4292
198.4964
0
0
-
86.3645
-
41.5035
5889.3216
HCS-LSAL [38]
0.8125
0.4375
42.09844
176.6366
-
2.01
E
-
09
-
0.0331588
-
0.002495
-
63.3634
6059.7143
NM-PSO [39]
0.8036
0.3972
41.6392
182.412
3.65
E
-
05
3.79
E
-
05
-
1.5914
-
57.5879
5930.3137
CSA [41]
0.8125
0.4375
42.098445
176.636598
-
4.02
E
+
09
-
0.0358808
-
7.12
E
+
04
-
63.3634
6059.7144Table 13
Statistical results of different approaches for pressure vessel design problem.
Algorithm
Best
Mean
Worst
Std.
EOCSO
5885.3328
5885.3328
5885.33279
5.14
E
-
06
EBA [35]
6059.71
6173.67
6370.77
142.33
CS  [7]
6059.714
6447.736
6495.347
502.693
ISA [36]
6059.71
6410.08
7332.84
384.6
MBA [32]
5889.32
6200.64
6392.5
160.34
IACO [37]
6059.73
6081.78
6150.13
67.2418
HCS-LSAL [38]
5930.3137
5946.7901
5960.0557
9.1614
NM-PSO [39]
6059.7143
6087.3225
6137.4069
2.21
E
-
02
IAPSO [40]
6059.7143
6068.7539
6090.5314
14.0057
CSA [41]
6059.714363
6342.499106
7332.841621
384.9454163


## 4.7.1. Speed Reducer Design Problem

Speed reducer design problem is proposed by the famous scholar Mezura-Montes, which is a classic constrained optimization and is used to verify the design engineering constrained optimization algorithm performance, as shown in Figure25. There are 7 variables and 11 inequality constraints of the problem. The mathematical model is represented as follows:(13)minfx=0.7854x1x223.3333x32+14.9334x3-43.0934-1.508x1x62+x72+7.477x63+x73+0.7854x4x62+x5x72s.t.g1x=27x1x22x3-1≤0,g2x=397.5x1x22x32-1≤0,g3x=1.93x43x2x3x64-1≤0,g4x=1.93x53x2x3x74-1≤0,g5x=745x4/x2x32+16.9×1061/2110x63-1≤0,g6x=745x5/x2x32+157.5×1061/285x73-1≤0,g7x=x2x340-1≤0,g8x=5x2x1-1≤0,g9x=x112x2-1≤0,g10x=1.5x6+1.9x4-1≤0,g11x=1.5x6+1.7x5-1≤0,where 2.6≤x1≤3.6, 0.7≤x2≤0.8, 17≤x3≤28,7.3≤x4,x5≤8.3, 2.9≤x6≤3.9, and 5.0≤x7≤5.5. This case study has been previously solved using other methods such as PSO-DE [31], MBA [32], HEAA [33], and HGA [34]. The best results of the various methods for solving the speed reducer design problem are shown in Table 7. Table 8 shows the statistical results of the different methods.Figure 25
Speed reducer design problem.It can be seen from Table10 that the optimal solution of the EOCSO algorithm is better than the other 4 kinds of algorithms. From Table 11, the optimal value, the average value, and the worst value of EOCSO are better than those of the other 4 algorithms. In terms of stability, the standard deviation of EOCSO is smaller than those of MBA and HEAA algorithms by 6 and 4 orders of magnitude, respectively.Table 10
Comparison of the best solutions obtained by different methods for speed reducer design problem.
EOCSO
PSO-DE [31]
MBA [32]
HEAA [33]
HGA [34]
x
1
3.500000000000003
3.5000000
3.500000
3.5000228993
3.500000
x
2
0.700000000000000
0.700000
0.700000
0.7000003924
0.700000
x
3
17.00000000000000
17.000000
17.000000
17.0000128592
17
x
4
7.300000000000000
7.300000
7.300033
7.3004277414
7.300000
x
5
7.715319911478278
7.800000
7.715772
7.7153774494
7.71533234
x
6
3.350214666096451
3.350214
3.350218
3.3502309666
3.35021511
x
7
5.286654464980222
5.2866832
5.286654
5.2866636970
5.28666404
g
1
(
x
)
-
0.073915280397874
-
0.07391528
-
0.07391528
-
0.07392283
-
0.07391528
g
2
(
x
)
-
0.197998527141950
-
0.19799853
-
0.19799853
-
0.19800568
-
0.19799853
g
3
(
x
)
-
0.499172248102422
-
0.49917185
-
0.49916745
-
0.49909455
-
0.49917251
g
4
(
x
)
-
0.904643904556068
-
0.90147170
-
0.90462711
-
0.90464255
-
0.90464413
g
5
(
x
)
-
3.33066907388
e
-
15
5.96466298
e
-
07
-
2.93020139
e
-
06
-
1.3895524
e
-
05
-
3.97499993
e
-
07
g
6
(
x
)
0
1.68865328
e
-
08
3.505341306
e
-
07
-
5.2295669
e
-
06
-
5.43110185
e
-
06
g
7
(
x
)
-
0.702500000000000
-
0.7025000
-
0.70250000
-
0.7024996
-
0.70250000
g
8
(
x
)
-
8.881784197001
e
-
16
0
0
-
6.1140457
e
-
06
0
g
9
(
x
)
-
0.583333333333333
-
0.58333333
-
0.58333333
-
0.58333078
-
0.58333333
g
10
(
x
)
-
0.051325753541825
-
0.05132589
-
0.05132936
-
0.05137799
-
0.05132566
g
11
(
x
)
-
4.21884749358
e
-
15
-
0.01085237
-
5.86590687
e
-
005
-
6.14133272
e
-
06
-
2.45744437
e
-
07
f
(
x
)
2994.341315684048
2996.348167
2994.482453
2994.499107
2994.47Table 11
Comparison of statistical results for speed reducer design problem by the various algorithms.
Algorithm
Best
Mean
Worst
Std.
EOCSO
2994.3413156840
2994.3413169852
2994.341318013
8
1.201279
1
E
-
06
PSO-DE [29]
2996.348167
2996.348174
2996.348204
6.4
E
-
06
MBA [34]
2994.482453
2996.769019
2999.652444
1.56
HEAA [30]
2994.499107
2994.613368
2994.752311
7.0
E
-
02
HGA [25]
2994.47
N
A
N
A
N
A


## 4.7.2. Pressure Vessel Design Problem

Another benchmark structural optimization problem is the pressure vessel design problem proposed by Kannan and Kramer. Figure26 shows a pressure vessel design problem which has four variables (x1,x2,x3,x4) and four nonlinear inequality constraints (g1,g2,g3,g4). The objective function of the problem can be expressed as follows:(14)minfx=0.6224x1x3x4+1.7881x2x32+3.1661x12x4+19.84x12x3s.t.g1x=-x1+0.0193x3≤0g2x=-x2+0.0095x3≤0g3x=-πx32x4-43πx33+129000≤0g4x=x4-240≤0,where 0≤x1,x2≤100 and 10≤x3,x4≤200. The approaches have previously been applied to solve this problem including many different numerical optimization techniques, such as the new modification approach on bat algorithm (EBA) [35], the cuckoo search (CS) algorithm  [7],  interior search algorithm (ISA) [36], the mine blast algorithm (MBA) [32], the improved ant colony optimization (IACO) [37], an effective hybrid cuckoo search algorithm for constrained global optimization (HCS-LSAL) [38], the hybrid Nelder-Mead simplex search and particle swarm optimization (NM-PSO) [39], an improved accelerated PSO algorithm [40], and crow search algorithm (CSA) [41].Figure 26
The pressure vessel design problem.The best solutions obtained by the various methods are reported in Table12. Table 13 shows the statistical results. It is shown in Tables 12 and 13 that EOCSO performance for the pressure vessel design problem surpassed the other 10 methods in terms of the minimum obtained value, solution average, and the standard deviation.Table 12
Comparison of the best solutions for pressure vessel design problem by different algorithms.
Algorithm
x
1
x
2
x
3
x
4
g
1
x
g
2
x
g
3
x
g
4
x
f
x
EOCSO
0.778169
0.384649
40.31962
200
-
7.20
E
-
11
-
1.12
E
-
09
-
4.29
E
-
04
-
40
5885.3328
CS  [7]
0.8125
0.4375
42.098446
176.636596
—
—
—
—
6059.714
MBA [32]
0.7802
0.3856
40.4292
198.4964
0
0
-
86.3645
-
41.5035
5889.3216
HCS-LSAL [38]
0.8125
0.4375
42.09844
176.6366
-
2.01
E
-
09
-
0.0331588
-
0.002495
-
63.3634
6059.7143
NM-PSO [39]
0.8036
0.3972
41.6392
182.412
3.65
E
-
05
3.79
E
-
05
-
1.5914
-
57.5879
5930.3137
CSA [41]
0.8125
0.4375
42.098445
176.636598
-
4.02
E
+
09
-
0.0358808
-
7.12
E
+
04
-
63.3634
6059.7144Table 13
Statistical results of different approaches for pressure vessel design problem.
Algorithm
Best
Mean
Worst
Std.
EOCSO
5885.3328
5885.3328
5885.33279
5.14
E
-
06
EBA [35]
6059.71
6173.67
6370.77
142.33
CS  [7]
6059.714
6447.736
6495.347
502.693
ISA [36]
6059.71
6410.08
7332.84
384.6
MBA [32]
5889.32
6200.64
6392.5
160.34
IACO [37]
6059.73
6081.78
6150.13
67.2418
HCS-LSAL [38]
5930.3137
5946.7901
5960.0557
9.1614
NM-PSO [39]
6059.7143
6087.3225
6137.4069
2.21
E
-
02
IAPSO [40]
6059.7143
6068.7539
6090.5314
14.0057
CSA [41]
6059.714363
6342.499106
7332.841621
384.9454163


## 5. Conclusion

In this paper, an improved chicken swarm algorithm based on elite opposition-based learning is proposed to overcome the disadvantages of the deficiencies of lack of population diversity, being easy to stick to “premature,” and low searching precision in later stage of evolution in chicken swarm algorithm. Search mode of dynamic adaptivet distribution is applied for cock swarm to balance the global development ability and local exploitation ability of the algorithm. Search mode of elite opposition-based learning is used to enrich population diversity for hen swarm. Greedy dimension-by-dimension searching is used for individual of optimal chicken swarm to do local search, which improves the search precision and convergence rate of the algorithm. Numerical experiments of 18 standard test functions and 2 engineering structure optimization problems are conducted to verify the availability and feasibility of the algorithm. The search precision, convergence rate, and robustness are all better than those in other typical intelligent algorithms.



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*Source: 2734362-2017-03-26.xml*